Electromagnetic Techniques for Analysis
and Design of Ultra Sensitive Receivers
for Far-Infrared Astronomy
Presented by
Colm Bracken, B.Sc.
A thesis submitted in partial fulfilment of the requirements for the degree of
Doctor of Philosophy
Department of Experimental Physics
National University of Ireland, Maynooth
Maynooth, Co. Kildare, Ireland
January 2015
Head of Department
Prof. J. Anthony Murphy, M.Sc., M.S., Ph.D.
Research Supervisor
Prof. J. Anthony Murphy, M.Sc., M.S., Ph.D.
Contents
Introduction
8
1 Far-IR Astronomy and Instrumentation
1.1
1.2
11
Science Case for Far Infra-Red Astronomy . . . . . . . . . . . . . . . 12
1.1.1
Star and Planetary System Formation . . . . . . . . . . . . . 14
1.1.2
Birth and Evolution of Galaxies and AGN . . . . . . . . . . . 18
SPICA Space Observatory . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.1
SPICA Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.2
The SAFARI Instrument . . . . . . . . . . . . . . . . . . . . . 25
1.3
Far-Infrared Imaging Fourier Transform Spectroscopy . . . . . . . . . 29
1.4
VNA Measurement Techniques
1.5
Direct Detection Techniques . . . . . . . . . . . . . . . . . . . . . . . 33
1.6
Transition Edge Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . 31
2 Computational Methods for Electromagnetic and Quasioptical Design and Analysis
39
2.1
Overview of Computational Methods . . . . . . . . . . . . . . . . . . 40
2.2
Gaussian Beam Mode Analysis . . . . . . . . . . . . . . . . . . . . . 41
2.2.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.2
Gaussian Beam Mode Analysis of Uniformly Illuminated Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1
2.3
2.4
2.2.3
Gaussian Beam Mode Analysis of EM Horns and Waveguides
2.2.4
Propagation Through Quasioptical Systems: Mirrors and Lenses 56
Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.2
Rectangular Waveguide . . . . . . . . . . . . . . . . . . . . . . 59
2.3.3
Cylindrical Waveguide . . . . . . . . . . . . . . . . . . . . . . 61
Electromagnetic Mode Matching . . . . . . . . . . . . . . . . . . . . . 63
2.4.1
2.5
49
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Finite Element Method: COMSOL
Multiphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.1
2.6
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3 The Analysis of the SAFARI Multimode Horns
3.1
71
Rectangular Horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1.2
Approximate Method . . . . . . . . . . . . . . . . . . . . . . . 74
3.1.3
Example Horn (Approximate Method) . . . . . . . . . . . . . 76
3.1.4
Example Horn (Mode Matching Approach) . . . . . . . . . . . 78
3.1.5
Example Horn (FEM Approach) . . . . . . . . . . . . . . . . . 80
3.1.6
Inclusion of a Conducting Ground Plane . . . . . . . . . . . . 84
3.1.7
Comparing Horn to Waveguide . . . . . . . . . . . . . . . . . 85
3.1.8
Directional Gain of Example Horn
3.1.9
External Back Scatter . . . . . . . . . . . . . . . . . . . . . . 89
. . . . . . . . . . . . . . . 86
3.1.10 Experimental Verification . . . . . . . . . . . . . . . . . . . . 89
3.2
SAFARI Horns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2.1
Far-field Radiation Patterns of SAFARI Horns . . . . . . . . . 91
3.2.2
Optical Cross-talk in SAFARI Horn Array . . . . . . . . . . . 97
2
3.2.3
3.3
3.4
Experimental Verification . . . . . . . . . . . . . . . . . . . . 105
Characterising Throughput of SAFARI-Like Multi-Moded Feed Horns 107
3.3.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.3.2
Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . 112
3.3.3
Manual Optimisation of Set-Up . . . . . . . . . . . . . . . . . 115
3.3.4
Alignment of the System Inside the Vacuum Tubes . . . . . . 118
3.3.5
Efficiency of FTS Beam Splitters . . . . . . . . . . . . . . . . 120
3.3.6
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.3.7
Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . 129
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4 Conical Mode Matching for Shaped Smooth-Walled CMB Horns 133
4.1
4.2
Smooth Walled Conical Horns . . . . . . . . . . . . . . . . . . . . . . 134
4.1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.1.2
Derivation of Spherical Harmonic Fields . . . . . . . . . . . . 136
4.1.3
Calculation of Eigenvalues . . . . . . . . . . . . . . . . . . . . 142
4.1.4
Evanescence of Modes . . . . . . . . . . . . . . . . . . . . . . 148
4.1.5
Cartesian Field Components . . . . . . . . . . . . . . . . . . . 150
4.1.6
Example Horn for Comparison . . . . . . . . . . . . . . . . . . 151
4.1.7
Limitations of Approximate Method . . . . . . . . . . . . . . 152
Scattering at Conical Horn Junctions
(Spherical SCATTER) . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.3
4.2.1
Derivation of S Parameters . . . . . . . . . . . . . . . . . . . . 157
4.2.2
Coupling Between Two Conic Sections . . . . . . . . . . . . . 162
4.2.3
Coupling Between Multiple Conic Sections . . . . . . . . . . . 168
Spherical SCATTER Program . . . . . . . . . . . . . . . . . . . . . . 171
4.3.1
Description of Spherical SCATTER Code . . . . . . . . . . . . 171
4.3.2
Test Horn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
3
4.3.3
4.4
Experimental Verification . . . . . . . . . . . . . . . . . . . . 178
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5 The Electromagnetic Design and Analysis of Cavity Coupled Absorbers for SAFARI
180
5.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.2
Modelling Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.3
5.4
5.5
5.6
5.2.1
Mode Matching Approach . . . . . . . . . . . . . . . . . . . . 183
5.2.2
3D FEM Approach . . . . . . . . . . . . . . . . . . . . . . . . 188
5.2.3
2D FEM Approach . . . . . . . . . . . . . . . . . . . . . . . . 192
5.2.4
Comparison of Results . . . . . . . . . . . . . . . . . . . . . . 193
Optimisation of Absorber Position and Cavity Geometry . . . . . . . 202
5.3.1
Optimising Absorber Position for T E11 Mode . . . . . . . . . 202
5.3.2
Optimisation of Absorber Position for an Over-Moded System 203
5.3.3
Cavity and Absorber Size . . . . . . . . . . . . . . . . . . . . 205
5.3.4
Cavity Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
SAFARI L-Band Receivers . . . . . . . . . . . . . . . . . . . . . . . . 212
5.4.1
Vacuum Gap between Neighbouring Pixel Cavities . . . . . . . 213
5.4.2
COMSOL Model of SAFARI L-Band Detectors . . . . . . . . 224
5.4.3
Improving the SAFARI L-Band Design . . . . . . . . . . . . . 228
SAFARI S-Band Receivers . . . . . . . . . . . . . . . . . . . . . . . . 234
5.5.1
Single Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . 237
5.5.2
Analysis of Higher Order Modes . . . . . . . . . . . . . . . . . 238
5.5.3
Experimental Verification . . . . . . . . . . . . . . . . . . . . 240
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6 Summary and Conclusions
251
4
Abstract
This thesis focuses on EM (electromagnetic) modelling methods for simulating the
behaviour of receivers and detectors for far-IR (far-infrared) astronomy and CMB
(cosmic microwave background) applications, toward an optimum design for SAFARI (the Spica Far-Infrared Instrument) on the proposed space based telescope
SPICA (Space Infrared Telescope for Cosmology and Astrophysics). The computational methods required for modelling SAFARI-like receivers are described, and
sophisticated conical waveguide mode matching theory is derived. Generic far-IR
systems are investigated, including analysis of SAFARI-like multi-mode feed horns,
and the absorber coupled superconducting TES (transition edge sensor) bolometers
that are typically used in far-IR receivers. Special attention is given to the integrating cavity backshorts that serve to increase pixel efficiency, and new designs
for improved cavity geometries, which provide high coupling between the detector
and the incoming signal while reducing cavity losses and crosstalk, are presented.
Where more than one simulation approach is employed for a particular problem, the
agreement between the results served to mutually validate the approaches. Where
necessary, experimental measurements were carried out for verification of the simulations, and the corresponding results are presented. The results of a measurement
campaign carried out at SRON (Netherlands Institute for Space Research) during
an extended visit during the summer of 2013 are presented, where a SAFARI-like
horn antenna was characterised in terms of its frequency dependent throughput at
THz (terahertz) frequencies. A series of cavity mounted absorber measurements are
also described, where the detector components are appropriately scaled up in size to
suit the frequency range of the in-house VNA (vector network analyser) system at
NUIM (National University of Ireland, Maynooth). The main drivers for the work
presented in this thesis were two Technology Research Programmes, both commissioned by ESA (European Space Agency).
List of acronyms used throughout the text:
Far-IR (far-infrared), CMB (cosmic microwave background), TES (transition edge
sensor), Ta (tantalum), THz (terahertz: 1012 Hz), NEP (noise equivalent power),
EM (electromagnetic), VNA (vector network analyser), SRON (Netherlands Insti5
tute for Space Research), RAL (Rutherford Appleton Laboratory), NUIM (National
University of Ireland, Maynooth), SPICA (Space Infrared Telescope for Cosmology
and Astrophysics), SAFARI (Spica Far-Infrared Instrument), FEM (finite element
method), FIT (finite integration technique), GBM (Gaussian beam mode), ESA
(European Space Agency), JAXA (Japane Aerospace Exploration Agency), ISAS
(Institute of Space and Astronautical Science), TE (transverse electric), TM (transverse magnetic), AGN (active galactic nuclei), FTS (Fourier transform spectrometer), FOV (field of view), SFR (star formation rate).
6
Acknowledgements
The work presented in this thesis would not have been possible without the support
of many people. I would like to thank my research supervisor Prof. Anthony Murphy,
for all his support and assistance, who went above and beyond what is expected of
a supervisor in terms of his time and effort in helping me in this pursuit.
I would also like to thank Dr. Neil Trappe, who always made time to help me
with research issues, and with my teaching. Many thanks to Dr. Créidhe O’Sullivan
and Dr. Marcin Gradziel, for their time and effort in organising our team meetings
for the THz Optics Group, and for providing help any time it was needed. I give a
special thank you to Ms. Gráinne Roche, the department secretary, for all her help
and good advice over the years. I also thank Mr John Kelly and Mr Derek Gleeson
for all of their support with technical issues. I also appreciate the help provided to me
by the other postgraduate students, in particular Niall Tynan, Darragh McCarthy,
and Niamh Tobin, for their help with measurement procedures. Thank you also, to
Dr Valerie Heffernan of the German department at N.U.I. Maynooth for sharing her
car with me each day, and all of her great advice.
Many thanks to Dr. Gert de Lange and Dr. Damien Audley at SRON, Groningen, for hosting me during my work placement there, and helping me and my family
settle into our new surroundings in the Netherlands. And thank you to Mr WillemJan Vreeling for his help with measurements at SRON, and for entertaining my kids
during their visit to the lab.
For the financial supports provided to me throughout my studies, I thank the
John and Pat Hume Scholarship fund, the Department of Experimental Physics at
N.U.I. Maynooth, and ESA.
Finally, to Ciara, Hollie, and Kayla, thank you so much for your love and
patience through this intensive period. Ciara, your support always made it possible
for me to balance my time between work and being a dad; you have been my
foundation. I am so excited about having my weekends free now to have fun with
you and the girls. Thank you too, to my Mother, Susan, and my mother-in-law,
Pat, for all of your love and support. And thank you to my father-in-law, Sean;
you have been like a Father to me, and your guidance is what helped me to see the
importance of physics, and the true beauty of Nature.
7
Introduction
“Imagination is more important than knowledge. For knowledge is limited to all
we now know and understand, while imagination embraces the entire world, and all
there ever will be to know and understand.”
- Albert Einstein
8
The ability to be able to predict accurately the full optical and EM performance
of typical receiver systems has involved a significant research effort over the last
two decades. Many previously released theses from the THz Optics Group of the
Experimental Physics Department of NUIM focussed on modelling the optics that
comprise the free-space path described above. Much has also been carried out on the
beam patterns of corrugated feed horns, leading to the development of the highly
efficient SCATTER mode matching program [Murphy et al., 2010]. However, what
is now required for a more complete understanding of such systems is a thorough
analysis of the horn/cavity/absorber structures using EM simulation techniques.
The main reason for the lack of progress in this area to date is that, without resorting
to super computers, the computational power for modelling these electrically large
structures was simply not available in the past. Only now is it possible to carry
out a complete vector EM analysis of the integrated receiver and detector structures
with a moderately powerful PC. And by employing optimisation techniques, and
combining results from EM mode matching, FEM (finite element method) and FIT
(finite integration technique) we can design systems with the highest coupling to
the incoming signal possible, limited only by manufacturing constraints.
Due to the extremely small dimensions of the feed horns and cavities required
for detection of signals in the THz frequency range, manufacturing tolerances are
the main limiting factor in the design and construction of these types of detector
systems. It is worth noting that, in as early as 1999, corrugated feeds had been built
by RAL (Rutherford Appleton Laboratory) in the U.K. for operation at frequencies
up to 2.5 THz [Wilsher et al., 1999]. However, producing large numbers of such
horns in array format would undoubtedly prove extremely difficult and expensive.
For this reason particular attention was given to smooth walled structures in this
thesis. Chapter 3 focusses on the analysis of smooth walled rectangular feed horns
for the SAFARI instrument, where the far-field beam patterns are calculated using
an efficient mode matching simulation method, and horn-to-horn crosstalk effects
are predicted using finite element analysis.
It has been shown that, with very careful optimisation of the geometry of
smooth walled feeds, similar performance to corrugated feeds can be achieved over
reasonable bandwidths [McCarthy et al., 2013], [Tan et al., 2012]. In fact a relatively simple profile consisting of just a few straight smooth walled conic sections
9
can be enough to achieve the required levels of beam symmetry, low-level side lobes
and low cross-polar levels, although with some restrictions of the horn profile dimensions. A complete chapter of this thesis was devoted to creating a new conical mode
matching code, with the aim of producing a more exact model of smooth-walled
conical feed horns. It was also hoped that the code would be more efficient than
the alternative cylindrical mode matching approach. The work is also interesting
from an academic point of view as there is little literature available on the subject
of conical modes and conical mode matching. This work is described in detail in
chapter 4.
Another main focus of the thesis is the investigation and development of new
and efficient methods for modelling the interaction between the cavity and absorber
of the bolometers. More efficient techniques were required for the modelling of this
component, in particular due to the electrically large size of the cavity which is
typically at least a few wavelengths in diameter as one moves into the THz regime.
Waveguide mode matching techniques proved very efficient, and the results were
checked against FEM and FIT results for verification of the approach. However the
waveguide mode matching code was also limited since many real systems employ a
vacuum gap between the horn array block and the detector array (see Figure 5.1
in chapter 5), which cannot be accurately modelled with waveguide modes. For
this reason a discussion on the possibility of extending the NUIM in-house mode
matching code “SCATTER” to account for this vacuum gap is presented in chapter
5.
Although the systems that are investigated in this thesis are for generic instruments for far infrared astronomy and CMB detection, it is mostly toward SAFARI
instrument that the research is primarily aimed. The SAFARI instrument will be
part of the SPICA spacecraft that is due to be launched by ESA and JAXA (Japane
Aerospace Exploration Agency) in 2022/2024. The primary mission objectives of
SPICA are to understand the origin and evolution of galaxies, stars and planets
and for this reason some background information is given on each of these topics in
sections 1.1.1 and 1.1.2. An overview of the SPICA mission as well as the SAFARI
instrument are given in sections 1.2 and 1.2.2 respectively.
10
Chapter 1
Far-IR Astronomy and
Instrumentation
“We are just an advanced breed of monkeys on a minor planet of a very average
star. But we can understand the Universe. That makes us something very special.”
- Stephen Hawking
11
1.1
Science Case for Far Infra-Red Astronomy
It has now been accepted that in order to have a more complete understanding
of our Universe and the workings within it a full panchromatic (multi-waveband)
astronomical view is going to be required. Many different physical processes relating to star formation, galaxy evolution, chemical abundance variations etc. can be
observed in vastly different wavelength ranges and therefore detecting radiation at
all these wavelengths and performing science with the corresponding data will be
essential for building a complete picture of nature on both the small and extremely
large astronomical and cosmological scales. As is illustrated in Figure 1.1, a significant portion of all the photons in the universe fall within the far-IR band, thus
this waveband of the EM spectrum will undoubtedly reveal many secrets of how the
universe works.
Figure 1.1: Schematic spectral energy distributions of the extragalactic background. The blue area is the cosmic
optical background, the red area is the cosmic infrared background, and the grey area is the cosmic microwave
background. The units of the specific brightness are Watts per metre squared per steradians [Dole et al., 2006].
In October 2005 ESA announced its plans for the decade spanning 2015 to 2025
in what it called ESA’s Cosmic Vision program. This “vision” posed four questions
that were to be the drivers for the objectives of future space science missions over
the time period of 2015 to 2025. These questions are as follows:
1) What are the conditions for planet formation and the emergence of life?
2) How does the Solar System work?
12
3) How did the Universe originate and what is it made of?
4) What are the fundamental physical laws of the Universe?
The first three questions set out by ESA can be addressed with observations
in the mid-IR and far-IR. As such, a next generation space based mid-to-far infrared observatory will be key in helping to fulfil this ambitious Cosmic Vision.
SPICA would address the first question with its state of the art Mid-IR coronagraph
[Enya et al., 2008] that would provide the first direct imaging and spectroscopy of
young massive planets. SPICA’s far-IR spectrometer SAFARI, which will be more
than ten more sensitive times than any previous far-IR spectrometer, will be able
to probe deeper into a wider range of star forming regions. SPICA will thus reveal previously hidden protostars and protoplanetary discs and allow astronomers
to determine what types of stars, and which regions of the Galaxy, are the most
favourable for planetary system formation.
SPICA can address the second question (how our Solar System works) by studying asteroids and comets in our Solar System. The chemical make-up of these primitive remains of the Solar nebula can provide information on the chemical mixture
and initial conditions of the early nebula from which the planets formed. SPICA’s
unprecedented high sensitivity spectroscopy will reveal the chemistry of these objects
and will allow astronomers to compare these spectra to those of protoplanetary discs
around nearby stars. By studying the formation of neighbouring stellar systems we
can constrain various models of how our own Solar system formed.
The third question set out in ESA’s Cosmic Vision will also be best answered
by SPICA. Understanding how the Universe originated and what is it made of will
be unveiled through further resolving the far-IR background into discrete sources
beyond the level achieved with Herschel [Oliver et al., 2010], and by revealing the
star-formation activity hidden at optical wavelengths by dust clouds. The combination of mid-IR and far-IR spectroscopy on SPICA will be essential to “Trace the
formation and evolution of the super-massive black holes at galactic centres – in
relation to galaxy and star formation – and trace the life cycles of chemical elements
through cosmic history” [Ferlet et al., 2010].
The following subsections will outline the strong science case for far-IR observations and science with a SAFARI-like instrument. The science case will be explained
in terms of specific astronomical processes such as star and planetary system for13
mation, and birth and evolution of galaxies and AGN (active galactic nuclei).
1.1.1
Star and Planetary System Formation
Star formation is the process by which large clouds of gas (usually molecular clouds)
collapse under gravity and coalesce, increasing in density and temperature. It is
thought that the initial collapse of the previously stable gas clouds is due to two
possible mechanisms. It is possible for two molecular clouds to collide with each
other and this collision would cause instability leading to collapse. Another possible mechanism involves the pressure wave of a nearby supernova event causing the
instability. One way or another, once the instability is generated and there is sufficient mass according to the Jeans criterion [Jeans, 1928] there is a runaway effect.
As the material is compressed at the centre of collapse the density and temperature
continue to rise. Eventually the density and pressure become sufficiently large that
fusion of hydrogen to helium can take place. The object present just as nuclear
fusion begins, and the time leading up to this, is called a protostar. A diagram
depicting how the star formation process is somewhat cyclic is shown in Figure 1.2.
Figure 1.2: The life cycle of stars and nebulae [Kulesa, 2011].
Observing newly formed protostars and the clouds they are embedded in is crucial for understanding both how are own Solar system was formed, and the general
14
processes involved with star and planet formation. Although some disk shaped objects had been observed with the Very Large Array (VLA) radio telescope and some
optical telescopes, the first confirmed images of protoplanetary disks that showed
structure were recorded with the Hubble Space Telescope (HST) in 1992, described
in a paper released on June 20th 1993 [O’Dell et al., 1992]. Further ground-breaking
work was done in 2003 when it was shown that dust grains within these protoplanetary disks were significantly larger than what would be expected in the interstellar
medium, implying that grain growth had occurred [McCabe et al., 1992]. This was
achieved by analysing scattered light from the protoplanetary disk around the star
HK Tau B using the 10 metre W. M. Keck telescope in Mauna Kea, Hawaii.
The far-IR and submm bands of the EM spectrum can reveal these protostars
and protoplanetary disks due to the relatively long wavelength of radiation compared
to the size of typical dust grains. Also the circumstellar dust and gas, that makes
up the disks, emit strongly at these wavelengths. The ALMA facility will be able to
image these protostellar environments in the mm and submm continuum and line
emissions with unmatched angular resolution (sub-arc second) [Zapata et al., 2013].
This is possible due to ALMA’s large array of 12 metre antennas that can be formatted with baselines up to 16 km. However, the far-IR, which is out of reach for
ALMA due to atmospheric attenuation of signal as already indicated, can inform on
younger disks that are richer in gas and less-rich in dust. This is due to transitions
in H2 , H2 O and hydrides that correspond to emissions in the far-IR, and thus in
order to study the youngest protostellar disks a space based far-IR observatory will
be essential.
Although ALMA has the highest angular resolution of any submm facility it
is not designed to map distributions of gas and dust in giant molecular clouds
(GMCs) on large scales. SPICA-SAFARI would be capable of mapping these large
faint regions from both the dust continuum and transition lines in the gas. The
2’×2’ large FOV (field of view) of SAFARI would allow astronomers to trace the
molecular outflows in GMCs on parsec-scales as they impact the so-called “interclump medium” [Ferlet et al., 2010]. Ground based mm and sub-mm observatories
such as JCMT can map low energy emissions from molecules such as CO, CN, and
C2 H, and the dust continuum emission over large spatial scales in GMCs at submm wavelengths [Plume et al., 2007]. However, the brightest gas cooling lines [Si
15
II], [OI], and [C II], at 34, 63, and 158µm, respectively, are beyond the minimum
wavelength range of ground based instruments such as JCMT-SCUBA 2.
Our understanding of the chemical conditions and the physical processes involved in the evolution of protostellar disks is relatively limited, in particular the
transition for a stage I protostar, embedded in a substantial envelope, to a stage
II configuration, with only a star and protoplanetary disk. The PACS instrument
aboard the Herschel Space Observatory has recently been used to make a leap in
progress by investigating the physical and chemical structure of the planet-forming
regions of such disks, unveiling the major carbon and oxygen chemical species
(CO, C+, H2O, OH, O) and the mineralogy of the dust grains in these regions
[Van Kempen et al., 2010]. Figure 1.3 shows a spectrum that was obtained of the
protostar DK Chamaeleontis with Herschel/PACS.
Figure 1.3: The spectrum from 55–190µm of the gas line emission spectrum of protostar DK Chamaeleontis. The
inserts show expanded views of parts of the spectrum [Van Kempen et al., 2010].
Although this type of spectral analysis is a significant step in our understanding
of protoplanetray disks, the PACS instrument on Herschel has only been able to
survey about twenty of the brightest, relatively close disks around intermediate
mass stars, and due to the limited sensitivity of Herschel performing surveys of
large numbers of protoplanetary disks was not possible. Furthermore disks around
Sun-like stars proved to be insufficiently bright for PACS to measure. With the huge
increase in sensitivity that would be provided by SPICA/SAFARI large surveys of
hundreds of protoplanetary disks around T-Tauri stars could be performed. The
chemistries of T-Tauri stars have been shown to be much richer than the Herbig
16
Ae/Be disks, with higher abundances of Polycyclic Aromatic Hydrocarbon (PAH)
for example [Geers et al., 2006]. Thus far-IR spectral surveys of protoplanetary
disks around Sun-like stars would undoubtedly reveal many surprises in terms of
the chemistry of these planet forming regions.
The importance and spectral richness of the mid to far-IR region is illustrated in
Figure 1.4 where the detailed spectrum from the Infrared Space Observatory (ISO)
is shown [Rosenthal et al., 2007]. A simulated spectrum of a typical Milky Way-like
galaxy undergoing modest SFR is then shown in Figure 1.5 where the region of the
spectrum that would be covered by SPICA/SAFARI is shown together with the
spectral coverage of similar facilities.
Figure 1.4: Measured spectrum from the Infrared Space Observatory (ISO) [Rosenthal et al., 2007].
As can be seen in Figure 1.5 the majority of emitted radiation falls within either
in the optical or in the mid to far-IR. The multitude of atomic, ionic, molecular, and
fine-structure lines corresponding to carbon and oxygen, and ionic lines of neon and
sulphur together with with rotational transitions of molecules such as molecular
hydrogen, water, carbon monoxide, and hydroxyl demonstrate the importance of
the mid and far-IR ranges for astronomy. These powerful diagnostic features can
be used to study both star and planetary formation very locally, and star formation
and the influence of AGN in the very distant Universe [Ferlet et al., 2010]. SAFARI
will be capable of measuring radiation from the proto-planetary disks of new stars
17
Figure 1.5: Left: A synthetic spectrum of a typical spiral galaxy undergoing modest rates of star formation showing
the parts of the spectrum that will be covered by SPICA and other similar facilities [Ferlet et al., 2010]. The symbol
L⊙ in the y-axis units represents one solar luminosity. Right: Measured spectra from the Infrared Space Observatory
(ISO) [Rosenthal et al., 2007].
at almost all galactic distances [Goicoechea et al., 2009]. Due to its unique spectral
range SAFARI will perform spectral observations that include fine structure lines
of atomic Oxygen (63µm and 145µm), water vapour rotational lines (50µm and
above), and features of water ice (44µm and 62µm). The water ice spectral data
that will be acquired by SAFARI is essential to constrain current models of planet
formation[Mayer et al., 2005].
1.1.2
Birth and Evolution of Galaxies and AGN
The huge variety of galaxy morphologies has puzzled astronomers ever since the discovery that the Milky Way is but one of the hundreds of billions of galaxies populating the universe. In 1926 astronomer Edwin Hubble created a classification scheme
for this morphology that separated galaxies into three basic classes [Hubble, 1926].
The diagram that demonstrates this has come to be known as Hubble’s tuning fork
diagram and can be seen in Figure 1.6.
The cosmological study of galaxies has undergone important changes over recent
years as observational evidence is increasingly showing that galaxy formation and
18
Figure 1.6: Hubble’s ”tuning fork” diagram illustrating the three broad classes of galaxies.
evolution is more complex than first assumed. In 1962 Eggen et al. put forward
a theory that spiral galaxies form in a “top-down” process through a monolithic
collapse of a huge gas cloud [Eggen et al., 1962] in a manner similar to the collapse
of a GMC in the formation of a protostar. However, more recent theories of how
the first galaxies formed is based on a cold dark matter (CDM) dominated universe
in which ordinary matter collected around areas of dark matter and through the
process of accretion began to clump together and flatten out into a spiral shape
in a “bottom-up” formation scenario. Computer simulations of galaxy formation
further support the CDM, bottom-up based theories. They seem to indicate that
disk galaxies such as spiral and barred spiral galaxies form as a result of accretion
of intergalactic material whereas spherical and elliptical galaxies are a result of
mergers between two or more separate galaxies [Steinmetz and Navarro, 2003]. The
SFR (star formation rates) within galaxies across a range of cosmological distances
z (where z = (λobserved − λrest )/λrest ) can inform on how galaxies evolve. Thus,
studying SFR across all values of z is clearly crucial for understanding the evolution
of the Universe as a whole.
Far-IR emission is now an established tool that can be used to determine SFR
within galaxies. The idea of using far-IR wavelengths to study the evolution of
galaxies and to try to constrain current formation theories can be traced back to
the IRAS mission that was launched on January 25th 1983. A paper released in
May 1987 described how source counts with IRAS at 60µm were used to show
modest levels of evolution within starburst galaxies [Hacking et al., 1987]. Then in
1990 Carol J. Lonsdale confirmed the steep slope of the faint 60µm source counts
discovered by Hacking et al. with the IRAS data, thereby re-enforcing the idea of
19
cosmological evolution at a rate higher than that predicted by conventional passively
evolving models [Lonsdale, 1990]. The Spitzer Space Telescope then provided a big
leap by allowing mid-IR surveys and spectroscopy of faint distant galaxies at high-z
and revealing the star formation history of the universe to z ≈ 3 [Soifer et al., 2008].
However, the Herschel Space Observatory was again at the forefront in terms of the
cutting-edge of the science being performed in the area of galaxy formation and
evolution, and the breakthroughs provided by Herschel are discussed in the following
paragraphs.
The infrared, and particularly the far-IR, is one of the most powerful tracers of
star formation as IR luminosity directly scales with SFR [Seymour et al., 2010]. The
total IR output (luminosity) of a galaxy can provide an straightforward observational
measure that can be used to estimate the total SFR provided care is taken when considering obscuration by dust, and AGN and recently quenched galaxies are excluded
from the data [Hayward et al., 2014]. The Herschel Space Observatory made it possible to directly measure the total IR luminosity of distant galaxies using rest-frame
far-IR measurements of individual galaxies at wavelengths at which the IR spectral
energy distribution (SED) peaks [Elbaz et al., 2011]. Previous studies had relied
on extrapolations from mid-IR or sub-mm photometry ([Melbourne et al., 2005] for
example), or indirect analysis such as stacking of far-IR data from individually undetected sources ([Zheng et al., 2006] for example). The direct far-IR measurements
by Elbaz et al. helped to clear up misconceptions of drastic evolution of IR SED due
to previous incorrect extrapolations of the total IR luminosity from mid-IR (24µm)
observations at z ≥ 1.5, and the associated claim for a mid-IR excess. They demonstrated that the past incorrect extrapolations were a result of confusion between
two separate star forming regimes which exhibit quite different IR SEDs; that of
galaxies with extended star formation and that of galaxies with compact starbursts
[Elbaz et al., 2011]. Again, this was only possible through direct measurement of
radiation within the far-IR band.
As already indicated in the previous section 1.1.1, the sensitivity of the Herschel
Space Observatory was ultimately limited due its relatively warm 80K primary mirror. Thus Herschel was limited in the number of galaxies it could characterise using
the ground-breaking far-IR measurement techniques described above. Furthermore
due to the limited frequency ranges of the instruments on Herschel galaxies at very
20
large z were simply out of reach. The spectral coverage SPICA/SAFARI over the
full THz regime will allow the above described techniques to be applied to galaxies
at medium and high redshifts, thus allowing the characterisation of even the most
distant galaxies. The background limited sensitivity of SPICA, and thus the much
shorter integration times, coupled with the wide FOV of SAFARI (2’×2’) will make
it possible to perform the first all-sky spectroscopic surveys of galaxies. These maps
will be wide and deep enough to unveil the underlying physical processes that drive
galaxy evolution for all sources out to z ≤ 4, and at even higher z for the most
luminous objects [Ferlet et al., 2010].
Another way the SAFARI instrument can extend on the work of Herschel in
determining the evolutionary mechanisms in galaxies and AGN for example is based
on measurements of massive molecular outflows and negative feedback in ultraluminous infrared galaxies (ULIRGs) performed with the Herschel/PACS instrument [Sturm et al., 2011]. The authors describe how mass outflows driven by stars
and AGN may produce the observed black hole-galaxy mass relation, and regulate
and quench both star formation and black hole accretion in the host galaxies or
AGN. Sturm et al. also describe how such observational evidence of these feedback
processes are still scarce. Again, SAFARI could be used measure these mass outflow
and feedback processes out to much higher redshifts, thus providing more extensive
data spanning all cosmological epochs.
1.2
1.2.1
SPICA Space Observatory
SPICA Satellite
The SPICA space observatory is a proposed mission that had been scheduled for
launch sometime in 2022, and is a joint project between JAXA and ISAS (Institute
of Space and Astronautical Science) of Japan, and ESA. It will have a primary mirror
of 3.5 metre diameter cooled to less than 6K which will allow background limited observation over a wavelength range of 34-210µm (mid to far-IR). The large aperture,
monolithic mirror will provide diffraction limited performance at 34µm, and will enable high spatial resolution with a reasonably well controlled point spread function.
SPICA, as proposed, will carry a mid infrared camera, mid-IR spectrometers and
21
a mid-IR coronagraph (built by JAXA institutes), a far-IR imaging spectrometer
(SAFARI, provided by ESA), and a far-IR/sub-millimetre spectrometer (proposed
by the U.S.A.). SPICA is planned to have a lifetime of 5 years with a 3 year minimum, and will orbit the second Lagrange point (L2) of the Earth-Sun system. It
is proposed that SPICA will be launched on the H-IIA 204 in 2022 from the JAXA
Tanegashima Space Centre in Tanegashima, Japan.
European involvement in SPICA was first proposed in 2007 in response to a
call for missions that fit the ESA Cosmic Vision 2015-2025 programme. SPICA
was selected for assessment as a candidate M-class mission, and was termed a ”Mission of Opportunity”. The assessment phase ran for two years from 2007 to 2009
[Ferlet et al., 2010], and it included an internal ESA study of the mission concept,
and 1-year industrial study programme of the SPICA cryogenic telescope assembly
by both Thales Alenia Space in France and at Astrium, also in France. A study
of one of the proposed main instruments for SPICA, a far-IR imaging spectrometer
(SAFARI), was also carried out over the same period by a European consortium.
This SAFARI study included contributions from the author of this thesis as described in chapter 5.
An updated proposal for European involvement in the SPICA project, based
on the outcome of the ESA assessment study, was put forward in late 2009/early
2010. This proposal called for ESA to assume a partner agency role in SPICA, with
JAXA to lead the mission ([Swinyard et al., 2009] or [Nakagawa, 2010] for example),
and ESA would provide the SAFARI instrument to be developed under a European
Consortium led by the SRON. Then in 2013 ISAS/JAXA and ESA agreed that the
scheme for SPICA was not compatible with a timely and robust implementation
of the mission. Both JAXA and ESA decided that an increased ESA contribution
to the mission would lead to a lower risk and more robust mission implementation [IRAP, 2014]. However, any significant extension of ESA’s contribution to the
SPICA mission would bring the mission into the M-Class (medium mission) range,
meaning that the mission would need to be proposed to an ESA call for M-Class
missions where it would need to be peer-reviewed against other proposals for the
same opportunity. It was therefore decided to stop all ESA support activities on
SPICA in early Autumn 2013. In spite of this, research continues on the detector
technology required for a SAFARI-like instrument but under a more general far-IR
22
detector research effort. Strong collaboration exists between SRON in the Netherlands and the NUIM in terms of pushing the development of SAFARI-like receivers
and detectors as described in chapter 5 of this thesis, and as can be seen by recent
publications such as [Audley et al., 2013] or [Bracken et al., 2014] for example.
Figure 1.7: An artist’s impression of SPICA (left) and a graph showing the superior sensitivity of SPICA due to
its < 6K primary mirror operating temperature (right).The blue lines on the graph represent the diffuse infrared
radiation emitted by some astronomical sources while the red lines show infrared radiation emitted by the telescopes
themselves due to their temperature (image credit: ESA)
As already indicated the SPICA mission as proposed will build on previous
work carried out using the Herschel Space Observatory (mission now finished), and
will complement future work of the Atacama Large Millimetre Array (ALMA) and
the James Webb Space Telescope (JWST) which is due to be launched in 2018. It
will “fill in the gaps” between the lower frequency limit of JWST’s near-IR bands
and the upper frequency limit of ALMA’s sub-millimetre bands. It will achieve
this by extending the frequency range of Herschel to cover the full far-IR band
while providing a huge improvement in sensitivity that will be at least an order of
magnitude greater than Herschel.
SPICA is an ambitious project as it promises to help us understand the origin
and evolution of galaxies, stars, planets, our Earth and perhaps even of life itself.
To answer these questions good spatial resolution together with high sensitivity is
required, but achieving both of these requirements has been a major problem in the
past when observing at the mid and far-IR wavelengths. For good spatial resolution
a large primary mirror is necessary, but a larger mirror means a larger observatory which would be much more difficult to cool to very low temperatures to reach
23
(a)
(b)
Figure 1.8: (a): H-IIA 204 launcher vehicle that will take SPICA into Earth orbit, from where SPICA will eventually
transit to L2. (b): The Tanegashima Space Center (TNSC) located on Tanegashima, an island located 115km south
of Kyushu, from where the H-IIA 204 rocket will launch, carrying SPICA into orbit (image credit: ESA)
.
the high sensitivities also required. The ESA Infrared Space Observatory (ISO),
launched in November 1995, had excellent sensitivity for a facility of its time due to
the relatively small 0.6m primary mirror being cooled to 1.7K by means of superfluid helium. For example the Long Wave Spectrometer (LWS) aboard ISO had a
√
sensitivity of ≈ 10−16 W Hz [Leeks et al., 2003]. The ESA Herschel Space Observatory (retired as of August 2013) had a primary mirror with a diameter of 3.5m and
therefore had very good spatial resolution, but the sensitivity across the 55-210µm
range was limited as the primary mirror could only be cooled to 80 Kelvin (passive
cooling to ambient temperature of space). The significant improvement of SPICA
compared to previous infrared observatories like Herschel comes from the combination of a large diameter mirror combined with active cooling using new technology
mechanical cryocoolers [Sugitaa et al., 2008] to a very low operating temperature.
Thus, the primary mirror of 3.5m combined with cooling to temperatures below 6K
promises to allow 10 to 100 times the sensitivity of Herschel in the far-IR band. This
huge increase in sensitivity will allow integration times orders of magnitude shorter
than could be achieved with Herschel thus enabling quick mapping capabilities right
across the far-IR band.
24
1.2.2
The SAFARI Instrument
SAFARI is being developed by an international consortium led by SRON in Groningen, the Netherlands, including Cambridge University (UK), Cardiff University
(Wales), and NUIM (Ireland) [Mauskopf et al., 2010] or [TRP AO/1-5922/08/NL/EM, 2011].
The SAFARI detectors are split into three distinct bands corresponding to the highest frequencies that SAFARI will measure, the intermediate frequencies, and the
the lowest frequencies, and the three bands have been given the names S-Band,
M-Band, and L-Band, respectively. Table 1.1 presents a breakdown of the spectral
coverage of each of the three SAFARI focal plane arrays, together with the number
of pixels required to fully sample SAFARI’s 2’×2’ FOV. SAFARI will have various
spectral resolution modes with R ranging from 10 to 10,000, for example R ≈ 2, 000
at 100µm [Goicoechea and Nakagawa, 2011], where R is defined as λ/∆λ. NUIM is
involved in both the L-Band which is being investigated by Cardiff University and
Cavendish Laboratory, Cambridge, [Mauskopf, 2011] and the S-Band which is under
development at SRON Groningen [Audley et al., 2013].
Band
Range
Fλ/2 Pixel Size Number of Pixels
[µm]
[µm]
S
34 - 60
480
64×64
M
60 - 110
840
38×38
L
110 - 210
1600
20×20
Table 1.1: Specifications of the three SAFARI spectral bands. Adapted from [Mauskopf et al., 2010]
The baseline optical configuration of SAFARI is a Mach-Zehnder imaging FTS
(Fourier transform spectrometer) [Zetie et al., 2000], [Naylor et al., 2006], which will
provide high mapping speeds due to spatial multiplexing. It will have the ability
to incorporate a photometric imaging mode and will have operational flexibility to
tailor the spectral resolution to the desired science. The parts of SAFARI that were
examined and modelled by the author, as described throughout this thesis, are the
L-Band and S-Band detection boxes that are visible in the top-right of Figure 1.9,
where each detection box consists of an array of superconducting TES bolometers
mounted above an array of integrating backshorts, and fed by an array of feed horns.
This part of the optical beam path is referred to as the “Waveguide Path”, and it
25
was discussed above in the introduction section at the beginning of this thesis. The
remainder of the optics that are visible in Figure 1.9 are referred to as the “FreeSpace Path”, also described above. Expanded views of the “Waveguide Path” are
shown in Figure 1.10 where each sub-figure shows further zooming toward the TES
bolometer.
Figure 1.9: A schematic diagram of the SAFARI instrument. The detection boxes in the upper right area are the
primary concern of this report (image credit: SRON).
(a)
(b)
(c)
(d)
Figure 1.10: Expanded view of the “Waveguide Path” section of the SAFARI optical system. Further zooming going
from (a) to (d) (image credit: SRON).
26
The SAFARI instrument on SPICA will exploit the power of MIR/FIR imaging
spectroscopy and photometry. It will analyse radiation from processes involving
galaxy evolution that are usually hidden from view due to dust. A selection of the
important atomic, ionic, and fine-structure lines that will be accessible by SAFARI
(and other SPICA instruments) are shown in Figure 1.11(b). SAFARI will be able
to cover the full 34-210µm band in a few thousandths of the time that the photo
detector array camera and spectrometer (PACS) on Herschel took to obtain a deep
spectrum of comparable resolution. This is due to its broad instantaneous spectral
coverage and high sensitivity. It also has a particularly wide FOV of 2′ × 2′ making
it possible to carry out spectroscopic surveys that are both wide and deep enough
to analyse the underlying processes that drive galaxy evolution [Ferlet et al., 2010].
(a)
(b)
Figure 1.11: (a): Spectroscopic mapping speeds of the SAFARI Instrument compared to Herschel PACS. The
mappings are shown superimposed on an output from the Millennium simulation at z ≈ 1.4 [Springel et al., 2006],
[Ferlet et al., 2010]. The small blue box in the centre of the image represents the footprint of the instantaneous
spectroscopic FOV of Herschel-PACS, the yellow box surrounding the blue box represents the footprint of the
instantaneous spectroscopic FOV of SAFARI (2′ × 2′ ), and the large green box shows the area that could be
mapped by SAFARI in a 900 hour full wavelength spectral survey (≈ (1o )2 ). (b): A selection of the atomic,
ionic, and fine-structure lines detectable with the propsed instruments on SPICA (including SAFARI), plotted as a
function of critical density and ionisation potential [Ferlet et al., 2010].
Figure 1.11(a) illustrates the mapping capabilities of SPICA-SAFARI. The
large green box that marks out roughly ≈ (1o )2 represents the area that could be
mapped by SAFARI in a 900 hour full wavelength spectral survey simultaneously
covering the full 34-210µm band. The large scale cosmological structure shown in
27
the image was taken from the Millennium project output [Springel et al., 2006]. The
Millennium project, or “Millennium Run”, was an N-body simulation performed at
the Max Planck Society’s Supercomputing Centre in Garching, Germany, with the
facility’s principal supercomputer over a one month period. It used more than 10
billion particles to trace the evolution of the matter distribution in a 2 billion lightyears cubed volume of the Universe. By applying sophisticated modelling techniques
to the 25 Tbytes of stored output, the team of scientists (VIRGO Consortium) were
able to model the evolutionary histories of the roughly 20 million galaxies which
populate the huge cube of the simulated universe. The simulations also involved the
evolution of supermassive black holes which lie at the heart of most galaxies and
quasars [Springel et al., 2006], [Springel et al., 2005].
As described in sections 1.1.1 and 1.1.2 above, SAFARI will be able to detect the
key mid-IR and far-IR lines from distant galaxies by using mid-IR/far-IR diagnostics
that have been revealed and calibrated by Spitzer and Herschel. At a redshift
of z = 1 the rest-frame bandwidth of 11-35µm, which is very rich in ionic fine
structure as well as H2 rotational lines, moves to 22-70µm which is partially in the
spectral range of SAFARI. For more distant galaxies corresponding to z = 2 this
same bandwidth moves to 33-105µm which can be completely covered by SAFARI.
In fact galaxies out to a redshift of z = 6 will still be in the SAFARI spectral
range. SAFARI will detect the brightest mid-IR and far-IR lines out to z = 2 in
intermediate luminosity objects in a 1 hour integration which is a great improvement
on Herschel-PACS, which only has the sensitivity to detect the very brightest midIR and far-IR lines in several hours of integration per line on the most luminous
objects at z ≈ 1 [Ferlet et al., 2010].
SAFARI will have enough photometric sensitivity to undertake very deep largearea surveys. At 70µm, for example, it will detect all galaxies with LIR (infrared
luminosity) of LIR ≥ 1011 L⊙ at a redshift of z = 2 (where L⊙ is the luminosity
of the sun), all those with LIR ≥ 5 × 1011 L⊙ at a redshift of z = 3 and the most
luminous IR galaxies (≥ 1012 L⊙) out to z = 4.
28
1.3
Far-Infrared Imaging Fourier Transform Spectroscopy
Clearly photometric techniques can reveal the spatial and morphological details of
planets, stars, and galaxies. However, spectral imaging techniques must be employed
if the chemical composition and physical conditions of the same astronomical sources
are to be determined. There are a number of measurement approaches that can be
used for obtaining astronomical spectral information such as the familiar diffraction
gratings commonly found in optical telescope systems. However, such diffractive
techniques are usually quite limited in terms of the spectral coverage that can be
achieved with them.
Fourier transform spectroscopy involves splitting the incoming signal beam into
two, and by varying the optical path length of one of the beams before the two are
recombined, an interferogram can be constructed that contains the spectral information of the signal beam. By subsequently applying sophisticated data analysis
techniques the Fourier transform of the spatial intensity data can be extracted,
thus revealing the signal intensity as a function of spatial frequency (wavenumber).
With the high throughput, broad spectral coverage, and variable resolution possible,
Fourier transform spectroscopy has been the most commonly employed spectroscopic
method when dealing with the far-IR band [Naylor et al., 2013].
There are in fact multiple techniques that have been demonstrated for performing Fourier transform spectroscopy. One such approach is described in chapter 3
of this thesis, where a Michelson interferometer was used to characterise a multimoded SAFARI-like feed horn. The choice of a Michelson interferometer in that
case was based on the relatively short path length that can be employed with such
a technique, since the entire optical system had to be mounted in a vacuum system
due to the far-IR wavelengths that were under study. The drawback of the Michelson interferometer is that 50% of the signal beam will inevitably be reflected back
toward the source and thus be lost (see Figure 1.18 in section 1.5 for illustration). A
significant improvement on the Michelson design, in terms of efficiency, is the MachZehnder (MZ) interferometer [Zetie et al., 2000], [Naylor et al., 2006], as illustrated
in Figure 1.12. Due to its two separate output ports the full signal beam is available for analysis at the output. Figure 1.13 shows three Fourier transform spectra
29
across a frequency range of 1.25 to 1.56 THz measured with the SPIRE Mach-Zender
interferometer on Herschel [Wesson et al., 2010]. Due to the high throughput and
unprecedented sensitivity required of the SAFARI instrument, the MZ design is the
best suited interferometer.
Figure 1.12: Schematic of a Mach-Zender FTS. BS1: beamsplitter1, BS2: beamsplitter 2, MM: moveable plane
mirror, CML: left collimating mirror, CMR: right collimating mirror, FML: left focussing mirror, FMR: right
focussing mirror. It is also possible to have a second input that would be positioned symmetrically to the left of
BS1 with respect to Input 1. Input 2 would then trace out two similar beam paths to those shown for Input 1.
Figure 1.13: Continuum-subtracted Fourier transform spectra of three sources measured with the SPIRE instrument
on the Herschel Space Observatory. The black line is from NGC 7027, the red line is from AFGL 618, and the blue
line is from AFGL 2688 [Wesson et al., 2010].
Previous sub-mm and far-IR instruments that have employed the MZ inteferometer format include the spectrometer of the SPIRE instrument on-board Herschel
30
[Swinyard et al., 2003] and the spectrometer that is used with the SCUBA-2 instrument on JCMT [Naylor et al., 2006]. The Astronomical Instrumentation Group at
the University of Lethbridge, Canada has a long heritage in producing FTS’s for
astronomical applications at far-IR wavelengths [Naylor et al., 2013] and had large
involvement in the two instruments just mentioned.
1.4
VNA Measurement Techniques
The modelling approaches presented throughout this thesis often required verification, and where multiple simulation techniques are applied to the same problem the
results can act to verify mutually one approach against another, assuming there is
reasonable agreement between the results. Experimental verification is of course also
important, and thus a number of experimental measurements were carried out as
part of the work of this thesis as further verification of the modelling approaches described throughout the thesis. Section 1.5 describes the direct detection techniques
that were involved in the characterisation of multi-moded SAFARI-like feed horns.
In this section a description is given of the VNA and frequency extension heads that
were employed for both measurement of the far-field beam pattern of a rectangular
feed horn, which was used as a verification of the modelling approaches presented in
chapter 3, and for cavity mounted absorber measurements, as a verification of the
modelling work presented in chapter 5.
The VNA measurement approach made use of the in-house network analyser
system in the department of experimental physics at NUIM. A VNA is an instrument that can perform coherent measurements of the network parameters of electrical networks, and can be used to analyse a device under test (DUT) by measuring its affect on the amplitude and phase of the coherent signal, across a sweptfrequency range. In the application for the work of this thesis it was the s–parameters
[Patel et al., 2009] that were analysed since these parameters can provide information about the reflection and transmission of a DUT at high frequencies. Network
analysers can be used to characterize networks with an arbitrary number of ports.
