hv
photonics
Article
A Single-Element Plane Grating Monochromator
Michael C. Hettrick
Received: 24 November 2015; Accepted: 24 December 2015; Published: 11 January 2016
Hettrick Scientific, Ho Chi Minh City 70000, Vietnam; hettrickscientific@gmail.com; Tel.: +1-909-962-8745
Abstract: Concerted rotations of a self-focused varied line-space diffraction grating about its groove
axis and surface normal define a new geometric class of monochromator. Defocusing is canceled,
while the scanned wavelength is reinforced at fixed conjugate distances and horizontal deviation angle.
This enables high spectral resolution over a wide band, and is of particular advantage at grazing
reflection angles. A new, rigorous light-path formulation employs non-paraxial reference points to
isolate the lateral ray aberrations, with those of power-sum ď 3 explicitly expanded for a plane grating.
Each of these 14 Fermat equations agrees precisely with the value extracted from numerical raytrace
simulations. An example soft X-ray design (6˝ deviation angle and 2 ˆ 4 mrad aperture) attains
a resolving power ą 25, 000 over a three octave scan range. The proposed rotation scheme is not limited
to plane surfaces or monochromators, providing a new degree of freedom in optical design. Grating
rotation about its third (meridional) axis may be employed to cancel vertical deflection of the diffracted
beam while maintaining the above aberration correction. This enables a simpler (pure rotary) motion
for the exit slit and a fixed beam direction both horizontally and vertically.
Keywords: optical design; grazing incidence; diffraction gratings; monochromators; geometrical
aberrations; soft X-ray; varied line-spacing; VLS; in-focus; single-element
1. Introduction
The reflection of light at wavelengths λ À 100 nm is encumbered by losses to absorption and
scattering, whose reduction favors few and simple (plane or spherical) optical surfaces. This design
philosophy inspired the invention of two prior self-focusing grating monochromators suitable for use
at grazing incidence [1–3]. The first combined rotation and translation of a varied line-space (VLS)
concave grating, while the second introduced surface-normal rotation (SNR) and initially employed
a constant line-space (CLS) concave grating. To date, these have been the only single-element solutions
which remain “in-focus” (no spectral aberration linear versus aperture) with scanned wavelength, yet
employ slit positions and ray directions fixed in the direction of dispersion. The endpoint in this
progression towards minimization would be reflection from a single plane grating surface, which
can exhibit near-invariance of the focal length with graze angle and provide access to the most
accurate fabrication methods. These include float-polishing and Silicon crystal cleaving to produce
atomically-smooth plane surfaces and short-wavelength lithographies which offer a new generation of
flat-substrate gratings having ultra-low scatter and unconstrained two-dimensional line patterns [4].
Existing fixed-slit plane grating monochromators require additional (mirror) reflections to focus
the incident beam or to maintain this focus as the grating is rotated to scan wavelength. For example,
the classical Czerny–Turner design is theoretically free of geometrical aberrations, but requires two
concave (in principle, parabolic) mirrors for collimation and refocusing [5]. Designs in which the
grating is illuminated by uncollimated light require some form of effective aberration correction
to maintain fixed slits upon rotation of the grating, of which five distinct geometric solutions have
previously been devised: (I) a CLS grating plus a fixed concave (ideally, elliptical) mirror and a rotating
plane mirror [6]; (II) a CLS grating plus a rotating concave (spherical) mirror [7]; (III) a VLS grating
plus a fixed concave mirror [8,9], spherical or otherwise; (IV) a VLS grating plus a rotating-translating
Photonics 2016, 3, 3; doi:10.3390/photonics3010003
www.mdpi.com/journal/photonics
Photonics 2016, 3, 3
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plane mirror [10]; and (V) a CLS grating rotating about its surface normal plus a fixed (ideally elliptical)
concave mirror [3]. The cited references are the original disclosures, with the defining (minimum)
optical geometries and characteristic imaging properties being unchanged in numerous reformulations,
optimizations, augmentations, rebranding and other derivatives.
Presented here is a new monochromator geometry, in which a self-focusing VLS grating scans
wavelength between slits at fixed distances and horizontal deviation angle, without the need for other
optics. This paper reports the detailed imaging characteristics of the basic (astigmatic, single-element,
plane grating) configuration, particularly the spectral resolution as a function of aperture and scan
range. In Section 2, an approximate light-path formulation provides a cogent algebraic and geometric
understanding of the new focusing principle (first degree correction) and derives an advantageous
conjugate distance ratio to correct the spectral aberration of second degree. Section 3 introduces
a rigorous general light-path formulation and Section 4 applies this to obtain the expansion equations
for the present dual-rotation plane VLS grating. As analyzed in Section 5, these equations provide to
an exacting degree the focusing condition, the required tilts of the object (or entrance slit) and image
(or exit slit) and the lateral ray aberrations in both directions. In Section 6, these spatial aberrations are
converted to the geometrical spectral resolution and are exemplified using the dimensional parameters
of an ultra-high resolution soft X-ray monochromator. In Section 7, independent simulations (numerical
raytracings) are performed and quantitatively compared to the light-path calculations. As the present
introductory work lays the theoretical foundation on which subsidiary performance characteristics
may be added, Section 8 briefly indicates some prospects for future practical enhancements. The new
results disclosed in this paper, including the proposed geometry and two precise methods of deriving
the component geometrical aberrations, are summarized in Section 9.
2. The Basic Scheme
Figure 1 illustrates a minimal astigmatic configuration of the proposed optical geometry.
The coordinate systems are Cartesian and right-handed, with the general frame fixed in the laboratory
being (y ´ n̂o ´ z). The object and image plane frames are also fixed, being perpendicular to the plane
of the figure and using coordinate systems A n̂ ´ x ´ z and B n̂ ´ x1 ´ z1 , respectively. The coordinate
system of the grating frame is ω ´ n̂ ´ σ, with its origin at the grating pole (P) and the σ-axis being
coincident with the laboratory z-axis (pointing towards the viewer). On the grating surface, G is
a general point pω, σq and E is at an extreme corner p˘ω̆, ˘σ̆q of a pole-centered rectangular aperture.
The distances r “ Oo P and r1 “ P P Io are measured along the principal incident and diffracted rays,
which intersect at the grating pole. There the principal angles of incidence (α) and diffraction (β) are
measured relative to the grating tangent plane. Their sum is the in-plane angular deviation 2γ and
their difference equals 2δ. Thus α “ γ ´ δ, β “ γ ` δ and γ is the effective graze angle.
At an initial wavelength λo , the grating surface normal n̂o is oriented at an angle δo relative to its
zero order direction (for which α “ βq, the grooves are oriented “in-plane” (at θ “ 0, thus parallel to
the sagittal σ-axis) and self-focusing is provided by VLS positioning of the grooves. The object point Oo ,
the grating pole P and the image point P Io for θ “ 0 define the “horizontal” plane of Figure 1, whose
intersection with the grating tangent plane at its pole forms the meridional ω-axis. The “Gaussian”
image plane is shown, which is at the focal distance r1 for which the resulting horizontal defocus
(first-degree) term 20 x1 “ 0 at θ “ 0. As shown in Figures 1 and 2b, the (horizontally) “paraxial” image
point P Io shall refer to the intersection of the exiting principal ray (for any θ) with this image plane,
being independent of the pupil coordinates (“non-aberrant”).
The scanning to longer wavelengths (λ) consists of a conventional rotation (about the σ-axis) to the
angle δ, coordinated with a second (and larger) rotation to the angle θ about the grating surface normal
(n̂-axis). If (as illustrated in Figure 1) the first rotation axis lies on the grating surface, it intersects
a stationary grating pole which views the in-plane object point (Oo ) and its paraxial image point (P Io )
along principal ray directions whose projections onto the horizontal plane are fixed. The σ-axis rotation
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results in a strong defocusing of the spectral width which, as will be shown, may be canceled by the
defocusing of opposite sign resulting from the n̂-axis rotation.
