arXiv:1605.07972v2 [physics.optics] 31 Jul 2016
Topical Review
Nonlinear and Quantum Optics with Whispering
Gallery Resonators
Dmitry V Strekalov1,2 , Christoph Marquardt2,3 , Andrey B
Matsko4 , Harald G L Schwefel2,3,5 and Gerd Leuchs2,3
1 Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, CA
91108, USA
2 Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße
1/Building 24, 90158 Erlangen, Germany
3 Institute for Optics, Information and Photonics, University Erlangen-Nürnberg,
Staudtstr.7/B2, 90158 Erlangen, Germany
4 OEwaves Inc., 465 N. Halstead Str., Suite 140, Pasadena, California 91107,
USA
5 Department of Physics, University of Otago, Dunedin, New Zealand
E-mail: dmitry.v.strekalov@jpl.nasa.gov
2 August 2016
Abstract. Optical Whispering Gallery Modes (WGMs) derive their name from
a famous acoustic phenomenon of guiding a wave by a curved boundary observed
nearly a century ago. This phenomenon has a rather general nature, equally
applicable to sound and all other waves. It enables resonators of unique properties
attractive both in science and engineering. Very high quality factors of optical
WGM resonators persisting in a wide wavelength range spanning from radio
frequencies to ultraviolet light, their small mode volume, and tunable in- and
out- coupling make them exceptionally efficient for nonlinear optical applications.
Nonlinear optics facilitates interaction of photons with each other and with other
physical systems, and is of prime importance in quantum optics. In this paper
we review numerous applications of WGM resonators in nonlinear and quantum
optics. We outline the current areas of interest, summarize progress, highlight
difficulties, and discuss possible future development trends in these areas.
PACS numbers: 42.65, 42.50, 42.79.Nv
Keywords: Microresonators, optical wave mixing, non-classical light
Nonlinear and Quantum Optics with Whispering Gallery Resonators
1. Introduction
1.1. The role of nonlinearity in optical science
Light does not interact with light in everyday life. This
interaction can only be induced by means of nonlinear
physical systems and is only observable with strong
enough electromagnetic fields. It is hardly possible
to achieve such strong fields using incoherent light
sources, so the research in nonlinear optics commenced
with the advent of high power light sources, namely
the invention of lasers. Powerful pulsed lasers are
often needed to observe nonlinear optical phenomena
in bulk materials. Methods involving optical fibers [1],
plasmons [2], hollow-core photonic crystal fibers [3],
as well as metameterials [4–6] allow reducing the light
power requirement. This reduction is still insufficient
for many applications where it is desirable to achieve
a strong nonlinear response with faint light, ultimately
at a few- or single-photon level.
Large nonlinear optical susceptibility is needed to
observe nonlinear interactions at low light levels. There
are two major ways of producing such a susceptibility:
i) usage of a resonant, either natural or artificial,
nonlinear media, and ii) usage of nonresonant but
highly transparent nonlinear media embedded into
optical cavities. Each of these approaches has its
own advantages and disadvantages, and each should be
evaluated in the context of the particular problem to
be solved. In this contribution we focus on the second
approach and discuss applications of monolithic optical
microresonators in quantum and nonlinear optics.
Since their inception, monolithic microresonators
became tools of nonlinear optics. Their major use
is in the enhancement of the efficiency of nonlinear
interactions occurring in transparent optical media.
Unlike other types of optical cavities, monolithic
ones can be integrated on a chip and multiplexed,
which makes them indispensable in creation of chipscale nonlinear optical devices able to generate optical
harmonics, produce nonclassical states of light, process
quantum information and so on. These resonators
allow not only reducing the footprint of nonlinear
optics experiments and moving them from the lab
to industrial applications, but also facilitate nonlinear
interaction at the single-photon level, representing one
of the major goals of optical science nowadays.
The merit of a nonlinear optical system is
often judged with respect to its optical attenuation
2
introducing unwanted optical loss and decoherence.
Resonant nonlinear media, such as atoms or plasmons,
may have huge optical nonlinearity in a relatively
narrow frequency band enabling interaction among
single photons. However, the attenuation is also
resonantly enhanced. Nonresonant nonlinear media,
on the contrary, typically have relatively small optical
nonlinearity, but also very small attenuation. As the
result, photons confined in such a media for a long
time have a better chance to interact without being
absorbed. To support such a long interaction time
optical cavities are utilized.
Cavity enhancement of the nonlinear interaction
depends on the quality factor Q and mode volume
V of the cavity. The quality factor defines the
interaction time, while the mode volume characterizes
the magnitude of the electric field of the confined
photons. The smaller the cavity and the larger the
quality factor, the stronger is the interaction. It is
hard to reach the desirable values of both parameters
at the same time. Usually, reduction of mode volume
results in a decrease of the quality factor. For instance,
the mode volume as small as (λ/n0 )3 (where λ is
wavelength and n0 is the refractive index of the
cavity host material) was achieved in photonic bandgap
nanocavities [7–9], but quality factor does not exceed
2 × 106 . Numerical optimization of the cavity shape
show that Q-factor of a photonic bandgap cavity can
reach 2 × 107 [10] if the fabricated cavity has ideal
quality, but further increase is unlikely. This limitation
arises from fundamental as well as from technical
reasons. The confinement of light within a dielectric
structure is reduced as the structure becomes smaller,
while the tolerances of fabrication of such a structure
become more stringent. It is possible to imagine a
3D photonic crystal nanocavity with a subwavelength
mode volume, but fabrication of the cavity is out of
reach of the existing technology.
Whispering gallery mode resonators (WGMRs)
[11–14], on the other hand, allow achieving three
to four orders of magnitude higher quality factors
at the expense of increased mode volumes. Since
enhancement of the nonlinear conversion efficiency is
usually proportional to Q2 /V or Q3 /V , it may be
reasonable to trade the small mode volume for larger
Q-factor. On the other hand, higher Q results in
narrower linewidth and consequently in slower devices,
so an optimization is usually needed. Another salient
advantage of WGMRs is related to efficient coupling
Nonlinear and Quantum Optics with Whispering Gallery Resonators
techniques developed for the resonators, which are
extremely important in all the applications.
Here we review recent progress in both nonlinear
and quantum optics applications of WGMRs characterized with relatively small (< 106 × (λ/n0 )3 ) mode
volumes and ultra-high Q-factors (Q > 107 ). These
resonators have been widely utilized for efficiency enhancement of nonlinear optical processes such as threeand four-wave mixing, and also for observation of quantum effects associated with these processes. Long interaction time together with strong spatial confinement
of light in nonlinear resonators result in a significant
decrease of the thresholds of both lasing and stimulated scattering processes. Resonators made from
optical crystals are excellent candidates for observation of broadband nonlinear optical phenomena, since
these materials are highly transparent in the wavelength range of 150 nm to 10 µm. Finesse of these resonators can be very large. The highest demonstrated
finesse in optics (F > 107 ) has been achieved with a
fluorite (CaF2 ) resonator [15].
The spectral properties of WGM resonators are
well understood nowadays, and many means of design
and control of WGM spectra are known (see sections
2 and 3). There are also many efficient techniques
for coupling light in and out of these structures
(see section 2). As all spatially extended systems,
WGM resonators require phase matching for nonlinear
interactions to be efficient. Phase matching in rotationsymmetric geometry, however, is very different from
that in Cartesian geometry, and allows for greater
flexibility.
This topic is discussed in sections 4
and 5 for the second- and third-order nonlinearoptical processes, respectively. Interaction of optical
photons with solid-state excitations such as phonons
is discussed in section 6. Finally, in section 7 we
review quantum-optical phenomena accessible with
monolithic microresonators. In the rest of this section,
we briefly review the history and distinctive properties
of WGM resonators.
1.2. Discovery of WGM phenomena
The first observation of a whispering gallery has
faded from memory. In the realm of acoustics a
smooth and circular wall features the characteristic
that a soft spoken whisper can propagate along the
wall to be heard by a listener far away from the
source but close to the wall.
Examples include
the Gol Gumbaz in India, the Temple of Heaven in
Beijing, and the St. Paul Cathedral in London. It
was in London that Lord Rayleigh first described
the phenomenon scientifically [16–18], noting that the
spherical wall reflects and continuously refocuses the
sound wave and thus a whispering gallery is formed.
An experimental observation of this effect was reported
3
by Raman [19]. This concept of confining waves in the
vicinity of a curved boundary was soon demonstrated
in the scattering of electromagnetic radiation from
gold spheres, which led to the establishment of Mie
scattering theory [20]. It was furthermore found to be
applicable to dielectric interfaces, which was important
for understanding radar scattering from rain and
hail. Debye simplified the underlying mathematical
description [21] and applied it to the waves propagating
along dielectric wires. For the next decades whispering
gallery modes were only occasionally studied, such as
by Richtmyer who considered dielectric wires bent into
loops, i.e. ring resonators [22]. One of the first solid
state optical lasers was implemented in in a samarium
doped fluorite sphere [23]. On a much different length
scale, the Earth troposphere and ionosphere can act
as a closed waveguide supporting modes similar to
WGMs [24].
The WGM concept was picked up again in
the 1980s and developed in two different directions
comprising the study of properties of liquid droplets
[25] in the optical domain, and properties of properly
shaped pieces of high permittivity dielectrics [26] in the
microwave domain. Let us discuss these directions in
some more detail.
1.3. Historic investigations of microwave WGM
resonators
Microwave WGM resonators were studied in parallel with the optical ones. Spherical [27, 28], cylindrical [26, 29–31], and more complex resonator morphologies [32] were considered. While there is no fundamental difference between the optical and microwave structures, they are practically dissimilar because of i) the
nature of attenuation in the resonator host material,
and ii) the WGM excitation techniques.
The maximum achievable Q factor of a microwave
structure is determined by the dielectric loss tangent
defined as tan δ = ǫ′′ /ǫ′ , where ǫ = ǫ′ + iǫ′′ is
the complex permittivity of the material. Typically
the loss tangent increases with microwave frequency
[33,34]. As a result, the product Q×f = f /tan δ, where
f is the frequency of the microwaves, is considered as
a constant. It means that available Q-factors are not
very large at high microwave frequencies. For instance,
the highest Q-factor attainable at f = 9 GHz in a
sapphire resonator at room temperature is 2 × 105 [35].
This number can be improved to 109 and possibly
further at cryogenic temperatures [36].
Since the WGM quality factors are rather large
compared to the quality factors of other types of
dielectric structures, the microwave WGM resonators
found use as filters [37]. They also are utilized
as energy storage elements in ultra-stable microwave
oscillators producing spectrally pure signals [38–42].
Nonlinear and Quantum Optics with Whispering Gallery Resonators
As a benchmark, a 9 GHz oscillator with phase
noise of −160 dBc/Hz at 1 kHz offset frequency was
demonstrated using microwave WGMs [40]. These
oscillators are widely used. For instance, they were
proposed for the tests of local Lorentz invariance by
searching the difference in the speed of light in the
directions parallel and perpendicular to the direction
of the Earth motion around the Sun [43–45]. Another
application of the high-Q structures is related to the
study of weak attenuation of microwaves in nominally
transparent materials [34, 42, 46, 47] and the material
permitivity [48, 49]. A variety of nonlinear microwave
phenomena can also be observed in WGM resonators
[50].
The evanescent field of a microwave dielectric
resonator may extend by as much as a millimeter. It
means that a metallic antenna can be used to excite
the WGMs [42]. For high radio frequency (RF) and
THz radiation [51], the excitation can be realized
using dielectric [52] or metal waveguides. Low order
modes in the dielectric cavities are coupled to the free
space and have significant radiative losses which allows
applications of the structure as dielectric RF antennas
[53].
1.4. Historic investigations of optical WGM
resonators and their close relatives
The requirement for a smooth resonator surface is
much more stringent in optical than in the microwave
domain. One approach to form a nearly perfect
interface and reduce the scattering from the boundary
roughness is offered by surface tension. That is why
WGMs have been initially studied in liquid droplets
either caught in optical or ion-traps [54] or planted
onto hydrophobic surfaces [55]. Their Q-factors were
of the order of 104 − 106 , thus stimulated phenomena
such as Raman lasing was readily observed [55–61].
Conventional lasing was achieved as well by including
dyes such as Rhodamine 6G [62–64] or quantum dots
[65] into the solution.
Surface tension also defines shape and surface
quality of microsphere resonators created by thermal
reflow of glass. In 1989, Braginsky and co-workers [66]
were the first to realize that nearly perfect spheres
can be formed by melting high-grade silica glass fibers.
The potential of WGM resonators in quantum optics
was emphasized already in this pioneering research:
“With possible reduction of controlling energy of optical
switching down to a single quantum and employment of
the monophotonic states of light, the whispering-gallery
microresonators can open the way to realize Feynman’s
quantum-mechanical computer.”
These microspheres are by now quarter-century
old, and although they could not so far reach the
parameters required to operate as quantum computer
4
nodes, they allow observing essentially quantum
effects. For instance, cavity QED experiments were
performed with microspheres doped with nanoparticles
[67, 68]. The resonators also find such applications as
single virus detection [69, 70] and, by including rare
earth dopants into the glass, lasing [71–74].
Vahala and co-workers managed to combine the
benefits of surface tension induced smoothness and
lithographic production by creating micro-toroids out
of silica on silicon [75]. These resonators showed an
interaction of their optical WGMs and mechanical
resonances via the radiation pressure [76]. Such
opto-mechanical coupling allowed for cooling of a
single mechanical mode of an optical WGM resonator
close to the quantum-mechanical ground state [77].
Toroidal fused silica cavities were used to achieve
frequency comb generation [78], and very recently, to
demonstrate the octave-wide comb operation [79–81].
Along the same line is the fabrication of a different
kind of resonator, the bottle resonator [82, 83]. These
resonators are shaped by compressing a locally molten
silica fiber in the axial direction, similarly to the first
microtoroids [84]. The large tunability and potentially
very large mode volume of bottle resonators make
them interesting for coupling to atomic transitions
[85]. Efficient nonlinear switching, envisioned for
microspheres [66], was successfully demonstrated in
bottle-shaped fused silica microresonators [86, 87].
If a capillary is used instead of a fiber,
hollow bottles or microbubble resonators [88–90]
can be created. They show promise for chemical
sensing applications.
Note that the whispering
gallery waveguides realized as straight (unstructured)
capillaries also have been used for this purpose [91,92].
Heating fibers by a CO2 laser can be utilized
for fabrication of Surface Nanoscale Axial Photonics
(SNAP) resonators [93] featuring sub-atomic precision
of the surface profile control. The SNAP technology
makes use of the frozen stress in the fiber, which, when
locally heated, results into nanoscale deformation.
Laser fabrication can be also applied to larger
silica rods, yielding the resonators used for high
quality frequency combs generation [94].
Precise
shaping of optical rods can furthermore allow for
experimental verification of interesting and counterintuitive behaviors of WGM light which can form
quasi-modes even in open systems [95, 96].
Crystalline materials with very low absorption can
only be formed into high quality WGM resonators
by mechnical polishing. One of the first solid state
lasers reported in 1961 was a doped CaF2 :Sm++
crystal sphere [23]. Record quality factors of over
1010 have been achieved in undoped fluorite resonators
[15, 97–99]. These resonators also feature frequency
combs [100–105] and opto-mechanical interaction [106].
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Another important aspect of crystalline materials is
that they can be anisotropic and therefore support
second order nonlinearities, paving the way for
applications in quantum optics [107]‡ and photonics
[73].
Generation of optical squeezing [108] and
entanglement, coherent frequency conversion [109,110],
and realization of quantum memories [111] are all
within reach to be implemented in the cavities. WGM
resonators also can be useful for creation of efficient
narrowband single photon sources [112] for quantum
information protocols.
Lithographic fabrication of crystalline resonators
of nearly any geometry and any material [113] has the
benefit of mass-production, but still fails at achieving
the surface qualities achieved with mechanical polishing. However with isotropic materials such as silica
it was possible to achieve surface qualities en par with
polishing and melting by chemical etching of wedge resonators [114]. The flexibility in the geometry opened
the whole field of asymmetric resonant cavities, demonstrating a variety of interesting dynamic behaviors such
as chaos and scar-type instabilities [115–117]. Asymmetric resonators have been used in high power quantum cascade lasers [118] and sub-wavelength lasers
[119]. Theoretical descriptions of modal structures in
deformed cavities cannot be done in the framework of
Mie analysis, due to the non-integrability of the geometric system, leading to wave-chaotic formulations
[120]. The full interaction of such non-trivial resonance
structures with non-uniform gain regions was only recently solved in the framework of a self-consistent lasing theory [121].
Besides the conventional WGM resonators that
are typically small, there exist larger monolithic
crystalline resonators that are also based on total
internal reflection (TIR) [122]. They have been used for
second harmonic generation [123] and parametric down
conversion [124]. Finally, low-order WGMs can be
observed around an irregularity in photonic bandgap
crystals [125, 126].
1.5. What is special about WGM resonators
In summary, the WGM resonators are attractive
because of their
• high Q-factor. WGM resonators that are large
with respect to the wavelength of light and have
high enough surface quality suffer extremely low
radiative losses. Depending on the resonator
material they can also have low intrinsic losses.
• wide spectral range.
WGM resonators
provide high-Q resonances throughout the entire
‡ In Eq.(5) of [107], both instances of S(1 + S) should be read
as S(2 + S).
5
transparency range of the dielectric they are made
of.
• low mode volume. WGMs are localized close
to the rim of the resonator and therefore have
small volumes. A mode volume is conventionally
introduced as the spatial integral over the field
intensity normalized to the intensity maximum
[127].
• mechanical stability. WGM resonators are
monolithic and small, and therefore suffer only
minimal mechanical instabilities.
• tunable wavelength. WGM spectrum can be
engineered as well as dynamically tuned by a
variety of techniques.
• variable coupling. Coupling to WGM resonators is usually achieved via frustrated TIR
whereby the coupling rate depends on the distance
between the resonator and the evanescent coupler.
This allows for a simple control over the coupling
rate.
• strong nonlinear interaction WGM resonators
made out of nonlinear material allow for achieving
strong nonlinear interaction at low light levels.
In the next section we formulate the mathematical
description of the WGM spectrum and provide a strong
footing of the special properties described above.
2. Spectrum of WGM and ring resonators
2.1. Mode structure and dispersion equation
Monolithic dielectric resonators have a refractive index
larger than that of their surrounding n > no (or,
equivalently, the relative index of refraction n̄ ≡
n/no > 1) and confine the light by total internal
reflection (TIR), see Fig. 1. Here the complex angle
of incidence and, therefore, the complex wave vector
corresponds to the evanescent field in the less optically
dense medium. As TIR only depends on the angle of
incidence and the refractive indices, it is a very broadband process.
In a resonator with geometric path length L and
constant index of refraction n the resonant frequency
follows from the argument that an integer number of
wavelengths m needs to fit into the optical path length
nL: λm = nL/m, or in terms of the wave number,
km = 2πm/(nL). The optical path of large enough
WGM resonators (L ≫ λm ) can be approximated by
expression L = 2πR, were R is the resonator radius.
This expression does not take the geometric dispersion
of the WGM spectrum into account.
In order to understand the nature of this
dispersion and to find a more accurate approximation
for WGM spectrum the Helmholtz equation needs
6
Nonlinear and Quantum Optics with Whispering Gallery Resonators
in this study can be simplified by realizing that only
two orthogonal potentials are independent [21]. This
yields two sets of solutions commonly known as the
transverse (to the equatorial planek ) electric field (TE)
and transverse magnetic filed (TM) polarization mode
families. The main field component for each family is
given by expression
Emlq (r, θ, ϕ) ∼ jm (nk0 r) × Plm (cos θ)eimϕ
= jm (nkq r) × Ylm (θ, ϕ),
Figure 1. Schematic of a dielectric whispering gallery mode
resonator. In the ray-dynamical description, light incident at an
angle χ can be either partially refracted out (if χ < χcrit ) or
totally internally reflected (if χ > χcrit ).
to be solved for the boundary conditions set by the
resonator geometry. The physical solutions are selected
by requiring the continuity and the smoothness of
the tangential derivatives of the electric field at the
boundary. Furthermore, the so-called Sommerfeld
boundary conditions should be selected at infinity,
limiting the solutions to outgoing fields only. Under
these conditions the solutions become complex, calling
for some probing/external energy flow in order to excite
and probe the resonances.
A WGM resonator belongs to a class of the socalled open resonators. Exact mathematical solution
shows that modes of such a resonator include unbound
spherical waves running away from the resonator. The
modes are fundamentally unconfined, and defining
volume of the mode is not straightforward. The
problem is usually circumvented by neglecting the
radiated wave part and considering the associated
complex eigenvalue of the mode as radiative loss. The
value of the loss is usually much smaller than the other
types of attenuation. A calculation of the radiative
Q-factor for a 100 µm water droplet results in 1073
at λ = 600 nm [128]. As a rule, the radiation loss
can be safely neglected in any resonator with the
circumference exceeding a couple dozens of wavelength
and a high enough refractive index contrast with the
environment.
The most straightforward assumption is to
consider a spherical resonator and thus introduce a
spherical coordinate system §.
The transformation
of the Laplacian into spherical coordinates introduces
a centrifugal potential which provides a condition for
bound states [131,132]. The interaction of plane waves
with such bound states was studied by Mie [20]. The
analysis of six independent field components arising
§ This is not the only possible choice of a coordinate system.
Other coordinate systems, such as cylindrical, spheroidal,
toroidal and ellipsoidal may also be used [129, 130].
(1)
where jm is the spherical Bessel function of order
m, Plm are the associated Legendre polynomials, or
respectively, Ylm are the spherical harmonics [131].
Figure 2. Field distribution of whispering gallery modes found
analytically and numerically. The resonator rotation axis is
vertical, and the dashed line indicates the resonator boundary.
The top row represents the fundamental equatorial WGM with
q = 1, p = 0. In the bottom row, q = 2, p = 2. The right-hand
panel shows the field cross section along the lines shown on the
left-hand panel. Note again that the analytical solution does not
take the evanescent field into account. The numerical however
does. Reprinted from [130].
Here a nomenclature for the different integer
separation constants is at order. The spherical Bessel
function is an oscillating function of the radius r, whose
zeros can be numbered by the integer q = 1, 2, 3, . . .,
where q = 1 designates the fundamental WGM.
The angular momentum m describes the number of
wavelengths that fit around the equator. It can also
be negative, pointing to a counter clockwise rotating
mode. A positive integer p = l − |m| corresponds to
the number of nodes in the polar direction (see Fig. 2),
indicating a transverse mode structure similar to that
of a Fabry-Perot resonator. Finding the eigenvalues
and eigenfunctions for modes in large resonators, where
the m ≈ l > 10, 000 is not trivial but can be made
k TE and TM are sometimes defined in the opposite way, as
transverse to the resonator surface.
Nonlinear and Quantum Optics with Whispering Gallery Resonators
feasible by suitable approximations to the high-index
Bessel functions [133, 134] and Legendre Polynomials
[130].
These approximations are useful for describing
spherical resonators. However many WGM resonators
are better described as spheroids with the major
and minor semi-axes a = R and b, respectively.
Eigenfrequencies of such resonators can be found in
the so-called semi-classical limit by the eikonal method
[129, 135]. In [129] we find the following dispersion
equation:
νn
= nklpq R ≃ l − αq (l/2)1/3
(2)
2πR
c
2
3αq
2p(R − b) + R
ζn
+
−√
(l/2)−1/3
+
2b
20
n2 − 1
2nζ(2ζ 2 − 3n2 )
αq 2p(R3 − b3 ) + R3
−2/3
+
−
(l/2)
,
12
b3
(n2 − 1)3/2
where αq are the negative zeros of the Airy function,
and ζ equals 1 for TE and n−2 for TM modes. The
leading term (∼ l1 ) of this equation’s right hand side
clearly corresponds to the Fabry-Perot type resonance
condition derived from the simple ray model. The
second term (∼ l1/3 ) is a correction taking into
account change of the WGM diameter depending on
its wavelength as well as q number. It may be said
that higher-q modes effectively see a smaller resonator,
hence the positive frequency shift. In the third term
(∼ l0 ) a correction for different curvatures in polar and
azimuthal direction is taken into account. The fourth
term (∼ l0 ) arises in the eikonal method from the
polarization-dependent Fresnel phases and implicitly
accounts for the evanescent field of the resonator (hence
the factor ζ distinguishing TE and TM modes). Note
that this term explodes as the resonator material index
of refraction n approaches that of the surrounding
media (no = 1 is assumed in Eq. (2)) and WGMs
become poorly confined.
In large resonators the evanescent field can be
neglected for the sake of simplicity. This is done by
setting the metallic boundary conditions at the rim of
the resonator: E(r = R) = 0. Then introducing a local
coordinate system such as shown in Fig. 3 it is rather
straightforward to find approximate expressions for the
eigenfunctions [130]:
2
2
E ∼ Ai(u/um − αq ) × e−θ /2θm Hp (θ/θm ) × eimϕ ,
3/4
R
1
R
√ , and um = 1/3 2/3 .
(3)
θm =
ρ
m
2 m
Here Hp is the Hermitian polynomial of the order p,
e.g. H0 = 1, H1 = 2θ/θm , H2 = 4(θ/θm )2 − 2, and so
on. Ai is the Airy function with its negative zeros αq .
Note the relation between the local radius of
curvature r in Fig. 3 and the √
minor semi-axis of the
approximating ellipsoid: b = Rr. The results (3)
7
Figure 3. Surface shape of realistic WGM resonators may
significantly differ from a sphere. In such cases introducing a
local coordinate system such as shown here is more suitable.
Reprinted from [130].
agree well with numerical simulations that take the
evanescent tail of the mode properly into account, see
Fig. 2.
There are other ways to approximatly solve
Laplase equation for large WGM resonators of an
arbitrary shape.
WGMs whose wavelengths are
much smaller than the resonator size are localized in
the vicinity of the equator of the resonator. Here
cylindrical coordinates can be used for resonators of
any shape, including spherical. The shape of the
resonator can be presented in form R = R0 + L(z),
where R0 ≫ L(z). The corresponding wave equation
can be solved using separation of variables [136].
Dispersion relation (2) has a nontrivial dependence on the mode numbers (l, p, q) and resonator parameters. This leads to complicated spectra that are
difficult to interpret for larger resonators. However,
identifying a WGM numbers, especially q, is critical for finding the phase matching conditions of various nonlinear-optical processes, as shown in section
4.2. Various techniques of optical WGM identification have been developed, including the free spectral
range (FSR) measurement by sideband spectroscopy
[137,138], far-field emission pattern analysis [139–141],
or a combination of these techniques [142]. The detailed spectrum of a WGM resonator can be measured
using a reference optical frequency comb [143].
Up to now we only considered WGM resonators
that had isotropic material properties.
These
properties are inherent to amorphous WGM resonators
such as droplets [25, 55–57, 59–61, 144–148], molten
silica spheres [66], toroids [75], or wedge shaped
resonators [114]. Moreover, all crystals with cubic
symmetry like CaF2 are isotropic. Most of the crystals
fall, however, in one of the four other symmetry
classes, where the refractive index varies depending
on polarization and propagation direction of the light.
Among crystals of these four symmetry classes, the
group of uniaxial crystals shows only one direction
of light propagation, i.e. one optic axis, where the
refractive index is independent of the polarization. The
propagation of light in the plane perpendicular to the
optic axis is then governed by the ordinary index of
8
Nonlinear and Quantum Optics with Whispering Gallery Resonators
refraction no for perpendicular polarization, while the
extraordinary index of refraction ne is valid for the
polarization parallel to the optic axis.
WGM resonators made out of uniaxial materials
are very important in our following discussion. Usually
they are made such that the optic axis is parallel
to the rotational symmetry axis, i.e. in the so-called
z-cut geometry. In this special case the TM (TE)
polarized mode is mainly influenced by the ordinary
(extraordinary) refractive index. Therefore the two
mode families can be tuned independently, as the
thermo-refractive, electro-optical, etc. effects all scale
with the respective refractive indices, see sections 3.2
and 3.3.
A less common is the x-cut geometry, when the
optic axis lies within the equatorial plane of the WGM
resonator, i.e. is perpendicular to the axis of rotation.
In this case the TE polarized mode is governed
by the constant ordinary refractive index, however
the TM polarized mode experiences the refractive
index oscillating approximately harmonically between
the ordinary and extraordinary values. We discuss
applications of such resonators in section 4.4.
The general case when the optic axis makes an
arbitrary angle with the symmetry axis is highly
non-trivial. It is not clear if there still exist two
mode families in such resonators. The refractive
index experienced by the light varies along the path
of the WGM, and thus its polarization becomes
position dependent.
Position-dependent walk-off
further complicates the analysis. In spite of several
theoretical and experimental investigations of WGM
properties in the resonators with arbitrary orientation
of the optical axis [141, 149–153], their behavior is not
yet fully modeled and understood.
2.2. Geometrical dispersion and effective index
approximation
Comparing the WGM dispersion equation (2) to
the dispersion equation describing an ideal onedimensional resonator of the optical length L = 2πRn:
2πR
νn
= m,
c
(4)
we see that Eq. (2) includes higher-order terms that
can be attributed to the geometrical, or waveguide,
dispersion. This dispersion arises from the boundary
conditions and the resonator curvature.
It is
often convenient to treat large WGM resonators as
one-dimensional resonators described by dispersion
equation (4) with the effective index of refraction n →
ñ which incorporates all the geometrical and material
dispersion effects [154, 155]. Note that ñ depends on
a variety of parameters, including the wavelength, the
resonator semi-axes a = R and b, and the WGM family
specified by the polarization and mode numbers q and
p. To find ñ, we first numerically solve (2) for l, then
substitute m = l − p into the right-hand side of (4),
and solve it for ñ. In doing so we may give up the
requirement for l and m to be integer, which means
that we treat the resonator spectrum as continuous.
The effective index approximation is convenient
for finding the phase matching conditions for nonlinear
frequency conversion, e.g. spontaneous parametric
down conversion (SPDC) νp → νs + νi , supported in
resonators with nonzero χ(2) nonlinearity. In this case
the phase matching condition takes a simple form
νp ñp = νs ñs + νi ñi ,
(5)
which allows us to find the suitable temperature and
wavelengths numerically. The benefits and limitations
of this approach are further discussed in section 4.5.
This approach has been successfully used for
evaluating the phase matching in WGM resonators
for second harmonic generation (SHG) [156], SPDC
[157, 158], and direct third harmonic generation from
a fused silica microsphere [159]. In [160]¶ it was used
to infer the parameters for the double phase matching,
such that the signal mode excited in the SPDC process
can generate its own second harmonic.