The work presented in chapter 3 employed a two-port network, whereas the work
presented in chapter 5 made use of just a one-port network.
The particular VNA used was a Rohde and Schwarz ZVA-24 system which
31
can generate a swept frequency range from 10-24 GHz. The VNA is operated in
combination with two ZVA-Z110 W-band frequency extension heads that increase
the frequency range to 75-110 GHz by using a series of frequency multipliers. This
higher frequency range then allows coherent millimetre wave measurements of feed
horns, cavities, and absorbers to be carried out.
In order to obtain the most accurate measurement results possible, a careful
calibration must be carried out using a calibration kit that was provided by the
manufacturer. The kit consists of three parts, the first of which, the short (see
Figure 1.14(a)), allows for almost perfect reflection of the signal, and hence for the
S11 and S22 reflection parameters to be correctly calibrated to unity (or 0 dB).
The shim (Figure 1.14(b)), used in combination with the short, then contains a
short transmission line causing a λ/4 phase shift around the center frequency of
the converters operating range and a 180 deg phase shift of the reflection coefficient.
Finally the match component (Figure 1.14(c)) allows for almost perfect absorption of
the signal. The standard calibration used for a one-port network is the “off-set short,
short, match” (OSM), which requires each of the above described components to be
attached separately to the converter head being used. The analyzer’s “Calibration
Wizard” then provides a straightforward procedure for calibration of the head. For
a two-port network the same procedure is applied to both heads, one at a time, and
then a “through” calibration step is performed, where the two converter heads are
directly connected by a waveguide section.
(a) Short
(b) Shim
(c) Match
Figure 1.14: The VNA calibration kit showing the three components.
The far-field beam pattern measurements used for verification of the modelling
approaches described in chapter 3 also required the use of an x − y plane raster
scanner. The DUT was mounted on head 1, and placed on a translation stage that
32
could move in the direction of propagation z, while kept stationary in x − y. A
WR-10 waveguide probe was then mounted on head 2, which itself was mounted on
the raster scan assembly. The VNA scan system is shown in Figures 1.15 and 1.16.
Figure 1.15: Photograph of the VNA scan system for measuring far-field beam patterns.
(a) Horn Antenna
(b) Waveguide Probe
Figure 1.16: VNA scan system showing converter heads and raster scanner.
1.5
Direct Detection Techniques
During an extended visit by the author to the SRON, Groningen, from June to
September 2013, an analysis of the frequency dependent throughput of SAFARI-like
33
horns was carried out. The particular horn assembly that was analysed consisted
of a back-to-back horn structure with a rectangular waveguide connecting the two
horns. This particular format provided good beam control while also ensuring well
defined cut-off frequencies for the propagating waveguide modes. The measurements
and results are discussed in section 3.3 of this thesis. In this section a description
of the FTS system that was used for the measurements is presented.
Fourier transform spectroscopy is the best method to analyse the frequency response of a device under test (DUT) at THz frequencies, when a broadband source
is being used. An FTS based on a Michelson interferometer was set up in combination with a data acquisition device (DAQ) and a PC, and a bolometer cooled to
4K using a cryostat with liquid helium in the inner stage and liquid nitrogen in the
outer stage. The FTS and cryostat are shown in Figure 1.17. A program called the
Integrated Receiver Test & Control System (IRTECON) [Shitov et al., 1998] was
used for the control of the stepper motor in the FTS, and for compilation and initial
analysis of the data corresponding to the detected power by the bolometer. The
Michelson interferometer employed replaceable Mylar beam-splitters. Two different
beam splitters were used separately; one with a thickness of 56µm, and one with a
thickness of 12µm.
(a) FTS
(b) Detector Cryostat
Figure 1.17: The FTS and cryostat that were used for the horn throughput experiments.
Figure 1.18 shows a schematic of an FTS and how it functions by moving one
of the mirrors through a series of steps in order to vary the path length travelled by
one part of the split beam, thus varying the phase of the signal beam at the end of
that path. If the two parts of the beam are then combined at the detector they will
34
interfere in a manner that depends on the path length difference between the beam
paths, and the frequency distribution within the original beam. If the path length
difference is incrementally changed in uniform steps of ∆x a number N times across
a total scan path of X = N ∆x, and the intensity of the recombined beam f (n) is
measured at each point, an interferogram like the one shown in Figure 1.19(a) can
be created.
Figure 1.18: Schematic of a FTS and how it functions. The position of the moveable mirror is incrementally varied
by a stepper motor which is controlled by the IRTECON software. The different colours of the separate beams is
just for clarity, and should not be mistaken as different wavelengths. As is indicated, half of both beam 1 and beam
2 are lost from the system as they encounter the beam splitter for the second time.
(a) Spatial Data
(b) Spectral Data
Figure 1.19: (a): Interferogram for a back-to-back horn illuminated by a glow bar in a Michelson FTS. (b): Spectral
data for a back-to-back horn derived by applying a discrete Fourier transform (DFT) to the interferogram data.
The spatial data that is shown plotted in the interferogram (Figure 1.19(a)) can
35
then be used to obtain the frequency spectrum F (k) (Figure 1.19(b)) by applying a
discrete Fourier transform operation to the data as follows:
F (k) =
N
−1
X
f (n) exp (−2πink/N )
(1.1)
n=0
where n is the nth position of the moveable mirror when it is at n∆x from zero, and
f (n) is the intensity at position n.
The highest frequency that can be computed from the spatial data depends on
the size of steps ∆x. The smaller the steps the higher the frequencies that can be
extracted, and the maximum frequency is determined by fmax = fN Q =
1
.
2∆x
fN Q is known as the Nyquist frequency.
The bolometer that was housed in the cryostat could be coupled to one of three
low pass filters. The 312cm−1 and 100cm−1 filters were used for the measurements.
For further details of the measurements and results see section 3.3.
1.6
Transition Edge Sensors
A TES, or transition edge sensor, is essentially an extremely sensitive thermometer
which is made from a material that can exhibit superconducting behaviour. Its
principle of operation exploits the strong temperature dependent resistance of a
superconducting material when it is at its phase transition from a superconductor
to a normal metal. The material is held at its transition temperature Tc by a bias,
where a very small increase in temperature causes a large increase in resistance. As
can be seen by the steep curve in Figure 1.20, an increase in temperature of less
than 10mK gives rise to an increase in resistance of 20mΩ.
The fundamental phonon noise in a TES is directly related to temperature, as
described by the equation governing NEP (5.27) (see section 5.4.3). Since sensitivity
is inversely related to phonon noise and thus, temperature, TES’s for the far-IR
are typically operated at sub-Kelvin temperatures [Khosropanah et al., 2011]. The
TES is usually coupled to a thin piece of metal such as Ta which is deposited on
a substrate that is transparent to the radiation being detected. The thin sheet of
absorbing material is ideally impedance matched to free-space in order to absorb
as much radiation as possible. The TES/absorber pair is then usually thermally
36
Figure 1.20: Temperature dependent resistance profile for a Mo/Au bilayer TES [Figueroa-Feliciano, 2001].
isolated from the bath by mounting it on thin silicon nitride (SiN) legs, as shown in
Figure 1.21.
Figure 1.21: A micrograph of a TES coupled to a Ta absorber, and thermally isolated from its surroundings by SiN
legs [Khosropanah et al., 2011].
Over many years TES’s have been shown to be powerful detectors of single photons at wavebands ranging from infrared to gamma rays [Irwin and Hilton, 2005].
More recently they have been demonstrated to operate well in close-packed array format [Iyomoto et al., 2008] which will be required for the SAFARI instru-
37
ment. Research into the characterisation of TES’s specifically for far-IR wavelengths is ongoing with projects such as the SAFARI test-bed at SRON, for example
[Audley et al., 2013].
1.7
Conclusions
In this chapter a strong science case for far-IR astronomy was made, where it was
explained that three out of the four questions set out by ESA’s Cosmic Vision could
be answered by performing ultra-sensitive observations in the mid-IR and far-IR. It
was shown that a next generation space based mid-to-far infrared observatory will
be key in fulfilling this ambitious Cosmic Vision. The proposed JAXA/ESA space
observatory SPICA was described, with particular attention given to SPICA’s main
instrument SAFARI, a far-IR imaging FTS. It was explained that SPICA’s cooled
main mirror would be key in achieving background limited sensitivity for SPICA’s
suite of instruments, including SAFARI, and how SAFARI’s large format focal plane
arrays would provide the high mapping speeds necessary for carrying out the first
all sky spectroscopic surveys in the far-IR.
The principles of far-IR Fourier transform spectroscopy were outlined, including
a description of the Mach-Zender interferometer format that SAFARI will be based
upon. A description of the Michelson interferometer that was used for measurements
presented in this thesis was given, as well as a brief overview of the VNA system
that was employed for further measurements as a verification of simulation methods
presented throughout this thesis.
38
Chapter 2
Computational Methods for
Electromagnetic and Quasioptical
Design and Analysis
“Somewhere, something incredible is waiting to be known.”
- Carl Sagan
39
2.1
Overview of Computational Methods
In this chapter the various computational modelling approaches and techniques used
in the analysis of SAFARI-like receivers and detectors will be explained. Due to
the electrically large multi-moded structures that make up such systems, computationally efficient methods such as EM mode matching [Wexler, 1967] and GBM
(Gaussian beam mode) analysis [Wylde and Martin, 1993] are required. However
due to the relatively restrictive nature of such efficient modelling techniques more
flexible and exact simulation methods are also required, such as FEM employed by
the COMSOL Multiphysics software [COMSOL, 2011]. The drawback of the FEM
approach is that it is relatively inefficient compared to mode matching, for example.
Section 2.2 describes the GBM method which is very efficient at modelling
propagation through free-space and beam transformation by optical elements such
as lenses, mirrors, etc., and it was used in the analysis of the lens system that
was employed for the throughput experiments carried out at SRON, as described in
chapter 3. It provided a method to model the beam width of the THz radiation as
it propagated from the source to the feed horn, where the model was then compared
to the measured spot size. However GBM analysis is not suited to modelling fields
inside conducting structures such as waveguides, horns and cavities. Thus, other
techniques are required in order to characterise fully the SAFARI detectors.
Section 2.3 of this chapter summarises the theory for modelling the behaviour
of EM fields inside conducting structures such as waveguides and cavities. The
separated TE (transverse electric) and TM (transverse magnetic) field components
are tabulated for reference in later chapters. Derivations of the modal fields can be
found in [Olver et al., 1994] or [Clarricoats et al., 1984], for example.
The TE and TM waveguide mode field descriptions were required in many circumstances during the work presented in this thesis. For example the COMSOL
software package that was used for modelling horns, cavities and absorbers (chapters 3, 4 and 5) required the mathematical descriptions of modes as inputs for port
boundary conditions as a means of exciting EM fields through the structures. Although COMSOL does have some predefined mode options, if symmetry conditions
in the E or H-planes are exploited in order to reduce the problem size, then the user
must provide analytical field descriptions at the input ports.
40
The waveguide mode descriptions were also used in combination with a spherical phase cap [Clarricoats et al., 1984] as a means of describing approximate solutions for aperture fields of smooth walled conical and rectangular horns for comparison with exact solutions. Examples of these approximate methods are given in
chapters 3 and 4.
Throughout this thesis the waveguide mode solutions introduced above are used
in EM mode matching simulation methods, so a brief description of the approach is
given in section 2.4 of this chapter. The theory is described in much greater detail
in chapter 4 for the specific case of conical mode matching. In general EM mode
matching is a technique used to model EM fields inside conducting structures. It is
similar to GBM analysis in the sense that the electric and magnetic fields are again
defined as being a superposition of a set of modes, but the primary difference is the
boundary conditions due to the walls of the conducting structure.
The final section 2.5 of this chapter describes the COMSOL Multiphysics package and the FEM technique upon which it is based. COMSOL Multiphysics is used
throughout the work of this thesis to model a wide variety of problems relating to
SAFARI-like THz receiver systems. The COMSOL software provides an interactive
environment based on the proven FEM technique that can be used for modelling
and solving many kinds of scientific and engineering problems. The finite element
analysis is computed in combination with adaptive meshing and error control and
provides multiple direct and indirect numerical solvers. The software also offers a
wide range of post-processing tools for plotting data, making animations and carrying out other post-processing operations.
2.2
2.2.1
Gaussian Beam Mode Analysis
Background
Recent years have seen a fast growing interest in the application of millimetre and
sub-millimetre waves (THz waves). Some examples of these applications were given
in the introduction of this thesis. There are several different techniques for guiding
these waves from one point or plane to another, such as EM waveguides and stripline
systems. However, due to the difficulty in fabricating ever smaller components as
41
the desired operational frequency increases, and mass limitations for space based
applications, alternative wave guiding techniques are needed. To date fibre optics
techniques for the control of beams have not been possible due to no known suitable
materials being available. One such alternative is quasioptics [Goldsmith, 1992], and
it involves the free space propagation of narrow, quasi-collimated beams, sometimes
called pencil beams when the beam opening angle is very small. This quasioptical approach relies on using EM components such as waveguides and feed horns in
combination with optical elements such as lenses and mirrors. These narrow pencil beams can be mathematically described as a set of independently propagating
Gaussian beam modes which are solutions to the wave equation.
GBM analysis is an alternative way of mathematically analysing the evolution
of beams of EM radiation as they propagate through space. The classical approach
uses Fresnel diffraction integrals to examine the radiation pattern of light at a particular location in space. This process can be computationally inefficient and is very
abstract in terms of giving a physical feeling for how the beam evolves. GBMA provides a more efficient and more intuitive means of modelling such beams in terms
of both their near and far field diffraction patterns. The same limitations of the
paraxial approximation [Hyde, 2000] that are assumed for Fresnel diffraction apply
to GBM analysis also [Murphy and Egan, 1993]. The basic approach is based on
decomposing the electric or magnetic field into a set of independently propagating
modes by performing an overlap integral with the field and each mode to determine
the component mode amplitudes at a plane where the field is well defined. The
component modes can then be independently propagated through the system or
through free space, and then simply summed in a linear fashion to reproduce the
evolved field at any other point or plane. The technique can be used for propagation through free-space, propagation through lenses, mirrors, polarizing grids, etc.,
and can be applied to more complex diffraction phenomena as described in many
publications by the THz Optics group at NUIM, such as classical phase aberrations [Trappe et al., 2003], GBM beam shaping [Lavelle and O’Sullivan, 2010], the
quasi-optical analysis of Bessel beams in the far infrared [Trappe et al., 2005].
In the following sections some examples of free-space propagation using GBM
analysis are presented to demonstrate the power of the technique and to validate
the GBM code written by the author. The derivation of the GBM theory can be
42
found in many publications, including [Goldsmith, 1998].
For a normalised field (i.e.
R~ 2
~ field, the mathe|E| dA = 1), for example the E
matical description can be written as [Wylde and Martin, 1993]:
s
(x2 + y 2 )
(x2 + y 2 )
2
exp −jk
exp −
E(x, y, z, t) =
πW (z)
W (z)2
2R(z)
(2.1)
exp [jφ(z)] exp [−jkz] exp [jωt]
~ ω = 2πν, where ν is the frequency
where E represents one of the components of E,
of radiation, k = 2π/λ, where λ is the wavelength of radiation, and R(z), W (z),
and φ(z) have the following definitions:
"
2 #
πW02
R(z) = z 1 +
(Phase Radius of Curvature)
λz
"
λz
W (z)2 = W02 1 +
πW02
λz
φ(z) = arctan
πW02
2 #
(Beam Width)
(Phase Slippage)
(2.2)
(2.3)
(2.4)
where W0 is called the beam waist, whose radius will depend on the wavelength of
radiation, and corresponds to the minimum value that W (z) will take. In (2.1) to
(2.4) above, it is assumed that propagation is in the z direction. Figure 2.1 illustrates
some of the terms in the mathematical description of a quas-optical Gaussian beam.
Figure 2.1: Illustration of a Gaussian beam showing the beam width and phase radius of curvature as a function
of propagation distance. The propagation distance is normalised to the Rayleigh range Zc , which is defined as
Zc = πW02 /λ. The Rayleigh range is also the distance from the position of the waist z0 , in the direction of
√
propagation, to the place where the value of the beam width is 2W0 . A related parameter is called the confocal
distance (sometimes called the depth of focus), and is defined as 2Zc . Image credit: Dr. Ronan Mahon
43
Equation (2.1) describes the propagation of an ideal, purely Gaussian beam.
This is called the fundamental Gaussian, or fundamental mode as higher order modes
are usually also present to some degree in a propagating beam. To model accurately
quasioptical systems it is usually necessary to include some of these higher order
modes. This is achieved by combining (2.1) with a set of appropriate polynomials,
the form of which depends on the geometry of the problem (i.e. Cartesian or cylindrical polar). Each higher order polynomial governs the structure of one particular
mode while multiplication by the Gaussian term (Gaussian envelope common to
all modes) “kills off” the amplitude in the transverse direction (i.e. the field has
finite extent). The form of these modes in the different coordinate systems will be
introduced in the simple examples considered in the following sections. It should be
noted that the phase slippage term φ(z) is mode dependent as there is a different
amount of phase slippage for each mode (the higher order modes fall out of phase
faster than the lower order modes, relative to an infinite plane wave travelling in
the same direction).
2.2.2
Gaussian Beam Mode Analysis of Uniformly Illuminated Apertures
The field that describes a uniformly illuminated circular aperture can be defined
as being unity across the aperture and zero everywhere else. This is the form of
the field, produced by a distant point source, at the aperture of a telescope such as
SPICA for example. The image that is formed on the detector array plane then is
called the point spread function [Cox and Hodge, 2006].
The current SAFARI test-bed being used at the SRON to characterise a single
SAFARI pixel employs a small circular aperture between the thermal source and
the conical horn that feeds the cavity coupled TES detector. A basic schematic
of the source, aperture and horn layout is shown in Figure 3.35, and a detailed
description of the SAFARI test-bed is given in chapter 3, section 3.3. The source,
which illuminates the circular aperture, is actually a conical blackbody radiator
but for simplicity in the model it will be assumed that the aperture is uniformly
illuminated with monochromatic light.
It is most convenient and efficient to decompose the aperture field into modes
44
with similar symmetry to that of the field, i.e. cylindrical polar or Cartesian. Thus,
for the example of a small circular aperture, the best suited basis set is the LaguerreGauss mode set. The rotationally symmetric (about the axis of propagation) higher
order Laguerre-Gauss modes, travelling in the z direction, have the following form
(in the appropriate cylindrical polar coordinate system) [Murphy et al., 1997]:
s
(r2 )
(r2 )
2
En (r, z) =
L
(r,
z)
exp[−
]
exp[−jk(z
+
) + jφ(z)] (2.5)
n
πW (z)2
W (z)2
2R(z)
where Ln (r) is a Laguerre polynomial that is a function of the propagating mode
number (n), and the off-axis distance (r) perpendicular to the direction of travel of
the wave. The Laguerre polynomials have the form [Pampaloni and Enderlein, 2004]:
Ln (r) =
er dn
(e−r rn ).
n! dr n
In order to calculate the mode coefficients An , an overlap integral with the
aperture field and each mode was calculated [Murphy and Egan, 1993]. Since the
waist position of the modes was unknown it was assumed that the aperture plane is
the waist position for each mode, therefore z = 0, φ(z) = 0 and R(z) = R(0) = ∞.
Mathematically, this is expressed as:
s
Z a
2
2r2
r2
An (W0 ) =
(L
(
)
exp[−
]2πr)dr
n
πW02 0
W02
W02
(2.6)
where n is the mode number and is an integer that goes from 1 up to the required
number of modes. Note that W0 was left as a variable so that it could be varied
in order to minimise the number of modes required [Murphy and Egan, 1993]. The
integration is over 0 to a where a is the radius of the circular aperture. The choice
of waist radius and position is somewhat arbitrary, and non-ideal values will not
necessarily result in an incorrect field description. Rather, unrealistic or non-ideal
values will simply result in a larger mode set being required for a given level of
accuracy.
By incrementally varying the beam width of the modes from 0 to a in steps of
0.01a, and examining the reconstructed aperture field at each iteration, it was found
that a value of W0 = 0.2a provides a reasonably accurate description of the field with
relatively few modes. Figure 2.2 shows slices through the aperture “top-hat”-like
truncated field for three cases, each calculated with a different number of modes. It
can be seen that as more modes are used the field becomes more accurately defined
with the ringing being reduced and the edges of the beam becoming sharper. The
on-axis magnitude is normalised to unity in each case.
45
(a) 8 modes
(b) 16 modes
(c) 32 modes
Figure 2.2: Slices through the aperture “top-hat” electric field intensity for increasing number of modes.
32 modes (n = 32) were then propagated to, and combined at, various locations
in order to investigate how the diffraction pattern varied as propagation distance
z was increased. 3D plots of the electric field are shown on Figure 2.3 for four
propagation distances. As can be seen the electric filed experiences a lot of change
in the near field (Fresnel region) as would be expected. The far-field is reached by
z=
2a2
,
λ
as can be seen by the roughly sinc profile in Figure 2.3(d). Contour plots
of off-axis intensity as a function of increasing propagation distance are then shown
in Figure 2.4.
Similar in a way to the example of a uniformly illuminated circular aperture, the field that describes a uniformly illuminated rectangular aperture can be
defined as being unity across the aperture and zero elsewhere. Such an illuminated rectangular aperture was used in the SAFARI-like horn throughput experiments that are described in chapter 3. The equation for a Gaussian beam mode
that has rectangular symmetry travelling in the z direction has the following form
[Pampaloni and Enderlein, 2004]:
s
E(x, y, z) =
2
exp[−
2
1
πW (z)2 2m+n−1 n!m!
2
Hn (y, z)Hm (x, z)
(2.7)
2
(x + y )
(x + y )
] exp[−jk(z +
) + j(m + n + 1) + jφ(z)]
2
W (z)
2R(z)
where Hm (x, z) and Hn (y, z) are Hermite polynomials that are functions of the
propagating mode number (m,n), and the off-axis distance (x,y) perpendicular
to the direction of travel of the wave. The Hermite polynomials have the form
[Pampaloni and Enderlein, 2004]: H(m, x) = (−1)n ex
2
dn −x2
e .
dxn
Following similar steps as before, the field at a small square aperture was decomposed in terms of such Hermite-Gauss modes, thus allowing the aperture field
46
Intensity vs. Off -Axis Distance
1.0
0.5
0.0
-0.2
0.2
0.0y
0.0
x
-0.2
0.2
(a) z=0
(c) z =
a2
3λ
(b) z =
a2
6λ
(d) z =
2a2
λ
Figure 2.3: Diffraction patterns for various propagation distances from a circular aperture. 32 Laguerre-Gauss
modes were used in the analysis.
(a)
(b)
Figure 2.4: Diffraction pattern for increasing propagation distance from a circular aperture. An expanded view of
the Fresnel region bounded by the red box in (a) is shown in (b). In both plots the colour legend represents electric
field intensity. 32 Laguerre-Gauss modes were used.
to be reconstructed as shown in Figure 2.5(a), and the diffraction pattern to be
computed at any distance from the aperture. 100 modes (m = 10, n = 10) were
used for the analysis and the diffraction patterns for three propagation distances are
47
shown in Figures 2.5(b) to 2.5(d). It is clear that the far-field (Fraunhofer region)
has been reached by z =
2a2
λ
as the sinc profile is clearly visible through the x and y
centres in Figure 2.5(d). Figure 2.6 shows contour plots, illustrating how the electric
field varies rapidly in the near-field before settling down in the far-field.
(a) z=0
(c) z =
a2
3λ
(b) z =
a2
6λ
(d) z =
2a2
λ
Figure 2.5: Diffraction patterns for various propagation distances from a rectangular aperture. 100 modes were used
in the analysis.
(a)
(b)
Figure 2.6: Diffraction pattern of electric field intensity for increasing propagation distance from a square aperture.
An expanded view of the Fresnel region bounded by the red box in (a) is shown in (b). 100 modes were used.
48
2.2.3
Gaussian Beam Mode Analysis of EM Horns and Waveguides
The field at the aperture of a corrugated conical horn antenna can be approximated
by a circularly symmetric zero order Bessel function [Wylde and Martin, 1993] as
|E(r)| ∝ J0 (kr) for r ≤ a, where k =
2.405
,
a
a is the radius of the horn aperture,
and J0 is a zero order Bessel function of the first kind. The first root of such a
Bessel function is 2.405; hence the field goes smoothly to zero at the aperture edge
(r = a). The exact aperture fields of both rectangular and conical horns are discussed in chapters 3 and 4 and are generally calculated with rigorous mode matching
calculations. For the current purposes however the Bessel function approximation
is sufficient.
Again due to the circular symmetry of the aperture field, the most convenient
free space modes to use are Laguerre-Gauss modes, and as always we aim to minimise
the number of modes required for the analysis. In this case, because the field is quasiGaussian, we maximise the amplitude of the fundamental mode and thus only a few
higher order modes are required to make up most the remaining field. This gives
W0 = 0.6435a at the aperture of the horn, leading to a fraction of 0.9792 of the
total power in the fundamental mode. For the higher order modes we again carry
out overlap integrals to compute the mode coefficients, similar to the integrals used
for the uniformly illuminated aperture examples.
An (W ) =
s
2
πW02
Z
a
(E0 J0 (2.405r)Ln (2
0
r2
r2
)
exp[−
]2πr)dr
W2
W2
f or n = 1, 2, 3, ..N
(2.8)
where E0 is the amplitude of the aperture field, and N is the desired number of
modes.
Table 2.1 shows the coefficients for the first seven modes and the corresponding
fraction of power carried by each mode for an example corrugated conical horn with
dimensions of a = 1mm, L = 5mm, and λ = 0.5mm. A schematic of the horn is
shown in Figure 2.7(a).
Figure 2.8 shows the aperture field of the example horn reconstructed using
various numbers of Laguerre-Gauss modes. It was determined that using 15 such
49
Ln
n=0
n=1
n=2
n=3
n=4
n=5
n=6
n=7
An
1.1297
-0.0001
-0.1375
-0.0491
0.0224
0.0389
0.0228
0.0002
Pn
0.9792
6.9 × 10−9
0.0145
0.0019
0.0004
0.0012
0.0004
3.3 × 10−8
Table 2.1: Electric field and power coefficients of a reconstructed beam from a corrugated conical horn.
modes gave rise to a reconstructed field with 99.9% accuracy.
(a)
(b)
Figure 2.7: (a): Diagram of example corrugated conical horn showing dimensions. (b): Illustration of the equiphase
surface of the electric field inside the horn.
(a)
(b)
(c)
Figure 2.8: Bessel function aperture field and reconstructed aperture field for (a): 5 modes, (b): 10 modes, and (c):
15 modes.
In this example it was assumed that the beam waist and phase centre were
positioned at the aperture/mouth of the horn (z=0), whereas in reality, the phase
centre typically exists inside the horn [Goldsmith, 1998]. According to Goldsmith,
for a given conical corrugated horn, both the position of the beam waist z0 and its
radius W0 can be calculated by two relatively simple equations ((2.9) and (2.10),
respectively). Thus, for a corrugated horn with the values of a = 1mm, L = 5mm,
and operating at a wavelength of λ = 0.5mm, the corresponding values for the
position and radius of the waist are: z0 = 1.0654mm and W0 = 0.5708mm. In fact,
50
as demonstrated by Lucci et al., the position of the phase centre of the horn can be
optimised to suit the particular optical system being used [Lucci et al., 2004].
L
z0 =
1+
W0 = q
λL
π(0.6435a)2
0.6435a
1+
(2.9)
2
(2.10)
2
)2
( π(0.6435a)
λL
The near, mid and far-field radiation patterns (in dB) are shown in Figure 2.9
with the normalised patterns also displayed. By comparing Figure 2.9(c) to 2.9(d)
it can be seen that the structure of the field does not significantly change over a
distance from z = 10mm to z = 100mm, with only the side-lobes appearing more
defined. This indicates that the beam reaches the far-field within 10mm of the
aperture. In fact the confocal distance (the distance from the waist at which the
edge of the Fresnel region is being approached) is given by πW02 /λ ≈ 2mm.
(a)
(b)
(c)
(d)
Figure 2.9: Power patterns (in dB) of a corrugated conical horn for various propagation distances. (a): Aperture
field, (b): intermediate field, (c): beyond Fresnel region, and (d): far-field.
For other data that is also suited for analysis in polar cylindrical coordinates
like the small circular aperture or corrugated conical horn examples above, but
without the rotational symmetry that was present in those examples, a higher order
51
Laguerre-Gauss basis set can be used. The associated Laguerre-Gauss mode basis
set Lαn , appropriate for decomposition of fields that are not rotationally symmetric
are similar to the Laguerre-Gauss modes Ln described above, but they also have
azimuthal dependence governed by the order α in the superscript of Lαn . In fact the
case of the rotationally symmetric Laguerre-Gauss mode set can be considered as
a special case of the more general associated Laguerre-Gauss mode set, where it is
the associated Laguerre polynomial of order 0 that is used (L0n ). The description of
these free-space associated Laguerre-Gauss modes is as follows [Trappe et al., 2005]:
~ αm
Ψ
(W, R, r, φ, z) =
α
ψm
(W, R, r, φ) exp (−ikz + iφm,α (z))
cos αφ
sin αφ
î ±
ĵ
− sin αφ
cos αφ
(2.11)
where
α
ψm
(W, R, r, φ) =
q
2m!
πW 2 (m+α)!
exp
−r 2
W2
2r2
W2
α2
Lαm (2r2 /W 2 ) exp (−ik (r2 /2R)).
Such a basis set, 2.11, is well suited for the decomposition, reconstruction, and
propagation of smooth walled conical horn aperture fields such as those presented
in chapter 4 of this thesis. Even though smooth walled conical horn structures have
rotationally symmetric apertures (generally), the fields at the horn apertures (T E11
mode for example) are typically asymmetric due to the different boundary conditions
in the E-plane and H-plane. Of course this can be overcome by introducing carefully
designed step discontinuities or change in flare angles in order to excite higher order
modes (typically the T M11 ) so that mode-mixing can build symmetry in the aperture field. Of course profiled corrugated horns can also produce symmetric fields,
as demonstrated by Ade et al., for example, when designing the ultra-Gaussian feed
horns for the CLOVER experiment [Ade et al., 2009]. Examples of EM field analysis with associated Laguerre-Gauss modes applied to multi-moded structures are
presented in section 5.4.1 of chapter 5. As is shown in that section, this type of
analysis is required for modelling part of the SAFARI pixel with mode matching
techniques.
As an example here, the T E11 mode that is present at the aperture of a cylindrical waveguide feed, such as the guide that feeds the SAFARI L-Band detectors
~ 0n
is now decomposed using the free-space associated Laguerre-Gauss mode sets Ψ
~ 6 for n = 0 to n = 40. The waveguide that was modelled was given a radius
to Ψ
n
of 1mm, the wavelength of radiation was set to 2mm (f = 150GHz), and the opti52
mum beam waist radius was found to be 1.65mm. The technique for calculating the
mode coefficients is the same as in the previous examples, where overlap integrals
are calculated for the waveguide aperture field with each of the free-space modes.
Of course in this case the coefficients must be worked out for each mode of radial
degree n and azimuthal order α. The first ten values for each azimuthal order are
shown in Table 2.2.
Aαn
α=0
α=1
α=2
α=3
α=4
α=5
α=6
n=0
1.0908
0
0.3592
0
0
0
0
n=1
0.3530
0
0.3821
0
0
0
0
n=2
0.0096
0
0.3177
0
0
0
0
n=3
0.1109
0
0.2279
0
0
0
0
n=4
0.1178
0
0.1439
0
0
0
0
n=5
0.0771
0
0.0794
0
0
0
0
n=6
0.0261
0
0.0378
0
0
0
0
n=7
0.0163
0
0.0170
0
0
0
0
n=8
0.0431
0
0.0123
0
0
0
0
n=9
0.0539
0
0.0184
0
0
0
0
Table 2.2: Coefficients of for the first 10 radial modes of various azimuthal order for decomposed cylindrical waveguide
aperture field.
What is clear from Table 2.2 is that the T E11 waveguide mode only couples to
~ 0 and Ψ
~ 2 free-space associated Laguerre-Gauss modes. This is simply due to
the Ψ
n
n
the orthogonality of the modes in azimuthal dependence α, [Trappe et al., 2005].
This property of the associated Laguerre-Gauss modes is very useful as it significantly reduces the number of calculations that are required for analysing the
free-space propagation of the fields from multi-moded systems (see section 5.4.1 of
chapter 5).
~ 0 and 5
Figure 2.10(a) shows the reconstructed waveguide aperture field for 5 Ψ
n
~ 2n modes with visible differences in the data compared to the original field, which
Ψ
is shown for comparison. Figure 2.10(b) then shows the reconstructed waveguide
~ 0 and 20 Ψ
~ 2 modes with much better agreement now clearly
aperture field for 20 Ψ
n
n
displayed. Accepting this as verification that the correct mode descriptions were defined in the code, the modes were then propagated to the far-field by simply entering
53
a large value for the propagation distance z. The modes were then linearly summed
in the usual manner and the resulting far-field patterns are shown in Figure 2.11.
A FEM analysis of a similar model was carried out using COMSOL for comparison,
and the FEM results are also shown in Figure 2.11. Although there is reasonable
agreement in the general trend of the curves, there are some differences that require
explanation.
10 Modes
40 Modes
1.0
1.0
0.8
0.8
Recon H
Recon H
0.6
0.6
Recon E
Recon E
0.4
0.4
WG H
WG H
0.2
0.2
WG E
0.0
0.2
0.4
0.6
0.8
1.0
0.0
~ 0n and 5 Ψ
~ 2n
(a) 5 Ψ
WG E
0.2
0.4
0.6
0.8
1.0
~ 0n and 20 Ψ
~ 2n
(b) 20 Ψ
Figure 2.10: Reconstructed aperture field of a cylindrical waveguide for (a): 10 modes, (b): 40 modes.
Figure 2.11: Far-field radiation patterns of a cylindrical waveguide using 40 free-space associated Laguerre-Gauss
modes. Cuts are shown through φ = 0o (E-Plane), φ = 90o (H-Plane), and the cross-polar computed across the
φ = 45o plane.
The main reason for the differences in Figure 2.11 is based on the fact that
the COMSOL model includes the effects of the domain around the aperture of
54
the waveguide, whereas the modal analysis neglects the behaviour of the fields in
that domain. The COMSOL results will vary depending on whether the waveguide
aperture is surrounded by a conducting ground plane or simply free-space, and in
this case a conducting ground plane was used in the simulations. The effects of
including such a ground plane in the FEM models is explained in detail in section
3.1.6 with more examples presented, and, as will be shown, the effect of including a
conducting ground plane at the aperture of a radiating structure is only significant
for a uniform waveguide. For this reason a smooth walled conical horn antenna was
also modelled with the free-space associated Laguerre-Gauss modes for verification
of the code. Since a ground plane has no significant effect on the radiation patterns
of such a feed, better agreement between the FEM approach and modal approach
was expected.
The conical horn was given a length of 3mm, an exit aperture of diameter 1mm,
and an entry aperture of diameter 0.25mm, leading to a half angle of 7.2o . Due to the
small entrance aperture the frequency was set to 600GHz to allow the fundamental
mode to propagate through the horn. Figure 2.12 shows the far-field calculation
that was achieved using 40 free-space modes with a beam waist of 1.8mm. The
results of a similar simulation performed using the FEM approach with COMSOL
(FEM approach to be introduced in section 2.5) are shown for comparison. As can
be seen there is reasonably good agreement for the two approaches in both co-polar
planes and the cross-polar distribution, thus further verifying the modal code.
The above examples of expanding asymmetric waveguide and horn fields in
terms of associated Laguerre-Gauss modes are important because in order to model
the full SAFARI-pixels (both S-Band and L-Band) with modal techniques the current version of NUIM’s mode matching code SCATTER will need to be extended
to include such free-space modes. The reason for this is due to a vacuum gap that
must be included in the pixel design to allow sufficient space between the horn array
block and the delicate TES detector array chip. This is explained in more detail in
chapter 5, and Figure 5.1 can be referred to for an illustration.
55
(a) Far-field comparisons
(b) E-Plane comparisons
(c) H-Plane comparisons
(d) X-Polar comparisons
Figure 2.12: Far-field radiation patterns of a smooth walled conical horn antenna computed with 40 associated
Laguerre-Gauss free-space modes, and FEM results shown for comparison. (a): All three cuts, (b): E-Plane cut
(φ = 0o ), (c): H-Plane cut (φ = 90o ), and (d): X-Polar cut (φ = 45o ).
2.2.4
Propagation Through Quasioptical Systems: Mirrors
and Lenses
In the experimental system set up at SRON to probe the multi-mode nature of
SAFARI-like horns (see chapter 3) a beam guide system was used to control the
free-space propagation of THz beams through a vacuum system. This required the
design of a quasi-optical lens based system. We now explain how to model such
quasi-optical lens systems using the so-called ABCD matrices from ray optics.
The radius of curvature of a geometrical optics beam R is analogous to the
complex beam parameter q of a Gaussian beam, so it is reasonable to assume that
the effects of optical components on a Gaussian beam can be analysed in terms of the
effects of the corresponding ABCD matrices on the parameter q [Goldsmith, 1998].
qout =
A · qin + B
C · qin + D
56
(2.12)
Furthermore, a single ABCD matrix representing a complete quasioptical system can be constructed by cascading the ABCD matrices representative of each
element of the system. The final matrix can then be applied to the complex beam
parameter at the input plane, qin , with the result being the complex beam parameter at the output plane, qout . This output complex parameter can then be used
to calculate the beam width W , the phase radius of curvature R, and the phase
slippage φ for each mode at the output plane. From these values the complete field
can be reconstructed.
Since the complex beam parameter q is related to the propagation distance z
and the confocal distance z0 through the following relationships:
1
1
λ
2
=
= −j
=−
2
q0
jz0
πW0
jkW02
(2.13)
1
1
1
λ
1
=
=
=
−j
2
z + q0
z + jz0
R(z)
πW (z)
q(z)
(2.14)
and
where the definition of the confocal distance was used: z0 =
πW02
,
λ
then the phase ra-
dius of curvature, beam width, and the phase slippage can be expressed as functions
of q by the following relationships [Goldsmith, 1998]:
R(q) =
−1
1
Re
q
!1/2
λ
πIm(− 1q )
1
φout − φin = ∆φ(q) = − Arg A + B
qin
W (q) =
(2.15)
(2.16)
(2.17)
Based on the above definitions the GBM analysis code that was written by the
author, and used to model the examples of radiating apertures and feed horns presented earlier in this section, was extended to include analysis of lenses, mirrors, and
any other optical components that can be defined by an ABCD matrix. The code
requires an initial complex beam parameter as an input, together with the A, B, C,
and D values for each optical element that is to be modelled. A piecewise function is
then defined to take account of the changing overall matrix as the propagation distance increases, and as more optical elements are encountered by the beam. Each
time a new section of free-space or an optical element is encountered, the corresponding matrix is cascaded with the overall matrix up to that point. A graphical
57
user interface (GUI) was built into the code to allow the user to vary the parameters
of the optical elements, for example the positioning or focal lengths of lenses in a
system. An example of a system that was modelled with the code is presented in
section 3.3, where a three lens focussing system was used in the experimental set-up
for measurements of the throughput of a SAFARI-like horn antenna.
2.3
Waveguides
2.3.1
Background
The TE and TM field descriptions of EM waves in waveguide structures were required for multiple simulations as part of the work carried out for this thesis (see
chapters 3 and 4). For example the FEM based software COMSOL requires the user
to define the field structure of the particular mode to be excited at the input port
boundary if symmetry conditions are used to reduce the problem size computationally. Also the sections of mode matching code written by the author required the
normalised mode definitions to be defined so that analysis of mode-to-mode scattering could be correctly calculated (see for example chapter 4). The description of
how EM fields propagate through conducting waveguide structures can be found in
many sources, for example [Olver et al., 1994] and [Clarricoats et al., 1984]. Since
the theory is well known, only a basic summary of the modal field descriptions will
be outlined in this section, and some example plots of the fields will be shown to
demonstrate the validity of the code written by the author. The step-by-step mode
derivations are not included since they can be found in most electrical engineering
textbooks [Anwane, 2007], for example.
The solutions to Maxwell’s equations with PEC boundary conditions can be
classified into different groups, depending on whether the electric and magnetic fields
have zero or non-zero longitudinal components. The possible mode types are listed
in the table below.
58
Electric Field
Magnetic Field
Mode Notation
Ez = 0
Hz 6= 0
TE (Transverse Electric)
Ez 6= 0
Hz = 0
TM (Transverse Magnetic)
Ez = 0
Hz = 0
TEM (Transverse Electromagnetic)
Ez 6= 0
Hz 6= 0
HE or EH (Hybrid Modes)
Table 2.3: Definitions and notation of waveguide fields.
2.3.2
Rectangular Waveguide
Excluding the case of TEM modes as they do not exist as harmonic modes of a waveguide, and excluding HE and EH modes as they can be considered as combinations of
TE and TM modes, the Cartesian components of the possible transverse modes in a
rectangular waveguide are listed in Table 2.4. The rectangular waveguide has width
a in the x direction, and height b in the y direction. The m and n values describe
the order of the T Emn modes. β is the propagation constant and it is related to the
free-space wavenumber k, and the waveguide wavenumber γ by k 2 = kc 2 − γ 2 , where
γ = jβ and kc is the cut-off wavenumber [Clarricoats et al., 1984] (although slightly
ω
different notation is used in the referenced text). ZTE = Z βc
and ZTM = Z βc
, where
ω
Z is the impedance of free space (≈ 377Ω). Finally, ω is the angular frequency of
radiation (ω = 2πf ).
TE Modes
TM Modes
jα
Ex = ZTE kjβc 2 nπ
cos mπx
e
sin nπy
b
a
b
jα
sin mπx
e
cos nπy
Ey = −ZTE kjβc 2 mπ
a
a
b
jα
Ex = − kjβc 2 mπ
cos mπx
e
sin nπy
a
a
b
sin mπx
ejα
cos nπy
Ey = − kjβc 2 nπ
b
a
b
jα
nπy
sin
e
Ez = sin mπx
a
b
jα
1 jβ nπ
sin mπx
e
cos nπy
Hx = ZTM
a
b
kc 2 b
jα
nπy
1 jβ mπ
mπx
Hy = − ZTM
cos
e
sin
2 a
a
b
kc
Ez = 0
jα
sin mπx
e
cos nπy
a
b
nπ
Hy =
cos mπx
ejα
sin nπy
b
a
b
jα
cos nπy
e
Hz = cos mπx
a
b
Hx =
jβ
kc 2
jβ
kc 2
mπ
a
Hz = 0
Table 2.4: Mode expressions in a rectangular waveguide where α = (ωt − βz).
Figure 2.13(a) shows the lowest order TE mode for a rectangular waveguide of
width a = 30µm and height b = 15µm, operating at a frequency of 5THz. Some
higher order propagating modes are also shown in Figure 2.13(b) - 2.13(f), where the
59
frequency was increased for each simulation to ensure it was above cut-off frequency
for each mode (ν = 29THz for T E42 for example). Figure 2.14(a) shows the lowest
order TM mode for the same rectangular waveguide of width a = 30µm and height
a = 15µm. Some higher order propagating T M modes are also shown in Figure
2.14(b) - 2.14(f).
(a) T E01 E − F ield
(b) T E10 E − F ield
(c) T E31 E − F ield
(d) T E33 E − F ield
(e) T E40 E − F ield
(f) T E42 E − F ield
Figure 2.13: Plots of electric field intensity for TE modes of a rectangular waveguide of width a = 30µm and height
a = 15µm. (a): T E10 , (b): T E01 (c): T E31 (d): T E33 (e): T E40 (f): T E42
.
(a) T M11 E − F ield
(b) T M21 E − F ield
(c) T M31 E − F ield
(d) T M33 E − F ield
(e) T M41 E − F ield
(f) T M42 E − F ield
Figure 2.14: Plots of electric field intensity for TM modes of a rectangular waveguide of width a = 30µm and height
a = 15µm. (a): T M11 , (b): T M21 (c): T M31 (d): T M33 (e): T M41 (f): T M42
.
60
2.3.3
Cylindrical Waveguide
The components of the transverse modes that can exist in a cylindrical waveguide
are listed in Table 2.5. The p′νl values in the T E mode descriptions correspond to the
ν th roots of the derivative of the lth order Bessel functions. The pνl values in the T M
mode descriptions correspond to the ν th roots of the lth order Bessel functions. For
the lowest order mode which is the T E11 mode, (ν = 1 and l = 1), p′11 = 1.8412, and
for the lowest order TM mode (T M11 ) p11 = 2.405. The roots of Bessel functions and
their derivatives corresponding to higher modes can be found widely in the literature,
for example [Beattie, 1958]. All other symbols (β, kc , etc.) have the same definitions
as for the rectangular modes described above. The sin(νφ), cos(νφ), etc. terms in
the mode descriptions in Table 2.5 represent two separate polarizations of the fields.
TE Modes
TM Modes
Er = ZTE kjβν
2 Jν
c r
Eφ =
p′νl r
p′
ZTE kjβc 2 aνl Jν ′
Ez = 0
Hr =
Hφ =
a
jα − sin(νφ)
e
p′νl r
a
cos(νφ)
ejα cos(νφ)
sin(νφ)
′
′
−jβ pνl
′ pνl r
jα cos(νφ)
e
2 a Jν
a
kc
′ sin(νφ)
pνl r
sin(νφ)
jβν
ejα −cos(νφ)
J
a
kc 2 r ν
H z = Jν
p′νl r
a
ejα
cos(νφ)
sin(νφ)
ejα cos(νφ)
sin(νφ)
jα − sin(νφ)
pνl r
−jβν
E φ = kc 2 r Jν a e
cos(νφ)
Ez = Jν pνla r ejα cos(νφ)
sin(νφ)
sin(νφ)
pνl r
1 jβν
Hr = ZTM kc 2 r Jν a ejα −cos(νφ)
1 jβ pνl
J ′ pνla r ejα cos(νφ)
Hφ = − ZTM
sin(νφ)
kc 2 a ν
Er =
−jβ pνl
J ′
kc 2 a ν
pνl r
a
Hz = 0
Table 2.5: Mode expressions in a cylindrical waveguide where again α = (ωt − βz).
Figure 2.15(a) shows a cross-sectional cut through the lowest order mode (T E11 )
for a cylindrical waveguide of radius a = 50µm, operating at a frequency of 1.8THz
which is just above the cut-off frequency of 1.76THz. Figure 2.15(b) shows E-plane
and H-plane cuts through the same T E11 mode. The electric field intensity for
some higher order propagating T E modes is shown plotted in Figure 2.16, where
the frequency was increased in each simulation to ensure propagation of the mode
(18.4THz for the T E54 mode, for example). The electric field intensity for some
higher order propagating T M modes is shown plotted in Figure 2.17. Again, the
frequency was increased for each simulation to ensure propagation of the mode being
analysed (19.9THz for the T M54 mode, for example).
61
(a)
(b)
Figure 2.15: Plots of T E11 electric field intensity for a cylindrical waveguide of radius a = 60µm. (a): 2D planar
slice plot, and (b): 1D cuts through E and H-planes.
(a) T E21 E − F ield
(b) T E22 E − F ield
(d) T E33 E − F ield
(e) T E52 E − F ield
(c) T E31 E − F ield
(f) T E54 E − F ield
Figure 2.16: Plots of electric field intensity for higher order TE modes of a cylindrical waveguide of radius a = 6µm.
(a): T E21 , (b): T E22 (c): T E31 (d): T E33 (e): T E52 (f): T E54 .
As was already indicated, the theory presented in this section is widely accessible in the literature [Olver et al., 1994], [Clarricoats et al., 1984]. The simulation
work presented throughout this thesis (chapters 3, 4 and 5) required code to be
written that was capable of modelling the modal fields, thus the mode descriptions
were required to be defined computationally. In contrast to the readily available
62
(a) T M11 E − F ield
(b) T M22 E − F ield
(c) T M31 E − F ield
(d) T M33 E − F ield
(e) T M52 E − F ield
(f) T M54 E − F ield
Figure 2.17: Plots of electric field intensity for higher order TM modes of a cylindrical waveguide of radius a = 6µm.
(a): T M11 , (b): T M22 (c): T M31 (d): T M33 (e): T M52 (f): T M54 .
rectangular and cylindrical coordinate solutions, conical mode solutions were much
harder to find in the literature. Thus, conical EM modes are derived in chapter 4.
2.4
2.4.1
Electromagnetic Mode Matching
Background
In the mode matching technique a corrugated or profiled waveguide structure is
regarded as a sequence of short uniform waveguide segments which step in radius
(cylindrical) or width and height (rectangular) at the junctions between the segments. For each segment the natural modes of propagation are the usual TE and
TM modes of a uniform waveguide, and the mathematical description of such fields
for both polar cylindrical and Cartesian geometries were presented in section 2.3
of this chapter. The theory of EM mode matching is already well established and
it is described in detail in many texts [Olver et al., 1994], [Clarricoats et al., 1984].