Figure 1. The Single-Element Plane Grating Monochromator (SEPGM), comprising (minimally)
δ and θ of a VLS grating maintain precise
one grazing reflection. Coordinated on-center rotations
self-focusing as the wavelength is scanned. Projection is onto the horizontal plane, except the angle
label for θ is viewed from a slight elevation for clarity. The extreme ray aberration (separation of the
image points) is shown greatly exaggerated.
For generality and clarity, most equations will use dimensionless variables in italics, where the scale
≡ r /r
factor is r. These include the conjugate distance ratio η ” r1 {r, the meridional pupil coordinate ω, the
sagittal pupil coordinate σ, the grating ruled width coordinate υ, the object plane horizontal coordinate
x and vertical coordinate z, the image plane horizontal coordinate x1 and vertical coordinate z1 and the
rotated image plane coordinates x2 (in the spectral or “slit width” direction) and z2 (in the “spatial”
or “slit length” direction). By convention, the meridional plane is “horizontal” and the sagittal plane
is “vertical”, though this directional labeling need not correspond to the ground-level orientations
of either the physical instrument or the figures drawn
in this paper. Meridional and sagittal shall
(
refer to the positions pω and σ, respectively) on the grating surface, whereas horizontal
and vertical
′
′
1
1
shall refer to the lateral coordinates (x or x and z or z , respectively) within the object or image planes.
“Longitudinal” shall nominally refer to the direction of ray propagation. The meridional projection
+ )r
of the(1physical
distance travelled by the principal ray is p1 ` ηq r, which is the nominal length of
the monochromator relative to the object point.
≡
/ properties at grazing incidence are largely determined by the
As the spectral imaging
(1 +
/
)/(1 −
/
)
dimensionless ratio ρ ” sinβ{sinα, this variable will appear in=the
light-path
equations
rather
than
tan /
= ( − 1)/( + 1)
the rotation angle δ. Conversion between ρ and δ is given by ρ “ p1 ` tanδ{tanγq { p1 ´ tanδ{tanγq
>1
>0
m<0
<1
<0
and tanδ{tanγ “ pρ ´ 1q { pρ ` 1q. The European sign convention is adopted, designating the inside
m>0
spectral orders (ρ ą 1, δ ą 0) as negative (m ă 0) and the outside orders (ρ ă 1, δ ă 0) as positive
(m ą 0).r Assuming
no horizontal
optic follows the grating, and the monochromator transmits
>0
> focusing
0
1
the scanned
wavelength
by
use
of
a
spatial
filter (an exit slit), the grating image must
r>0
<0
r <be
0 real (r ą 0).
This results in η ą 0 when the grating is mounted in a diverging incident beam (a real object, r ą 0),
and η ă 0 in the case of a converging incident beam (a virtual object, r ă 0). The former corresponds
to a self-focusing monochromator, for which the linear axes arrows in Figures 1 and 2 point to positive
L = OG
L = GI
dimensionless values. However
in the latter case (a virtualmobject), the negative value of the scale
m
factor r results in negative dimensionless values in the directions of these arrows.
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2.1. The Standard Light-Path Formulation
The wavefront path length F is the sum of the physical path lengths (L “ OG and L1 “ GI) and
the interference shift of mλN between N grooves, where m is the spectral order and λ is the physical
value of the wavelength. As a powerful tool in the development and performance analysis of optical
designs, the path lengths and groove number are usually mathematically de-composed by power
series expansion in the pupil coordinates, providing (in dimensionless units):
F pω, σ, µq “
ÿ
i,j
Fij pµq ω i σ j ; where Fij pµq “ µNij ` Lij pµq ` L1ij pµq
(1)
where µ ” mλ{do is the dimensionless wavelength variable given a physical line spacing do at
the grating pole. In what shall hereinafter be called “the standard formulation”, the path-length
coefficients derive from the series expansion of the distances to conjugate points (O and I) which
have no dependence upon the pupil coordinates ω or σ. In particular, I is equated to the paraxial
image point P I. Thus, the series expansion yields equations for Lij and L1ij which depend only upon
parameters at the grating pole, namely the grating orientation (α, β and θ), its position (η) relative to
the object and image points and the shape of its surface (e.g., its radius of curvature).
Employing Fermat’s principle, the image plane lateral ray positions x1 pω, σ, µq and z1 pω, σ, µq are
also power series obtained by differentiation of Equation (1) w.r.t. the grating pupil coordinates:
x1 pω, σ, µq “ ´
hη BF pω, σ, µq ÿ
hη
1 i ´1 j
”
σ where ij x1 pµq “ ´i
F pµq
ij ij x ω
sinβ
Bω
sinβ ij
z1 pω, σ, µq “ h3 η
BF pω, σ, µq ÿ
1 i j ´1
where ij z1 pµq “ jh3 ηFij pµq
”
ij ij z ω σ
Bσ
(2)
(3)
where the inclination factor h » 1 (given more precisely later). In Section 3, the above standard
formulation of the light-path aberrations is shown to be mathematically flawed, and is replaced by
a rigorously correct theory. However, to introduce the essential geometrical principles underlying the
new design, this complication is temporarily neglected and a concise initial analysis is presented based
on Equations (1)–(3) and other simplifying (though not flawed) approximations. For brevity, functional
dependences upon the pupil coordinates and the wavelength shall hereinafter be understood without
their explicit notation.
The power series terms of first degree will be referred to as de-focus (20 x1 ), astigmatism (02 z1 ),
horizontal (sagittally-induced) image tilt (11 x1 ) and vertical (meridionally-induced) image tilt (11 z1 ).
Though semantical distractions may be discouraged by referencing the higher-degree terms according
to only their (i, j) subscripts and lateral direction, the commonly accepted descriptions (based largely
on the image shape) will also be used when this facilitates the discussion. Thus, “meridional coma”,
“horizontal coma” or simply “coma” is the (3,0) term (30 x1 ), “sagittal coma”, “astigmatic coma” or
“astigmatic curvature” is the horizontal (1,2) term (12 x1 ), “spherical aberration” is the (4,0) term (40 x1 )
and “mixed spherical aberration” is the horizontal (2,2) term (22 x1 ).
2.2. Surface-Normal Rotational Transformation of the Varied Line-Space Coefficients
A varied line-space (VLS) grating is a design in which the groove positions are relatively
unconstrained yet possess sufficient symmetry to permit a mechanical ruling. The most common of
these symmetries is where the grooves are straight and parallel, thus the groove number (N) may be
expressed as a 1D power series in the ruled width coordinate υ:
N “ pr{do q
ÿ
k
Nk υk
(4)
where N1 “ 1 and Nk (for k ą 1) are the dimensionless VLS ruling coefficients. Equation (4) corresponds
ř
k´1 and to dimensional coefficients M “
to a relative local groove density do {d pυq “
k kNk υ
k
k
´
1
Nk {pdo r ).