Geometrical dispersion contribution ∆n ≡ ñ − n
is always negative in a spheroidal resonator. This is
evident from the mode structure shown in Fig. 2. Here
the optical field is concentrated inside the resonator,
some distance away from its rim. The resonator optical
path is shorter than 2πRn because of that. As a
function of wavelength, the geometrical contribution
to dispersion is normal : d∆n(λ)/dλ < 0. In part,
this is because at longer wavelengths more of the
optical power is carried by the evanescent field outside
the resonator, and consequently the effective index
of refraction is lower. However, the major reason is
that the power is localized further from the resonator
surface at longer wavelength, so the effective radius of
the mode is smaller there.
In small resonators geometrical dispersion can be
significant. This opens up an interesting opportunity
for the WGM dispersion engineering by varying
parameter b, or more profoundly, by machining the
resonator rim into various non-spheroidal shapes. Such
shapes usually do not allow for analytic dispersion
equations such as (2), and finite-element numerical
techniques are required to find their spectra, see e.g.
[161, 162].
The effect of the resonator rim shape on the
Kerr optical comb properties has been experimentally
studied [81, 104, 163, 164]. In particular, the “belt”
resonators of rectangular crossection have been shown
¶ In the left-hand sides of Eqs. (1) and (2) in [160], the factor
2π should be in numerator rather than in denominator.
9
Nonlinear and Quantum Optics with Whispering Gallery Resonators
to compensate the chromatic dispersion of the the
group velocity in magnesium fluoride, resulting in
generation of extended-range optical frequency combs
[165]. The group velocity dispersion can also be
optimized by engineering the waveguide cross section
dimensions [166], adding a proper material cladding
[167], or using slotted waveguide structures [168–170].
2.3. Coupling, loss and quality factors
Coupling WGM resonators to external light is achieved
by frustrated TIR. This coupling can be realized via
optical waveguides, prisms, gratings, other resonators,
etc. Coupled in this way resonators have been used to
build efficient narrow-band add/drop filters, including
those enabling narrow linewidth light sources [171–
173], division multiplexers, and other optical devices
[174, 175].
The most efficient coupling to date was realized
with tapered fiber couplers [132, 176, 177]. Fiber
taper is a single mode bare waveguide with diameter
optimized for phase matching with selected WGM.
The coupling efficiency can approach 99.96% with an
adiabatically tapered fiber [178].
Prism coupling to WGMRs was investigated
both theoretically and experimentally [139, 179–181],
reaching some 80% efficiency with microspheres [181].
The efficiency can be optimized by adjusting the
shape of the optical beam as well as resonator
morphology. Coupling efficiency exceeding 97% was
achieved in elliptical LiNbO3 resonators [182], and
nearly perfect coupling (more than 99% criticality,
[183]) was achieved in a LiNbO3 WGM resonator with
optimized shape of the rim.
Planar integration of WGMRs with waveguides is
deemed to be the most practical approach [184–192].
For example, strip-line pedestal antiresonant reflecting
waveguides have been utilized for robust coupling to
microsphere resonators as well as to photonic circuits
[193–196]. Integration of a lithium niobate resonator
with a planar waveguide made with proton exchange
in X-cut lithium niobate substrate was realized [197].
Strong coupling between a resonator and a waveguide
integrated on the same chip but lying in different planes
was realized as well [198].
Theoretically coupling of WGM resonators can
be described via a simple transfer matrix scheme
[199] illustrated in Fig. 4. In this approach the
normalized single mode outside fields a1 , b1 are related
to the internal fields a2 , b2 by the complex coupling
parameters κ and t = |t| exp(−iφ):
b1
t
κ
a1
=
,
(6)
−κ∗ t∗
b2
a2
where |κ|2 + |t|2 = 1.
Figure 4. Schematic of the WGM resonator coupling. The
incoming field a1 is related to the reflected field b1 and the
internal cavity field b2 . The coupling coefficients are r and κ.
The internal field amplitudes are additionally
related by a2 = α exp(iθ)b2 , where α ≪ 1 is the loss
per round trip and θ the phase change due to the
round trip. The system is said to be in resonance
and a stationary mode can form if the round-trip
phase increment is an integer multiple of 2π. The
system is critically coupled and no field exits the cavity
(|b1 |2 = 0) if the internal losses equal the coupling
losses: α = |κ|. At the critical coupling, the full width
at half maximum of WGM resonance is given as
δω =
2|κ|2 c
,
nL
(7)
where L = 2πR is the length of the resonator
circumference.
The finesse of a critically coupled resonator is the
loss per round trip: F = π/(1−|t|2 ) = π/|κ|2 . It is also
equal the ratio of the FSR Ω = 2πc/nL = c/nR to the
linewidth δω (full width at the half maximum) of the
mode. The quality factor is related to the linewidth
and finesse as
ω
Q=
= F · m.
(8)
δω
There are many various loss channels in a WGM
resonator that account for the loss factor α:
α = αss + αmaterial + αrad .
(9)
The terms of (9) represent the surface scattering
(αss ), material absorption (αmaterial ), and radiative loss
(αrad ). They can all be lumped into the resonatorlimited (i.e. independent of coupling) quality factor
Q0 :
1/Q0 = 1/Qss + 1/Qmaterial + 1/Qrad.
(10)
Strictly speaking, a curved boundary does not
permit TIR [131], so, as we already mentioned, a WGM
10
Nonlinear and Quantum Optics with Whispering Gallery Resonators
resonator always radiates into space. The radiative
loss is usually negligible and becomes observable
only if the radius of curvature of the boundary
becomes comparable to the wavelength of the light
confined within the resonator [23, 131, 200–202]. The
corresponding quality factor of a sphere is given by [23]:
2πR ζ e2T
√
,
(11)
λ
n̄2 − 1
p
2πR
T
=
cosh−1 n̄ − n̄ − n̄−1
λ
where n̄ is the relative index of refraction (n̄ = n/no in
terms of Fig. 1) and ζ equals 1 for TE and n−2 for TM
modes defined according to our convention. This loss
channel is irrelevant for crystalline WGM resonators
which are usually much larger than the wavelength.
Surface roughness, on the contrary, can be an
important loss channel. The corresponding quality
factor scales with the root mean square (rms) size of
the surface inhomogeneity s, and the correlation length
of the roughness at the resonator surface B as [203]
Qrad ≈
Qss ≈
3λ3 R
,
8π 2 n̄B 2 s2
(12)
The fact that this term depends on λ3 can be used to
test if the quality factor is mainly limited by the surface
quality simply by testing it at different wavelengths.
Material absorption is usually characterized by the
loss coefficient αmaterial describing power attenuation
per unit length, which leads to the following expression:
Qmaterial =
2πn
.
λαmaterial
(13)
The transparency of an ideal dielectric in the optical
domain can be defined by the tails of multi-phonon
absorption on the long wavelength side, and by
electronic transitions absorption (the so called Urbach
tail) on the short wavelength side [98, 204]
αmaterial (λ0 ) = αuv eλuv /λ0 + αir e−λir /λ0 .
(14)
Here the coefficients αuv , αir as well as λuv , λir are
experimentally found values. The importance of the
multiphonon absorption in the mid-IR was realized
recently [205, 206].
3. Tuning the resonator spectrum
In section 2.2 we noticed that WGM spectra can be
engineered by modifying the resonator rim profile. We
also mentioned that the engineering and reversible
dynamical tuning of the WGM spectra are important
for achieving the phase matching in various nonlinear
conversion processes. In this section we review some
other available techniques to achieve both permanent
and dynamical WGM frequency tuning.
3.1. Mechanical stress and deformation
Mechanical deformation of resonators can change their
spectral properties rapidly and, with a due caution,
reversibly. The leading-order effect here is the physical
change of the resonator length, although the pressure
dependence of the refraction index should also be taken
into account. Higher-order effects may include stressinduced birefringence, heating, and piezoelectric effect
in some materials.
In an early experiment by Ilchenko et al. [207],
a fused silica microsphere (R = 80 µm) was squeezed
between two copper pads actuated by a piezo. The
spectrum was shifted by 150 GHz (approximately
3/8 of the FSR) while maintaining the quality factor
over 108 . A small variation of the FSR was also
observed. The same technique was later applied [98]
to a crystalline magnesium fluoride resonator with
R = 250 µm. In this case multiple-FSR tuning was
easily achieved. Note that a continuous tuning of the
WGM spectrum over one FSR allows to couple the
resonator to any optical wavelength. Such a resonator
is called fully tunable. Pressure-tuning study of highly
non-equatorial WGMs in polystyrene microspheres is
reported in [208].
Bubble resonators offer a rather unique way of
dynamic tuning: they can be not only stretched
or compressed [89], but also inflated [90].
All
these techniques allow for achieving the fully tunable
operation.
3.2. Thermal
Tuning a resonator frequency by changing its temperature is perhaps the most common tuning technique. It
broadly applies not only to WGM, as in e.g. [209, 210],
but also to practically all other kinds of resonators.
This type of tuning is based on the combination of two
major effects: the thermal expansion and the temperature dependence of the resonator index of refraction.
While the thermal expansion usually leads to a negative frequency shift with increasing temperature (as the
resonator becomes larger), the thermorefractive contribution can be either positive or negative depending on
the resonator material and temperature. Remarkably,
the ordinary and extraordinary thermorefractive coefficients in birefringent materials can significantly differ. This allows for the differential tuning of the TE
and TM WGM spectra, a capability very important
+
for some sensor applications [211, 212] and particularly for achieving the phase matching in various nonlinear optics applications [213]. Practically all nonlinear WGM optics studies that we cite below relied on
the temperature tuning.
+ In the left-hand part of Eq. (1) in [211], 2π should be in the
numerator rather than in denominator.
Nonlinear and Quantum Optics with Whispering Gallery Resonators
High-Q WGM resonator spectra can be very
sensitive to variations of the temperature. It is easy
to see that the frequency tuning rate given in the units
of the WGM linewidth δω is related to the material
thermal expansion and thermorefraction coefficients µL
and µn as
1 dω
= −Q(µL + µn ).
(15)
δω dT
For the typical parameters of Q = 108 and µL + µn =
10−5 K−1 , a milidegree temperature change can easily
shift the spectrum by a full linewidth. This makes the
consideration of the optical power dissipation inside
the resonator relevant even for very weak light. For
a critically coupled resonator all in-coupled optical
power is dissipated and eventually converted to heat.
Therefore a rapid thermal tuning can be achieved
by controlling the input optical power. This control
parameter has been put to use in a temperature
stabilization application [211, 212, 214], allowing to
reach a few nK temperature stability at above room
temperature set point. In very high-Q resonators,
the spectral response to the injected optical power via
the mode volume heating may lead to thermal noise
and instability which we will discuss in more detail in
section 6.2.
3.3. Electro-optical
Because of high temperature sensitivity of the WGM
spectra, the temperature tuning range can be very
large, reaching a few to several tens of nanometers.
However, this method is not very convenient because it
is slow. Much faster tuning can be achieved leveraging
the electro-optical effect. The electro-optical tuning is
only possible in materials where such effect is present.
All WGM modulators described in section 4.1 below
were electro-optically tuned. Like the thermal, the
electro-optical tuning leads to the differential TETM frequency shift, due to different values of the
electro-optical tensor components. Differential tuning
among different q-families of the same polarization can
be enabled by creating circuar patterns of inverted
domains near the resonator rim [215], which may
be conveniently achieved in lithium niobate using
the calligraphic poling technique based on “drawing”
inverted domains with a sharp electrode [216, 217].
Compared to the thermal, the electro-optical
tuning has the advantage of higher speed. On the down
side, it is difficult to control the quasi-static electric
field in the region of the optical mode, because the
electrodes cannot be applied to this region without
compromising the optical quality factor. As a result,
both the magnitude and the direction of the tuning
field is poorly controlled in the very region where it
matters, and the tuning rates observed in experiment
are rarely consistent with the theoretical estimates.
11
Ferroelectric materials, such as lithium niobate,
have additional problems associated with electrooptical tuning. Visible light induces photocurrents in
such materials [218, 219], which means mobilizing the
charges that may screen the optical field area from the
control electric field. As a result, the WGM spectrum
follows a rapid bias field variation for a short period
of time but then returns to its original state, with the
time constant ranging from sub-seconds to hundreds of
seconds, depending on the injected optical power and
wavelength [220].
External electric fields can affect the WGM
spectrum not only via the electro-optical, but also
via the electrostriction effect.
This effect does
not require nonlinear response and is present in all
materials. It has been observed with hollow solid
polydimethylsiloxane microspheres [221]. Let us point
out that both electrosctriction and electro-optical
effects can be used not only for configuring the WGM
spectra, but also for sensitive measurements of quasistatic electric fields by monitoring the spectral changes
[222].
3.4. Other methods of WGM spectrum engineering
Dispersion engineering can also be realized by coating
a resonator with another dielectric that has a different
index of refraction [167, 223].
Different WGMs
penetrate into the coating layer differently, which gives
rise to the differential TE-TM frequency tuning, as
well as differential tuning of different q-familes of
the modes that have the same polarization. Making
the coating optically active opens up the possibilities
for quick spectral tuning by means of an optical
control. For example, coating a microsphere with
three bacteriorhodopsin protein monolayers allowed
for a reversible high-speed control of a target WGM
frequency, thereby achieving all-optical switching of
a 1310 nm with 532 and 405 nm control beams in
50 µs [224]. Optical tuning can be also implemented
in complex WGM structures comprising a solid-core
microstructured optical fiber with magnetic fluids
[225].
A dynamically controlled, polarization discriminating tuning can be achieved with a dielectric probe
moving in the resonator’s evanescent field [226]. In
some configurations they may impart the anomalous
blue shift to one polarization family, while the other
experiences the normal red shift [227]. This technique
has been used for fine frequency-tuning of WGM-based
SPDC source [228].
Closely related to dielectric coating is immersing a
resonator in a lower-index liquid [229]. This technique
allows for index engineering, and also forms a basis
for various sensing techniques, including those used for
detection of organic molecules and viruses [69, 70].
12
Nonlinear and Quantum Optics with Whispering Gallery Resonators
3.5. Mode crossing
where the modes interaction strength is given by κ.
This Hamiltonian has two eigenfrequencies:
ω1,2 =
1p
ωb + ωd
(ωb − ωd )2 + 4κ2 .
±
2
2
(17)
An example of avoided-crossing is shown in Fig. 5.
Here the asymptotes correspond to the standard
change of the unperturbed resonances with the external
parameter, which may be the temperature, exposure
time in a photorefractive experiment, bias voltage,
and so on. At the degeneracy point, the modes are
separated by twice the coupling constant 2κ.
WGM spectrum tuning by avoided mode crossing
does not usually offer the versatile control or large
tuning range available from other techniques we
discussed earlier. However it may be efficient in the
situations when the quasi-equidistant character of the
WGM spectrum needs to be altered, e.g. for the design
of single sideband (SSB) electro-optical modulators
[110]. Furthermore, in non-Hermitian systems where
1.0
30
25
0.8
20
0.6
2κ
15
0.4
10
0.2
5
0
-20
bright mode
dark mode
0
-15
-10
-5
0
5
10
15
20
Parameter detuning
Figure 5.
Theoretical description of interaction between
a bright and dark modes.
The bright mode is critically
coupled, and the external parameter controlling the modes’
relative dispersion is tuned. Near the degeneracy the coupling
rate κ detunes the resonance frequencies, and a characteristic
avoided-crossing pattern is seen. Asymptotes correspond to the
unperturbed modes.
1 – coupling depth
Large WGM resonators have very dense spectra.
Occasionally, modes from different families become
nearly degenerate [234, 235]. Ideally, these modes
would not interact. However even a minute scattering
can facilitate their energy exchange comparable with
the intrinsic loss rate. In such a case the two modes
can interact, which leads to the so-called avoidedcrossings. As we have seen earlier in this section,
modes from different families respond differently to
external parameters such as temperature, pressure,
electric field, or presence of scatterers. This opens one
path to reaching a degeneracy, which is observed in e.g.
photorefraction experiments [231–233].
Usually in such experiments one of the crossing
modes is coupled to a probing light source more
efficiently than the other, which may be due to the
different polarizations or q’s. Therefore these modes
can be named a bright mode (designated photon
annihilation operator b̂) and a dark mode (designated
ˆ respectively, with
photon annihilation operator d),
their unperturbed resonance frequencies ωb (ωd ).
There is a number of mode-coupling mechanisms,
such as e.g. scattering by sub-wavelength features
[203, 236–238], that may be realized as impurities or
refractive index inhomogeneities. Scattering may also
occur between modes with different polarizations. This
type of scattering has been studied in silicon nitride
micro resonators [239, 240]. It also was observed [211]
and studied in magnesium fluoride [241].
Regardless of how the mode coupling is physically
realized, a straightforward approach to describing it is
given by the following Hamiltonian:
(16)
Ĥ = ~ωb b̂† b̂ + ~ωb dˆ† dˆ + ~κ b̂† dˆ + b̂dˆ† ,
Shift of resonance position [MHz]
Permanent or quasi-permanent spectral changes
can be induced in WGM resonators by various
photochemical processes in the host material that
affect the index of refraction. For example, spectra
of germanium-doped silica micropheres have been
tailored by a UV-exposure for an optical filtering
application [230]. The advantage of this approach is
that the spectral changes can be monitored in real
time, and the process can be stopped when the desired
frequencies are reached.
Photorefractivity in lithium niobate and tantalate
offers another method for controlled modification of a
WGM spectrum [231–233]. Unlike with the UV curing
of Ge-doped silica, here one can selectively pump a
chosen mode at a relatively high optical power (even
in the infrared wavelength range [231]) shifting its
frequency with respect to the other modes. The process
is surprisingly mode-selective, considering a significant
overlap between the low-order WGMs. Once the
first mode is sufficently tuned, the process can be
repeated with a different mode, and so on. One can in
principle arrange a number of WGMs into a spectral
pattern that could be used e.g. for “fingerprinting” of
complex atomic or molecular spectra. The underlying
photorefractive patterns “engraved” in the resonator
material are only quasi-permanent and can be erased
by e.g. UV exposure [232].
13
Nonlinear and Quantum Optics with Whispering Gallery Resonators
there is a loss for each mode, the avoided crossing near
the degeneracy allows for realizing the so-called parity
time symmetric (PT −Symmetric) systems [242–244],
and even more generally, to study the exceptional
points [238, 245–247]. This research leads to such
counter intuitive observations as the onset of lasing by
inducing loss [242].
4. Second-order nonlinear processes
Second-order nonlinear optical processes arise from the
quadratic response of the media polarization P~ to the
~ [248]:
external electric field E
X (2)
(2)
Pi =
χijk Ej Ek ,
(18)
j,k
where χ(2) is the nonlinear susceptibility tensor, and
i, j, k index the crystallographic axes x, y, z.
Second-order polarization (18) gives rise to a
nonlinear term of the optical energy
Z
X (2)
1
H (2) =
dV
χijk Ei Ej Ek .
(19)
3
i,j,k
Here the factor 1/3 is necessary to account for the index
permutations that preserve the frequency sum relation.
The χ(2) tensor elements involved in such permutations
are postulated to be equal by the Kleinman symmetry
convention [248]. In many cases of interest polarization
of each field is fixed with respect to the χ(2) tensor axes,
and then the sum in (19) can be omitted along with
the factor 1/3.
Following the canonical quantization of scalar
single-mode electromagnetic field [249] inside a dielectric [250, 251], we write
√
Ê(~r, t) = in−1 2π~ω âψ(~r)e−iωt − ↠ψ ∗ (~r)eiωt ,
(20)
where n is the index of refraction for given optical
frequency ω and polarization, ↠and â are creation
and annihilation operators for a photon in the given
mode, [↠, â] = 1, and ψ(~rR) is the mode’s eigenfunction
normalized to unity:
dV |ψ(~r)|2 = 1.
Note
that in the strict sense such normalization would
be impossible because the integral diverges as r →
∞. Indeed, the radial part of ψ(~r) for r > R
is given by Hankel functions that asymptotically
correspond to diverging spherical waves representing
radiative loss, as discussed in section 2.1 and in [252].
The unity normalization therefore can be introduced
only approximately, enforcing the metallic boundary
condition (see section 2.1) and limiting the integral to
r < R.
Substituting (20) into (19) we derive a quantummechanical interaction Hamiltonian which governs all
three-wave mixing processes in resonators:
Ĥint = ~g(â1 â2 â†3 + â†1 â†2 â3 ),
(21)
p
(2π)3 ~ω1 ω2 ω3
σ123
n1 n2 n3
(22)
where
g = χ̃
(2)
is the nonlinear coupling rate for interacting photons,
ω3 = ω1 + ω2 , the effective nonlinear susceptibility χ̃(2)
is determined by the fields polarizations, and
Z
σ123 = dV ψ1 (~r)ψ2 (~r)ψ3∗ (~r)
(23)
is the WGM overlap integral. In (22) we used cgs units,
with the following conversion of the standard secondorder nonlinearity:
χ(2) [cgs units] =
3 × 10−8
d[pm/V].
4π
(24)
Besides the overlap integrals (23), a different notation is often used, see e.g. [107, 253–257], apparently
ascending to the tradition of field quantization in the
plane waves. In this notation the eigenfuctions ψ(~r) are
normalized to the quantization volume (mode volume)
V rather than to unity. The electric field expression
(20) acquires in this case an extra factor V −1/2 and
the mode overlap (23) is measured in the units of volume, e.g. cm3 , instead of cm−3/2 . Just like the unity
normalization, the volume normalization can only be
done approximately by assuming the metallic boundary condition.
The disadvantage of normalizing WGM eigenfunctions to the mode volume is that, except for the case
of plane waves quantization in a box, this volume
lacks a rigorous first-principles definition. Usually it
is defined as the volume integral of the optical intensity distribution normalized to the maximum intensity
value [127,258]. With this definition equivalence of the
volume- and unity-normalized approaches can be easily
proven, however it may not be a good definition for the
eigenfunctions that have multiple nearly-equal maxima. On the other hand, the advantage of the volumenormalized approach is the ability to directly compare
the overlap integral (23) to the mode volume and therefore to easily quantify the coupled modes overlap in
space as the ratio of these volumes. Other possible approaches to eigenfunctions normalization and defining
the mode volume are discussed Kristensen et al. [259],
who also prove them to be equivalent.
Interaction Hamiltonian (21) leads to a set of
ordinary differential equations (ODEs) describing the
nonlinear processes and the input-output relations in
a resonator. For any mode labeled j we have
i
â˙ j = −(γj + iωj )âj + [Ĥint , âj ]+ F0 e−iω0j t δj0,j , (25)
~
Nonlinear and Quantum Optics with Whispering Gallery Resonators
where δj0,j is the Kronecker’s delta, j0 labels the
externally pumped mode, γj = γcj + γij is the half
width at the half maximum for the optical modes,
and γcj and γij stand for coupling and intrinsic losses,
respectively. F0 represents the external pumping at a
frequency ω0j :
r
2γc Pin iφin
(26)
e
F0 =
~ω0
where Pin is the input power and φin is the phase of
the pump.
Below we discuss various three-wave mixing
processes described by Hamiltonian (21). Let us also
mention a very recent review [260] dedicated to this
subject.
4.1. Electro-optical phenomena and applications of
WGMRs
We start our discussion of the second-order nonlinear
optical phenomena from reviewing the interaction of
optical and static or quasi-static (on the optical cycle
time scale) electric fields. Such interaction enables
a variety of important optical modulation, sensing
and frequency conversion applications. Perhaps the
most important of these applications is the electrooptical modulator (EOM). Operation of an EOM can
be considered as based on a strongly nondegenerate
parametric process [253]. It upshifts or downshifts the
frequency of a pump photon by the frequency of a
microwave photon. The upshift (anti-Stokes) process
corresponds to absorption of the pump photon and
microwave photon and emission of the higher frequency
photon. The downshift (Stokes) process corresponds to
absorption of the pump photon and emission of both
the microwave and lower frequency photon.
While the efficient modulation usually requires
an external microwave field, a spontaneous Stokes
parametric frequency conversion has been predicted
[253] and demonstrated for the frequency shifts ranging
from sub-THz to approximately 20 THz [255]. When
such frequencies are efficiently out-coupled from the
modulator, this process may be used as a narrow-band
all-optical THz source. A similar type of THz source
has been realized in a conventional cavity-assisted
single-resonant optical parametric oscillator (OPO)
[261]. Let us also point out that the spontaneous
Stokes process results in unavoidable quantum noise
background present in EOMs [262].
Utilizing microwave and optical resonances can
enhance the parametric process efficiency at the cost
of limiting the bandwidth [263–270]. Efficiency of
the existing commercial EOMs still remains very low
compared to the theoretical limit of combining each
microwave photon with an optical photon.
The
14
reason is relatively short interaction length as well
as insufficient spacial overlap of the interacting fields.
Recent developments of highly efficient resonant EOMs
based on WGM resonators allow to circumvent these
problems.
Electro-optically active WGM resonators are
attractive for the EOM application because they can
provide a good overlap between the microwave and
optical fields, which usually requires an additional
microwave cavity. They also have low loss and high
quality factors in a wide range of optical as well as
microwave frequencies determined by the transparency
window of the resonator material [187,271–291]. These
modulators operate either within the mode bandwidth
[283] or involve different high-Q optical modes of the
same [273,277,285] or different polarizations [292–294].
The efficiency and directionality of the modulation
process can be regulated by the phase matching
conditions. Realization of the phase matching between
light and microwaves is complicated because the index
of refraction of the electro-optical materials is very
different at the optical and microwave frequencies. It
can be achieved by optimizing the geometrical shape of
both the microwave and optical parts of the modulator
as well as using different materials for these parts.
The significant dissimilarity of the optical and
microwave wavelengths gives a lot of flexibility for such
optimization. For WGM resonators, this approach was
initially proposed in [271, 272]. It was shown that it
is possible to confine the microwave field in a metal
resonator built on a top of an optical resonator to
achieve the desirable phase matching [271–280]. In a
similar way it is possible to control the modulation
process with high flexibility, e.g. to suppress the
Stokes process nearly completely and create an SSB
modulator [293] that is able to upconvert a microwave
photon to the optical frequency domain with nearly
100% efficiency [295]. This type of modulator can
be utilized for counting microwave or THz photons at
room temperature [109, 110, 183, 296].
A WGM EOM can be characterized by a modulation coefficient, defined as the ratio of the output power of the first optical harmonic and the optical pump power. The modulation coefficient is pro2
portional to Pmw Q2 Qmw rij
[278], where Q and Qmw
are the loaded quality factors of the optical and mi2
crowave modes respectively, rij
is the relevant electrooptical coefficient of the material, and Pmw is the applied microwave power. Therefore, the higher the quality factors and the nonlinearity, the lower is the microwave power required to achieve the same modulation efficiency. WGM resonators made out of crystalline LiNbO3 and LiTaO3 , characterized by the optical bandwidth ranging from hundred kilohertz (weakly
coupled) to gigahertz (fully loaded) as well as by large
Nonlinear and Quantum Optics with Whispering Gallery Resonators
electro-optical coefficients [297], are particulary attractive for WGM EOMs. Tunable and multi-pole filters, resonant electro-optical modulators, photonic microwave receivers, opto-electronic microwave oscillators, and parametric frequency converters were realized
using such EOMs.
Let us compare the efficiency of a conventional
running wave electro-optical phase modulator and
a WGM-based modulator. The optical field Eout
emerging from a phase modulator is related to the
input field Ein as
Eout
V
= exp iπ
cos ωmw t ,
(27)
Ein
Vπ
where V is the voltage of the RF signal at the
modulator electrode, Vπ is the characteristic voltage of
the modulator imparting a π phase shift to the optical
carrier, and ωmw is the modulation frequency. The
relative power of the first modulation sidebands is
Pmw
P±
=
,
Pin
Psat
(28)
where the characteristic (saturation) power can be
expressed via the Vπ and resistance of the microwave
circuitry R as
4 V2
Psat = 2 π .
(29)
π 2R
For a WGM EOM based on z-cut lithium niobate
resonator we have [297]
Psat =
n2mw Vmw ωmw
2 σ2 ,
32πQmw Q2 n4e r33
(30)
where nmw and ne are the extraordinary refractive
indices at the microwave and optical frequencies,
respectively, Vmw is the microwave mode volume, r33
is the electro-optical coefficient, and σ is the overlap
integral for the process. It is easy to verify that for a
typical running wave modulator with Vπ = 3 V and
R = 50 Ohm the saturation power exceeds 100 mW,
while for a lithium niobate WGM EOM with 10 MHz
bandwidth it is only about 2.5 µW [110,298]. It means
that the equivalent Vπ of such WGM EOM is less than
20 mV.
In addition to the light modulation leading to
generation of optical harmonics, resonant WGM EOMs
can be used as quadratic receivers of the microwave
field [279, 280].
The operation principle of such
devices is based on a nonlinear absorption of the
input light, which increases proportionally to the
microwave signal power. The increase of the optical
loss changes the coupling conditions towards undercoupled and reduces the coupling contrast. Such
a behavior resembles operation of an opto-electronic
transistor-like device, where a low-power (a few tens
15
of microwatts) microwave signal changes transmission
of a higher-power (a few milliwatts) optical signal.
Similarly a modulated microwave input results in
modulation of the light passing through the resonator.
Optimally modified WGM EOMs can be furthermore used as efficient electric field sensors [221, 299–
303]. A sensor based on a microwave dielectric cylindrical antenna concentrating the microwave field within
a lithium niobate WGM resonator was originally proposed in [304, 305]. This sensor operates on similar
principles as the fiber-based running wave [306] and
resonant [307] electric field sensors. A more efficient
all-resonant WGM configuration of a dielectric E-field
sensor was introduced theoretically [308] and validated
experimentally [309]. High sensitivity of these devices
has inspired the application proposals in areas such as
e.g. biomedical studies [310].
Concluding the section on electro-optical phenomena in WGM resonators, we should also mention the
possibility of magneto-optical phenomena. Such a possibility has been discussed theoretically [311, 312], and
experimentally explored in ferromagnetic microspheres
made from yttrium iron garnet [313–316]. The conventional Faraday effect as well as non-reciprocal sidebands generation at the magnon frequency is observed
in these experiments. Although most typical materials
with significant Faraday effect are lossy, and their Qfactor is limited to approximately 106 [313], the emerging field of WGM magneto-optics holds a great promise
for building ultra-sensitive, room-temperature compact magnetometers as well as for efficient microwaveto-optics conversion.
4.2. Natural phase matching and selection rules for
second-order processes
Now let us turn to discussing second-order nonlinear
interaction of optical fields. The overlap integral (23)
is responsible for all resonator-specific features of the
three-wave mixing, including the mode selection rules.
These rules reflect the symmetries inherent to the
resonator and its eigenfunctions. The fundamental
WGM symmetry is associated with rotation, which is
expressed in the eimϕ term in the eigenfunction (1).