NUIM has a strong heritage in the subject of mode matching, where previous graduate students have developed code that is suited for modelling conical feed horn
63
structures, both smooth walled and corrugated [Gleeson, 2004], [Colgen, 2001]. The
result was a highly efficient simulation tool called Cylindrical SCATTER, and it
has been used to model complex horn structures such as the HIFI (High Frequency
Instrument) horns on the Planck observatory [Murphy et al., 2010]. More recently
a Cartesian based mode matching code was written [Doherty, 2012] that is suited
to modelling the behaviour of smooth walled or corrugated rectangular horns, and
is flexible enough to account for both straight or profiled geometries.
The topic of EM mode matching is introduced here as it is used throughout
this thesis as a means of comparing results with other simulation methods such as
the FEM approach. Also, the author was involved in verifying a new section of
cylindrical mode matching code that was developed with the aim of including cavity
mounted absorbers in the models. This work is discussed in chapter 5. The mode
matching theory was then extended further by the author through the development
of a new mode matching program. The new Spherical SCATTER code was aimed
at modelling smooth walled conic section horns in a more exact way compared to
Cylindrical SCATTER. It was also aimed at increasing the computational speed at
which conical horn can be modelled. Work pertaining to conical modes and Spherical
SCATTER is presented in chapter 4.
Figure 2.18 shows an example of a typical EM feed horn structure that is
ideally suited for analysis with the mode matching technique. It is a profiled “superGaussian” corrugated feed horn from the CLOVER instrument [Ade et al., 2009],
and due to the many small corrugations throughout its geometry, modelling such
a structure with the FEM, for example, proves very computationally demanding.
The tiny mesh that is required for a FEM simulation, in order to correctly map the
sub-wavelength corrugations, results in a huge number of degrees of freedom, and
thus a large problem size. The mode matching approach is a much more efficient
simulation tool for such problems due to the well behaved fields in each uniform
section, and the cylindrical symmetry of the overall structure. Figure 2.19 shows
the far-field radiation patterns for the CLOVER horn, calculated with the mode
matching code SCATTER, and with FEM software COMSOL.
The FEM approach required just over four hours to calculate the radiation
patterns, compared to just 10.5 minutes for the mode matching method. Clearly,
mode matching is hugely advantageous over FEM techniques when dealing with
64
(a)
(b)
Figure 2.18: FEM screen-shot of CLOVER profiled corrugated horn. (a): Full Horn, (b): Zoomed view of section
highlighted in (a) by black box.
(a) E-Plane cut through far-field
(b) E-Plane cut through far-field
Figure 2.19: Far-field radiation patterns of a corrugated feed horn at an operational frequency of 100GHz.
structures with high symmetry such as the CLOVER horn. However, when the
structure to be modelled is lacking such symmetry then other techniques (such as
FEM) will clearly be required. The following section (2.5) presents a brief description
of the FEM method, with an example model described to illustrate the approach.
2.5
Finite Element Method: COMSOL
Multiphysics
2.5.1
Background
The FEM technique is a numerical analysis approach used by many commercially
available software packages for modelling a range of science and engineering problems. COMSOL Multiphysics uses the FEM approach and any FEM analysis that
is presented in this thesis was carried out with the COMSOL Multiphysics package.
The main benefit of using the FEM approach is its versatility in terms of the types
65
of problems that can be modelled with it.
While some progress is being made in terms of combining waveguide and freespace techniques, as presented in chapter 5 of this thesis, the FEM approach allows
more complex models with multiple sub-domains, each with different boundary conditions, to be solved. There is also virtually no restriction on the shape of the
problem domain. There is of course the disadvantage that the FEM approach is
relatively inefficient, and compared to waveguide and free-space modal techniques,
FEM typically has computation times that are orders of magnitude larger.
The FEM approach is based on finding approximate solutions to boundary
value problems for differential equations, for example Maxwell’s equations for EM
field calculations. FEM employs variational methods to produce a stable solution
by minimizing an error function. Similar to the way a circle can be approximated
by a series of small straight connected lines, a problem domain Ω can be broken into
many smaller elements Ωe and then reconnected through nodes as shown in Figure
2.20. There are a number of different mesh types available with COMSOL but the
most commonly used is a tetrahedral mesh in the 3D modelling environment, and a
triangular mesh in the 2D environment (shown in Figure 2.20).
Figure 2.20: Triangular mesh that is used to approximate a problem domain.
The PDEs (partial differential equations) that describe the nature of the problem must then be discretised over the meshed problem domain. The manner in
which Maxwell’s equations are handled by COMSOL is now described as an example of the general COMSOL FEM approach. Assuming harmonic time dependence
66
(ejωt ) Maxwell’s equations can be written as:
~ = ǫ′ ǫ0 E
~
D
r
~ = µr µ0 H
~
B
~ ×E
~ = −jω B
~
∇
(2.18)
~ ×H
~ = J~S + σ E
~ + jω D
~
∇
~ and H
~ can then be written as:
The vector wave equations for E
1 ~
~
~
~ = −jk0 η0 J~S
∇ × E − k02 ǫr E
∇×
µr
1~
1 ~
2
~
~
~
~
∇×
∇ × H − k 0 µr H = ∇ ×
JS
ǫr
ǫr
(2.19)
′
where J~S is the source current density and ǫr = ǫr − j ωǫσ0 .
From (2.19) above the weak form [Tio et al., 2004] of the vector wave equations
~ is:
can be formulated, which for E
Z
Z
Z
1 ~
~ W
~ dV = −jk0 η0
~ ×W
~ dV − k0
~ · ∇
~ dV
ǫr E·
∇×E
J~S ·W
V
V
V µr
~
for all W
(2.20)
It is these weak form vector equations that are formulated for each element in the
problem domain, resulting in a set of simultaneous equations that must then be
solved.
Using COMSOL’s built-in meshing functions, the problem domain is broken up
into the finite mesh elements, where the number of elements depends on the physics
of the problem. For example an EM problem will typically require a minimum of 5
mesh elements per wavelength to reach a solution, however at least 6 mesh elements
per wavelength are recommended in the COMSOL literature [COMSOL, 2011] for
a reasonably high level of accuracy. An advantage of the FEM approach over other
finite analysis techniques, such as the finite difference method for example, is that
there is very little restrictions on the type of geometry that the problem domain
represents. The finite difference method is limited to rectangular geometries due to
the fact that the mesh must be uniform. The mesh of the FEM problem domain
can be irregular and take almost any closed form. Care should be taken however of
very small sections of a problem domain, ensuring that the mesh size is sufficiently
small to map these sections. This is particularly important for EM devices such as
corrugated horn structures as the corrugations are typically on scales corresponding
to a fraction of a wavelength.
67
The COMSOL user must specify material properties and boundary conditions
for each sub-domain of the overall problem domain, and any scalar variables such as
the frequency of radiation to be modelled, etc. must also be defined. The software
has multiple built-in solvers, both direct solvers such as PARADISO, SPOOLES, and
MUMPS, and iterative solvers such as Conjugate Gradient, GMRES, and FGMRES
for example, and COMSOL will usually select the optimum solver based on the
chosen space dimension, the physics of the problem and the type of study being
carried out, however the user can alternatively select any of the other solvers that
are available.
An example FEM model is illustrated in Figure 2.21, where the model was built
using the COMSOL Multiphysics computer aided design (CAD) package (Figure
2.21(a)). The T E11 mode was excited at the entry port, and a perfectly matched
layer was defined at the output port to simulate propagation into free-space. Both
ports were given the same dimensions as the WR-10 waveguides that were used
in combination with the VNA for measurements described in chapters 3 and 5:
a = 1.27mm and b = 2.54mm. The walls of the structure were defined as being
perfect electric conductors (PEC) and a tetrahedral mesh was then defined for the
entire geometry, as shown in Figures 2.21(b) and 2.21(c), with an element size of
0.45mm corresponding to λmin /6, where λmin is the minimum wavelength of the
frequency range to be analysed. The time harmonic wave equation (E ∝ eiωt )
was then solved for the electric field over the frequency range of the VNA system
(75-110GHz) in steps of 1GHz.
COMSOL allows a frequency sweep to be easily performed due to its ability
to automatically increment the frequency in user defined steps while saving the solutions for each step. COMSOL also allows similar parametric sweeps for other
variables such as changing spatial distances, etc. to be carried out. Figure 2.21(d)
shows a screen shot of the COMSOL post-processing environment, where the z component of the electric field is depicted as a raised colour plot. The post-processing
module can also be used for evaluating parameters such as the transmitted signal
(S21 ), the return signal (S11 ), integrated Ohmic losses across a particular surface,
etc.
68
(a) CAD Model
(b) Meshed Model
(c) Mesh (Close-Up)
(d) Electric Field
Figure 2.21: Steps in solving a FEM problem.
2.6
Conclusions
In this chapter the computational approaches that were employed for the work
presented throughout this thesis were described. The first section 2.2 of the chapter
presented methods for modelling free-space propagation of radiation, with examples
provided for both Cartesian and polar cylindrical reference frames. GBM analysis
provides an efficient means of modelling free-space propagation when dealing with a
signal with relatively low divergence, such as that from a horn antenna with shallow
flare angle. This limitation is based on the paraxial approximation from which the
theory is derived. Relatively good agreement was found for a conical horn antenna
with a half angle of 7.2o when compared to a similar FEM model. An alternative
approach for modelling fields with significant divergence, and which is not based
upon the paraxial approximation, is spherical wave expansion [Olver et al., 1994].
This vector based approach is similar to the scalar GBM analysis in the manner
in which the field is decomposed into a basis set of modes. When dealing with
radiation with significant beam divergence or horns with large flare angles (low Fnumber optics) spherical wave expansion would likely be required in place of GBM
methods [Clarricoats et al., 1984],[Olver et al., 1994].
In sections 2.3 and 2.4, simulation methods specific to fields in conducting
structures (PEC boundaries) were described. The mode matching code SCATTER
was shown to be a very efficient tool for modelling corrugated feed horn structures
in terms of analysing far-field radiation patterns, throughput, and return power, etc.
However, it was explained that SCATTER is relatively limited in terms of the types
of problems it can be used to model.
69
In section 2.5 a description of the FEM approach was given, which is the basis
of the commercially available COMSOL Multiphysics package and which was used
extensively in the work presented throughout this thesis. FEM was shown to be
a versatile simulation tool that is capable of modelling problems relating to both
free-space propagation and waveguide propagation, as well as a host of other multiphysics problems. This versatility comes at a cost however, and the approach is
relatively inefficient computationally. It was shown that solving a problem with
the FEM approach can take orders of magnitude longer that the mode matching
approach.
It will be shown throughout this thesis that in order to accurately model the receivers and detectors of far-IR instruments, such as the SAFARI S-Band for example,
a combination of the computational approaches presented in this chapter is required.
The efficiency of mode matching techniques is essential for SAFARI receivers due to
the large electrical size of the structures, and the flexibility of the FEM method is
necessary when dealing with asymmetric structures or problems involving multiple
sub-domains, where free-space propagation may be required in one sub-domain and
PEC boundary conditions may be present in an adjacent sub-domain.
70
Chapter 3
The Analysis of the SAFARI
Multimode Horns
“I hear and I forget.
I see and I remember.
I do and I understand.”
- Confucius
71
3.1
Rectangular Horns
3.1.1
Background
Smooth walled horns, both conical and rectangular, are becoming increasingly utilised
in sub-millimetre and far-IR systems. As we continue to design instruments with
feed horn coupling structures for higher operational frequencies the size of the horns
and waveguides become increasingly smaller. Manufacturing these feed structures
with dimensions on the order of a few microns is proving difficult for the manufacturers, particularly when arrays of such feeds are required. In the true THz regime
(approximately 3 to 10 THz) the components typically have dimensions on the order of tens of microns, and building corrugated structures on these scales is simply
not possible with current manufacturing techniques. For this reason smooth walled
feeds are proving to be the only feasible option.
The current design of the SAFARI horns are two sectioned smooth walled
rectangular horns consisting of a short 0.54mm length section with a flare angle of
20o , and a longer 4.84mm section with a softer flare angle of 4.5o (see Figure 3.1 (a)).
The horns are sufficiently wide to allow higher order modes to propagate through
them and therefore the overall throughput will be increased compared to a single
mode horn. However due to both the smooth walled nature of the horns and the
fact that they are multi-moded, the quality of the beam patterns will not compare
to corrugated conical feeds such as those typically used in CMB experiments. This
will not be an issue however since qualities such as high beam symmetry will not
be required as the SPICA telescope will not be aimed at performing ultra-sensitive
polarisation experiments like those of, say, the Planck CMB telescope.
Extremely low side-lobes are not required either as the focal plane arrays will
be contained in an extremely low temperature (dark) area, illuminated only by a
pinhole as shown in Figure 3.1 (b) during testing. This means that the side-lobes
in the beam patterns of the horns will only “see” a cold baffle. Therefore the sidelobes of the SAFARI horns will essentially only see a cold baffle and will not pick
up artefacts that more exposed horns arrays may detect in other telescope systems.
The priority of the SAFARI horn design is for maximum throughput which led
to the multi-moded design. Each propagating mode is capable of carrying power
through the horn and onto the TES detectors, so a multi-moded system allows for
72
(a)
(b)
Figure 3.1: (a): Layout of the Safari horn array block showing dimensions and angles. Only five horns from one
row are shown (b): Part of the SRON SAFARI testbed for characterising single pixels for the future detector array.
A conical horn and hemispherical backshort are shown in the expanded window (image credit: SRON).
a significant increase of total throughput and therefore an increase in sensitivity.
Careful design of the smooth walled horn structures can still produce relatively
good quality beams in terms of beam symmetry and low side-lobe levels.
In the first section of this chapter various techniques for modelling smooth
walled pyramidal horns will be described. The results from the different approaches
will be compared to each other and to measurements performed with the VNA
facility. Three methods for calculating far-field beam patterns of pyramidal horns
will be discussed, and it will be shown that there is excellent agreement between
the three approaches. The driving force behind these experiments was to examine
how far the simulations could be taken, taking into account aspects of the radiating
horns that have thus far not been included in the models. Such progressions in the
models include analysis of the fields at the mouth of the horn in order to determine if
significant power propagates behind the horn, including a conducting ground plane
between horn apertures to examine if induced surface currents can influence the farfield patterns of the horns, and quantifying the levels of optical cross-talk between
horns in a SAFARI-like horn array.
As will be demonstrated the best modelling approach for these SAFARI-type
horns depends on the particular problem being considered. For example analysis of
the far-field radiation patterns at the high operating frequencies of the SAFARI in-
73
strument requires the rectangular mode matching technique, as EM FEM approaches
are simply too inefficient for such electrically large, multi-moded structures. On the
other hand if optical cross-talk between horns, or induced surface currents on the
horn array block are being investigated, then the FEM approach is required as the
current mode matching software is relatively restricted in terms of these broader
questions. By employing multiple simulation methods, antenna feed horns such as
the current SAFARI design can be accurately modelled in terms of their far-field
radiation patterns, and more subtle effects such as optical cross-talk.
3.1.2
Approximate Method
In this section the fields inside, and at the aperture, of smooth walled pyramidal
horns are described. The derivation is almost identical to the case of a rectangular waveguide with the inclusion of a phase slippage term to account for the extra
distance covered by the field at the horn’s sides. The equiphase surface of the
field forms a spherical cap (quadratic phase cap) that extends beyond the mouth
of the horn. This quadratic approximation works well for horns of small flare angle [Milligan, 2005]. To analytically derive the fields at the mouth of such horns,
Maxwell’s equations can be solved in spherical coordinates with the appropriate
PEC boundary conditions. This method is discussed in section 4.1.2, where the
fields inside conical horn structures are derived.
The coordinates and parameters of a typical pyramidal horn are shown in Figure
3.2(a), and Figure 3.2(b) shows the extra distance ∆L travelled by the fields which
results in the quadratic phase error at the aperture. The equiphase surface at the
mouth of the horn is shown as a dotted line.
The structure of the field amplitude at the mouth of a rectangular/pyramidal
feed horn is defined by the input waveguide mode. The phase distribution is approximately quadratic across the aperture, with each aperture coordinate having
a separate quadratic phase distribution constant (RX and RY in this case for the
H-plane and E-plane respectively). The constant RX for the H-plane is formulated
as follows:
By applying Pythagoras’ theorem to the sides L, a2 and LX of figure 3.2(b), the
q
p
2
2
2
length LX can be defined as LX = (L) + (ra ) , or L = (LX )2 − a2 . It is
74
(a)
(b)
Figure 3.2: Pyramidal horn diagrams showing typical parameters used to describe such horns. (a): Primary horn
and waveguide parameters. (b): Illustration of the equiphase surface in the H-plane of the horn.
q
clear from figure 3.2 that ∆L = LX − L, therefore: ∆L = LX − (LX )2 −
⇒ ∆ L = LX
1−
s
(a/2)2
1−
(LX )2
!
≈ LX
⇒ ∆L =
(a/2)2
1− 1−
2(LX )2
a2
8LX
=
a2
8LX
a 2
2
(3.1)
(3.2)
where ∆L is the extra distance travelled by a field near the walls of the horn in
the H-plane (with respect to the on-axis field). Multiplying both sides of (3.2) by
the free-space wavenumber k gives the dimensionless constant RX of the quadratic
phase distribution:
∆L k =
ka2
= RX
8LX
(3.3)
By applying the same reasoning to the E-plane the dimensionless constant of the
quadratic phase distribution for the E-plane was calculated as:
kb2
= RY
8LY
(3.4)
Using the two constants RX and RY in combination with the definitions for the
transverse fields of rectangular waveguides given in section 2.3.2, the aperture fields
for pyramidal/rectangular horns can be completely defined. Each field component
for both TE and TM modes is defined as the rectangular waveguide field component
h
i
2y 2
2x 2
multiplied by exp −j RX a + RY b
, for example:
h
i
2y 2
nπy
2x 2
mπx
Ex = ZTE kjβc 2 nπ
cos
exp
(j(ωt
−
βz))
exp
−j
R
+
R
] .
sin
X
Y
b
a
b
a
b
75
3.1.3
Example Horn (Approximate Method)
The aperture field and far-field distributions are calculated for an example horn with
a rectangular waveguide section acting as a mode filter (as for example if a waveguide
feeds an integrating cavity) with dimensions such that only the fundamental mode
(T E10 ) can propagate. The waveguide has transverse dimensions of a standard WR10 guide section, a′ = 2.54mm, b′ = 1.27mm, and its length is defined as l = 5mm.
The horn section has aperture dimensions of a = 15.08mm, and b = 10.80mm, with
an axial length of 20.69mm, giving rise to slant lengths of Lx = 24.88mm, and
Ly = 23.06mm.
This particular small horn was chosen because its beam pattern can be experimentally measured with the VNA and it is small enough to be modelled through
FEM analysis on a reasonably powerful PC. In fact the size of the example horn
with respect to the operational wavelength is just within the limits of what can be
solved with the FEM technique on a PC with 16 gigabytes of RAM if symmetry
conditions are exploited to minimise the problem size. Symmetry conditions are put
in place for the E-plane and H-plane of the model resulting in a 75% reduction in
the geometry size and therefore a 75% reduction in the number of mesh elements
and the number of degrees of freedom.
While there are no limitations on the size of the horn that can be modelled
with the approximate method, the FEM approach is limited by the power of the
PC being used and the amount of RAM available. The mode matching approach
is only limited by the amount of computation time available since it calculates in a
stepwise fashion and each section can be calculated independently.
The central operating frequency of the VNA system (ν = 100GHz) is below
the cut-off of any higher order modes. So considering the expressions in Table 2.4
for the x and y components of the fundamental TE mode, where the subscripts
1 and 0 denote that m = 1 and n = 0, then Ex and Ey can be calculated over
a cut through the aperture of the horn by keeping y constant, say at y = b/2.
Similarly to calculate Ex and Ey through a cut perpendicular to the previous one
then the x coordinate should be kept constant, say at x = a/2. The magnitude of
the total electric field can of course be calculated across either of the aperture cuts
p
~ = Ex2 + Ey2 and keeping either x or y constant.
by calculating |E|
76
Now using the two expressions for RX and RY defined in 3.3 and 3.4, and combining all constants such as ZTE , β, etc. that appear in the T E10 mode description
into one arbitrary amplitude E0 (since only the general structure of the field is of
interest at this point), then the aperture field for the T E10 mode of a rectangular
horn can be written as:
Ey (x, y) = sin
πx
a
k x2
y2
exp −j
+
2 Lx L y
(3.5)
Equation 3.5 above corresponds to a reference frame where the aperture is
defined as going from 0 to a which is the usual convention in textbooks. A similar
expression is defined for the aperture field going from − a2 to a2 as:
πx
k x2
y2
exp −j
+
Ey (x, y) = cos
a
2 L x Ly
(3.6)
The far-field radiation pattern can then be calculated from the aperture field by
deriving the Fourier transform of either of the two equations 3.5 and 3.6 above. For
the case of evaluating the field from − a2 to
k
(1+cos θ)
EF F (θ, φ) =
4πr
Z
b
2
− 2b
Z
a
2
− a2
a
2
the result is:
EN F (x, y) exp [jk(x sin θ cos φ + y sin θ sin φ)]dxdy
(3.7)
Using the expressions just outlined for the aperture field and far-field distributions
the following graphs were plotted.
(a)
(b)
Figure 3.3: Small rectangular VNA horn modelled with approximate theory, where the aperture field phase is
approximated as quadratic. (a): The magnitude of the normal electric field across the aperture. (b): The phase of
the electric field across the aperture.
77
Figure 3.4: Far-field of small rectangular VNA horn modelled with approximate theory, where the aperture field
phase is approximated as quadratic. Cuts are shown for 0 and 90 degrees about the azimuthal angle φ corresponding
to the E and H-planes, respectively. The magnitude was normalised to the on-axis value.
3.1.4
Example Horn (Mode Matching Approach)
The general mode matching theory will not be discussed in great detail in this
section, as it is explained in depth for spherical geometries in chapter 4 and an
overview of the general theory is given in chapter 2. The theory specific to rectangular/Cartesian geometries is detailed in Stephen Doherty’s PhD thesis mentioned
above [Doherty, 2012] and the approach is based on approximating the smooth
walled pyramidal geometry as a series of rectangular sections with increasing size
in a stepwise fashion, as is shown in Figure 3.5. The fields in each uniform section
are defined as the Cartesian modal solutions set out in chapter 2 of this thesis and
mentioned again in the discussion on the approximate method above.
Using the basis set of modes appropriate for Cartesian geometries, scatter matrices are formulated for analysing how much power scatters between modes on
either side of each junction so that complex power is conserved, and then the scatter matrices are cascaded for the total number of sections. The term complex power
is used since the scattering is calculated by integrating the product of the complex
electric and magnetic fields over the cross-section of the waveguide junction. A detailed description of complex power is given in [Morgenthaler, 2011]. The scatter
matrices are constructed from three power coupling integrals (or overlap integrals)
~ or
P,Q, and R which are derived from taking cross products between either the E
78
(a)
(b)
Figure 3.5: Simplified illustration of a smooth walled rectangular horn showing stepwise approximation. (a): A
smooth walled horn with regular flat walls. (b): A cut through the horn when it is representad in a stepwise fashion.
The steps are shown exaggerated for clarity.
~ field on one side of the junction with the complex conjugate of either the H
~ or
H
~ field on the opposite side of the junction for cross-coupled power scattering, or
E
else cross products between fields on the same side of the junction for self-coupled
power scattering. The exact formulation of the P,Q, and R integrals are described
in chapter 2.
By running the Rectangular SCATTER program with an input geometry file
corresponding to a horn and waveguide with the same dimensions described in the
approximate method, the following graphs were generated for the aperture field and
far-field patterns. The horn’s geometry was split into 400 steps each with a length
of 0.05173mm, or λ/58. 58 steps per wavelength was most likely over-kill but the
program is very fast so calculation time was not an issue. A study into the minimum
requirement for step number was not investigated in this study.
Examining the figures corresponding to both the approximate solutions of the
fields and the mode matching results it is clear that there is relatively good agreement in the E-plane for the aperture fields, particularly for the magnitude. The
SCATTER results for the H-plane are very different than the constant unity magnitude shown across the aperture in the approximate method. The phase also exhibits
large differences in the H-Plane, however the far-field patterns of both techniques
agree very well in both planes. A more detailed comparison will be shown after the
next section and it will include results from yet another approach, namely the FEM
technique.
79
(a)
(b)
Figure 3.6: Small rectangular VNA horn modelled with mode matching algorithms. The X-plane (blue) corresponds
to the H-plane, and the Y-plane (green) corresponds to the E-plane, in both graphs. (a): The magnitude of the
~ across the aperture. (b): The phase of the electric field across the aperture.
electric field |E|
Figure 3.7: Far-field of small rectangular VNA horn modelled with mode matching algorithms. Two cuts through
the far-field pattern are shown for 0 and 90 degrees about the azimuthal angle φ corresponding to the E and H-planes
respectively. The gain (dBi) is relative to an isotropic radiator.
3.1.5
Example Horn (FEM Approach)
Figure 3.8 shows the model that was created in COMSOL to simulate the behaviour
of the same feed horn that was analysed with both the approximate and the mode
matching methods. The quarter section of the horn is shown in (a) and the model
after meshing is shown in (b). The symmetry boundary in the E-plane was defined
as a perfect magnetic conductor (PMC), while the symmetry boundary in the Hplane was defined as a perfect electric conductor (PEC). The input waveguide was
80
excited with the lowest order rectangular waveguide mode (T E10 ), and the quarterhemispherical cap that extends beyond the mouth of the horn and the quartercylindrical section that runs adjacent to the horn were defined as the far-field domain.
(a)
(b)
Figure 3.8: Small VNA horn modelled with COMSOL. (a): A quarter of the full horn was sufficient to model due
to the symmetry of the geometry and the field in the E and H-planes. (b): The model completely meshed.
Figure 3.9 shows the default model solution. It is a surface plot for the log of
the electric field for the full model at two separate viewing angles. 2D slice plots and
1D line plots can also be generated for the elecric and magnetic fields, for example,
using post-processing. The post-processing environment in COMSOL also allows
analysis of many other parameters such as current densities, resistive heating, S
parameters, etc. and the latest version of COMSOL (version 4.3) also provides a
polar plot option for graphing data such as far-field patterns.
(a)
(b)
Figure 3.9: Electric field of a small VNA horn modelled with COMSOL.
~ A and H
~ A across the surface that
Following calculation of the aperture fields E
makes up the aperture of horn, and the plane surrounding the aperture, S, the
81
Stratton-Chu formula [Volakis and Sertel, 2012] is applied to the near field data in
~ F F at each point on a sphere infinitely far
order to calculate the far electric field E
from the aperture of the horn. The Stratton-Chu formula for calculating the far-field
at each point P on the sphere has the following form:
Z h
i
ik
FF
~
~
~
~r0 ×
~n × EA − η~r0 × ~n × HA exp (ik~r · ~r0 ) dS
EP =
4π
(3.8)
~ A and H
~ A are the tangential fields on the surface S across the aperture of
where E
the horn, ~n is the unit normal to the surface S, η is the impedance, and k is the
wavenumber (k =
2π
).
λ
Finally, ~r0 is the vector pointing from the centre of the horn
aperture to the field point P . The geometry of the problem is illustrated in Figure
3.10.
Figure 3.10: Illustration of COMSOL far-field calculation [Orfanidis, 2002].
According to Volakis and Serte, the Stratton-Chu integral formulae for field representations are in fact among the most popular in scattering and antenna related
problems [Volakis and Sertel, 2012]. And the COMSOL support literature states
that the COMSOL far-field calculation is identical to what is known as the “scattering amplitude” in an EM scattering problem [COMSOL, 2011]. In deriving the
waveguide modes in chapter 3 the starting point was the homogeneous Helmholtz
~ r) = 0, since we
equation, usually just called the Helmholtz equation (∇2 + k 2 ) ψ(~
were dealing with the source free case. However since the aperture of the horn acts
as a source for the far-field the inhomogeneous Helmholtz equation is required as a
starting point for deriving the relationship between the aperture field and far-field:
~ r) = −f (x) in Rn
∇2 + k 2 ψ(~
82
(3.9)
where n = 1, 2, 3, and f is some function yet to be defined.
The way to solve the above equation uniquely is to specify a boundary condition
at infinity. The boundary condition typically used is the Sommerfeld radiation condition [Schot, 1992]. The solution then to the inhomogeneous Helmholtz equation
is:
(3.10)
∇2 + k 2 G ~r, r~′ = −δ ~r − r~′ in Rn
where δ ~r − r~′ is the Dirac delta function, and G ~r, r~′ is the Green’s function,
the form of which depends on the dimension of the space n. For n = 3 the Green’s
function has the following form:
eik~r
G ~r, r~′ =
4π~r
(3.11)
where the boundary condition that the Green’s function is an outgoing spherical
wave for |~r| ⇒ ∞ was chosen.
Finally, using this particular Green’s function, and applying Green’s theorem
to the spherical surface that bounds the far-field domain in the model, we obtain an
important expression relating the values of the fields in the interior of the surface
to those at the closed surface, that of equation 3.8.
There are two reasons for allowing a small amount of propagation beyond the
mouth of the horn. The first is to allow the field to freely propagate away from the
aperture of the horn. This serves to avoid unwanted reflections that could occur if
the model geometry was truncated right at the aperture of the horn. If the surface of
the quarter sphere that bounds the propagation region is given appropriate boundary
conditions (Scattering Boundary or Perfectly Matched Layer) then no reflections will
occur. The second reason is to create a surface that has points with well-defined
angular position (θ, φ) with respect to the horn aperture for the far electric field to
be analysed across.
This transformation process using the Stratton-Chu formula generates a new
variable (Ef ar ) with three components (Ef ar r̂, Ef ar θ̂ and Ef ar φ̂) that represent the
~ F F across the spherical surface.
collection of far-field data points E
P
The results of the COMSOL model are shown together with the results from
the other two methods discussed previously.
As can be seen in Figure 3.11, there is excellent agreement between the three different simulation techniques particularly out to about 40 degrees which corresponds
83
(a)
(b)
Figure 3.11: Comparisons of the far-field beam patterns for three simulation methods, the approximate phase-cap
method, the rectangular mode matching method, and the FEM approach.(a): The beam pattern in the plane with
the azimuthal angle set equal to zero (φ = 0, or E − plane), (b): The beam pattern in the plane with the azimuthal
angle set equal to 90 degrees (φ = 90, or H − plane)
.
to approximately −20dB in the E-plane and −32dB in the H-plane. However beyond
40 degrees there is still reasonably good agreement considering the FEM technique
is a very different computational approach than the mode matching technique.
3.1.6
Inclusion of a Conducting Ground Plane
The COMSOL results displayed so far have assumed free-space surroundings at
the aperture of the horn. The mode matching technique does not consider the
surroundings of the horn at all, and the same is true for the phase-cap approximate
method. However the horns on the SAFARI instrument will be combined in an array
with conducting material (gold coated) between them (see Figure 3.1(a) at the start
of this chapter). Any induced surface currents across this adjoining plane could
possibly act as sources for the far-field radiation leading to increased side-lobes for
example. For this reason COMSOL simulations were run again with the boundary
adjacent to the horn aperture defined as a perfect electric conductor (PEC) boundary
condition. These results were again compared to the mode matching results and they
are displayed in Figure 3.12.
As can be seen there are small differences between the COMSOL results when
the conducting ground plane is included, however the effect is so small it would be
considered negligible. What is interesting though is that the small deviation is in the
84
(a)
(b)
Figure 3.12: Comparisons of the far-field beam patterns for the effect of inclusion of a conducting ground plane.
(a): The beam pattern in the plane with the azimuthal angle set equal to zero (φ = 0o , or E-plane), (b): The beam
pattern in the plane with the azimuthal angle set equal to 90 degrees (φ = 90o , or H-plane)
.
direction of bringing the results into better agreement with the SCATTER results,
in both planes. Because there was a small visible change in the beam patterns an
analysis of the induced surface currents across the ground plane was carried out.
(a) X direction (H-Plane)
(b) Y direction (E-Plane)
Figure 3.13: 1D cuts of the surface current density across the conducting ground plane adjacent to the horn aperture.
The off-axis distance is from the edge of the horn aperture to the edge of the propagation zone
.
3.1.7
Comparing Horn to Waveguide
In order to investigate the most extreme case of the possible effect of the conducting
ground plane on the far-field, a simple waveguide was modelled. The far-fields were
85
calculated for both the case of a free standing waveguide and one embedded in a
ground plane. A straight waveguide was chosen because it has poor directionality
and would therefore likely lead to significant induced currents across the ground
plane. The waveguide was given the same transverse dimensions as the waveguide
section of the horn model (WR-10), and the same length as that of the horn.
Figure 3.15 shows the far-field radiation patterns for both a waveguide in freespace and a waveguide connected to a ground plane. As can be seen there are
significant differences in both far-field planes when the ground plane is included. The
effect is much more obvious in the case of a waveguide than it was for a rectangular
horn, as would be expected. The data corresponding to the surface current density
is also shown plotted in Figure 3.14. When compared to the surface current density
generated across the ground plane from the horn it is clear that it is much greater
for a waveguide. These induced currents act as sources for the far-field.
(a) X direction (H-Plane)
(b) Y direction (E-Plane)
Figure 3.14: 1D cuts of the surface current density across the conducting ground plane adjacent to waveguide
aperture. Off-axis distance is from the edge of the waveguide aperture to the outer edge of the propagation zone.
3.1.8
Directional Gain of Example Horn
In the far-field patterns shown for the COMSOL FEM simulations and those of the
mode matching approach, the data on the vertical axis was directional gain. While
the aim of plotting the far-field data was to compare the structure of the far-field
patterns for the different computational approaches, the calculation of gain can also
be a very useful parameter for describing the performance of a horn antenna. The
86
Figure 3.15: Far-field patterns of a waveguide in free-space and connected to a conducting ground plane.
measure of directional gain provides important information about the directivity of
the beam compared to an isotropic radiator. However there is no built-in option
in the COMSOL software to calculate the gain so the author derived an expression
that would allow it to be calculated.
The directivity of the horn can be defined as 10Log10 (I (θ, φ) /Iiso ), where
I (θ, φ) is the power flux or power density at a point P , and Iiso is the power flux of
an ideal isotropic radiator at the same point P on a sphere with radius r, which is of
course is defined as Iiso = P0 /4π|r|2 , where P0 is the total input power at the horn
~ 2 /2Z, where Z is the impedance of
aperture. Now I (θ, φ) can also be defined as |E|
free space and is equal to 377Ω, so the directivity can be defined as:
!
~ 2
4π|r|2 |E|
G = 10Log10
2ZP0
(3.12)
Now the default value for the input power P0 in the RF module of COMSOL
is 1 Watt. Also, strictly speaking the value of |r| in the far-field is infinity, however
if this was allowed to be the case then the fields would all go to zero in the far-field.
For this reason COMSOL defined r to be unity in the definition of the far-field. So
setting P0 = 1 and |r| = 1 the above expression for the directive gain reduces to:
!
~ 2
2π|E|
G = 10Log10
dBi
(3.13)
Z
where the i in dBi denotes that the gain is with respect to an ideal isotropic radiator.
87
To be more precise the directive gain should really be computed with respect
the an ideal isotropic radiator with power equal to that at the mouth of the horn
rather than the power input at the entry port. If there is no significant back scatter
then the result will be almost identical, but if there is significant losses due to back
scatter (S11 ) then the correct definition should be:
G = 10Log10
!
2
~
2π|E|
1 − |S11 |2 dBi
Z
(3.14)
It was Equation 3.14 that was used for calculating the gain in the previous farfield plots. Then using the same definition an analysis was carried out to investigate
the gain of the example horn across the frequency range of the VNA system, namely
65 to 110GHz, in steps of 6GHz. The same analysis was also performed for the
waveguide described above so that the benefit of using a horn instead of a waveguide
would be highlighted. Figure 3.16 shows the gain plotted as a function of frequency
for both the horn and waveguide.
(a)
(b)
Figure 3.16: Gain plotted as a function of increasing frequency for (a): a small pyramidal horn, and (b): a waveguide
with similar dimensions.
The gain is significantly higher for the horn, as can be seen in Figure 3.16. The
trend of the gain graphs is similar for both cases, with the gain increasing steadily
as frequency increases. However the value at each frequency point is much higher
for the horn. It can be concluded that using a horn in place of a simple waveguide
provides greater directionality with less back scatter. This would clearly be expected
based on λ/d considerations, where d is the size of the aperture which is significantly
larger for the horn, and λ is the wavelength of radiation.
88
3.1.9
External Back Scatter
The ability to include a propagation analysis domain with COMSOL made it possible
to model external back scatter around the back of the horn. It has always been
assumed that the power levels in this direction would be insignificant, but it is
a question of interest all the same. In fact the Planck CMB satellite had some
problems when an artefact in the detector data could not be explained. The source
of the problem was eventually traced to some of the horns being able to see a warm
part of the satellite behind the horns. Being able to model the beam patterns of
horns right through the full 360o field will allow more careful design of systems that
can avoid such problems.
By extending the propagation region through 180o it was possible to perform a
far-field transformation on the field data behind the horn using the same StrattonChu method described above. Figures 3.17(a) and 3.17(b) show the co-polar and
cross-polar far-field beam patterns for both a WR-10 waveguide and the example
pyramidal horn from before. As can be seen for the case of the horn, the gain does
begin to increase for angles greater than 90o , but not to significant levels. However,
for very sensitive receiver horns such as those required for CMB polarisation experiments, these levels could become significant. A very strong source behind the horns,
such as the Sun for example, could become noticeable if sufficient shielding is not
employed.
The 180o plot for the waveguide is also interesting, particularly the cross-polar
field. It shows a very smooth distribution that is almost symmetric about the 90o
angle. Careful analysis of the behaviour of the fields in the vicinity of a waveguide
aperture may be important for modelling the VNA set-up in the future as short
waveguide sections are being increasingly used as probes for examining the far-field
patterns of radiating feed horns. Tests to date seem to show that while a waveguide
probe is well suited for measuring the co-polar fields of radiating components, they
are not ideal for analysing cross-polar distributions.
3.1.10
Experimental Verification
In order to verify the computational approaches presented in this section, a series
of beam pattern measurements were carried using the VNA system at NUIM. Neil
89
(a) Horn
(b) Waveguide
Figure 3.17: Cuts through the far-field beam patterns of a small pyramidal horn and a WR-10 wavguide in free
space. The far-field data is plotted over the full 180o , thus including the space behind the radiating component.
Tynan of the Department of Experimental Physics at NUIM provided assistance
with the experimental set-up and the alignment procedures. The experimental setup is illustrated in Figures 1.15 and 1.16 in section 1.4 of chapter 1. The particular
pyramidal horn that was analysed had the same dimensions as the example horn
described in this section, and the details can be found in subsection 3.1.3 above.
A two-port network was used for the analysis, where port one was used in
combination with a frequency converter head (head one) as the transmitter. Port
two, also used in combination with a frequency converter head (head two), was
employed as the receiver. The waveguide probe was mounted head one of the VNA
system, which was fixed to the x − y raster scanner. The pyramidal horn was then
mounted head two, which was fixed and a z translation stage. The distance (∆z)
between the horn aperture and the waveguide probe aperture was set to 450mm
using the translation stage. The maximum off-axis angle at which the beam pattern
could be measured was limited by the range of the raster scanner in the x − y
plane. The limits of the scan in x and y were both -140mm to +140mm, leading
to a maximum off-axis angle range of −17.28o to +17.28o in both the E-plane and
H-plane.
Figure 3.18 shows the E-plane (x = 0) and H-plane (y = 0) cuts through
the measured far-field beam pattern. The COMSOL FEM and SCATTER mode
matching simulation results are shown again for comparison, and as can be seen
there is very good agreement between measurement and model.
90
(a) E-Plane
(b) H-Plane
Figure 3.18: Far-field beam pattern measurements of a pyramidal horn using a VNA.
3.2
SAFARI Horns
The current feed horn being used in the SAFARI test bed at SRON is a smooth
walled conical design. However the current design for the SAFARI feed horns in
array format is a two angled pyramidal horn as shown in Figure 3.21. The reason
for the choice of a pyramidal geometry over a conical one is down to limitations in
the manufacturing process. For horns on these micrometer scale the most accurate
method of manufacture is to start with a block of material (usually aluminium)
and to first fashion rows of horns through one dimension as shown in Figure 3.20
(b). The same process is then carried out in the perpendicular direction to reveal an
array of pyramidal mandrels as shown in Figure 3.20 (c). RAL in the U.K. have built
monolithic arrays of 388 pyramidal horns as shown in Figure 3.19 (b) and (c). The
horns are electroformed in copper on wire-eroded aluminium mandrels (see Figure
3.19 (a)). During the wire erosion process it becomes difficult to transfer heat away
from the tip of the mandrels due to the extremely small dimensions of the horn
apertures. The sudden change from the softer angle (4.5o ) to a much steeper angle
(20o ) solves this problem and this is what leads to the design shown in Figure 3.21.
3.2.1
Far-field Radiation Patterns of SAFARI Horns
The frequency range the SAFARI S-Band instrument is designed for is 4.3THz to
10THz (λ = 30 − 70µm) meaning that the dimensions of the horns correspond
to many wavelengths. Standard computers will not be able to fully model these
91
(a)
(b)
(c)
Figure 3.19: SAFARI horn array and mandrel. (a): A mandrel for construction of an 8 × 8 pyramidal horn array,
(b): An array of 388 pyramidal horns with a hand to demonstrate the small scale of the structures, and (c): The
same array of 388 pyramidal horns shown enlarged an face-on.
(a)
(b)
(c)
Figure 3.20: Illustration of the wire-erosion process for micro horn array manufacture. (a): A solid block of
material. (b): The block following wire-erosion in one dimension, and (c): The block following wire-erosion in two
perpendicular directions revealing pyramidal horn mandrels.
Figure 3.21: left: 3D model of curent SAFARI horn design. right: Curent SAFARI horn design showing the
dimensions of the horns in a slice through the array, where all dimensions are in mm.
large horn structures using FEM or FIT techniques, particularly at the highest
frequencies. The pyramidal mode matching approach will be essential in order to
characterise these feed horns across the desired frequency range.
Using a particularly powerful computer with 64 gigabytes of RAM, and taking
advantage of symmetry conditions in both the E-plane and H-plane, it was possible
92
to use COMSOL to perform a FEM simulation of a SAFARI S-Band horn at the
lowest operational frequency of 4.3THz. The beam patterns at azimuthal cuts of
φ = 0o and φ = 90o are shown in Figure 3.22. The same horn was simulated using the
Pyramidal SCATTER program for comparison. As can be seen in the comparison
plot there are large differences in the far-field patterns calculated by COMSOL and
SCATTER.
Figure 3.22: Far-field pattern of a SAFARI horn at 4.3THz (70µm) modelled with COMSOL and SCATTER.
The main beam in the SCATTER data is clearly much wider than the COMSOL
data. Upon comparing the approaches of COMSOL and SCATTER in terms of
mode excitation at the input waveguide feed, the large differences are expected.
COMSOL only allows a single mode to be defined for each simulation (the T E01
mode in this case with the electric field polarised in the y direction, or φ = 90o ,
leading to the higher side-lobes in the φ = 90o far-field cut), thus the higher order
propagating modes were not included. In contrast, SCATTER is including the
power contribution from all possible propagating modes. The symmetry in the Eplane and H-plane for the SCATTER results is due to the horn being square rather
than rectangular, and since all propagating modes are included in the calculation,
then both polarisations of each mode will also be included, for example the T E10
and T E01 modes, leading to the observed symmetry.
COMSOL does account for power scattering from the input mode to higher
order modes when a step discontinuity or change in flare angle is encountered, for
example the change in angle in the SAFARI horn in this case. This scattering
93
behaviour is clear in the example of the spline fit conic section horn modelled with
COMSOL in chapter 4 of this thesis. In that case the horn is fed by a rectangular
WR-10 waveguide excited with only the fundamental T E01 mode with the resulting
horn aperture field exhibiting a combination of modes that is evident from the
symmetric Gaussian field observed at the output. As can be seen in the analysis
presented in section 4.3.2 both COMSOL and SCATTER in cylindrical coordinates
agree very well for that particular example, and this is due to the fact that the
first section of the spline fit conic horn supports only a single mode. However,
the difference in the present case of the SAFARI horn is that the first waveguide
section is itself over-moded, meaning that the large number of modes generated by
the SCATTER code will each carry power through the horn and contribute to the
far-field pattern.
An attempt to encourage the FEM and mode matching results to agree was
made by independently simulating higher order modes with COMSOL and then
summing them during post-processing analysis. Only modes with cut-off frequencies
below 4.3 THz were considered since any modes with higher cut-off frequencies would
be highly attenuated through the horn. Table 3.1 shows the cut-off frequencies for
some of the higher order rectangular modes, and thus the modes that were considered
in the FEM analysis. The far-field data corresponding to each propagating mode
was summed in a linear fashion and the resulting far-field pattern is shown in Figure
3.23 with the SCATTER result shown again for comparison.
Mode
Cut-Off (THz)
Behaviour
Mode
Cut-Off (THz)
Behaviour
T E10
1.875
Propagating
T E21
4.193
Propagating
T E01
1.875
Propagating
T E12
4.193
Propagating
T E11
2.652
Propagating
T M21
4.193
Propagating
T M11
2.652
Propagating
T M12
4.193
Propagating
T E20
3.750
Propagating
T E22
5.303
Evanescent
T E02
3.750
Propagating
T M22
5.303
Evanescent
Table 3.1: Cut-off frequencies of some of the higher order modes at the input aperture of a SAFARI horn.
As can be seen there is now much better agreement between the two results,
but there are still some differences in far-field distributions for the SCATTER and
COMSOL models. In the COMSOL analysis each propagating mode was excited
94
Figure 3.23: Far-field pattern of a SAFARI horn at 4.3THz (70µm) modelled with COMSOL and Pyramidal SCATTER. In this example the T E10 , T E01 , and T M11 modes were analysed separately with COMSOL and then
summed.
with equal power at the input port while no consideration was given to evanescent
modes. The SCATTER mode matching code excites all modes (up to the number
defined by the user) including evanescent modes, some of which may eventually become significant since the SAFARI horn widens relatively quickly and therefore these
modes may actually contribute to the far-field, thus leading to differences between
the two results. It may also be possible that simulating each mode independently
with COMSOL is missing more subtle modal cross-coupling effects that SCATTER
is so good at capturing.
Following the analysis at the lowest frequency of 4.3THz, and having pushed
the limits of the COMSOL FEM approach with the highest powered PC available,
the SAFARI horns were then modelled at both an intermediate frequency (7.15 THz)
and the highest operational frequency (10 THz) with the SCATTER mode matching
code in order to examine the aperture fields and far-field patterns of the current horn
design. The calculation time at 7.15 THz and 10 THz was 21.49 hours and 92.70
hours respectively. The near field and far-field patterns are shown in Figures 3.25
and 3.24 below for the three frequencies that were examined, corresponding to the
start, middle, and end of the operational frequency range of the SAFARI S-Band
horn array.
The aperture field plots shown in Figure 3.25 show interesting field distribution
95
(a)
(b)
Figure 3.24: Far-field patterns of a SAFARI horn at three spot frequencies. (a): The E-plane (φ = 0o ) cut at 4.3,
7.15, and 10.0THz, and (b): The H-plane (φ = 90o ) cut at 4.3, 7.15, and 10.0THz. It is clear that the E and H-plane
cuts are identical, as would be expected.
(a)
(b)
Figure 3.25: Aperture fields of a SAFARI horn at three spot frequencies. (a): The E-plane (φ = 0o ) cut at 4.3, 7.15,
and 10.0THz, and (b): The H-plane (φ = 90o ) cut at 4.3, 7.15, and 10.0THz. It is clear that the E and H-plane
cuts are identical, as would be expected. The off-axis distance is is plotted for the extent of the aperture of the horn
(-0.325 to 0.325 mm).
across the aperture of the horn compared to a typical single mode rectangular horn.
The aperture field distributions vary significantly for the various frequencies. The
far-field patterns however show much more similarity for different frequencies but
are again very different from what would be seen by a single mode horn. The main
beam has a large width which indicates the higher levels of throughput that are
possible with such multi-moded horns.
As efficient as the mode matching approach is compared to the FEM approach
for example, the computation times for modelling the SAFARI horns are still very
96
long for the higher frequencies. A brief discussion is now given on a possible method
for significantly increasing the efficiency of the mode matching technique used.
A huge reduction in computation time would be possible if conical modes could
be employed in the mode matching code in place of the rectangular modes. The
difficulty that arises of course is that the SAFARI horns are rectangular in nature
meaning that conical modes do not fit naturally with the horn geometry. However
it would still be possible to describe the fields by conical waveguide modes by means
of coordinate transformations since a rectangular horn with a spherical phase cap
at the aperture is just as much a section of a sphere as a conical horn.
The reason this would be useful is because the only place where scattering of
power occurs is at the plane where there is a change in flare angle, meaning that
power scattering overlap integrals would only be required to be calculated at one
junction rather than a few hundred. In fact the only modes that can actually carry
significant power through the horn are the modes that can propagate through the
cross sectional area at the plane of angle change. The current rectangular mode
matching code continues performing overlap integrals for the sections making up
the entire length of the horn, and since the cross section of the horn becomes very
large toward the aperture a huge number of modes are included in the calculation
that in reality will carry little to no power.