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Figure 2a shows that a surface normal rotation θ (positive = clockwise viewed from above) of
ruling coordinate υ relative to the grating meridional coordinate ω results in the transformation
υ “ ω cosθ ` σ sinθ. The groove number coefficients Nij in the grating frame (ω, σ), as used in
Equation (1), are thus obtained by that substitution in Equation (4):
Nij “ cij Nk cosi θ sin j θ
(5)
where cij “ 1 for pure meridional terms (j “ 0); “ 1 for pure sagittal terms (i “ 0); “ i ` j for the
mixed terms (1,1), (1,2), (2,1), (3,1) and (1,3); “ 6 for the mixed term (2,2) and where k “ i ` j. The (i,0)
terms provide a meridionally-projected groove density of
do {d pωq “
ÿ
i
”
ı
iNi0 ω i´1 “ 1 ` ωC2 ` ω 2 C3 ` ω 3 C4 ` ... cosθ
(6)
where Ci ” iNi cosi´1 θ, revealing that the relative magnitudes between successive coefficients have
decreased by cosi´1 θ. This results in diminished focusing power (i “ 2) and progressively smaller
correction of the higher-degree aberrations (i “ 3, 4, . . .). Additionally, Equation (5) reveals that
nonzero values of θ create new coefficients of sagittal (i “ 0, j ‰ 0) and mixed (i ‰ 0, j ‰ 0)
powers. These new terms result in significant 3D imaging characteristics, including image tilts and the
introduction of a dominant mixed aberration.
Figure 2. (a) The grating ω ´ σ plane viewed from above (+ n̂), detailing the rotational transformation
of the varied line spacing between the ruling axis and the meridional axis for any point G on the
grating surface, as given by Equation (5); (b) Frontal view (as seen by a distant observer upstream of
the entrance slit), detailing the slit rotations in the object and image planes. The paraxial (non-aberrant)
deflection out of the meridional plane is shown both originating from the center of the entrance
slit (_______ ) and originating from one end of this slit (__ __ __ ). The (dimensionless) length of the
principal ray is hη. The presence of aberrations, notably astigmatism, for the extreme non-principal
ray (__ ¨__ ) results in the lateral image position summed from all the power terms given in Section 4.
Slit rotation angles ψ (about A n̂) and ψ1 (about B n̂q, given in Section 5, are relative to the grating frame.
Thus, an equivalent optical geometry can maintain the entrance slit parallel to z in the laboratory frame,
but rotate the grating and exit slit frames by –ψ about A n̂.
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2.3. The Principal Ray Terms
`
`
˘
˘
These are not aberrations (pupil-dependent), but are the horizontal x1 10 and vertical z1 01
image coordinates, whereby the wavefront from this point to the grating pole is stationary. Using
Equation (1), this condition (BF{Bω “ BF{Bσ “ 0) is expressed as:
b
`
˘
µ cosθ ` pcosα ´ z tanψ sinαq { 1 ` z2 1 ` tan2 ψ
˙ b
ˆ 1
x 10
sinβ ´ cosβ { 1 ` px1 10 {ηq2 ` pz1 01 {ηq2 “ 0
`
η
ˆ 1 ˙ b
b
`
˘
z 01
2
2
“ µ sinθ ´ z{ 1 ` z 1 ` tan ψ ´
{ 1 ` px1 10 {ηq2 ` pz1 01 {ηq2 “ 0
η
F10 “
F01
(7)
(8)
The leading terms in Equations (7) and (8) are the differential phase shifts, namely N10 and N01 as
given by Equation (5). The middle terms are the differentials of the object distance OG/r about G “ P,
obtained by simple trigonometry using Figures 1 and 2b (or Equations (32)–(34)) with O being a point
along a line (e.g., an entrance slit) tilted by an angle ψ from the vertical. The last terms are differentials
of the image distance G P I{r about G “ P, also obtained trigonometrically (Equations (35)–(37)) with P I
being the paraxial image point. Simultaneously meeting Equations (7) and (8) yields quadratics with
the following series solutions:
ˆ
˙
3
1
3
z1 01 {η “ ´ 1 ` µ2 sin2 θ z ` p1 ` Γ{2q pµ sinθq z2 ` µ sinθ ` µ3 sin3 θ ` µ5 sin5 θ
2
2
8
µ cosθ
x1 10
tanβ »
η
˙
1 2 2
1
µ sin θ ` µ4 sin4 θ cosβ
2
8
˘
1 2
1 3 2
1 4`
2
4
2
“ ε ´ ε cosβ tan θ ` ε cos β tan θ ` ε 5cos β tan2 θ ` 1 cosβ tan4 θ
2
2
8
˘
1 `
` ε5 5cos2 β tan2 θ ` 3 cos2 β tan6 θ
8
`
˘
“ f β cosβ ´ cosα “ f β ´ Γ cosβ “ ε ´
„
`
ˆ
"
‰
µ2 sin2 θ 1 “
µ cosθ
tanψ z ´
`
1 ` pΓ ` 3q tan2 β
2β 2
2cosβ
tan
ˆ
˙
*
˘
µ sinθ `
1 1
2 ψ z2
`
´
Γ
tan
1 ` 3tan2 β tanψ `
ρ tanβ
2 ρ2
˘
1 ` µ2 sin2 θ µ sinθ `
ˆ
tanβ
ρ
˙
(9)
(10)
(11)
where Γ ” cosα{cosβ and ε ” cosβ ´ cosα. Equation (9) is a “law of sagittal reflection” with a vertical
object position (z) and generalized to include a surface-normal rotation angle (θ). As illustrated
in Figure 2b, the latter causes a deflection of the image center out of the horizontal plane and is
dominated by the µ sinθ term, which scales linearly with wavelength. A possible method of canceling
this (undesired) movement is proposed in Section 8.1, however this term will be included in all the
aberration equations to be derived in Sections 3–5. Equation (10) is a generalized “grating equation”
for the in-plane object point (z “ 0q, including both the cosθ projection of the groove spacing and
1
1
the fine-correction factor f β “ 1 ´ µ2 sin2 θ ´ µ4 sin4 θ. The result is essentially exact agreement
2
8
with the numerical raytracings (discrepancy ă 10´13 radians). The last equality in Equation (10) is
a five-term series solution which retains an accuracy of ă 4.2 ˆ 10´11 radians. Alternatively, if the µ4
term is excluded, Equation (10) is quadratic in µ with the following simple solution being accurate to
ă 3.2 ˆ 10´9 radians (~ 0.036 microns at the grating focal distance of ~ 11 meters):
a
1 ` 2ε cosβ tan2 θ ´ 1
µ cosθ “
tan2 θ cosβ
(12)
In the absence of groove axis rotation, Equations (9) and (10) are found to collapse to the results
reported previously for a pure surface-normal rotation monochromator [2,3], namely z101 » r1 µo tanθ
and µcosθ “ µo . Equation (11) is the horizontal image position of off-plane points z along an entrance
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slit, revealing that a vertically displaced object point (z ‰ 0q results in a horizontally-displaced image
point (x1 ‰ 0). The term linear with z corresponds to image tilt, while the quadratic term accounts for
image curvature of a straight slit; further analysis is given in Sections 5.4 and 5.5.
2.4. Pure Meridional Aberrations and their Graze Angle-Invariant Approximation
Using the paraxial horizontal image position (x1 “ 0q to construct the path lengths and
neglecting the effect of off-plane vertical image positions (z1 ), the grating Equation (10) simplifies to
µcosθ “ ε ” cosβ ´ cosα and the lowest three meridional-only (j “ 0) wavefront terms of Equation (1)
may be written concisely in the following form:
Fi0 “ µNi cosi θ `
ı
1”
bi0 ρ2 {η i´1 ` ai0 p´1qi sin2 α
2
(13)
ff
ρ2 {η i´1 ` p´1qi
µ cosθ
θ´
ρ2 ´ 1
(14)
5
5
where i “ 2, 3 or 4, b30 “ cosβ, a30 “ cosα, b40 “ 1 ´ sin2 β, a40 “ 1 ´ sin2 α, b20 “ 1 and a20 “ 1.