Therefore the fundamental selection rule arising from
(23) corresponds to the angular momenta conservation:
m1 + m2 = m3 . Further selection rules are discussed
in [317–320]∗ . In particular, it is shown [318–320] that
in large resonators the overlap integral σ123 factors
into the radial and angular parts as a very good
approximation. The angular part yields ClebschGordan coefficients hl1 , l2 ; m1 , m2 |l3 , m3 i that describe
the photon’s angular momenta conversion and invoke
∗ Note that [320] required a Corrigendum [321].
16
Nonlinear and Quantum Optics with Whispering Gallery Resonators
well-known selection rules:
m1 + m2 = m3 ,
|l1 − l2 | ≤ l3 ≤ l1 + l2 ,
l1 + l2 + l3 = 2N ,
(31)
where N is a natural number. For equatorial modes
(li = mi for i = 1, 2, 3) in large resonators, one can use
the following asymptotic relation [319]♯:
hm1 , m2 ; m1 , m2 |m3 , m3 i ≈ 0.5(m3 /π 3 )1/4 ,
(32)
which indicates a slow increase of the angular overlap,
scaling as approximately the fourth-power root of the
radius-to-wavelength ratio.
The radial part of the overlap integral given
by the Airy functions overlap does not lead to any
strict selection rules. However for large radial mode
numbers qi and respective wavelengths λi it favors such
conversion channels that q1 /λ1 + q2 /λ2 ≈ q3 /λ3 [320],
see Fig. 6. This occurs because for large q the radial
part Ai(u/um − αq ) of a WGM eigenfunction (3) takes
on oscillatory form, and the selection rules begin to
resemble those for coupled harmonic waves.
Analyzing the overlap integrals (23) one can
also learn how the nonlinear conversion efficiency
depends on the resonator size. For example in [160],
this dependence was theoretically studied for the
frequency doubling of several pump wavelengths via
various equatorial channels in a lithium niobate WGM
resonator. Curiously, in all studied cases the same
scaling law was found: |σ|2 ∝ R−1.8 , see Fig. 7.
♯ In Eq. (6) of [319], the numerical factor should be 4, not 8.
27
24
21
18
15
12
21
18
15
12
q780
9
6
3
1
2
3
4
5
6
7
8
9
q520
9
6
3
10
q1560
Figure 6. Absolute-square of the overlap integral (23) radial
part is represented by the dots size for various mode numbers
combinations {q1 , q2 , q3 }, for the 780 nm + 1550 nm → 520 nm
frequency-sum generation in a Lithium niobate WGM resonator
of 0.65 mm radius. Reprinted from [320].
Figure 7. Overlap integrals σ for λp = 1 µm to λSH = 500 nm
SHG are shown as functions of the resonator radius for various
conversion channels [qp , qSH ]. Reprinted from [160].
The same power scaling law was established, although
not reported, for the 780 nm + 1550 nm → 520 nm
frequency-sum generation [320]. Moreover, this is
consistent with the Vp−1 ∝ R−1.83 WGM volume
scaling in micro spheres [322, 323]†† .
4.3. Quasi-phase matching in periodically-poled WGM
resonators
Similarly to bulk crystals and straight waveguides,
phase matching in WGM resonators can be modified by
periodical poling. Here the “periodic” means a radial
pattern such as shown in Fig. 8 rather than a series
of parallel lines. Periodical poling in crystals such
as lithium niobate changes the local direction of the
(2)
crystallographic z-axis and the sign of those χijk tensor
components that have only one or all three indices i, j
and k matching z. As a result, the χ̃(2) factor in (21)
becomes coordinate-dependent and enters the overlap
integral (23).
Although various radial poling patterns have
been discussed [325], it is easy to see that what
really matters is the Fourier-transform of this pattern
with respect to the azimuthal angle ϕ. The most
efficient quasi-phase matching (QPM) is therefore
achieved with a radially-poled structure consisting of K
equidistant (in φ) lines, see Fig 8. In this case the first
phase matching condition (31) is modified as follows:
m1 + m2 = m3 ± N K,
N = 0, 1, 2, ...,
(33)
where in close analogy with periodically poled straight
waveguides and bulk crystals N is the the phase
†† Ref. [323] quotes Vp ∝ R1.83 as calculated in [322], but [322]
only provides a plot for Vp (R) in its Fig.3. We assume that in
[323] the power-law fitting of this plot for large R was performed.
17
Nonlinear and Quantum Optics with Whispering Gallery Resonators
technique [216, 217].
4.4. Crystal symmetry based and “cyclic” quasi-phase
matching
Figure 8. Radial poling pattern on a lithium niobate wafer,
visualized by etching. Reprinted from [324].
matching order. It should be mentioned that a
fabrication error leading to eccentricity between the
resonator and poling pattern may have a considerable
phase matching broadening effect that may be
desirable or undesirable depending on the specific
application [157].
Let us also point out that the early and some of
the later demonstrations of QPM in WGM resonators
were proposed [254] and carried out [107, 326] with
the poling patterns consisting of parallel, rather than
radial, lines. Phase matching in such resonators allows
interesting interpretation in terms of the effective index
approach discussed in section 2.2. In this approach a
large WGM resonator can be “unfolded” into a straight
waveguide with a modified index of refraction. A linear
equidistant poling pattern then becomes variable, with
the period ranging from its nominal value (where the
poling lines are radial) to practically infinity (where
the poling lines are tangential). If the nominal period is
sufficiently small, there will be four (in degenerate case,
two) locations where the poling period is just right and
the QPM is locally achieved for the desired process at
the desired wavelengths. Only these narrow segments
of the resonator will contribute to the nonlinear
conversion. Importantly, their contributions will add
coherently and may lead to either constructive, or
destructive interference. A wide range of wavelengths
can be nonlinearly converted in such a resonator, each
at its own four locations. The cost of the wavelength
versatility is the reduced conversion efficiency due to a
limited interaction length at each location.
Using the linearly-poled resonators was enabled
by the commercial availability of periodically poled
lithium niobate (PPLN)wafers. Since then, several
lithography-based radial poling techniques have been
developed [324] specifically for WGM resonators,
including the already mentioned calligraphic poling
QPM in WGM resonators may arise even without
artificial domain inversion. As an optic field propagates
around the resonator, its polarization orientation
generally changes with respect to the crystallographic
axes. This may lead to ϕ-dependent modulation of
the effective nonlinearity χ̃(2) and/or of the index of
refraction. The former effect can be observed e.g. in
the SHG experiments when χ̃(2) couples the x and
y projections of the optic field. This process has
been predicted [327–329] and observed [330] in the
4̄ symmetry crystals, such as GaAs, GaP, ZnSe and
others. In this case harmonic variation of the field
projections leads to the following modification of the
azimuthal phase matching condition:
m1 + m2 = m3 ± 2.
(34)
This modification may have an appreciable effect on
the phase matching wavelength in small resonators
where the FSR is large.
In a birefringent resonator whose optical axis
does not match the axis of symmetry, the index of
refraction for TM-polarized WGMs also becomes a
periodic function of ϕ, as we discussed in section
2.1. This leads to a more complicated situation than
just the ϕ-dependent χ̃(2) . Because this situation is
not analytically tractable as yet, only a special case
of a uniaxial crystal with the optical axis lying in
the resonator equatorial plane has been utilized in
nonlinear optics. This type of resonators, usually
called the x-cut or xy-cut resonators, manufactured
from crystal quartz [286], lithium niobate and tantalate
[331], and BBO [331–333] have been shown to support
high-Q modes that can be identified as TE and TM at
least at the coupling location.
An SHG cyclic phase matching in an x-cut BBO
resonator has been demonstrated in [332, 333]. In
such resonators, the refraction index for TM modes
nTM oscillates between the ordinary and extraordinary
values no and ne according to
2
2
2
cos(ϕ)
sin(ϕ)
1
=
+
,
(35)
nTM (ϕ)
ne
no
where we chose to measure the azimuthal angle ϕ
from the optical axis.
Since this resonator has
no (infinite-order) rotation symmetry, the quantum
numbers q, L, m cannot be introduced in the strict
sense, and selection rules (31) no longer apply. Phase
matching in this resonator can be formulated following
the effective index of refraction approach as
β3 = β1 + β2 ,
(36)
18
Nonlinear and Quantum Optics with Whispering Gallery Resonators
where
2π
β(λ) =
ñ(λ)
(37)
λ
is the effective wave number for each mode. The local
phase detuning for up-conversion of a TE polarized
pump into a TM polarized second harmonic ∆β(ϕ) =
2βp − βSH (ϕ) depends on the azimuthal angle ϕ or
on local coordinate z = Rϕ measured along the
“unfolded” waveguide. This detuning governs the
propagation equation for the second harmonic field
[332]:
Z z
βSH (z) (2)
dESH
2
′
′
=i 2
χ̃ (z)Ep exp i
∆β(z )dz .
dz
nSH (z)
0
(38)
As in the previously discussed example of linearly
poled WGM resonators, only four short waveguide
regions may contribute to the second harmonic
field build-up.
However now these regions are
determined not by the local QPM, but by a stationaryphase condition ∆β(zpm ) = 0. Other z-dependent
parameters of equation (38) can be evaluated at
z = zpm as a very good approximation. Again,
contributions from different regions zpm may interfere
constructively or destructively, and a wide-range
cyclic phase matching comes at the price of reduced
conversion efficiency.
4.5. Exotic phase matching in WGM resonators
The phase matching conditions, or mode selection
rules (31) in WGM resonators are significantly more
relaxed compared to free space. Indeed, the usual
phase matching requirement ~k1 +~k2 = ~k3 is represented
by a set of three equations imposing constraints,
whereas (31) has only one such equation. This
flexibility allows for achieving in WGM resonators
such types of phase matching that cannot be achieved
in bulk crystals. In particular, Type 0 and Type
II SPDC was predicted in lithium niobate, lithium
tantalate and BBO z-cut crystals in a wide range
of pump wavelengths [160], although the overlap
penalty for using high-order modes with large q
may be significant.
It is even possible to find
a double phase matching, e.g. for nondegenerate
SPDC with simultaneous frequency-doubling of the
signal, idler or both [160]. Double phase matching
in strongly nonlinear systems is interesting in the
context of quantum-optics applications. It can lead
to multipartite entanglement [334] and to control of
the photon pair statistics in SPDC via quantum Zeno
blockade.
A search for exotic phase matching in WGM
resonators can be conveniently accomplished using
the effective index of refraction approach introduced
in section 2.2. This approach leads to a simple
but approximate analytical expressions for the phase
matching conditions in the wavequide form (36). They
are approximate because the WGM frequencies are
treated as continuous, whereas in an actual resonator
they are discrete. Therefore the accuracy of this
method is limited by the resonator’s free spectral
range. The simplicity of this approach comes from
the same approximation, which allows us to solve
a continuous-value set of equations (36) rather than
to look for the discrete-value solutions to the WGM
dispersion equations set (2) constrained by selection
rules (31).
An important observation regarding the effective
index phase matching is that it strongly depends on the
radial mode numbers q, especially when the resonator
is small. In section 2.2 we already mentioned that the
geometric correction to the refractive index is negative:
∆n = ñ−n < 0. Moreover, |∆n| becomes progressively
larger for larger q, because the optical field of a higherorder mode effectively has a shorter path to travel, see
Fig. 2. In small resonators this effect can be quite
strong even in comparison with natural birefrigence,
which explains the unusual phase matching. We should
point out that while the double phase matching can
be relatively easily found by the effective index of
refraction method, its realization in a fully-resonant
system such as a WGM resonator is difficult to achieve
in practice. It depends on such control parameters that
cannot be easily tuned, e.g. the resonator radius and
rim shape.
4.6. Dynamics of the second-order processes in
triply-resonant systems
Analysis of nonlinear optical processes dynamics in
phase-matched WGM resonators has been reported for
the SHG [107,213,256,317,335] and OPO [213,253–256,
336], as well as for the sum-frequency generation (SFG)
[320, 321, 337, 338] and difference-frequency generation
(DFG) [339]. Usually this analysis is carried out in
terms of coupled ODEs such as given by Eq. (25). It
shows the dynamics similar to other triply-resonant
systems with second-order nonlinearity, see e.g. [340–
345]. Its details strongly depend on the in/out coupling
regime for the nonlinearly coupled modes, as well as
on the relation between the resonator linear loss and
nonlinear conversion rates. There are however some
universal features. For example, the SHG conversion
efficiency reaches a maximum at a saturation pump
power Ps , and then drops. The OPO has a pump
power threshold Pth . For a degenerate OPO these two
parameters are closely related: Ps = 4Pth [319], where
Pth =
c
λp Qp
n3
16πσχ(2) Qs
2
,
(39)
19
Nonlinear and Quantum Optics with Whispering Gallery Resonators
1.0
SFG efficiency (a.u.)
where λp is the pump wavelength and n = ne (λp ) =
no (λs ) is the phase-matched refractive index.
Degenerate Type I SPDC in WGM resonators
cannot be described with just a pair of coupledmode equations(25).
Indeed, because the WGM
spectral lines are locally equidistant, degenerate
down conversion of the pump mode with an orbital
momentum 2m0 into the signal (and idler) mode with
ms = mi = m0 also implies the possibility of the same
pump conversion into non-degenerate modes ms =
m0 ± ∆m, mi = m0 ∓ ∆m. Here ∆m = 1, 2, ..., and up
to some maximum number determined by the WGM
linewidths and the group velocity dispersion of the
resonator. Conversion channels with small ∆m will be
nearly on-resonance and hence almost equally efficient.
A SPDC optical comb is therefore expected [346] to
form around the degenerate wavelength.
As a higher-order effect in a multi-mode neardegenerate SPDC, the tresholdless sum-frequency
generation process among the parametric comb lines
may lead to building up another comb around the
pump mode. For this secondary conversion to be
efficient, the FSRs at the pump and degenerate SPDC
wavelengths must be nearly equal, which is granted
by the phase matching. This kind of a double-comb
structure has been observed [347, 348] and studied
theoretically [349] in straight waveguide cavities.
The presence of multiple near-equidistant modes
also makes the conventional WGM SHG description
based on two coupled-modes equations insufficient
beyond the low-power linear regime.
Once the
circulating SHG power exceeds the OPO threshold,
the non-degenerate multiple-line down conversion
commences. This phenomenon has been predicted
to lead to self-pulsing in triply-resonant [350], and
observed in single-resonant SHG processes [351]. It
may be responsible for the anomalous SHG signal
behavior observed in [318].
Similarly to the SHG process, conversion efficiency
saturation also occurs in SFG and DFG. Above
the saturation power, efficiency of these processes
decreases, asymptotically approaching zero as shown
in Fig. 9. In this Figure we plot the SFG conversion
efficiency (defined as the ratio of the SFG signal to
the input probe power) vs. the input pump power
calculated using the equations from [320].
Both
parameters are given in arbitrary units. The actual
peak conversion efficiency depends on the relation
between the linear and non-linear coupling rates of
all involved modes. Three curves in Fig. 9 illustrate
the effect of increasing the nonlinear coupling rate g
introduced in Eq. (22).
The saturation behavior of SFG in triply-resonant
structures is different from the non-resonant structures
with undepleted pump [352], where the oscillations of
0.8
g
0.6
2g
0.4
4g
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Pump power (a.u.)
Figure 9. A qualitative dependence of the SFG conversion
efficiency on the pump power for various nonlinear coupling rates
g. All three fields are assumed to be resonant.
the probe and sum-frequency powers are expected. In
high-Q, strongly nonlinear resonators the saturation
opens up the possibility of realization an efficient lowpower all-optical switch, which has been both predicted
[337–339] and implemented in Fabri-Perot [353] as well
as in WGM resonators [354].
4.7. Experimental observations of the second-order
processes
Optical WGM and ring resonators with the secondorder nonlinearity have been successfully used for
optical frequency doubling [107, 156, 257, 318, 330, 332,
333,355–362], tripling [356,363], and quadrupling [326],
as well as parametric down conversion above [108, 157,
158, 255, 319, 364, 365] or below [112, 228, 366, 367] the
OPO threshold. Generation of optical frequency sum
[320] and difference [368] also have been demonstrated.
Some of these experiments [108, 110, 112, 158, 228, 255,
257, 318–320, 357, 360, 361, 366, 367] have been relying
on the natural phase matching whereas QPM with
periodical domain inversion was harnessed in others
[107,157,326,363–365]. Crystal symmetry based QPM
[330,368] and the cyclic phase matching [332,333] have
also been experimentally realized.
WGM resonators have been shown to have a
high nonlinear conversion efficiency, which is due to
the exceptionally high Q-factor and small interaction
volume. For SHG of the pump wavelength λp in lowpower regime the conversion efficiency is proportional
to the pump power, and therefore can be characterized
by the slope efficiency η. At a higher power regime
saturation occurs and the conversion efficiency reaches
its limit ηm . These parameters are listed in Table 1
for the SHG experiments discussed here. In the
absence of the saturation power data, ηm is evaluated
20
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Ref.
[318]
[257]
[332]
[333]
[107]
[156]
[330]
[362]
[359]
[358]
[357]
[360]
[361]
λp nm
1064
490
634-1560
634
1550
1560
1985
1550
800
1546
1550
1580
1548
Crystal
LiNbO3
LB4
BBO
BBO
PPLN
GaN
GaAs
GaP
LiNbO3
LiNbO3
AlN
AlGaAs
ZnSe
η W−1
4500
37
0.6-18
7.4
1.5
0.15
0.5
0.38
0.135
0.135
0.001
7e-4
8e-4
Ps
3.2
300
0.1
3
ηm ‰
90
22
0.7-16
7.4
500
0.03
0.005
0.133
0.08
0.005
0.025
0.002
0.001
Table 1. Summary of reported SHG observations in WGM
resonators: pump wavelength λp in nanometers, resonator
material (PPLN is periodically poled lithium niobate, LB4 is
lithium tetraborate), slope efficiency η, saturation power Ps in
mW, and maximum observed pump power conversion ηm .
at the maximum pump power used in a particular
experiment.
WGM OPOs can be characterized by the threshold power Pth and maximum observed conversion efficiency ηm . These parameters are summarized in Table 2. All these experiments have been carried out
with lithium niobate resonators: not poled for [319],
and periodically poled in radial direction for the others. Strong variation in the listed threshold power is a
consequence of the trade-off inherent to WGM OPOs:
for strongly out-coupled resonators the maximum observed conversion efficiency increase, but so does the
threshold, due to reduction of the Q-factors. In different experiments different choices have been made
regarding this trade-off.
It should be mentioned that in many cases,
especially among the most efficient SPDC and SHG
processes observed in WGM resonators, the measured
conversion efficiency falls considerably short of the
theoretical predictions. The same applies to the SFG
observations [320, 321]. Usually this is attributed to
a sub-optimal selection of the conversion channel, e.g.
coupling non-equatorial modes, or to photorefractive
Ref.
[319]
[364]
[158]
[365]
[217]
λp → λs + λi , nm
532 → 1010 + 1120
1037 → 2011 + 2140
488 → 707 + 865
1040 → 2080 + 2080
1040 → 1800 + 2500
Pth , µW
7-28
200
66-330
86-2200
21
ηm %
18
30
7
55
45
Table 2. Summary of reported OPO observations in WGM
resonators made from lithium niobate: pump, signal and idler
wavelengths, threshold power Pth and maximum observed pump
power conversion ηm .
damage. A rigorous investigation of this discrepancy
has not been carried out.
5. Third-order nonlinear processes
Third-order nonlinear optical processes involve four
interacting optical fields and generally described by
the term four-wave mixing (FWM). In this section
we focus on two major FWM processes observed
in microresonators – hyper-parametric oscillation and
third harmonic generation. Weak cubic nonlinearity of
transparent optical solids requires high peak intensity
of the pump to observe these processes in bulk. Such
an intensity level can be reached with powerful pulsed
lasers. Interaction length can be increased by using
optical fibers, which allows to reduce the laser power
requirements [1, 369]. Usage of resonators allows to
reduce it even further due to the intracavity field buildup, making the FWM effects observable with CW low
power lasers.
5.1. Interaction Hamiltonian and phase matching
Third-order nonlinear optical processes result from the
cubic response of the media polarization P~ to the
~ [248], similarly to quadratic
external electric field E
response (18):
X (3)
(3)
Pi = 4
χijkl Ej Ek El ,
(40)
j,k,l
where χ(3) is the third-order nonlinear susceptibility
tensor. The third-order polarization (40) corresponds
to the interaction energy
Z
X (3)
H (3) =
dV
χijkl Ei Ej Ek El .
(41)
i,j,k,l
This Hamiltonian can be simplified for the case
of a hyper-parametric or third harmonic generation
process in isotropic homogeneous medium. The hyperparametric process involves transformation of a pair
of photons at some frequencies ω1 and ω2 to another
pair of photons at frequencies ω3 and ω4 , so that
ω1 + ω2 = ω3 + ω4 . Often ω1 = ω2 pertain to the same
optical pump field, which suggests an analogy with the
previously discussed parametric down conversion and
justifies the term “hyperparametric”. In the case of
a narrowband process (|ωi − ωj | ≪ |ωi + ωj |, i, j =
1, . . . , 4) the Hamiltonian (41) in the rotating wave
approximation for this process leads to
ĤHP = −
~g X
âi âj â†k â†l ,
2
(42)
i,j,k,l
where g is the coupling parameter.
Under the
assumption of complete spacial overlap of the resonator
21
Nonlinear and Quantum Optics with Whispering Gallery Resonators
modes
~ω02 c
n2 .
(43)
Vn2
Here V is the effective mode volume [100], n is
the linear index of refraction and n2 is the cubic
nonlinearity of the material at the carrier frequency
ω0 . We assumed that the medium is isotropic and
homogeneous, so that the nonlinearity is polarizationindependent and (see [248])
g=
χ(3) =
n2 c
n2 .
12π 2
(44)
Resonant third harmonic generation (THG) can
be described using a similar interaction Hamiltonian
presented in the rotation wave approximation
ĤT HG = −
~g X 3 †
(â â + â†3
i âj ),
3 i,j i j
(45)
where g is defined by Eq. (43).
Interaction
Hamiltonians (42) and (45) lead to the same general
form ODEs (25) that are already familiar to us from
the second-order interaction discussion.
5.2. Self-Phase and Cross-Phase Modulation
The simplest effect related to the FWM is the effect
of self-phase modulation (SPM). It involves only
one mode evolving in time in accordance with the
Hamiltonian
g
ĤSP M = −~ ↠↠ââ.
2
(46)
SPM results in power-dependent frequency shift and
corresponding distortion of the WGM resonance curve.
At higher pump power it leads to the bistability with
respect to the optical pump. It is easy to show
using Eq. (46) that for a lossless mode populated with
N̂ = ↠â photons at frequency ω0 the state evolution
is described by the operator
â(t) = e−i(ω0 −gN̂ )t â(0).
(47)
Therefore, g represents an SPM frequency shift of the
mode per photon.
Among the WGM resonators, the effect of
bistability was initially studied in microspheres
[66, 370].
A significant number of bistabilityrelated research papers was published very recently.
For instance, optical bistability in Er-Yb co-doped
phosphate glass microspheres was investigated in
[371].
A regenerative pulsation arising from the
competition between Kerr nonlinearity and thermal
nonlinearity, when the two nonlinearities with very
different timescales are comparable in magnitude
but opposite in sign, was studied in cryogenically
Figure 10. A Mach-Zehnder interferometer coupled to an
AlGaAs microring. Reprinted with adaptations from [374].
cooled microspheres [372]. Signal processing and alloptical switching by means of Kerr-based bistability
in silica bottle microresonators with Q > 108 was
studied in [86, 87]. It was predicted that coupled
microresonators can demonstrate stronger SPM effect
than a single resonator in the chain [373]. This was
later demonstrated in a setup combining an on-chip
Mach-Zehnder interferometer with a microring [374], as
shown in Fig. 10. Here, a nonlinear phase shift induced
by the microring has lead to self-switching of an optical
pulse from one Mach-Zehnder interferometer output to
the other depending on the pulse power.
SPM can also lead to the quantum phenomena of
photonic blockade and quadrature squeezing, discussed
in sections 7.
Cross-Phase Modulation (XPM) is another simple
FWM effect that involves interaction of two modes.
The interaction is represented by the Hamiltonian
ĤXP M = −2~g↠b̂† âb̂.
(48)
XPM results in frequency shift of each mode that
depends on the photon number in the other mode, as
described by the evolution equations
â(t) = e−i(ω0 −2gN̂b )t â(0),
b̂(t) = e−i(ω0 −2gN̂a )t b̂(0).
(49)
XPM was directly observed in an amorphous
silicon carbide microdisk with R = 6 µm using a
pump-probe technique [375]. This effect can be useful
in quantum nondemolition measurements of photon
number in WGM resonators [376].
5.3. Hyper-parametric oscillation
Hyper-parametric optical oscillations [377] involve, at
the fundamental level, three optical modes populated
with four photons: two pump, one signal, and one idler.
This system is also prone to the SPM and XPM. Its
complete Hamiltonian is
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Ĥ
= Ĥ0 + ĤHP ,
†
(50)
~ω+ b̂†+ b̂+
+
~ω− b̂†− b̂− ,
Ĥ0
= ~ω0 â â +
ĤHP
= ĤSP M + ĤXP M + ĤP ,
g
= − ~ ↠↠ââ + b̂†+ b̂†+ b̂+ b̂+ + b̂†− b̂†− b̂− b̂− ,
2
= − 2~g b̂†− b̂†+ b̂+ b̂− + ↠b̂†+ b̂+ â + ↠b̂†− b̂− â ,
= − ~g b̂†− b̂†+ ââ + ↠↠b̂+ b̂− ,
ĤSP M
ĤXP M
ĤP
where ω0 , ω+ , and ω− are the eigenfrequencies of
the pump, signal, and idler optical cavity modes,
respectively, â, b̂+ , and b̂− are the annihilation
operators for these modes, and g is the coupling
constant (43).
ĤSP M , ĤXP M , and ĤP stand
for the SPM, XPM and hyper-parametric processes,
respectively.
The SPM and XPM terms change
spectral properties of the system, while the hyperparametric term results in the oscillation process, in
which the signal and idler optical sidebands grow at
the expense of the pumping wave.
Using the same coupling constant g in all terms
of (50) is an approximation which is based on the
mode volume concept and on the assumption of
complete spacial overlap made while deriving (43).
A more accurate analysis requires evaluation of the
overlap integrals similar to (23) but with four WGM
eigenfunctions instead of three. The results may be
different for different processes as well as for different
modes. These overlap integrals determine the selection
rules arising for each of the third-order processes.
The orbital selection rule only arises for the hyperparametric process: 2ma = mb+ + mb− . Other
selection rules associated with the angular momentum
conservation of four photons are not as broadly used
as those arising in the three-wave mixing case from the
Clebsch-Gordan coefficients, see Eq. (31). The radial
part of these selection rules is even less studied. We
will not invoke this cumbersome analysis here and stay
with the approximation (43) throughout the rest of this
paper.
A WGM resonator supporting hyper-parametric
interactions may become unstable and begin to oscillate [378]. The WGM hyper-parametric oscillations
were first observed in liquid droplets [59] and then studied in solid state WGM resonators [379,380]. Similarly
to the case of χ(2) -based OPO discussed in section 4,
the onset of the hyper-parametric oscillations occurs
when the stimulated conversion dominates the spontaneous conversion, which requires the pump power to
exceed a certain threshold. The pump threshold of resonant hyper-parametric oscillation [378] is
PHP ≃ 0.4
ω 0 n2 V
.
cn2 Q2
(51)
22
Equation (51) is derived under the assumptions that
parameters of all interacting modes are identical, and
that the pump mode is critically coupled. It is easy to
see that the threshold decreases with the quality factor
increase and mode volume decrease.
The hyper-parametric oscillations are different
from the parametric ones. The parametric oscillations
i) are based on χ(2) nonlinearity coupling three photons, and ii) involve far separated optical frequencies.
The hyper-parametric oscillations i) are based on χ(3)
nonlinearity coupling four photons, and ii) usually involve nearly-degenerate optical frequencies, although
strongly non-degenerate hyper-parametric oscillations
have been also observed [381].
Three-mode hyper-parametric process can be
observed experimentally. Suitable mode structure can
be created in a WGM resonator either by intentional
engineering of its spectrum, as discussed earlier in
sections 2 and 3, or by properly selecting the optical
pump frequency so that only three modes become
phase matched. Different mode families can be used
to achieve this type of the hyper-parametric oscillation
[381]. Unless these special measures are taken, multiple
modes of the same WGM family become phasematched leading to a multi-mode hyper-parametric
oscillation and formation of an optical comb structure.
5.4. Cascaded hyper-parametric oscillation as
frequency comb generation
Frequency combs are important in contemporary
physics.
Two-point stabilization of a frequency
comb produced in a mode-locked laser resulted in a
revolution in metrology and many other fields [382],
providing a precise link between oscillators operating in
optical and microwave frequency domains. A stabilized
frequency comb is an important part of optical clocks
[382] and of high spectral purity microwave photonic
oscillators [383]. Frequency combs are used in the
search for extraterrestrial planets [384], in optical
communications [385] and sensors [386].
In WGM resonators, frequency combs can be
generated via hyper-parametric processes. Similarly to
the degenerate SPDC discussed in section 4.6, hyperparametric conversion of a monochromatic pump may
occur simultaneously into multiple modes, forming
a frequency comb structure. Microresonator-based
optical frequency combs have rapidly become a
subject of extensive research as a simple alternative
to the conventional frequency combs produced with
femtosecond modelocked lasers [78, 100, 385, 387–
393].
It was proven experimentally [78] that
the microresonator-based combs can have excellent
uniformity as well as high repetition rate, which makes
them ideal for many practical applications.
Similarly to the simple three-mode hyper-
Nonlinear and Quantum Optics with Whispering Gallery Resonators
parametric oscillation, generation of Kerr frequency
combs occurs above a certain power threshold. While
expression (51) for the hyper-parametric oscillation
threshold power PHP can be used as an estimate,
the actual threshold value may also depend on the
material group velocity dispersion (GVD) and thermorefractive properties, as well as on the excitation
regime [102,394]. Two different excitation regimes can
be realized. In the “soft” regime, the oscillations may
start from either the vacuum fluctuations of the field or
nonzero initial conditions, while in the “hard” regime,
the oscillations can only start from nonzero initial conditions.
Under certain conditions the lines of the generated
frequency combs are phase locked. Such frequency
combs are considered coherent. The phase of the
mutually locked comb lines depends on the phase of
the optical pump. While the relative phase stability
of the harmonics can be much higher than the pump
phase stability, it is still prone to fundamental diffusion
processes.
Generation of a Kerr frequency comb is a multimode nonlinear process that does not need an optical
amplifying medium to sustain itself. This process can
be conservative, in the sense that the same optical
power exits the resonator as enters it, for the case
of an overloaded WGMR with vanishing attenuation.