The highest mode index at the intermediate frequency of 7.15 THz is m =
31. This gives rise to 1, 984 modes that must be accounted for in the scattering
N
P
calculations, and is determined by 4 ×
i since each index i can correspond to
i=1
T Emi , T Ein , T Mmi , or T Min . The highest mode index at the high end of the
frequency band (10 THz) is m = 45, giving rise to 4, 140 modes.
A detailed description of conical and spherical modes, and the theory of conical
mode matching is given in chapter 4. It is the same techniques that could be applied
to modelling the SAFARI rectangular horns but with some alterations to account for
the rectangular pyramidal PEC boundaries in place of the conical PEC boundaries.
3.2.2
Optical Cross-talk in SAFARI Horn Array
In this section the term ‘optical cross-talk’ will define the amount of power detected
at the entry aperture of a horn (smaller aperture of the SAFARI horns) that is
97
adjacent to a transmitting horn. More specifically it is defined as a ratio of the total
detected power at Port 2 to the total input power at Port 1 (see Figure 3.26). The
SAFARI instrument that is being investigated in this thesis is purely a receiving
system. However, it is still desirable to quantify the crosstalk effects between feed
horns in the SAFARI horn array in a manner that assumes one feed horn to be
transmitting and the rest of the neighbouring feed horns to be in the receiving
mode, as this represents an extreme case that can be taken as an upper limit on
cross-talk levels. This is because any receiving system will always reflect a small
amount of the incident power, effectively making it a weak transmitter.
Figure 3.26: Screen shot of COMSOL model of optical cross-talk in the SAFARI horn array.
Figure 3.26 shows a screen shot of a 2D model of the transmit/receive problem
just described that was built with COMSOL multiphysics modelling software. As
can be seen from the image the COMSOL software is ideal for this analysis due
to the ability to include a propagation domain that serves to link neighbouring
feed horns together through free-space. However, as mentioned already the FEM
approach that COMSOL employs is very computationally demanding, so modelling
multiple horns even at the lowest frequencies would be impossible without some sort
of super computer. In fact, as was shown in the previous section, modelling even
one SAFARI horn at the lowest frequency of 4.3 THz was just about manageable if
the PEC and PMC symmetry boundaries were used to reduce the problem size to
one quarter of the original full horn model.
The most computationally efficient way to model this optical cross-talk problem
98
for the SAFARI horns using FEM analysis, and without resorting to the use of a
super computer, was to take advantage of the 2D modelling environment available
with COMSOL. This 2D simulation format is very computationally efficient but it
was unclear how accurate it would be. In order to investigate the accuracy of the
2D approach two smaller horns were modelled in both the 3D and 2D COMSOL
environments and the results were compared. Figure 3.27 shows a cut through the
E-plane of the 3D COMSOL model. The same VNA example horn from the previous
section was used again for this analysis in which the distance between the radiating
and receiving horns was incrementally increased.
It was considered important to examine how the cross-talk levels changed with
separation distance in both the E-plane and the H-plane as the fields behave very
differently in each plane unless some sort of mode mixing is carried out by using corrugations or some sort of smart profile. In a straight, non-profiled, smooth
walled horn the electric field will smoothly approach zero toward the boundary of
the horn in the H-plane, whereas the field will be significantly greater than zero at
the horn boundary in the E-plane; usually approximately 65% of the on axis intensity. Therefore it would be expected that the levels of cross-talk will be greater in
the E-plane both through radiative processes and possibly through induced surface
currents across the boundary joining the apertures of the two horns.
Figure 3.27: Screen shot of the example horn cross-talk model showing a cut through the E-plane.
The level of cross-talk can be calculated in the post-processing mode of COMSOL following the completion of the solver. It is calculated by integrating the power
99
across the surface of the detector port 2 and then dividing the result by the total input power; usually 1 Watt in the form of a particular TE or TM mode. This method
will give an upper limit on the cross-talk level for the particular arrangement since
not all this power will couple into the waveguide that will typically join the horn to
the detector cavity. A more accurate method for calculating the cross-talk is to examine how much of the power detected at port 2 actually couples to the eigenmodes
of the waveguide. This is the S21 parameter and is defined as follows:
R
(Ec · E2 ∗ )dA2
S21 = RP ort2
(E2 · E2 ∗ )dA2
P ort2
(3.15)
where Ec is the total electric field across port 2, and E2 is the electric field eigenmode
of the rectangular waveguide that has the same transverse dimensions as the exit
aperture of the receiving horn. The overlap integral is then normalised with respect
to E2 . The subscript 21 represents power that will couple to port 2 from the input
field at port 1.
The value of the return power S11 can also be calculated in a similar manner
by:
S11 =
R
((Ec − E1 ) · E1 ∗ )dA1
(E1 · E1 ∗ )dA1
P ort1
P ort1
R
(3.16)
The subscript 11 represents power detected at port 1 from port 1.
Figure 3.28 shows the predicted results of incrementally increasing the separation distance between the two example feed horns. As was expected the levels of
cross-talk are indeed significantly higher in the E-plane data. What is also clear
is that there is relatively good agreement between the 2D and 3D data, at least in
terms of the general trend. There are some differences between the two but the general trends are in agreement, and they are of the same order of magnitude. Based
on these results it can be assumed that a 2D analysis of optical cross-talk for the
SAFARI horns can be relied upon to give an approximate result that is of the correct
order of magnitude.
As was done with the analysis of the effect of induced surface currents on farfield patterns, the most extreme possible case was examined so that an upper limit
on cross-talk could be calculated. As before a uniform waveguide was considered to
have to the lowest directionality of any waveguide component (ignoring instruments
such as isotropic radiators). A similar cross-talk analysis to that performed for the
small horn was carried out for two uniform waveguides in an array. As was expected
100
Figure 3.28: Predicted optical cross-talk levels between two small pyramidal VNA horns at 100GHz. It is clear that
the levels are significantly greater in the E-plane.
the levels of detected power by the neighbouring waveguide were significantly greater
than the case of a horn, and a neighbouring waveguide in the E-plane again gave rise
to much higher levels than one in the H-plane. Figure 3.29 shows how the cross-talk
between two rectangular waveguides varied over a range of separation distances in
both the E-plane and H-plane.
(a)
(b)
Figure 3.29: Predicted optical cross-talk levels between two WR-10 waveguides at three test frequencies in (a): the
H-plane, and (b): the E-plane. The 70GHz frequency was first modelled with a conducting plane separating the
horns, and then a free-space plane.
An interesting feature of the waveguide cross-talk plots is how much smoother
the fall-off of detected power is in the E-plane compared to the H-plane as the
waveguides are separated. The maximum levels were found at the lowest frequency
101
of 70GHz, so this frequency was chosen for an investigation into the difference between including a conducting boundary between horns or a free-space boundary.
Employing a PEC boundary is representative of waveguides in an array, whereas a
free-space boundary represents the waveguides in free-space and therefore not connected through currents. The aim of this analysis was to try to determine if the
cross-talk effects were the result of radiative processes through coupling of EM fields,
or induced surface currents across the boundary that separates the guides.
The plot shown in Figure 3.29(a) for a neighbouring waveguide in the H-plane
seems to indicate that a conducting boundary can affect the levels of cross-talk
by as much as 5dB when the horns are very close together. As the separation
distance is increased the effects of the conducting boundary decrease. This would
be expected as the electric field goes to zero at the aperture of the guide in the Hplane meaning that few surface currents would be expected to be induced across the
adjoining boundary. Figure 3.29(b) shows that as the separation distance between
two waveguides in the E-plane is increased the conducting boundary has a greater
effect. Again this would be expected due to the relatively large magnitude of the
electric field at the aperture of the waveguide in the E-plane. The electric field can
excite surface currents across the adjoining boundary meaning that removing the
PEC condition reduces the levels of cross-talk detected.
With a good understanding of the processes governing cross-talk between neighbouring waveguide structures, two SAFARI horns were then modelled in a similar
manner to that described above but only in the 2D COMSOL environment due to
electrical size of the horns. Three spot frequencies corresponding to the lowest, intermediate, and highest end of the S-Band were examined for various TE and TM
modes. The structure of the field in the third (unaccounted for) dimension, which
would be in and out of the screen in the z direction since the model plane on-screen
represents the x − y plane, is assumed to be constant. The transverse fields can be
thought of as having subscript m = 0 in a T E0n mode or m = 1 in a T M1n mode
that would exist in a three dimensional model. However it is again an approximation
since with the real 3D horn the presence of conducting walls bounding the fields in
the z direction will have an effect on the field structure in the other two directions,
particularly if the mode is close to cut-off.
Another issue with the 2D environment is that for any T E0n mode analysed
102
in the on-screen (x − y) plane, the analysis is always carried out in the H-plane.
In other words the 2D model of T E modes represents two horns side-by-side in the
H-plane. However the 3D cross-talk study performed on the example horn in the
previous section demonstrated that the cross-talk levels are significantly higher in
the E-plane. In order to analyse cross-talk in the E-plane using the 2D environment
T M modes were considered as this allowed post-processing analysis of power flow
in the E-plane. Before the analysis was carried out a separate investigation was
performed in order to determine if different mode orders gave rise to varying levels
of cross-talk. If the cross-talk proved to be independent of mode order then 2D
analysis of the T M11 mode would provide a means for determining an approximate
value of the SAFARI horn cross-talk in the E-plane, accurate to within an order
of magnitude. Likewise, 2D analysis of the T E01 mode would provide a means of
evaluating the cross-talk in the H-plane to within an order of magnitude.
The first simulations involved examining cross-talk levels for multiple propagating modes in the SAFARI horn array. Three frequencies were simulated at two
values of separation distance corresponding to the most extreme case of d = 0mm,
and the value of the current SAFARI horn array design of d = 0.19mm. Figure 3.30
shows that the T E01 , T E02 and T E03 modes gives rise to almost identical levels of
cross-talk at each of the test frequencies. The T M modes are also in very good
agreement with the largest difference of about 2dB at the intermediate frequency.
What is clear is that analysis of the TM modes results in much larger levels of
cross-talk as was expected since the TM mode analysis represents two horns in the
E-plane. The post-processing analysis of the TE modes carried out in the H-plane
shows significantly lower levels of spill over as expected.
With confidence that there is relatively low dependence on mode number for
cross-talk levels, the next analysis that was performed was a parametric sweep for
separation distance between two SAFARI horns for the lowest frequency of 4.3 THz
and the highest frequency of 10 THz. Figure 3.31(a) shows the results. Again it
can be seen that there are significantly higher levels in the E-plane data compared
to the H-plane data as would be expected, however even when the horn apertures
are allowed to touch the levels do not exceed -30dB. The cross-talk levels are then
lower for the higher frequency data which again would be expected since bore-sight
gain generally increases with frequency, and since gain is related to directivity then
103
(a)
(b)
Figure 3.30: Levels of cross-talk in SAFARI horn array for various propagating modes. Three test frequencies were
examined in each instance. (a): Cross-talk levels for zero spacing between horn apertures. (b): Cross-talk levels for
0.19mm spacing between horn apertures.
the higher frequency beams would be expected to have less power off-axis.
Figure 3.31(b) shows the results of a frequency sweep that was analysed across
the full operational range of the SAFARI S-Band array. The sweep was across 4.3
to 10THz for 50 spot frequencies at 0.114THz intervals. Again there are significantly higher cross-talk levels in the E-plane, but it is below -30dB right across the
band. It should be again pointed at that these results only provide an approximation. This is due to the analysis being carried out in the 2D COMSOL modelling
environment which was shown to sometimes yield inaccurate results, some of which
can be significant as can be seen at certain frequency values in Figure 3.28. A final
verification using the 3D COMSOL package on a super computer would be required
for complete confidence that cross-talk levels would not be an issue in terms of the
required sensitivity of SAFARI.
The levels of cross-talk that were calculated from the simulations described in
this section are not considered to be of concern for a SAFARI-like instrument. Even
though -30dB is a significant level for some aspects of the instrument such as the
magnitude of the far-field beam at large angles, the optical cross-talk will only be
due to reflections of power from a horn rather than a radiating horn. Therefore the
initial emitted power at the ’radiating’ horn will already be very small, so -30dB of
an already low-level source will have little to no effect compared to the incoming
source signal.
104
(a)
(b)
Figure 3.31: Predicted cross-talk levels in SAFARI horn array. (a): Predicted levels for various separation distances.
(b):Predicted levels for a frequency sweep from 4.3 to 10THz.
3.2.3
Experimental Verification
In order to verify the optical cross-talk simulations presented in this section a series
of measurements were carried out using the VNA at NUIM. Two WR-10 waveguides
were chosen rather than two horns since the poor directionality of uniform waveguides allows for a stronger detectable signal, as was shown in the simulations in this
section. A single separation distance was chosen with its value constrained by the
dimensions of the waveguide flange sizes, as shown in Figure 3.32.
A frequency sweep from 75 to 100 GHz was carried out in both the E-plane
and H-plane. The results of the measurements are shown in Figure 3.33, where the
results of a corresponding FEM model are also shown for comparison. It is clear that
there is excellent agreement in the E-plane, and relatively good agreement in the
H-plane considering the very low levels measured. What should be noticed though
is that the H-plane data in the FEM simulation appears to indicate an upper limit
on the level of crosstalk since the levels in the measured data are consistently lower
than the model across the band. This indicates that the models can be trusted as
they appear to represent a worst case scenario.
The optical crosstalk tests presented here are clearly trivial measurements to
perform. However, they were important nevertheless since they served to verify
the simulations presented throughout section 3.2.2. It would also have been possible to cut the waveguides, and sacrifice the flanges. However, because this would
have rendered the waveguides useless afterwards, and because of the previous good
105
(a)
(b)
(c)
(d)
Figure 3.32: Photographs of WR-10 waveguides (a) to (c), and CAD model of WR-10 waveguide face. The holes
and connecting dowels were included in the COMSOL simulations so as to match the experimental set-up (d).
Figure 3.33: Optical crosstalk between two WR-10 waveguides measured with the VNA at NUIM. The simulation
results shown were performed using COMSOL.
agreement reported, it was chosen not to take the destructive route.
106
3.3
Characterising Throughput of SAFARI-Like
Multi-Moded Feed Horns
3.3.1
Background
The SAFARI research team at the SRON is currently performing a detailed analysis of a single SAFARI-like pixel in what they are calling the SAFARI testbed
[Audley et al., 2013]. Figure 3.34 shows a schematic that highlights the main components of the sysem. The testbed is an ultra-low background test facility that
consists of a multi-stage refrigerator that is pre-cooled by a Cryomech PT-415 pulsetube cooler which is attached to the 50-K and 3-K stages. The SRON team report
base temperatures below 8 mK on the mixing chamber, and describe operating detectors in the dark with bath temperatures as low as 15 mK. The primary aim of the
SAFARI testbed is to characterize and qualify the SAFARI focal plane arrays and
readout before they are integrated into the SAFARI instrument, and the testbed is
now being used routinely for measurements of prototype SAFARI detectors.
(a)
(b)
Figure 3.34: (a): A schematic of the SRON SAFARI S-Band testbed that is currently being used to characterise a
single SAFARI-like pixel. (b): A micrograph of the TES coupled to the Ta absorber in front of the hemipsherical
backshort. The white circle in the lower left of the image represents the footprint of the 46µm exit aperture of the
conical horn (image credit: SRON).
The measurements involve characterising the optical response of the TES detectors designed for SAFARI’s short-wave band (S-Band) focal plane array (FPA)
that is designed for operation across a frequency range of 30—70 µm. However if
the optical response of the TES is to be precisely characterised, then the effects of
107
Figure 3.35: Shematic showing the positioning of the horn, aperture, source, etc. of the SRON SAFARI S-Band
testbed (image credit: SRON).
all other components in the optical system must be taken into account. For example
the spectrum of the blackbody source must be known and the throughput of the
horn as a function of frequency should be well understood.
In this section an experiment to try to characterise the frequency dependent
throughput of a SAFARI-like horn is described. The aim of the experiment was
to prove the concept that such a method, as will be discussed, could be used to
measure the increase in throughput as a function of increasing frequency as more
modes begin to propagate as the ’cut-on’ frequencies of the higher order modes are
reached. The detectors of the SAFARI imaging FTS will require such multi-moded
feed horns (i.e. feed horns fed by oversized waveguides) to direct the signal toward
the TES’s that will be housed inside integrating cavities.
These feed horns will have their own frequency profile along with the profile of
the imaging FTS and the detector itself. This is because the number of modes that
can propagate through a waveguide structure depends on the transverse dimensions
of the structure relative to the wavelength of the radiation. The larger the cross
sectional area of the guide compared to the wavelength, the more modes it will
allow to propagate through it. And since each propagating mode can carry power
through the structure then we would expect the throughput of the horn to increase
as a function of increasing frequency. It will therefore be important to take account
of the effect of all the components of the instrument as a function of frequency to
fully understand the detected power by the TES’s. In this regard it is the frequency
108
response of SAFARI-like feed horns that is discussed in this section.
What can be expected for a multi-moded horn with a straight smooth walled
waveguide section is that as the frequency is increased the amount of power passing
through the horn should also increase. In fact due to the waveguide section the
increase should occur in steps as each higher order mode ’cuts on’. This is because
each propagating mode has a particular cut-off frequency, below which the mode
will not propagate. The cut-off frequency fco for a given mode depends on the
dimensions of the waveguide and is described by 3.17
r
c nπ 2 mπ 2
+
fco =
2π
a
b
(3.17)
where c is the speed of light, a and b are the width and height of the waveguide in
the transverse direction, n is an integer corresponding to the variation of the electric
field in the x direction, and m is an integer corresponding to the variation of the
electric field in the y direction.
The SAFARI testbed described above is currently using a smooth walled conical feed horn with no waveguide section for coupling the radiation to the detectors
at frequencies ranging from 4.3THz to 9THz. The throughput characterisation experiment described in this section is concerned with the analysis of a smooth walled
back-to-back rectangular horn configuration with a rectangular waveguide section
of width 250µm and height 125µm between the two horns, but in principle the
same technique could be applied to any type of feed horn. Figure 3.36(a) shows
the dimensions of the horn that was chosen for characterisation. Figure 3.37 is a
micrograph taken of the waveguide section of the back-to-back horn structure. As
the right side of the image shows there is an excellent finish on the structure even
at these extremely small scales. Figure 3.36(b) displays a human hair that is approximately 100µm in width which is just slightly smaller than the width of the
waveguide section of the back-to-back structure.
Figure 3.38 shows a simple geometrical analysis of propagating mode number
as a function of frequency for a waveguide with the dimensions described above. The
expected number of modes based on a 1/λ2 approximation is shown for comparison.
In fact if there was no uniform waveguide section to act as a mode filter the expected
throughput as a function of frequency would be expected to look more like the
smoothly increasing 1/λ2 profile. The reasons for this are explained in detail in
109
(a) Horn dimensions
(b) Human hair
Figure 3.36: (a): Dimensions of back-to-back horn that was characterised at SRON. (b): Micrograph of a human
hair for a reference of micron scales.
Figure 3.37: Dimensions of waveguide section of horn that was characterised at SRON. The expanded image on the
right shows how good the etching process is at achieving a smooth finish on the structure. Two separate precision
measurements were made of the height of the waveguide section showing 125.84µm ±10nm and 126.51µm ±10nm.
chapter 4 and it arises due to the partially evanescent behaviour of conical modes
when they are close to cut-off. The presence of the waveguide section between the
two horn sections is expected to yield a much more quantized or stepped profile,
similar to the geometric solution in the plot.
It can be seen that the waveguide has a ’switch-on’ frequency of 600GHz, and
if it is assumed that each propagating mode carries equal power then the graph can
be interpreted as throughput vs. frequency. The frequency of the sources ranged
from 200GHz to 4THz, and the cut-on frequencies of the first 20 modes within this
range are shown in Table 3.2.
Some of the issues that inevitably arise during an analysis of such tiny feed
horns at THz frequencies are the high absorption due to water vapour in the air, the
noise generated by the detector itself at temperatures above liquid Helium levels,
and the tiny size of the actual horns leading to difficulties in alignment and reduced
throughput levels. The solutions to all of these issues are discussed in the next
110
Cut-Off Freq.
Mode
(THz)
Total No.
Cut-Off Freq.
of Modes
(THz)
Mode
Total No.
of Modes
0.6
T E10
1 (’Switch-On’)
2.4
T E02
11
1.2
T E20
2
2.4
T E40
12
1.2
T E01
3
2.474
T E12
13
1.342
T E11
4
2.474
T M12
14
1.342
T M11
5
2.683
T E22
15
1.697
T E21
6
2.683
T M22
16
1.697
T M21
7
2.683
T E41
17
1.8
T E30
8
2.683
T M41
18
2.163
T E31
9
3.0
T E32
19
2.163
T M31
10
3.0
T M32
20
Table 3.2: Cut-on frequencies of the first 20 modes in the waveguide linking the back-to-back horn. The ’switch-on’
frequency, based on the transverse dimensions of the rectangular waveguide section, is 600GHz.
Figure 3.38: Increasing number of propagating modes as a function of frequency. If each mode is considered to
carry equal power then the graph can be thought as throughput as a function of frequency also. A scaling factor of
2 × 10−7 m2 was used to plot the 1/λ2 fit.
subsection.
111
3.3.2
Experimental Set-Up
An FTS based on a Michelson interferometer was used for the analysis of the horn.
A detailed description of the FTS system is presented in section 1.5 of this thesis.
Two different sources were employed; a high voltage mercury (Hg) arc lamp and
a glow bar [Shumyatsky and Alfano, 2011]. Hg arc lamps are commonly used in
a wide range of scientific experiments ([Yuana et al., 2014], for example), so no
description was deemed necessary. A glow bar (or globar) is a resistively heated
rod, typically made of silicon carbide. It is somewhat similar to a tungsten filament
in an incandescent light bulb. Of course any object will emit radiation with a peak
intensity at a wavelength that depends on its temperature. Wien’s displacement
law can be used to show that for a light bulb aimed at producing visible light
(λ ≈ 500nm) the blackbody temperature required for the tungsten filament is TBB ≈
5, 800K. Based on Wien’s displacement law then, it would appear that for the desired
THz frequencies, say for example, 3THz (λ ≈ 0.1mm), the temperature of the glow
bar should be set to TBB = 28.98K for the emission spectrum to peak at 3THz and
exhibit an overall intensity distribution that follows a blackbody spectrum as that
shown by the green line in Figure 3.39. However, this assumes that the surroundings
are cooler than the globar, which is not typically the case. Rather, the globar is
usually operated at a high temperature, and then emits over a very wide spectrum,
some of which covers the THz regime, but with a relatively weak output power at the
THz frequencies [Shumyatsky and Alfano, 2011]. This is the scenario under which
the globar was operated for the experiment described in this section. Research into
the spectra of globars can be found in [Pozela et al., 2013], for example.
The spectrum of the Hg lamp was not accurately known, but it was assumed to
have more discrete features and also having high emission in the ultra violet (UV)
requiring protective glasses to be worn. Both sources produced a large amount of
low frequency noise so a beam chopper and lock-in amplifier were used to cut out
these low frequencies. The beam chopper was set to 16.67Hz and the lock-in amp
was set to match the frequency of the chopper.
To overcome the absorption due to water vapour in the air a vacuum system
was used that completely housed the source, the FTS, the horn, the detector, and
the lenses (which will be discusses shortly) as illustrated in the schematic in Figure
3.40. The vacuum tube system, FTS, and pump are shown in Figure 3.41.
112
Figure 3.39: Predicted blackbody emission for radiating sources of various temperatures.
Figure 3.40: Schematic of the vacuum system used to house the horn, lenses, and FTS for throughput experiment.
Each section of steel tubing was connected together with air-tight seals before pumping commenced.
To deal with the thermal emission of the bolometer [Hulbert and Jones, 1955]
(which would lead to noise in the signal detected) it was housed in a two-stage
cryostat and cooled to near absolute zero. Liquid nitrogen was first used to cool the
central stage of the cryostat to 77 K. The liquid nitrogen was then pumped out of the
central stage and liquid helium was pumped in, thereby reducing the temperature
to 4.3 K. Liquid nitrogen was then pumped into the second (outer) stage of the
cryostat to maintain the low temperature of the detector by reducing the boil-off
rate of the liquid He. The signal detected by the bolometer was fed to an amplifier
which amplified it by a factor of either 200 or 1,000 (depending on the setting). The
cryostat also had three low pass filters built into it which cut off frequencies above
113
either 3THz, 9.4THz, or 18THz, depending on the setting chosen.
(a) FTS and vacuum tubes
(b) FTS close-up
(c) Vacuum pump
Figure 3.41: FTS and vacuum system.
In order to focus the source onto the aperture of the horn a three lens telescope
design was decided upon, where high-density polyethylene (HDPE) lenses were chosen as the HDPE material is well suited for control of radiation at THz frequencies
due to its excellent clarity, chemical stability, and the fact that it is easily machinable
[Lee, 1979]. HDPE is also particularly suited for THz radiation due to t The design
was comprised of three lenses with three different focal lengths, two of which were
positioned between the source and the horn, and one positioned between the horn
and the FTS (see Figure 3.42). The optical modelling package ZEMAX was used for
the design of the lens system. The optimisation tool was used to find the optimal
foci of the lenses and their positioning. The result was an 80mm focal length lens
positioned 80mm (d1) from the source, a 180mm lens at 40mm (d2) from the first
lens, the horn positioned 180mm from the 180mm lens (d3), and finally a 60mm
lens located 60mm from the exit aperture of the horn (d4). It should be noted that
the 180mm lens is focussed on the entry aperture of the horn as the phase centre
was unknown. All the lenses had diameters of 40mm.
The first lens (L1) was a relatively fast lens (f/2) that served to parallelise
quickly the source. The 180mm lens (f/4.5) labelled as L2 was designed to slowly
focus the beam to match the slow angle of the feed horn. The fast 60mm (f/1.5) lens
labelled as L3 was aimed to capture the light emerging from the horn and parallelise
the beam before it entered the FTS; (if a good interferogram is to be observed it is
important to have a parallel beam entering the FTS as this is assumed by optical
design). The entry port of the FTS was positioned at 70mm from the third lens.
The cryostat housing the detector was connected to the output port of the FTS.
114
Figure 3.42: Diagram showing the placement of the horn and lenses.
3.3.3
Manual Optimisation of Set-Up
Before the system was put under vacuum it was decided to experiment with the
positioning of the lenses and horn to verify that the prescribed separation distances
based on the lens focal lengths were indeed optimised. This was to ensure that
there were no minor errors in the curvature of the lenses, etc. due to manufacture
tolerances. If the curvature of any lens deviated from the prescription then the focal
length would also deviate from the expected value. Also the refractive index of the
lens material was not accurately known so again the focal lengths of the lenses came
into question to some degree.
An optical bench was set up with the horn and lenses positioned at the distances
calculated by ZEMAX. The set-up was the same as described above but without
the vacuum system. Also the FTS was not used as at this point the aim was just
to optimise the positions of the components for maximum throughput of the horn.
The glow bar that was described above was used as a source and placed 80mm from
the first lens L1. 5 V was supplied to the glow bar with a current of 3.92 A. The
cooled detector was placed at 60mm from the third lens L3. The initial positions of
the back-to-back horn and lens L2 were as shown in Figure 3.42.
The lenses were attached to metal screens with apertures that had the same
dimensions as the lenses, and the horn was attached to a similar screen with an
aperture the same size as the horn opening. The optical bench was set up such
that the transverse positions (x) of the horn and lenses were aligned and fixed by
keeping the edges of the lens/horn holders tight against a metal straight edge that
115
was itself aligned perpendicular to the direction of axial propagation. The optical
components were free to move in the direction of axial propagation (z). The heights
(y) of the components were set equal to each other and kept fixed by manufacturing
the lens/horn holders appropriately. Part of the set-up can be seen in Figure 3.43,
although the transverse alignment bar (straight edge) and the detector are not in
the image at that point. In retrospect some absorber material should have been
used to coat the metal parts of the screens as standing waves could possibly arise in
such a set up.
As the lenses and horn were incrementally moved, one at a time, the value of
the detector voltage was monitored. An audio device was connected to the output
of the lock-in amplifier so that as the detector power increased an audio signal could
be heard to increase in pitch. This allowed the positions to be varied with real time
audible feedback indicating increase or decrease of received signal.
Through the method described above it was discovered that indeed there were
errors on the assumed focii of the lenses. The 60mm lens was found to have a focal
length of 64mm, the 80cm lens had a focal length of 82mm, and the largest deviation
was on the 180mm lens which actually had a focal length of 167mm. However it
should be noted that a small amount of the apparent error on the 180mm lens (L2)
is likely due to the phase centre of the horn being located inside the horn. It would
therefore make sense that the lens would be better positioned closer to the horn
since the focal point/plane should ideally be situated at the phase centre of the
horn. The same might be true for the error on the 60mm lens L3 since this was
directly coupled with the horn also.
Another test that was carried out at this point was an analysis of the spot
size of the beam at the position of the entry aperture of the horn. Obviously the
aim was to have as small a spot size as possible in order to focus the maximum
amount of power into the horn. Although the method used did not give a direct
measure of the spot size, it did give a good indication of the order of the size of the
beam width. The method involved replacing the horn with a pinhole (small circular
aperture (r ≈ 0.5mm) in a metal screen) located at the previous position of the
entry aperture of the horn. This was also the position of the beam waist which was
verified in the procedure just described above. The pinhole was at the same height
as the horn aperture and this height was kept fixed.
116
Figure 3.43: Set-up of lenses and horn for verification of optimal positioning.
Starting at the central location the position of the pinhole was varied in the
horizontal transverse direction in increments of 0.5mm, out to a distance of 20mm.
The pinhole was then returned to the central location and then incrementally varied
in the opposite direction. At each position the detector voltage was measured and
recorded and the results can be seen in the graph in Figure 3.44. The power fell
to
1
e2
of the on-axis power at a distance of approximately 6mm on each side of the
maximum, indicating a beam width of approximately 12mm at the beam waist. This
was not ideal since the aperture of the horn was only 3.45mm×3.45mm, but it was
the best that could be achieved.
Figure 3.44: Beam width of a focussed broadband beam at the focal point of the 180mm lens.
117
The lens system with the pinhole in place of the horn was modelled with the
GBM analysis code described in chapter 2. It was found that by including a source
field corresponding to a horn with a wide flare angle a spot size of approximately
10mm in diameter was formed at the location of the pinhole. The model assumes
monochromatic light as a source, and the input beam width and phase radius of
curvature (input q) are somewhat arbitrary. Thus, as it stands the model is only an
approximation. Further work will be required to account more accurately for the
spacial and spectral distribution of the source. Figure 3.45 shows two screen shots
from the Gaussian beam program that was developed by the author for modelling
free-space propagation.
(a) Beam width vs. propagation distance
(b) G.U.I. for varying lens focal lengths
Figure 3.45: (a): Screen shot of beam width calculation from GBM analysis code. (b): Graphical user interface
(G.U.I.) for varying the focal lengths of two of the lenses. The separation distances are automatically recalculated
when the focal lengths are changed.
3.3.4
Alignment of the System Inside the Vacuum Tubes
An issue that arose when packing the horn and lenses into the vacuum tubes was
how to rigidly fix each component in the correct position such that it would remain
there and stay completely perpendicular to the direction of propagation. An earlier
attempt had been made that used three thin threaded guide wires that were inserted
through each component, with small nuts tightened at each surface to hold the
components in place. This set-up had too much movement in it and it was very
difficult to ensure that the lenses and horn were completely perpendicular to the
beam once the vacuum system was sealed. An exaggerated example diagram is
shown in Figure 3.46.
An improvement on this system was achieved using PVC plumbing pipes. A
length of pipe was found that had almost exactly the same outer diameter as the
118
inner diameter of the vacuum tubing system. The diameter of the PVC pipe was
just slightly too large so it was trimmed down in the workshop using a lathe. The
result was a pipe that fit snugly inside the vacuum tubes. The pipe was then cut
into various lengths relating to the various spacings that were required between the
lenses and horn. The lenses and horn mount had diameters just slightly less than
the inner vacuum tube diameter so they could be held in position by being pressed
between sections of the PVC piping and then inserted into the vacuum system (see
Figure 3.47).
Figure 3.46: An exaggerated diagram of how the guide wires might bend as the conponents are pushed through the
vacuum tubing.
(a) PVC piping cut to sizes
(b) Vacuum tube and PVC
(c) Tube with ring insert
pipe
(d) Pipe section in tube
(e) Optical components shown (f) Optical components shown
separated
snugly clamped
Figure 3.47: Some components of the set-up showing how the system was aligned and kept fixed.
119
3.3.5
Efficiency of FTS Beam Splitters
As will be shown in the results of the throughput experiments described in the next
section, a particularly strong spectral effect in any FTS data can arise from the beam
splitter if the material is too thick compared to the wavelength of the signal. A beam
splitter of any thickness will give rise to reduced efficiency at certain frequencies due
to destructive interference between different parts of the beam travelling different
path lengths within the beam splitter material. Any signal radiation that reflects off
the inner boundary of the beam splitter twice will interfere with signal propagating
straight through the material. Since the phase of the doubly reflected part will be
shifted with respect to the direct part of the beam some sort of interference will
occur either constructive, destructive, etc.
The type of interference described above is known as the Fabry-Perot effect.
Figure 3.48 shows the transmission profile of a 12µm and a 56µm Mylar beam
splitter based on this Fabry-Perot effect. The data for the graphs were calculated
from equations given in a paper by James and Ring [James and Ring, 1967] (see
also page 256 of Goldsmith [Goldsmith, 1998]). The analysis by James and Ring
was for a thin film of Melinex but the same principles can be applied to the Mylar
thin film used for this experiment by simply using the correct refractive index.
The efficiency of a beam splitter in a Michelson interferometer is given by the
depth of modulation of the interference fringes:
η(σ) = 4RT
(3.18)
where R and T are the reflection and transmission intensity coefficients respectively.
If absorption of the Mylar material is neglected then T = 1 − R, leading to:
η(σ) = 4R(1 − R)
(3.19)
James and Ring (see also page 62 of [Born and Wolf, 1999]) showed that the
intensity reflection coefficients could be described by [James and Ring, 1967]:
Rs =
2
2r1s
(1 − cos2δ)
4
2
1 + r1s − 2r1s
cos2δ
Rp =
2
2r1p
(1 − cos2δ)
4
2
1 + r1p − 2r1p
cos2δ
(3.20)
where the subscripts s and p refer to the two planes of polarisation of the electric field
relative to the plane of incidence. δ is the relative phase difference between adjacent
120
emerging beams and is defined as δ =
2π
ndcosφ1 ,
λ
where d is the film thickness, n is
the refractive index and φ1 is the angle of the beam inside the film relative to the
surface normal. Finally r1s and r1p are the well known Fresnel amplitude reflection
coefficients, and are defined as [James and Ring, 1967]:
r1s =
n0 cosφ0 − n1 cosφ1
n0 cosφ0 + n1 cosφ1
r1p =
n0 cosφ1 − n1 cosφ0
n0 cosφ1 + n1 cosφ0
(3.21)
where φ0 is the angle of incidence, φ1 is the angle of refraction, and n0 and n1 are the
refractive indices of the initial medium and the final medium, respectively. Figure
3.48 shows the transmission T of a Mylar beam splitter (n ≈ 1.76 at 1THz) for a
thickness of 12µm and 56µm. It is clear that the 12µm film has significantly less of
an effect on the throughput of the signal over the frequency range of interest.
Figure 3.48: Power transmission for Mylar beam splitters of different thicknesses.
For the experimentalist interested in performing Fourier transform spectroscopy
across the entire IR range using such Mylar (PET) beam splitters, a variety of beam
splitter films would be required, each with a different thickness in order to access a
series of discrete frequency regions, which could then be merged to form a continuous
spectrum. In a paper by Homes et al. they explain how thick silicon beam splitters
can offer numerous advantages over Mylar films [Homes et al., 2007].
121
3.3.6
Experimental
With the lenses and horn sandwiched snugly between the pipe sections and inserted
into the vacuum tubes the pump was switched on to evacuate the system. The
power supply for the glow bar was switched on and ten minutes was allowed for
the bar to heat up. The chopper and lock-in amplifier were switched on and the
phase of the lock-in amp was locked such that it and the chopper were completely
in phase (as described above). A series of scans were performed with the FTS with
varying resolution and varying frequency range. Performing a scan over a longer
distance range results in higher resolution in the Fourier transformed data, whereas
performing a scan with smaller step size ensures higher frequencies are reached in
the Fourier transformed data (see Nyquist frequency [Oppenheim et al., 1999]). A
number of different beam splitters were used also with the best results being obtained
with the 12µm Mylar beam splitter.
Each scan that was performed for the back-to-back horn was repeated keeping
all the parameters the same except with a metal screen with a small aperture in
place of the horn. The aperture was given the same dimensions as the horn, and an
extra length of PVC pipe was cut and placed in the vacuum system to account for
the length of the horn.
The aim of doing the scans with an aperture in place of the horn was to obtain a
calibration data set for each horn scan. Then by dividing the data from the aperture
scans into the data from the horn scans, any spectral features not specific to the
horn would be cancelled out. Such effects include the frequency response of the
detector itself, the spectrum of the source, the efficiency of the FTS beam splitter,
and any frequency dependent absorption profile of the PVC lenses. For example the
spectra of both the glow bar and the Hg arc lamp were not accurately known, and if
they were to be characterised then a well understood measurement system would be
required with a detector whose spectral response was accurately characterised. The
response of the bolometer that we were using was not well understood so it could
not be used to properly investigate the spectra of the sources.
Figure 3.49(a) shows a plot of the back-to-back horn throughput data, and
the corresponding aperture throughput data that would be used for calibration is
shown in Figure 3.49(b). Figure 3.50 then shows the result of dividing the back-to-
122
back horn data by the aperture data in an attempt to remove all undesired spectral
artefacts. The recorded noise levels in the bolometer were approximately 0.1mV,
where the source was removed but everything else remained the same, including the
level of vacuum.
(a) Uncorrected Data
(b) Calibration Data
Figure 3.49: (a): Uncorrected throughput for a back-to-back horn antenna illuminated by a glow bar in a Michelson
FTS. (b): Throughput of a small rectangular aperture illuminated by a glow bar in a Michelson FTS. This data
was used to correct the back-to-back horn throughput data.
Figure 3.50: Corrected throughput for a back-to-back horn antenna illuminated by a glow bar in a Michelson FTS.
The rectangular aperture data was used to correct the data plotted in Figure 3.49(a).
What should be noticed when comparing Figures 3.49(a) and 3.49(b) is that
the aperture data in 3.49(b) displays a continuous increase in throughput as the
frequency is increased whereas the same is not rue for the back-to-back horn data
123
in 3.49(a). This results in the corrected throughput profile shown in Figure 3.50
exhibiting a general fall-off of power as frequency increases. As will be shown in the
following results this was an ongoing phenomenon that was observed throughout the
measurement campaign. The reason for this is still unclear, however some discussion
on the matter is provided at the end of this section.
Performing identical scans with the rectangular aperture is analogous to using
a flat field in astronomy imaging with a CCD. The flat field is aimed at correcting
the image for differences in pixel efficiency from pixel to pixel. Of course we were
attempting to account for a lot more variables with the aperture scans used in this
experiment, however it was deemed the most accurate method with this particular
set up for removing the unwanted spectral features described above.
Scan 1:
The first scan performed was with the horn in the centre of the evacuated tubing
system without any lenses (see Figure 3.42). The glow bar was used as a source and
the beam chopper was set to 16.67Hz with the sensitivity on the lock-in amplifier
at 20mV and the time constant set to 1 second. The 56µm Mylar beam splitter
was used in the FTS and a scan of 10mm was carried out with a step size of 7.5µm.
The 100cm−1 filter was selected on the detectors filter wheel, with the gain set to
200×. The Fourier transformed data are shown in Figure 3.51. As can be seen
there is very little transmission, resulting in mostly noise in the plot. As before, the
recorded noise levels (source removed) were approximately 0.1mV.
Scan 2:
The second scan carried out included the three lenses, with two of them serving to
focus the source radiation onto the horn aperture and the third one positioned after
the horn with the aim of forming a parallel beam for entry into the FTS (see again
Figure 3.42). All the scan parameters were the same as the previous scan (Scan 1).
The Fourier transformed data are shown in Figure 3.52. It is clear that there is much
higher throughput when the lenses are used, and there is a sudden jump at 600GHz
which is exactly the position of the ’switch-on’ frequency of the waveguide section
between the two horn sections. The data shown have not yet been corrected for the
124
Figure 3.51: Throughput of the back to back horn with no lenses.
efficiency of the Mylar beam splitter, or any of the other undesirable effects outlined
above (i.e. the data has not yet been divided by the corresponding aperture scan
data). Nonetheless the ’switch-on’ frequency is quite obvious and even some of the
higher order propagating modes can be seen to ’switch on’, for example the T E20 &
T E01 at 1.2THz, and some indication of the T E11 & T M11 at 1.4THz. The positions
(in terms of frequency) of the cut-on frequencies of some of the higher order modes
are shown superimposed on the same graph.
Figure 3.52: Throughput of the back to back horn with three lens optical system included.
125
The next graph shown in Figure 3.53 has the efficiency function of a 56µm Mylar
beam splitter shown superimposed on the previous plot data. This demonstrates
the strong modulating effect that such a beam splitter will have on the result. As
can be seen the null in the throughput data at 1.8THz coincides exactly with the
position of complete destructive interference in the beam splitter.
Figure 3.53: Throughput of back to back horn with three lens system, and 56 mum beam splitter efficiency.
Scan 3:
The 56µm beam splitter film was replaced with the 12µm film and another scan
with the same parameters as before was carried out. A second identical scan was
then performed but with the rectangular aperture replacing the horn. The back-toback horn throughput data was then divided by the aperture throughput data for
calibration. The corrected Fourier transformed data is shown in Figure 3.54 and
this represents the highest quality result that was achieved using the 100cm−1 filter
and the glow bar source during the measurement campaign at SRON.
Again the switch-on of the fundamental mode can be seen at 600GHz and
some of the higher order propagating modes can be seen to ’switch on’. The T E20
& T E01 are again recognisable at 1.2THz, and the T E11 & T M11 are assumed to be
the source of the increase in detected signal at 1.4THz. In fact there is also some
indication of the cut-on of the T E21 & T M21 modes at about 1.7THz (both should
126
Figure 3.54: Normalised throughput of the back to back horn with a 12µm beam splitter replacing the 56µm one.
cut-on at 1.697THz), and the T E31 & T M31 modes at about 2.2THz (these two
should cut-on at 2.163THz).
While it can be seen that the reduced beam splitter film thickness has improved
the result, there are still some details in the data that do not agree with what is
expected for the throughput as a function of increasing frequency for a horn with
a smooth walled waveguide section acting as a mode filter. Figure 3.54 shows that
there is a steady fall-off of detected signal beyond approximately 1.5THz. This is
possibly a result of the aperture having better coupling to the FTS than the backto-back horn does. A sophisticated modelling software that could simulate the full
optical system would likely be very informative on the observed spectral profiles.
Such a computational tool, capable of modelling the FTS, the quasioptical lens
beam path, the broadband source, and the horn and aperture would make for an
interesting and very useful thesis.
Scan 4:
A final set of measurements was performed with the aim of achieving higher detectable frequencies. The 9.4THz filter was selected on the filter wheel (in front of
the bolometer), the step size of the scan was set to 4µm allowing frequencies up to
almost 6THz to be resolved, and the source was changed to a Hg arc lamp as it
was thought that it would have stronger emission at the higher frequencies than the
127
glow bar. The thinner 12µm beam splitter was again used.
The uncorrected Fourier transformed data of the final scan for the back-to-back
horn can be seen in Figure 3.55, where the calibration data corresponding to the
scan for the rectangular aperture is also shown in the plot. What is clear straight
away is that there appears to be better agreement between the horn and the aperture
in terms of throughput at the higher frequencies. The previously observed higher
throughput for the aperture is not present in this scan.
Figure 3.56 shows the corrected throughput plotted as a function of frequency.
As can be seen the switch on at 600GHz is not clear and in fact there is a high level
of noise at frequencies lower than 600GHz that was not observed in the previous
scans. It is interesting however that the high levels of noise seem to dissipate once
the switch on frequency is reached.
Even though the there is not a clear rise in throughput at the expected 600GHz,
there is an evident rise in detected signal between 1.2 and 1.3THz which can be
attributed to the T E20 & T E01 modes which have a switch on frequency at 1.2THz,
and the T E11 & T M11 which are expected to switch on at 1.342THz. The next jump
that is observed at approximately 1.7THz is attributed to the T E21 & T M21 modes
(fcut−on = 1.697THz) and some contribution from the single T E30 mode (fcut−on =
1.8THz). There is then another significant step in signal just before 2.2THz which
again fits with two more modes with switch on frequencies at 2.163THz, namely
the T E31 & T M31 . The next modes are expected to cut-on at 2.4THz but again
there is a fall off of signal at these higher frequencies so these modes are not visible
switching on.
The final plot shown in Figure 3.57 is the same data as that in 3.56 but the
theoretical throughput profiles are also shown for comparison. The geometrical
analysis based on the transverse dimensions of the waveguide assumes that each
mode carries equal power and contributes 0.25mV to the corrected signal. This
appeared to provide a good fit with the measured data. The continuous red line is
again the expected profile based on a 1/λ2 approximation.
128
Figure 3.55: Throughput of the back to back horn and aperture for the Hg arc lamp using the 312cm−1 filter.
Figure 3.56: Corrected throughput of the back to back horn for the Hg arc lamp using the 312cm−1 filter.
3.3.7
Discussion of Results
The results presented in this section show clear evidence that the multi-mode behaviour of tiny feed horns can be measured at THz frequencies. There are still some
effects in the system that have not been taken care of as can be seen in each of
the graphs, particularly the general fall off of power as higher frequencies are approached especially when using the glow bar and 100cm−1 filter. Future work will
aim to characterise accurately the spectrum of the glow bar source, the Hg arc lamp
129
Figure 3.57: Corrected throughput of the back to back horn with theoretical plots shown also. It was assumed that
each mode carrys equal power and contributes 0.25mV to the corrected signal. This scaling provides a good fit to
the measured data.
source, the bolometer, and the frequency dependence of the absorption of the lens
material. Once these factors are known the data can be more accurately reduced
and corrected.
Another option for future work would be to redesign the optical system with
mirrors replacing the lenses. The benefits of this type of set-up would be easier
alignment and focus of the system, and there would be a large reduction in the
losses that occur due to absorption by the lens material. A mirror system would
be more difficult to manufacture due to the limited space within the vacuum tubes,
but the benefits would be worth it.
Although the back to back horn structure analysed in this report was larger
than the horn currently being used in the SAFARI testbed, the principle of the
technique has been successfully demonstrated. With sufficient care and precision
the same method could be applied to characterising the smaller horns presently
being employed by the testbed.
130
3.4
Conclusions
In the first section of this chapter three simulation methods were compared for
modelling smooth walled rectangular horns and calculating the corresponding farfield beam patterns. The COMSOL model included a propagation region beyond the
mouth of the horn that allowed for analysis of a conducting ground plane adjacent
to the mouth of the horn. It was shown that the effects of such a ground plane are
negligible unless the extreme case of a poor directional antenna is considered, such
as a uniform waveguide for example.
Through the comparison study the results of the Pyramidal SCATTER program were verified which is significant as the SCATTER program will be necessary
for modelling large SAFARI horn designs in the future. This necessity for SCATTER
is because the mode matching technique allows for the analysis of electrically large,
multi-moded structures in a very computationally efficient manner. High frequency,
multi-mode analysis of SAFARI-like horns was shown not to be possible with the
FEM approach on a computer with moderate processing power.
The optical cross-talk between neighbouring SAFARI horns in an array was
shown to be negligible. This is an important result due to the unprecedented sensitivity requirements of SAFARI. It was shown that the cross-talk levels are considerable only for instruments with poor directivity, such as two uniform waveguides
which are quite close together, for example.
The final section of this chapter detailed the measurement work that was carried
out by the author at SRON during a three month visit there from June 2013 to
September 2013. The work was aimed at measuring the throughput of SAFARI-like
rectangular feed horns as a function of frequency. The measurement technique was
shown to be successful, and results were presented which displayed clear evidence
of modes ‘switching on’ at the correct frequencies leading to increasing throughput
as frequency was increased. The experimental difficulties with such measurements
were discussed, and solutions to some of these issues were described. However, the
data presented was still quite noisy, and the falloff of signal at the higher frequencies
was difficult to explain. For these reasons some ideas on how to improve the system
in the future were given at the end of the section.
The techniques discussed in this chapter for characterising multi-moded feed
131
horns, both the experimental measurements and the modelling techniques, will be
essential for understanding and predicting the behaviour of future SAFARI horn designs and similar multi-moded structures, and will help in maximising the sensitivity
of these far-IR receivers.
132
Chapter 4
Conical Mode Matching for
Shaped Smooth-Walled CMB
Horns
“An investment in knowledge pays the best interest.”