4
4
1
At grazing angles (γ À 15˝ ), the small-angle approximations psinα » α, sinβ » β, cosα » 1 or 1 ´ α2
2
1 2
2
and cosβ » 1 or 1 ´ β ) provide accuracies À 1%. This yields ai0 » bi0 » 1, sin α »
2
`
˘
´2 pµcosθq { ρ2 ´ 1 and a simplified (γ-invariant) expression for the wavefront:
«
Fi0 » Ni cos
i ´1
at nonzero spectral orders (ρ ‰ 1). At the initial wavelength (prior to scanning), ρ “ ρo and θ “ 0 and
all horizontal aberrations vanish (Fi0 » 0) by choice of the following VLS coefficients:
Ni »
ρ2o {η i´1 ` p´1qi
ρ2o ´ 1
(15)
As the geometrical imaging properties of VLS plane gratings contrast with intuitions fostered
in classical optics, it is emphasized that the required coefficients Ni are (for all i) nearly independent
of γ at grazing incidence, with this invariance becoming (asymptotically) exact as γ approaches zero.
This is opposite to the behavior of classical (curved surface) methods of focusing, which are strongly
dependent on a precise γ at grazing incidence, and become increasingly so as γ decreases. As with the
prior VLS self-focusing plane grating geometry (IV in Table 1), this invariance enables a given grating
to provide aberration correction at any graze angle given fixed values for ρo , r and r1 . This offers
a flexibility in graze angle and wavelength coverage not available with concave gratings.
For the author’s original VLS plane grating converging-beam geometry (III in Table 1), the
conjugate distance ratio is η » ´ 1, simplifying Equation (15) to Ni » p´1qi´1 (dimensionally, this is
` ˘ 1´ i
Mi » r1
{do ). Though this grating mount requires a horizontal focusing mirror to provide the
incident converging beam, Equation (14) with η » ´1 reveals the yet stronger γ-invariance whereby the
aberration correction is also independent of ρo . This allows the angles of incidence and diffraction to be
changed independently (while still maintaining the fixed focal length r1 ) given only that the dimensional
factor r is unchanged. For example, the grating may be configured with this mirror into an erect-field
(varied β) spectrograph [8] with fixed α, a constant-deviation (fixed α ` β) monochromator [9] or any
other desired combination of the 2 angles. A derivative of this geometric class (i.e., having unaltered
imaging properties) thus provides a fixed difference pα ´ βq for “on-blaze” diffraction efficiency to
a moveable slit (or add a conventional rotating plane pick-off mirror to redirect this β to a fixed slit).
However, when η ą 0 (r ą 0), the focal length is a strong function of ρ. This sensitivity is a general
characteristic of a self-focusing grating geometry at grazing incidence, whether it be a classical or VLS
grating, and whether the surface is plane or curved. In contrast, it has been previously shown [2,3] that
a surface-normal (n̂) rotation does not change the focal length of a CLS grating, given fixed meridional
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angles and a fixed axis-symmetric (e.g., plane or spherical) surface curvature. However, in the case of
a VLS grating, an n̂-rotation angle diminishes the interference term by the factor cosi´1 θ, as seen in
Equation (14). This would result in an under-correction of all aberrations initially corrected at θ “ 0.
The new focusing condition presented here arose by considering if this under-correction could balance
the strong defocus induced by the change to ρ upon a conventional (groove axis) rotation.
2.5. The New Focusing Condition
Combining Equations (14) and (15) in the case of no defocusing (F20 “ 0) yields:
cos rθ pρqs »
ˆ
ρ2o ´ 1
ρ2 ´ 1
˙ˆ
ρ2 ` η
ρ2o ` η
˙
(16)
As ρ » p1 ` δ{γq { p1 ´ δ{γq, Equation (16) provides a mathematical relationship between the two
rotation angles (θ and δ). However, this is of no physical relevance unless 0 ă cosθ ă 1. Fortuitously,
this is always the case, as may be seen by making the substitution ρ2 “ ρ2o ` ∆ρ2 . Equation (16) is then
˘‰
˘‰ “
`
“
`
rewritten as 1 ` ∆ρ2 { ρ2o ` η { 1 ` ∆ρ2 { ρ2o ´ 1 . For negative spectral orders, ρo ą 1 and a scan
towards longer wavelengths requires ∆ρ2 ą 0. Since η ą 0 in the present (self-focusing) geometry,
the value of this equation must be between 0 and +1. By the principle of optical reversibility, this
must also be the case for positive orders, for which the photon reverses its direction of travel and ρ
becomes 1/ρ. The potential to cancel the defocusings induced by each of the 2 rotations (the 2 terms in
Equation (14)) may also be understood geometrically. As illustrated by the first animation sequence of
Figure 3, scanning towards longer wavelengths (shown visually going from “blue” to “orange”) by
a σ-axis rotation always decreases the focal length. However, due to the cosθ reduction in VLS focusing
power, the n̂-rotation (away from θ “ 0) always increases the focal length.
Though Equation (16) relies upon the small-angle approximation and also accounts only for
pure horizontal defocusing (the effect of image tilt is addressed in Section 5), it confirms the physical
relevance of this scheme. Additionally, it provides some insight by considering the case of unit
magnification at η “ ρo . By use of Equation (16) and the small-angle approximation to the generalized
grating Equation (10), the in-plane rotation and the surface-normal rotation are each seen to increase
the scanned wavelength by the same amount. In the more general cases of η ‰ ρo or ρ ‰ ρo (away
from λo ), though the contributions are not equal they are comparable, thus the scanning range (λ{λo )
scales approximately as the square of that provided by each rotation. Due to this (second) fortuity, the
proposed dual rotation is not only the key to a focused single-element plane grating monochromator,
but also enables a wider scan in wavelength than does a single rotation for the same change in ρ (the
latter affecting grating imaging, magnification and efficiency). For the monochromator parameters
exemplified in Section 2.6, a factor 8 in scan range results from a factor 2.54 due to the in-plane rotation
(changing ρ by only a factor of 1.89) times a factor 3.12 due to the surface-normal rotation.
Photonics 2016, 3, 3
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Figure 3. Concerted rotations of a varied line-space (VLS) self-focused plane grating about two
axes provide in-focus scanning of the wavelength (please see the animation of this figure in the
supplementary material). Tilt and off-plane deflection of the image are not shown here. Slits may be
placed at the object and image.
2.6. Two-Point Coma-Correction: Optimization of the Conjugate Distance Ratio
With defocusing eliminated at all scanned wavelengths, the next higher degree aberration to
= 0
be considered is given by F30 in Equation (1). At grazing angles, this aberration is typically large
and thus determines whether a system which is technically “in-focus” (F20 “ 0) can in fact deliver
high resolution at a usable aperture. It is tempting to first consider unit magnification pη{ρ “ 1q,
which would eliminate this aberration in the absence of a varied line-density term N3 . This is similar
to the classical concave grating being free of this aberration on the (unit magnification) Rowland
circle. Setting N3 “ 0 would also remove this interference term from Equation (14) and thus avoid its
multiplication by cos2 θ (which would re-introduce coma upon surface-normal rotation). However,
because α and β are changing in opposite directions as the wavelength is scanned in the first (groove
axis) rotation, the resulting change in ρ ~ β/α allows η{ρ “ 1 at only one wavelength. Even if this
correction wavelength is optimally selected (by choice of N3 ‰ 0 alone) to be near the center of
the scanned spectrum, the growth in coma away from a single correction point is rapid, making it
ineffective for all but a very narrow wavelength range.