This energy conservation law does not prevent power
redistribution among the comb lines.
In some cases both hyper-parametric oscillations
that involve only a few frequency harmonics [379, 380]
and broad frequency combs [78, 79, 388] observed in
WGM resonators can be considered as generalized
hyper-parametric oscillations because the high order
harmonics smoothly grow from the lower order ones,
as described in [395]. This allows to define three stages
of Kerr comb development. At the first stage, a hyperparametric oscillation starts. At the second stage,
each oscillation harmonic forms its own, secondary,
independent oscillation. At the third stage, all the
secondary harmonics phase lock due to the XPM effect.
The locking occurs if the primary oscillation spacing is
nearly an integer multiple of the secondary oscillation
spacing. This simplified model has several deficiencies.
It does not describe the general hard excitation regime
[102] of the oscillations when the harmonics cannot be
generated from quantum fluctuations. Furthermore,
the frequency combs corresponding to optical solitons
confined in the resonators have a different growth
mechanism, in which case all harmonics are formed
simultaneously.
5.5. Mode-locked Kerr-comb generation
Observations. The major difference between the
cascaded hyper-parametric oscillation and the mode-
23
locked Kerr frequency comb generation is the natural
formation of high-contrast optical pulses within the
microresonator pumped solely with CW light, in the
latter case. The pulse generation is usually associated
with the hard excitation regime in which the pulses
[102] emerge as dynamic normal modes of the nonlinear
structure. Duration of the pulses is much shorter
than the round trip time of the cavity. In contrast,
the hyper-parametric oscillation corresponds to low
contrast pulses with duration comparable with the
round trip time [396].
Anomalous GVD is considered optimal for the
mode-locked Kerr frequency combs corresponding
to formation of bright optical solitons. The first
experimental demonstration of the mode-locked Kerr
comb in a WGM resonator was reported in [397].
The pulses with 35.2 GHz FSR and 450 kHz
loaded bandwidth were generated in a magnesium
fluoride resonator and recorded using frequencyresolved optical gating technique.
Existence of
the mode-locked (soliton) regime was independently
confirmed for magnesium fluoride [143], silicon nitride
[398] and fused silica [399] resonators. A broad, 2/3
octave, coherent frequency comb was observed in a
silicon nitride resonator [400]. The optical spectra
resembling the mode-locked regime were observed by
other groups as well [401, 402].
Normal GVD resonators can support generation of
dark solitons. This phenomenon was predicted theoretically [394,403–405] and observed experimentally [406].
Both narrow [407,408] and broad Kerr frequency combs
were demonstrated for normal GVD [404, 406]. The
stability range of the mode-locked combs in normal
GVD resonators is rather narrow [403,409]. A different
type of stable pulses can exist in this case, enabled by
irregularities of the WGM spectrum due to the modes
anti-crossing [410].
A resonator can support one or several solitons
as independent solutions of the corresponding set of
Hamilton equations. It is possible to excite a preferable
number of solitons by selecting proper initial conditions
or by dynamically manipulating the pumping light
parameters [411–413]. Each of these independent
solutions corresponds to a separate frequency comb
with no fixed phase relation to others. Multiple
solitons are not always independent.
They can
interact via e.g. Raman scattering [414] which leads
to their synchronization. In [400] two- and threesoliton combs were experimentally investigated and
coherence properties of the two-soliton combs were
demonstrated. In this case the solitons propagating
around the resonator rim are separated by exactly
180◦ .
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Theoretical Models. Kerr frequency combs generation
can be described in terms of the ODEs such as (25),
with the number of equations equal to the number of
the comb lines. This approach was proposed in [101]
considering the theoretical model of three-mode hyperparametric oscillation [378]. Numerical simulations of
the multimode Kerr frequency comb were advanced
in [100, 101, 415] and then followed by other groups
[102, 138, 397, 416–418].
Another, equivalent to the ODE [419], approach
proposed for theoretical study of the Kerr frequency
comb generation involves Lugiato-Lefever (LL) equation. The original LL equation [420] has been adapted
for the description of frequency combs. Similarly to
the ODE set, the LL equation is valid at times longer
than the ring-down time of the resonator. Solutions of
this equation have been studied, and existence of the
stationary mode-locked regimes of the Kerr frequency
comb has been confirmed [421–425]. The analyisis predicts that optical pulses can emerge from the nonlinear
microresonator pumped with CW light, without any
pulse seeding [392, 397, 419, 426, 427]. The power loss
of the pulses is compensated by their nonlinear interaction with the background within the resonator [424].
The LL equation for the optical field inside a
nonlinear resonator with constant second-order GVD
reads [419, 426]
∂2A
i
∂A
β2Σ 2 =
(52)
+
∂τ
2τ0
∂t
ig|A|2 A − [γ0c + γ0 + i(ωj0 − ω)] A + F0 ,
where A(τ, t) is the slowly varying envelope of the
electric field, τ is the slow time, t = t − z/vg is
the retarded time, vg is the group velocity, β2Σ =
(2ωj0 − ωj0+1 − ωj0−1 )(τ0 /Ω2 ) is the GVD parameter,
Ω ≃ (ωj0+1 − ωj0−1 )/2 is the FSR of the resonator. By
definition, time scale τ is much longer than the round
trip time τ0 . Eq. (52) can be used to find the time
dependent amplitude of light exiting the resonator.
The similarity between the LL and ODE approaches was shown in [392, 419]. An optimization of
the numerical algorithm makes the computation time
for the LL and ODE approaches also equivalent [418].
The possibility of the direct pulse generation in the
resonator was demonstrated by the numerical solution
of the ODE set [102]. Analytical solutions of the LL
equation describing the mode-locked Kerr comb regime
were provided in [397,427,428] and technical as well as
fundamental quantum noise associated with the repetition rate of the pulses was analyzed in [428–430].
Impact of the modes anti-crossing. The first observed
phase-locked Kerr frequency comb, obtained in an onchip fused silica cavity [78], was most likely impacted
by the mode GVD modification due to the anticrossings [138]. A similar assumption can be made
24
regarding other frequency comb observations [395,431,
432]. The envelope shape of these frequency combs
is significantly different from that of a mode-locked
(soliton) frequency comb. Modes anti-crossing results
in an asymmetry of the frequency comb envelope.
It can suppress or enhance generation of comb lines
within some frequency bands. Introduction of the
avoided mode crossings seems to be important to
explain observation of the majority of Kerr frequency
combs, and a further study is needed for a complete
understanding of this process.
The first observation of the low repetition rate
Kerr frequency combs in fluorite resonators was also
based on modes crossings [388]. Initially a primary
comb emerged, and then a broad frequency comb with
irregular envelope was generated. The repetition rate
of the primary comb was determined by the interaction
between different families of the WGMs. Similar
results were obtained with another fluorite resonator
[433]. A truly mode-locked frequency comb generated
in a normal GVD resonator has a significantly different
envelope [404]. A controlled anti-crossing of modes
from two different coupled resonators [434] can be used
to adjust the repetition rate and achieve mode locking
of a WGM comb [435, 436].
Unsolved problems. There are many unsolved problems related to Kerr frequency combs. Low efficiency of
the nonlinear process involving the mode-locked Kerr
comb is one of them [437]. The efficiency, defined
as the ratio of the pump power and averaged pulse
train power, degrades with growth of the comb spectral width, and is inversely proportional to the number of comb lines. A solution has to be developed to
circumvent this restriction and to enable generation of
the spectrally broad, high power Kerr frequency combs
with a relatively low repetition rate. Other problems
include generation of a coherent octave-spanning Kerr
frequency comb, self-stabilization of the comb, generation of frequency combs in visible and ultra violet parts
of the optical spectrum (mid-IR frequency combs were
recently demonstrated [438, 439]), as well as complete
integration of the comb generator with the pump laser
on a chip. The complete understanding of the coherence properties of a Kerr frequency comb is also yet to
be achieved.
5.6. Forced frequency combs
It is possible to generate thresholdless frequency combs
using not monochromatic, but bichromatic [440] or
multi-chromatic [441] pumping light. The repetition
rate of such forced combs is determined by the
modulation frequency if the pump power is low enough.
However, if the power exceeds a certain value, the comb
becomes unstable. The major signature of this process
Nonlinear and Quantum Optics with Whispering Gallery Resonators
is that the generated comb lines are no longer locked to
the frequencies of the pumps [442]. The forced comb
can also become chaotic.
The impact of the pump spectrum on the comb
formation was studied under different conditions [410–
412, 443, 444]. These include flat-topped dissipative
solitonic pulse generation [410] and parametric seeding
for stabilization of a conventional Kerr frequency comb
[411, 412]. The parametric seeding can be used to
improve the combs stability and achieve low frequency
and phase noise. It was also suggested that advantage
may be taken of the injection locking properties of
the Kerr frequency comb oscillator to create an optoelectronic oscillator that involves an active Kerr comb
oscillator as a part of the optical loop [445].
5.7. Frequency dependent absorption in mode locking
As we mentioned above, the mode-locked Kerr
frequency combs can be generated in a WGM resonator
with either anomalous or normal GVD. In the case of
anomalous GVD the intracavity pulses corresponding
to the combs are bright, and in the case of normal
GVD – dark. It was shown both theoretically and
experimentally [398] that this situation can change if
the resonator modes have wavelength dependent loss.
Such loss can play a role of a built-in bandpass filter
that mode locks the Kerr frequency comb.
In this experiment [398] a Si3 N4 ring resonator
with 115.6 GHz TE11 and 111.2 GHz TE21 FSRs
was used.
The group velocity dispersion of the
fundamental mode family (TE11 ) was normal, as was
verified by coherent wavelength interferometry and
numerical simulations. The Q-factors of the TE21
mode family was an order of magnitude lower than
of the fundamental TE11 mode family (1.2 × 105
versus 1.1 × 106 ). The TE21 mode family also had
a larger mode volume. Other higher-order mode
families had even lower Q-factors and larger mode
volume. Therefore, these modes did not support
efficient generation of Kerr frequency comb and also
did not couple with the fundamental modes in a
way that would sufficiently alter the GVD. Still, the
resonator produced short (74 fs) bright pulses when
pumped with CW light.
This result was explained by the presence of the
wavelength-dependent attenuation in the resonator.
The intrinsic bandpass filtering imposed by the Hbond absorption of Si3 N4 in the short wavelength
range and the increased coupling loss in the long
wavelength range served to achieve a clean and short
bright pulses in spite of the globally normal GVD
of the fundamental mode family. This is a different
mechanism of mode locking of Kerr frequency comb
compared to the conventional early experiments. A
simple analytical model confirms this conclusion [398].
25
5.8. Third harmonic generation and up-conversion
via four-wave mixing
It is more difficult to achieve phase matching for
the resonant THG than for the resonant hyperparametric process because of chromatic dispersion
of the resonator material. This dispersion usually
is not critical in a hyper-parametric process when
all the involved fields have nearly the same optical
wavelengths. However the wavelengths involved in
the THG process differ by a factor of three. The
consequent difference of the index of refraction can be
compensated by the geometrical dispersion discussed
in section 2.2, similarly to how it is done in optical
microfibers [446–452].
The pioneering WGM THG observations were
carried out in various organic and inorganic liquid
micro-droplets [25, 144–148]. Recently, efficient third
harmonic generation was demonstrated in silica micro
toroids [159], Si3 N4 micro rings [356], and in silica
micro spheres [453].
All these resonators were
pumped in a low-power CW regime at 1550-1560 nm
wavelength, producing visible (green) third harmonic.
The phase matching in WGM resonators [159, 453]
was achieved between the fundamental pump and
higher-order third harmonic modes. Because of the
small resonator size, the geometrical dispersion of
these modes was sufficient to compensate for chromatic
dispersion of the resonator material.
Observing the third harmonic emission from
evanescently-coupled resonators is not trivial. Because
of different evanescent field decay lengths, a significant
output coupling of the third harmonic wavelength
leads to strong overcoupling of the pump. This
causes a loss of conversion efficiency, which scales
as the cube of the loaded Q-factor for the pump
[159]. For this reason surface scattering was observed
instead of the waveguide output to confirm THG in
[159, 453]. Evidently, to make practical application
of the third harmonic WGM sources one needs to
solve a problem of the selective coupling, such that
allows the pump in-coupling rate to be adjusted
independently from the signal out-coupling rate. This
type of coupling based on polarization dispersion has
been demonstrated in a monolithic cavity [123]. It is
also possible to use waveguides optimized for coupling
at particular wavelengths. For example, a “pulley”
shaped waveguide bending around a short segment of
a resonator may show strongly suppressed coupling at
a desired wavelength [454].
Along with the third harmonic, series of discrete
blue-shifted emission peaks arising from four-wave
mixing of the input radiation and the stimulated
Raman-scattered radiation were reported [144, 453]. A
similar phenomenon was observed in a large, 7 mm in
diameter, crystalline MgF2 resonator [381], in addition
26
Nonlinear and Quantum Optics with Whispering Gallery Resonators
to the hyper-parametric comb and stimulated Raman
scattering. In a more controlled way, up-conversion
via four wave mixing was studied by injecting two
different pump wavelengths (1553 nm and 1674 nm)
in a silica micro toroid [159]. Emission at 542 nm
was observed, which corresponds to combining a 1553
nm photon with two 1647 nm photons. A similar
experiment was perfromed in a doped high-index silica
glass microring, combining a 1558.02 nm photon with
two 1553.38 nm photons [455]. We are not aware of
any quantitative analysis of such processes in WGM
resonators, although the ODEs describing this process
in waveguides, see e.g. [456], can be relatively easily
adapted for this purpose.
An added degree of freedom makes the four-wave
mixing based up-conversion more flexible compared
to a direct third harmonic generation. One may
expect this approach to allow for efficient conversion
of quantum states from infrared to visible range, or
from visible to ultraviolet range. As a motivation
for this research, let us mention that a narrowline, tunable and efficient ultraviolet source is highly
desirable for compact and low-power spectroscopy
applications. In section 7.5 we will also see how this upconversion technique may be useful in quantum optics
applications.
5.9. χ(2) -χ(3) processes
Generation of Kerr frequency combs in microresonators
made of materials characterized with both quadratic
and cubic nonlinearity is especially interesting since
it allows observing both the second and third order
nonlinear effects in the same microcavity. In addition,
the noncentrosymmetric materials characterized with
nonzero χ(2) usually demonstrate electro-optic effect,
which enables simple electric manipulation of the Kerr
frequency combs. Generation of the Kerr combs was
demonstrated in aluminium nitride (AlN) microrings
[457–459]. However it was not achieved in crystalline
quartz [286], nor in lithium tantalate [309] resonators
for reasons yet unknown.
AlN is a wide-band semiconductor that has significant second order nonlinearity and approximately
two orders of magnitude larger thermal conductivity
than Si3 N4 . It has both strong Kerr nonlinearity and
electro-optic Pockels effect. The microring resonators
with Q-factor approaching 6 × 105 at 1.5 µm wavelength [460] allow for strong power enhancement, leading to Kerr frequency comb generation and cascaded
frequency conversions in the visible range. Three frequency comb sets in IR, red and green were simultaneously generated in an AlN microring pumped by a
single telecom IR pump laser [458]. These combs arise
from a combination of the three- and four-wave mixing
in the same resonator. High-resolution spectroscopic
study of the visible frequency lines indicates matched
free spectrum range over all the bands. Therefore, the
observed process simplifies self locking of the frequency
combs. Electro-optic switching of the frequency comb
also becomes feasible [459].
6. Other nonlinear processes
6.1. Raman and Brillouin scattering
Light scattering by various solid state excitations,
such as optical and acoustic phonons and polaritons,
leads to nonlinear-optical phenomena known as Raman
and Brillouin scattering. Usually these excitations do
not form modes with any specific spatial symmetry,
and the corresponding processes do not require phase
matching.
However they still have to conserve
energy, and furthermore can only occur for the optical
frequencies supported by the resonator.
The threshold of a resonant Raman or Brillouin
laser [461, 462] is
PR,B ≃
π 2 np ns Vp
,
Gλp λs Qp Qs
(53)
where λp = 2πc/ωp is the pump wavelength and G
is the Raman or Brillouin gain. Here the oscillation
threshold is proportional to the factor Vp /Qp Qs just
as for the hyper-parametric oscillations (51).
It
means that these three processes compete. While
PHP ≃ PR [378], the threshold of Brillouin laser is
usually much lower than the other two [462], and
only phase mismatch can make it less favorable in a
microresonator.
Spontaneous and stimulated Raman scattering in
liquid micro-droplets [55–57, 59–61] was among the
earliest nonlinear-optical WGM experiments. Raman
scattering was also observed in solid crystalline
[463–466] and amorphous [323, 461, 467–469] WGM
resonators, as well as studied theoretically [470]. The
interest in this process is stimulated by a wide range of
potential application for Raman lasers. In Table 3 we
summarize parameters and performance of such lasers
implemented in solid WGM resonators.
An important feature of WGM Raman lasers is
their high gain, which can be further enhanced by a
special coating [471–474] or by making the resonator
out of chalcogenide glass [469]. The high Raman
gain allows for a cascaded scattering, when the lowerorder Stokes signal serves as a pump for the higherorder process. In [467] such cascaded Raman lasing
up to the fifth order was demonstrated in a fused
silica microsphere. It was also demonstrated in a large
fluorite resonator [463], where a record threshold as low
as 1 µW was reported. The cascaded Raman scattering
was furthermore shown to form a mode-locked optical
27
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Ref.
[463]
[464]
[465]
[466]
[461]
[467]
[323]
[468]
[469]
[471]
[472]
[473]
[474]
λp → λR nm
1064→1102
1320→1378
1064→1102
1532→1610
1555→1670
976→1021
1550→1660
1550→1660
1550→1636
485→535
685→850
765→796
1561→1679
Media
fluorite
fluorite
fluorite
fluorite
silica
silica
silica
silica
As2 S3
J-aggr.
PDMS
silica:Ti
TEOS
Pth µW
1
1600
78
300
62
56
62
74
13
ηm %
24
65
50
35
6.5
45
45
10.7
1300
52.6
640
Table 3. Summary of reported Raman WGM lasers: pump
and Stokes wavelengths in nanometers, Raman-gain media,
threshold power Pth in µW, and maximum observed pump power
conversion ηm . In cases when multiple Raman lines are observed,
the strongest conversion is shown.
comb [466]. Cascaded hyper -Raman conversion was
demonstrated in a lithium niobate resonator [475].
This is a higher-order process when instead of one,
two pump photons are absorbed to generate a (higherfrequency) Stokes photon and to excite a phonon.
Distinctly from Raman scattering, generalized
Brillouin scattering involves acoustic phonons that typically have lower frequencies and narrower bandwidths,
but larger wave numbers than the optical photons or
polaritons. Brillouin interactions are responsible for
one of the strongest known optical nolinearities. However they are difficult to observe in microresonators
whose large FSRs are incompatible with small frequency shifts arising from Brillouin scattering. In other
words, it is difficult to find a pair of modes with close
enough frequencies and yet vastly different local wave
number (i.e., different effective index of refraction). In
WGM resonators, two solutions to this problem are
possible: i) backward scattering, or ii) forward scattering into a different mode family, e.g. accompanied by
a large change of the radial mode number q.
Both these approaches have been explored. In
Table 4 we summarize the key parameters for backward
[462, 476, 477] and forward [478–480] scattering
experiments in resonators ranging from approximately
100 µm [476, 479, 480] to 5-6 mm [462, 477] in
diameter.
Many of these experiments also show
cascaded Brillouin scattering.
The backward Brillouin scattering reported in
[462, 476, 477] relies on bulk acoustic phonons.
In contrast, the forward scattering relies on a
surface acoustic wave which may form its own
high-Q WGM [482–485].
In this respect the
forward Brillouin scattering is closely related to optomechanical processes. Its phase matching conditions
Ref.
[114]
[462]
[476]
[477]
[478]
[479]
[480]
[481]
Material
silica
fluorite
silica
silica
BaF2
silica
silica
silica
fB , GHz
10.8
9-20
10-11
21.7
8.2
0.06-1.4
0.04
∆fB , kHz
700
0.01-0.2
27
0.58
4
0.1-3.3
Pth , µW
50
3
26
7000
22.5
100
Table 4. Summary of reported Brillouin WGM lasers: resonator
material, Brillouin frequency shift fB , linewidth ∆fB , and pump
threshold power Pth . The linewidth in [477] is reduced by a
feedback loop; in [478] it is measured at -20 dB level.
are similar to (31).
The relatively low frequency of the surface
acoustic waves allows to observe Brillouin scattering on
the wing of the optical pump WGM line. Alternatively
and more efficiently, an “overmoded” resonator may
be used [478]. Such resonators have very high spectral
density of modes and can potentially function as highQ white-light resonators [486], capable of supporting
optical fields at practically any wavelength.
Brillouin scattering enables very narrow-line and
low-noise, low-threshold lasers [481], which explains a
rapidly growing interest in studying this process in
high-Q WGM resonators. These studies also lead to
an exciting and very active topic of optomechanics.
Perhaps the most remarkable aspect of this topic is
the possibility to optically cool a mechanical oscillator
to the quantum ground state [487, 488]. Finally, let
us note that Brillouin scattering of light may occur
not only on phonons but also on other solid-state
excitations, such as magnons that can form their own
WGMs [313, 315, 316].
6.2. Thermal nonlinearity
The thermo-optical frequency shift is a fundamental
property of WGM resonators, important in many
applications [97, 489–491]. As we mentioned in section
3.2, this property originates from the increase of the
resonator temperature due to absorption of the light
confined in a resonator mode. The frequency shift
results from thermorefraction and thermal expansion
of the resonator as described by Eq. (15). For a
resonator with Q = 1010 , one degree temperature
change usually leads to the frequency shift exceeding
the WGM resonance bandwidth by as much as five
orders of magnitude [97]. Given extremely small modal
volume, even a small absorbed optical energy may be
sufficient to “heat the WGM out of resonance”. This
effect is clearly nonlinear with respect to the optical
power, and may be compared to SPM. An important
distinction, however, is that unlike Kerr nonlinearity,
Nonlinear and Quantum Optics with Whispering Gallery Resonators
thermal nonlinearity is very slow on the optical time
scale. Also note that this effect is solely due to heating
of the mode volume by the optical power absorbed in
the material; scattering does not lead to the thermooptical effect.
In the simplest approximation the evolution of a
single-mode thermo-optical system can be described by
a pair of equations [491]
Ė + E [γ + i (ω + δ)] = F (t),
δ̇ + Γδ = Γξ|E|2 , (54)
where E is the complex amplitude of the field confined
in the resonator mode, γ is the mode bandwidth,
ω = 2πν is the unperturbed mode frequency, δ is the
thermal frequency shift, F (t) stands for the external
pump, Γ characterizes the thermal relaxation rate, and
ξ is the thermal nonlinearity coefficient.
Equations (54) describe a variety of interesting
behaviors. An immediate and most direct manifestation of the thermo-optical effect is the non-Lorentzian
asymmetric line shape of WGM resonances such as
shown in Fig. 11. Here the frequency of a strong probe
laser is slowly scanned through a WGM resonance in
two directions. The observed line shape depends on the
input optical power and the sweep speed, but most dramatically on the laser frequency sweep direction [491].
For a sufficiently high laser power, thermo-optical line
distortion leads to bistability and hysteresis [127, 378].
Both these phenomena are clearly present in Fig. 11.
The speed of the probe laser sweep is important
because a fast sweep does not lead to a significant
heating of the resonator mode [492,493]. If furthermore
the scan period is much longer than the thermo-optical
1.0
Tranmission
0.9
0.8
0.7
0.6
0.5
0.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Optical frequency sweep (MHz)
Figure 11. A typical signature of the thermo-optical effect is
the line shape dependence on the direction of the probe laser
frequency sweep. In this case a 120 mW pump was swept across
a WGM resonance of an R = 4 mm, Q = 109 magnesium fluoride
resonator in 34 ms in directions indicated by the arrows. This
measurement was performed as a part of experiment [211] but
not previously reported.
28
time constant Γ−1 (which is typically of the order of a
millisecond), the mode volume can efficiently exchange
heat with the bulk of the resonator. Therefore to
ensure that the true shape of the resonance is observed,
it is important to reduce the input power and sweep the
laser frequency fast enough and with low duty cycle to
minimize the pulling effect of the thermal nonlinearity.
Besides the bistability, numerical modeling of
the systems described by equations (54) predicts
the oscillatory instability [491]. This instability is
commonly observed in high-Q resonators, see e.g.
[97, 491], where it competes with soliton formation.
Even in Fig. 11 we see a trace of this effect which
appears as the noise ripples on the most gradual slope
of the resonance curve. To minimize the thermo-optical
effect one needs to reduce the absorption of light in
the resonator or to adjust its temperature [494]. On
the other hand, it is possible to find a regime when
the interplay between the thermorefractive and Kerr
effects leads to very strong thermo-optical relaxation
oscillations [495].
The thermo-optical effects enable the thermooptical locking of a WGM frequency to the pump
laser frequency [490, 494]. This technique is used
in cavity opto-mechanics, resonant nonlinear optics
and microresonator lasers, thermo-optical cooling
experiments, resonant optical sensors, and many other
applications and systems. It relies on the frequency
self-tuning of the resonator pumped with a CW
light. Let us consider the normal thermo-optical effect
corresponding to the decrease of the resonator mode
frequency with the resonator temperature increase,
and assume that the laser frequency is initially set
at the blue slope of the WGM resonance, see Fig. 11.
The WGM frequency may slowly drift due to ambient
temperature variations or other reasons. A blue drift
would shift the mode towards the laser. Coupling,
circulating power, and the consequent mode volume
heating would increase, and the rising temperature will
tune the mode frequency back to the red. Likewise,
a red drift of the WGM frequency will cause the
compensating blue tuning due to the temperature
decrease. In this way the mode frequency can be locked
to the frequency of the laser, and the temperature of
the resonator can be stabilized. It is worth noting, that
this locking strategy depends on the power stability of
the laser. The locking point changes with the pump
power.
There are numerous studies devoted to the
thermo-optical effect. For some applications it would
be desirable to reduce this effect. This can be achieved
by dynamically varying the optical power during
the sweep [496], or by using composite resonators.
By precisely controlling the optical field overlap
with the polymer film coating a silica resonator,
Nonlinear and Quantum Optics with Whispering Gallery Resonators
an environmentally stable devices were demonstrated
whose resonant frequencies were independent of the
input power [497, 498]. Various polimer coatings
reducing or enhancing the thermo-optical effect as
well as the Kerr SPM effect were studied in [499].
Interplay between the negative thermo-optical effect,
thermal expansion, and the Kerr effect may result
in stable thermo-optomechanical oscillations, as was
demonstrated with high-Q ZBLAN WGM resonators
[500].
6.3. Photorefraction
The term “photorefraction” describes the change of
the refractive index resulting from light-mediated
redistribution of charges within some optical materials.
There are multiple mechanisms of photorefractivity.
In the most common one, absorption of a photon
excites an electron from a donor site to the conduction
band. The electron diffuses from the region of high
light intensity to the regions of low light intensity and
becomes trapped at an ionized deep donor site. Each
photo-electron leaves behind a positive ionic charge.
Therefore a nonuniform space charge is created when
the electron is trapped at a different site. This space
charge generates a position-dependent electric field
that changes the refractive index of the material via
the electro-optic effect. The maximum photorefractive
change of the extraordinary refractive index of the
material (for instance, lithium niobate) arising from
this mechanism is determined by the diffusion field
ED [219]:
1
∆ne = − n3e r33 ED .
(55)
2
Here the electric field ED is linearly proportional to the
light intensity at the beginning of the exposure. The
saturated value of ED also depends on the intensity,
but this dependence is not necessarily linear [501].
The maximum value of ED is limited by the domain
flipping.
Similarly to the SPM and thermal nonlinearity,
photorefraction is a nonlinear optical phenomenon
leading to the refractive index change in response to
the optical field. However while the thermorefractive
response is much slower than Kerr response, the
photorefractive response is much slower than the
thermorefractive response and also can be saturated.
The magnitude of photorefractivity depends on
the energy of photons that induce the charge
redistribution, one one hand, and on the dark current
that reverces this process, on the other.
The
dark curent depends on the temperature, crystal
composition, and other factors. Photorefractivity of
lithium niobate, quite strong in the UV and visible
parts of the spectrum, diminishes in the infrared and
far infrared. High sensitivity of WGM spectra allows
29
for measuring this effect with low-power infrared light.
Photorefractivity was observed with a continuous wave
780 nm and 1550 nm light in WGM resonators
made from lithium niobate (including the MgO-doped
samples) as well as lithium tantalate [231–233]. WGM
resonators allowed to discover that not only light,
but also low-power radio-frequency electromagnetic
radiation can result in a significant modification of the
refractive index of strontium barium niobate (SBN),
one of widely used photorefractive materials [502]. The
effects observed in SBN cannot be explained using
existing theories of photorefractivity in bulk material.
The photorefractive changes observed in WGM
resonators share the common features of other
photorefractive experiments: i) the observed optical
modification of the WGM spectrum does not disappear
if the light is switched off; ii) the changes can be
removed by illuminating the resonators with UV light;
iii) the time scale of the observed effect is in the
range of hours; and iv) the observed effects are more
pronounced for shorter wavelengths of light but are
present in the near infrared as well. Because of
their high sensitivity, WGM experiments provide more
insight into the long wavelength properties of the
impurities of the photorefractive crystals than those
with bulk samples. These observations support the
point of view that the photorefractivity does not
have a distinct red boundary in wavelength. This
understanding is important for various applications of
lithium niobate resonators, and it points at a possible
source of “aging” of the telecom devices that use
lithium niobate elements.
7. From nonlinear to quantum optics
7.1. General prerequisites for efficient quantum optics
processes
Nonlinear optics is a cornerstone in the generation
and manipulation of quantum states of light. The
generation and processing of non-classical states such
as single-photon states, squeezed states or entangled
states requires efficient nonlinear interactions [503,
504]. Both second- and third-order nonlinear processes
have been used to generate non-classical light in WGM
or ring resonators and will be discussed below. The
requirements on these processes are more stringent
than in the classical applications because quantum
states are very fragile.
There are three main
concerns that make the generation of non-classical
light particularly demanding: loss, unwanted nonlinear
processes, and addition of noise.
Loss destroys the fragile quantum states. In the
case of continuous-variable squeezed and entangled
states it will lead to a convolution of their Wigner
functions with that of a vacuum state.
Hence
Nonlinear and Quantum Optics with Whispering Gallery Resonators
the quantum correlations will be reduced or lost.