- Benjamin Franklin
133
4.1
Smooth Walled Conical Horns
4.1.1
Background
Smooth walled sectional horns are becoming increasingly popular due to the simplicity of their geometry which leads to easier manufacture and therefore lower cost
compared to corrugated structures, for example. In fact at the operational wavelengths of the SAFARI short wave (SW) band (λ = 30µm to 70µm) manufacturing
arrays of corrugated structures is not possible with current technologies. Therefore
high performance smooth walled horns, that can have similar performance to corrugated horns, are currently in high demand for far-IR and sub-millimetre systems.
Of course smooth walled geometries will never completely match the excellent
performance of corrugated feeds, particularly for wide-band systems, but with optimisation of the profile similar performance can be achieved for most of the important
parameters across smaller bandwidths. An excellent example of this is described in
a paper by the THz Optics Group at NUIM [McCarthy et al., 2013]. The genetic
algorithms that can be used for optimisation of the profile of a smooth walled feed
horn are outlined and the authors then go on to describe a final design that was optimised, modelled, and then built for testing. The optimisation routines were based on
an adapted version of the in-house mode matching code SCATTER. The details of
the EM theory for mode matching in waveguides that SCATTER is based upon are
given in a number of publications ([Murphy et al., 2010] and [Murphy et al., 2001]
for example) and the theory was discussed in chapter 2 of this thesis.
The cylindrical mode matching theory is well established and the SCATTER
code for modelling systems in cylindrical coordinates that is based on this theory
has proved accurate and efficient for modelling a wide range of horn antennas. Due
to its high efficiency and short computational time for modelling even electrically
large structures, SCATTER was a good choice for the optimisation of the profile of
smooth walled horns due to the requirement that the calculations must be performed
a very large number of times. Although SCATTER is primarily aimed at simulating
the behaviour of corrugated horns it can be adapted for modelling smooth walled
horns quite easily by simply approximating the smooth-wall geometry by a large
number of very small steps. A screen shot of an example horn is shown in Figure
4.1.
134
Figure 4.1: Diagram showing how a smooth wall of a feed horn can be approximated by a series of small steps. The
green boxed-off section on the left is shown magnified on the right hand side so that the small steps are visible.
As efficient as this approach is, as demonstrated in chapter 2 where it was compared to the FEM approach, it was thought that there is still room for improvement
in the speed of computation. The solution is quite obvious and it involves ignoring
the steps that approximate the smooth walls. This can be achieved by deriving a
basis set of conical modes through solving Maxwell’s equations in spherical coordinates and by applying appropriate boundary conditions. Theoretically the only
scattering of power occurs when there is a change of flare angle, and there should be
no scattering of power occurring throughout a section of constant flare angle. This
assumption is based on the spherical Helmholtz equation having analytical solutions
for conic sections with PEC walls, where it will be shown in the following section
4.1.2 that the modal solutions, and the corresponding eigenvalues, depend directly
on the flare angle of the conic section. This implies that power coupling integrals
need only be performed at the relatively few junctions corresponding to change of
angle rather than the few hundred steps required for approximating even a small
horn.
It is this approach that is described in this chapter, starting with a derivation
of the conical modes. A description is then given of the new software entitled
Spherical SCATTER that was coded by the author using Mathematica. Finally,
the results for comparisons between the already existing Cylindrical SCATTER
code and the Spherical SCATTER will be shown, and then both methods will be
compared to experimental data corresponding to far-field beam measurements of a
smooth walled sectional horn performed using a VNA system. Excellent agreement
will be demonstrated between both theoretical approaches and measured data.
135
There were many drivers for this work with the primary motivation being the
requirement to design and produce a smooth walled Gaussian horn that compared
well with similar corrugated horns, as part of an ESA Announcement of Opportunity
(AO) programme [RFQ 3-6418/11/NL/CBi]. Another driver was based on the fact
that when the author began to investigate the theory for conical waveguide modes
it became clear that the existing literature on the subject is sparse and incomplete.
A number of publications discuss the ideas of conical mode matching to some degree, most notably those of [Olver et al., 1994] and [Clarricoats et al., 1984], but the
mathematical description of the spherical modes are only stated, with no derivation
of the theory in these texts. Furthermore the book by Clarricoats for example limits
the discussion to conical waveguide modes of degree 1 (m = 1) and gives no description on how to deal with modes of higher azimuthal dependence (m > 1). The
theory required to account for the higher order modes is essential for the design and
analysis of multi-moded feeds and it was therefore necessary to develop the theory
further.
4.1.2
Derivation of Spherical Harmonic Fields
The electric field of a particular mode of degree m = 1 and order l at the mouth of a
smooth walled conical horn antenna for the T E1l mode is described by [Gleeson, 2004]
as having the following structure:
~
E(x,
y) =
r
r
r
−jkr2
J0 p′1,l
+ J2 p′1,l
cos(2φ) ı̂ + J2 p′1,l
sin(2φ) ̂ exp
a
a
a
2L
(4.1)
where p′1,l is the lth root of
∂
J (z)
∂z 1
= 0. For the T E11 mode p′1,1 = 1.841.
Equation 4.1 is an approximation that has proven to be very accurate for horns
of moderate flare angle, however it is not completely analytically derived. It is
calculated by taking the analytically derived solution of the field inside a cylindrical
waveguide (see chapter 2.3) and adjusting the phase based on the flare angle of the
horn. It is assumed that the amplitude profile is unaffected when the waveguide
develops a uniform flare angle, meaning that the only effect is on the phase front
which becomes spherical, with its apex coinciding with the apex of the cone of the
horn. An extra term is thus included in the field definition to account for this effect
2
(exp −jkr
). A similar quadratic approximation can be exploited for smooth walled
2L
136
rectangular horns, as shown in chapter 3.
Figure 4.2: Smooth walled conical horn showing the extension back to its virtual apex where there will typically be
a cylindrical waveuide section adjoining the horn. The spherical phase cap is also shown at the aperture.
To derive an analytical solution without approximations that includes the phase
automatically, the fields must be expanded in terms of conical modes which have
the natural symmetry of a cone. These modes can be derived by looking for separable solutions to Maxwell’s equations in spherical coordinates, and considering the
boundary condition at the walls of the horn, i.e. that the tangential component of
the electric field must be zero (perfect electric conductor (PEC) boundary). The
resulting solutions can be partitioned into the usual TE modes and TM modes.
The starting point is to solve the electric Helmholtz equation (∇2 E + k 2 E = 0) in
spherical coordinates. One such solution is:
E(r, θ, φ) =
ν
∞ X
X
ν
γνm [jν (kc r) + yν (kc r)] Pνm (θ, φ)
(4.2)
m=−ν
where γνm is just some constant, Pνm (θ, φ) are Legendre functions of order ν and
degree m, and jν (kc r) and yν (kc r) are spherical Bessel functions of the first and
second kind respectively, and they are related to the radial Bessel functions Jν (kc r)
and Yν (kc r) by:
jν (kc r) =
r
π
J 1 (kc r)
2kc r ν+ 2
and yν (kc r) =
r
π
Y 1 (kc r)
2kc r ν+ 2
(4.3)
There are in fact a number of other solutions to the spherical Helmholtz equation, but the most general solutions that are convenient here are:
E(r, θ, φ) =
E(r, θ, φ) =
ν
∞ X
X
ν m=−ν
∞
ν
X X
ν
m=−ν
γνm [jν (kc r) + iyν (kc r)] Pνm (θ, φ)
(4.4)
γνm [jν (kc r) −
137
iyν (kc r)] Pνm (θ, φ)
where i is just the imaginary unit.
jν (kc r)+iyν (kc r) is sometimes called a Bessel function of the third kind, or more
(1)
commonly a Hankel function of the first kind denoted by hν (kc r), and jν (kc r) −
iyν (kc r) is sometimes called a Bessel function of the fourth kind, or more commonly
(2)
a Hankel function of the second kind denoted by hν (kc r) (see figure 4.3). Using
this notation for Hankel functions together with the relations in (4.3) gives:
r
r
∞ X
ν
X
π
π
Jν+ 1 (kc r) + i
Yν+ 1 (kc r) Pνm (θ, φ) (4.5)
E(r, θ, φ) =
γνm
2
2
2kc r
2kc r
ν m=−ν
⇒ E(r, θ, φ) =
where
(1)
hν (kc r)
ν
∞ X
X
m
γνm h(1)
ν (kc r)Pν (θ, φ)
(4.6)
m=−ν
ν
= jν (kc r) + iyν (kc r).
HankelHrL
HHrL
Re@HankelHrLD
0.4
Im@HankelHrLD
0.2
10
20
30
40
50
r
-0.2
-0.4
Figure 4.3: Real and imaginary parts of the Hankel function of the first kind. Notice that as r goes to zero the
amplitude of the imaginary part of the Hankel function goes to minus infinity, i.e. a singularity.
Equation 4.6 represents a random component of the electric field that is neither
specifically TE or TM. However, it is more convenient to begin with the radial
component of either the TE or TM modal field. Nagelberg gives a definition for the
radial component of the electric field for the TM modes, assuming harmonic time
dependence (e−iωt ), as [Nagelberg, 1967]:
Er = A
µ(µ + 1) (1)
hµ (kr)Pµm (cos θ)eimφ and Hr = 0
ikr2
(4.7)
and for the TE modes the radial component of the magnetic field [Nagelberg, 1967]:
Hr = B
ν(ν + 1)
m
imφ
y0 h(1)
and Er = 0
ν (kr)Pν (cos θ)e
ikr2
(4.8)
The above expressions are valid for waves travelling away from the apex and
toward the aperture of the horn (forward going waves). For waves moving in the
138
opposite direction back toward the apex of the cone (backward going waves) the
mathematical descriptions are very similar but with the radial component being
governed by a Hankel function of the second kind:
Er = A
µ(µ + 1) (2)
hµ (kr)Pµm (cos θ)eimφ and Hr = 0
ikr2
(4.9)
and for the TE modes:
Hr = B
ν(ν + 1)
m
imφ
y0 h(2)
and Er = 0
ν (kr)Pν (cos θ)e
ikr2
(4.10)
In fact the above field components can be further separated into two polarizations. For the forward going TM modes:
µ(µ + 1) (1)
cos(mφ)
m
hµ (kr)Pµ (cos θ)
and Hr = 0
Er = A
ikr2
sin(mφ)
(4.11)
And for the forward going TE modes:
sin(mφ)
ν(ν + 1)
(1)
m
y0 hν (kr)Pν (cos θ)
Hr = B
and Er = 0
ikr2
− cos(mφ)
(4.12)
A similar separation of polarisations holds for the backward going waves also.
Knowing the radial components of the electric and magnetic fields should allow the
remaining components of the fields to be calculated. This is straightforward in the
case of constant radius cylindrical waveguides, where the equivalent component is
the longitudinal ’z’ component, due to the e−γz dependence, but it becomes much
more difficult for the case of a conical horn due to the spherical geometry. Since
the radius continually changes with length the e−γz dependence has been replaced
with Hankel functions of either the first or second kind that account for the diffusion
or concentration of the field as it fills ever increasing or decreasing cross sections,
respectively, as r increases.
The transverse components (θ and φ) of the TE and TM modes can be derived
from Maxwell’s curl equations:
~
~
~
~ ×E
~ = −∂ B = −µ ∂ H = −iωµH
∇
∂t
∂t
~
~ ×H
~ = ǫ ∂ E = iωǫE
~
∇
∂t
(4.13)
(4.14)
~ for example, in spherical coordinates is:
And the curl of H,
1
~ ×H
~ =
∇
det
2
r sin θ
r̂
rθ̂
r sin θφ̂
∂
∂r
∂
∂θ
∂
∂φ
Hr rHθ r sin θHφ
139
(4.15)
Thus 4.14 yields:
1 ∂(Hθ )
1 ∂Hr 1 ∂(rHφ )
1 ∂(sin θHφ )
~
iωǫE =
r̂ +
−
−
θ̂
r sin θ
∂θ
r sin θ ∂φ
r sin θ ∂φ
r ∂r
(4.16)
1 ∂(rHθ ) 1 ∂(Hr )
+
−
φ̂
r ∂r
r ∂θ
~ are then given by:
And rearranging the above equation, the components for E
1 ∂(Hθ )
1 ∂(sin θHφ )
1
−
(4.17)
Er =
iωǫ r sin θ
∂θ
r sin θ ∂φ
1
1 ∂Hr 1 ∂(rHφ )
Eθ =
(4.18)
−
iωǫ r sin θ ∂φ
r ∂r
1 1 ∂(rHθ ) 1 ∂(Hr )
(4.19)
−
Eφ =
iωǫ r ∂r
r ∂θ
Applying similar analysis to equation 4.13, which relates the magnetic field to
~
the curl of the electric field, yields for the components of H:
1
1 ∂(Eθ )
1 ∂(sin θEφ )
Hr = −
−
iωµ r sin θ
∂θ
r sin θ ∂φ
1
1 ∂Er 1 ∂(rEφ )
Hθ = −
−
iωµ r sin θ ∂φ
r ∂r
1 1 ∂(rEθ ) 1 ∂(Er )
−
Hφ = −
iωµ r ∂r
r ∂θ
(4.20)
(4.21)
(4.22)
Substituting the expression 4.22 for Hφ into the expression 4.18 for Eθ yields:
rEθ = −
iωµ ∂Hr
1 ∂ ∂(Er )
1 ∂ 2 (rEθ )
+
−
k 2 sin θ ∂φ
k 2 ∂r2
k 2 ∂r ∂θ
(using k 2 = ω 2 µǫ)
(4.23)
which can be rewritten as:
iωµ ∂Hr 1 ∂ ∂(Er )
∂ 2 (rEθ )
=− 2
rEθ −
+
∂(kr)2
k sin θ ∂φ
k ∂θ ∂kr
(4.24)
Similar analysis applied to substituting the expression 4.21 for Hθ into the expression
4.19 for Eφ yields the result:
∂ ∂(Er )
∂ 2 (rEφ )
iωµ ∂Hr
1
rEφ +
= 2
+
∂(kr)2
k ∂θ
k sin θ ∂φ ∂kr
(4.25)
TE Mode Solutions
If at this point only the TE modes are considered then Er = 0. Assuming equation
(1)
y0 hν (kr)Pνm (cos θ) −sin(mφ)
(see (4.12))
4.24 for Hr of the form Hr = B ν(ν+1)
ikr2
cos(mφ)
implies:
∂ 2 (rEθ )
ωµy0 B ν(ν + 1) (1)
cos(mφ)
m
⇒ rEθ +
=− 2
hν (kr)Pν (cos θ)m
(4.26)
∂(kr)2
k r sin θ
kr
sin(mφ)
140
Using the relationship on page 445 of Abramowitz and Stegun, the spherical Hankel
functions satisfy the following differential equation [Abramowitz and Stegun, 1972]:
(1)
ν(ν + 1)
∂ 2 (z.hν (z))
(z.h(1)
+ 1−
ν (z)) = 0
∂z 2
z2
where
z.h(1)
ν (z)
=
r
πz (1)
H
(z)
2 ν+0.5
(4.27)
Then with some rearranging and substitution it can be shown that:
ν(ν + 1) (1)
1 ∂ 2 (1)
rh
(kr)
hν (kr) = rh(1)
(kr)
+
ν
ν
r
k 2 ∂r2
(4.28)
Substituting the right side of the above expression into equation 4.26:
!
"
#
(1)
2
∂ 2 (rEθ )
mωµy0 B
∂
rh
(kr)
cos(mφ)
ν
=−
Pνm (cos θ)
rh(1)
rEθ +
ν (kr) +
∂(kr)2
k sin θ
∂(kr)2
sin(mφ)
(4.29)
and therefore we obtain
cos(mφ)
mωµy0 B (1)
m
hν (kr)Pν (cos θ)
⇒ rEθ = −
sin(mφ)
k sin θ
pµ
Finally by using the fact that ωµ
=
= y10 , this simplifies to:
k
ǫ
cos(mφ)
mB (1)
m
h (kr)Pν (cos θ)
Eθ = −
r sin θ ν
sin(mφ)
(4.30)
(4.31)
Applying the same TE conditions to equation 4.25 for Eφ , and with similar
analysis and substitutions, an expression for Eφ can be derived that is independent
of any other field components.
rEφ +
1 ∂2
iωµ ∂Hr
(rEφ ) = 2
2
2
k ∂r
k ∂θ
(4.32)
giving
ν(ν + 1)
iωµ
∂Pνm (cos θ) sin(mφ)
∂ 2 (rEφ )
(1)
rEφ +
= 2 B
y0 hν (kr)
∂(kr)2
k
ikr2
∂θ
− cos(mφ)
(4.33)
which yields in a similar manner that:
B (1)
∂Pνm (cos θ) sin(mφ)
⇒ Eφ = hν (kr)
r
∂θ
− cos(mφ)
(4.34)
In order to derive the transverse components of the magnetic field (Hθ and Hφ )
for the TE solutions the expressions above derived for Eθ and Eφ (4.31 and 4.34) can
be substituted into equations 4.22 and 4.21 respectively, setting Er = 0 to satisfy
the TE conditions. Starting with 4.22:
141
(1)
1 1 ∂(rEθ )
mkB ∂hν (kr) m
cos(mφ)
Hφ = −
=
Pν (cos θ)
iωµ r ∂r
iωµr sin θ ∂kr
sin(mφ)
(1)
cos(mφ)
imBy0 ∂hν (kr) m
Pν (cos θ)
⇒ Hφ = −
r sin θ
∂kr
sin(mφ)
(4.35)
(4.36)
Similarly, for Hθ :
(1)
iBy0 ∂hν (kr) ∂Pνm (cos θ) sin(mφ)
1 1 ∂(rEφ )
=−
Hθ =
iωµ r ∂r
r
∂kr
∂θ
− cos(mφ)
(4.37)
TM Mode Solutions
The transverse components of both the electric and magnetic fields for the case of
TE modes have been derived above. If the same starting point of equations 4.24
and 4.25 are taken, but this time allowing the radial component of the magnetic
field to be zero (Hr = 0) thereby satisfying the conditions of TM modes, then a
similar analysis can be applied to derive the transverse field components of the TM
solutions. The resulting expressions are:
(1)
iA ∂hν (kr) ∂Pνm (cos θ) cos(mφ)
Eθ = −
r
∂kr
∂θ
sin(mφ)
(1)
sin(mφ)
imA ∂hν (kr) m
Pν (cos θ)
Eφ =
r sin θ ∂kr
− cos(mφ)
sin(mφ)
mAy0 (1)
m
h (kr)Pν (cos θ)
Hθ = −
r sin θ ν
− cos(mφ)
Ay0 (1)
∂Pνm (cos θ) cos(mφ)
Hφ = −
hν (kr)
r
∂θ
sin(mφ)
(4.38)
(4.39)
(4.40)
(4.41)
Expressions for the complete set of electric and magnetic field components have
now been derived for both the TE and TM mode solutions and they are summarised
in Table 4.1 below. Note that the eigenvalues for the TM modes are now denoted
by subscript µ rather than ν as before. This is to differentiate them from the TE
mode eigenvalues. This notation will be used from this point onwards.
4.1.3
Calculation of Eigenvalues
One of the complications that arises when trying to calculate the fields of a smooth
walled conical horn in spherical coordinates is that the eigenvalues of the Legendre
142
TE Modes
Er = 0
cos(mφ)
(1)
m
h
(kr)P
(cos
θ)
Eθ = − rmB
ν
ν
sin θ
sin(mφ)
∂Pνm (cos θ) sin(mφ)
B (1)
Eφ = r hν (kr) ∂θ
− cos(mφ)
(1)
ν(ν+1)
Hr = B ikr2 y0 hν (kr)Pνm (cos θ) −sin(mφ)
cos(mφ)
(1)
sin(mφ)
iBy0 ∂hν (kr) ∂Pνm (cos θ)
Hθ = − r
− cos(mφ)
∂kr
∂θ
(1)
imBy0 ∂hν (kr) m
Hφ = − r sin θ ∂kr Pν (cos θ) cos(mφ)
sin(mφ)
TM Modes
(1)
hµ (kr)Pµm (cos θ) cos(mφ)
Er = A µ(µ+1)
ikr2
sin(mφ)
(1)
m
iA ∂hµ (kr) ∂Pµ (cos θ) cos(mφ)
Eθ = − r ∂kr
sin(mφ)
∂θ
(1)
imA ∂hµ (kr) m
Eφ = r sin θ ∂kr Pµ (cos θ) −sin(mφ)
cos(mφ)
Hr = 0
h
sin(mφ) i
(1)
m
0
h
(kr)P
(cos
θ)
Hθ = − mAy
µ
µ
− cos(mφ)
r sin θ
h
i
∂Pµm (cos θ) cos(mφ)
(1)
Ay0
Hφ = − r hµ (kr) ∂θ
sin(mφ)
Table 4.1: Mode expressions in spherical coordinates for a conical feed horn.
functions are not integers as is the case for spherically symmetric systems such as
the electron in the spherically symmetric potential of the proton (Hydrogen). This
is because the conical horn is only a conic section of a sphere, rather than a complete
sphere, so the boundary condition on the polar coordinate θ is determined by the
flare angle of a cone instead of the requirement that the solution be finite everywhere
from θ = 0 to θ = π. Therefore the values for ν and µ must be calculated by
considering the boundary conditions at the walls of the horn where (θ = θh ). Thus,
for TM modes, Er = 0 and Eφ = 0 when θ = θh , leading to Pµm (cos θh ) = 0, while
d m
[Pν (cos θ)] θ=θ = 0.
for TE modes, Eφ = 0 when θ = θh , leading to dθ
h
Values for ν and µ were calculated for the case of m=1 by finding the roots
of the two equations above for particular values of horn half angle θh . The values
are listed in tables 4.2 and 4.3, and the values for a horn of half angle of θh = 5◦
are highlighted as this is appropriate for the example horn that is most commonly
discussed in this section. It is clear from the tables that as the half angle of the horn
gets larger the eigenvalues get smaller. In fact the eigenvalues approach integers for
very large values of half angle as would be expected as one goes smoothly over to
143
the free space case.
Horn Half
µ1
µ2
µ3
µ4
µ5
1◦
219.041
401.464
582.397
762.891
943.198
2◦
109.271
200.482
290.949
381.196
471.349
3◦
72.6819
133.489
193.800
253.964
314.066
4◦
54.3874
99.9921
145.225
190.348
235.425
5◦
43.411
79.8943
116.081
152.179
188.240
6◦
36.0935
66.4958
96.6508
126.733
156.784
7◦
30.8669
56.9255
82.7725
108.557
134.315
8◦
26.9471
49.7479
72.3638
94.9252
117.463
9◦
23.8985
44.1654
64.2682
84.3227
104.356
10◦
21.4598
39.6995
57.7918
75.8408
93.8711
Angle (θh )
Table 4.2: Tabulated results for the first five eigenvalues for TM modes for horns of various opening half angles.
Horn Half
ν1
ν2
ν3
ν4
ν5
1◦
104.995
304.970
488.596
670.205
851.121
2◦
52.253
152.237
244.049
334.853
425.311
3◦
34.674
101.326
162.533
223.070
283.375
4◦
25.887
75.871
121.776
167.178
212.407
5◦
20.616
60.599
97.322
133.643
169.826
6◦
17.103
50.417
81.019
111.287
141.439
7◦
14.594
43.145
69.375
95.318
121.163
8◦
12.714
37.692
60.642
83.342
105.955
9◦
11.252
33.450
53.849
74.027
94.128
10◦
10.084
30.057
48.416
66.575
84.666
Angle (θh )
Table 4.3: Tabulated results for the first five eigenvalues for TE modes for horns of various opening half angles.
If a mode matching simulation tool based on these field definitions was to be
developed it would obviously require a technique for automatically calculating the
eigenvalues of horn sections with arbitrary flare angles. Various approximations can
144
be applied for the Legendre functions and their derivatives, including one described
on page 62 in [Clarricoats et al., 1984]. That particular approximation states that
for small angles (and therefore large µ and ν) the Legendre function for m=0, and
its derivative, can be written as:
1
θ
(4.42)
µ+
Pµ (cos θ) ≈ J0
2
1
d
1
ν+
J1
θ
(4.43)
[Pν (cos θ)]
≈− ν+
dθ
2
2
θ=θh
d
But Pµ1 (cos θh ) = dθ
[Pν (cos θh )] therefore the Legendre function for m=1, and its
derivative, can be written as:
Pµ1 (cos θ)
d 1
[P (cos θ)]
dθ ν
θ=θh
1
≈−
2
1
≈− µ+
2
1
ν+
2
2
J0
J1
1
µ+
2
1
ν+
2
θ
(4.44)
1
ν+
θ − J2
θ
2
(4.45)
where J0 , J1 and J2 are the zero order, first order and second order Bessel functions,
′
respectively, and the relationship Jn (x) = 21 (Jn−1 (x) − Jn+1 (x)) was used.
The above approximations are very convenient since the roots of the Bessel
functions are well known and they can be automatically generated in some mathematical programming environment such as Mathematica. The required eigenvalues
of the Legendre functions (and their derivatives) can then be calculated by a “find
root” function with the approximation just described as a starting point for the
search.
Comparisons for the Legendre functions and the Bessel function approximations are shown below for a range of angles. It is clear from figure 4.4 that the
approximation (4.44) only begins to deviate for angles approaching 50◦ . There is
greater deviation at smaller angles for the approximation (4.45) (≈ 30◦ ), but since
these approximate solutions will only be used as an initial search point for the
Legendre function eigenvalues then the agreement is considered sufficiently good.
Furthermore, feed horns for the purposes of SAFARI-like multimode systems will
generally consist of relatively shallow flare angles that would certainly not exceed
30◦ . In fact for horns that lie within these limits the approximations could actually
be used for describing the fields and could therefore be employed in the mode matching algorithms. Analytical solutions for power coupling integrals involving Bessel
145
functions are what are used in the Cylindrical SCATTER program so they are well
understood. Of course this close relationship between the Legendre functions and
Bessel functions is related to the good approximation of representing the fields of
a conical horn by a wavegide mode. The Pν1 (cos θ) function for the conical horn
determines the amplitude as it varies with θ, but with a spherical wavefront that is
(1)
represented by the Hankel function hµ (kr), which is constant as θ varies. This is
2
taken care of by the exp −ikr
term in the waveguide mode approximation.
2L
(a)
(b)
(c)
(d)
Figure 4.4: Comparison of Legendre function (Pµ1 (cos θ)) and first order Bessel function (− µ +
for increasing angle. (a): θh = 10◦ , (b): θh = 30◦ , (c): θh = 50◦ , and (d): θh = 60◦ .
1
2
J1
µ+
1
2
θh )
Equations 4.42 to 4.45 show that if the zeros of the corresponding Bessel functions can be calculated then the dependence of the Legendre function eigenvalues
on the horn opening angle can be calculated analytically and plotted. For the case
m = 0 for example, the dependence of ν for the T E1l modes, on the horn half angle
can be written as:
νl1
≈
′
p1l
θh
146
−
1
2
(4.46)
(a)
(b)
(c)
(d)
h
i
d
1 (cos θ )]
and Bessel approximation
[P
Figure 4.5: Comparison of derivative of the Legendre function
h
ν
dθ
θ=θh
1
1 2
1
1
−2 ν + 2
. (a): θh = 10◦ , (b): θh = 20◦ , (c): θh = 30◦ , and (d): θh = 60◦ .
J 0 ν + 2 θ h − J2 ν + 2 θ h
′
′
where p1l is the lth root of the derivative of the first order Bessel function (J1 ), and
θh is the horn half flare angle in radians. The ν values can then be plotted as a
function of horn flare angle as shown in Figure 4.6.
For the case of higher order modes, i.e. those with azimuthal dependence
m > 1, a recursive relationship can be applied to equations 4.44 and 4.45 to yield:
m
1
1
m
m
µ+
Pµ (cos θ) ≈ (−1)
θ
(4.47)
µ+
Jm
2
2
and
d m
[P (cos θh )]
dθ ν
θ=θh
(m+1)
m
1
1
(−1)
1
ν+
ν+
≈
ν+
Jm−1
θ − Jm+1
θ
2
2
2
2
(4.48)
A more useful approximation however is found on page 362 of Abramowitz and
147
Ν
TE11
80
TE12
TE13
TE14
60
TE15
TE16
40
TE17
TE18
20
TE19
TE
degrees1,10
Θ
0
20
40
60
80
Figure 4.6: Graph of
(ν) as a function of horn half angle (θ) for the first ten modes assuming the
eigenvalue
′
p1l
1
1
approximation νl = θ
− 2 . The azimuthal order is fixed at m = 1.
h
Stegun [Abramowitz and Stegun, 1972] of the form:
θh ∼
m −m
cos
µ Pµ
= Jm (θh )
µ
(4.49)
which leads to the relationship:
m
Pµm (cos θh ) ∼
= (−1) µm Jm (θh µ)
(4.50)
where the (−1)m is brought in to account for the fact that Pµ−m = −Pµm .
Equation 4.50 then leads to
pml
θh
pml
=
θh
Pµm (cos θh ) ∼
= 0 when µ =
⇒ µml
(4.51)
where pml correspond to the lth root of the mth order Bessel function, which are
always the same and therefore are easily tabulated for functions to call upon. Tabulated roots of Bessel functions and their derivatives can be found widely in the
literature, for example [Beattie, 1958], and Figure 4.7 illustrates the trends of some
of the calculated roots. It is these values that can be used to estimate initial search
points for the eigenvalues of the Legendre functions.
4.1.4
Evanescence of Modes
A very interesting feature of the conical mode approach is how the evanescence
of modes is governed. When analysing modal power flow and power coupling in
148
(a)
(b)
′
Figure 4.7: Point plot of Zeros of (a) Bessel functions (pnl ), and (b) derivatives of Bessel functions (pnl ). Each dot
corresponds to the lth root of the nth order Bessel function (as in (a)) or its derivative (as in (b)). The value of the
root is represented on the y axis and roots are approximately separated by π for each order and its derivative.
cylindrical waveguide sections we first determine whether each mode is purely propagating or purely evanescent for a particular cylindrical section radius and operating
frequency. If the frequency is above a certain critical value (cut-off frequency) or
equivalently, if the radius of the section is above a certain critical value that is proportional to the frequency, then the mode will propagate (propagating mode). If on
the other hand the radius is below the critical value or the frequency is below the
cut-off value for the particular section radius then the mode will decay exponentially
as it propagates through the section (evanescent mode). The length of the section in
the direction of propagation will determine how much power, if any, will be present
in that mode at the end of the section.
The cylindrical mode matching program SCATTER works in a way that analyses power coupling between modes at a junction where there is a change in section
radius, then the behaviour of each mode is considered over the length of the section, i.e. whether it is propagating or evanescent. Following propagation through
the length of the section the power coupling integrals are then performed again at
the next junction. This process is repeated for all sections that make up the horn
structure.
The approach taken with conical modes is quite different. No mode is purely
propagating or purely evanescent since in a conic section the radius of the cone
149
is continuously changing, either smoothly increasing or decreasing, depending on
whether the wave is propagating toward or away from the apex of the cone. So if a
wave is moving away from the apex, and it starts out in a part of the cone where the
radius is sufficiently small such that the evanescent behaviour dominates, then the
amplitude of the wave will begin to decay exponentially as it travels. But as it does
so the radius of the cone is steadily increasing and the rate of decay continuously
decreases. If the section is long enough, and if there was enough initial power in the
mode, then there will be a certain point where the decay in amplitude will cease and
the evanescent behaviour will no longer dominate. As the evanescence continues to
decline with increasing radius the mode will begin to resemble a true propagating
mode.
This inclusion of evanescence in the propagation of conical modes is a result of
the behaviour of the Hankel functions. Figure 4.3 at the beginning of this section
shows how the real and imaginary parts of the Hankel function of the second kind
behave as the length (r ) of the conic section increases. As r goes to very small values
(close to the apex) the imaginary part of the Hankel function goes toward minus
infinity. This is what leads to the evanescent behaviour of the EM modes when the
radius of the conic section is small compared to the wavelength of radiation.
The relationship between wavelength and conic section radius is governed by
the fact that the Hankel function operates on kr rather than just r, where k is
the free-space wavenumber k =
2π
.
λ
As the length of the section increases, and
therefore the radius continuously increases, the Hankel function smoothly changes to
a sinusoidally varying waveform that exhibits overall exponential decay (exponential
envelope) and then it eventually settles down to a sinusoid of constant amplitude
that represents a true propagating mode. In this way the Hankel functions are
capable of analytically describing the true nature of an EM field as it propagates
through a smooth walled conic section.
4.1.5
Cartesian Field Components
In order to compare the analytical modal fields with the approximate field expressions set out at the beginning of this chapter it was required that the spherical modes
be defined in Cartesian coordinates so a comparison could be carried out across a
150
plane. To find the x, y and z components of the electric field the dot product of
the field with the corresponding unit vectors is taken (see Figure 4.2 for coordinate
system), which gives:
Ex = Er sin θ cos φ + Eθ cos θ cos φ − Eφ sin φ
(4.52)
Ey = Er sin θ sin φ + Eθ cos θ sin φ − Eφ cos φ
(4.53)
Ez = Er cos θ + Eθ sin θ
(4.54)
The above relationships hold true for both the TE modes and the TM modes
described earlier. Each of the Cartesian field components are made up of a combination of the spherical components. The electric fields for the TE modes expressed in
Cartesian form are shown below. The electric fields for the TM modes and the magnetic fields for both the TE and TM (Hr = 0) mode sets can easily be transformed
into Cartesian form in the same way.
m
∂Pν (cos θ)
B (1)
m
m
imφ
ı̂
(P (cos θ)) cos θ cos φ −
sin φ
H
(kr)e
Ex =
sin θ ν
∂θ
kr ν+0.5
(4.55)
Ey =
m
(P m (cos θ)) cos θ sin φ −
sin θ ν
∂Pνm (cos θ)
∂θ
cos φ
B (1)
imφ
̂
H
(kr)e
kr ν+0.5
(4.56)
mB
(1)
m
imφ
k̂
Ez =
(P (cos θ)) sin θHν+0.5 (kr)e
kr sin θ ν
4.1.6
(4.57)
Example Horn for Comparison
Figure 4.8 below shows the E-plane and H-plane cuts of the electric field intensity
for a horn of half angle equal to 5 degrees or 0.087 radians. The length of the horn
(r ) was set to 2 mm, making the radius at the mouth of the horn a = 0.175mm.
To compare the new results with the approximate method, the same example
of a smooth walled horn with θh = 5◦ and r = 2 mm was analysed using equation
4.1. The comparisons of the E-plane and H-plane cuts of the electric field intensity
for the conical modes vs. the approximate description are shown in Figure 4.9,
and it is clear that there is complete agreement between the two descriptions of
151
Intensity
Intensity
1.0
E-Plane
40
H-Plane
40
X-polar
3.0 ´ 10
2.5 ´ 10
E-Plane
0.8
H-Plane
0.6
2.0 ´ 1040
1.5 ´ 1040
0.4
1.0 ´ 1040
0.2
5.0 ´ 1039
Θ
Θ
-0.05
0.05
-0.05
0.00
(a)
0.05
(b)
Figure 4.8: (a): E-Plane, H-plane and cross-polar cuts of the electric field at the aperture of a smooth walled conical
horn of half angle (θh = 5◦ = 0.087 radians), and (b): The E-Plane and H-plane cuts with the on-axis intensity
normalised to unity.
the electric field intensity. This is expected since the horn analysed contained a
relatively small half angle of 5 deg. Horns with larger half angles are considered in
the next subsection.
(a)
(b)
Figure 4.9: Comparison of magnitude of electric field at the aperture of a smooth walled conical horn antenna for
analytically derived and semi-analytically derived solutions. (a): E-plane comparison and (b): H-plane comparison.
4.1.7
Limitations of Approximate Method
Spline profiled conical horns have multiple sections with varying flare angles; some
of which angles are quite large as can be seen in Figure 4.10, where an example
of a multiple conic section spline horn is illustrated. Therefore the limitations of
the approximate description for the electric fields must be considered if this method
is to be used to model these horns. Multiple examples were analysed and the re152
sults compared for horns with increasing half angle. The example of a horn with
a half angle of 5◦ was already shown to have good agreement for both the approximate and spherical mode approaches. in Figure 4.11 below three cases are shown
corresponding to horns with half angles of 10◦ , 20◦ and 30◦ .
Figure 4.10: Example of a multiple conic section horn. As can be seen some of the section flare angles can be quite
large, particularly the final section.
Although only two modes were considered when comparing the approximate
and spherical mode approaches, in principle many more higher order modes could
propagate in the forward direction through the larger sections of a typical spline
horn as the radii of the biggest sections are significantly larger than the wavelength
of radiation that the horns are designed for. However the higher order modes will
only be present if they are generated through scattering processes since the input
waveguide will typically only support the fundamental mode (assuming a single
mode system). All higher order modes near the throat of the horn are effectively
evanescent and therefore will be attenuated significantly, if not severely, before they
reach a radius where they can propagate. As can be seen in 4.11 (a) and (b) there
is still excellent agreement for a horn with half angle of 10◦ . However for 20◦ there
is a small difference in the field structure and for 30◦ the difference is significant.
Therefore to accurately model spline horns with sections of flare angle greater than
20◦ (half angle = 10◦ ) the full spherical harmonic description will be required. Also,
it cannot be guaranteed that even for the case where all flare angles θh < 20◦ that
there won’t be some level of scattering into higher order modes at such an interface
of changing flare angle. However it should again be pointed out that the angle θh is
the half angle so θh = 20◦ actually represents a full horn flare angle of 40◦ which is
a relatively large angle in terms of typical horns of this type.
153
half angle = 10°
half angle = 10°
Intensity
1.0
Intensity
1.0
Analytical
0.8
Approximate
Analytical
0.8
Approximate
0.6
0.6
0.4
0.4
0.2
0.2
10 °
10 °
0.00
0.05
0.10
0.00
0.15
0.05
0.10
(a)
0.15
(b)
half angle = 20°
half angle = 20°
Intensity
1.0
Intensity
1.0
Analytical
0.8
Approximate
Analytical
0.8
Approximate
0.6
0.6
0.4
0.4
0.2
0.2
10 °
10 °
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.35
0.05
0.10
0.15
(c)
0.20
0.25
0.30
0.35
(d)
half angle = 30°
half angle = 30°
Intensity
1.0
Intensity
1.0
Analytical
Analytical
0.8
0.8
Approximate
Approximate
0.6
0.6
0.4
0.4
0.2
0.2
10 °
10 °
0.0
0.1
0.2
0.3
0.4
0.0
0.5
(e)
0.1
0.2
0.3
0.4
0.5
(f)
Figure 4.11: Comparison of T E11 electric field patterns inside a smooth walled conical horn antenna for analytically
derived and semi-analytically derived solutions. (a) H-plane comparison for θh = 10◦ , (b) E-plane comparison for
θh = 10◦ , (c) H-plane comparison for θh = 20◦ , (d) E-plane comparison for θh = 20◦ , (e) H-plane comparison for
θh = 30◦ , and (f) E-plane comparison for θh = 30◦ .
154
half angle = 10°
half angle = 10°
Intensity
1.0
Intensity
1.0
Analytical
0.8
Analytical
0.8
Approximate
0.6
0.6
0.4
0.4
0.2
0.2
Approximate
Θ
0.00
0.05
0.10
Θ
0.15
0.00
0.05
0.10
(a)
0.15
(b)
half angle = 20° HTE12 L
half angle = 20°
Intensity
1.0
Analytical
0.8
Approximate
Intensity
1.0
Analytical
0.8
Approximate
0.6
0.6
0.4
0.4
0.2
0.2
Θ
0.05
0.10
0.15
0.20
0.25
0.30
Θ
0.35
0.00
0.05
0.10
0.15
(c)
0.20
0.25
0.30
0.35
(d)
half angle = 30° HTE12 L
half angle = 30° HTE12 L
Intensity
1.0
Analytical
0.8
Approximate
Intensity
1.0
Analytical
0.8
Approximate
0.6
0.6
0.4
0.4
0.2
0.2
Θ
0.0
0.1
0.2
0.3
0.4
Θ
0.5
0.0
(e)
0.1
0.2
0.3
0.4
0.5
(f)
Figure 4.12: Comparison of T E12 electric field patterns inside a smooth walled conical horn antenna for analytically
derived and semi-analytically derived solutions. (a) H-plane comparison for θh = 10◦ , (b) E-plane comparison for
θh = 10◦ , (c) H-plane comparison for θh = 20◦ , (d) E-plane comparison for θh = 20◦ , (e) H-plane comparison for
θh = 30◦ , and (f) E-plane comparison for θh = 30◦ .
155
4.2
Scattering at Conical Horn Junctions
(Spherical SCATTER)
With analytical forms now derived for the various components of the TE and TM
modes in a conical shaped waveguide or horn, analysis on scattering processes at
the interface between horn sections with different flare angles such as in a spline fit
horn is now considered in this section.
For constructing the scattering matrices between two conic horn sections the
same logic is followed as that for the cylindrical polar and rectangular Cartesian
approaches used in the cylindrical and rectangular SCATTER code. The main
difference, as will be discussed, is that the power coupling overlap integrals are
calculated over a spherical cap, rather than a transverse plane, as was done with
the other two cases. The difficulty that arises however is if two conic sections have
different flare angles then they both also have different equiphase caps, leading to
the question of which phase cap to integrate over. And regardless of which cap is
employed for the calculation, the natural mode set on the other section will need
to be defined in the coordinate system of the selected one (the two sections have
different centres of curvature - or apices in our case). As will be shown this requires
a coordinate transformation so that the modes on both sides of the junction can be
defined across the same surface.
Another difficulty that arose during this investigation was finding analytical
forms for the type of integrals that arise in the power coupling calculations. The integrals involve products of Legendre functions which in free space do have analytical
solutions that are defined in much of the literature [Abramowitz and Stegun, 1972],
[Lebedev, 1972]. However products of Legendre functions with non-integer eigenvalues (due to the boundary conditions) are not described in any of the literature
studied by the author. For this reason all of the integrals over the angle θ were
calculated numerically. This process is very computationally demanding, leading to
long calculation times for problems involving even a moderate number of modes.
156
4.2.1
Derivation of S Parameters
The scattering or “S” parameters for the scattering processes involved in EM structures like the conic section spline fit horns discussed above have been explored in
some of the texts already mentioned. The texts by both Olver ([Olver et al., 1994])
and Clarricoats ([Clarricoats et al., 1984]) provide some results and definitions for
the power coupling integrals and S parameters but the derivations are either not
given or are unclear. For the development of the Spherical mode matching code
(Spherical SCATTER) it was necessary to revisit their derivations to ensure complete mathematical integrity. Furthermore only the case of single mode systems was
considered in any literature that was found on the subject, including the two texts
just referenced.
To make the Spherical SCATTER code as flexible as possible the theory was
extended to include multi-moded systems. As already mentioned in the introduction to this thesis, multi-moded horn structures are increasingly in demand due to
higher throughput requirements for faint signal detection and due to manufacturing
limitations on how small components can be made. Therefore multi-moded capability was considered an essential part of the new Spherical SCATTER code requiring
that the existing mode matching theory be extended.
It should be made clear that the term multi-moded refers to calculations involving EM modes with higher φ dependence. This is represented in the theory by
the azimuthal dependence of the associated Legendre functions. When single mode
horns are discussed in terms of mode matching only modes of order 1 (m = 1 in
the associated Legendre functions Pνm (cos θ)) are considered. Even though there
may be multiple modes with higher degree (governed by the ν in the Pνm (cos θ))
involved in the calculations of power coupling, it is still referred to as a single mode
system since the modes are coherent. The main reason for this is due to the fact
that modes of a particular order (m = 1 for example) do not couple to modes of any
other order (m 6= 1). This is simply down to the symmetry of the field structure of
the modes, and if power coupling overlap integrals are performed between modes of
different order the result will always be zero. Due to this orthogonality condition
we say that a horn that can only support modes with m = 1 (but multiple modes
with varying ν) is a single mode system, and a horn that can support modes with
m > 1 is a truly multi-moded system. Each azimuthal order (m) can be treated
157
completely separately and then summed in quadrature for calculation of the total
field. The sum of squares is used since the azimuthal modes are incoherent, i.e.
excited independently of one another.
Considering a particular azimuthal mode order, say m = 1, the total electric
and magnetic fields within a particular section of a smooth walled conical horn, or
conic section of a spline horn, can be defined as being a sum of the spherical modal
fields as:
~ =
E
∞
X
~ =
An~en + Bn e~′ n and H
n=1
′
∞
X
n=1
An~hn − Bn h~′ n
(4.58)
where en and e n represent forward going and backward going waves, respectively,
and An and Bn are the corresponding amplitude coefficients. The same is true for
the magnetic fields hn and h′n . Then at the interface (yet to be precisely defined)
between two conical sections, the two sets of fields for the two sections are:
~L =
E
~R =
E
∞
X
~L =
An~enL + Bn e~′ nL and H
n=1
∞
X
~R =
Cn e~′ nR + Dn~enR and H
n=1
∞
X
n=1
∞
X
n=1
An~hnL − Bn h~′ nL
(4.59)
Cn h~′ nR − Dn~hnR
where An and Bn are the forward and reflected amplitude coefficients, respectively,
of mode n on the left side of the junction. Cn and Dn are the backward and reflected
amplitude coefficients, respectively, of mode n on the right side of the junction. enL
and enR represent forward going waves on the left and right sides respectively, while
e′ nL and e′ nR represent backward going waves. The same is true for hnL , hnR , h′ nL
and h′ nR (see Figure 4.13).
Figure 4.13: Fields on either side of a conical horn flare angle discontinuity.
158
The SCATTER code calculates the aperture fields of a horn by cascading the
S parameters for each section of the horn. The derivation of these parameters
begins by considering the modal fields on each side of a spherical surface (yet to be
precisely defined) that interfaces two horn sections. The electric field on the left side
of the interface must equal the electric field on the right side since the field must
be continuous unless there is electric charge present on the interface, which clearly
there is not. The same is true for the magnetic field since it too must be continuous
across the boundary unless there are electric currents present on the interface, which
clearly there is not.
~L = E
~ R and H
~L = H
~ R , the expressions in 4.59 yields:
Allowing E
∞
X
An~enL + Bn e~′ nL =
n=1
∞
X
n=1
An~hnL − Bn h~′ nL =
∞
X
Cn e~′ nR + Dn~enR
(a)
n=1
∞
X
n=1
(4.60)
−Cn h~′ nR + Dn~hnR
(b)
The next step in this process involves performing an overlap integral for both
sides of equation 4.60(a) with the complex conjugate of the magnetic field on the
right side of the junction. The requirement is to relate the coefficients Bn and Dn of
the transmitted and reflected fields at the two sides of the interface to the incident
coefficients An and Cn .
Z
Z
∞
∞
X
X
∗
′
∗
~
~
~
~
~
~enL × h mR · dA +
An
e nL × h mR · dA
Bn
=
n=1
∞
X
n=1
Cn
Z
e~′ nR × h~∗ mR
~ +
· dA
n=1
∞
X
Dn
n=1
Z
~enR × h~∗ mR
(4.61)
~
· dA
Using the Dirak bra-ket notation this can be re-written as:
∞
X
n=1
An h~enL , ~hmR i∗ +
∞
X
n=1
Bn he~′ nL , ~hmR i∗ =
∞
X
n=1
Cn he~′ nR , ~hmR i∗ +
∞
X
n=1
Dn h~enR , ~hmR i∗
(4.62)
Similarly for 4.60(b), but with the complex conjugate of the electric field on the
right side of the junction gives:
∞
X
n=1
An h~emR , ~hnL i +
∞
X
n=1
−Bn h~emR , h~′ nL i =
∞
X
n=1
−Cn h~emR , h~′ nR i +
∞
X
n=1
Dn h~emR , ~hnR i
(4.63)
For concise presentation we use some notation for the various overlap integrals just
159
defined above. It should be noted that the notation used from this point on is
the same as that in section 4.7 of Olver [Olver et al., 1994], however there are no
derivations of how these authors arrived at the final scatter matrices. Furthermore,
the work in that publication deals with hybrid spherical modes (i.e. a corrugated
conical section) as opposed to the separated TE and TM modes discussed in this
work. The following notation will now be used for the various cross products:
P1mn = h~enL , ~hmR i∗
P2mn = h~emR , ~hnL i
Q1mn = he~′ nL , ~hmR i∗
Q2mn = h~emR , h~′ nL i
R1mn = he~′ nR , ~hmR i∗
R2mn = h~emR , h~′ nR i
T1mn = h~enR , ~hmR i∗
(4.64)
T2mn = h~emR , ~hnR i
⇒
T2mn = T1∗mn
Using these definitions equations 4.62 and 4.63 can be rewritten as:
P1mn An + Q1mn Bn = R1mn Cn + T1mn Dn
(4.65)
P2mn An − Q2mn Bn = −R2mn Cn + T2mn Dn
(4.66)
Or, to account for all modes m and n the above can be written in matrix form as:
[P1 ] [A] + [Q1 ] [B] = [R1 ] [C] + [T1 ] [D]
(4.67)
[P2 ] [A] − [Q2 ] [B] = − [R2 ] [C] + [T2 ] [D]
(4.68)
We now wish to relate [P1 ], [P2 ], [Q1 ], [Q2 ], etc. to the usual scattering matrices associated with interface junctions. The equation at the start of the previous
section describing a two port scattering system is shown again below. Equation
4.69 is equivalent to this and it describes the possible scattering processes that can
occur across the spherical boundaries depicted in Figures 4.15 and 4.16 in the next
subsection. The transmitted and reflected fields are related to the incident fields at
the two boundaries by:
~
B
~
D
~
=S
~
A
~
C
=
S~11 S~12
S~21 S~22
~ = S~11 A
~ + S~12 C
~
B
(a)
~ = S~21 A
~ + S~22 C
~
D
(b)
160
~
A
~
C
(4.69)
~ and B
~ are now column matrices representing the forward and reflected
where A
amplitude coefficients, respectively, for all the modes on the left side of the junction.