The high level of coma correction required to maintain ultra-high spectral resolution over wide
scanning ranges is realized by determining optimal values for the two free parameters (η and N3 )
available from Equation (14) when i “ 3 to correct F30 at two wavelengths. In general, the simultaneous
solution is a quartic, however it collapses to a quadratic if the two correction points are (tentatively)
chosen to be at θ “ 0 (cos θ “ 1) and θ “ 60˝ (cos θ “ ½), resulting in the following closed-form solution:
ηc “
´
ρ2o
¯ b
2
´ 1 ` pρ2o ´ 1q ` ρ2o
(17)
The corresponding N3c is obtained by setting F30 “ 0 in Equation (14) at θ “ 0 and η “ ηc .
Minimization of F30 over the scan range is then obtained by adjusting both η and N3 from these
tentative values, such that the two correction points straddle the spectrum center and are symmetrically
inset from the edges. In this case, |F30 | reaches the same maximum at the extreme ends of the range and
at a wavelength between the 2 correction points. Such optimization for a scanning range of λmax {λo ~
8 (θmax ~ 71.25˝ ) resulted in η “ 1.275 ηc and N3 “ 1.08 N3c (if the scan range were reduced to θmax
~ 60˝ (λmax {λo ~ 3.6), optimization of this coma-correction would yield η “ ηc and N3 “ 1.04 N3c ).
This and the preceding meridional analysis provide the following set of dimensionless parameters
for an example soft X-ray SEPGM: 2 γ “ 6˝ , ρo “ 1.4458939, η “ 3.699998982, N2 “ 1.432972878,
N3 “ ´0.8378590, N4 “ 1.28655, where N4 results from using Equation (14) to cancel spherical
Photonics 2016, 3, 3
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˝
aberration (F40 “ 0) at θ „ 32 , thereby minimizing its average magnitude over the three-octave
scan range.
2.7. A Classification of Plane Grating Geometries
The simple form of Equation (14) also enables a concise comparison of the fundamental differences
between various aberration-corrected plane grating geometries. In Table 1, the present class is listed
alongside the previous five, comparing the defining categories of the minimum number of reflections
(grating + mirror(s)), the grating line density function (the VLS coefficients Ni ), the functional value of
η defining the grating mount, the grating parameter which is fixed upon a focused (F20 “ 0) scan of
wavelength and the corrected horizontal aberrations (Fi0 “ 0).
Table 1. Plane grating aberration-corrected monochromators with fixed slits.
Geometry
Optics (min)
Approx. VLS Ni
(i “ 2, 3, 4)
Grating
Mount (a)
Fixed
Parameter(s)
Corrected
Aberrations
I. Petersen [6]
3
N{A pCLSq
η “ ´ρ2
ρ
1
x20
II. Lu [7]
2
N{A pCLSq
η “ ´ρ2
r1 only pbq
1
x20
III. Hettrick [9]
2
» p´1q
i´1
IV. Harada [10]
2
»
V. Hettrick [3]
2
N{A pCLSq
1
ρ2 {η i´1 ` p´1qi
» o
ρ2o ´ 1
VI. This Work
ρ2o {η i´1 ` p´1qi
ρ2o ´ 1
η » ´1
α ` β (c)
ηą0
ρ
η “ ´ρ2
α, β pdq
ηą0
α ` β (e)
1
x20
1
x30 p1 λq
1 p1 λq
x40
1
x20
1 (1 λq
x30
1 (1 λq
x40
1
x20
1
x20
`
˘
1
x30 2 λ1 s
1 (1 λq
x40
Notes: (a) η “ ´ρ2 is the Monk–Gillieson mount and η ą 0 is a “self-focusing” mount; (b) the grating object
distance r changes to be confocal with the spherical mirror image; (c) given η » ´1, ρ is unconstrained at
grazing angles (allowing fixed α, fixed α ` β, fixed α ´ β, etc.); (d) type “SNR-III” grating system of 8th figure in
the referenced patent; and (e) detailed in the present work for comparison to other monochromator designs
with horizontally fixed slit positions; however, other combinations of α and β may be used by changing the
focusing condition.
3. A Rigorous Theory of Light-Path Expansion
The purist of wave aberration methods [11] constructs the physical light-path from object point to
actual (floating) image point and employs Fermat’s principle by setting the wave aberration differential
to zero for the sum of all terms. This approach does not evaluate the individual aberrations, other than
those of lowest degree in each direction (i.e., defocusing and astigmatism). This avoids the difficult
task of constructing an accurate reference wavefront, and is convenient for optimizing (“fine-tuning”)
the parameters of existing geometries by minimizing the numerical variation in the image point
position as the incident ray wanders over the pupil surface. Unfortunately, this provides little intuitive
information conducive to algebraic or geometrical understanding, and is therefore not suited to the
development or qualitative improvement of new geometries. Indeed, even the (numerical) data from
the precise raytracings performed as part of the present work (Section 7) are mathematically reduced
to extract the individual coefficients of the algebraic power series, and are thus more convenient for
basic design development than is the total wave aberration formalism.
More insight and fundamental progress in optical design is facilitated by isolating and controlling
successive expansion terms ωi´1 σ j (and ωi σ j´1 ) of the lateral ray aberrations. Especially in the case of
large mixed terms (neither i nor j being zero), this requires a more rigorous light-path formulation than
previously presented in the literature. Though complicating the derivation, the resulting separation
of the individual terms provides a precise and consistent analytical method indispensible to the
Photonics 2016, 3, 3
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development of new designs. In this section, the general equations are given for rigorously expanding
the lateral aberrations of a reflection grating. In Section 4, these will be simplified for the case of a plane
grating and used to obtain the explicit aberration terms for the basic astigmatic configuration of the
new monochromator geometry.
3.1. Flaw in the Standard Theory
Relative to any chosen point in space, the optical wavefront (a surface of constant phase) is
a (physically) measureable quantity, as is the lateral image position of a ray diffracted from any
point on the pupil. However, their expansion into power series component terms is (though very
useful) only a mathematical abstraction. In expanding the image path length relevant to an individual
(x1 ij or z1 ij ) lateral aberration term, a consistent formulation requires the use of a correspondingly
abstract reference image point. This must be chosen to remove the other lateral aberration terms which,
upon expansion and pupil differentiation of the light-path, may erroneously form the same power
dependence as the intended term. Only in this way do the (mathematically-constructed) terms become
separate incremental components, the sum of which is the total (physical) aberration.
In general, the reference point depends not only upon the (i, j) term in the light-path function,
but also upon whether a given (i, j) term is differentiated relative to the meridional pupil coordinate
to determine (via Fermat’s principle) the x1 ij lateral ray position or relative to the sagittal pupil
coordinate to determine z1 ij . Thus, a single light-path function may not be expanded (e.g., into a power
series) applicable to both directions. Nonetheless, the individual derivatives (B{Bω and B{Bσ) of the
appropriate functions may be expanded for rigorous determination of the separate lateral ray positions,
as given in Section 3.3.
Unfortunately, the standard formulation (briefly summarized in Section 2.1) systematically
employs only the paraxial image as the reference point (in wavefront terminology, it is usually
asserted to use the “Gaussian reference sphere”). The resulting expanded (inferred) power terms are
therefore subject to cross-contamination, providing incorrect results for both the individual aberrations
and their sum, the latter thereby not matching the actual (measurable) lateral ray position. In the
analysis of classical optics, such contamination is usually small, due to these designs either being
rotationally symmetric, having an in-plane dispersion geometry, or being anastigmatic and (ideally)
absent of second-degree (e.g., meridional coma) lateral aberrations, and is neglected. An historical
exception appears to have been in determining the mixed power term of astigmatic coma (12 x1 ), where
Beutler [12] correctly added the aberration of astigmatism (02 z1 ) to form the image reference point for
a concave grating geometry (particularly on the Rowland circle). Though such special consideration
of the lowest-degree vertical aberration is consistent with the image reference being “paraxial”, it
contradicts the assertion that the wave aberration is evaluated at the (single point) intersection of the
principal ray with the (horizontal) Gaussian image plane [13]. As astigmatism separates the horizontal
and vertical image planes, the actual wavefront becomes toroidal. Moreover, inclusion of this one
aberration in the reference does not provide a correct determination of all the lateral aberrations.