In discrete-variables measurements, loss will reduce
counting and coincidence rates. As we have mentioned
in section 1.1, strong optical nonlinearity is often
accompanied by a considerable loss. When designing
an efficient source for quantum states one has to take
special care of the balance between high nonlinearity
and loss involved.
Unwanted nonlinear processes are another source
of degradation of quantum states. These processes
include stimulated Raman and Brillouin scattering,
photorefractive effects and nonlinear processes of a
different order. Second-order nonlinear interactions are
much stronger than the third-order ones, and hence are
more immune to such parasitic processes.
Even when all of the above concerns are resolved,
the system can still suffer from spontaneous scattering
or from fluorescence. Spontaneous scattering (Raman,
Brillouin, and Rayleigh) can lead to additional
noise that masks the features of the generated
quantum states. Depending on the pump wavelength,
fluorescence can also have an essential impact.
Therefore the resonator materials and the optical
wavelengths have to be chosen carefully. In very
small resonators thermorefractive noise can also have
a considerable impact [505].
Let us first discuss the sources of non-classical
light that use second-order nonlinear processes. The
interaction part of the Hamiltonian for three-wave
mixing (21) can lead to the variety of nonlinear
processes introduced in section 4. All these classical
processes can be used to generate or process quantum
states.
The most widely used process in quantum optics
is SPDC. In terms of quantum optics this process can
be interpreted as a pump photon annihilation and two
photons in the signal and idler modes creation. A
more detailed study shows that these signal and idler
beams are entangled in their field variables (i.e. are
in a continuous-variable entangled state), which leads
to correlations in photon number measurements and
phase-sensitive measurements performed in the signal
and idler beams. The dynamical behavior of SPDC
depends on the pump power. In resonators SPDC may
lead to the OPO regime if the pump power exceeds the
threshold (39). Depending on the operation regime,
different quantum states can be generated. Very low
pump power leads to the regime of cavity assisted
spontaneous parametric down conversion [506]. As a
good approximation, such a system may be regarded as
emitting photon pairs. The efficiency and bandwidth
of this process are modified by the resonator. At
higher pump power the generation of multiple photon
pairs has to be considered [507]. Further increasing
the pump power to just below the threshold is widely
30
used to generate the squeezed vacuum beams [504,508].
Above the threshold, the resulting OPO is known to
emit bright beams which are intensity-correlated (twin
beams) [509].
In addition to varying the pump power, there
is a possibility to seed the system with light from
an external source, leading to an optical parametric
amplification. This approach has been used to generate
squeezed states in conventional resonators as well as
in bulk crystals. Second harmonic generation is also
known to generate squeezed states of light, both in
the up-converted [510–512] and the pump frequency
[513, 514]. Not all of the above mentioned possibilites
have been tested with microresonators yet, although
the trend of recent years shows that such applications
may become common.
Third-order nonlinear processes can also generate
non-classical states of the optical field.
These
processes are described by the interaction part of the
Hamiltonian (50). Note that with a classical pump
field the hyper-parametric conversion term of this
Hamiltonian is effectively reduced to the three-wave
mixing discussed above, leading to similar ODEs and
similar quantum states that can be generated. For this
reason in quantum optics papers the parametric and
hyper-parametric processes are sometimes intermixed,
disregarding the difference in the underlying optical
nonlinearity.
Although the third-order processes are weaker
than the second-order processes, they have some important advantages. They can be realized in amorphous materials and materials that are compatible with
nano-manufacturing. They also do not usually lead to
a very large wavelength difference between the pump
and the non-classical light, which is typical in the
second-order processes. On the downside, in most of
the quantum experiments involving third-order nonlinearities, scattering and parasitic nonlinear processes
can have a severe impact on the purity of the states.
Usually one has to carefully design the experimental
parameters to minimize the detrimental contribution
of these effects.
In the following we will investigate different
experiments where microresonators and micro-rings
have been used to generate or process non-classical
states of light.
7.2. Second-order processes above threshold: squeezing
One of the earliest experiments on producing quantumcorrelated beams of light used an optical parametric
oscillator operating above the threshold [509]. The
generated in this experiment beams of light were bright
and quantum-correlated in intensity. The latter means
that the intensity fluctuations of these two beams are
correlated stronger than could be achieved by any
Nonlinear and Quantum Optics with Whispering Gallery Resonators
classical modulation of coherent states originating from
a laser.
As intensity correlations can be easily measured,
this is a good test for the generation of quantum
states through a three-wave mixing process. With this
technique the first generation of squeezed light inside a
nonlinear crystalline whispering gallery resonator was
demonstrated [108].
The experiment [108] employed a lithium niobate
WGMR that was pumped at 532 nm and operated
as a non-degenerate OPO above the threshold. The
Type I phase matching was controlled via temperature
and voltage tuning as described in sections 3.2 and
3.3. The signal, idler and pump wavelengths were
separated, and signal and idler beams were coupled
into fast photodetectors. Their photo currents were
subtracted and added, and the resulting noise power
was measured at a suitable sideband frequency with
an electronic spectrum analyzer. A reduction of the
difference photo current below the shot noise level was
observed, see Fig. 12.
The two-mode photon-number squeezing, such as
measured in the experiment [108], does not depend
on the pump power. The interaction Hamiltonian
(21) always induces correlation between signal and
idler photon numbers by generating them strictly in
pairs. Other parameters however do significantly alter
the measured squeezing. Since the conversion only
occurs to the light inside the resonance bandwidth, the
squeezing depends on the ratio between measurement
sideband frequency and this bandwidth.
Best
squeezing is achieved at low frequencies that are well
inside the total resonator linewidth. The observed
squeezing also depends on the ratio between the
parametric light out-coupling rate and total resonator
linewidth, that is, on the role of dissipative loss inside
Figure 12. Measurement of twin-beam intensity correlations.
The sum and difference of the intensity noise of the signal and
idler beams of a WGM OPO are plotted vs. the pump detuning
from the WGM resonance. The difference noise falls below
the shot noise level (SNL), proving the quantum correlations.
Reprinted from [108].
31
the resonator relative to the out-coupling. Therefore
one should try to employ high-Q resonators where the
coupling rate can easily dominate the internal loss rate,
i.e., the resonator can be strongly over-coupled. As we
will see later, this optimization is also beneficial for
generation of other quantum states.
A very low pump power threshold of the WGM
OPOs helped to directly investigate the phenomena
that previously were outside the experimental capabilities. It had been known that far above the pump
threshold, in addition to the quantum correlations in
intensity of the twin beams, each signal and idler beam
is also amplitude squeezed [343,510,515]. However this
phenomenon eluded a direct observation with conventional OPOs because the combination of the required
high pump power and accessible linewidths caused relaxation oscillations [516–518] hiding the effect. The
OPO inside the WGMR made it possible to measure
this behavior directly [108].
7.3. Second-order processes below threshold: photon
pairs generation
OPO pumped below the threshold can be used
to generate photon pairs.
More accurately, the
generated quantum state can be described, to a
good approximation, as a superposition of vacuum
states and a pair of single photons in the signal
and idler beams. This cavity-assisted [506] SPDC is
a very efficient tool in optical quantum information
processing. The consequences of enhancing SPDC
with a cavity are manifold. One is that the required
pump power is dramatically reduced, especially within
the triply-resonant WGM systems. In the experiment
[112] the required pump power was in the range of
100 nW. Furthermore, the bandwidths of the generated
photon pairs are governed by the bandwidths of their
respective WGMs. In the case of lithium niobate
WGM resonators this may range from a few to a
few hundred MHz in the visible and near-infrared
wavelength range. Such bandwidths together with
the continuous wavelength tuning capability allow for
efficient coupling of SPDC photon pairs to various
atomic transitions. This coupling was used to perform
single-photon time-resolved spectroscopy of cesium D1
transition, allowing to directly probe the selected
transition lifetime [228]. Fig. 13 shows how the
two-photon correlation function of the parametric
light emitted from the resonator is transformed
by the atomic re-emission process, which includes
approximately a 10 ns peak delay and asymmetric
broadening of the correlation function. The solid line
is a theoretical model describing the resonator ringdown for panel (a) and its convolution with the atomic
ring-down for panel (b).
Photon pair sources can also be used as
Nonlinear and Quantum Optics with Whispering Gallery Resonators
32
Another important benefit of using a WGM
resonator as a photon pair source is the tunability
of its central wavelength and bandwidth. Continuous
tuning of these parameters was demonstrated and its
significance was discussed in [112, 228].
7.4. Third-order processes for generation of
non-classical light
Figure 13. A two-photon correlation function of the SPDC light
emitted from a WGM resonator (a) is modified by re-emission
from an atomic transition (b). The new correlation function
is fit to a model combining both the resonator and the atomic
transition ring-down times. Reprinted from [228].
heralded single photon sources with e.g. detection
of an idler photon heralding the arrival of a
signal photon [112].
Photon heralding is a key
to many quantum information protocols based on
conditional measurements, in particular in linear
quantum computing. Flexibility in the signal and idler
wavelengths available with the WGM approach can be
leveraged to optimize the quantum efficiency of the
heralding detection.
The resonator defines the electromagnetic modes
in which signal and idler are generated. It has been
found that the phase matching conditions discussed
earlier in section 4 can be selective enough to constrain
the photon pair source to strictly a single pair of modes
for the signal and idler [366]. Single-mode operation
is important in quantum information processing when
photon pairs from different sources have to interfere,
or when the mode-selective measurements such as
homodyne detection are required. Furthermore, as we
have mentioned in section 4.6, near degeneracy the
parametric down conversion in the WGM resonators
leads to a comb structure of pair-wise quantumcorrelated signal and idler modes. Such combs may
be used for creating multipartite entangled states,
similarly to Fabry-Perot or bow-tie resonators [519,
520]. Multipartite entanglement is a resource highly
desired in many quantum information applications, e.g.
in linear quantum computing.
The χ(3) hyperparametric processes have been widely
used for generation of squeezed light and correlated
photon pairs in nonlinear optical fibers, see e.g. the
recent review [521] and references therein. With this
process one can generate photon pairs in the materials
compatible with on-chip integration techniques. This
provides an opportunity to use microfabricated high-Q
resonators that can be easily overcoupled to achieve a
large heralding efficiency of single photons.
Recently there has been a number of experimental
demonstrations of photon pairs generated via four
wave mixing in silicon on-chip fabricated microring
and microdisk resonators [522–530]. All these devices
operate in the telecom wavelength range and required
very low (sub- to ten milliwatts) pump power,
providing high pair-production rate within a narrow
linewidth. In [526, 527] time-energy entanglement
of the generated photon pairs was demonstrated by
violating Bell’s inequality, and in [528] time-energy and
polarization hyper-entanglement was achieved, also
verified by Bell’s inequality violation.
Two-mode squeezing among multiple pairs of
modes was demonstrated in a microfabricated Si3 N4
ring [531]. In this experiment the resonator FSR
was large enough to select a single pair of squeezed
modes by spectral filtering. These modes were found
to be squeezed at the level of 1.7 dB, or 5 dB when
corrected for losses. Broadband quadrature squeezing
based on SPM in the same kind of resonator has been
theoretically predicted [532].
All on-chip devices described in this section are
integrated with the coupling waveguides using on-chip
processing technology. The efficiency of the overall onchip device is tied to technological progress in surface
smoothness and overall loss in the photonic guidance.
7.5. Quantum-coherent frequency converters
Converting a quantum state of an optical mode from
one frequency to another is an important problem in
quantum information processing and communications.
This problem usually arises when the optical frequency
of a quantum processing node (e.g., a trapped atom,
ion, color center or an opto-mechanical system),
defined by the system’s physical nature, is poorly
compatible with the communication or distribution
channel, e.g. an optical fiber, quantum light source, or
33
Nonlinear and Quantum Optics with Whispering Gallery Resonators
a high-efficiency and low-noise detector. To overcome
these problems, SFG was proposed [352] and realized
for frequency-conversion of squeezed states [533] and
single photons [534, 535] in PPLN waveguides. With
strong enough pump, the conversion efficiency was
shown to reach 46% in an open wave guide [535], and
over 90% in a cavity [534]. Non-degenerate four-wave
mixing in a photonic cystal fiber used for a similar
purpose has reached 29% single-photon conversion
efficiency [536]. This process involves two classical
pump fields at two different frequencies. Annihilating
a photon at one frequency while generating one at
the other shifts a signal photon frequency by the
difference of the pump frequencies. Such conversion
was also demonstrated in an on-chip Si3 N4 microring
(R = 40 µm), in which case the conversion efficiency
reached over 60% [454].
A practical quantum-coherent frequency converter
needs to have nearly unity efficiency, and also be free
of spontaneous conversion processes and of other noise
and decoherence mechanisms. The unity efficiency
requirement is very difficult to achieve in nonlinear
optics with weak fields. On the other hand, using
a high-power optical pump as in the experiments
mentioned earlier may lead to excessive noise. In this
situation, the extremely strong enhancement of the
nonlinear processes rate and prolonged interaction time
offered by WGM resonators may provide a solution.
For example, the four-wave mixing up-conversion
approach has been suggested [537] and demonstrated
[454]. The second-order SFG [320,354] can also be used
for this purpose.
A special case of quantum-coherent frequency
conversion is up-conversion of microwave photons into
optical domain.
This case is important because
many quantum systems proposed for quantum-logic
implementation operate at such frequencies [538, 539].
It is also important for sub-mm wave astronomy [540],
where individual microwave photons detection may be
desirable. Performing such detection efficiently and
with low noise is very difficult due to low photon
energy, while it presents a much lesser problem in
the optical domain. A microwave-to-optics coherent
converter can be realized as a unity-efficient, antiStokes singe sideband electro-optical modulator, such
as we discussed in section 4.1.
7.6. Direct coupling of WGM light with quantum
systems
WGM resonators can facilitate strong interaction of
their optical fields with quantum systems such as
quantum dots, atoms, color centers, or condensed
matter exitations such as phonons or plasmons. The
enhancement factor of spontaneous emission rate of a
dipole interacting with a resonator mode is [541]
Fp =
3 Qλ3
,
4π 2 Vp n3
(56)
where the mode volume Vp is defined slightly differently
from the previously defined V :
R 2
n (~r)|E(~r)|2 dV
Vp = V
.
(57)
max[n2 (~r)|E(~r)|2 ]
In (57) n(~r) is the spatial distribution of the refractive
index, and |E(~r)| is the electric field amplitude.
Eq. (56) shows that the spontaneous emission
rate can be enhanced by factor proportional to Q/Vp .
The enhancement scales linearly with Q because only
one mode of a linear optical resonator is used. It is
worth noting, though, that enhancement of stimulated
Rayleigh scattering scales as Q2 /Vp , even though this
scattering is linear in the sense that it does not depend
on the power of the scattered wave [203].
In order to reach the strong coupling regime,
both high quality factor and small mode volume are
required at the same time, see (56), which means a
high finesse. WGMs fulfill both requirements allowing
for demonstration of strong coupling of the light field to
various quantum systems. Optimization of resonator
parameters for cavity QED applications is discussed in
more detail in e.g. [322, 542]. Theoretical analyis and
experimental investigations have been performed with
WGM coupled to atoms [543–550], molecules [551],
quantum dots [67,552–554] and nitrogen vacancy (NV)
centers in diamonds [68, 541, 555–558].
Alkali atoms were the first quantum systems
coupled with the evanescent field of WGM resonators.
In 1994, Mabuchi and Kimble proposed a scheme for
trapping atoms in the evanescent field of a microsphere
using dipole forces of two WGMs blue- and red-detuned
from an atomic transition [543]. Conveniently, the
wavelength scaling of the evanescent fields in this
case creates a repulsive potential near the resonator
boundary and an attractive potential farther out. Best
to our knowledge, this kind of atom trapping near
WGM resonators has not been realized yet. But
four years later the same group at Caltech realized
a simpler experiment, demonstrating velocity-selective
interactions between WGM field of a fused silica
microsphere (Q = 5 × 107 , V = 10−8 cm3 ) and
cesium atomic vapor [544]. More recently the Caltech
group repeated this experiment with a microtoroidal
resonators and cold cesium atoms that were released
from a magneto-optical trap to fall on the chip
supporting the microtoroids [545]. A strong coupling
regime between an individual atom and the WGM field
was reached in this experiment.
A similar technique was later used with cold
rubidium atoms and a bottle WGM resonator by
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Junge et al. [548]. In this experiment the focus
was made on the study of polarization properties of
the evanescent WGM field. Such fields can have
a strong longitudinal polarization component, which
fundamentally affects the interaction between light and
atoms. This component was detected, and its effects
studied by coupling the TE and TM WGM modes
to different Zeeman levels of the F = 3 → F ′ = 4
transition in rubidium.
Coupling a single atom to a microtoroidal
resonator, Dayan et al. [546, 547] have converted
Poissonian laser light to single-photon state manifested
by strongly sub-Poissonian statistics, thereby realizing
a photon turnstile device. Later a similar device using
an optical microsphere coupled with a single rubidium
atom was shown to function as an all-optical switch,
almost reaching the single-photon switching limit [549].
Specifically, a transition from the highly reflective (R
= 65%) to the highly transmitting (T = 90%) state
was achieved with an average of approximately 1.5
(3 with the linear losses included) control photons
per switching event. A variation of this technique
has allowed for deterministic single-photon subtraction
from an optical pulse [550]. Remarkably, this photon
is not absorbed but re-routed into a different optical
channel and can be used e.g. for heralding of the
modified pulse arrival.
Fluorescence of a single organic molecule attached
to a WGM microsphere and excited by its near field
was directly imaged in [551]. This experiment required
cryogenic temperatures to suppress inhomogeneous
broadening of the molecular spectrum.
Similar
coupling of a single GaAs [553] or InAs [552, 554]
quantum dots also required low temperatures, and has
lead to the strong coupling regime. This approach
has enabled a pulsed single-photon source with
strongly suppressed high photon number amplitudes,
characterized by the Glauber correlation function
reaching as low as g (2) (0) = 0.16 [559].
In contrast with molecules and quantum dots,
NV centers in diamond provide the most stable
quantum emitters at room temperature. Most of
the experiments coupling NV centers to WGM and
similar resonators aimed at the increase of the
emission rate due to Purcell factor (56) and achieving
discrete emission spectra. Quantum properties of
this emission have been demonstrated as well. For
example in [555] emission from a single NV center
in a diamond nanocrystal was coupled to a tiny
(4.84 µm in diameter) polystyrene microspherical
WGM resonator. The non-classical character of the
single quantum emitter light was verified by Glauber
correlation function measurement g (2) (0) < 0.5, and
the coupling to the WGMs by a discrete spectrum of
the emission. Coupling NV centers to WGM resonators
34
has been proposed for quantum information transfer
[557], decoherence-free quantum gate implementation
[558], and generating W-states [560].
7.7. Quantum Zeno effect.
The Quantum Zeno effect [561] arises when frequent
state projections, usually realized as a series of
measurements, inhibit continuous evolution of a
quantum system. To understand this effect it is
instructive to consider the following example from
classical optics. A series of half-wave plates inserted
in an optical beam can be aligned so as to rotate the
beam polarization each by a small angle, adding up to
a large angle. If however each plate is followed by a
polarizer set to transmit the original polarization and
to absorb the other, the final polarization obviously
will not change. Moreover, in the limit of very small
rotation angle steps, the total optical loss in this system
will asymptotically vanish.
Quantum Zeno effect leads to a very interesting
concept of interaction-free manipulation of a system’s
evolution. This means that e.g. a photon can be
affected not by a measurement (which would annihilate
it), but by a possibility of the measurement. From the
practical perspective, this approach may be useful for
manipulating fragile quantum states, e.g. in quantum
logic implementations [563–565]. Such applications
have been proposed based on electromagnetically
induced transparency of WGM resonator’s evanescent
field in surrounding atomic vapor [566], or its twophoton absorption [562], see Fig. 14.
Either of
these processes can be regarded as a “measurement”
sensitive to the number of photons in the optical
mode. Even though there is no observer present to gain
knowledge from such a “measurement”, it still enables
Figure 14. A Si3 N4 microdisk used in the all-optical switch
based on two-photon absorption in surrounding rubidium vapor.
Reprinted from [562].
35
90
16
80
14
70
12
60
10
50
8
40
6
30
4
20
10
0
2
(a)
-40 -20
0
20
40
60
0
80 100 120
Second harmonic power ( W)
Laser frequency tuning (MHz)
Prametric signal ( W)
120
400
100
300
80
60
200
40
100
20
(b)
0
0
-100
-50
0
50
Transmitted pump power ( W)
quantum Zeno effect.
Not only a dissipative, but also a reversible
Hermitian process can, under certain conditions,
serve as a “measurement” in quantum Zeno effect.
Difference- and sum-frequency generation have been
considered for this purpose [338, 339, 354]. In this
context the term “Quantum Zeno Blockade” (QZB)
is frequently used, implying that the presence of a
control photon in a cavity prevents, via strong photonphoton interaction, the signal photon from entering the
cavity. The exact mechanism of the QZB depends on
the modal structure and decay time of the mediating
(difference- or sum-frequency) field. It can range
from incoherent to fully coherent QZB [338, 565]. A
simplistic explanation of the incoherent QZB is the
cross-phase modulation of the signal photon by the
control photon. When this phase is shifted by π,
interference of the amplitudes for the signal photon
reflecting from the cavity and exiting it becomes
constructive instead of destructive, and switching
between the “drop” and the “through” ports (see
Fig. 14) occurs. The coherent QZB, on the other hand,
can be explained in terms of Autler-Townes splitting
[339] which leads to equivalent switching operation.
A QZB single-photon switch is the quantum limit
of the earlier mentioned all-optical switch. It requires,
in the SFG-based version, that even a single photon
should saturates the SFG process yielding near-zero
SFG conversion efficiency, i.e. operates on the farright side of Fig. 9. Furthermore, operation of the
single-photon optical switch depends on deterministic
coupling of a single photon into a resonator, which
requires special pulse shaping [338].
The single-photon optical switches have not yet
been demonstrated. In the next section we will discuss
their feasibility with the best available resonators. Let
us point out here that the onset of Zeno blockade-like
behavior can be observed in the all-optical switches
operating in the low-power but still classical regime. In
the WGM SFG-based switch [354], counter intuitively,
the SFG emission decreases for higher control power,
which means that the switching loss for the signal beam
is reduced. Let us also recall the pump resonance
distortion observed in efficient SHG [318], SPDC [319,
567] and SFG processes [320] which may be due to a
similar effect of self-decoupling the resonator from the
pump (possibly aided by formation of a comb in the
case of SHG), see Fig. 15.
Quantum Zeno blockade can not only facilitate
the quantum gates functionality, but also modify the
statistics of a mode occupancy by photons. In [568]
antibunched emission of photon pairs is predicted to
occur via spontaneous hyper-parametric conversion
in a microcavity coupled to rubidium vapor. Here
the emission of multiple pairs will be suppressed due
Transmitted pump power ( W)
Nonlinear and Quantum Optics with Whispering Gallery Resonators
100
Laser frequency tuning (MHz)
Figure 15. Distortion of the transmitted pump and emitted SH
(a) and SPDC (b) resonances indicates the presence a dynamical
processes effectively decoupling the pump from the resonator
and impeding the conversion efficiency. Plot (a) is reprinted
from [318], plot (b) shows previously unreported result of the
experiment described in [319].
to the strong two-photon absorption in the vapor.
Similarly, antibunched pairs can be expected to emerge
from an SPDC process when the signal or idler is
simultaneously phase matched for the SHG [160].
7.8. Feasibility of strong coupling in the single-photon
limit and of quantum logic with photons
In the idealized case of just two WGMs coupled by the
SHG process in a lossless resonator, the first nontrivial
solution of the Schrödinger equation generated by
Hamiltonian (21) is
|Ψi = A(t)|2ip |0is + B(t)|0ip |1is ,
(58)
where the indices p and s stand for the pump
and second harmonic, respectively. The quantummechanical amplitudes A and B oscillate as
√
√
A(t) = sin( 2gt + φ), B(t) = cos( 2gt + φ), (59)
and the phase φ is determined by the initial conditions.
Nonlinear and Quantum Optics with Whispering Gallery Resonators
The meaning of this solution is that starting
at some point of time with a photon pair at the
fundamental pump frequency (|A|2 = 1, B = 0), we
expect this pair to up-convert to a single photon at
double frequency
with certainty (A = 0, |B|2 = 1) after
√
∆t = π/(2 2g), then return to the initial state after
2∆t, and so on ad infinitum.
In a resonator with a finite decay rate γ this
oscillation eventually decays to the ground state (A =
B = 0). In a realistic nonlinear resonator the
oscillation period 2∆t is longer than the resonator ringdown time 1/γ. Then the probability of up-conversion
of a photon pair, or of down-conversion of a doublefrequency photon, is
√
√
ps↔p = sin2 ( 2g/γ) = sin2 ( 2gQ/ω).
(60)
It is easy to find that for a millimeter diameter lithium
niobate resonator with Q = 4 · 108 and λp = 1.5 µm,
p1↔2 ≈ 0.1. This is a very impressive probability for
interaction of individual photons, and it raises certain
optimism regarding quantum-optical applications of
nonlinear WGM resonators.
As we have seen in section 4.6, a two-mode model
for SHG is an oversimplification, and a more elaborate
analysis is required. However the main message of
the above example holds: the feasibility of singlephoton all-optical switches and photonic quantum logic
gates hinges on reaching the strongly coupled regime
with single photons, when the nonlinear coupling rate
defined in (22) exceeds the resonator decay rate, g > γ.
Let us turn to the discussion of an SFG-based
QZB switch in [338]. Here equations of motion are
derived for the signal, pump, and sum-frequency field
operators inside and outside of the resonator using
a Hamiltonian whose interaction part is equivalent
to (21). The resonator is assumed to be strongly
over-coupled, which means that the cavity decay is
dominated by the outcoupling while the absorption and
scattering losses are negligible. The initial conditions
for these equations of motion are set by the external
signal and pump pulse shape, which may be Gaussian
or the ringdown-matching exponential [569].
Interaction inside the resonator entangles the
signal and pump states. The output joint signal-pump
wave function can be decomposed into Schmidt modes,
which allows to define the gate fidelity as the overlap
of the input signal pulse with its first Schmidt mode
at the output. It is shown that in the absence of the
pump, the resonator-matching signal pulse has a very
high fidelity, if one takes into account the time reversal
and sign change of the output pulse [569]. Moreover,
in the strongly coupled regime in presence of the pump
pulse, signal fidelity is also high. In this case the sign
change and time reversal do not occur.
The analysis [338] predicts that fidelity of 0.99
can be reached in a lithium niobate resonator with
36
Q = 108 for g > 400 MHz, which requires the resonator
radius R < 25 µm. The required combination of Q
and R does not appear entirely unrealistic, at least
in theory. But we have to note that the expression
for the nonlinear coupling rate Υ ≡ g provided in the
Supplementary material to [338] is equivalent to the
incorrect expression for the coupling rate g in [320], and
may be also incorrect. The correct expression has an
extra factor (ns np nf )−1 [321]. Therefore the estimates
for g provided in [338] may be unrealistically high, and
the benchmark fidelity would in fact be much harder
to reach.
The dependence of the coupling rate Υ on the
resonator radius R computed in [338] is approximately
captured by the expression log10 Υ ≈ 0.073(log10 R)2 −
0.77 log10 R + 1.74. If we limit the fitting function by
the linear term of log10 R, the approximation would
be log10 Υ ≈ −0.86 log10 R + 1.73, which is again
consistent with the “magic” scaling of the overlap
integral Υ ∝ |σ| ∝ R−0.9 discussed in section 4.2.
Concluding this section, we need to mention a
specific concern regarding using single-photon XPM
for realization of quantum logic operations. Such
operation would require very strong Kerr interaction
that is not presently achieved. However, even if it
is achieved, it has been shown [570, 571] that delayed
χ(3) response of realistic Kerr media (which is a direct
consequence of causality) will induce enough phase
noise to make high-fidelity operation of a quantum
logic gate impossible.
8. Summary and conclusion
WGM and ring resonators found applications in nearly
every branch of nonlinear and quantum optics. Their
advantages come from the exceptionally high quality
factor maintained within a very large wavelength
range, small mode volume, continuously tunable
coupling, and inherent mechanical stability. On the
other hand, complexity of the optical spectrum and
its strong temperature dependence make some of
nonlinear optics applications challenging, while the
exponential behavior of the evanescent field makes
simultaneous optimal coupling of different wavelength
difficult. We have discussed some of the approaches
allowing to solve or circumvent these problems.
Perhaps the most important limitation of WGM
resonators is their fabrication process. We will discuss
this issue specifically in the end of this section.
8.1. Applications and phenomena not covered in this
review
Some of nonlinear-optical phenomena observed in
WGM resonators and associated with them applications have been left out from this paper because they
37
Nonlinear and Quantum Optics with Whispering Gallery Resonators
were covered in recent reviews in a great detail. Here
we would like to mention them briefly and to direct the
readers to these reviews.
One such prominent application, dating back to
1960s, is WGM lasers [23, 572]. Their history and
state of art is discussed in a recent review [573]. It
is worth mentioning that in such lasers the resonator
can be either entirely made from the gain media
[23, 71, 72, 74, 113, 572, 574–578], or coated with it
[474, 579, 580].
The coating technique reminds us of an important
role surface physics plays in WGM optics. Besides
introducing laser gain media, various molecular
coatings can be used to facilitate SHG [356, 581] as
well as SPDC [582] in resonators made from inversionsymmetric materials. Theoretically, even a perfectly
clean surface of a WGM resonator is expected to enable
such processes [317], as it too breaks the inversion
symmetry. Optical response of molecules and microobjects inadvertently attached to WGM resonators
immersed in liquids or gases gives rise to efficient biosensing techniques recently reviewed in [69,70,252,583].
Discussing the interaction of WGM photons with
acoustic phonons, we alluded to the new and rapidly
developing field of optomechnics. One of the main
goals in this field is to reach the quantum regime
with mechanical oscillators, which is normally hard
to achieve because of low phonon energy and strong
coupling to a thermostat. Many spectacular results
have been demonstrated in this field lately, see reviews
[77,584,585]. Opto-mechanically induced transparency
[586] and light storage [587], as well as storing optical
information as a mechanical excitation [588], were
demonstrated in silica microresonators. Formation
of opto-mechanical dark modes is reported in [589].
Resolved-sideband and cryogenic cooling of an optomechanical resonator is reported in [590]. Generation
of high quality radio frequency signals by optical means
is another significant achievement in the field [591]. Of
the most direct relevance to the scope of our review
is the proposal [592] to couple mechanical oscillations
of a WGM resonator with the squeezed light internally
generated in it as discussed in section 7.2.
8.2. Anticipated development of the field
One of the most attractive goals that may be
achieved using WGM resonators is strong interaction
between individual photons.