~ and D
~ are column matrices representing the backward and reflected
Similarly C
amplitude coefficients, respectively, for all the modes on the right side of the junction. S~11 , S~12 etc. are sub-matrices that contain the information regarding power
coupling between all modes on the input side with all modes on the output side of
the interface.
Taking equations 4.67 and 4.68 as a system of simultaneous equations, the
coefficient D can be eliminated from the system.
[P1 ] [A] + [Q1 ] [B] = [R1 ] [C] + [T1 ] [D]
(a)
[P2 ] [A] − [Q2 ] [B] = − [R2 ] [C] + [T2 ] [D]
(b)
(4.70)
Some rearranging gives:
−1
[B] = − [T1 ]−1 [Q1 ] + [T2 ]−1 [Q2 ]
[T1 ]−1 [P1 ] − [T2 ]−1 [P2 ] [A]
−1
+ [T1 ]−1 [Q1 ] + [T2 ]−1 [Q2 ]
[T1 ]−1 [R1 ] + [T2 ]−1 [R2 ] [C]
(4.71)
Comparing equation 4.71 above with equation 4.69(a) it is clear that:
−1
S~11 = − [T1 ]−1 [Q1 ] + [T2 ]−1 [Q2 ]
[T1 ]−1 [P1 ] − [T2 ]−1 [P2 ]
−1
S~12 = [T1 ]−1 [Q1 ] + [T2 ]−1 [Q2 ]
[T1 ]−1 [R1 ] + [T2 ]−1 [R2 ]
(4.72)
Starting again with equations 4.65 and 4.66 as a system of simultaneous equations, this time the coefficient B can be eliminated from the system. Again, some
simple rearranging gives:
−1
[D] = [Q1 ]−1 [T1 ] + [Q2 ]−1 [T2 ]
[Q1 ]−1 [P1 ] + [Q2 ]−1 [P2 ] [A]
−1
− [Q1 ]−1 [T1 ] + [Q2 ]−1 [T2 ]
[Q1 ]−1 [R1 ] − [Q2 ]−1 [R2 ] [C]
(4.73)
Comparing equation 4.73 above with equation 4.69(b) it is clear that:
−1
S~21 = [Q1 ]−1 [T1 ] + [Q2 ]−1 [T2 ]
[Q1 ]−1 [P1 ] + [Q2 ]−1 [P2 ]
−1
S~22 = − [Q1 ]−1 [T1 ] + [Q2 ]−1 [T2 ]
[Q1 ]−1 [R1 ] − [Q2 ]−1 [R2 ]
(4.74)
So the four scattering parameters S~11 , S~12 , S~21 , and S~22 have now been derived in
terms of combinations of integrals of vector triple products. The integrals represent,
in some sense, the flow of complex power across the spherical interface.
161
Figure 4.14: Schematic showing the power scattering possibilities for a two port system containing a discontinuity.
4.2.2
Coupling Between Two Conic Sections
As was discussed at the outset of this section if modal power scattering between
conic sections of different flare angles is to be analysed then an interface needs to be
defined so that the integrals can be calculated over that surface. The appropriate
coordinate transformations that are required to enable the fields to be defined over
such an interface are discussed in this sub-section for the two possible situations
that can arise:
1) Going from a section with a smaller flare angle to a section with a larger
flare angle.
2) Going from a section with a larger flare angle to a section with a smaller
flare angle.
The expectation was that the two conditions would be equivalent, but this was
checked for verification. So starting with the example of going from a section with
a smaller flare angle to a section with a larger flare angle the spherical coordinate
systems of both cones were defined as (RL , θL , φL ) for the cone on the left and
(RR , θR , φR ) for the cone on the right. For this situation (θL < θR ) it was decided to
perform the power coupling integrations over the equiphase surface of the cone on the
right (RR , θR , φR ). This required expressing the coordinates of the left cone in terms
of the coordinates of the right cone, i.e. RL (RR , θR ) and θL (RR , θR ) so they could
be inserted into the expressions for the spherical mode fields of the left hand conic
section. The phi component should remain unchanged under the transformation.
As can be seen from Figure 4.15, many parameters require definition due to the
virtual extensions of the conic sections back toward the apex of such a full cone for
′
example. Considering the triangle ∠aRR RL in Figure 4.15 and how it would change
′
as the point at the apex of the sides RL and RR traced out the surface CR . The side
162
′
RR remains constant, the horizontal length a remains constant, and the angle θR
rotates through θR degrees. The parameter RL will also slowly vary as θR changes,
as will the angle θL vary from θL to zero. The relationship between θR and RL was
′
derived through use of the cosine rule applied to the triangle ∠aRR RL , giving:
RL =
q
′
′
a2 + RR2 + 2aRR cos θR
(4.75)
where the relationship cos(180◦ −A) = − cos(A) was used. The relationship between
θR and θL was then derived through use of the sine rule applied to the triangle, giving:
"
#
′
R
sin(θ
)
R
R
θL = sin−1 p
(4.76)
′
′2
2
a + RR + 2aRR cos θR
The situation of going from a larger to a smaller flare angle was dealt with in a
′
very similar manner to that described above. Considering the triangle ∠aRL RR in
′
Figure 4.16 and how it would change as the point at the apex of the sides RR and
RL traced out the surface CR (this time CR is the inner surface). Again the side
′
RR remains constant, and the horizontal length a remains constant but this time it
is negative since a = LL − ∆LR and LL < ∆LR for θL > θR . Again the angle θR
rotates through θR degrees. The parameter RL will again slowly vary (but this time
it will get shorter instead of longer) as θR changes, as will the angle θL vary from θL
to zero. The relationship between θR and RL was derived through use of the cosine
′
rule applied to the triangle ∠aRL RR , giving:
q
′
′
⇒ RL = a2 + RR2 − 2aRR cos θR
(4.77)
which is almost identical to the case of going from smaller to larger angles except
for the minus sign. The relationship between θR and θL was then derived through
use of the sine rule applied to the triangle, giving:
#
"
′
R
sin
θ
R
R
θL = sin−1 p
′
′2
2
a + RR − 2aRR cos θR
(4.78)
which is also almost identical to the transformation derived for the case of smaller
to larger angles except for the minus sign.
163
Figure 4.15: Cone-to-cone coordinate transformation for the case of going from a smaller to a larger cone flare angle.
The heavy blue and red lines represent the actual horn section boundaries while the dashed black lines trace the
virtual extensions of the sections back to their apices.
164
Figure 4.16: Cone-to-cone coordinate transformation for the case of going from a larger to a smaller cone flare angle.
Again the heavy blue and red lines represent the actual horn section boundaries while the dashed black lines trace
the virtual extensions of the sections back to their apices.
165
The coordinate transformations just derived allowed the mode sets on both
sides of the junction to be defined across a common surface. This made it possible
to perform the power coupling integrals described in section 4.2.1 that are required
~ matrices account must
for construction of the S matrices. In deriving the various S
be taken of all possible left to right and right to left mode couplings including
between the TE and TM modes. Table 4.4 shows the possible coupling between the
various mode types.
Transverse Electric
Transverse Magnetic
Transverse Electric
TE → TE
TM → TE
Transverse Magnetic
TE → TM
TM → TM
Table 4.4: Possible combinations of modal coupling.
The power coupling integrals listed in 4.64 were defined for each combination in Table 4.4. They each had a similar form, for example for a TE mode in
E
the left section coupling to a TE mode in the right section P1mn = h~eTnLE , ~h∗T
mR i =
i
h
R 2π R θR T E
E
· r̂ r2 sin θdθdφ, where the fields ~eTnLE in the left section were
~enL × ~h∗T
mR
0
0
defined as a function of the coordinates of the right section using the above transformations:
Z 2π Z
P1mn =
0
θR
h
i
E
~eTnLE (RL (RR , θR ) , θL (RR , θR )) × ~h∗T
(R
,
θ
)
· r̂ r2 sin θdθdφ
R
R
mR
0
(4.79)
Instructions were then defined that stated:
(a) If n ≤
N
2
and m ≤
N
2
then P1mn = PT E→T E
(b) If n ≤
N
2
and m >
N
2
then P1mn = PT E→T M
(c) If n >
N
2
and m ≤
N
2
then P1mn = PT M →T E
(d) If n >
N
2
and m >
N
2
then P1mn = PT M →T M
where n is the mode number on the left side of the junction, m is the mode number
on the right side of the junction, and N is the total number of modes to be calculated
and it is split into
N
2
TE modes and
N
2
TM modes.
Following this the full N × N power coupling matrix P1 was constructed in a
manner where it is essentially partitioned into four quadrants with each quadrant
166
representing one of the four power coupling possibilities.
T E11
T E11
T E12
T E13
T M11
T M12
T M13
T E12
T E13
T M11
T M12
T M13
(P1 )11 (P1 )12 (P1 )13 (P1 )14 (P1 )15 (P1 )16
(P1 )21 (P1 )22 (P1 )23 (P1 )24 (P1 )25 (P1 )26
(P1 )31 (P1 )32 (P1 )33 (P1 )34 (P1 )35 (P1 )36
(P1 )41 (P1 )42 (P1 )43 (P1 )44 (P1 )45 (P1 )46
(P1 )51 (P1 )52 (P1 )53 (P1 )54 (P1 )55 (P1 )56
(P1 )61 (P1 )62 (P1 )63 (P1 )64 (P1 )65 (P1 )66
Similar matrices were constructed for P2 , Q1 , Q2 , etc. But each of the S matrices
(S11 , S12 , etc.) are are made up of products of various combinations of the P, Q, R,
and T matrices. However the S matrices still have the same partitioned form as
that shown above. For example the S~21 matrix that governs how much power will
be present in each mode on the right side of the junction (output) for a given input
mode set on the left side will have the form:
T E11
T E11
T E12
T E13
T M11
T M12
T M13
T E12
T E13
T M11
T M12
T M13
(S21 )11 (S21 )12 (S21 )13 (S21 )14 (S21 )15 (S21 )16
(S21 )21 (S21 )22 (S21 )23 (S21 )24 (S21 )25 (S21 )26
(S21 )31 (S21 )32 (S21 )33 (S21 )34 (S21 )35 (S21 )36
(S21 )41 (S21 )42 (S21 )43 (S21 )44 (S21 )45 (S21 )46
(S21 )51 (S21 )52 (S21 )53 (S21 )54 (S21 )55 (S21 )56
(S21 )61 (S21 )62 (S21 )63 (S21 )64 (S21 )65 (S21 )66
where each element (S21 )ij now represents the quantity of the input mode (along
the top of the matrix) that scattered to the output mode (along the vertical of the
matrix). For example the amplitude coefficient shown in bold (S21 )52 represents
how strongly the input T E12 mode on the left side of the junction scattered into
the output T M12 mode on the other side of the junction. It should be pointed out
that the above scatter matrix represents only one azimuthal order and only one
junction. Modes of different azimuthal order are treated completely separately as
was mentioned previously. The three remaining S matrices (S~11 , S~12 , and S~22 ) were
constructed in the same format using the relations given in 4.72 and 4.74.
167
4.2.3
Coupling Between Multiple Conic Sections
For sectional spline horns consisting of multiple conic sections, each with a different
flare angle, scatter matrices of the form above must be calculated for each junction
of the horn. These scatter matrices take care of mode-to-mode power scattering
at each of the section-to-section junctions, but so far there has been no mention of
power flow through the length of each conic section. The behaviour of the spherical
mode fields through a conic section of constant flare angle is relatively straight
forward compared to the scattering that occurs at a discontinuity. In fact as the
fields propagate through the section there is no scattering that occurs, so one only
needs to consider how each mode changes between the start of the section and the
end of the section.
The Legendre functions that govern how the fields behave in the θ and φ directions remain unchanged since θh remains unchanged and φ is always analysed
over 2π rad. So the only parameter that changes is the Hankel function since it is
dependent on kr, where r is the radial distance along the conic section. Of course
the phase front of each mode will change as it propagates through a section, and it
would be expected that the amplitude would decrease for the forward going waves
as the power spreads out to fill the widening cone.
As was mentioned in the previous subsection, unlike the case for cylindrical
waveguides where modes are either purely propagating or purely evanescent, conical modes that begin with dominant evanescent behaviour can become dominantly
propagating modes as they move from the narrower to the wider end of a conic
section. The opposite is also true for the case of a backward moving wave, i.e.
the evanescent behaviour of a mode can begin to dominate as the horn narrows.
What is true, however, is that for both forward and backward moving waves the S~11
and S~22 matrices are zero for propagation through a section of constant flare angle,
meaning there is no back scatter in either direction, which is also true in the case of
cylindrical waveguide mode propagation. In fact the only transformations required
for propagation through the uniform sections is to change the value of r in the Hankel function terms contained in the mode descriptions. The r coordinate must be
changed from its value relating to the start of the conic section, which corresponds
to the length of the full cone including the virtual trace back to the apex, to the
value relating to the end of the conic section, also including extension to the apex,
168
as illustrated in Figure 4.17.
Figure 4.17: Illustration of field transformation from one conic section to the next.
The final step in the scattering process was then to cascade the matrices corresponding to each scattering junction. This is carried out by first cascading the scatter
matrices for the first and second sections [S a ] and S b resulting in a third scatter
matrix [S c ] that represents the amplitude coefficients in the succeeding section. The
algorithm then defines that S b = [S a ] and [S c ] = S b and then calculates a new
[S c ] from the new [S a ] and S b for the next succeeding section. The algorithm
continues this loop until the last section is reached. The final [S c ] that is calculated
from the last two sections is defined as [Stotal ]. The method for calculating [S c ] from
[S a ] and S b is as follows:
b a −1 b a
a
c
a
]
[S22 ]
S11 [S21 ] + [S11
[S11
] = [S12
] [I] − S11
−1
a
b
c
a
b
[S22
]
S11
[S12
] = [S12
] [I] − S11
b
b −1 a
c
a
[S21
] = S21
[I] − [S22
] S11
[S21 ]
b −1 a b b
b
a
c
[S22 ] S12 + S22
[I] − [S22
] S11
[S22
] = S21
(4.80)
The above cascading processes are not derived as they have appeared in many
publications including for example [Olver et al., 1994] and [Doherty, 2012]. Again,
the above process applies to modes of only one azimuthal order, say m = 1. For
a single mode system this would be the end of the mode matching process and in
order to calculate the aperture field of the sectional spline horn the absolute values
of the elements of the [S21 total ] matrix are used to weight the corresponding TE and
TM modes that are defined for the final horn section. The weighted modes are then
169
simply summed linearly to produce the total aperture field as follows:
~ Ap (r, θ, φ) =
E
{n=N/2
j=N/2 }
X
TE
[S21 total ]1j E~1n
(r, θ, φ) +
{n=N/2
j=N }
X
{j=n=1
N +1}
2
{n=1
j=1 }
TM
[S21 total ]1j E~1n
(r, θ, φ)
(4.81)
If a multi-moded system is being considered, where there is more than one
azimuthal order being accounted for, then all of the above described steps must
be carried out for each order m independently. The behaviour of the fields in the
azimuthal direction (φ) for the higher order modes is governed by both the order
of the Legendre function m for the particular mode and a trigonometric function
either of the form cos mφ or sin mφ. In fact it is a product of these two functions
that describes the behaviour of the fields, which is clear from Table 4.1. Once the
final S matrices have been calculated for each azimuthal order (including that of
m = 0 for multi-moded horns) then the aperture fields corresponding to each of
m
these independent power channels can be calculated from the [S21
total ] matrix that
corresponds to that value of m.
In order to calculate the full aperture field for all modes of degree l and order
m that contribute significant power to the fields in the final section of the horn the
m
modes, now weighted by the coefficients encoded in the final [S21
total ] matrices, can
be combined. Care must taken however in the manner in which they are combined.
Assuming the horn is illuminated by a blackbody source, and thus each set of azimuthal modes are excited independently, then the modes are clearly incoherent and
should be summed in quadrature [Gleeson, 2004]. If modes of common azimuthal
dependence are spatially coherent then they should be summed linearly. The full
description for the aperture field is then:
~ M M (r, θ, φ) =
E
Ap
n=N/2
j=N/2
2 21
{
}
{X}
M
X X
~
T
E
T M (r, θ, φ)
[S21 total ]1j E~mn
[S21 total ]1j Emn (r, θ, φ) +
n=1
n=1
m=0
{ j=1 }
{j= N2 +1}
n=N/2
j=N
(4.82)
The total return power of the horn can also be calculated in a very similar
m
manner but using the [S11
total ] matrices. In this case the actual field structure of
170
the modes is not required as the field amplitude for each mode (and therefore the
power information) is contained in the [S11 total ] matrices.
4.3
Spherical SCATTER Program
4.3.1
Description of Spherical SCATTER Code
With the full theory for multi-moded conical mode matching described, an account
is now given of the structure of the code that was developed by the author. The
code is based on heritage from the cylindrical geometry mode matching software
SCATTER [Gleeson, 2004], [Colgen, 2001]. The new Spherical SCATTER code was
written with the primary aim of calculating aperture field patterns and far-field
radiation patterns, specifically for smooth walled structures, in the most computationally efficient way possible. The section-to-section matrix cascading algorithms
were similar to those used in cylindrical SCATTER, but the actual power scattering integrals were of a very different form. Spherical SCATTER was written as an
alternative solution for modelling smooth walled conical horns with a more exact
description of the modal fields. It also has the potential to be much faster computationally due to the fact that the number of calculations is greatly reduced compared
to the step-wise smooth wall approximation that cylindrical SCATTER employs.
The first process that is performed in Spherical SCATTER is reading in a
geometry file of “*.i.text” format that contains information on operational frequency,
maximum azimuthal order, number of conic sections, the length of each section, and
the final radius of each section. Table 4.5 shows the layout of the geometry file that
is read-in by the code. Figure 4.18 then shows what the r and L values in Table 4.5
represent in terms of the geometry of the horn antenna to be modelled.
The next step involves calculating necessary parameters from the data in the
geometry file. These parameters include the flare angle θh of each conic section, the
slant length R of each section (including the virtual extensions back to the apex),
the difference in radial position ∆zLR of the apices of each pair of sections, etc. The
next section of code then uses these parameters to plot a 2D slice of the horn so the
user has a visual representation of the structure.
Using the flare angles θh for the various sections the eigenvalues νlm and µlm are
171
Line 1
Frequency (GHz)
From: Line 2
Conic Section
To: Line N+1
Lengths Ln (mm)
From: Line N+2
To: Line 2N+1
Max. Azimuthal
Number of Conic
Order (m)
Sections (N )
Conic Section
Total Number of
End Radii rn (mm)
Modes of Degree l
Table 4.5: Format of input text file for Spherical SCATTER.
Figure 4.18: Parameters of sectioned spline horn geometry file. The other required parameters for definition of the
EM mode fields are derived from rN and Ln using arithmetic and trigonometric relationships.
then calculated for both the TE and TM modes, respectively. The eigenvalues µlm
correspond to the zeros of the Legendre functions in the mathematical descriptions of
the T Mlm modes while the eigenvalues νlm correspond to the zeros of the derivatives
of the Legendre functions in the descriptions of the T Elm modes. The calculated
eigenvalues are tabulated allowing them to be called by the mode functions during
the mode matching calculations. The k parameter is then defined and its value is
calculated from the operational frequency that was included in the geometry file.
The mathematical description of the TE and TM modes is then defined for
both the forward and backward moving waves. The general definitions are the same
for all conic section profiles and therefore are constant for all models. The variables
in the TE mode descriptions are ν, m, r, θ, φ, and R, where ν is the eigenvalue of
mode of degree l, m is the azimuthal order, r is the radial distance along the conic
section, θ is the off-axis alt. angle, φ is the off-axis az. angle, and R is the full
length of the cone from the virtual apex to the spherical cap of the section. The
same variables are left free for the TM mode definitions where ν is of course replaced
with µ.
172
The overlap integrals are then defined for each of the power scattering possibilities P1 , P2 , Q1 , etc. for TE to TE, TE to TM, etc. Using these definitions the scatter
matrices are then constructed for each cone-to-cone junction. The mode transformation matrices corresponding to propagation through the uniform conic sections
are also now calculated. The cascading process can then be performed comprising of
alternating scatter and transformation matrix cascading to account for propagation
through a section followed by scattering at a discontinuity, followed by propagation
through the succeeding section, etc. until the final section is reached. If a multimoded horn is being analysed then at this point the value of m is incremented and
the processes described in this paragraph are repeated. The processes are further
repeated for each value of m including m = 0.
This is where the mode matching section of the Spherical SCATTER code ends.
All further calculations performed by the code such as aperture field description,
~ matrices.
return power, and horn efficiency involve using the elements of the final S
An overview of the structure of the Spherical SCATTER code is shown in Figure
4.19.
Figure 4.19: Overview of how the Spherical SCATTER code performs its algorithms.
173
4.3.2
Test Horn
As a verification test of the Spherical SCATTER code an example horn was modelled and the resulting aperture fields were compared to those calculated with the
well established mode matching code Cylindrical SCATTER which has shown excellent agreement with experiment over many tests and was used to design and
analyse the feed horns for the HFI instrument on the PLANCK CMB system
[Murphy et al., 2010]. The particular horn consisted of five conic segments, each
with a different flare angle, and one initial straight waveguide section (see Figure
4.20). The lengths of the conic sections were all the same at 8.781 mm, the maximum azimuthal order was set to one (single-mode horn), and the frequency was set
to 100 GHz.
Figure 4.20: Automatated plot from the Spherical SCATTER program. A plot of this type (excluding the table of
values on the right) is automatically generated after the geometry file is ’read in’ to the program.
The design of this horn was optimised through the work of the THz Optics
Group at NUIM, with the aim of developing a smooth walled horn that is capable of producing similar beam qualities to that of corrugated structures, as already
outlined at the beginning of this chapter [McCarthy et al., 2013]. The particular
beam qualities demanded by the design were high Gaussicity and good beam symmetry. High Gaussicity ensures low side-lobe levels in the far-field pattern, while
a good level of rotational symmetry in the beam allows for accurate polarisation
measurements.
What should be noticed in the table of values in Figure 4.20 is that there
are only six lengths and six radii. In contrast, the Cylindrical SCATTER code
174
typically requires a few hundred lengths and radii for definition of the horn geometry
due to the manner in which the smooth walls are approximated by small uniform
waveguide sections in a “stairs-like” format. The six sections of the test horn and
waveguide indicate that power scattering integrals will need to be evaluated at just
five junctions for this example. Table 4.6 shows the eigenvalues for the first six TE
and the first six TM modes.
Section (n)
W.G.
1
2
3
4
5
Angle (θh )
0o
11.67o
0.72o
13.86o
13.71o
30.29o
ν11 (T E11 )
∞
8.5828
146.487
7.15735
7.24223
3.08647
ν12 (T E12 )
∞
25.6963
425.117
21.549
21.7962
9.6135
ν13 (T E13 )
∞
41.4324
680.967
34.7899
35.1858
15.6634
ν14 (T E14 )
∞
56.9981
934.008
47.8883
48.4313
21.6532
ν15 (T E15 )
∞
72.5052
1186.08
60.9377
61.6271
27.6223
ν16 (T E16 )
∞
87.985
1437.71
73.9641
74.7998
33.5818
µ11 (T M11 )
∞
18.3257
305.391
15.3448
15.5224
6.76452
µ12 (T M12 )
∞
33.9601
559.565
28.5005
28.8259
12.7783
µ13 (T M13 )
∞
49.4686
811.663
41.5511
42.023
18.7479
µ14 (T M14 )
∞
64.94
1063.15
54.5706
55.1886
24.7044
µ15 (T M15 )
∞
80.3956
1314.37
67.5768
68.3408
30.6552
µ16 (T M16 )
∞
95.8428
1565.46
80.5761
81.486
36.6032
Table 4.6: Eigenvalues for the first six TE and TM modes in each of the five horn sections. If the table is analysed
closely it will be clear that the larger angles produce smaller eigenvalues while the smaller angles produce the larger
eigenvalues.
Figure 4.21 shows the final aperture field through the E-plane and H-plane cuts
calculated with the new Spherical SCATTER code. Figure 4.21(a) shows the fields
compared to the original input field that was excited through the uniform waveguide
section. It is clear that the output aperture field has changed significantly from the
input T E11 mode, which demonstrates the level of scattering that occurs due to the
multiple changes in flare angle throughout the horn. Both the E-plane and H-plane
cuts show a high level of Gaussicity, and there is good symmetry in the field, as
required. Figure 4.21(b) shows the same aperture field results, this time compared
to data that was calculated using the more established Cylindrical SCATTER to
175
model the same horn. As can be seen there is good agreement between the two
approaches.
(a)
(b)
Figure 4.21: Aperture field patterns for the E-plane and H-plane cuts of the electric field intensity through a smooth
walled conic section horn. (a): The aperture fields compared to the input T E11 waveguide field, which demonstrates
the level of scattering that occurs in the horn. (b): The new Spherical SCATTER results compared to Cylindrical
SCATTER results.
Since the conic section horn that was used for verification of the new Spherical
SCATTER code was a real design that was built for testing, further analysis of
the horn using the FEM approach was carried out. A simulation of the induced
currents across the adjoining boundary of an array of such horns, similar to the
analysis that was performed for the SAFARI horns as presented in chapter 3, was
analysed with COMSOL. The effects of any induced currents on the far-field beam
pattern were investigated for the conic section test horn. Figure 4.22 shows a screenshot of the FEM model, where the magnitude of the electric field is indicated by
the colour variation. Both the E-plane and H-plane symmetry conditions were used
to reduce the problem size by a factor of four. The spheric section connected to the
exit aperture of the horn, which represents the propagation region, is also visible
in screen-shot. Figure 4.23(a) shows E-plane and H-plane linear cuts through the
surface current density across the conducting ground plane (GP) that lies in the
plane of the exit aperture of the horn. Figures 4.23(b) to 4.23(d) show comparisons
of the far-field pattern with and without the conducting ground plane, i.e. the horn
in free-space compared to the horn in an array block. It is clear that any induced
currents across the ground plane have negligible effects on the far-field pattern.
The cross-polar distribution shows some differences for the two cases, although the
176
maximum levels are roughly the same.
Figure 4.22: Screen-shot of FEM model of smooth walled conic section horn.
(a)
(b)
(c)
(d)
Figure 4.23: Results for a profiled smooth walled conic section horn. (a): The induced surface current density across
the conducting ground plane that would separate neighbouring horns, (b): E-plane cut through the far-field pattern,
(c): H-plane cut through the far-field pattern, and (d): cross-polar far-field pattern through the φ = 45o cut.
177
4.3.3
Experimental Verification
A rigorous set of measurements was carried out by members of the THz Optics
Group at NUIM in terms of testing the performance of the test horn described
above. The work was primarily driven by Darragh McCarthy, and it involved measuring both the co-polar and cross-polar radiation patterns at various propagation
distances, including the far-field. The experimental measurements agreed extremely
well with the Cylindrical SCATTER results, and thus Spherical SCATTER, and
the full analysis is due to be presented at the SPIE: Millimeter, Submillimeter, and
Far-Infrared Detectors and Instrumentation for Astronomy VII conference in Montreal, Canada, in June 2014. A paper on the horn measurements and results will be
included in the corresponding conference proceedings [McCarthy et al., 2014].
4.4
Conclusions
In this chapter the theory for EM propagation through smooth walled conical horns
and conic sections was described in detail in its full analytical form. Comparisons
were made between the exact solutions for TE and TM modes and the approximate
solutions that employ cylindrical waveguide modes with a spherical cap correction
factor. Good agreement between the two approaches was demonstrated for horns
of modest flare angle, and it was shown that the approximations were sufficiently
accurate for use as a starting point for a search of roots of the Legendre functions
and their derivatives, which were required for determining the eigenvalues of the TE
and TM mode solutions.
The results of the new mode matching code, Spherical SCATTER, which was
developed by the author, was shown to be in good agreement with the already
established and verified Cylindrical SCATTER code, and the FEM software package
COMSOL. The verified results of the Spherical SCATTER code demonstrates that
it is possible to model the novel smooth walled conic section horns with two orders of
magnitude fewer power scattering integrals, compared to the Cylindrical SCATTER
approach.
It was explained that some of the theory in this chapter pertaining to spherical
mode matching has already been reported on in the literature, most notably by
178
[Olver et al., 1994] and [Clarricoats et al., 1984]. However since there was little to
no derivation of the mathematics it was necessary to revisit the theory with a full and
detailed derivation of the most important aspects to ensure mathematical integrity
of the new Spherical SCATTER code. Furthermore, the theory in the literature
only describes examples of single-mode systems, thus requiring an extension of the
theory to account for multi-moded horns. This extended theory was presented in
section 4.2.3 of this chapter.
This novel approach to modelling smooth walled conic section spline horns with
conical mode matching is an exact solution that relies on approximations only for
initial ‘guess’ values for the eigenvalues of the modal fields. All of the fields were
derived analytically from Maxwell’s equations. The Spherical SCATTER code, in its
present state, ultimately proved to be computationally slower than the Cylindrical
SCATTER code. This is due to the numerical integrations used in the analysis
of the power scattering calculations. Although modelling a smooth walled horn
with Spherical SCATTER requires much fewer power scattering calculations than a
similar Cylindrical SCATTER model, the analytical forms employed by Cylindrical
SCATTER outweighs the large number of calculations.
Potential future work in this area is the optimisation of the code to increase
computational speed. This could be achieved by deriving analytical forms for the
integrals involving products of the Legendre functions and their derivatives, and
implementing them in place of the numerical integrals used presently. Alternatively
the Bessel function approximations that were shown to be accurate for horns of
moderate flare angle could be used in place of the Legendre functions. Efficient
analytical solutions to integrals involving Bessel functions are well understood and
are already used in the Cylindrical SCATTER code.
179
Chapter 5
The Electromagnetic Design and
Analysis of Cavity Coupled
Absorbers for SAFARI
“We are at the very beginning of time for the human race. It is not unreasonable
that we grapple with problems. But there are tens of thousands of years in the future.
Our responsibility is to do what we can, learn what we can, improve the solutions,
and pass them on.”
- Richard P. Feynman
180
5.1
Background
The detectors of the SAFARI instrument will employ a TES [Khosropanah et al., 2012]
in each pixel housed in an integrating cavity. Each TES is mounted on a square
piece of absorbing material that is relatively large compared to the TES. For example the current design of the SAFARI Short-Wave Band (S-Band) pixel employs
a 200 × 200µm square of Ta (tantalum) film that is only a few nanometres thick
coupled to the TES. The TES/absorber combination is cooled to a few milliKelvin
and therefore the Ta operates in a superconducting state. A SAFARI test-bed being studied by the SRON currently uses a hemispherical backshort and the system
is fed by a smooth walled conical horn. Radiation from the blackbody source is
focussed onto the absorber by the feed horn, while the shaped backshort behind
the TES/absorber serves to reflect any radiation that is not initially absorbed back
toward the absorber (see Figure 5.1). The cavity thus set up between the absorber
and the backshort provides a matched impedance for the incoming radiation in order to maximise the radiation that is absorbed. From an optical viewpoint standing
waves are set up in the cavity between the absorber and the backshort. A detailed
description of the SRON SAFARI S-Band test-bed is given in section 3.3.
(a)
(b)
Figure 5.1: Illustration of a SAFARI TES mounted on the Ta absorber. (a): Aerial view showing how the TES
and absorber are monted over the cavity by using thin niobium (Nb) legs. (b): A cut through the SAFARI pixel
showing the current design of a hemispherical integrating cavity.
Clearly it is important to be able to predict the optical efficiency of such cavity coupled detectors for the high sensitivities required of far infra-red astronomy
and CMB experiments. Efficient computational models are essential for the design
181
and analysis of these detector systems and the results must be reliable. Modelling
such cavity mounted absorbers requires sophisticated EM modelling which tends
to be computationally intensive and in the past it was generally not possible to
model electrically large cavity structures (in terms of wavelength) without the use
of supercomputers.
By analogy with large integrating spheres in infrared systems (based on ray
tracing), the assumption was generally made that a hemispherical shaped backshort
would refocus in some sense the radiation not absorbed by the absorber in the first
pass through it back to the absorber. Now, with more powerful PCs available and
employing more efficient simulation techniques, it is possible to include these absorbing cavities in the models (see for example simulations for the SPIRE instrument
on the Herschel Space Observatory [Glenn et al., 2003]).
Another assumption during the design of previous systems was often that such
cavities (integrating cavities) could be assumed to behave like blackbody cavities,
and if feeding multi-mode horn antennas they would excite all the waveguide modes
equally (Planck HFI for example [Maffei et al., 2010]). Clearly for achieving the
highest sensitivities possible, a more complete understanding of the absorbing cavity
is required. Again this is particularly important for multi-mode systems as then it
is possible to predict both the multi-mode beam pattern of the horn based on the
correct coupling of the individual waveguide modes to the absorbing cavity, and
predict an overall optical efficiency or throughput for the system as a whole for both
point like and extended source illumination, or indeed the predicted illumination by
the FTS output beam for the system on the telescope.
In this chapter we first consider the computational simulation tools available
and discuss the validity of various approximate approaches. The original SAFARI
Long-wave Band (L-Band) cavity design proposed (based on the hemispherical backshort) is then discussed and simulations using the FEM with the commercially available COMSOL package are described. The redesign of the backshort configuration
for improved coupling of the absorber to the cavity is then considered with both
FEM simulations and a novel mode matching approach. Finally the SAFARI SBand is discussed and a redesign of the S-Band multi-moded cavity for improved
absorber efficiency is described. The theory of including the Ta absorber in the
models is explained, and potential methods for the future for including the vacuum
182
gap in the mode matching models are discussed.
Modelling cavity structures is very similar to the way waveguides and feed
horns are handled as described in chapters 3 and 4, and the absorber material is
approximated as an infinitely thin, two dimensional resistive sheet with units of
Ohms per square (Ω/). The TES itself in omitted from the models as it is the
Ta absorbing film that couples to the incoming radiation, and the TES essentially
acts as a thermometer to detect the rise in temperature of the Ta. A research team
at SRON are attempting to model the TES and the SiN legs on which the TES
and absorber are mounted in terms of the heat conductance and the heat capacity
in the SiN legs, and the impedance and noise in the TES itself (see for example
[Khosropanah et al., 2012]).
The validity of the resistive sheet approximation for the piece of Ta absorber
is supported by work carried out by a research group at Cambridge University U.K.
[Withington et al., 2013]. They showed that, for a THz beam within the frequency
range of approximately 1.4 - 8.8 THz incident on a thin sheet of Ta at moderate
angles of incidence, the reactive part of the complex impedance is almost three orders
of magnitude smaller than the resistive part, and frequency-independent absorption
is possible. They explain that this is true once the thickness of the Ta sheet is less
than 10 nanometres (d < 10nm)(See also [Thomas and Withington, 2013]).
5.2
Modelling Approaches
5.2.1
Mode Matching Approach
The theory for modelling waveguide and horn structures with mode matching techniques was discussed in detail in the previous chapter 4. It is an extended version of
the mode matching program SCATTER (for cylindrical systems) that was employed
in this section to model closed cavities housing resistive sheet absorbers. A summary
of the theory that was required for the extension of the code is presented in this section, however the author was not involved in developing the theory or implementing
it into the original SCATTER code. The author carried out verification analysis of
the code by using it to model a variety of systems, some of which are presented in
this chapter, and comparing the results to the same systems modelled with FEM
183
techniques. The code was validated based upon the good agreement observed in the
comparison analysis.
In order to model a closed cavity little modification to the existing code is
required. If for example a hemispherical cavity is desired the input geometry text
file representing the geometry of the structure is constructed in a manner where the
spherical shape is approximated by a series of waveguide sections with increasingly
smaller raddi where the rate at which the radii reduce is just governed by the
equation for a circle. Figure 5.2 illustrates an example of a hemisphere. In the
illustration there are only five sections making up the structure for clarity, however
there would be a lot more sections in a real model in order to build a smoother,
less jagged profile; the minimum number of steps depends on the wavelength of the
radiation being simulated.
Figure 5.2: Diagram showing how a profiled waveguide structure can be approximated by a series of uniform
cylindrical sections. This particular example is for a hemisphere.
Building up cavity geometries in this stepwise manner is clearly a natural extension of how horn structures are modelled with the SCATTER code. The real
modification of the SCATTER code was required for including the resistive sheet
absorber, and it was the author’s role to validate the code by using it to model
some simple examples of cavity housed absorbers and then to compare the results
to those of FEM models of similar examples. Following validation through testing
the author then employed the extended SCATTER code to model various proposed
systems such as the SAFARI L-Band, and then to design improved systems as will be
shown throughout the chapter. The theory for including a resistive sheet absorber
in the SCATTER mode matching code is now summarized.
The TE and magnetic fields in a smooth walled cylindrical waveguide section
184
were derived and tabulated in chapter 2. The first step in describing the mode
matching approach that also takes account of an absorbing resistive sheet begins,
as before, by regarding a propagating signal as a sum of such transverse waveguide
modes. And considering first the case where the cylindrical symmetry is not broken
the modes of the same azimuthal order do not mix. For the sections of the cavity
or horn structure where there is no absorber material present, the theory is the
same as that described in the mode matching description given in section 2.4. If an
absorber is then to be modelled at a particular position/junction between the horn
and cavity, or anywhere within the horn or cavity, it can be considered as a resistive
current sheet with the electric field constant across the boundary and a drop in the
magnetic field due to induced surface currents, leading to the following boundary
conditions:
~L = E
~R
E
~L = H
~R − K
~ × k̂
H
(5.1)
~
~
~ = EL = ER
with K
η
η
~ is the induced current density, η is the sheet resistance defined in units of
where K
Ω/ and k̂ is just a unit vector in the direction of propagation (z).
The sheet resistance described in (5.1) is valid for a two-dimensional system
when the thickness of the absorber material is negligible. The sheet resistance
implies that there is only induced currents in the plane of the sheet and no currents
in the perpendicular direction. In S.I. units the more familiar bulk resistivity ρ has
units of Ωm (Ohm metres) or alternatively Ωm2 /m since the overall bulk resistance
R will depend on a product of the bulk resistivity ρ, the length L in m, and the
reciprocal of the area A in m−2 of the medium, i.e. R =
ρL
.
A
If we then consider
the area A being made up of a width W and a thickness δ as shown in Figure 5.3
then we can write the resistance as R =
ρL
A
=
ρL
.
Wδ
Then combining ρ with 1/δ and
calling this RS = ρ/δ, then the resistance can be written as:
R = RS
where RS =
1
,
σδ
L
W
(5.2)
since ρ = 1/σ.
Since both L and W have units of metre then the dimensions really cancel out,
however the term Ohms per square is used since if L = W (i.e. a square) then such
185
Figure 5.3: Illustration comparing bulk resistance to sheet resistance.
a square with sheet resistance, say 50 Ohms/square, has an actual resistance of 50
Ohms, regardless of the size of the square.
If we want to consider the sheet resistance in terms of induced surface currents
due to the jump in the magnetic field as stated in the boundary conditions in 5.1
then we begin by stating Ohm’s law:
~
J~ = σ E
(5.3)
~ ×H
~ = J~ and integrating it over an area
Then taking Maxwell’s fourth equation ∇
corresponding to the width W and thickness δ from Figure 5.3:
Z Z
Z Z
~
~
~
~
∇ × H · dA =
J~ · dA
(5.4)
~ for J~ (Ohm’s
Applying Stoke’s theorem to the left side of 5.4 and substituting σ E
law) in the right side of 5.4, and then equating both sides yields:
Z Z
Z
~
~
~ · dA
~
E
H · dl = σ
(5.5)
Integrating the magnetic field around the loop (Figure 5.4) dl, and again, assuming the thickness δ is negligible, then the left side of 5.5 just becomes HL W −
HR W . Then, assuming E is continuous across the boundary the right side of 5.5
becomes σEδW , thus:
HL W − HR W = σEδW
E
⇒ HL − HR = σδE =
RS
since it was shown above that RS =
(5.6)
1
.
σδ
Following analysis presented in chapter 2, if the incident and reflected fields on
the input side of a wageguide junction have mode coefficients An and Bn respectively,
186
and the mode coefficients on the output side for the fields are Cn and Dn for the
nth mode, and if a symmetrically placed absorbing disc partially fills the waveguide
junction, and assuming z = 0 at the position of the absorber, then applying the
condition in 5.1 yields:
N
X
′
[An + Bn ] ~en =
n=1
N
X
n=1
′
[An − Bn ] ~hn =
N
X
n=1
N
X
[Dn + Cn ] ~en
(a)
n=1
[Dn − Cn ] ~hn − [η(r)]
N
X
−1
n=1
[An + Bn ] ~en × k̂
(5.7)
(b)
where η(r) = ∞ outside the boundary of the absorber (r > R) and η(r) = η across
the absorber (r ≤ R), where R is the radius of the absorbing disk.
Taking the same approach as with scatter matrices at waveguide junctions as
presented in chapter 2 we can solve these equations using conservation of complex
power to yield the following scatter matrices:
−1
Γ
Γ
∗ −1
∗ −1
· [Q ]
S11 = − I + [Q ]
2η(r)
2η(r)
(5.8)
−1
Γ
∗ −1
S12 = I + [Q ]
2η(r)
R
where Qnm = S ~e∗n × ~hm dS and the integration is over the full area of the guide
R′
R′
S, Γmn = S (~e∗m × ~en ) dS ′ , and Γnm = S (~e∗n × ~em ) dS ′ , where S ′ in the integrals is
now the area of the absorber.
Since [Q] is a diagonal orthogonal waveguide mode set with elements Qmm =
R
∗
~
~em × hm dS = Z1m , where Zm is the impedance of mode m, then 5.8 can be
S
written as:
S11
−1
Z·Γ
Z·Γ
=− I+
·
2η(r)
2η(r)
−1
Z·Γ
S12 = I +
2η(r)
(5.9)
where Z is a diagonal matrix of mode impedances. The Γmn are real valued as
R
R
R
| Sabsorber ~em∗ · ~en dS| = | Sabsorber em,r en,r + em,φ en,φ dS| = | Sabsorber ~em · ~en dS|. It
should also be noticed that S11 = S22 and S12 = S21 by symmetry, which was not
the case for a junction between two waveguide sections, although the S11 and S12
were very similar to S22 and S21 respectively, in that case.
We now consider the case of a closed cavity such as the hemispherical example
shown in Figure 5.2. Assuming no other losses, the amount of power absorbed by
187
the resistive sheet can be inferred by analysing the return power through the S11
parameter. For no absorber material present the return power will equal the input
power so this clearly represents no power absorbed. If the absorbing disk is then
included in the closed cavity structure the power absorbed can be calculated by:
!
N X
M
X
PAbs = (1 − P owerReturned) = 1 −
S11 [i, j]2 δij
(5.10)
i=1 j=1
where N is the maximum radial mode degree that is considered, M is the maximum azimuthal mode order that is excited, and it is assumed unity power is evenly
distributed between all propagating modes i, j.
Care must be taken though when dealing with modes that are close to cut-off.
If the frequency is below the cut-off frequency of a particular mode then it will
decay rapidly through evanescent propagation, thus in that scenario there will be
no power reflected back and it will appear that all the power was absorbed. If the
return power for a propagating mode is close to zero, say 0.01, then the amount
of power absorbed would be (1 - 0.01) = 0.99 of the total power contained in that
mode. When analysing horns similar concerns with evanescent modes exist at the
horn aperture.
5.2.2
3D FEM Approach
The general theory involved in FEM simulation was described in detail in chapter
2 of this thesis, and the specific approaches required for modelling waveguide and
horn structures with the COMSOL FEM package was explained in chapters 4 and 3.
Thus the only discussion on theory required in this section is how the Ta absorber
was treated in the 3D COMSOL models that will be described in this chapter.
The method of instructing COMSOL 3.3 to treat the absorber as a resistive
sheet was by using the predefined ’Transition Boundary Condition’. This allows the
user to input a value for ’Surface Impedance’ and an optional externally applied
electric field E~s (see equation 5.11). The magnetic field H is analysed on both sides
of the transition boundary, and the amount of power absorbed by the sheet can be
calculated from the jump in the magnetic field HL − HR due to the surface currents
this induces. The electric field is continuous across the boundary. The induced
current density across the resistive sheet can also be calculated in the post-processing
188
environment and the ’Surface Resistive Heating’ parameter can then be calculated
from current density and integrated over the full disk to calculate the amount of
power absorbed directly. If we now consider such a resistive sheet situated in the
x − y plane and positioned at z = 0, and if the field is propagating in the z direction,
then the ’Transition Boundary Condition’ is defined by COMSOL as:
~
~
~
~
η k̂ × HL − HR − k̂ × E × k̂ = k̂ × Es × k̂
(5.11)
where η is the surface impedance, H~L and H~R represent the magnetic field on the
output and input side of the boundary, respectively (for a wave travelling from right
to left), k̂ just denotes a unit vector normal to the surface of the resistive sheet, and
E~s is the optional externally applied electric field which was set equal to zero for
these simulations as no external field was applied.
Figure 5.4: Schematic showing the absorber situated in the x − y plane positioned at z = 0 with the magnetic field
on both sides of the boundary shown. The direction of the induced surface currents is indicated by the blue arrows.
Since E~s = 0 then,
~ × k̂
η k̂ × H~L − H~R = k̂ × E
(5.12)
Or,
⇒
H~||L = H~||R −
k̂ × H~L − H~R =
~ × k̂
k̂ × E
η
~
E|| × k̂
(5.13)
η
where the subscript || denotes the component of the field parallel to the surface of
the absorber.
189
If we now label
~
E
η
~ the boundary conditions for the resistive sheet can be
as K,
written as:
~ × ~k
H~R = H~L − K
~R = E
~L
and E
(5.14)
~ is the surface current density which is similar to bulk current density
where K
~ =
J~ = σ E
~
E
.
ρ
The only difference is that the bulk resistivity ρ is replaced with sheet
resistance η.
Clearly the boundary conditions in 5.14 for a resistive sheet as defined in the
COMSOL 3D models are identical to those defined in the mode matching approach
5.1. The fraction of power that is absorbed by the resistive sheet in the COMSOL
models can be determined in a similar fashion to the method applied with SCATTER
where the S11 return power was analysed. The post processing environment can be
used to analyse the S11 parameter, and assuming no other losses the power absorbed
by the resistive sheet then is:
2
Pabsorb = 1 − S11
(5.15)
where in this case S11 will be a single value in contrast to the S11 matrix in the
SCATTER mode matching analysis.
This S11 analysis is also the approach that was used by Glenn et al. through the
use of the Hewlett-Packard High-Frequency Structure Simulator (HFSS) to calculate
the fraction of power absorbed by the SPIRE absorbers on the Herschel Space Observatory [Glenn et al., 2003]. Similarly it is the approach taken by Stephen Doherty
with the use of the CST Microwave Studio package for similar absorber analysis
[Doherty, 2012].
An issue that arises with this indirect approach occurs when there are losses in
the system besides the power absorbed by the resistive sheet. This is appropriate in
an array system such as SAFARI in which several detectors are fabricated on a single
chip and there is an open gap between the detector backshort structure and the horn
antenna array block which feeds the detectors. This open gap or ’vacuum gap’ as
it has come to be called is clearly a source of loss from the cavity structure, and
therefore simply analysing the S11 parameter will not be sufficient for determining
precisely how much power is absorbed by the resistive sheet.
190
Figure 5.5: left: Schematic of three SPIRE pixels for the Herschel Space Observatory. right: A single cavity backed
detector expanded for clarity. As can be seen a gap must be left between the horn array block and the delicate
detector wafer [Glenn et al., 2003].
The solution that both Doherty and Glenn et al. applied to this problem was to
run the simulation twice; first without the absorber present, and then again with the
absorber. The difference in the value of S11 for the two separate solutions will then
yield an approximate value for the fraction of power absorbed by the Ta. There are
of course two assumptions that must be made with this model; first, the presence of
the absorber does not affect the amount of power lost through the vacuum gap, and
second, any higher-order modes generated as the wave propagates into the cavity
are evanescent.
As an alternative approach a method for directly calculating the amount of
power absorbed by the Ta sheet was formulated by the author. Any induced surface
currents on the two dimensional resistive sheet will cause heating (surface resistive
heating) across the sheet. Integrating this resistive heating parameter across the
area of the absorber will yield the amount of power absorbed by the Ta as follows:
Z Z
Z Z
1 ~ ~
Pabsorb =
Qsrh · dS =
(5.16)
Re JS · E · dS
2
This direct technique allows a more precise calculation of the power absorbed
by the resistive sheet. A similar approach is applied for determining the precise
amount of power lost through the vacuum gap and this will be discussed in more
detail in section 5.4.1 of this chapter.