In the more general sense, this approach leaves uncorrected an underlying problem that the
standard formulation is based on the methods developed for the treatment of normal incidence and
rotationally symmetric classical optics systems. For such, the horizontal and vertical image planes
coincide, neither astigmatism nor coma exist when using an on-axis object point and a single light-path
function may be used to determine the lateral aberrations in both directions. Using such methods, the
lateral terms which can dominate grazing incidence and asymmetrical systems are not systematically
removed from the expansion of higher-degree aberrations. The neglected terms include all of the
1 (such as horizontal coma) and the mixed terms x1 (in which both i
pure meridional aberrations xi0
ij
and j are nonzero). Though a comprehensive discussion of the errors resulting from widespread use
of the standard formulation is beyond the scope of this paper, Appendix A shows that the standard
result for the spherical aberration term of even a (classical) spherical mirror is incorrect at non-unit
Photonics 2016, 3, 3
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magnification. This illustrates that the new formulation yields more accurate results even in the
absence of surface-normal rotation, varied line-spacing, grazing incidence or even grating diffraction.
The light-path expansion theory developed below reveals that the isolation of different power
series terms is generally more complex than previously explained by use of paraxial reference points.
The new formulation employs a pupil-dependent (“aberrant”) reference image position tailored to the
expansion of each power term, and a Fermat differentiation of infinitesimal pupil variables (∆ω and
∆σ) which are independent of this image position and the pupil coordinates (ω and σ).
3.2. The Reference Path Lengths
The reference path-lengths are Aij (ω, σ; xij , zij ) as the distance between a reference point (xij , zij )
in the object plane and the grating pupil coordinate (ω, σ), and Bij (ω, σ, ξij , ζ ij ) as the distance between
(ω, σ) and a reference point (ξij , ζ ij ) in the image plane. The general equations in this section include
a grating surface of revolution with a radius of curvature R ” R{r at the pole, where R ą 0 is concave
(positive focusing power) and R ă 0 is convex (negative focusing power). In the below equations,
S “ 0 provides the exact results for a paraboloidal surface, while a spherical surface is treated (correct
to the 5th degree in path-length) by substituting S “ 1. Given the coordinate systems shown in
Figures 1 and 2:
¯1{2
´
Aij “ 1 ` A tij
(18)
in which A tij is the sum of the squares of the two lateral (”transverse”) segments:
A tij
where,
“
`
˘ ‰`
˘
“ 2ω cosα ´ 2xij ω sinα ´ 2zij σ ` xij2 ` z2ij ` 1 ´ sinα ` xij cosα {R ω 2 ` σ2
`
˘ ‰`
˘2
1“
` 1 ´ S sinα ` xij cosα {R ω 2 ` σ2 {R2
4
xij “ ij x ω i´1 σ j and zij “ ij z ω i σ j´1
(19)
(20)
In Equation (19), z is the vertical distance of an object point from the horizontal (meridional)
plane. The terms x10 and z01 have no dependence upon the grating pupil coordinates and thus
define a source point whose emitted rays fill the grating aperture. Making x10 a function z01 (or
vice-versa) can define the two-dimensional curve on which such points may lie in the object plane.
For example, a parabolic entrance slit illuminated by a spatially diffuse upstream source may be
specified by x10 “ z01 tanψ ` z201 { p2Rslit q , where ψ is the slit’s tilt angle towards the x-axis and Rslit
is its curvature radius at z01 “ 0. In the case of a linear slit 1/Rslit “ 0; if ψ is also zero, then
the slit is aligned with the vertical z-axis. Equations (19) and (20) also allow power series terms
(i ` j ą 1) to accommodate dependences between the object point position and the grating pupil
position. For example, consider a vertically aligned entrance slit being illuminated by a horizontally
aligned linear source at an upstream distance Y. This longitudinal separation of the effective horizontal
and vertical object conjugate planes results in z02 » σ{p1 ` r{Y). Nonzero values for xij or zij can also
be constructed to include the geometrical aberrations of optics preceding the grating (or the object
plane), e.g., those of a mirror which reduces the (vertical) astigmatism and/or provides horizontal
re-focusing of a distant source.
Again referring to Figures 1 and 2 one formulates:
Bij “ η
¯1{2
´
1 ` B tij
(21)
Photonics 2016, 3, 3
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in which,
˙
ˆ
˙ˆ ˙
ˆ ˙ ˆ ˙ ˆ ˙2 ˆ ˙2
ζ ij
ξij sinβ
ζ ij
ξij
ωcosβ
ω
σ
`
`2
´2
`
B tij “ ´2
η
η
η
η
η
η
η
«ˆ ˙
ˆ ˙2 ff
2
“
`
˘ ‰
σ
ω
`
` 1 ´ ηsinβ ` ξij cosβ {R
η
η
«ˆ ˙
ˆ ˙2 ff2
`
˘ ‰
1 ´ η ¯2 “
σ
ω 2
`
1 ´ S ηsinβ ` ξij cosβ {R
`
4 R
η
η
ˆ
(22)
where,
ξij “
ř
p I,J q‰pi,jq
x1I J “
ř
p I,J q‰pi,jq I J x
1
ω I ´1 σ J and ζ ij “
ř
p I,J q‰pi,jq
z1I J “
ř
p I,J q‰pi,jq I J z
1
ω I σ J ´1
(23)
The coordinate system of the image plane has as its origin the principal ray position for the
in-plane configuration, namely where the object point is at px “ 0, z “ 0) and where the grating
rotation angle θ “ 0. However, all other object points and all nonzero values of θ will result in an image
∦
position which does not strike this origin in one or both lateral directions. Therefore, ξij “ ξij ` x1 p0, 0q
∦
and ζ ij “ ζ ij ` z1 p0, 0), where the superscript ∦ denotes the component due to aberrations. In the case
of the minimal astigmatic configuration of the SEPGM geometry (whose detailed expansion is given in
Section 4), the aberration terms assume an object at px “ 0, z “ 0). This constrains x1 p0, 0q “ 0 for all
∦
rotation angles, thus ξij contains only the aberration terms (ξij ). However, z1 p0, 0q ‰ 0 when θ ‰ 0, as
∦
given by Equation (9). Thus, ζ ij is the sum of an aberration component (ζ ij ) and the off-plane position
of the principal ray.
3.3. Fermat Derivation of the Lateral Ray Aberrations
Mathematical separation of the individual aberration power terms requires strict adherence to
Fermat’s principle, which specifies the light-path (i.e., the direction along which the optic provides
constructive interference) to be the one for which the phase is stationary relative to small offsets in the
pupil coordinates. Deviations from this can be converted to the lateral ray deviations (aberrations)
from a reference image point. In mathematical terms, the first derivative must be taken relative to
the pupil coordinates, while maintaining a fixed reference image point. To insure a proper formulation,
it is therefore convenient to add small offsets (∆ω and ∆σ) to the two pupil coordinates (ω and σ)
appearing in Equations (19) and (22), but not to the pupil coordinates appearing in Equations (20) and
(23). The Fermat derivatives of Equations (18) and (21) are then taken with respect to ∆ω and ∆σ. In the
absence of these effectively separate variables, i.e. if one simply differentiated with respect to ω and σ,
any aberration (a ray position which depends on the pupil coordinates) included in the construction of
the reference image point would move that point during the differentiation, improperly distorting the
(spherical) reference wavefront. The same situation occurs in the presence of an aperture-dependent
object position (discussed in Section 3.2).