Achieving this goal
would constitute a breakthrough in quantum logic and
quantum computing with photons. With the state-ofart nonlinear WGM resonators this goal appears to be
close enough but not yet within reach. Its feasibility
hinges on the relation between the linear loss rate γ =
ω/Q and non-linear conversion rate g. To make this
relation favorable (g > γ) one can increase the overlap
integral σ, the Q-factor, or the nonlinear susceptibility.
Unfortunately, only limited resources are available for
the progress in either of these directions.
The overlap integral has almost a universal power
scaling with the resonator size, which is captured in
Fig. 7. Therefore making smaller resonators is the
most direct way to increase their nonlinear response.
But very small resonators suffer radiative loss. From
Eq. (11) we find that in the near infrared Qrad
of lithium niobate spherical resonators drops to the
absorption-limited value of 108 when R ranges from
15 to 20 µm (the exact radius value depends on the
wavelength and polarization), which appears to set
a limit for the minimal useful resonator size. For
smaller resonators Q is limited by Qrad which drops
exponentially with the radius. However, microring
resonators have been reported with the Qs significantly
exceeding the radiative limit found from Eq. (11),
as can be established by substituting the resonator
parameters from e.g. [330, 360, 361] into this equation.
Evidently, the model underlying Eq. (11) does not work
well for non-spherical shapes, and a more accurate
(perhaps numeric) approach is required to establish
the radiative loss and find the minimal useful resonator
size.
The other two optimization parameters, the
ultimate absorption-limited Q-factor and nonlinear
susceptibility, are both determined by the resonator
material. Therefore it is important to explore new
optical materials that are strongly nonlinear and at
the same time very transparent. These materials also
need to be compatible with a stringent surface quality
requirements, which includes chemical and mechanical
stability, low solubility, etc. Since the second-order
susceptibility χ(2) is much stronger than the thirdorder susceptibility χ(3) , let us focus on optical crystals
with quadratic nonlinearity. A summary of resonator
parameters achieved with such crystals is given in
Table 5 for WGM resonators and in Table 6 for ring
resonators made from thin crystalline films. Some
of these materials have been explored in multiple
works. Here we only quote those with the highest
demonstrated Q.
Ref.
[286]
[151]
[309]
[107]
[257]
[151]
[502]
Crystal
Quartz
BBO
LiTaO3
LiNbO3
LB4
BBO
SBN
λ, nm
1550
1560
1550
1310
490
370
1550
Q, 106
5000
740
570
200
200
150
75
dij , pm/V
0.311
2.322 , 0.1631
1633 , 2.731 , 1.622
2533 , 4.931 , 2.722
0.5533 , 0.07331
2.322 , 0.1631
Table 5. Summary of WGM resonators machined from various
second-order nonlinear optical crystals.
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Ref.
[593]
[594]
[362]
[361]
[330]
[156]
[360]
Film
LiNbO3
AlN
GaP
ZnSe
GaAs
GaN
AlGaAs
λ, nm
1550
1555
1550
1548
1985
1560
1580
Q, 103
1190
600
100
50
30
10
5
dij , pm/V
2533 , 4.931 , 2.722
2.533
6836
3336
9436
3.833 , 2.531 , 2.515
Table 6. Summary of on-chip ring resonators fabricated from
various second-order nonlinear optical crystalline films. Note
that for GaP, ZnSe and GaAs d36 = d14 .
In Tables 5 and 6 we list the most significant
quadratic susceptibility tensor components dij =
(2)
χij /2. A combination of these components specific for
a given type of phase matching and crystal orientation
will determine the χ̃(2) .
Quality factors reported in Tables 5 and 6 are
vastly different. Note that in Table 5 Q is given in
millions, and in Table 6 in thousands. The inferior
quality of the “large” (R = 40 µm) ring resonators
[156, 594] is clearly unaffected by the radiative losses
and is most likely due to the poor surface quality. This
means that there is a potential for a significant increase
of Q by improving the fabrication process. There is also
a possibility that the absorption-limited Q in complex
crystals such as lithium niobate can be increased by
optimizing the crystal composition and reducing the
impurities concentration [595].
Besides facilitating a strong interaction between
two photons, development of strongly nonlinear highQ resonators is motivated by the possibility to access
multipartite and multiphoton entangled states, such
as optical GHZ states [596, 597], W-states [560, 598,
599], cluster [600] and graph [601] states, Smolin
states [602], and others. Theses states are very
important in the context of quantum information
applications as well as in fundamental quantum optics
[603–605], but due to weak optical nonlinearity, their
realization has been so far only possible at low photon
rate with very strong pump pulses, a combination
usually leading to noisy data. One may expect that
nonlienar WGM resonators will advance this field of
research towards higher photon numbers (and higherdimensional Hilbert space) while significantly reducing
the requirements to the pump lasers.
While yielding the best Q factors, fabrication of
crystalline resonators by diamond turning and polishing unfortunately remains an art. Meanwhile, many
research and industrial applications involving classical
optical fields could benefit from this technology if a robust and scalable fabrication technique was available.
It may be expected that developing such techniques,
even at the cost of the quality factor, will be one of the
38
thrust directions in the WGM research in the nearest
future.
8.3. Fabrication challenges and scalability quest
The complexity of WGM resonators fabrication and
integration appears to be the main factor limiting
their broad infusion into nonlinear and quantum optics
technology. While WGM resonators remain objects
of research, it is acceptable to invest considerable
time in their individual fabrication, mounting on
a mechanically and thermally stable platform, and
coupling to the optical input and output. However
using a resonator as an instrument component requires
repeatable and scalable fabrication process, ideally
chip-based. This is partially accomplished with fused
silica microdisks made from the naturally formed
oxide layer on a silicon substrate. The layer is
photolithographically patterned with disk preforms,
then the substrate is dry-etched to form pedestals, and
the elevated preforms are either reflown into toroidal
shapes by laser heating, e.g. [78,323,379,387,468,477],
or chemically etched into wedge shapes [114, 399].
The quality factor of toroids routinely exceeds 108 , in
wedge-shaped resonators it can almost reach 109 [114].
Such a technologically important material as
lithium niobate presents a harder challenge, which has
been addressed only most recently. In one approach a
lithium niobate wafer was bound on a silicon surface,
polished down to one micron thickness and milled with
a focused ion beam into a disk shape. The disk was
elevated from the silicon substrate by etching the later
with XeF2 [606]. This fabrication technique yielded
R = 35 µm resonators with Q = 4.84 × 105 .
An alternative technological approach involving
femtosecond laser ablation followed by focused ion
beam milling, HF etching and finally high temperature
annealing yielded 0.7 µm thick, R = 41 µm resonators
with Q = 2.5 × 105 [359, 607].
A combination of chemical (HF) and reactive
ion etching of a lithographically patterned lithium
niobate - silica - lithium niobate “sandwich” allowed for
fabrication of 0.4 µm thick, R = 28 µm resonators with
the quality factor reaching 105 [358]. This technology
was later perfected by a different group, allowing to
reach Q = 1.19 × 106 with R = 39.6 µm resonators of
similar thickness [593].
Diamond is another technologically appealing
material with remarkable optical properties. Very
small (R = 2.4 µm) diamond ring-resonators with
Q = 5000 have been fabricated by bonding diamond
film onto a silica-on-silicon substrate, followed by a
series of lithographic patterning and dry etching steps
[541]. These resonators have been used to enhance NV
centers emission at low temperature.
A versatile method that may be applicable to a
Nonlinear and Quantum Optics with Whispering Gallery Resonators
Figure 16. Top: an array of microdisk resonators fabricated
from a bulk diamond. Bottom: a magnified view of a single
resonator. Reprinted from [610].
39
silicon-on-insulator microfabrication techniques [454,
522–528, 611, 612]. Besides the earlier mentioned fourwave-mixing based applications, such devices have
been used as all-optical switches [611, 612]. This
application is based on the free carrier induced optical
bistability, and leads to extremely low optical pulse
switching energy of just several pJ.
These techniques enables production of coupled
WGM devices, potentially with multiple resonators
and a network of couplers and waveguides on the same
chip. One drawback of this approach is the fixed
coupling rate: the gaps between the resonators and
couplers cannot be changed, although the coupling rate
may be adjusted by immersing the device in variable
index fluids. Besides, as we already mentioned, at
the present technology level the Q-factors of etched
microring resonators is usually limited, although a
very respectable value of Q ≈ 1.2 × 106 was reported
in a doped high-index silica glass waveguide-coupled
microring at 1544 nm wavelength [393]. Nearly as high
Q-factors in the same optical band were measured in
single-crystal diamond waveguide-coupled microrings
[613], that were also used to generate Kerr combs.
In χ(2) materials, the quality factors are far more
modest. For a lithium niobate microring (R = 100 µm)
resonator Q ≈ 4000 has been achieved [614]; for a
waveguide-coupled GaAs microdisk (R = 3.7 µm),
Q ≈ 104 [615].
Acknowledgments
variety of materials has been developed by Burek et al.
for fabrication of various free-standing nanostructures,
including disk, ring, and racetrack resonators, out
of single-crystal diamond [608, 609]. The method is
based on the angle-etching with ions, a technique
when the ions trajectories are bent by a Faraday cage,
allowing to undercut and eventually to fully separate
the lithographically defined structures from the bulk
material. With this technique, Q ≈ 1.5 × 105 was
achieved for the racetrack resonators [609]. Etching
bulk diamond was also shown to produce arrays of
R = 3.95 µm on-chip disk resonators with Q ≈ 105
[610], see Fig. 16. This process was quite complicated
and consisted of seven coating/etching steps.
The above technologies produce well controlled
resonators, but the couplers (typically, tapered fibers
or waveguides) need to be fabricated and mounted
separately, which limits its scalability.
On-chip
fabrication of resonator and waveguide coupler arrays,
such as shown in Figs. 10 and 14, is possible using
reactive ion etching of various resonator material layers
grown on a substrate [156, 356, 357, 374, 385, 393,
455, 531, 594], sometimes complete with a protective
layer. In the case of silicon, the resonator-waveguide
structures have been fabricated using the standard
D.V.S. acknowledges financial support from Alexander
von Humboldt Foundation and the DARPA Quiness
program, and would like to thank Dr.
Maria
Chekhova for useful discussions. H.G.L.S. would
like to thank Florian Sedlmeir and Alfredo Rueda
for useful discussions. The authors appreciate Drs.’
Kartik Srinivasan, Serge Rosenblum and Hailin Wang
valuable feedback on this paper preprint.
The
research was partly carried out at the Jet Propulsion
Laboratory, California Institute of Technology, under
a contract with the National Aeronautics and Space
Administration.
References
[1] Agrawal G P 1995 Nonlinear fiber optics (New York, NY
USA: Academic Press)
[2] Maier S A 2007 Plasmonics: fundamentals and applications (Springer Science & Business Media)
[3] Soukoulis C M 2012 Photonic crystals and light localization in the 21st century vol 563 (Springer Science &
Business Media)
[4] Engheta N and Ziolkowski R W 2006 Metamaterials:
physics and engineering explorations (John Wiley &
Sons)
[5] Cai W and Shalaev V M 2010 Optical metamaterials vol 10
(Springer)
Nonlinear and Quantum Optics with Whispering Gallery Resonators
[6] Joannopoulos J D, Johnson S G, Winn J N and Meade
R D 2011 Photonic crystals: molding the flow of light
(Princeton university press)
[7] Akahane Y, Asano T, Song B S and Noda S 2005 Opt.
Expr. 13 1202–1214
[8] Noda S, Fujita M and Asano T 2007 Nat. Phot. 1 449–458
[9] Dharanipathy U P, Minkov M, Tonin M, Savona V and
Houdr R 2014 Appl. Phys. Lett. 105 101101
[10] Song B S, Noda S, Asano T and Akahane Y 2005 Nature
materials 4 207–210
[11] Chang R K and Campillo A J 1996 Optical processes in
microcavities vol 3 (World scientific)
[12] Vahala K J 2003 Nature 424 839–846
[13] Vahala K 2004 Optical microcavities vol 5 (World
Scientific)
[14] Matsko A B and Ilchenko V S 2006 IEEE J. Sel. Top.
Quantum Electron 12 3
[15] Savchenkov A A, Matsko A B, Ilchenko V S and Maleki L
2007 Opt. Expr. 15 6768–6773
[16] Rayleigh J W S B 1896 The Theory of Sound (Macmillan)
[17] Rayleigh L 1910 Philosophical Magazine Series 6 20 1001–
1004
[18] Rayleigh L 1914 Philosophical Magazine Series 6 27 100–
109
[19] Raman C V and Sutherland G A 1921 Nature 108 42
[20] Mie G 1908 Annalen der Physik 330 377–445
[21] Debye P 1909 Annalen der Physik 335 57–136
[22] Richtmyer R D 1939 J. Appl. Phys. 10 391
[23] Garrett C G B, Kaiser W and Bond W L 1961 Phys. Rev.
124 1807
[24] Wait J R (ed) 1962 Electromagnetic Waves in Stratified
Media (Pergamon)
[25] Benner R E, Barber P W, Owen J F and Chang R K 1980
Phys. Rev. Lett. 44 475
[26] Vedrenne C and Arnaud J 1982 Microwaves, Optics and
Antennas, IEE Proceedings H 129 183–187
[27] Gastine M, Courtois L and Dormann J L 1967 Microwave
Theory and Techniques, IEEE Transactions on 15 694–
700
[28] Affolter P and Eliasson B 1973 Microwave Theory and
Techniques, IEEE Transactions on 21 573–578
[29] Wait J R 1967 Radio science 2 1005–1017
[30] Tobar M E and Mann A G 1991 Microwave Theory and
Techniques, IEEE Transactions on 39 2077–2082
[31] Krupka J, Tobar M E, Hartnett J G, Cros D and Le Floch
J M 2005 Microwave Theory and Techniques, IEEE
Transactions on 53 702–712
[32] Eremenko Z E, Filipov Y F, Kharkovsky S N, Kutuzov
V V and Kogut A E 2002 Microwave Theory and
Techniques, IEEE Transactions on 50 2647–2649
[33] Fiedziuszko S J, Hunter I C, Itoh T, Kobayashi Y,
Nishikawa T, Stitzer S N and Wakino K 2002 Microwave
Theory and Techniques, IEEE Transactions on 50 706–
720
[34] Hartnett J G, Tobar M E, Ivanov E N and Krupka J 2006
Ultrasonics, Ferroelectrics, and Frequency Control,
IEEE Transactions on 53 34–38
[35] Hartnett J, Tobar M E, Ivanov E N and Luiten A N 2013
Ultrasonics, Ferroelectrics, and Frequency Control,
IEEE Transactions on 60 1041–1047
[36] Braginsky V B, Ilchenko V S and Bagdassarov K S 1987
Phys. Lett. A 120 300–305
[37] Jiao X H, Guillon P, Bermudez L A and Auxemery P 1987
Microwave Theory and Techniques, IEEE Transactions
on 35 1169–1175
[38] McNeilage C, Searls J, Ivanov E, Stockwell P, Green D and
Mossamaparast M 2004 A review of sapphire whispering
gallery-mode oscillators including technical progress
and future potential of the technology Frequency
Control Symposium and Exposition, 2004. Proceedings
40
of the 2004 IEEE International pp 210–218
[39] Boudot R, Gruson Y, Bazin N, Rubiola E and Giordano
V 2006 Electron. Lett 42 929–931
[40] Ivanov E N and Tobar M E 2006 Microwave Theory and
Techniques, IEEE Transactions on 54 3284–3294
[41] Locke C R, Ivanov E N, Hartnett J G, Stanwix P L and
Tobar M E 2008 Rev. Sci. Instrum. 79 051301
[42] Le Floch J M, Fan Y, Humbert G, Shan Q, Férachou D,
Bara-Maillet R, Aubourg M, Hartnett J G, Madrangeas
V, Cros D et al. 2014 Rev. Sci. Instrum. 85 031301
[43] Stanwix P L, Tobar M E, Wolf P, Susli M, Locke C R,
Ivanov E N, Winterflood J and van Kann F 2005 Phys.
Rev. Lett. 95 040404
[44] Hartnett J G, Locke C R, Ivanov E N, Tobar M E
and Stanwix P L 2007 Cryogenic sapphire oscillator
with exceptionally high long-term frequency stability
Frequency Control Symposium, 2007 Joint with the
21st European Frequency and Time Forum. IEEE
International (IEEE) pp 1028–1031
[45] Tobar M E, Ivanov E N, Stanwix P L, Le Floch J M G
and Hartnett J G 2009 Phys. Rev. D 80 125024
[46] Krupka J, Derzakowski K, Abramowicz A, Tobar M E and
Geyer R G 1999 Microwave Theory and Techniques,
IEEE Transactions on 47 752–759
[47] Krupka J, Derzakowski K, Tobar M, Hartnett J and Geyer
R G 1999 Meas. Sci. and Tech. 10 387
[48] Tobar M E, Krupka J, Ivanov E N and Woode R A 1998
J. Appl. Phys. 83 1604–1609
[49] Luiten A N, Tobar M E, Krupka J, Woode R, Ivanov E N
and Mann A G 1998 J. Phys. D: Appl. Phys. 31 1383–
1391
[50] Nand N R, Goryachev M, Floch J M l and Creedon D L
2014 J. Appl. Phys. 116 134105
[51] Damm C, Schwefel H G L, Sedlmeir F, Hartnagel H,
Preu S and Weickhmann C 2015 Selected emerging
thz technologies Semiconductor Terahertz Technology
ed Carpintero G, Garcı́a Muñoz L E, Hartnagel H L,
Preu S and Räisänen A V (Chichester, UK: John Wiley
& Sons, Ltd) pp 340–382
[52] Rivera-Lavado A, Preu S, Garcı́a Muñoz L, Generalov
A, Montero-de Paz J, Döhler G, Lioubtchenko D,
Méndez-Aller M, Sedlmeir F, Schneidereit M, Schwefel
H, Malzer S, Segovia-Vargas D and Räisänen A 2015
IEEE Transactions on Antennas and Propagation 63
882–890
[53] Long S A, McAllister M W and Shen L C 1983 IEEE
Transactions on Antennas and Propagation 31 406–412
[54] Arnold S and Hessel N 1985 Rev. Sci. Inst. 56 2066
[55] Sennaroglu A, Kiraz A, Dündar M A, Kurt A and Demirel
A L 2007 Opt. Lett. 32 2197–2199
[56] Snow J B, Qian S X and Chang R K 1985 Opt. Lett. 10
37–39
[57] Qian S X and Chang R K 1986 Phys. Rev. Lett. 56 926–
929
[58] Pinnick R G, Biswas A, Chylek P, Armstrong R L, Latifi H,
Creegan E, Srivastava V, Jarzembski M and Fernandez
G 1988 Opt. Lett. 13 494–496
[59] Lin H B and Campillo A J 1994 Phys. Rev. Lett. 73 2440–
2443
[60] Qian S X, Snow J B and Chang R K 1985 Opt. Lett. 10
499–501
[61] Kiraz A, Yorulmaz S Ç, Yorulmaz M and Sennaroglu A
2009 Phot. & Nanostr. 7 186–189
[62] Tzeng H M, Wall K F, Long M B and Chang R K 1984
Opt. Lett. 9 499–501
[63] Biswas A, Latifi H, Armstrong R L and Pinnick R G 1989
Opt. Lett. 14 214–216
[64] Campillo A J, Eversole J D, and Lin H B 1991 Phys. Rev.
Lett. 67 437–441
[65] Schäfer J, Mondia J P, Sharma R, Lu Z H, Susha A S,
Nonlinear and Quantum Optics with Whispering Gallery Resonators
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
Rogach A L and Wang L J 2008 Nano Lett. 8 1709–
1712
Braginsky V B, Gorodetsky M L and Ilchenko V S 1989
Phys. Let. A 137 393–397
Fan X, Palinginis P, Lacey S, Wang H and Lonergan M C
2000 Opt. Lett. 25 1600–1602
Park Y S, Cook A K and Wang H 2006 Nano Lett. 6 2075–
2079
Vollmer F and Arnold S 2008 Nat. Methods 5 591–596
Vollmer F and Yang L 2012 Nanophot. 1 267–291
Miura K, Tanaka K and Hirao K 1996 J. of Mat. Sci. Lett.
15 1854–1857
Sandoghdar V, Treussart F, Hare J, Lefevre-Seguin V,
Raimond J M and Haroche S 1996 Phys. Rev. A 54
R1777–R1780
Ilchenko V S, Yao X S and Maleki L 2000 Microsphere
integration in active and passive photonics devices
Symposium on High-Power Lasers and Applications
(International Society for Optics and Photonics) pp
154–162
Cai M, Painter O, Vahala K J and Sercel P C 2000 Opt.
Lett. 25 1430–1432
Armani D K, Kippenberg T J, Spillane S M and Vahala
K J 2003 Nature 421 925
Kippenberg T J, Rokhsari H, Carmon T, Scherer A and
Vahala K J 2005 Phys. Rev. Lett. 95 033901
Kippenberg T J and Vahala K J 2008 Science 321 1172–
1176
Del’Haye P, Schliesser A, Arcizet O, Wilken T, Holzwarth
R and Kippenberg T J 2007 Nature 450 1214–1217
Kippenberg T J, Holzwarth R and Diddams S A 2011
Science 332 555–559
Okawachi Y, Saha K, Levy J S, Wen Y H, Lipson M and
Gaeta A L 2011 Opt. Lett. 36 3398–3400
Del’Haye P, Herr T, Gavartin E, Gorodetsky M L,
Holzwarth R and Kippenberg T J 2011 Phys. Rev. Lett.
107 63901
Sumetsky M 2004 Opt. Lett. 29 8
Murugan G S, Wilkinson J S and Zervas M N 2010 Opt.
Lett. 35 1893–1895
Ilchenko V S, Gorodetsky M L, Yao X S and Maleki L
2001 Opt. Lett. 26 256–258
Louyer Y, Meschede D and Rauschenbeutel A 2005 Phys.
Rev. A 72 31801
Pollinger M and Rauschenbeutel A 2010 Opt. Expr. 18
17764–17775
O’Shea D, Junge C, Pöllinger M, Vogler A and
Rauschenbeutel A 2011 Appl. Phys. B 105 129–148
Sumetsky M, Dulashko Y and Windeler R S 2010 Opt.
Lett. 35 898–900
Sumetsky M, Dulashko Y and Windeler R S 2010 Opt.
Lett. 35 1866–1868
Han K, Kim J H and Bahl G 2014 Opt. Expr. 22 1267–
1276
White I M, Oveys H and Fan X 2006 Opt. Lett. 31 1319–
1321
Zamora V, Dı́ez A, Andrés M V and Gimeno B 2011
Photonics and Nanostructures - Fundamentals and
Applications 9 149–158
Sumetsky M 2012 Opt. Expr. 20 22537–22554
Del’Haye P, Diddams S A and Papp S B 2013 Appl. Phys.
Lett. 102 221119
Sumetsky M 2011 Opt. Lett. 36 145–147
Strekalov D V, Savchenkov A A, Savchenkova E A and
Matsko A B 2015 Opt. Lett. 40 3782–3785
Savchenkov A A, Ilchenko V S, Matsko A B and Maleki L
2004 Phys. Rev. A 70 051804
Grudinin I S, Matsko A B, Savchenkov A A, Strekalov D,
Ilchenko V S and Maleki L 2006 Opt. Comm. 265 33–38
Grudinin I S, Ilchenko V S and Maleki L 2006 Phys. Rev.
41
A 74 63806
[100] Chembo Y K, Strekalov D V and Yu N 2010 Phys. Rev.
Lett. 104 103902
[101] Chembo Y K and Yu N 2010 Phys. Rev. A 82 33801
[102] Matsko A B, Savchenkov A A, Ilchenko V S, Seidel D and
Maleki L 2012 Phys. Rev. A 85 023830
[103] Matsko A B, Savchenkov A A and Maleki L 2011
arXiv:1111.3907
[104] Grudinin I S, Baumgartel L and Yu N 2012 Opt. Expr. 20
6604–6609
[105] Wang C Y, Herr T, Del’Haye P, Schliesser A, Hofer J,
Holzwarth R, Hänsch T W, Picqué N and Kippenberg
T J 2013 Nat. Comm. 4 1345
[106] Hofer J, Schliesser A and Kippenberg T J 2010 Phys. Rev.
A 82 31804
[107] Ilchenko V S, Savchenkov A A, Matsko A B and Maleki L
2004 Phys. Rev. Lett. 92 43903
[108] Fürst J U, Strekalov D V, Elser D, Aiello A, Andersen
U L, Marquardt C and Leuchs G 2011 Phys. Rev. Lett.
106 113901
[109] Strekalov D V, Schwefel H G L, Savchenkov A A, Matsko
A B, Wang L J and Yu N 2009 Phys. Rev. A 80 033810
[110] Rueda A, Sedlmeir F, Collodo M C, Vogl U, Stiller B,
Schunk G, Strekalov D V, Marquardt C, Fink J M,
Painter O, Leuchs G and Schwefel H G L 2016 Optica
3 597–604
[111] Liu L, Kumar R, Huybrechts K, Spuesens T, Roelkens G,
Geluk E J, de Vries T, Regreny P, Van Thourhout D,
Baets R and Morthier G 2010 Nat. Phot. 4 182–187
[112] Förtsch M, Fürst J U, Wittmann C, Strekalov D, Aiello A,
Chekhova M V, Silberhorn C, Leuchs G and Marquardt
C 2013 Nat. Comm. 4 1818
[113] McCall S L, Levi A F J, Slusher R E, Pearton S J and
Logan R A 1992 Appl. Phys. Lett. 60 289–291
[114] Lee H, Chen T, Li J, Yang K Y, Jeon S, Painter O and
Vahala K J 2012 Nat. Phot. 6 369–373
[115] Lacey S and Wang H 2001 Opt. Lett. 26 1943–1945
[116] Rex N B and T
[117] Lacey S, Wang H, Foster D H and Nöckel J U 2003 Phys.
Rev. Lett. 91(3) 033902
[118] Gmachl C, Capasso F, Narimanov E E, Nöckel J U, Stone
A D, Faist J, Sivco D L and Cho A Y 1998 Science 280
1556–1564
[119] Song Q, Cao H, Ho S T and Solomon G S 2009 Appl. Phys.
Lett. 94 061109
[120] Türeci H E, Schwefel H G L, Jacquod P and Stone
A D 2005 Modes of wave-chaotic dielectric resonators
Progress in Optics, Vol 47 vol 47 ed Wolf E
(Amsterdam: Elsevier Science Bv) pp 75–137 ISBN 0444-51598-4
[121] Türeci H E, Stone A D and Collier B 2006 Phys. Rev. A
74 043822
[122] Schiller S, Yu I I, Fejer M M and Byer R L 1992 Opt. Lett.
17 378–380
[123] Fiedler K, Schiller S, Paschotta R, Kurz P and Mlynek J
1993 Opt. Lett. 18 1786–1788
[124] Schiller S and Byer R L 1993 JOSA B 10 1696–1707
[125] Ryu H Y, Kim S H, Park H G, Hwang J K, Lee Y H and
Kim J S 2002 Appl. Phys. Lett. 80 3883
[126] Lee P T, Lu T W, Fan J H and Tsai F M 2007 Appl. Phys.
Lett. 90 151125
[127] Collot L, Lefevre-Seguin V, Brune M, Raimnod J M and
Haroshe S 1993 Europhys. Lett. 23 327–334
[128] Datsyuk V V and Izmailov I A 2001 Physics-Uspekhi 44
1061–1073
[129] Gorodetsky M L and Fomin A E 2006 J. Sel. T. Q. El. 12
33–39
[130] Breunig I, Sturman B, Sedlmeir F, Schwefel H G L and
Buse K 2013 Opt. Expr. 21 30683–30692
[131] Oraevsky A N 2002 Quant. El. 35 377–400
Nonlinear and Quantum Optics with Whispering Gallery Resonators
[132] Little B E, Laine J P and Haus H A 1999 J. Lightwave
Tech. 17 704–715
[133] Lam C C, Leung P T and Young K 1992 JOSA B 9 1585–
1592
[134] Schiller S 1993 Applied Optics 32 2181–2185
[135] Schwefel H G L, Stone A D and Türeci H E 2005 JOSA B
22 2295–2307
[136] Ferdous F, Demchenko A A, Vyatchanin S P, Matsko A B
and Maleki L 2014 Phys. Rev. A 90(3) 033826
[137] Li J, Lee H, Yang K Y and Vahala K 2012 Opt. Expr. 20
26337–26344
[138] Savchenkov A A, Matsko A B, Liang W, Ilchenko V S,
Seidel D and Maleki L 2012 Opt. Expr. 20 27290–27298
[139] Gorodetsky M L and Ilchenko V S 1994 Opt. Comm. 113
133–143
[140] Dong C, Xiao Y, Yang Y, Han Z, Guo G and Yang L 2008
Chin. Opt. Lett. 6 300–302
[141] Sedlmeir F, Hauer M, Fürst J U, Leuchs G and Schwefel
H G L 2013 Opt. Expr. 21 23942–23949
[142] Schunk G, Fürst J U, Förtsch M, Strekalov D V, Vogl U,
Sedlmeir F, Schwefel H G L, Leuchs G and Marquardt
C 2014 Opt. Expr. 22 30795–30806
[143] Herr T, Brasch V, Jost J D, Mirgorodskiy I, Lihachev G,
Gorodetsky M L and Kippenberg T J 2014 Phys. Rev.
Lett. 113 123901
[144] Acker W P, Leach D H and Chang R K 1989 Opt. Lett.
14 402–405
[145] Leach D H, Acker W P and Chang R K 1990 Opt. Lett.
15 894–896
[146] Hill S C, Leach D H and Chang R K 1993 JOSA B 10
16–33
[147] Leach D H, Chang R K, Acker W P and Hill S C 1993
JOSA B 10 34–45
[148] Kasparian J, Krämer B, Dewitz J P, Vajda S, Rairoux P,
Vezin B, Boutou V, Leisner T, Hübner W, Wolf J P,
Wöste L and Bennemann K H 1997 Phys. Rev. Lett.
78 2952–2955
[149] Prokopenko Y V, Smirnova T A and Filippov Y F 2004
Tech. Phys. 49 459–465
[150] Ornigotti M and Aiello A 2011 Phys. Rev. A 84 013828
[151] Lin G, Fürst J, Strekalov D V, Grudinin I S and Yu N
2012 Opt. Expr. 20 21372–21378
[152] Sedlmeir F, Hauer M, Fürst J U, Strekalov D V and
Schwefel H G L 2013 proc. SPIE 8600 86001B
[153] Ornigotti M and Aiello A 2014 Physics Research
International 2014 1
[154] Hocker G and Burns W K 1977 Appl. Opt. 16 113–118
[155] Snyder A W and Love J 1983 Optical waveguide theory vol
190 (Springer Science & Business Media)
[156] Xiong C, Pernice W, Ryu K K, Schuck C, Fong K Y,
Palacios T and Tang H X 2011 Opt. Expr. 19 10462–
10470
[157] Beckmann T, Linnenbank H, Steigerwald H, Sturman B,
Haertle D, Buse K and Breunig I 2011 Phys. Rev. Lett.