191
5.2.3
2D FEM Approach
Such simulations, as described in the previous subsection, can be very time consuming using a 3D solver for an electrically large system such as SAFARI. We therefore
also consider a 2D approach that is now described. The approach taken to modelling an absorber in a waveguide with the 2D COMSOL modelling environments
is very similar to that outlined above for the 3D case. In the standard 2D option,
the 2D plane where the model is built is representative of a 2D planar cut of a
model that has translational symmetry in a direction perpendicular to the model
plane where the symmetry applies to the geometry and the physics. A depth of one
metre is assumed in the transverse direction which can be important if quantities
such as fluxes are to be calculated correctly, since the area over which the power will
be distributed will be 1 metre times the height that is defined in the model plane.
For example the one dimensional line shown in red in Figure 5.6 represents a two
dimensional sheet from the three dimensional model, and it has a length of 50µm in
the x-direction. If 1 Watt of power is defined to flow through the 1D line then the
real value for the flux will be
1W
(50.10−6 m×1m)
= 2 × 104 W m−2 .
Once such corrections are applied to the results of any 2D simulations (at least
for electrically large systems) we expect very similar results will be true for a 3D
environment. This is not strictly speaking true for small systems since Maxwell’s
equations do mix up the x and y directions. A clear example of this is a narrow
waveguide operating at a frequency close to cut-off. The equations governing the
waveguide fields in x, y andz are not separate, and changing the size of the guide in
one dimension will affect how the field behaves in all three dimensions.
Things are a little bit different in the COMSOL 2D axial symmetry environment. In this option the model is again built in a 2D plane but the geometry is
truncated at the axis of rotational symmetry. Rather than assuming a depth of 1
metre in the transverse direction as in the standard 2D option, the post processing
mode allows the user to calculate the equivalent values for rotation about the axis
of rotational symmetry. Integrals of variables can also be calculated over 2π radians
so that the equivalent 3D quantities can be calculated. It is assumed that both the
geometry and the physics are symmetric with respect to rotation about the axis
of symmetry. The interpretation of the 2D axial symmetry model is illustrated in
Figure 5.7.
192
Figure 5.6: Diagram showing what the COMSOL 2D planar symmetry environment represents. The 2D planar
model will always represent a 3D model with translational symmetry in the geometry and physics and with a length
of 1 metre in the direction perpendciular to the model plane.
Figure 5.7: Diagram showing what the COMSOL 2D axial symmetry environment represents. The 2D axial symmetry model will always represent a 3D model with rotational symmetry in the geometry and physics rotated through
2π radians.
5.2.4
Comparison of Results
In order to verify the mode matching approach and to investigate the accuracy of the
2D calculation, some simple simulations were run with these two techniques, as well
193
as with the standard 3D COMSOL mode. The 3D FEM approach is well established
and reliable so it served as a good benchmark for validating the other two methods
(SCATTER and 2D approach). Mode matching is inherently a different approach
to solving EM problems than FEM so agreement between results would be good
validation for both FEM and mode matching techniques.
Before considering an absorber in a cavity we began with an analysis of an
absorber in a waveguide terminated with a short circuit which could be considered
as the simplest kind of ”absorber cavity” to be analysed. For example this allowed
the multi-mode behaviour to be probed by increasing the waveguide size, and the
effect of the ”backshort” (i.e. the distance between the absorber and shorted end
of the guide) to be examined. Furthermore such a simple, well controlled example
provided a trusted set of results that allowed the inclusion of an absorber in the
SCATTER code to be verified.
A model was created of a simple circular waveguide structure that was sealed at
one end and with an absorber located at λ/4 from the shorted end of the waveguide.
The gap between the absorber and the closed end of the waveguide acted as a simple
cavity structure. The waveguide to be modelled was given dimensions of length L =
0.3mm, and radius r = 50µm as shown in Figure 5.8. The first modelling approach
applied was the 3D COMSOL RF module, with the waveguide being modelled as
a cylinder with perfect electric conducting boundaries, and the absorber material
was modelled as an infinitely thin resistive sheet. A similar model was then built
in the 2D COMSOL RF module, where the 2D waveguide was given a width equal
to the diameter of the 3D cylindrical waveguide (W = 100µm). An appropriate
geometry file was then written for an input for the mode matching model of the
same problem. The maximum absorption was expected to occur at λguide /4 which
can easily be calculated by the wave equation [Clarricoats et al., 1984]:
1
1
1
=
−
λ2g
λ20 λ2c
(5.17)
where λg is the waveguide wavelength, λ0 is the free-space wavelength, and λc is the
cut-off wavelength specific to the particular guide.
Figure 5.9 shows the guide wavelength (relative to the free-space wavelength)
plotted as a function of increasing frequency for the fundamental T E11 mode in a
cylindrical waveguide with the dimensions defined above.
194
Figure 5.8: Screen shot of a cut through the 3D COMSOL model of a simple shorted waveguide cavity and absorber
operating at 5THz.
Figure 5.9: Plot of guide wavelength as a function of increasing frequency. As the frequency approaches a few times
the cut-off value, the wavelength approaches the free-space value.
The first comparison analysis that was carried out using the waveguide cavity
example just described involved varying the surface impedance over a range of 0 to
Ω
while keeping both the frequency and the absorber position constant. The
750
frequency was just set to a sufficiently high value that the guide wavelength was
almost exactly the free-space wavelength, namely 5THz (essentially an oversized
guide meaning λ0 ≫ λc ). This frequency value clearly gives rise to a free-space
wavelength of 60µm and a guide wavelength approximately the same, therefore the
absorber was positioned at
λW G
4
= 15µm from the shorted end of the guide (Figure
5.8). The absorber was given the same radius as the waveguide thus completely
filling the guide. The results are shown in Figure 5.10 and as can be seen there is
excellent agreement between all three simulation methods.
195
Figure 5.10: Efficiency of a resistive sheet absorber for varying surface resistance. The absorber was placed 15µm
( λW4 G ) from the closed end of a cylindrical waveguide 100µm in diameter. The values of surface resistamce are
Ω
. The free-space wavelength (λ0 ) of the radiation was 60µm.
given in multiples of 377
The next step taken with each of the three computational techniques was to
simulate the same waveguide and absorber structure across a large frequency range.
The range studied was from 1.6THz (which is below the cut-off frequency of the
guide for the fundamental mode) to 10THz. What was expected was that maximum
absorption should occur when the wavelength was such that the distance between
the absorber and the closed end of the guide corresponded to λ4 ,
3λ 5λ
, 4,
4
etc, and that
a minimum amount of power be absorbed when the same distance was equivalent to
an integer number of half wavelengths (destructive interference). Any wavelengths
in between these should allow for some power to be absorbed following a Fabry-Perot
like curve. The results are shown in Figure 5.11.
It is clear that there is again good agreement for the various methods except for
the COMSOL 2D results based on axial symmetry, which are labelled as ’COMSOL
2D rot’ in the graph. This illustrates the issue with trying to approximate a three
dimensional model of this type with the axial symmetry mode in COMSOL. The
reason for the disagreement lies in the manner in which the EM modes are excited
across the input port boundary in the axial symmetry modelling environment. The
fundamental TE mode should have its peak power on axis and decay to zero at
the boundary of the waveguide as shown in Figure 5.12. The correct boundary
condition to be used for representing symmetry in the electric field is a perfect
196
Figure 5.11: Efficiency of a resistive sheet absorber fully filling the guide for increasing frequency. The absorber was
again placed 15µm ( λW4 G ) from the closed end of a cylindrical waveguide 100µm in diameter.
magnetic conductor (PMC) boundary which allows the electric field amplitude to
have a maximum at the boundary (whereas a conducting boundary forces the electric
field to zero at the boundary). However the COMSOL package only allows for one
definition per boundary and the axial symmetry boundary condition is required for
what should also be the PMC boundary. Therefore the field is forced to zero at
the axis of rotational symmetry which of course represents the central axis of a
3D circular waveguide, thus the field structure is actually representative of some
higher order mode like the T E21 or in fact a coaxial mode. The much higher cut-off
frequency of approximately 3.5THz is clearly evident in the graphed data showing
that at the lower frequencies the full mode structure cannot enter the waveguide,
except of course in an evanescent form which will decay rapidly within the guide.
For this reason the 2D axial symmetry option in the COMSOL package was not
used again for simulating these models as clearly it will lead to erroneous results.
Of course for the case when the resistive sheet completely fills the guide, the
power scattering integral Γ in the mode matching approach has a simple form.
So having found good agreement for this case, the next step taken was comparison
between the simulation techniques for the case of a resistive sheet only partially filling
the guide. Two examples were examined where the first case was for a sheet radius
(r) being 0.8 times that of the guide radius (R) and therefore almost completely
filling the guide, and the second example being the case of a sheet diameter only
197
(a)
(b)
Figure 5.12: Illustration describing the issue with simulating modal porblems with the rotational symmetry COMSOL mode. (a): Screen shot showing the axial symmetry boundary condition. (b): The structure of the electric
field across the input boundary of the model.
0.1 times that of the guide diameter and therefore allowing a large amount of space
for power to flow around the resistive sheet. This test probes how well the Γ term
takes this partially filled case into account.
The results for the case of
r
R
= 0.8 are shown in Figure 5.13 and as would be
expected there is not much degradation in the amount of power absorbed for any
of the simulation methods when compared to the case of the absorber completely
filling the guide. However the initial results for the case of
r
R
= 0.1 show a difference
of an order of magnitude between the 2D planar symmetry mode and the other two
methods. The 3D COMSOL and mode matching results are still in good agreement
with each other though, and for this reason it was assumed the error was in the
2D COMSOL planar symmetry method. The results showing the disagreement are
shown in Figure 5.14.
With some consideration of what the 2D model represents in the extruded 3D
model, the cause of the huge difference in results was obvious. If the 2D model
was to be extruded in the third dimension by a length of 1 metre as is assumed
by the software, then the ratio of the area of the absorber to the area of the full
waveguide cross-section ( Aa11 ) would be very different than the case of a circular
absorber in a circular waveguide ( Aa22 ). This scaling parameter is denoted by Λ and
198
Figure 5.13: Efficiency of a resistive sheet absorber almost completely filling the waveguide for increasing frequency.
The absorber was again placed 15µm ( λW4 G ) from the closed end of a cylindrical waveguide 100µm in diameter.
This time the absorber diameter was only 0.8 times that of the waveguide diameter, or Dabs = 80µm
Figure 5.14: Efficiency of a resistive sheet absorber for increasing frequency. The absorber was again placed 15µm
( λW4 G ) from the closed end of a cylindrical waveguide 100µm in diameter.
it was calculated as follows:
2
Λ=
a1
A1
a2
A2
=
πr1
πR12
(1m)(r2 )
(1m)(R2 )
=
r12 .R2
(1 × 10−4 )2 (10 × 10−4 )
= 0.1
=
r2 .R12
(1 × 10−4 )(10 × 10−4 )2
(5.18)
Applying this scaling factor of 0.1 to the 2D COMSOL data shown in Figure 5.14 a
new data set was produced and it is shown in Figure 5.15 with the SCATTER results
and 3D COMSOL results shown again for comparison. It is clear from the graph
199
that including the scaling factor brings the 2D results into much better agreement
with the 3D FEM and mode matching results, however the symmetry of the 2D and
3D cases is so different it is hard to draw conclusions from the 2D models.
Figure 5.15: Efficiency of a resistive sheet absorber for increasing frequency with the COMSOL 2D results corrected.
This graph is just zoomed in on the relatively low power.
It has been shown in this section that the extended mode matching code SCATTER that now allows for a resistive current sheet to be included in the simulations
is performing as expected. There is reasonable agreement between SCATTER and
the 3D COMSOL results for the case of an absorber filling the waveguide section,
and for the case of it only partially filling the guide which is more comparable to a
real cavity structure such as a SAFARI detector. An important point to be made
here is that the extended SCATTER code was found to be up to three orders of
magnitude faster than the 3D FEM approach. For example, the model of a simple
cavity containing an absorber that was presented above required 10.81 hours calculation time for one hundred spot frequencies with a FEM analysis. By contrast, the
mode matching analysis for the same one hundred spot frequencies took only 15.32
seconds.
It was also shown that the 2D planar symmetry models are not really applicable
to these cylindrical waveguide and cavity geometries due to the large differences in
symmetry, however with some corrections to the results the 2D approach could be
used as a quick method for initial simulations in order to gain approximate results.
200
The corresponding 3D models would then have to be investigated so that a more
reliable final result could be obtained. Of course the 2D axial symmetry approach
is a more accurate representation of the 3D cylindrical models in terms of geometry,
however the correct symmetry is not employed in the physics, and an example of
why it can not be trusted to yield reliable results was described above.
The final discussion in this section regards further verification of the SCATTER
absorber code that was carried out by Stephen Doherty as part of Technical Note
Three (TN3) for the Irish AO contract commissioned by ESA, and that the author
was also involved in [RFQ 3-6418/11/NL/CBi]. The approach taken by Doherty
involved simulating a range of different scenarios for an absorber inside a waveguide cavity using the adapted SCATTER code, and then comparing the results to
those of similar simulations with the commercially available CST Microwave Studio
package. The computational method employed by CST Microwave Studio is FIT,
and it is similar to the FEM approach that COMSOL is based on, but it is the
integral form of Maxwell’s equations that are solved within the meshed problem
geometry whereas it is the differential equations that are solved with FEM. As with
the good agreement shown in this section for the results from the 3D COMSOL
and SCATTER simulations, similarly good agreement was demonstrated for the
CST Microwave Studio FIT. An example of the CST/SCATTER results is shown
in Figure 5.16.
(a)
(b)
Figure 5.16: Results of CST verification of SCATTER code for modelling an absorber. (a): Absorbed power for
varying values of frequency in a single moded circular waveguide with the resistive sheet fully flling the guide and
at a fixed position of λg /4. (b): Absorbed power at 80 GHz for varying values of absorber/backshort distance ∆d
in a single moded circular waveguide with the resistive sheet fully flling the guide [Doherty, 2012].
201
5.3
Optimisation of Absorber Position and Cavity Geometry
Defining an optimum position for the resistive sheet absorber in a cavity is not
a trivial matter as there are a number of factors that one must consider, particularly for a large operational frequency range and an over-moded system such as
the case for SAFARI. Since maximum efficiency would be expected when the absorber/backshort distance ∆d is equal to a quarter of the guide wavelength as was
already demonstrated, then for a signal consisting of a wide band of wavelengths
there is no obvious optimum position. Furthermore, in an over-moded cavity structure where even a single frequency is considered, the ideal position of the absorber
again is not obvious since the guide wavelength of each mode can differ by significant amounts especially when they are close to cut-off. For these reasons a large
number of simulations were carried out in order to determine the absorber/backstop
distances that correspond to maximum coupling for both a broad frequency range
and for an over-moded system.
5.3.1
Optimising Absorber Position for T E11 Mode
With the mode matching code showing good agreement with the FEM results thus
far, it was chosen for the purposes of optimisation due to its high computational
speed. Both the SCATTER mode matching and COMSOL 2D approaches are highly
efficient, with COMSOL 2D being slightly computationally faster (same order of
magnitude). Also the mode matching code relies on fewer approximations, and the
2D FEM was shown to disagree with some previous results since strictly speaking
it really only applies to systems with Cartesian symmetry.
The aim was to optimise the position of the absorber relative to the backshort
for a broad frequency range. The optimum absorber/backshort position is of course
expected to vary for different frequencies, so the aim was to determine a position
that related to the overall average optimum value for the entire frequency range.
The same relatively simple example of a circular waveguide shorted at one end was
used for the initial analysis and the frequency range chosen was again 1.8 to 10THz
within which the full range of the SAFARI S-Band lies. The length and radius of the
202
guide were set to L = 0.3mm and r = 50µm respectively, as shown in Figure 5.17,
and the sheet impedance was set to 377Ω/. The optimisation routine consisted
of incrementally moving the absorber away from the backshort in increments of
∆d = 1µm out to a total of 40µm. At each absorber position the model was solved
for the entire frequency range in steps of 0.1THz.
Figure 5.17: Diagram showing the dimensions of a simple waveguide cavity model.
Figure 5.18 shows the contour plot that was generated from the optimisation
search procedure. As was expected the optimal position of the absorber clearly
depends on frequency. Figure 5.19 then shows the variation in efficiency of the
absorber across the band for each position, where the efficiency across the band
N
P
ef f (νi ), where νi is the ith spot frequency and N is the
was calculated by N1
i=1
total number of test frequencies. The result was that for a shorted cylindrical
waveguide fed by a single-mode waveguide over an operational frequency range of
1.8 to 10THz (30 to 167µm) the optimum position of the absorber is 11.5µm. The
central wavelength λcentral was 98.5µm, so the distance value of 11.5µm corresponds
to approximately ∆d =
λcentral
.
8.57
What is also evident in Figure 5.19 is that once
the absorber is placed at least 6µm from the backshort then the average efficiency
of the absorber never falls below about 64%. This is highlighted by the red cross
hairs placed on the graph and the distance corresponds to
λmin
5
where λmin is the
minimum wavelength of the band, i.e. 30µm.
5.3.2
Optimisation of Absorber Position for an Over-Moded
System
In the previous example it was assumed that only the T E11 cylindrical mode propagated, even though a large part of the frequency range was such that higher order
203
Figure 5.18: Contour plot showing the efficiency of a resistive sheet absorber for varying frequency and backshort
distance. The fractional power absorbed is represented by variation in colour with deep blue being minimum and
deep red being maximum.
Figure 5.19: Line plot showing the average efficiency across the band (∆f = 1.8 − 10THz) of a resistive sheet
absorber for varying backshort distance.
modes would propagate if excited. In this subsection the optimum position of an
absorber in an over-moded waveguide is investigated by examining the efficiency of
the absorber with respect to the various propagating modes. The same waveguide
structure as was analysed in the previous example was again used for this study,
204
i.e. L = 0.3mm and r = 50µm. It was assumed that each mode has equal weight
however, in a true blackbody cavity some modes may be more dominant than others.
Also, only the T E1,m modes were taken into account since the modes with zero phi
dependence are most often the dominant ones.
For this example a single frequency was chosen at the high end of the band
so that the higher order modes could also propagate. The frequency was kept at
10THz leading to a free-space wavelength of 30µm, and since the frequency was
so far above the cut-off frequency of 1.87THz then the approximation λW G = λ0
could be used. Therefore the expected optimum absorber position relative to the
backstop for the T E11 mode was
λ0
4
= 7.5µm. However for higher order propagating
modes, particularly those close to cut-off, the optimum position would be expected
to increase. The absorber/backshort distance was varied from 2 to 35µm in steps of
∆d = 1µm with the efficiency being calculated for each propagating mode at each
distance.
Figure 5.20 shows the results of the analysis and it is clear that indeed there is a
varying optimum absorber position depending on which mode is being analysed. By
calculating the average power absorbed across all modes at each particular distance
the plot in Figure 5.21 was generated. As more modes are included in the calculation
of average absorbed power the peak average efficiency shifts from
λ0
4
toward
λ0
.
3
Clearly the distribution of modes will affect precisely where the maximum absorption
should occur and this will need to be analysed for individual systems.
5.3.3
Cavity and Absorber Size
Having investigated the simplest possible cavity structure where a short circuit was
set up by sealing one end of a circular waveguide, thus creating a resonant cavity
between the resistive sheet absorber and the shorted end of the guide, an expanded
cavity is now considered where ’expanded’ refers to the cavity having a larger radius
than the waveguide feed. The waveguide is now assumed to be single moded however
the step into the larger cavity will almost certainly cause power to scatter into the
higher order modes supported by the cavity. Again, building up the complexity of
the models in a gradual manner, the first expanded cavity considered was cylindrical
in shape, and the radius of the cavity was varied incrementally in order to investigate
205
Figure 5.20: Block contour plot showing the efficiency of a resistive sheet absorber for higher order propagating
modes and varying backshort distance. The dotted line indicates the optimum absorber position.
Figure 5.21: Line plot showing the average efficiency of a resistive sheet absorber for varying backshort distance
when higher order propagating modes are includued. As more modes are considered the optimum distance shits to
slightly larger values.
the ideal size for such a cavity. Starting with a radius equal to that of the waveguide
feed (r = 50µm) the cavity radius was increased in steps of 5µm up to r = 250µm
while the absorber radius was kept fixed at r = 50µm. The distance between the
absorber and backshort was kept fixed at 18.2µm and the frequency range was from
206
4 to 10 THz, approximately the SAFARI S-Band.
Figure 5.22 shows the results of the analysis and it can be seen that the cavity
does not affect the absorber efficiency for frequencies that were already coupling
well to the absorber in a shorted circular waveguide. However, the cavity does
provide improvement in the amount of power absorbed at the frequencies that were
experiencing destructive interference due to the standing waves that were set up
between the absorber and backshort. At least at some of these frequencies there
are particular values for the cavity radius that allow higher coupling of power to
the absorber, and what is clear is that a larger cavity is not necessarily better. In
fact based on the average power absorbed across the band for each cavity radius the
ideal size is 110µm, or 2.2 times the waveguide feed radius.
Figure 5.22: Block contour plot showing the efficiency of a resistive sheet absorber for cavities of varying radius.
The highest average efficiency is found for a cavity with a radius of 110µm, or 2.2 times the input waveguide radius
(rW G = 50µm)
By fixing the radius of the cylindrical cavity to the optimum value of 110µm, the
radius of the absorber was then varied from 5µm (0.1 times the waveguide radius) to
110µm (completely filling the cavity) in steps of 2.5µm. Again the distance between
the absorber and backshort was kept fixed at 18.2µm and the frequency range was
from 4 to 10 THz. The results of this analysis are presented in Figure 5.23 and
the average absorbed power across the frequency band for each absorber radius is
shown in Figure 5.24, where it is clear that while a larger absorber does increase
207
the amount of power absorbed, there is a point where increasing the radius further
has no effect. This value occurs for an absorber radius of approximately 50µm, or
a radius equal to that of the waveguide feed. This is an important result as it is
preferred to have a smaller Ta absorber where possible because the level of noise in
the absorber is proportional to its volume, and the speed of response (time constant
τ ) is also related to the absorber volume [Audley et al., 2013]. In fact a discussion is
given in section 5.4.3 on techniques for removing portions of the absorber material
while maintaining its optical cross section, therefore an understanding of the effects
of absorber size in terms of optical coupling is clearly important. It should again
be pointed out that these results are specific to the examples chosen for this study,
and while the same techniques developed here could be applied to other systems,
the results may differ based on the parameters of those systems.
Figure 5.23: Block contour plot showing the efficiency of a resistive sheet absorber in a cavity with a radius of
110µm (based on the previous optimisation). The radius of the absorber was varied from 5µm to 110µm in steps of
2.5µm.
5.3.4
Cavity Shape
So far in this section only cavities with cylindrical shapes have been considered
as they are the simplest geometries due to their similarity to uniform cylindrical
waveguides. In fact the standard type of cavity geometry that has been most com208
Figure 5.24: Average efficiency across the frequency band for a resistive sheet absorber in a cavity with a radius of
110µm. The radius of the absorber was varied from 5µm to 110µm in steps of 2.5µm.
monly chosen for cavity coupled bolometer type detectors in previous instruments
has been the hemispherical cavity since it was assumed to be the best choice for
maximising coupling to the bolometer. This assumption is based on ray tracing and
the assumption that the cavity is large electrically (many wavelengths in radius).
Thus by placing the detector close to the centre of curvature of the hemisphere any
power that was transmitted through the absorber was refocussed back onto the absorber. Because of the complex calculations required, clearly the assumption that
ray tracing is a valid approach may not be true for such cavities since interference
effects need to be taken into account and the cavities are only a modest number of
wavelengths in scale size. Until recently modelling the entire pixel with more accurate EM techniques simply was not possible without resorting to super computers.
For this reason an investigation was carried out in order to determine if there is a
more optimum design for the geometry of the integrating cavity in terms of optical
coupling.
A range of different cavity designs were examined and the amount of power
absorbed by the resistive sheet was calculated across a frequency range of 1.8THz
(just above cut-off for the waveguide) to 10THz. The results for three of these designs
are shown in Figure 5.26 and the average absorber efficiency predicted for each design
is presented in Table 5.1, together with the maximum and minimum efficiency for
each of the three designs. The results of the simple shorted waveguide are shown
209
again for comparison and the input feed to the cavity was defined as having the
same dimensions as that of the cylindrical waveguide used in the comparison study,
namely a radius of 50µm and a length of 0.30mm. The cylindrical cavity was given
a radius twice that of the waveguide radius R = 2RW G and a depth d = 2RW G , the
hemisphere cavity was also designed with a radius of 2RW G which of course results
in the same depth of d = 2RW G , and the sine profiled cavity was designed with an
initial radius of Ri = 2RW G that varied as R(x) = sin π2 1 − A Rxi
where x is the
distance from the start of the cavity up to a depth of d = 2RW G (if A = 1) meaning
that each of the designs had the same depth. Of course the value of A could be
allowed to vary in an optimisation analysis for example, however the depth of the
cavity was kept the same as the hemisphere and cylinder in this case so that the
comparison was balanced. Clearly setting A = 0.5 for example will create a cavity
with a depth twice that (d = 4RW G ) as for A = 1, with a slower curving profile as
shown in Figure 5.25(a). Figure 5.25(b) shows the approximate shapes of the other
cavity backshorts that were examined. The absorber was given a radius of 50µm
and sheet resistance of 377Ω/, and was positioned at 20µm from the start of the
cavity for each case.
(a)
(b)
Figure 5.25: Schematic of various cavity backshort geometries. (a): Geometries for sine function profiles with
different weighting factors (A), which determines the strength of the curve and the depth of the cavity relative to
its radius. (b): Three of the different cavity geometries that were investigated.
What should be clear from both Figure 5.26 and Table 5.1 is that the shape
of the cavity does indeed have a significant effect on the coupling of power to the
absorber. Both the hemisphere and sine profiled cavity shapes give rise to an average
absorber efficiency across the band of 87%, but the sine profiled shape exhibits
210
Figure 5.26: Absorber efficiencies for various cavity geometries across the frequency band of 1.8 − 10THz for T E11
mode only.
Cavity Design
Simple Waveguide
Maximum
Minimum
Average
Power Absorbed
Power Absorbed
Power Absorbed
1.0
0.0
0.69
1.0
0.2
0.76
1.0
0.55
0.87
1.0
0.78
0.87
(R = 2RW G )
Cylindrical
(R = 2RW G )
Hemisphere
(R = 2RW G )
Sine Profile
(Ri = 2RW G )
Table 5.1: Absorber efficiency values across the frequency band of 1.8 − 10THz, for T E11 mode only, for cavities
with varying geometry.
much less variance giving rise to a much smoother frequency dependence compared
to the hemisphere example. The conclusion drawn therefore would be that for the
cavity examples examined in this case, fed by a single mode waveguide, the ideal
cavity geometry for maximising power coupling between the incoming signal and the
absorber is the sine profiled cavity. This is surprising considering the almost straight
211
walls of the cavity in this configuration. There are of course many other geometries
that could be examined such as a paraboloid or sin2 profile for example, and some
of these designs could possibly yield further improved coupling of power to the
absorber, however the aim here is to demonstrate the techniques for modelling such
systems, and to show that these simulations are important for designing optimised
systems.
5.4
SAFARI L-Band Receivers
Having investigated generic systems to gain an understanding of the effects of bandwidth, absorber size, and backshort configuration on the coupling efficiency of a
cavity mounted absorber, the next step taken was a model of a realistic system
with realistic dimensions and properties. The system chosen was the SAFARI LBand design that was under investigation by Dr Phil Mauskopf’s research team at
Cardiff University, Wales where they were carrying out a measurement campaign of
the system while also using the software package HFSS (High Frequency Structural
Simulator) to model the system [Mauskopf, 2011]. They report very good agreement
between the measurements and simulations.
Another member of the NUIM THz Optics research group, Stephen Doherty,
carried out a similar analysis using the CST Microwave Studio package [Doherty, 2012]
and the results published were in excellent agreement with those of the Cardiff group.
Therefore the Cardiff results served as a good comparison for the results of the current study reported in this thesis. At the time of writing the current design for the
SAFARI L-Band pixel [Goldie et al., 2012] consisted of a square absorber contained
within a hemispherical cavity with a diameter of 500µm, fed by a single mode cylindrical waveguide with a radius of 60µm. The absorber was located at 50µm from
the exit aperture of the input waveguide and had dimensions of 320µm ×320µm
(see Figure 5.27). As before the absorber was modelled as an infinitely thin sheet
Ω
and given a surface resistance of 377
. The frequency range analysed was from 1.5
to 3.0THz.
Ω
The free-space matched sheet resistance of 377
used in the L-Band simulations
is an idealised value, however the actual sheet resistance in the Cardiff measurement
Ω
Ω
system was thought to be about 90
. The ideal 377
is not unachievable though,
212
and the SAFARI S-Band research group at SRON, Groningen describe such possible
Ω
[Audley et al., 2013]. If the Ta is below its
sheet resistances up to about 367
transition temperature and thus is operating in a superconducting state then the
surface resistance will of course be a lot lower. The SRON team report an apparent
transition temperature for the film of about 700 mK, which they claim is lower than
the expected value of 900 mK [Audley et al., 2013]. The nitride island on which the
Ta absorber sits is expected to operate at a higher temperature than this, therefore
Ω
achieving a 367
surface resistance should not be a problem.
Figure 5.27 illustrates the SAFARI L-Band pixel and its various components.
As can be seen there is a vacuum gap between the waveguide and cavity which
is highlighted in green. This corresponds to a distance of 100µm from the exit
aperture of the waveguide to the entrance of the cavity backshort. The absorber
was placed at the halfway point between the waveguide and cavity. The reason
for this vacuum gap arises from the fact that the cavities/backshorts of the pixel
array are manufactured on a single substrate (see bottom gold section in Figure
5.1(b) at the beginning of this chapetr) with the TES detector array chip then fixed
onto it (green section in Figure 5.1(b)). The waveguide/horn array (in the top gold
section of the image) that feeds the detector array is a separate block that must
be fitted to the detector and backshort array but with a certain amount of space
required to be left between the two so as not to damage the delicate electronics and
TES detectors. Progress is being made in terms of minimising this gap in order
to minimise the amount of power that can potentially leak out through it, but for
these simulations a relatively large gap was applied in order to match the Cardiff
L-Band set-up, and presumably to consider a worst case scenario. In fact the SRON
group have succeeded in reducing the original 230µm vacuum gap to 50µm in the
S-Band test-bed [Audley et al., 2013] so similar reductions should be possible for
the SAFARI L-Band also.
5.4.1
Vacuum Gap between Neighbouring Pixel Cavities
The vacuum gap just described brought new challenges in terms of modelling the
behaviour of power flow through the pixel with the three simulation techniques described so far, in particular estimating the power lost through the gap. Including the
gap with COMSOL was possible with two different approaches. The first approach
213
Figure 5.27: Diagram of waveguide fed cavity coupled detector with a vacuum gap of 100µm. The green coloured
section is the vacuum gap.
consisted of employing a boundary condition known as a ‘perfectly matched layer’
where the impedance is perfectly matched at the boundary and therefore simulating
propagation into free-space with no reflections.
The second method utilises a ‘scattering boundary condition’ where the scattering coefficients are just set to zero so that no power is reflected and free-space
propagation is again simulated. The type of wave expected to be incident on the
scattering boundary, say a plane wave or spherical wave, must be defined in the
model before calculation of the propagating fields, and an instruction to then reflect zero intensity for any incident plane or spherical waves on the boundary is
defined. In both approaches the amount of power lost through the gap is calculated
by integrating the power outflow across the surface of the boundary. In the 2D environment the same two techniques are applied but with the integral being of course a
line integration across the 1D free-space boundary. The issue again arises as to how
to interpret the 2D results. It is a bit more complicated now that there are three
surfaces over which power can flow, namely the input waveguide entrance (including
back scatter through it (S11 )), the surface of the absorber (power absorbed due to
Joule heating), and the surface that bounds the vacuum gap. Previously the ratio
of power flow surface areas was used to correct the differences in the 2D results with
some success so a similar approach was applied again. However, this is not expected
to yield an accurate result, only an order of magnitude estimate. The results will
be discussed following a description of the mode matching approach.
Modelling the vacuum gap with the mode matching code SCATTER proved
214
much more difficult. In his PhD thesis Stephen Doherty of NUIM attempts to model
a vacuum gap with the mode matching technique [Doherty, 2012]. The ‘annular trap’
method he attempted involved defining a waveguide section with a radius larger than
the cavity radius and with a length equal to that of the vacuum gap as illustrated
in Figure 5.28. Then, by considering what modes were possible to propagate in this
annular section based on the cut-off frequencies of the modes, an upper limit to the
amount of power that could radiate out through to free-space was calculated. This
was achieved by investigating what fraction of power in each mode was contained in
the annular section. The results from that study showed that the amount of power
that coupled into the annular section was strongly dependent on the radius of the
section. It was also shown that as the radius was increased to large values relative
to the cavity radius the results tended toward an asymptote. It was concluded that
this method would allow for an upper limit to be calculated for the power lost to
free-space, but the required large annular radius allowed for a large number of modes
and therefore long computation times, and of course this was just an approximation.
Figure 5.28: Illustration of the ‘annular trap’ method for modelling a vacuum gap in a SAFARI-like detector.
An aim of future work is to model propagation across the vacuum gap using
free-space Gaussian beam modes and waveguide modes for the backshort cavity and
waveguide entrance aperture. The following theory and discussion relied on heritage
from a similar approach taken by Emily Gleeson to model a similar problem in the
design of the feed horn array for the HFI instrument on the PLANCK satellite
[Gleeson, 2004]. In that case the vacuum gap related to the space between the backto-back horns and the detector horns and it was the location of the filter stack, as
shown in Figure 5.29. The aim here is to allow the free-space modes to propagate
215
freely across the vacuum gap region until they encounter the cavity entrance. The
electric field of the free-space EM waves propagating across the vacuum gap can
be described mathematically as a sum of such free-space modes based on the GBM
theory that was presented in chapter 2. Of course if the frequency is such that
the waveguide feeding the vacuum gap is single-moded then the decomposition of
the waveguide field in terms of free-space modes will be straightforward and will be
similar to how the corrugated horn aperture fields were decomposed in the examples
in chapter 2. However, the waveguide feeds in the SAFARI detectors will be overmoded, and the integrating backshort cavities will certainly be multi-moded, and
since each waveguide mode that is present at the waveguide aperture needs to be
decomposed in terms of free-space modes then the most convenient approach is to
define a transformation matrix [T] that can account for all possible mode-to-mode
coupling. We can represent the loss at the edge of the pixel then as absorbing walls
as illustrated in Figure 5.30.
Figure 5.29: Schematic of the optical chain for the HFI channels on the PLANCK satellite. The front horn determines
the light collected from the telescope. The back horn launches the CMB radiation through a set of filters into the
100 mK horn which concentrates it onto the bolometer in the detector cavity.
The TE and TM waveguide modes that were described in chapter 2 can be
rewritten in terms of combinations of î and ĵ components so that the terms containing
Bessel functions of order n − 1 and trigonometric functions of (n − 1) φ are grouped
together and similarly terms with Bessel functions of order n + 1 and trigonometric
functions of (n + 1) φ also grouped together. The normalised waveguide modes
T E(n−1)
are then written as [Trappe et al., 2005] ~enlT E = ~enl
216
T E(n+1)
+ ~enl
and ~enlT M =
Figure 5.30: Illustration of the waveguide-to-free space transformation method of modelling a vacuum gap.
T M (n−1)
~enl
T M (n+1)
+ ~enl
where:
s
(2 − δn0 )
(1 − (n/qnl )2 ) Jn2 (qnl )
q r cos(n − 1)φ
sin(n − 1)φ
nl
î −
Jn−1
ĵ
a
− sin(n − 1)φ
cos(n − 1)φ
T E(n−1)
~enl
=
πa2
s
(2 − δn0 )
πa2 (1 − (n/qnl )2 ) Jn2 (qnl )
q r
cos(n + 1)φ
sin(n + 1)φ
nl
î +
ĵ
Jn+1
a
− sin(n + 1)φ
cos(n + 1)φ
T E(n+1)
~enl
=
=
(5.21)
s
(5.22)
(2 − δn0 )
2
πa2 Jn+1
(pnl )
p r
cos(n + 1)φ
sin(n + 1)φ
nl
î +
ĵ
Jn+1
a
− sin(n + 1)φ
cos(n + 1)φ
T M (n+1)
~enl
=
(5.20)
s
(2 − δn0 )
2
πa2 Jn+1
(pnl )
p r
cos(n − 1)φ
sin(n − 1)φ
nl
î −
ĵ
Jn−1
a
− sin(n − 1)φ
cos(n − 1)φ
T M (n−1)
~enl
(5.19)
Clearly, because of the orthogonality of the φ dependent components of the
modes, the waveguide modes couple to normalised vector combinations of two free
space Laguerre modes of the form:
~α
Ψ
m
(W, R, r, φ, z) =
α
ψm
(W, R, r, φ) exp (−ikz + iφm,α (z))
217
sin αφ
cos αφ
î ±
ĵ
− sin αφ
cos αφ
(5.23)
where
α
ψm
(W, R, r, φ) =
q
2m!
πW 2 (m+α)!
exp
−r 2
W2
2r2
W2
α2
Lαm (2r2 /W 2 ) exp (−ik (r2 /2R)),
which are the scalar Associated Laguerre-Gauss modes that were presented in chapter 2, and α is an integer that represents the degree of the Laguerre polynomials. The
computation time involved in converting between waveguide modes and free space
modes with this method is greatly reduced because when the azimuthal order, n, of
T E(n−1)
the waveguide modes is non-zero both the ~enl
T M (n−1)
and ~enl
waveguide mode
T E(n+1)
~ n−1 (r, φ, z) free-space modes, and the ~e
components only couple to the Ψ
m
nl
T M (n+1)
~enl
and
~ n+1 (r, φ, z). Thus, for each waveguide
components only couple to the Ψ
m
mode of radial degree l and azimuthal order n we can write:
~enlT E =
M
X
(n)
(n)
~ n−1
~ n+1
Tm,l Ψ
m + Tm+M,l Ψm
m
~enlT M =
M
X
(n)
(n)
(a)
(b)
~ n−1
~ n+1
Tm,l+L Ψ
m + Tm+M,l+L Ψm
m
(5.24)
(n)
where the Ti,j can be regarded as the elements of a 2M × 2L transformation matrix
T~ (n) for waveguide modes of azimuthal order n. 2M is the number of Laguerre
modes used in the analysis for a given waveguide mode, and 2L is the number of
modes of radial degree l that are used for each n (half T E and half T M ).
The reason for the m + M in the subscripts in the right hand side terms of
5.24 is related to how the matrix T~ (n) is ordered, i.e. with the first half of the rows
~ n−1 coefficients and the second half of the rows corresponding
corresponding to the Ψ
m
~ n+1
to the Ψ
coefficients. The reason then for l + L subscripts in 5.24(b) for the TM
m
waveguide modes is again related to how the matrix T~ (n) is ordered, i.e. with the first
half of the columns corresponding to the ~enlT E TE coefficients, and the second half of
the columns corresponding to the ~enlT M TM coefficients. An example corresponding
to waveguide modes with azimuthal order n = 3 and radial degree l = 1 to l = 3
represented by 8 Laguerre-Gauss free-space modes (4 of azimuthal order 2 and 4 of
azimuthal order 4) is shown below in the T~ (3) matrix.
218
~ 20
Ψ
T E31
T E32
3
T11
3
T12
3
T13
3
T14
3
T15
3
T16
3
T22
3
T23
3
T24
3
T25
3
T26
3
T32
3
T33
3
T34
3
T35
3
T36
3
T42
3
T43
3
T44
3
T45
3
T46
3
T52
3
T53
3
T54
3
T55
3
T56
3
T62
T363
T364
T365
T366
3
T72
3
T73
3
T74
3
T75
3
T76
3
T82
3
T83
3
T84
3
T85
3
T86
2
3
~
Ψ1 T21
~2
Ψ
T3
2
31
3
~ 23
T41
Ψ
~4
Ψ
T3
0
51
~4
Ψ
T3
1
61
3
~ 42
T71
Ψ
3
~ 43
T81
Ψ
T E33
T M31
T M32
T M33
3
) as an example,
One of the elements in the T~ (3) matrix is highlighted in red (T62
and it corresponds to the amount of the T E32 waveguide mode that couples to the
~ 4 free-space mode. Clearly the Ψ
~ 4 free-space mode will also carry power from the
Ψ
1
1
other waveguide modes if they are present, and the contributions of each mode to
~ 41 are represented in the sixth row of T~ (3) (shown in bold face). It is these
the Ψ
~ n fields that carry the power across the vacuum gap as far as the cavity entrance
Ψ
m
plane, and due to their formalism they diffract as they propagate. This diffraction
causes the intensity of the free-space modes to spread out meaning that not all of
the power will be available for coupling to the waveguide modes of the cavity, which
are of course truncated at the edge of the cavity. Therefore performing the power
coupling integrals over the area corresponding to the cavity aperture will result in
a total power available for coupling to the absorber less than the power present
before propagation across the vacuum gap. Converting the EM field represented
by Laguerre-Gauss modes back into cylindrical waveguide modes appropriate for a
cylindrical backshort cavity would be the next step, and the theory is very similar
to that just described for the reverse case. Each transformation matrix for each
azimuthal order n of waveguide modes will have the form of the complex transpose
of the 2L × 2M T~ (n) matrices described above, i.e 2M × 2L T~ †(n) .
219
~ 20
Ψ
T E31
T E32
T E33
T M31
T M32
T M33
~ 21
Ψ
~ 22
Ψ
~ 23
Ψ
~ 40
Ψ
~ 41
Ψ
~ 42
Ψ
~ 43
Ψ
3
T11
3
T12
3
T13
3
T14
3
T15
3
T16
3
T17
3
T18
3
T21
3
T22
3
T23
3
T24
3
T25
3
T26
3
T27
3
T28
3
T31
3
T32
3
T33
3
T34
3
T35
3
T36
3
T37
3
T38
3
T41
3
T42
3
T43
3
T44
3
T45
3
T46
3
T47
3
T48
3
T51
3
T52
3
T53
3
T54
3
T55
3
T56
3
T57
3
T58
T361
3
T62
T363
T364
T365
T366
T367
T368
The technique described above would be a first order approximation for accounting for losses due to the gap between the horn array block and the detector array. A higher order approximation could include multiple reflections of the
free-space Laguerre-Gauss modes within the vacuum gap in a manner similar to
modelling a Fabry-Perot etalon environment, and even multiple transformations
from waveguide-to-Laguerre-Gauss modes and Laguerre-Gauss-to-waveguide modes
to account for the effects of the vacuum gap on return power from the cavity. This
could be implemented with a scattering matrix formalism.
Taking account of the resistive sheet absorber would be either straightforward or
require even more extra code, depending whether the absorber was to be positioned
within the cavity structure or in the vacuum gap between the horn and cavity (see
Figure 5.30). If the design consisted of the absorber being located at the aperture of,
or within the cavity backshort, then the code to model it already exists as presented
in section 5.2.1, where the absorber was treated as an infinitely thin resistive current
sheet with a two dimensional sheet resistance in units of Ω/. If on the other
hand it was located within the vacuum gap (as is the case with past designs), then
the absorber could be modelled with the appropriate scattering matrix, where the
interaction of free-space Laguerre-Gauss modes with a resistive sheet would need to
be considered.
Assuming the absorber is located just after the cavity backshort and not in the
vacuum gap, then the overall scattering matrix for an entire pixel system could be
calculated by cascading the scatter matrices for each sub-domain of the system as
follows:
220
h i
~
S
z
T otal
=
0
V~ (n)
V~ †(n)
0
(n)
(n)
S~11
S~12
(n)
S~21
z
S~22
(n)
z
0
T~ †(n)
T~ (n)
0
0
T~ (n)
†(n)
~
T
0
(n)
(n)
S~11
S~12
z
(n)
(n)
~
~
S21
S22
where the middle matrix represents propagation across the vacuum gap, and the z
symbol indicates that the scattering matrices are cascaded [Olver et al., 1994], as
opposed to multiplied. The ABCD matrix for propagation through free-space could
be used to transform each Laguerre-Gauss free-space mode:
1 L
0 1
where L is the length of the vacuum gap.
As a first order approximation the T E11 mode that can propagate in the waveguide feed of the SAFARI L-Band (see Figure 5.27 above) was decomposed in terms
of associated Laguerre-Gauss free-space modes. This T E11 mode is the field that
is present at the waveguide exit, at the start of the vacuum gap. The higher order T E12 mode will only propagate at the highest frequencies of the SAFARI LBand range, and since the blackbody source used in the measurements at Cardiff
[Mauskopf, 2011] produced only faint signal levels at the T E12 switch-on frequency,
it was ignored in the analysis. Twenty Laguerre-Gauss free-space modes were used
~ 0 and ten of Ψ
~ 2 ).
in the analysis (ten of Ψ
n
n
The Laguerre-Gauss free-space modes were then propagated across the vacuum
gap by simply using a value for propagation distance equal to that of the gap depth.
The power coupling integrals for converting back from Laguerre-Gauss modes to
the T E11 waveguide mode were then calculated, where the T E11 mode is now that
corresponding to a cylindrical waveguide with a radius the same as the cavity backshort.
By summing the normalised mode coefficients An corresponding to the transformation from the waveguide T E11 mode to free-space modes (this of course was
equal to unity), and then comparing this to the sum of the normalised mode coefficients A∗n corresponding to the transformation from free-space modes to the cavity
T E11 mode, a first order approximation was deduced for the amount of power lost
221
as a result of the vacuum gap:
Loss =
N
X
n
An −
N
X
n
A∗n
=1−
N
X
A∗n
(5.25)
n
The analysis was carried for the SAFARI L-Band frequency range from 1.5 to
3.0THz in steps of 0.1THz, with the results shown in Figure 5.31. The results from
a FEM analysis of the SAFARI L-Band detector (to be discussed in the next section
5.4.2) are shown for comparison. As can be seen there is large differences between
the two results. In fact it appears that the modal analysis may provide a lower
limit on the power radiated through the gap. Since the free-space Laguerre-Gauss
mode simulation does not include multiple reflections, and the FEM model of the
SAFARI L-Band does, a second FEM model was created which attempted to match
the scenario modelled with the modal approach. In this case the vacuum gap was
terminated with a perfectly matched layer boundary condition, as shown in Figure
5.32, and as can be seen there is a significant improvement in terms of agreement
between the FEM and modal approaches.
Figure 5.31: Fractional power radiated through the vacuum gap between the horn array and the pixel array chip
~ 0 and 10 of Ψ
~ 2 ). The
for the SAFARI L-Band. The analysis involved 20 Laguerre-Gauss free-space modes (10 of Ψ
n
n
results from a FEM analysis of the same problem are shown for comparison (see Figure 5.36 in section 5.4.2).
The method described above is a very approximate approach since there are
a number of factors not accounted for in the analysis. First, while only the T E11
mode was present at the single-mode waveguide feed aperture (at 1.5 THz), the
222
(a)
(b)
Figure 5.32: Illustration of the matched model for Laguerre-Gauss mode verification. (a): The SAFARI L-Band
FEM model illustrates how radiation can be lost due to the hemispherical PEC walls directing signal toward the
gap. (b): The matched FEM model depicts the perfectly matched layer that was used in order to try to match what
is being simulated in the Laguerre-Gauss mode model.
radius of the cavity aperture is over eight times that of the waveguide meaning
that higher order modes can propagate if excited. The Laguerre-Gauss free-space
modes with azimuthal dependence α = 0 and α = 2 can couple to waveguide
modes with higher order radial dependence (T E1m ), thus an unknown fraction of
the observed loss may be coupling to these higher order modes. In fact the LaguerreGauss modes with azimuthal dependence α = 2 can also couple to waveguide modes
with azimuthal dependence l = 3 (as discussed earlier in this section), including all
possible radial modes of this azimuthal symmetry T E3m . The second point to be
made is that multiple reflections will need to be considered for an accurate result,
as described above. The hemispherical backshort in the SAFARI L-Band design is
likely to direct a large amount of the signal toward the gap (based on ray tracing),
leading to the significantly greater levels of loss shown in Figure 5.31 for the FEM
model of the SAFARI L-Band detector. Correctly accounting for the vacuum gap
using the mode matching approach will clearly require a significant extension of the
existing SCATTER code. The FEM modelling approach was therefore employed for
the analysis of the SAFARI L-Band and S-Band pixels presented in the following
sections.
Ideally the SAFARI receivers would have no gap between the feed aperture
and the cavity, but the absorber is typically situated at the entrance of the cavity
structure, or even a small distance in front of the entrance, and a minimum distance
must be left between the feed horn array block and the sensitive TES detector array
223
chip. The value for this minimal distance is quoted by the SRON team as 30µm
and therefore a vacuum gap of this length is required in the current designs. A new
design that would allow for the removal of the vacuum gap is discussed in section
5.5 of this chapter, where simulations of the SAFARI S-Band are presented.