The proper formulation of Fermat’s principle thereby yields:
BAij {B p∆ωq “
ı´
¯´1{2
“
`
˘ ‰
1”
B A tij {B p∆ωq 1 ` A tij
“ cosα ´ xij sinα ` 1 ´ sinα ` xij cosα {R ω
2
ˆ
¯´1{2
`
˘ ‰`
˘
(
1 “
1 ´ S sinα ` xij cosα {R ωσ2 ` ω 3 {R2 1 ` A tij
`
2
BAij {B p∆σq “
ı´
¯´1{2
“
`
˘ ‰
1”
B A tij {B p∆σq 1 ` A tij
“ ´zij ` 1 ´ sinα ` xij cosα {R σ
2
ˆ
¯´1{2
`
˘ ‰`
˘
(
1 “
1 ´ S sinα ` xij cosα {R ω 2 σ ` σ3 {R2 1 ` A tij
`
2
(24)
(25)
Photonics 2016, 3, 3
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ı´
¯´1{2
η”
BB tij {B p∆ωq 1 ` B tij
2
"
ˆ ˙
“
`
˘ ‰ ω
ξij
“ ´cosβ `
sinβ ` 1 ´ ηsinβ ` ξij cosβ {R
η
η
«
«ˆ ˙ ˆ ˙
ˆ ˙3 ff ´ ¯ + ´
¯´1{2
˘ ‰
1
ω
σ 2
η 2
ω
`
1 ´ Spηsinβ `ξij cosβ {R
1 `B tij
`
2
η
η
η
R
BBij {B p∆ωq “
BBij {B p∆σq “
ˆ ˙
ı´
¯´1{2 " ζ
“
`
˘ ‰ σ
η”
ij
BB tij {B p∆σq 1 ` B tij
“ ´
` 1 ´ sinβ ` ξij cosβ {R
2
η
«ˆ ˙η ˆ ˙ ˆ ˙ ff
*
¯´1{2
1
ω 2 σ
σ 3 ´ η ¯2 ´
` r1 ´ S pηsinβ ` ξij cosβq{Rs
`
1 `B tij
2
η
η
η
R
(26)
(27)
The reference coordinates ξij and ζ ij in the above equations are each sums of all the other lateral
ray aberration power terms x1I J ‰ xij1 and z1I J ‰ z1ij , respectively, which when combined and expanded
with the other terms give rise to the desired power term.
?
The reciprocal radicals in Equations (24)–(27) are to be expanded by Taylor series (1{ 1 ` t «
1
3
5
35 4
1 ´ t ` t2 ´ t3 `
t q to isolate the different power terms. The horizontal lateral aberration x1
2
8
16
128
is obtained by Fermat conversion of the wavefront error by summing the ω i´1 σ j terms, retained in the
∆ω-derivatives of the three path-length components (N, A and B), and multiplying by the “lever arm”
distance to the image :
x1 pω, σ, µq “
ÿ
ij
xij1 “
ÿ
ij x
1
ij
ω i ´1 σ j “
ÿÿ
ij
ijk x
1
µ k ω i ´1 σ j
where,
#
BBij
BAij
1
`
ij x “ ´hη iµNij `
B p∆ωq B p∆ωq
„
(28)
k
ω i´1 σ j coe f f icient
+
{sinβ ´
´
i´1,j x
R
1
¯
cotβ
(29)
1
and h “ 1 ` µ2 sin2 θ is the inclination factor due to an off-plane image coordinate of µ sinθ. As hη is
2
thereby the distance between the optic center and the paraxial image point, this conversion is accurate
to order ηω 7 , as given by Born and Wolf [14]. The trailing term in Equation (29) accounts for the linear
variation of 1/sinβ with ω when the optical surface is not flat (1{R ‰ 0).
For the vertical ray aberration (summing the ω i σ j´ 1 terms), the inclination factor h enters a second
time at the exit pupil (in the same manner as the 1/sinβ factor in the horizontal equation) and a third
time in projecting the transverse aberration (normal to the direction of propagation) onto the image
15
3
pµsinθq4 ` . . ., as employed below:
plane. This results in a net factor of h3 – 1 ` pµsinθq2 `
2
8
ÿ
ÿ
ÿ ÿ
1
1 i j ´1
1 k i j ´1
z1 pω, σ, µq “
“
(30)
ij zij “
ij ij z ω σ
ij
k ijk z µ ω σ
where,
ij z
1
!
‰
“
“ h3 η jµNij ` BAij {B p∆σq ` BBij {B p∆σq ωi σ j´1
coe f f icient
)
(31)
The above general equations provide the foundation for an accurate mathematical decomposition
of the lateral ray aberration into power terms xij1 (horizontally) and z1ij (vertically). The improvement
over the standard formulations lies in the systematic inclusion of all other relevant (i.e., able to form
expansion terms of power ω i´1 σ j (horizontally) or ω i σ j´ 1 (vertically)) aberrations x1I J and z1I J in
constructing the reference image point (ξij , ζ ij q for calculation of the geometrical path lengths Bij used
in determining both xij1 and z1ij .
One could integrate Equations (29) and (31) to form “wavefront”
¯ (path-length) coefficients; simply
´
being Equations (2) and (3) in reverse, yielding H Fij ” ´ ij x1 sinβ { pihηq for the horizontal direction
`
˘
and V Fij ” ij z1 { jh3 η for the vertical direction. Though such an exercise adds no new information,
it confirms that the individual terms in the decomposition of the wavefront (and thus lateral ray
Photonics 2016, 3, 3
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positions) are mathematical abstractions. If they were physical quantities, then H Fij ω i σ j and V Fij ω i σ j
would not “know” the direction in which they may be differentiated and thus would be equivalent
( H Fij ” V Fij ” Fij ). However, for the present astigmatic design, the required reference point (ξij , ζ ij )
for xij1 is different from that for z1ij at all (i, j) pairs (see Table 2). Thus, H Fij ‰ V Fij for all i and j (when
i + j ą 1), indeed with the ratios V F11 { H F11 , V F12 { H F12 and V F22 { H F22 calculated to be of significant
magnitude (102 to 103 ).
4. The Aberration Equations for an Astigmatic Plane Grating Monochromator
def
In this section, the lateral aberration terms of “power-sum” n “ (i + j ´ 1) “ 1, 2 and 3 are
explicitly expanded for a planar (R “ 8) grating surface.
4.1. The Fermat Derivatives
Setting the curvature radius to infinity simplifies Equations (19), (22) and (24)–(27) to the following:
BAij {B p∆ωq “ pcosα ´ xij sinα ` ωqp1 ´
BAij {B p∆σq “ pσ ´ zij qp1 ´
where,
8
A tij
3
5 8 3
35 8 4
18
t ` 8t 2 ´
t `
t q
2 A ij 8 A ij
16 A ij
128 A ij
38 2
5 8 3
35 8 4
18
t q
A tij `
A tij ´
A tij `
2
8
16
128 A ij
¯
´
“ 2ω cosα ` ω 2 ` σ2 ´ 2xij ω sinα ´ 2zij σ ` xij2 ` z2ij
BBij {B p∆ωq “ p´cosβ `
BBij {B p∆σq “ p
ξij
ω
1
38 2
5 8 3
35 8 4
sinβ ` qp1 ´ 8
t q
B tij `
B tij ´
B tij `
η
η
2
8
16
128 B ij
σ ζ ij
1
38 2
5 8 3
35 8 4
´ qp1 ´ 8
t q
B tij `
B tij ´
B tij `
η
η
2
8
16
128 B ij
(32)
(33)
(34)
(35)
(36)
where,
8
B tij
“ ´2
ˆ
ωcosβ
η
˙
`2
ˆ
ξij sinβ
η
˙ˆ
ω
η
˙
´2
ˆ
ζ ij
η
˙ ˆ ˙ ˆ ˙2 ˆ ˙2 ˆ ˙2 ˆ ˙2
ζ ij
ξij
σ
ω
σ
`
(37)
`
`
`
η
η
η
η
η
For the lateral aberrations, the expansions will use the in-plane object point xij “ zij “ 0.