106 143903
[158] Werner C S, Beckmann T, Buse K and Breunig I 2012 Opt.
Lett. 37 4224–4226
[159] Carmon T and Vahala K J 2007 Nat. Phys. 3 430–435
[160] Strekalov D V, Kowligy A S, Velev V G, Kanter G S,
Kumar P and Huang Y P 2016 J. Mod. Opt. 63 50–63
[161] Grudinin I S and Yu N 2012 JOSA B 29 3010–3014
[162] Kaplan A, Tomes M, Carmon T, Kozlov M, Cohen O,
Bartal G and Schwefel H G L 2013 Opt. Expr. 21 14169–
14180
[163] Grudinin I S, Baumgartel L and Yu N 2013 Opt. Expr. 21
26929–26935
[164] Grudinin I S and Yu N 2014 Opt. Eng. 53 122609
[165] Grudinin I S and Yu N 2015 Optica 2 221–224
[166] Moss D J, Morandotti R, Gaeta A L and Lipson M 2013
Nat. Phot. 7 597–607
42
[167] Riemensberger J, Hartinger K, Herr T, Brasch V,
Holzwarth R and Kippenberg T J 2012 Opt. Expr. 20
27661–27669
[168] Zhang L, Yue Y, Beausoleil R G and Willner A E 2010
Opt. Expr. 18 20529–20534
[169] Zhang L, Bao C, Singh V, Mu J, Yang C, Agarwal A M,
Kimerling L C and Michel J 2013 Opt. Lett. 38 5122–
5125
[170] Bao C, Yan Y, Zhang L, Yue Y, Ahmed N, Agarwal A M,
Kimerling L C, Michel J and Willner A E 2015 JOSA
B 32 26–30
[171] Sprenger B, Schwefel H G L and Wang L J 2009 Opt. Lett.
34 3370–3372
[172] Sprenger B, Schwefel H G L, Lu Z H, Svitlov S and Wang
L J 2010 Opt. Lett. 35 2870–2872
[173] Collodo M C, Sedlmeir F, Sprenger B, Svitlov S, Wang L J
and Schwefel H G L 2014 Opt. Expr. 22 19277–19283
[174] Little B E, Chu S T, Haus H A, Foresi J and Laine J P
1997 J. Lightwave Tech. 15 998
[175] Monifi F, Ozdemir S K and Yang L 2013 Appl. Phys. Lett.
103 181103
[176] Knight J C, Cheung G, Jacques F and Birks T A 1997
Opt. Lett. 22 1129–1131
[177] Chin M K and Ho S T 1998 J. Lightwave Tech. 16 1433
[178] Cai M, Painter O and Vahala K J 2000 Phys. Rev. Lett.
85 74–77
[179] Schiller S and Byer R L 1991 Opt. Lett. 16 1138–1140
[180] Rowland D and Love J 1993 IEE Proceedings JOptoelectronics 140 177–188
[181] Gorodetsky M L and Ilchenko V S 1999 J. Opt. Soc. Am.
B 16 147–154
[182] Mohageg M, Savchenkov A and Maleki L 2007 Opt. Expr.
15 4869–4875
[183] Strekalov D V, Savchenkov A A, Matsko A B and Yu N
2009 Opt. Lett. 34 713–715
[184] Blom F C, Van Dijk D R, Hoekstra H J W M, Driessen A
and Popma T J A 1997 Appl. Phys. Lett. 71 747–749
[185] Rafizadeh D, Zhang J P, Hagness S C, Taflove A, Stair
K A, Ho S T and Tiberio R C 1997 Opt. Lett. 22 1244–
1246
[186] Little B E, Chu S T, Pan W, Ripin D, Kaneko T, Kokubun
Y and Ippen E 1999 IEEE Photonics Technology Lett.
11 215–217
[187] Rabiei P, Steier W H, Zhang C and Dalton L R 2002 J.
Lightwave Tech. 20 1968
[188] Tishinin D V, Dapkus P D, Bond A E, Kim I A K I, Lin
C K and O’brien J 1999 IEEE Photonics Technology
Lett. 11 1003–1005
[189] Poulsen M R, Borel P I, Fage-Pedersen J, Kristensen M,
Povlsen J H, Rottwitt K, Svalgaard M and Svendsen W
2003 Opt. Eng. 42 2821–2834
[190] Choi S J, Djordjev K, Choi S J, Dapkus P D, Lin W, Griffel
G, Menna R and Connolly J 2004 IEEE Photonics
Technology Lett. 16 828–830
[191] Tee C W, Williams K A, Penty R V and White I H
2006 IEEE Journal of Selected Topics in Quantum
Electronics 12 108–116
[192] Le T, Savchenkov A A, Tazawa H, Steier W H and Maleki
L 2006 IEEE Photonics Technology Lett. 18 859–861
[193] Little B E, Laine J P, Lim D R, Haus H A, Kimerling L C
and Chu S T 2000 Opt. Lett. 25 73–75
[194] Laine J P, Little B E, Lim D R, Tapalian H C, Kimerling
L C and Haus H A 2000 IEEE Photonics Technology
Lett. 12 1004–1006
[195] White I M, Oveys H, Fan X, Smith T L and Zhang J 2006
Appl. Phys. Lett. 89 191106
[196] White I M, Suter J D, Oveys H, Fan X, Smith T L, Zhang
J, Koch B J and Haase M A 2007 Opt. Expr. 15 646–651
[197] Conti G N, Berneschi S, Cosi F, Pelli S, Soria S, Righini
G, Dispenza M and Secchi A 2011 Opt. Expr. 19 3651–
Nonlinear and Quantum Optics with Whispering Gallery Resonators
3656
[198] Ghulinyan M, Ramiro-Manzano F, Prtljaga N, Guider R,
Carusotto I, Pitanti A, Pucker G and Pavesi L 2013
Phys. Rev. Lett. 110 163901
[199] Yariv A 2000 Electron. Lett. 36 321–322
[200] Nckel J U 1997 Resonances in nonintegrable open systems
Ph.D. thesis Yale University, New Haven, USA
[201] Braginsky V B, Gorodetsky M L and Ilchenko V S 1993
SPIE Laser Applications 2097 283–288
[202] Kippenberg T J 2004 Nonlinear optics in ultra-high-Q
whispering-gallery optical microcavities Ph.D. thesis
California Institute of Technology Pasadena, USA
[203] Gorodetsky M L, Pryamikov A D and Ilchenko V S 2000
JOSA B 17 1051–1056
[204] Grudinin I S, Matsko A B and Maleki L 2007 Opt. Expr.
15 3390–3395
[205] Grudinin I S, Mansour K and Yu N 2016 Opt. Lett. 41
2378–2381
[206] Lecaplain C, Javerzac-Galy C, Gorodetsky M L and
Kippenberg T J 2016 arXiv preprint arXiv:1603.07305
[207] Ilchenko V S, Volikov P S, Velichansky V L, Treussart F,
Lefevre-Seguin V, Raimond J M and Haroche S 1998
Opt. Comm. 145 86–90
[208] Wagner H P, Schmitzer H, Lutti J, Borri P and Langbein
W 2013 J. Appl. Phys. 113 243101
[209] Tapalian H, Laine J P and Lane P 2002 Photonics
Technology Lett., IEEE 14 1118–1120
[210] Rabiei P and Steier W H 2003 Photonics Technology Lett.,
IEEE 15 1255–1257
[211] Strekalov D V, Thompson R G, Baumgartel L M, Grudinin
I S and Yu N 2011 Opt. Expr. 19 14495–14501
[212] Baumgartel L M, Thompson R J and Yu N 2012 Opt.
Expr. 20 29798–29806
[213] Sturman B, Beckmann T and Breunig I 2012 JOSA B 29
3087–3095
[214] Weng W, Anstie J D, Stace T M, Campbell G, Baynes F N
and Luiten A N 2014 Appl. Phys. Lett. 112 160801
[215] Mohageg M, Savchenkov A, Strekalov D, Matsko A,
Ilchenko V and Maleki L 2005 Electron. Lett. 41 356
[216] Mohageg M, Strekalov D V, Savchenkov A A, Matsko A B,
Ilchenko V S and Maleki L 2005 Opt. Expr. 13 3408–
3419
[217] Meisenheimer S K, Fürst J U, Werner C, Beckmann T,
Buse K and Breunig I 2015 Opt. Expr. 23 24042–24047
[218] Gerson R, Kirchhoff J F, Halliburton L E and Bryan D A
1986 J. Appl. Phys. 60 3553–3557
[219] Peithmann K, Wiebrock A and Buse K 1999 Appl. Phys.
B 68 777–784
[220] Schunk G, Vogl U, Sedlmeir F, Strekalov D V, Otterpohl
A, Averchenko V, Schwefel H G L, Leuchs G and
Marquardt C 2016 J. Mod. Opt. 1–16
[221] Ioppolo T, Ayaz U and Ötügen M V 2009 Opt. Expr. 17
16465–16479
[222] Savchenkov A A, Liang W, Ilchenko V S, Dale E,
Savchenkova E A, Matsko A B, Seidel D and Maleki
L 2014 AIP Advances 4 122901
[223] Ilchenko V S, Savchenkov A A, Matsko A B and Maleki L
2003 JOSA A 20 157–162
[224] Roy S, Prasad M, Topolancik J and Vollmer F 2010 J.
Appl. Phys. 107 3115
[225] Lin W, Zhang H, Liu B, Song B, Li Y, Yang C and Liu Y
2015 Sci. Rep. 5 17791
[226] Teraoka I and Arnold S 2006 JOSA B 23 1381–1389
[227] Foreman M R, Sedlmeir F, Schwefel H G L and Leuchs G
2016 arXiv:1607.05098v1
[228] Schunk G, Vogl U, Strekalov D V and F
[229] Sedlmeir F, Zeltner R, Leuchs G and Schwefel H G L 2014
Opt. Expr. 22 30934–30942
[230] Savchenkov A A, Ilchenko V S, Handley T and Maleki L
2003 IEEE Phot. Tech. Lett. 15 543–544
43
[231] Savchenkov A A, Matsko A B, Strekalov D, Ilchenko V S
and Maleki L 2006 Phys. Rev. B 74 245119
[232] Savchenkov A A, Matsko A B, Strekalov D, Ilchenko V S
and Maleki L 2006 Appl. Phys. Lett. 88 241909
[233] Savchenkov A A, Matsko A B, Strekalov D, Ilchenko V S
and Maleki L 2007 Opt. Comm. 272 257–262
[234] Carmon T, Schwefel H G L, Yang L, Oxborrow M, Stone
A D and Vahala K J 2008 Phys. Rev. Lett. 100 103905
[235] Savchenkov A A, Matsko A B, Ilchenko V S, Strekalov D
and Maleki L 2007 Phys. Rev. A 76 023816
[236] Mazzei A, Götzinger S, de S Menezes L, Zumofen G,
Benson O and Sandoghdar V 2007 Appl. Phys. Lett.
99 173603
[237] Zhu J, Ozdemir S K, Xiao Y F, Li L, He L, Chen D R
and Yang L 2010 Nat. Phot. 4 46–49
[238] Wiersig J 2014 Phys. Rev. Lett. 112 203901
[239] Liu Y, Xuan Y, Xue X, Wang P H, Chen S, Metcalf A J,
Wang J, Leaird D E, Qi M and Weiner A M 2014 Optica
1 137–144
[240] Ramelow S, Farsi A, Clemmen S, Levy J S, Johnson A R,
Okawachi Y, Lamont M R E, Lipson M and Gaeta A L
2014 Opt. Lett. 39 5134–5137
[241] Weng W and Luiten A N 2015 Opt. Lett. 40 5431–5434
[242] Peng B, Özdemir S K, Rotter S, Yilmaz H, Liertzer M,
Monifi F, Bender C M, Nori F and Yang L 2014 Science
346 328–332
[243] Chong Y D, Ge L and Stone A D 2011 Phys. Rev. Lett.
106 093902
[244] Jing H, zdemir S, L X Y, Zhang J, Yang L and Nori F
2014 Phys. Rev. Lett. 113 053604
[245] Lee S B, Yang J, Moon S, Lee S Y, Shim J B, Kim S W,
Lee J H and An K 2009 Phys. Rev. Lett. 103 134101–4
[246] Liertzer M, Ge L, Cerjan A, Stone A D, Treci H E and
Rotter S 2012 Phys. Rev. Lett. 108 173901
[247] Wiersig J 2016 Phys. Rev. A 93 033809
[248] Boyd R W 2003 Nonlinear optics 3rd ed (New York:
Academic Press)
[249] Loudon R 2000 The quantum theory of light (Oxford
university press)
[250] Crenshaw M E and Bowden C M 2002 J. Mod. Opt. 49
511–517
[251] Crenshaw M E and Bowden C M 2002 Opt. Comm. 203
115–124
[252] Foreman M R, Avino S, Zullo R, Loock H P, Vollmer F
and Gagliardi G 2014 The European Physical Journal
Special Topics 223 1971–1988
[253] Matsko A B, Ilchenko V S, Savchenkov A A and Maleki L
2002 Phys. Rev. A 66 043814
[254] Ilchenko V S, Matsko A B, Savchenkov A A and Maleki L
2003 J. Opt. Soc. Am. B 20 1304–1308
[255] Savchenkov A A, Matsko A B, Mohageg M, Strekalov D V
and Maleki L 2007 Opt. Lett. 32 157–159
[256] Sturman B and Breunig I 2011 J. Opt. Soc. Am. B 28
2465
[257] Fürst J U, Buse K, Breunig I, Becker P, Liebertz J and
Bohatý L 2015 Opt. Lett. 40 1932–1935
[258] Vahala K 2004 Optical microcavities (World Scientific)
[259] Kristensen P T, Ge R C and Hughes S 2015 Phys. Rev. A
92 053810
[260] Breunig I 2016 Las. & Phot. Rev.
[261] Kiessling J, Sowade R, Breunig I, Buse K and Dierolf V
2009 Opt. Expr. 17 87–91
[262] Matsko A B, Savchenkov A A, Ilchenko V S, Seidel D and
Maleki L 2007 Opt. Expr. 15 17401–17409
[263] Gordon E I and Rigden J D 1963 Bell System Tech. J. 42
155–179
[264] Ho K P and Kahn J 1993 Phot. Tech. Lett. 5 721–725
[265] Kawanishi T, Oikawa S, Higuma K, Matsuo Y and Izutsu
M 2001 Electron. Lett. 37 1244–1246
[266] Gheorma I L and Osgood RM J 2002 Phot. Techn. Lett.
Nonlinear and Quantum Optics with Whispering Gallery Resonators
14 795–797
[267] Kato M, Fujiura K and Kurihara T 2004 Electron. Lett.
40(5) 299–301
[268] Benter N, Bertram R P, Soergel E, Buse K, Apitz D,
Jacobsen L B and Johansen P M 2005 Appl. Opt. 44
6235–6239
[269] Kato M, Fujiura K and Kurihara T 2005 Appl. Opt. 44
1263–1269
[270] Gan H, Zhang H, DeRose C T, Norwood R A, Fallahi M,
Luo J, Jen A K Y, Liu B, Ho S T and Peyghambarian
N 2006 Appl. Phys. Lett. 89 141113
[271] Ilchenko V S, Yao X S and Maleki L 2000 Proc. SPIE
3930 154–162
[272] Ilchenko V S and Maleki L 2001 Proc. SPIE 4270 120–130
[273] Cohen D A and Levi A F J 2001 Electron. Lett. 37(1)
37–39(2)
[274] Cohen D A, Hossein-Zadeh M and Levi A F J 2001
Electron. Lett. 37(5) 300–301(1)
[275] Cohen D A and Levi A F J 2001 Solid-St. El. 45 495 –
505
[276] Cohen D A, Hossein-Zadeh M and Levi A F J 2001 SolidSt. El. 45 1577 – 1589
[277] Ilchenko V, Savchenkov A, Matsko A and Maleki L 2002
Phot. Tech. Lett. 14 1602–1604
[278] Ilchenko V S, Savchenkov A A, Matsko A B and Maleki L
2003 JOSA B 20 333–341
[279] Hossein-Zadeh M and Levi A 2005 Solid-St. El. 49 1428 –
1434
[280] Hossein-Zadeh M and Levi A 2006 Microwave Theory and
Techniques, IEEE Transactions on 54 821–831
[281] Weldon M, Hum S, Davies R and Okoniewski M 2004 Phot.
Tech. Lett. 16 1295–1297
[282] Tazawa H and Steier W 2005 Electron. Lett. 41(23) 1297–
1298(1)
[283] Tazawa H and Steier W H 2006 Phot. Techn. Lett. 18
211–213
[284] Tazawa H, Kuo Y H, Dunayevskiy I, Luo J, Jen A K Y,
Fetterman H R and Steier W H 2006 J. Lightwave Tech.
24 3514–3519
[285] Bortnik B, Hung Y C, Tazawa H, Seo B J, Luo J, Jen
A K Y, Steier W H and Fetterman H R 2007 J. Sel. T.
Q. El. 13 104–110
[286] Ilchenko V S, Savchenkov A A, Byrd J, Solomatine I,
Matsko A B, Seidel D and Maleki L 2008 Opt. Lett.
33 1569–1571
[287] Gould M, Baehr-Jones T, Ding R, Huang S, Luo J, Jen
A K Y, Fedeli J M, Fournier M and Hochberg M 2011
Opt. Expr. 19 3952–3961
[288] Padmaraju K, Ophir N, Xu Q, Schmidt B, Shakya J,
Manipatruni S, Lipson M and Bergman K 2012 Opt.
Expr. 20 8681–8688
[289] Rabiei P, Ma J, Khan S, Chiles J and Fathpour S 2013
Opt. Expr. 21 25573–25581
[290] Kondratiev N and Gorodetsky M 2013 Bulletin of the
Russian Academy of Sciences: Physics 77 1432–1435
[291] Qiu F, Spring A M, Maeda D, aki Ozawa M, Odoi K,
Aoki I, Otomo A and Yokoyama S 2014 Opt. Expr. 22
14101–14107
[292] Savchenkov A, Liang W, Ilchenko V, Matsko A, Seidel
D and Maleki L 2009 Rf photonic signal processing
components: From high order tunable filters to high
stability tunable oscillators Radar Conference, 2009
IEEE pp 1–6
[293] Savchenkov A A, Liang W, Matsko A B, Ilchenko V S,
Seidel D and Maleki L 2009 Opt. Lett. 34 1300–1302
[294] Savchenkov A, Matsko A, Liang W, Ilchenko V, Seidel D
and Maleki L 2010 Microwave Theory and Techniques,
IEEE Transactions on 58 3167–3174
[295] Matsko A B, Strekalov D V and Yu N 2008 Phys. Rev. A
77 043812
44
[296] Strekalov D V, Savchenkov A A, Matsko A B and Yu N
2009 Las. Phys. Lett. 6 134
[297] Ilchenko V, Matsko A, Savchenkov A and Maleki L
2011 Electro-optical applications of high-q crystalline
wgm resonators Optical Processes in Microparticles and
Nanostructures, A Festschrift Dedicated to R.K. Chang
on His Retirement from Yale University ed Poon A and
Serpengüzel A (Singapore: World Scientific) pp 283–324
[298] Ilchenko V, Savchenkov A, Matsko A, Seidel D and Maleki
L 2010 SPIE Newsroom 10 002536
[299] Passaro V M N and De Leonardis F 2006 J. Sel. T. Q. El.
12 124–133
[300] Wang W C, Lotem H, Forber R and Bui K 1969 Opt. Eng.
45 4402
[301] Sun H, Chen A, Olbricht B C, Davies J A, Sullivan P A,
Liao Y and Dalton L R 2008 Opt. Expr. 16 6592–6599
[302] Passaro V M N, de Tullio C, Troia B, La Notte M,
Giannoccaro G and De Leonardis F 2012 Sensors 12
15558–15598
[303] Zhang X, Hosseini A, Subbaraman H, Wang S, Zhan Q,
Luo J, Jen A K Y and Chen R T 2014 J. Lightwave
Tech. 32 3774–3784
[304] Hsu R C J, Ayazi A, Houshmand B and Jalali B 2007 Nat.
Phot. 1 535–538
[305] Ayazi A, Hsu R C J, Houshmand B, Steier W H and Jalali
B 2008 Opt. Expr. 16 1742–1747
[306] Semertzidis Y, Castillo V, Kowalski L, Kraus D, Larsen R,
Lazarus D, Magurno B, Nikas D, Ozben C, SrinivasanRao T and Tsang T 2000 Nuclear Instruments and
Methods in Physics Research Section A: Accelerators,
Spectrometers, Detectors and Associated Equipment
452 396 – 400
[307] Runde D, Brunken S, Rüter C E and Kip D 2007 Appl.
Phys. B 86 91–95
[308] Matsko A B, Savchenkov A A, Ilchenko V S, Seidel D and
Maleki L 2010 J. Lightwave Tech. 28 3427–3438
[309] Savchenkov A A, Ilchenko V S, Liang W, Eliyahu D,
Matsko A B, Seidel D and Maleki L 2010 Opt. Lett.
35 1572–1574
[310] Tamee K and Yupapin P P 2013 J. Innov. Opt. Health
Sci. 06 1350044
[311] Smirnov A Y, Rashkeev S N and Zagoskin A M 2002 Appl.
Phys. Lett. 80 3503
[312] Deych L, Meriles C and Menon V 2011 Appl. Phys. Lett.
99 241107
[313] Zhang X, Zhu N, Zou C L and Tang H X 2015
arXiv:1510.03545v1
[314] Haigh J A, Langenfeld S, Lambert N J, Baumberg J J,
Ramsay A J, Nunnenkamp A and Ferguson A J 2015
Phys. Rev. A 92 063845
[315] Osada A, Hisatomi R, Noguchi A, Tabuchi Y, Yamazaki
R, Usami K, Sadgrove M, Yalla R, Nomura M and
Nakamura Y 2016 Phys. Rev. Lett. 116 223601
[316] Haigh J A, Nunnenkamp A, Ramsay A J and Ferguson
A J 2016 arXiv:1607.02985v1
[317] Kozyreff G, Dominguez Juarez J L and Martorell J 2008
Phys. Rev. A 77 43817
[318] Fürst J U, Strekalov D V, Elser D, Lassen M, Andersen
U L, Marquardt C and Leuchs G 2010 Phys. Rev. Lett.
104 153901
[319] Fürst J U, Strekalov D V, Elser D, Aiello A, Andersen
U L, Marquardt C and Leuchs G 2010 Phys. Rev. Lett.
105 263904
[320] Strekalov D V, Kowligy A S, Huang Y P and Kumar P
2014 New J. Phys. 16 053025
[321] Strekalov D V, Kowligy A S, Huang Y P and Kumar P
2015 New J. Phys. 17 099501
[322] Buck J R and Kimble H J 2003 Phys. Rev. A 67 033806
[323] Kippenberg T J, Spillane S M, Min B and Vahala K J 2004
J. Sel. T. Q. El. 10 1219–1228
Nonlinear and Quantum Optics with Whispering Gallery Resonators
[324] Breunig I, Beckmann T and Buse K 2012 proc. SPIE 8236
82360S
[325] Haertle D 2010 J. Opt. 12 5202
[326] Moore J, Tomes M, Carmon T and Jarrahi M 2011 Opt.
Expr. 19 24139
[327] Dumeige Y and Féron P 2006 Phys. Rev. A 74 63804
[328] Yang Z, Chak P, Bristow A D, van Driel H M, Iyer R,
Aitchison J S, Smirl A L and Sipe J E 2007 Opt. Lett.
32 826–828
[329] Kuo P S and Solomon G 2011 Opt. Expr. 19 16898–16918
[330] Kuo P S, Bravo-Abad J and Solomon G S 2014 Nat.
Comm. 5 3109
[331] Grudinin I S, Lin G and Yu N 2013 Opt. Lett. 38 2410–
2412
[332] Lin G, Fuerst J U, Strekalov D V and Yu N 2013 Appl.
Phys. Lett. 103 181107
[333] Lin G and Yu N 2014 Opt. Expr. 22 557–562
[334] Tan H T and Huang H 2011 Phys. Rev. A 83 015802
[335] Kozyreff G, Dominguez-Juarez J L and Martorell J 2011
Las. & Phot. Rev. 5 737–749
[336] Breunig I, Haertle D and Buse K 2011 Appl. Phys. B 105
99
[337] Huang Y P and Kumar P 2012 J. Sel. T. Q. El. 18 600–
611
[338] Sun Y Z, Huang Y P and Kumar P 2013 Phys. Rev. Lett.
110 223901
[339] Huang Y P and Kumar P 2010 Opt. Lett. 35 2376–2378
[340] Ashkin A, Boyd G D and Dziedzic J M 1966 J. Q. El.
109–124
[341] Graham G and Haken H 1968 Zeitschrift fur Physik 210
276–302
[342] Smith R G 1970 J. Q. El. 215–223
[343] Fabre C, Giacobino E, Heidmann A and Reynaud S 1989
J.de Phys. 50 1209–1225
[344] Debuisschert T, Sizmann A, Giacobino E and Fabre C 1993
JOSA B 10 1668–1680
[345] Berger V 1997 JOSA B 14 1351–1360
[346] Wu Z J, Ming Y, Xu F and Lu Y Q 2012 Opt. Expr. 20
17192–17200
[347] Ulvila V, Phillips C R, Halonen L and Vainio M 2013 Opt.
Lett. 38 4281–4284
[348] Ricciardi I, Mosca S, Parisi M, Maddaloni P, Santamaria
L, De Natale P and De Rosa M 2015 Phys. Rev. A 91
063839
[349] Leo F, Hansson T, Ricciardi I, De Rosa M, Coen S,
Wabnitz S and Erkintalo M 2016 Phys. Rev. Lett. 116
033901
[350] Marte M A M 1994 Phys. Rev. A 49 R3166–R3169
[351] Bache M, Lodahl P, Mamaev A V, Marcus M and Saffman
M 2002 Phys. Rev. A 65 033811
[352] Kumar P 1990 Opt. Lett. 15 1476–1478
[353] McCusker K T, Huang Y P, Kowligy A S and Kumar P
2013 Phys. Rev. Lett. 110 240403
[354] Strekalov D V, Kowligy A S, Huang Y P and Kumar P
2014 Phys. Rev. A 89 063820
[355] Yu J, Giulietti K, Sourgen F, Ross A, Wolf J P, Ferriol M,
Foulon G, Goutaudier C, Cohen-Adad M T and Boulon
G 1999 Opt. Lett. 24 394–396
[356] Levy J S, Foster M A, Gaeta A L and Lipson M 2011 Opt.
Expr. 19 11415–11421
[357] Pernice W H P, Xiong C, Schuck C and Tang H X 2012
Appl. Phys. Lett. 100 223501
[358] Wang C, Burek M J, Lin Z, Atikian H A, Venkataraman
V, Huang I C, Stark P and Lončar M 2014 Opt. Expr.
22 30924–30933
[359] Lin J, Xu Y, Fang Z, Wang M, Wang N, Qiao L, Fang W
and Cheng Y 2015 Science China Physic 58 114209
[360] Mariani S, Andronico A, Lemaı̂tre A, Favero I, Ducci S
and Leo G 2014 Opt. Lett. 39 3062–3065
[361] Vukovic N, Healy N, Sparks J R, Badding J V, Horak P
45
and Peacock A C 2015 Sci. Rep. 5 11798
[362] Lake D P, Mitchell M, Jayakumar H, dos Santos L F, Curic
D and Barclay P E 2016 Appl. Phys. Lett. 108 031109
[363] Sasagawa K and Tsuchiya M 2009 Appl. Phys. Exp. 2
122401
[364] Beckmann T, Buse K and Breunig I 2012 Opt. Lett. 37
5250–5252
[365] Werner C S, Buse K and Breunig I 2015 Opt. Lett. 40
772–775
[366] Förtsch M, Schunk G, Fürst J U, Strekalov D, Gerrits
T, Stevens M J, Sedlmeir F, Schwefel H G L, Nam
S W, Leuchs G and Marquardt C 2015 Phys. Rev. A
91 023812
[367] Förtsch M, Gerrits T, Stevens M J, Strekalov D, Schunk G,
Fürst J U, Vogl U, Sedlmeir F, Schwefel H G L, Leuchs
G, Nam S W and Marquardt C 2015 J. Opt. 17 065501
[368] Andronico A, Favero I and Leo G 2008 Opt. Lett. 33 2026–
2028
[369] Stolen R H and Bjorkholm J E 1982 IEEE J. Quant.
Electron. QE-18 1062
[370] Treussart F, Ilchenko V S, Roch J F, Hare J, LefevreSeguin V, Raimond J M and Haroche S 1998 Eur. Phys.
J. D 1 235–238
[371] Ward J M, O’Shea D G, Shortt B J and Chormaic S N
2007 J. Appl. Phys. 102 023104–023104
[372] Park Y S and Wang H 2007 Opt. Lett. 32 3104–3106
[373] Heebner J E, Chak P, Pereira S, Sipe J E and Boyd R W
2004 JOSA B 21 1818–1832
[374] Heebner J E, Lepeshkin N N, Schweinsberg A, Wicks G W,
Boyd R W, Grover R and Ho P T 2004 Opt. Lett. 29
769–771
[375] Lu X, Lee J Y, Rogers S and Lin Q 2014 Opt. Expr. 22
30826–30832
[376] Xiao Y F, Şahin Kaya Özdemir, Gaddam V, Dong C H,
Imoto N and Yang L 2008 Opt. Expr. 16 21462–21475
[377] Klyshko D N 1988 Photons and Nonlinear optics (New
York, NY USA: Taylor and Francis)
[378] Matsko A B, Savchenkov A A, Strekalov D, Ilchenko V S
and Maleki L 2005 Phys. Rev. A 71 033804
[379] Kippenberg T J, Spillane S M and Vahala K J 2004 Phys.