5.4.2
COMSOL Model of SAFARI L-Band Detectors
Since including the vacuum gap in the SCATTER mode matching code is still a work
in progress in terms of modelling the SAFARI L-Band, the current work focussed
on modelling the detector in the most computationally efficient manner achievable
with COMSOL. It was possible to model the system with the standard 3D COMSOL
approach due to the single mode nature and relatively modest size of the detector, so
this is the first method that was used to model the L-Band. However knowing that
the SAFARI S-Band is both a multi-moded system, and that the high frequencies
involved give rise to the structure being very large in terms of wavelength, the more
efficient (but less accurate) 2D COMSOL approach was also used to model the LBand so that it could be determined if the results could still be used to extrapolate
the approximate corresponding 3D behaviour for more complex structures.
A model was created in the COMSOL 3D environment with dimensions corresponding to the L-Band pixel, as illustrated in Figure 5.27 at the beginning of this
section. The T E11 mode was excited across the entry aperture of the waveguide
with a total power of 1 Watt. The surface impedance of the absorber sheet was
Ω
again set to the idealised value of 377
and a frequency sweep was simulated from
1.5 to 3.0THz for 200 spot frequencies. A similar model was then created in the
2D planar symmetry COMSOL environment that represented a 2D cut through the
centre of the 3D model in terms of geometry, however in terms of the physics the 2D
model actually represented a cut through the model illustrated in 5.33(b). The same
frequency sweep was analysed for the 2D model and the results are shown in graphs
below. Figure 5.34 shows the fractional power that is absorbed by the resistive sheet
for both models and the HFSS data that was produced by Dr. Phil Mauskopf from
Cardiff is also shown for comparison [Mauskopf, 2011]. The scaling factor calculated
for the 2D data, based on comparing the ratios of areas for the power flow surfaces
for both the 2D model and the extruded 3D Cartesian coordinate model, was 0.72
(ratio of absorber areas).
224
As can be seen there is excellent agreement between the COMSOL 3D and
HFSS data. There appears to be a higher level of structure in the COMSOL 3D
data compared to HFSS and this may be due to the time-domain approach (transient solver) used for the HFSS analysis in contrast to the frequency domain solver
used with COMSOL. If a sufficiently long time period was not allowed for the HFSS
calculations then this would transform into lower resolution in the Fourier transformed frequency spectrum and thus less structure. The Cardiff University report
good agreement between measurements and their HFSS models [Mauskopf, 2011]
and there is clearly very good agreement with the COMSOL results discussed here.
The corrected COMSOL 2D data is in agreement with the general trends seen
in the HFSS and COMSOL 3D data sets, with the largest differences occurring at
the higher frequencies and certainly less structure observed in the data. Again,
since the symmetry in the 3D models is cylindrical, then the 2D model is a gross
approximation and we would expect differences between the two models.
(a) 2D Model
(b) Symmetry Represented
(c) 3D model
Figure 5.33: Illustration of the 2D COMSOL model of the SAFARI L-Band and what it relates to in terms of its
3D analogue. (a): Illustration of the 2D model, (b): The one dimensional translational symmetry represented in
the 2D COMSOL model, and (c): The cylindrical 3D model of the SAFARI L-Band pixel.
It is clear from Figure 5.34 that with a vacuum gap included in the model the
efficiency of the absorber is significantly reduced when compared to a closed system,
such as those examined in the previous section. For this reason the two sources
of loss were separately examined in order to quantify how much power loss is due
leakage via the vacuum gap and how much is due to returned power back through
the input waveguide. Following the calculations across the frequency range, the
post-processing mode in COMSOL was used to first analyse the losses through the
225
Ω
in order to
Figure 5.34: Effiency of the SAFARI L-Band detector with the absorber surface impedance set to 400
match the Cardiff models. There is very good agreement between the COMSOL and HFSS results. The corrected
2D data displays some differences with lower efficiency at the higher frequencies, but the peaks and troughs still
allign to a good degree of accuracy.
vacuum gap. The time averaged power outflow parameter was integrated over the
area that contained the vacuum gap section. This was done for each frequency value
and the results are shown in Figure 5.36. Again there is clearly very good agreement
between the two 3D models in COMSOL and HFSS, although there is again more
structure in the COMSOL data, and there is surprisingly reasonable agreement with
the corrected 2D data also. The correction factor used for the vacuum gap loss data
was π which again arises due to the 2D model being representative of a slice through
a Cartesian geometry model with a length of 1 metre. The result is the vacuum gap
power loss being spread over the area of a rectangle of dimensions 2rh where r is the
distance from the centre to the edge of the cavity and h is the length of the vacuum
gap between the horn and cavity (the rectangle highlighted in green in Figure 5.27
shown earlier. The power loss should really be integrated over the area of a cylinder
(ignoring the top and bottom circular faces) with dimensions 2πr × h where again
r is the initial radius of the cavity and h is the vacuum gap length. The ratio of
the areas of the power flow surfaces for the two geometries is
2πr×h
2rh
= π as shown in
Figure 5.35.
The scaling factor for the amount of power returned back through the wavegide for the 2D model was determined to be 0.4 and it was calculated in a similar
226
(a) 2D Free-Space Boundary
(b) 3D Free-Space Boundary
Figure 5.35: Comparison of areas of the free-space boundaries in the 2D and 3D models.
manner to that described earlier for the closed waveguide cavity. The return power
is calculated from the S11 parameter that analyses the fraction of the electric field
that is reflected back onto the input waveguide port where the T E11 mode was
excited. Again Figure 5.37 shows good agreement between the 3D COMSOL and
HFSS results, and some level of agreement with the corrected 2D data. The next
step taken was to try to optimise the absorber/cavity configuration of the SAFARI
L-Band detector in order to maximise the coupling of power to the absorber.
Figure 5.36: Vacuum gap loss of the SAFARI L-Band detector. Again there is very good agreement for all three
approaches. The corrected 2D data again shows some differences with slightly higher levels of radiated power at a
few of the test frequencies. The general trend is again well aligned with the other two sets of results.
227
Figure 5.37: Return loss (S11 ) of the SAFARI L-Band detector. Once again there is very good agreement between
all three approaches.
5.4.3
Improving the SAFARI L-Band Design
In the simulations for generic systems presented in section 5.3.4 the cavity backshort
geometry that performed best in terms of maximising the amount of power absorbed
by the resistive sheet absorber was the sine profiled shape. Based on this result the
sine profiled backshort design was substituted for the hemisphere in the SAFARI LBand model, and using the full 3D COMSOL environment the system was simulated
for the same frequency range of 1.5 to 3.0 THz. All other parameters of the model
and dimensions of the structure were the same as for the model presented above, and
a screen shot of the E-plane cut of the electric field for the sine profiled backshort
at 2.2THz is shown in Figure 5.38.
Figure 5.38: Screen shot of the electric field through the E-plane of a SAFARI L-Band pixel at 2.2THz.
By employing the sine profiled cavity backshort the average amount of power
228
absorbed by the resistive sheet was increased from 0.54 to 0.59. Although this is
only an increase in efficiency of 10%, perhaps what is more important is that the
standard deviation in the range of efficiency values across the frequency band was
reduced from 0.22 to 0.15. This indicates a good improvement in consistency across
the band which is also highly beneficial for these detectors. The results are shown in
Figure 5.39 with the hemisphere cavity absorber data shown again for comparison.
Figure 5.39: Absorber efficiencies for two different SAFARI L-Band cavity designs. It is clear that the sine profiled
cavity has overall better performance with less variance across the band. Both systems were modelled with the full
3D EM COMSOL environment.
Figure 5.40 shows the losses through the vacuum gap that connects neighbouring pixels. Again there is clear improvement with the sine profile cavity design over
the hemisphere. The peak fractional loss is reduced by half from 0.7 to 0.35, and
there is again a significant reduction in how the loss varies across the band with
the standard deviation on the data being reduced by more than half from 0.185 to
0.077.
The change in design for the shape of the cavity backshort for the L-Band pixel
clearly has an impact on the efficiency of the Ta absorber sheet in terms of coupling
to the incoming signal. It should be pointed out that the redesign of the backshort
was limited in this case as the depth of the cavity was constrained to 250µm for
the purposes of maintaining a relative comparison to the hemispherical design of
the same depth. If the depth was permitted to vary, or alternatively the particular
sine function used was allowed to vary, then the improvements could possibly be
229
Figure 5.40: Losses through vacuum gap for two different SAFARI L-Band cavity designs. Both systems were
modelled with the full 3D EM COMSOL environment.
increased further.
Another improvement to the design of the L-Band receiver is brought about by
removing some of the absorber material in a manner that serves to reduce its volume
while not significantly impacting on the amount of power it can absorb across the
desired frequency range. The reason for wanting to reduce the volume of the absorber has to do with the speed of response of the detector and this was discusses by
Stafford Withington in a presentation given to the Virtual Institute of Astroparticle
Physics, Paris, at the Workshop on Microwave Spectral Polarimetry in December
2012 [Withington, 2012]. In his presentation he discusses two volume reducing designs they have been working with at his research institute at the Cavendish Laboratory, Cambridge University. One design consists of a mesh of absorber strips in the
two perpendicular directions while the other consists of strips in just one direction
(see Figure 5.41). The aim is to reduce the overall volume of the absorber whilst
maintaining the optical cross section. As the volume is decreased the maximum
speed of response of the detector is increased (time constant decreased) and this
can be understood by considering how the effective time constant τef f changes with
heat capacity, as can be seen from equation 5.26 [Mauskopf et al., 2010].
τef f =
C
G (1 + R)
230
(5.26)
(a) Solid
(b) Mesh
(c) Stripped
Figure 5.41: Photographs of TES’s coupled to the Ta absorbers. (a): A solid sheet of Ta, (b): A mesh design, and
(c): A striped design.
where C is the heat capacity of the TES and the absorber, G is the heat conductance
to the bath, and R is the electro-thermal feedback loop gain.
The heat conductance G also appears in 5.26 and the paper just cited by
Mauskopf et al. shows that in order to achieve the required detector NEP of 2 −
√
3 (×10−19 W/ Hz), with a detector operating at 100mK, then the heat conductance
must be G ≤ 2 × 10−13 W/K. This small value calculated for G means that the heat
capacity must be kept to a minimal value. The formula used to calculate this value
for G is shown in 5.27.
N EP =
p
4γkB T 2 G
(5.27)
where T is the operating temperature of the TES, kB is Boltzmann’s constant, and
γ is quoted as ≈ 0.5 [Jackson et al., 2012].
The new design for the Ta absorber sheet was basically a two dimensional
version of the fractal geometry Menger sponge [El Naschie, 2013]. This 2D format
is also known as a Sierpinski carpet [Crownover, 1995] and can be thought of as a
slice through the centre of the Menger sponge. Examples of both the Menger sponge
and Sierpinski carpet are shown in Figure 5.42. The rules for constructing the 2D
Sierpinski carpet are as follows:
(i) Divide any square into nine equal congruent sub-squares forming a 9×9 grid. (ii)
Remove the central sub-square. (iii) Repeat recursively for each of the remaining
sub-squares for as many iterations as desired.
A similar set of rules is defined for constructing a Menger sponge from an ini-
231
tial cube. An interesting result of the 3D Menger sponge is that as the number of
iterations increases to extremely large values the volume of the sponge approaches
zero while the surface area approaches infinity. There are many other examples of
fractal reduction topologies but it was the Sierpinski carpet that was the obvious
choice since the absorber in the SAFARI L-Band is square. In fact fractal geometry
structures have been shown to be the ideal solution for broadband receivers where
the efficient detection of a large range of frequencies is desired for a relatively small
antenna [Kadir, 2007]. Fractal designs can achieve this due to the self-similarity
that naturally arises from fractal structures [Hohlfeld and Cohen, 1999]. Some mobile/cell phones now use antennas with either Sierpinski carpet or Sierpinski triangle
geometries.
(a)
(b)
Figure 5.42: Examples of fractal geometry in 2D and 3D cases. (a) A cube having undergone 3 iterations of fractal
reduction resulting in what is generally known as a Menger sponge denoted as M3 . (b) A square having undergone
4 iterations of fractal reduction resulting in what is generally known as a Sierpinski carpet denoted by M4 . An
unaltered cube or square is denoted M0
The Sierpinski carpet absorber sheet was built in the COMSOL 2D environment
and then imported into the 3D model of the SAFARI L-Band in place of the solid 2D
absorber that was modelled previously. Absorbers with a range of iterations were
designed from one iteration (removal of the central square) to four iterations. Some
of the designs are shown in Figure 5.43. The results of the three and four iteration
absorbers are shown in Figure 5.44 along with the results from the solid sheet model
for comparison. The average power absorbed by the solid sheet across the band was
0.54. The average absorption across the band for the three and four iteration fractal
232
absorbers was 0.46 and 0.43, respectively. This is a clearly a reduction in absorber
efficiency but not a drastic decrease. The equation to calculate how much of the
n
absorber area remains after each iteration is simply A = 89 where n is the number
of iterations employed. So for the absorber with four iterations of fractal reduction
the remaining absorber area is 0.62 times that of the solid sheet. This method of
removing material from the absorber appears to be a better choice as the amount of
power absorbed by the striped design for example reduces by half [Mauskopf, 2011].
This 50% reduction is a result of only one polarisation of the signal being detected;
that which is polarised parallel to the absorber strips.
(a)
(b)
(c)
Figure 5.43: A CAD design of a 2D absorber with (a): Two levels of fractal subtraction, (b): three levels of fractal
subtraction, and (c): four levels of fractal subtraction.
Figure 5.44: Effiency of the SAFARI L-Band detector with a fractal absorber sheet with various fractal iterations.
233
5.5
SAFARI S-Band Receivers
A significant collaborative research effort has been ongoing at both NUIM and SRON
in terms of the multi-moded SAFARI S-Band (short-wave band) as was described
in section 1.2.2 of chapter 1. The research team at SRON led by Dr. Gert de
Lange have built a SAFARI testbed and they have undertaken a test campaign with
many cool down and warm up cycles where at each warm up they can modify the
system and/or add components and then upon re-cooling the measurements can begin again. The current detector configuration employs the traditional hemispherical
cavity backshort housing the bolometer being fed by a multi-moded smooth walled
conical horn [Audley et al., 2013]. Each bolometer consists of a superconducting
TES with a transition temperature of approximately 100 mK coupled to a Ta absorber on a thermally-isolated silicon nitride island. They report base temperatures
as low as 8 mK on the mixing chamber, and detector bath temperatures as low as
15 mK for dark measurements, which is significantly lower than the temperatures
required for testing the SAFARI detectors. This leaves room for heat loads such as
the blackbody source they use to illuminate the detector for optical measurements.
During the three months work placement at SRON the author of this thesis
attended regular weekly project progress meetings of the SAFARI testbed team
and gained a good understanding of the system. Some of the work performed by
the author at SRON during this period concerning characterisation of SAFARI-like
THz multi-moded feed horns is discussed in chapter 3. This section is concerned
with a new design for the cavity backshort for the S-Band detectors and results of
simulations for this new design.
The research team at NUIM has been making significant progress in modelling the current SRON SAFARI testbed, and collaboration between the two institutes is ongoing and funded through an ESA technical development contract
[RFQ 3-6418/11/NL/CBi]. Stephen Doherty (post-doctoral researcher at NUIM)
has been carrying out most of the simulation work on the current system and the
author of this thesis has contributed to this effort through the modelling of alternative geometries for the S-Band cavities.
During the period of work by the author at SRON a concept for a new cavity design was discussed with Dr. Damian Audley and Dr. Gert de Lange, both of SRON.
234
The concept was to use anisotropic silicon crystal etching as an efficient and accurate method for creating the integrating cavities required for the SAFARI S-Band
array. This process uses a chemical etching technique that relies on the different
chemical reactivities of different crystal planes in a silicon crystal [Alvi et al., 2008].
This plane dependent reactivity can be exploited through the use of specific chemical agents that etch faster is one direction over others, allowing the user to etch
geometrical shapes such as pyramids.
There is a large amount of literature on the topic of anisotropic silicon crystal
etching as it is regularly used in the bulk manufacture of micrometer scale electronic
components on integrated circuits (see for example [Kovacs et al., 1998]). Whereas
isotopic etching will result in a hemispherical shaped cavity being formed on a flat
silicon substrate due to the chemical agent reacting in all directions at the same rate,
anisotropic etching will typically result in either square shaped trenches being etched
into the flat substrate, or pyramidal shaped structures, depending on the nature of
the chemical agents, how dilute the reactants are, and a number of other factors such
as which crystal plane is exposed on the surface of the substrate, i.e the orientation
of the crystal. For example, in his PhD thesis, Samuel Pollock discusses that certain
alkali hydroxides exhibit an etch rate significantly higher for the (100) and (110)
directions than for the (111) direction [Pollock, 2010]. His work with this etching
process involves building atom traps with the aim of performing measurements with
cooled trapped atoms, which is obviously a very different application to the work of
this thesis. He states that potassium hydroxide (KOH) etches the (100) plane 400
times faster than the (111) plane, while the (110) plane is etched 600 times faster.
This results in the etching on the (111) planes being insignificant and the (111)
planes are easily revealed and expose themselves as the faces of a pyramidal shaped
cavity (see Figure 5.45). The walls of the pyramid make an angle of 54.74o with
the flat substrate resulting in a pyramid with an apex angle of approximately 70.5o .
The resulting pyramidal cavity has a surface roughness significantly less than can
be achieved with any mechanical milling process on these micrometer scales. This
is particularly critical given the the small dimensions of the cavities and the short
wavelengths involved. The manufacturing process is also a lot easier than other
methods.
Another benefit of employing this approach is the possibility of including a
235
Figure 5.45: Examples of geometries that result from various chemical etching techniques.
second pyramidal or square cavity section above the TES and absorber that would
link the integrating cavity with the feed horn array, thus removing the vacuum gap,
as shown in Figure 5.46. The overall cavity would therefore have an octahedron
geometry if two pyramidal structures were used, or another closed geometry if a
pyramid and cuboid were used. The wiring for the TES readout electronics could
be located within trenches that could be etched into the silicon wafer through the
chemical etching process. The straight, angled walls of the pyramidal cavity would
be similar to the almost straight angled sections of the sine profiled cavity design for
the SAFARI L-Band detectors discussed in section 5.4.3 of this chapter. In that case
the angled walls of the sine profile were shown to be a good solution for coupling
radiation to the resistive sheet; the result being an increase in absorber efficiency
and with a higher level of consistency across the band while reducing the losses
through the vacuum gap between the waveguide feed and the cavity backshort.
Before anything was built it was desired to know how much of an improvement
such a design would make over the current hemispherical design with the vacuum
gap. A 3D model of the double pyramid design was built in the COMSOL design
environment and a range of simulations were performed in order to investigate the
efficiency of the absorber sheet within this structure. Various absorber sizes were
modelled across the large frequency ranges corresponding to the operational range
of the SAFARI S-Band. A screen shot of the COMSOL model is shown in Figure
5.47 where the dimensions of the cavity structure are defined. These dimensions
give rise to the ≈ 71o angle that would result from appropriate anisotropic etching.
236
Figure 5.46: Comparison of an open hemipshere cavity and a closed double-pyramid cavity. This CAD drawing
clearly shows the vacuum gap in the hemispere design and how it would be removed in the double pyramid design.
Figure 5.47: Screen shot of pyramidal cavity model showing the dimensions of the structure.
5.5.1
Single Mode Analysis
As a first step the new S-Band cavity design was simulated using only the T E10
mode as an input via a square waveguide section with a length of 200µm and with
the same transverse dimensions as the cavity entrance, i.e. 100 × 100µm. Five
different absorber sizes were modelled, ranging from 50 × 50µm to 250 × 250µm,
with the absorber positioned in the centre of the cavity in each case. The fraction
237
of power absorbed as a function of frequency is shown for three of the absorber
sizes in Figure 5.48(a), and the average power across the band is shown in 5.48(b)
where it can be seen that there are two values for each absorber size. The blue points
represent the average efficiency across a range of 1.8THz (the cut-on frequency of the
square waveguide feed) to 10THz. The green points then correspond to the average
efficiency across the operational range of the SAFARI S-Band (4.3 - 10THz).
(a)
(b)
Figure 5.48: Effiency of the SAFARI S-Band resistive sheet absorber fed by a single mode (T E10 ) for various
absorber sizes. (a): The efficiency of a resistive sheet absorber, for a frequency range of 1.5 to 10THz, for three
different absorber sizes. The actual frequency range of SAFARI is indicated by the red box on the x-axis. (b): The
average efficiency across the band for five different absorber sizes.
It is clear that the removal of the vacuum gap from the design has a significant
impact on the amount of power that couples to the absorber. Also, for an absorber
size above 100 × 100µm there appears to be roughly a linear relationship between
absorber area and average power absorbed, while reducing the absorber sheet size
to 50 × 50µm results in a significant reduction in absorbed power. Figure 5.49 then
shows the efficiency profile for just two absorber sizes across the SAFARI S-Band
range where the frequency dependent behaviour of the absorbers is a bit clearer in
this expanded, less cluttered plot.
5.5.2
Analysis of Higher Order Modes
Having found that the resistive sheet absorber behaves well inside the double pyramid cavity when fed by the fundamental T E10 mode, the system was then inves238
Figure 5.49: Effiency of the SAFARI S-Band resistive sheet for two different absorber sizes. The behaviour across
the frequency range of the SAFARI S-Band (red box in Figure 5.48) is clearer in this graph.
tigated for excitation by the higher order modes that the waveguide feed could
support. Average efficiencies for each mode were calculated for an absorber size of
100×100µm across two frequency bands; the first corresponding to the full frequency
range of the SAFARI S-Band instrument, and the second range that accounted for
the cut-off frequency of each propagating mode. In the second instance the average
efficiency was calculated over the range from the switch-on frequency for the particular mode to the high end of the S-Band (10THz), and in this way the evanescent
behaviour of the higher order modes at the lower frequencies could be accounted
for. In fact this highlights an important question regarding multi-moded systems
such as SAFARI, where the efficiency can be ill-defined, and for this reason the band
average efficiency was calculated in the two ways just described.
As can be seen in Figure 5.50(b) and Table 5.2 the efficiency of the relatively
small 100 × 100µm absorber in the double pyramid cavity is very reasonable for each
higher order mode that the waveguide feed supports. There is of course symmetry
in the data with respect to the m = n diagonal of Table 5.2 simply due to the square
nature of the waveguide feed which would not be true for a rectangular guide.
239
T Emn
n=0
n=1
n=2
n=3
n=4
n=5
n=6
m=0
/
0.795
0.653
0.739
0.717
0.850
0.542
m=1
0.795
0.803
0.702
0.753
0.742
0.778
0.651
m=2
0.653
0.702
0.616
0.714
0.666
0.816
0.548
m=3
0.739
0.753
0.714
0.765
0.691
0.777
Cut-Off
m=4
0.717
0.742
0.666
0.691
0.636
0.621
Cut-Off
m=5
0.850
0.778
0.816
0.777
0.621
Cut-Off
Cut-Off
m=6
0.542
0.651
0.548
Cut-Off
Cut-Off
Cut-Off
Cut-Off
Table 5.2: Average absorber efficiencies across the propagation frequency range for higher order modes in a pyramidal
cavity. The remaining power is lost as return power back through the waveguide since there is only one port and
PEC boundary conditions were in place meaning that no Ohmic heating can take place in the walls.
(a)
(b)
Figure 5.50: Effiency of the SAFARI S-Band resistive sheet for higher order modes. The size of the absorber chosen
was 100 × 100µm. (a): The efficiency of each propagating mode averaged over the entire SAFARI S-Band range
(4.3 - 10THz). (b): The efficiency of each propagating mode averaged over a range from the cut-on frequency of
each specific mode to 10THz (propagation frequency range).
5.5.3
Experimental Verification
Having found the new S-Band cavity design to be well behaved as predicted by the
models, the next step involved designing and building a scaled model of the new
SAFARI S-Band pixel for testing at frequencies around 100GHz for verification of
the simulation approach. Experimental tests also serve to validate the modelling
approaches used for the SAFARI L-Band analysis presented in section 5.4 of this
chapter. The system used for the measurements was based on the VNA.
As already indicated in section 1.4 of chapter 1, the VNA system at N.U.I.
240
Maynnoth has a frequency range of 75 - 110 GHz, and a dynamic range that depends on the care taken with the calibration procedure, but with 70dB achievable.
The benefits of performing tests at these scaled frequencies is that there is little
atmospheric absorption of signal due to the atmospheric window around 100GHz,
and of course any thermal emission at these frequencies is greatly dominated by the
strength of the VNA signal, thus there is no need for vacuum systems or cryogenic
facilities. However, as a precaution, the temperature of the VNA laboratory was
kept constant to within ±0.5o . A further benefit of the scaled tests is based on the
dimensions of the scaled cavity and feed structures. Their sizes are clearly larger
at longer wavelengths meaning that standard machining processes can be employed
for their construction if care is taken and precision is maximised. Also, since the
ability of the thin deposit of Ta to absorb radiation is not frequency dependent it
was believed that the scaled measurements would provide valid data.
The disadvantage of this approach is that the results on the absorption of power
are not obtained by measuring the absorbed power directly. Rather, by measuring
the S11 parameter (return signal) of a one port VNA set-up, the efficiency of the
absorber is deduced. Essentially, any of the input power that is not observed to
return is assumed to have been absorbed by the Ta absorber. The Ta absorbers
used in the real SAFARI pixel will of course be coupled to ultra-sensitive TES’s,
which themselves will be coupled to superconducting quantum interference devices
(SQUIDs) for amplification, resulting in direct detection of the incoming signal.
Figure 5.51 shows a cut through the CAD design, and a photograph of the
scaled cavity that was designed by the author and then built by Mr. David Watson
in the departmental workshop at NUIM. The pieces of Ta absorber material shown
in Figure 5.52 were provided by the SRON. Each absorber consists of a layer of
Ta (≈ 6nm) deposited on a 0.25mm thick sapphire substrate. It was possible to
sandwich the largest absorber square between the top and bottom parts of the
cavity in order to fix it in position. The smaller pieces of Ta had to be mounted in
the cavity using fine thread as supporting legs, as shown in Figure 5.52.
Figure 5.53(a) shows the initial results for three different absorber sizes in the
cavity, together with a FEM simulation of the 5.0 × 5.0mm absorber. Figure 5.53(b)
shows the results of two measurements that were performed to investigate if the
thread had an effect in terms of absorbing power. Placing thread cross-hairs in the
241
(a)
(b)
Figure 5.51: The scaled pyramidal SAFFARI S-Band cavity that was designed and built at NUIM. (a): A cut
through the CAD design. (b): Some photographs showing how it was manufactured in separate pieces.
(a)
(b)
Figure 5.52: Pieces of Ta absorber that were provided by SRON for the scaled VNA measurements at NUIM.
(a): The absorber pieces with a scale bar. (b): The absorber pieces showing the thread that was attached to the
absorbers for mounting in the cavity.
cavity results in an almost identical profile to an empty cavity as can be seen from
the comparison plots.
Although the data in Figure 5.53(b) indicates that the thread has no significant
effect on the amount of power absorbed in the cavity, it is clear that the empty cavity
is causing significant losses as can be seen from the apparent absorbed power. One
would expect to observe almost 100% return power from an empty cavity since the
structure is closed apart from the excitation/detection port. The reason for the
low return power is thought to be due to the excitation of higher order modes,
caused by the shape of the cavity. Only the T E01 mode is excited in the input WR-
242
(a)
(b)
Figure 5.53: Measured efficiency of square Ta absorbers in a double pyramid cavity, and empty cavity excitation.
(a): Measured power absorbed for three different absorber sizes. The model results are for a 5.0 × 5.0mm absorber.
(b): Excitation of empty cavity with and without thread cross-hairs.
10 waveguide, however the break in symmetry inevitably causes power to scatter
into higher order modes inside the cavity. The higher order modes cannot return
back through the WR-10 guide as the dimensions of the guide give rise to a cutoff frequency of 118GHz for even the first higher order mode (T E02 ). Since the
analysis is being performed for a frequency range of 75 to 110 GHz even the highest
operational frequencies cannot propagate back through the guide in the form of
higher modes. The multiple reflections from the walls of the cavity for these modes
cause them to be absorbed.
The COMSOL model was based on PEC cavity boundaries, meaning that no
power could be absorbed by the walls, and an empty cavity returned 100% of the
power, as expected. The simulations were performed again, with the PEC cavity
walls replaced with boundaries having the properties of aluminium. The difference
in the return power with the more realistic boundary conditions was negligible,
meaning that the absorption of power observed in the measurement cavity must be
due to some other parameter, most likely the surface finish on the cavity walls.
An attempt was made to correct for the cavity losses by using the empty cavity
data to rescale the absorber results. It was somewhat successful, although there is
still significant disagreement between the simulations and the measured data. The
reason for this is due to how the absorber itself affects the behaviour of the fields
inside the cavity. The cavity losses in an empty cavity will not be the same as the
cavity losses when an absorber is present, since a significant amount of the signal
243
will be absorbed on its first pass through the Ta sheet. The higher order modes
are now preferably absorbed by the Ta absorber rather than the cavity walls. Thus
subtracting the losses observed in the empty cavity from the absorber measurements
is not an accurate approach.
Figures 5.54(a), 5.54(b) and 5.55(a) show the results for the apparent efficiency
of three different sized square Ta absorbers compared to corresponding FEM simulations for each absorber size. As can be seen in each case, the amount of power
absorbed in the measured data is significantly higher than that predicted by the
simulations. Figures 5.55(b), 5.56(a) and 5.56(b) show the corrected results, again
compared to the FEM simulations. In this case a scaled version of the bare cavity
data was used to flatten the results, where the scaling factor used was 0.45, which
gave rise to the best observed fit.
(a)
(b)
Figure 5.54: Efficiency of square Ta absorbers in a double pyramid cavity. (a): Fractional power absorbed for a
large absorber (8.0 × 8.0mm) filling the cavity cross section. (b): Fractional power absorbed for a medium sized
absorber (5.0 × 5.0mm) partially filling the cavity.
Since the shape of the pyramidal cavity was thought to be the cause of the
disagreement between simulation results and those of the particular single mode
measurement approach used, another set of absorber measurements were carried
out using a more symmetric cavity. A cylindrical cavity that was designed and built
previously by the THz Optics Group at NUIM was used, as it was believed that this
particular cavity would give rise to fewer higher order modes being generated, and
hence physical conditions that are more accurately suited to using the return power
as an indicator of power absorbed by the Ta sheet. The cylindrical cavity is shown
in Figure 5.57. A variety of similar rings as those shown in Figure 5.57(a) can be
244
(a)
(b)
Figure 5.55: Efficiency of a small square Ta absorber in a double pyramid cavity. (a): Fractional power absorbed
for a small absorber (2.5 × 2.5mm) partially filling the cavity cross section. (b): Fractional power absorbed for a
small absorber (2.5 × 2.5mm) corrected with data from an empty cavity measurement.
(a)
(b)
Figure 5.56: Normalised efficiency of square Ta absorbers in a double pyramid cavity normalised with data from
an empty cavity measurement. (a): Fractional power absorbed for a large absorber (8.0 × 8.0mm) filling the cavity
cross section. (b): Fractional power absorbed for a medium sized absorber (5.0 × 5.0mm) partially filling the cavity.
used to construct cylindrical cavities of a variety of depths and radii. The rings are
held in place in the holding tube also visible in the top left of Figure 5.57(a). The
absorber pieces can then be held in place by sandwiching the thread between two of
the rings as shown in Figure 5.57(b), where the carpet absorber [Crownover, 1995]
is shown mounted above a ring.
The dimensions of the particular cavity assembly that was used is illustrated in
Figure 5.58(a), and a screen shot of the COMSOL FEM model is shown in Figure
5.58(b) indicating the symmetry conditions used in order to reduce the problem size.
Figure 5.59 then shows the model following the meshing procedure, and finally the
245
(a)
(b)
Figure 5.57: Photographs of the cylindrical cavity used for the VNA absorber measurements. (a): The cavity, rings,
and front face showing the WR-10 waveguide input and screw holes for attachment. (b): The carpet absorber
mounted above one of the rings.
electric field magnitude through the structure. Figure 5.59(c) shows a comparison
between the apparent power absorbed by the empty pyramidal cavity and the empty
cylindrical cavity. It is clear that there are significantly lower losses in the cylindrical
cavity compared to the double pyramid cavity, although the levels are still relatively
high.
(a)
(b)
Figure 5.58: Schematic and screen-shot of the COMSOL FEM model of the cylindrical cavity and absorber set-up.
(a): Schematic showing dimensions of cavity and waveguide feed structures. (b): The geometry of the model showing
the E-plane and H-plane symmetry conditions that were employed.
The results of the absorber measurements in the cylindrical cavity are shown in
Figures 5.60 and 5.61, where Figure 5.60(a) shows the results of a circular absorber
completely filling the cavity cross section, and as can be seen there is a much higher
246
(a)
(b)
(c)
Figure 5.59: Screen-shot of the meshed model and the electric field from the cylindrical cavity and absorber FEM
model. (a): The model geometry following meshing, and (b): The electric field. (c): A comparison between the
apparent power absorbed by the empty pyramidal cavity and the empty cylindrical cavity.
level of agreement between measurement and model. Figures 5.60(b), 5.61(a), and
5.61(b) show the results of the 8.0 × 8.0, 5.0 × 5.0, and 2.5 × 2.5 absorbers, respectively, in the cylindrical cavity. Again, although there is some disagreement
between measurement and model, the agreement is significantly better than for the
pyramidal cavity, as predicted.
(a)
(b)
Figure 5.60: Efficiency of square Ta absorbers in a cylindrical cavity. (a): Fractional power absorbed for a large
circular absorber filling the cavity cross section. (b): Fractional power absorbed for a large square absorber (8.0 ×
8.0mm) partially filling the cavity.
The carpet absorber, where a central square (2.66 × 2.66mm) was removed
from the 8.0 × 8.0mm Ta square in its construction, was tested and compared to the
efficiency of the solid 8.0 × 8.0mm Ta square. The reasons for wanting to remove
some of the absorber volume were discussed in section 5.4.3 of this chapter. Figure
5.62(a) shows the measurement results, with the corresponding FEM simulation
247
(a)
(b)
Figure 5.61: Efficiency of square Ta absorbers in a cylindrical cavity. (a): Fractional power absorbed for a medium
square absorber (5.0×5.0mm) partially filling the cavity. (b): Fractional power absorbed for a small square absorber
(2.5 × 2.5mm) partially filling the cavity.
results also shown, and the measurement results for the solid absorber are shown
again for comparison. It is clear that removing some of the absorber material in
this manner has very little effect in terms of reducing the efficiency of the absorber.
This was also found in the simulations presented in section 5.4.3 of this chapter.
(a)
(b)
Figure 5.62: (a): Efficiency of a large square carpet absorber in a cylindrical cavity, where the carpet has dimensions
of 8.0 × 8.0mm with the central square (2.66 × 2.66mm) removed. (b): Average power absorbed across the band as
a function of absorber size.
Figure 5.62(a) then shows the average power absorbed across the frequency
band plotted against increasing absorber size. As is expected the average absorbed
power typically increases with absorber size, with the VNA measurements again
showing a higher level of apparent absorption, indicating other losses as already
248
discussed above. Interestingly the largest absorber, which was the circular piece
filling the cavity cross section (≈ 12mm in diameter) shows a reduction in measured
absorption compared to the smaller 8.0 × 8.0mm. This also brings the result into
closer agreement with the model than for the other absorber sizes. This is further
evidence that it is the break in symmetry that is causing the cavity losses, since the
largest absorber clearly has the same symmetry as the cylindrical cavity, whereas
the square absorber pieces of course differ from the symmetry of the cavity. Thus,
power losses in the walls of the cavity dominate when multiple reflections occur,
whereas the losses occur in the absorber when it is placed in the cavity.
5.6
Conclusions
It was shown in this chapter that in order to fully model a complete SAFARI L-Band
or S-Band pixel, multiple simulation methods are currently required. The progress
being made by the THz Optics Group at N.U.I. Maynnoth in terms of developing
an integrated mode matching software was discussed. Such an integrated program
should be able to account for the conducting waveguide structures of the pixel,
the resistive sheet absorber housed in an integrating cavity, and any vacuum gaps
within the pixel. The extension of the Cylindrical SCATTER mode matching code
to include a resistive sheet absorber in the models was shown to agree well with the
conventional FEM simulation approach. The extended SCATTER code was also
shown to be up to three orders of magnitude faster than FEM simulations.
The particular shape of the integrating cavity backshort in a far-IR pixel was
shown to be very important in terms of detector efficiency. Using the extended
SCATTER code, a cavity backshort with a sine profile was demonstrated as being a
good improvement compared to the conventional hemisphere design. The efficiency
of the SAFARI L-Band pixel was then shown to increase significantly when the
hemisphere backshort was replaced with the sine profiled backshort. A reduction
in power leakage through the vacuum gap was also observed with the new L-Band
design, and thus potential cavity cross-talk was reduced.
A new design for the SAFARI S-Band was shown to be a significant improvement over the current design. As was discussed in section 5.5, the primary cause
of the increased pixel efficiency was the removal of the cavity-to-cavity vacuum gap
249
from the design.
Finally, it was shown that it is possible to use a scaled measurement system
to verify the results of the simulations of cavity mounted Ta absorbers, although
there are restrictions in terms of using structures that do lead to too much power
being scattered into higher order propagating modes. Choosing a structure with
high symmetry and smoothly varying boundaries, such as a cylindrical cavity, was
shown to produce results that are in reasonable agreement with those from the FEM
modelling approach. The FEM results in turn were shown to agree well with the
more efficient mode matching approach in section 5.2.4 of this chapter.
250
Chapter 6
Summary and Conclusions
251
The primary aim of this thesis was to demonstrate that it is possible to model and
characterise the full structure of a SAFARI-like pixel comprising the over-moded
input feed, the absorbing resistive sheet, and the multi-moded cavity backshort,
using a single computer with moderate processing power. This was achieved by
employing multiple simulation techniques, combining the commercially available
packages such as COMSOL for FEM analysis with novel mode matching code. Use
was made of the available in-house mode matching code SCATTER, and further
mode matching code was developed by the author for increased efficiency for specific
horn geometries.
The work of this thesis was primarily driven by two technical research programs (TRPs). The first was a TES TRP funded by ESA, and led by SRON, with
involvement from Cambridge University and RAL in the UK, Cardiff University in
Wales, and NUIM in the Republic of Ireland [TRP AO/1-5922/08/NL/EM, 2011].
The TES TRP was a research program contract to investigate transition edge superconducting bolometers toward development of detectors for the SAFARI instrument
√
for SPICA with sensitivities approaching 1 × 10−19 W/ Hz. The TRP was from
2010 until 2012 and the author contributed to characterising optical cross-talk (hornto-horn) between SAFARI horns as required by the contract, and as presented in
chapter 3 of this thesis. Other work that was carried out by the author as required
by the TES TRP contract involved characterising cavity cross-talk between SAFARI pixels, and determining the optical efficiency of the Ta absorbers suggested
for SAFARI-like detectors. Although the TES TRP has now officially ended (as
of January 2012), a strong collaboration still exists between the research institutes
in the Netherlands, the UK, and Ireland, and work continues on the TES detector
technology toward the SAFARI instrument.
The second TRP that the author was involved in was an Irish AO (Announcement of Opportunity) and was again funded by ESA (Strategic Initiative by ESA
for Ireland) [RFQ 3-6418/11/NL/CBi]. This TRP was specifically aimed at the development of new technology horn antennas for CMB and far-IR astronomy, with
the goal of producing a profiled smooth walled horn antenna consisting of multiple
straight smooth walled conic sections that could match the beam control of simi-
252
lar corrugated structures. Producing such a horn antenna design required the use
of optimisation routines such as genetic algorithms, which must cycle through the
simulation code many hundreds or even thousands of times, to iteratively converge
on an optimum design solution [McCarthy et al., 2013]. As part of this work the
author developed and tested a new conical mode matching code in spherical coordinates that reduced the number of calculations required for each simulation loop
by up to two orders of magnitude. This work is presented in chapter 4 with a description of the new code given and the results of some examples shown. The Irish
AO contract also involved continued work on detector development with SRON for
SPICA/SAFARI, including an extended visit by the author to SRON, Groningen
from June to August of 2013. The measurement work carried out by the author
while at SRON is reported on in chapter 3 of this thesis, and it involves the characterisation of the multi-mode behaviour of SAFARI-like horns at THz frequencies.
Other modelling work performed by the author during the SRON visit is discussed
in chapter 5. This work involved the design and multi-mode analysis of a new twin
pyramid cavity for the highest frequency SAFARI band (SAFARI S-Band). Upon
returning to Ireland, the author then designed a scaled model of the new S-Band
cavity for testing at frequencies around 100GHz with NUIM’s in-house VNA apparatus. The scaled cavity was built by Mr. David Watson of the Experimental
Physics Department of NUIM, and samples of Ta absorber material were provided
for the tests by SRON, Groningen.
The majority of the work in this thesis was carried out by the author. Anywhere
others were involved is clearly indicated, and credit is given as such. The two-port
VNA set-up described in section 1.4 was already in place (see [Yurchenko et al., 2014],
for example), and the precise alignment procedures that were used in preparation
for horn antenna far-field beam pattern measurements described in section 3.1.10
were developed by Dr. Marcin Graziel and Mr. Niall Tynan, B.Sc., both of the department of experimental physics at NUIM. Assistance was provided by Mr. Niall
Tynan, B.Sc. and Miss Niamh Tobin, B.Sc., in performing the cavity and absorber
measurements described in section 5.5.3. Dr. Stephen Doherty provided assistance
with the use of the SCATTER absorber code that was used for analysis presented
in chapter 5, and with the use of the Rectangular SCATTER code that is discussed in section 3.2.1. Parts of the Spherical SCATTER mode matching code
253
developed by the author used the established Cylindrical SCATTER as heritage
[Murphy et al., 2010].
The series of horn throughput measurements that were carried out at SRON,
and described in section 3.3, were carried out under the supervision of Dr. Gert de
Lange, and large assistance was provided by Dr. Damien Audley and Mr. Willem
Jan Vreeling, B.Sc. The measurement system used at SRON was already in place,
and the work carried out by the author built on previous research by Mr. Chris de
Jong, B.Sc., and Dr. Manisha Ranjan. The design concept for the SAFARI S-Band
cavity, and the corresponding FEM simulations, discussed in section 5.5, were the
result of collaborative work by the author, Dr. Gert de Lange, and Dr. Damien
Audley.
In chapter 3 it was shown that the far-field beam patterns of the current SAFARI horn design can be efficiently calculated using the Pyramidal SCATTER mode
matching code [Doherty, 2012], since the electrically large size of the rectangular SAFARI horns leads to a FEM model that is very large computationally. However, the
characterisation of horn-to-horn optical crosstalk required the use of the FEM simulation approach, since the current mode matching code does not allow for analysis
in the near-field, nor for horn-to-horn interactions. It was shown that while the level
of crosstalk in the E-plane of two neighbouring horns can be orders of magnitude
larger than that in the H-plane, the predicted levels of crosstalk in the SAFARI
array are sufficiently low as to not overwhelm the detectors. The FEM simulation approach for crosstalk analysis was verified using experimental measurements
carried out using a two-port VNA system.
It was demonstrated that the frequency dependent throughout of SAFARIlike feed horns can be characterised using an FTS, provided care is taken with
proper calibration of the data. It was shown that the propagating modes can be
observed to switch-on in a step-like manner, as predicted by the geometrical analysis
of the switch-on frequencies of the rectangular waveguide section in the back-toback
horn configuration. Further work is required in terms of refining the measurement
technique, as described in section 3.3.7.
Chapter 4 focussed on the theory of conical waveguide modes, and by developing the new mode matching code Spherical SCATTER it was shown how new
technology smooth walled conic section horn antennas can be modelled in an al254
ternative manner to the approximate cylindrical mode matching approach, where
the number of power scattering integrals required to be performed can be reduced
by up to two orders of magnitude. To date, however, analytical solutions to the
scattering integrals have not been found in the literature or independently, leading
to long calculation times due to numerical integrals. Thus, the Spherical SCATTER
code is currently less efficient than the Cylindrical SCATTER approach. The theory
for the new code was verified by comparing the results of a test horn modelled with
both Spherical SCATTER and the established SCATTER for cylindrical geometries.
Good agreement was demonstrated between the two approaches, and similar good
agreement was shown for a FEM model of the same horn.
Finally, in chapter 5, new designs for far-IR cavity backshorts were investigated, with the aim of improving the SAFARI detector cavity design. By using a
combination of FEM and mode matching analyses the efficiency of the Ta absorbers
for both the SAFARI L-band and S-Band were predicted. It was shown that the
particular shape of the cavity backshort can have a significant effect on the amount
of power absorbed by the Ta sheet when housed in the cavity. The amount of power
lost through the vacuum gap between the horn array block and the detector array
chip was shown to be significant if the gap is too large, resulting in up to 65% loss
of signal for some frequencies of the L-Band. Again, the shape of the cavity was
shown to affect this parameter, with a sine profiled backshort providing a significant
reduction of this loss. Removal of the vacuum gap with a new cavity design was
shown to provide an improvement in the fraction of the signal absorber by the Ta
sheet, and of course removal of any free-space-gap losses, and thus elimination of
any cavity-to-cavity crosstalk. The simulation techniques were then verified through
experimental measurements using a one-port VNA set-up and small absorber pieces
mounted in conducting cavities. The measurement technique that was employed,
where analysis of the return power (S11 ) was used to infer the amount of power
absorbed by the Ta sheet, was shown to work only in special conditions. As was
discussed in section 5.6 care must be taken in choosing the profile of the cavity
structure to be used. The S11 parameter will only inform on the amount of power
returned in the fundamental T E01 mode, and therefore any power that is scattered
into higher order cavity modes will appear to have been completely absorbed by the
Ta, and hence lead to inaccurate inference of absorber efficiency.
255
A list of publications and conference proceedings related to the work presented
in this thesis, and contributed to by the author, can be found below. It is envisaged
that further papers relating to work presented in this thesis will be submitted for
publication in appropriate academic journals, following submission of this thesis for
examination.
256
Publications
[Audley et al., 2013] M. Audley, G. de Lange, and C. Bracken, Detector and Calibrationsource Models for the SAFARI Detector Test Facility, COMSOL Conference Proceedings, WTC Rotterdam, the Netherlands, October 23-25, 2013.
[McCarthy et al., 2013] D. McCarthy, N. Trappe, J.A. Murphy, C. Bracken,
S. Doherty, M.L. Gradziel, and C. O’Sullivan, Efficient Horn Antennas for NextGeneration THz and Millimeter-Wave Space Telescopes, Proc. SPIE 8624, Terahertz, RF, Millimeter, and Submillimeter-Wave Technology and Applications VI,
862413. doi:10.1117/12.2001268. March 27, 2013.
[Murphy et al., 2012] J.A. Murphy, S. Doherty, N. Trappe, C. Bracken, T. Peacocke, and C. O’Sullivan, New Developments in Waveguide Mode-Matching Techniques for Far- Infrared Astronomy, Proc. SPIE 8261, Terahertz Technology and
Applications V, 82610F. doi:10.1117/12.908068. February 9, 2012.
[Murphy et al., 2011] J.A. Murphy, S. Doherty, N. Trappe, C. Bracken, and C.
O’Sullivan, Efficient Modeling of Detectors for Far-IR Astronomy , SPIE Newsroom.
doi: 10.1117/2.1201112.004039. December 9, 2011.
[Trappe et al., 2012] N. Trappe, C. Bracken, S. Doherty, J. R. Gao, D. Glowacka,
D. Goldie, D. Griffin, R. Hijmering, B. Jackson, P. Khosropanah, P. Mauskopf,
D. Morozov, J.A. Murphy, C. O’Sullivan, M. Ridder, and S. Withington, Optical
Modeling of Waveguide Coupled TES Detectors Towards the SAFARI Instrument
for SPICA, Proceedings of the SPIE, Vol. 8452, pp. 84520L-84520L-6, 2012.
[Dirkx et al., 2013] D. Dirkx, T. Bocanegra, C. Bracken, et al.,, Unveiling the
Evolution and Formation of Icy Giants, EPSC Abstracts, Vol. 8, EPSC2013-745-2,
2013, European Planetary Science Congress. 2013.
[Bracken et al., 2014] C. Bracken, J.A. Murphy, S. Doherty, N.A. Trappe, Efficient Modeling of Cavity Coupled Detectors for Terahertz Frequencies, Proceedings
of the 24th International Symposium on Space Terahertz Technology, (In press,
2015).
[McCarthy et al., 2014] D. McCarthy, J.A. Murphy, C. O’Sullivan, N. Trappe,
M.L. Gradziel, S. Doherty, C. Bracken, N. Tynan, and Arturo Polegre, Efficient algorithms for optimising the optical performance of profiled smooth walled horns for
257
future CMB and Far-IR missions, Proceedings of SPIE 9153, Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII, 915316,
23 July 2014. doi: 10.1117/12.2056336.
258
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Adalbert Ding
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Institut Teknologi Bandung
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Centro Brasileiro de Pesquisas Físicas
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University of Nebraska Lincoln