Rev. Lett. 93 83904
[380] Savchenkov A A, Matsko A B, Strekalov D, Mohageg M,
Ilchenko V S and Maleki L 2004 Phys. Rev. Lett. 93
243905
[381] Liang W, Savchenkov A A, Xie Z, McMillan J F, Burkhart
J, Ilchenko V S, Wong C W, Matsko A B and Maleki L
2015 Optica 2 40
[382] Ye J and Cundiff S T 2005 Femtosecond optical frequency
comb technology (New York, NY, USA: Springer)
[383] Fortier T M, Kirchner M S, Quinlan F, Taylor J, Bergquist
J C, Rosenband T, Lemke N, Ludlow A, Jiang Y, Oates
C W and Diddams S 2011 Nat. Phot. 5 425–429
[384] Li C H, Benedick A J, Fendel P, Glenday A G, Kartner
F X, Phillips D F, Sasselov D, Szentgyorgyi A and
Walsworth R L 2008 Nature 452 610 – 612
[385] Levy J S, Gondarenko A, Foster M A, Turner-Foster A C,
Gaeta A L and Lipson M 2010 Nat. Phot. 4 37–40
[386] Coddington I, Swann W C, Nenadovic L and Newbury
N R 2009 Nat. Phot. 3 351–356
[387] Del’Haye P, Arcizet O, Schliesser A, Holzwarth R and
Kippenberg T J 2008 Phys. Rev. Lett. 101 053903
[388] Savchenkov A A, Matsko A B, Ilchenko V S, Solomatine
I, Seidel D and Maleki L 2008 Phys. Rev. Lett. 101
093902
[389] Grudinin I S, Yu N and Maleki L 2009 Opt. Lett. 34 878–
880
[390] Agha I H, Okawachi Y and Gaeta A L 2009 Opt. Expr. 17
16209–16215
[391] Braje D, Hollberg L and Diddams S 2009 Phys. Rev. Lett.
102(19) 193902
Nonlinear and Quantum Optics with Whispering Gallery Resonators
[392] Matsko A B, Savchenkov A A, Liang W, Ilchenko V S,
Seidel D and Maleki L 2009 Symp. Frequency Standards
and Metrology 7 539–558
[393] Razzari L, Duchesne D, Ferrera M, Morandotti R, Chu S,
Little B and Moss D 2010 Nat. Phot. 4 41–45
[394] Hansson T, Modotto D and Wabnitz S 2013 Phys. Rev. A
88(2) 023819
[395] Herr T, Hartinger K, Riemensberger J, Wang C Y,
Gavartin E, Holzwarth R, Gorodetsky M L and
Kippenberg T J 2012 Nat. Phot. 6 480–487
[396] Coillet A and Chembo Y 2014 Opt. Lett. 39 1529–1532
[397] Herr T, Brasch V, Jost J D, Wang C Y, Kondratiev N M,
Gorodetsky M L and Kippenberg T J 2014 Nat. Phot.
8 145–152
[398] Huang S W, Zhou H, Yang J, McMillan J F, Matsko A, Yu
M, Kwong D L, Maleki L and Wong C W 2015 Phys.
Rev. Lett. 114 053901
[399] Yi X, Yang Q F, Yang K Y, Suh M G and Vahala K 2015
Optica 2 1078–1085
[400] Brasch V, Geiselmann M, Herr T, Lihachev G, Pfeiffer
M H P, Gorodetsky M L and Kippenberg T J 2016
Science 351 357–360
[401] Grudinin I S and Yu N 2015 Proc. SPIE 9343 93430F–
93430F–9
[402] Liang W, Eliyahu D, Ilchenko V, Savchenkov A, Matsko
A, Seidel D and Maleki L 2015 Nat. Comm. 6 7957
[403] Matsko A B, Savchenkov A A and Maleki L 2012 Opt.
Lett. 37 43–45
[404] Liang W, Savchenkov A A, Ilchenko V S, Eliyahu D, Seidel
D, Matsko A B and Maleki L 2014 Opt. Lett. 39 2920–
2923
[405] Godey C, Balakireva I V, Coillet A and Chembo Y K 2014
Phys. Rev. A 89 063814
[406] Xue X, Xuan Y, Liu Y, Wang P H, Chen S, Wang J, Leaird
D E, Qi M and Weiner A M 2015 Nat. Phot. 9 594–600
[407] Coillet A, Balakireva I, Henriet R, Saleh K, Larger L,
Dudley J, Menyuk C and Chembo Y 2013 Photonics
Journal, IEEE 5 6100409–6100409
[408] Henriet R, Lin G, Coillet A, Jacquot M, Furfaro L, Larger
L and Chembo Y K 2015 Opt. Lett. 40 1567–1570
[409] Soltani M, Matsko A and Maleki L 2016 Las. & Phot. Rev.
10 158–162
[410] Lobanov V E, Lihachev G and Gorodetsky M L 2015 arXiv
preprint arXiv:1508.06850
[411] Papp S B, Del’Haye P and Diddams S A 2013 Opt. Expr.
21 17615–17624
[412] Taheri H, Eftekhar A, Wiesenfeld K and Adibi A 2015
Photonics Journal, IEEE 7 1–9
[413] Karpov M, Guo H, Lucas E, Kordts A, Pfeiffer
M, Lichachev G, Lobanov V, Gorodetsky M and
Kippenberg T 2016 arXiv:1601.05036
[414] Milián C, Gorbach A V, Taki M, Yulin A V and Skryabin
D V 2015 Phys. Rev. A 92 033851
[415] Chembo Y K and Yu N 2010 Opt. Lett. 35 2696–2698
[416] Matsko A B, Savchenkov A A and Maleki L 2012 Opt.
Lett. 37 4856–4858
[417] Savchenkov A A, Matsko A B, Liang W, Ilchenko V S,
Seidel D and Maleki L 2012 Phys. Rev. A 86 013838
[418] Hansson T, Modotto D and Wabnitz S 2014 Opt. Comm.
312 134–136
[419] Chembo Y K and Menyuk C R 2013 Phys. Rev. A 87(5)
053852
[420] Lugiato L A and Lefever R 1987 Phys. Rev. Lett. 58 2209
[421] Wabnitz S 1993 Opt. Lett. 18 601–603
[422] Blow K J and Doran N J 1984 Phys. Rev. Lett. 52(7)
526–529
[423] Ghidaglia J M 1988 Finite dimensional behavior for weakly
damped driven schrödinger equations Annales de l’IHP
Analyse non linéaire vol 5 pp 365–405
[424] Barashenkov I V and Smirnov Y S 1996 Phys. Rev. E
46
54(5) 5707–5725
[425] Afanasjev V V, Malomed B A and Chu P L 1997 Phys.
Rev. E 56(5) 6020–6025
[426] Matsko A B, Savchenkov A A, Liang W, Ilchenko V S,
Seidel D and Maleki L 2011 Opt. Lett. 36 2845–2847
[427] Coen S and Erkintalo M 2013 Opt. Lett. 38 1790–1792
[428] Matsko A B and Maleki L 2013 Opt. Express 21 28862–
28876
[429] Matsko A B and Maleki L 2015 J. Opt. Soc. Am. B 32
232–240
[430] Matsko A B and Maleki L 2015 Phys. Rev. A 91(1) 013831
[431] Ferdous F, Miao H, Leaird D E, Srinivasan K, Wang J,
Chen L, Varghese L T and Weiner A M 2011 Nat. Phot.
5 770–776
[432] Saha K, Okawachi Y, Shim B, Levy J S, Salem R, Johnson
A R, Foster M A, Lamont M R E, Lipson M and Gaeta
A L 2013 Opt. Express 21 1335–1343
[433] Savchenkov A A, Rubiola E, Matsko A B, Ilchenko V S
and Maleki L 2008 Opt. Exp. 16 4130–4144
[434] Miyazaki H and Jimba Y 2000 Phys. Rev. B 62 7976–7997
[435] Miller S A, Okawachi Y, Ramelow S, Luke K, Dutt A,
Farsi A, Gaeta A L and Lipson M 2015 Opt. Expr. 23
21527–21540
[436] Xue X, Xuan Y, Wang P H, Liu Y, Leaird D E, Qi M and
Weiner A M 2015 Las. & Phot. Rev. 9 L23–L28
[437] Bao C, Zhang L, Matsko A, Yan Y, Zhao Z, Xie G, Agarwal
A M, Kimerling L C, Michel J, Maleki L and Willner
A E 2014 Opt. Lett. 39 6126–6129
[438] Savchenkov A A, Ilchenko V S, Teodoro F D, Belden P M,
Lotshaw W T, Matsko A B and Maleki L 2015 Opt.
Lett. 40 3468–3471
[439] Lecaplain C, Javerzac-Galy C, Lucas E, Jost J D and
Kippenberg T J 2015 arXiv preprint arXiv:1506.00626
[440] Strekalov D V and Yu N 2009 Phys. Rev. A 79 041805(R)
[441] Fülöp A, Krückel C J, Castelló-Lurbe D, Silvestre E and
Torres-Company V 2015 Opt. Lett. 40 4006–4009
[442] Hansson T and Wabnitz S 2014 Phys. Rev. A 90(1) 013811
[443] Hu X, Liu Y, Xu X, Feng Y, Zhang W, Wang W, Song J,
Wang Y and Zhao W 2015 Appl. Opt. 54 8751–8757
[444] Lin G, Martinenghi R, Diallo S, Saleh K, Coillet A and
Chembo Y K 2015 Appl. Opt. 54 2407–2412
[445] Ilchenko V S, Byrd J, Savchenkov A A, Eliyahu D, Liang
W, Matsko A B, Seidel D and Maleki L 2013 Kerr
frequency comb-based k a-band rf photonic oscillator
European Frequency and Time Forum & International
Frequency Control Symposium (EFTF/IFC), 2013
Joint (IEEE) pp 29–32
[446] Akimov D A, Ivanov A A, Naumov A N, Kolevatova O A,
Alfimov M V, Birks T A, Wadsworth W J, Russell P S J,
Podshivalov A A and Zheltikov A M 1969 Appl. Phys.
B 76 515–519
[447] Grubsky V and Savchenko S 2005 Opt. Expr. 13 6798–
6806
[448] Grubsky V and Feinberg J 2007 Opt. Comm. 274 447–450
[449] Shahraam A V and Monro T M 2009 Opt. Expr. 17 2298–
2318
[450] Wiedemann U, Karapetyan K, Dan C, Pritzkau D, Alt W,
Irsen S and Meschede D 2010 Opt. Expr. 18 7693–7704
[451] Coillet A, Vienne G and Grelu P 2010 JOSA B 27 394–401
[452] Lee T, Jung Y, Codemard C A, Ding M, Broderick N G R
and Brambilla G 2012 Opt. Expr. 20 8503–8511
[453] Farnesi D, Barucci A, Righini G C, Berneschi S, Soria S
and Nunzi Conti G 2014 Appl. Phys. Lett. 112 93901
[454] Li Q, Davanço M and Srinivasan K 2016 Nat. Phot. 10
406–414
[455] Ferrera M, Razzari L, Duchesne D, Morandotti R, Yang
Z, Liscidini M, Sipe J E, Chu S, Little B E and Moss
D J 2008 Nat. Phot. 2 737–740
[456] Roussev R V, Langrock C, Kurz J R and Fejer M M 2004
Opt. Lett. 29 1518–1520
Nonlinear and Quantum Optics with Whispering Gallery Resonators
[457] Jung H, Xiong C, Fong K Y, Zhang X and Tang H X 2013
Opt. Lett. 38 2810–2813
[458] Jung H, Stoll R, Guo X, Fischer D and Tang H X 2014
Optica 1 396–399
[459] Jung H, Fong K Y, Xiong C and Tang H X 2014 Opt. Lett.
39 84–87
[460] Xiong C, Pernice W H, Sun X, Schuck C, Fong K Y and
Tang H X 2012 New J. Phys. 14 095014
[461] Spillane S M, Kippenberg T J and Vahala K J 2002 Nature
415 621–623
[462] Grudinin I S, Matsko A B and Maleki L 2009 Phys. Rev.
Lett. 102 043902
[463] Grudinin I S and Maleki L 2007 Opt. Lett. 32 166–168
[464] Savchenkov A A, Matsko A B, Mohageg M and Maleki L
2007 Opt. Lett. 32 497–499
[465] Grudinin I S and Maleki L 2008 JOSA B 25 594–598
[466] Liang W, Ilchenko V S, Savchenkov A A, Matsko A B,
Seidel D and Maleki L 2010 Phys. Rev. Lett. 105 143903
[467] Min B, Kippenberg T J and Vahala K J 2003 Opt. Lett.
28 1507–1509
[468] Kippenberg T J, Spillane S M, Armani D K and Vahala
K J 2004 Opt. Lett. 29 1224–1226
[469] Vanier F, Rochette M, Godbout N and Peter Y A 2013
Opt. Lett. 38 4966–4969
[470] Jouravlev M, Mason D R and Kim K S 2012 Phys. Rev.
A 85 013825
[471] Melnikau D, Savateeva D, Chuvilin A, Hillenbrand R and
Rakovich Y P 2011 Opt. Expr. 19 22280–22291
[472] Li B B, Xiao Y F, Yan M Y, Clements W R and Gong Q
2013 Opt. Lett. 38 1802–1804
[473] Deka N, Maker A J and Armani A M 2014 Opt. Lett. 39
1354–1357
[474] Yang L, Carmon T, Min B, Spillane S M and Vahala K J
2005 Appl. Phys. Lett. 86 091114
[475] Simons M T and Novikova I 2011 Opt. Lett. 36 3027–3029
[476] Tomes M and Carmon T 2009 Phys. Rev. Lett. 102 113601
[477] Li J, Lee H and Vahala K J 2013 Nat. Comm. 4 2097
[478] Lin G, Diallo S, Saleh K, Martinenghi R, Beugnot J C,
Sylvestre T and Chembo Y K 2014 Appl. Phys. Lett.
105 231103
[479] Bahl G, Zehnpfennig J, Tomes M and Carmon T 2011 Nat.
Comm. 2 403
[480] Dong C H, Shen Z, Zou C L, Zhang Y L, Fu W and Guo
G C 2015 Nat. Comm. 6 6193
[481] Loh W, Green A A S, Baynes F N, Cole D C, Quinlan F J,
Lee H, Vahala K J, Papp S B and Diddams S A 2015
Optica 2 225
[482] Matsko A B, Savchenkov A A, Ilchenko V S, Seidel D and
Maleki L 2009 Phys. Rev. Lett. 103 257403
[483] Savchenkov A A, Matsko A B, Ilchenko V S, Seidel D and
Maleki L 2011 Opt. Lett. 36 3338–3340
[484] Zehnpfennig J, Bahl G, Tomes M and Carmon T 2011 Opt.
Expr. 19 14240–14248
[485] Sturman B and Breunig I 2015 J. Appl. Phys. 118 013102
[486] Savchenkov A A, Matsko A B and Maleki L 2006 Opt.
Lett. 31 92–94
[487] Tomes M, Marquardt F, Bahl G and Carmon T 2011 Phys.
Rev. A 84 063806
[488] Bahl G, Tomes M, Marquardt F and Carmon T 2012 Nat.
Phys. 8 203–207
[489] Ilchenko V S and Gorodetsky M L 1992 Las. Phys. 2 1004–
1009
[490] Carmon T, Yang L and Vahala K J 2004 Opt. Expr. 12
4742–4750
[491] Fomin A E, Gorodetsky M L, Grudinin I S and Ilchenko
V S 2005 JOSA B 22 459–465
[492] Rokhsari H and Vahala K J 2005 Opt. Lett. 30 427–429
[493] Schmidt C, Chipouline A, Pertsch T, Tünnermann A,
Egorov O, Lederer F and Deych L 2008 Opt. Expr. 16
6285–6301
47
[494] Weng W, Anstie J D, Abbott P, Fan B, Stace T M and
Luiten A N 2015 Phys. Rev. A 91 063801
[495] Diallo S, Lin G and Chembo Y K 2015 Opt. Lett. 40 3834–
3837
[496] Grudinin I, Lee H, Chen T and Vahala K 2011 Opt. Expr.
19 7365–7372
[497] He L, Xiao Y F, Dong C, Zhu J, Gaddam V and Yang L
2008 Appl. Phys. Lett. 93 201102
[498] Choi H S and Armani A M 2010 Appl. Phys. Lett. 97
223306
[499] Murzina T V, Conti G N, Barucci A, Berneschi S,
Razdolskiy I and Soria S 2012 Opt. Mat. Expr. 2 1088–
1094
[500] Deng Y, Flores-Flores R, Jain R K and Hossein-Zadeh M
2013 Opt. Lett. 38 4413–4416
[501] Maleki L and Matsko A 2014 Lithium niobate whispering
gallery resonators: Applications and fundamental
studies Ferroelectric Crystals for Photonic Applications
(Springer) pp 337–383
[502] Savchenkov A A, Matsko A B, Ilchenko V S, Solomatine
I, Seidel D and Maleki L 2013 Rf-induced change of
optical refractive index in strontium barium niobate
Proc. SPIE 8600, Laser Resonators, Microresonators,
and Beam Control XV (International Society for Optics
and Photonics) p 86000O
[503] Gerry C C and Knight P L 2005 Introductory Quantum
Optics (Cambridge University Press)
[504] Bachor H A and Ralph T C 2004 A guide to experiments in
quantum optics (Weinheim: Wiley-VCH Verlag GmbH
& Co. KGaA)
[505] Gorodetsky M L and Grudinin I S 2004 JOSA B 21 697–
705
[506] Ou Z Y and Lu Y J 1999 Phys. Rev. Lett. 38 2556–2559
[507] Christ A, Laiho K, Eckstein A, Lauckner T, Mosley P J
and Silberhorn C 2009 Phys. Rev. A 80 033829
[508] Wu L A, Kimble H J, Hall J L and Wu H 1986 Phys. Rev.
Lett. 57 2520–2523
[509] Heidmann A, Horowicz R J, Reynaud S, Giacobino E,
Fabre C and Camy G 1987 Phys. Rev. Lett. 59 2555–
2557
[510] Collett M J and Walls D F 1985 Phys. Rev. A 32 2887–
2892
[511] Sizmann A, Horowitz R J, Wagner G and Leuchs G 1990
Opt. Comm. 80 138–142
[512] Kurz P, Paschotta R, Fiedler K, Sizmann A, Leuchs G and
Mlynek J 1992 Appl. Phys. B 55 216–225
[513] Drummond P D, McNeil K J and Walls D F 1981 Opt.
Acta 28 211–225
[514] Pereira S F, Xiao M, Kimble H J and Hall J L 1988 Phys.
Rev. A 38 4931
[515] Reid M D and Drummond P D 1988 Phys. Rev. Lett. 60
2731–2733
[516] Lee D H, Klein M E and Boller K J 1998 Appl. Phys. B
66 747–753
[517] Porzio A, Sciarrino F, Chiummo A, Fiorentino M and
Solimeno S 2001 Opt. Comm. 194 373–379
[518] Zhang Y, Kasai K and Hayasaka K 2004 JOSA B 21 1044–
1049
[519] Hage B, Samblowski A and Schnabel R 2010 Phys. Rev.
A 81 062301
[520] Pysher M, Miwa Y, Shahrokhshahi R, Bloomer R and
Pfister O 2011 Phys. Rev. Lett. 107 030505
[521] Andersen U L, Gehring T, Marquardt C and Leuchs G
2016 Physica Scripta 91 053001
[522] Clemmen S, Huy K P, Bogaerts W, Baets R G, Emplit P
and Massar S 2009 Opt. Expr. 17 16558–16570
[523] Azzini S, Grassani D, Strain M J, Sorel M, Helt L G, Sipe
J E, Liscidini M, Galli M and Bajoni D 2012 Opt. Expr.
20 23100–23107
[524] Engin E, Bonneau D, Natarajan C M, Clark A S, Tanner
Nonlinear and Quantum Optics with Whispering Gallery Resonators
[525]
[526]
[527]
[528]
[529]
[530]
[531]
[532]
[533]
[534]
[535]
[536]
[537]
[538]
[539]
[540]
[541]
[542]
[543]
[544]
[545]
[546]
[547]
[548]
[549]
[550]
[551]
[552]
[553]
[554]
[555]
[556]
[557]
[558]
M G, Hadfield R H, Dorenbos S N, Zwiller V, Ohira K,
Suzuki N, Yoshida H, Iizuka N, Ezaki M, O’Brien J L
and Thompson M G 2013 Opt. Expr. 21 27826
Guo Y, Zhang W, Dong S, Huang Y and Peng J 2014 Opt.
Lett. 39 2526–2529
Grassani D, Azzini S, Liscidini M, Galli M, Strain M J,
Sorel M, Sipe J E and Bajoni D 2015 Optica 2 88–94
Wakabayashi R, Fujiwara M, Yoshino K i, Nambu Y,
Sasaki M and Aoki T 2015 Opt. Expr. 23 1103
Suo J, Dong S, Zhang W, Huang Y and Peng J 2015 Opt.
Expr. 23 3985–3995
Rogers S, Lu X, Jiang W C and Lin Q 2015 Appl. Phys.
Lett. 107 041102
Jiang W C, Lu X, Zhang J, Painter O and Lin Q 2015
Opt. Expr. 23 20884–20904
Dutt A, Luke K, Manipatruni S, Gaeta A L, Nussenzveig
P and Lipson M 2015 Phys. Rev. Appl. 3 044005
Hoff U B, Nielsen B M and Andersen U L 2015 Opt. Expr.
23 12013–12036
Huang J and Kumar P 1992 Phys. Rev. Lett. 68 2153–2157
Albota M A and Wong F N C 2004 Opt. Lett. 29 1449–
1451
Langrock C, Diamanti E, Roussev R V, Yamamoto Y,
Fejer M M and Takesue H 2005 Opt. Lett. 30 1725–
1727
McGuinness H J, Raymer M G, McKinstrie C J and Radic
S 2010 Phys. Rev. Lett. 105 093604
Huang Y P, Velev V and Kumar P 2013 Opt. Lett. 38
2119–2121
Schoelkopf R J and Girvin S M 2008 Nature 451 664–669
Devoret M H and Schoelkopf R J 2013 Science 339 1169–
1174
Leisawitz D 2004 Adv. Space Res. 34 631–636
Faraon A, Barclay P E, Santori C, Fu K M C and
Beausoleil R G 2011 Nat. Phot. 5 301–305
Spillane S M, Kippenberg T J, Vahala K J, Goh K W,
Wilcut E and Kimble H J 2005 Phys. Rev. A 71 013817
Mabuchi H and Kimble J 1994 Opt. Lett. 19 749–752
Vernooy D W, Furusawa A, Georgiades N P, Ilchenko V S
and Kimble H J 1998 Phys. Rev. A 57 R2293–R2296
Aoki T, Dayan B, Wilcut E, Bowen W P, Parkins A S,
Kippenberg T J, Vahala K J and Kimble H J 2006
Nature 443 671–674
Dayan B, Parkins A S, Aoki T, Ostby E P, Vahala K J
and Kimble H J 2008 Science 319 1062–1065
Aoki T, Parkins A S, Alton D J, Regal C A, Dayan B,
Ostby E, Vahala K J and Kimble H J 2009 Phys. Rev.
Lett. 102 083601
Junge C, O’Shea D, Volz J and Rauschenbeutel A 2013
Phys. Rev. Lett. 110 213604
Shomroni X, Rosenblum S, Lovsky Y, Bechler O,
Guendelman G and Dayan B 2014 Science 345 903–
906
Rosenblum S, Bechler O, Shomroni I, Lovsky Y,
Guendelman G and Dayan B 2016 Nat. Phot. 10 19–22
Norris D J, Kuwata-Gonokami M and Moerner W E 1997
Appl. Phys. Lett. 71 297–299
Michler P, Kiraz A, Becher C, Schoenfeld W V, Petroff
P M, Zhang L, Hu E and Imamoglu A 2000 Science
290 2282–2285
Peter E, Senellart P, Martrou D, Lemaı̂tre A, Hours J,
Gérard J M and Bloch J 2005 Phys. Rev. Lett. 95
067401
Srinivasan K and Painter O 2007 Nature 450 862–865
Schietinger S, Schröder T and Benson O 2008 Nano Lett.
8 3911–3915
Larsson M, Dinyari K N and Wang H 2009 Nano Lett. 9
1447–1450
Li P B, Gao S Y and Li F L 2011 Phys. Rev. A 83 054306
Chen Q, Yang W L and Feng M 2012 Eur. Phys. J. D 66
48
238
[559] Ates S, Agha I, Gulinatti A, Rech I, Badolato A and
Srinivasan K 2013 Sci. Rep. 3 1397
[560] Jin G s, Wang C, Zhang Y and Jiao R z 2014 Int. J. Theor.
Phys. 53 3774–3779
[561] Misra B and Sudarshan E C G 1977 J. Math. Phys. 18
756–763
[562] Hendrickson S M, Weiler C N, Camacho R M, Rakich P T,
Young A I, Shaw M J, Pittman T B, Franson J D and
Jacobs B C 2013 Phys. Rev. A 87 23808
[563] Franson J D, Jacobs B C and Pittman T B 2004 Phys.
Rev. A 70 062302
[564] Franson J D, Pittman T B and Jacobs B C 2007 JOSA B
24 209–213
[565] Huang Y P, Altepeter J B and Kumar P 2010 Phys. Rev.
A 82 063826
[566] Clader B D, Hendrickson S M, Camacho R M and Jacobs
B C 2013 Opt. Expr. 21 6169–6179
[567] Breunig I, Sturman B, Bückle A, Werner C S and Buse K
2013 Opt. Lett. 38 3316–3318
[568] Huang Y P and Kumar P 2012 Phys. Rev. Lett. 108 30502
[569] Cirac J I, Zoller P, Kimble H J and Mabuchi H 1997 Phys.
Rev. Lett. 78 3221–3224
[570] Shapiro J H 2006 Phys. Rev. A 73 062305
[571] Dove J, Chudzicki C and Shapiro J H 2014 Phys. Rev. A
90 062314
[572] Walsh P and Kemeny G 1963 J. Appl. Phys. 34 956–957
[573] He L, Özdemir S K and Yang L 2013 Las. & Phot. Rev. 7
60–82
[574] Campillo A J, Eversole J D and Lin H B 1991 Phys. Rev.
Lett. 67 437–440
[575] Chang S, Rex N B, Chang R K, Chong G and Guido L J
1999 Appl. Phys. Lett. 75 166–168
[576] Polman A, Min B, Kalkman J, Kippenberg T J and Vahala
K J 2004 Appl. Phys. Lett. 84 1037
[577] Kippenberg T J, Kalkman J, Polman A and Vahala K J
2006 Phys. Rev. A 74 51802
[578] Murugan G S, Zervas M N, Panitchob Y and Wilkinson
J S 2011 Opt. Lett. 36 73–75
[579] Yang L and Vahala K J 2003 Opt. Lett. 28 592–594
[580] Yang L, Armani D K and Vahala K J 2003 Appl. Phys.
Lett. 83 825
[581] Dominguez-Juarez J L, Kozyreff G and Martorell J 2011
Nat. Comm. 2 254
[582] Xu Y, Han M, Wang A, Liu Z and Heflin J R 2008 Appl.
Phys. Lett. 100 163905
[583] Foreman M R, Swaim J D and Vollmer F 2015 Adv. Opt.
Phot. 7 168–240
[584] Schliesser A and Kippenberg T J 2010 Advances In
Atomic, Molecular, and Optical Physics 58 207–323
[585] Aspelmeyer M, Kippenberg T J and Marquardt F 2014
Rev. Mod. Phys. 86 1391–1452
[586] Dong C, Fiore V, Kuzyk M C and Wang H 2013 Phys.
Rev. A 87(5) 055802
[587] Fiore V, Dong C, Kuzyk M C and Wang H 2013 Phys.
Rev. A 87(2) 023812
[588] Fiore V, Yang Y, Kuzyk M C, Barbour R, Tian L and
Wang H 2011 Phys. Rev. Lett. 107(13) 133601
[589] Dong C, Fiore V, Kuzyk M C and Wang H 2012 Science
338 1609–1613
[590] Park Y S and Wang H 2009 Nat. Phys. 5 489–493
[591] Rokhsari H, Kippenberg T J, Carmon T and Vahala K J
2005 Opt. Expr. 13 5293–5301
[592] Peano V, Schwefel H G L, Marquardt C and Marquardt F
2015 Phys. Rev. Lett. 115 243603
[593] Wang J, Bo F, Wan S, Li W, Gao F, Li J, Zhang G and
Xu J 2015 Opt. Expr. 23 23072–23078
[594] Xiong C, Pernice W H P and Tang H X 2012 Nano Lett.
12 3562–3568
[595] Leidinger M, Fieberg S, Waasem N, Kühnemann F, Buse
Nonlinear and Quantum Optics with Whispering Gallery Resonators
K and Breunig I 2015 Opt. Expr. 23 21690
[596] Greenberger D M, Horne M A, Shimony A and Zeilinger
A 1990 Am. J. Phys. 58 1131–1143
[597] Pan J W, Bouwmeester D, Daniell M, Weinfurter H and
Zeilinger A 2000 Nature 403 515–519
[598] Dur W, Vidal G and Cirac J I 2000 Phys. Rev. A 62
062314
[599] Eibl M, Kiesel N, Bourennane M, Kurtsiefer C and
Weinfurter H 2004 Phys. Rev. Lett. 92 077901
[600] Briegel H J and Raussendorf R 2001 Phys. Rev. Lett. 86
910–913
[601] Hein M, Eisert J and Briegel H J 2004 Phys. Rev. A 69
062311
[602] Smolin J A 2001 Phys. Rev. A 63 032306
[603] Raussendorf R and Briegel H J 2001 Phys. Rev. Lett. 86
5188–5191
[604] Yeo Y and Chua W K 2006 Phys. Rev. Lett. 96 060502
[605] Agrawal P and Pati A 2006 Phys. Rev. A 74 062320
[606] Wang R and Bhave S A 2014 arXiv:1409.6351v1
[607] Lin J, Xu Y, Fang Z, Wang M, Song J, Wang N, Qiao L,
Fang W and Cheng Y 2015 Sci. Rep. 5 8072
[608] Burek M J, de Leon N P, Shields B J, Hausmann B J M,
Chu Y, Quan Q, Zibrov A S, Park H, Lukin M D and
Lončar M 2012 Nano Lett. 12 6084–6089
[609] Burek M J, Chu Y, Liddy M S Z, Patel P, Rochman J,
Meesala S, Hong W, Quan Q, Lukin M D and Lončar
M 2014 Nat. Comm. 5 5718
[610] Khanaliloo B, Mitchell M, Hryciw A C and Barclay P E
2015 Nano Lett. 15 5131–5136
[611] Xu Q and Lipson M 2006 Opt. Lett. 31 341–343
[612] Xu Q and Lipson M 2007 Opt. Expr. 15 924–929
[613] Hausmann B J M, Bulu I, Venkataraman V, Deotare P
and Lončar M 2014 Nat. Phot. 8 369–374
[614] Guarino A, Poberaj G, Rezzonico D, Degl’Innocenti R and
Günter P 2007 Nat. Phot. 1 407–410
[615] Baker C, Belacel C, Andronico A, Senellart P, Lemaitre
A, Galopin E, Ducci S, Leo G and Favero I 2011 Appl.
Phys. Lett. 99 1117
49