The resulting series from Equations (32) and (33) are the same as those of the standard
light-path formulation:
˘
˘
1`
1` 2 2
3ω sin α ` σ2 cosα `
4a40 ω 3 sin2 α ` 2a22 ωσ2
BAij {B p∆ωq “ ωsin2 α ´
2 ˙
ˆ 2
1
3 4
4
2
2
2
´
5a50 ω sin α ` 3a32 ω σ ´ σ cosα
2
4
˙ ˆ
˙
3
1 3
2
3
3
σ ´ a22 ω σ `
ωσ ´ a32 ω σ cosα
BAij {B p∆σq “ σ ´ ωσ cosα ´
2
2
ˆ
˙
ˆ
˙
ˆ
˙
ˆ
˙
5 2
3 2
7 2
5 2
“ 1 ´ sin α , a22 “ 1 ´ sin α , a50 “ 1 ´ sin α and a32 “ 1 ´ sin α
4
2
4
2
ˆ
a40
(38)
(39)
(40)
Due to the symmetry imposed by the in-plane object, there are no odd powers of σ in Equation (38)
and no even powers of σ in Equation (39).
Expansion of Equations (35) and (36) is considerably more involved, due to the presence of
an image reference point (ξij , ζ ij ) whose components, as given by Equation (23), are themselves
power series in both ω and σ. Equation (29) provides nine aberration terms in the horizontal
` 1
˘
1 , x1 , x1 , x1 , x1 , x1 , x1 , x1
direction x20
, x30
40 11 21 31 12 22 13 and Equation (31) provides nine aberration terms
in the vertical direction pz102 , z103 , z104 , z111 , z121 , z131 , z112 , z122 , z113 ). For a given frontal aperture aspect ratio
Photonics 2016, 3, 3
16 of 44
def
(g “ φs {φm ), the surface aperture ratio (σ̆{ω̆) scales linearly with the graze angle; the aberrations
thereby progressively decrease in magnitude within the nth -degree power-sum family along the
sequence ω n Ñ ω n´1 σ Ñ ω n´2 σ2 Ñ . . . Ñ σn . Thus, unless φs " φm , the least-significant horizontal
1 and x1 , and the least-significant vertical terms pz1 and z1 q are those which contribute
terms are x22
13
13
04
2
2
to the same spectral width terms p x22 and x13 ) following the image plane rotation given in Section 6.
Being exceedingly small for the example monochromator (quantitatively verified by the raytrace
extraction equations given in Section 7), these four terms are not explicitly expanded here. Each
of the remaining 14 aberrations employs image path-length derivatives derived from a dedicated
reference image coordinate pair (ξij , ζ ij ) summed from the other (non-(i,j)) lateral aberrations. The
resulting expansion coefficients of ω i´1 σ j in Equation (35) and ω i σ j´ 1 in Equation (36) complete the
determination of the Fermat-generated horizontal and vertical ray aberrations given by Equations (29)
and (31), respectively.
4.2. The Explicit Expansion Terms of Power-Sum ď 3
In accordance with the detailed manual algebraic procedures exemplified for the 21 x1 ωσ term
in Appendix B, the 14 aberration power series coefficients are given here in explicit form, following
a re-listing of the three Equations (12), (9) and (11) derived in Section 2.3 for the principal ray:
b
´
¯
µ “ r 1 ` 2 pcosβ ´ cosαq cosβ tan2 θ ´ 1s{ tan2 θ cosθ cosβ
z1 01
z1
1
3
3
Γ
“ 01 “ µ sinθ ` µ3 sin3 θ ` µ5 sin5 θ ´ p1 ` µ2 sin2 θqz ` p1 ` qpµ sinθqz2
η
η
2
8
2
2
ˆ
˙
"
„
‰
`
˘
x1 10
tanβ
µcosθ
µ2 sin2 θ “
1 ` pΓ ` 3q tan2 β
1 ` µ2 sin2 θ µ sinθ `
tanψ z ´
tanβ “
`
2
η
ρ ˆ
2cosβ
2tan β
˙
*
˘
µsinθ `
1 1
2
2
2
`
´ Γ tan ψ z
1 ` 3tan β tanψ `
ρtanβ
2 ρ2
1
1
3
3
15
` C2 µ sinθ tanθ ` µ2 sin2 θ ` C2 µ3 sin3 θ tanθ ` µ4 sin4 θ
η
2
2
8
ˆ
˙
1
C2
2
2
11 x
sinβ “ pC2 ` cosβq µsinθ ` pC2 cosβq µ sin θ tanθ `
` cosβ µ3 sin3 θ
´
η
2
˙
ˆ
˙
ˆ
1
3
cosβ
cosβ
11 z
µ sinθ `
µ3 sin3 θ
“ C2 ´
C2 ´
η
η
2
2η
˙
ˆ
1
`
˘
1
1
20 x
`
´
µcosθ ` pC2 cosβq µ2 sin2 θ ` µ4 sin4 θ
sinβ “ C2 `
2
η
Qη
Qρ ˙ ˆ
˙
ˆ
3 4 4
1 2 2
1
µ sin θ ` µ sin θ µcosθ
` C2 `
2
8
Qρ2
ˆ
˙
„
1
3
Q
03 z
“
` C3 tan2 θ µsinθ ` 3C2 ` pC2 ` cosβq2 µ2 sin2 θ tanθ
η
2
2
„
¯
15 3 ´ 2
C2 ` C3 tan2 θ µ3 sin3 θ
`
`
4
2
ˆ
˙ ˆ
˙
1
cosβ
cosβ
12 z
“ ´ cosα `
` 2C3 ´ C2
µ sinθ tanθ
η
ˆη
˙ ˆ η
˙
„
cosβ
1
3
cosα `
` QC2 ` 2 pC2 ` cosβq µ2 sin2 θ
` 3C2 ´
2
η
ρ
02 z
η
“ 1`
(41)
(42)
(43)
(44)
(45)
(46)
Photonics 2016, 3, 3
17 of 44
x1
´ 21 sinβ “
˙
ˆ
˙
ˆ
˙
„η
ˆ
1
1
cosβ
1
cosβ µsinθ
2C3 ` 1 ´ ` QC2 ` 2 C2 cosβ ` QC2 ` 2 cos2 β ´ cosα `
η
η
ρ
ρ
"
ˆ
˙
*
”´
¯
1
1
1
2
2 β tan2 θ µ2 sinθ cosθ
` C2 `
`
2C
`
C
´
cosβ
`
`
2QC
`
Qcosβ
C
cos
3
2
2
2
Qρ2
ρ2 η ˆ
˙
˙
„
ˆ
3
1
1
7
2
QC2 ` 2 cos2 β
` QC2 ` 2 C2 cosβ `
` C3 ` 2C2 `
2
2
ρ
ρ
ˆ
˙
´
¯
cosβ
2
3
2
3
3
´ cosα `
cosβ ` QC2 cos β tan θ µ sin θ
η
¸
˜
˙
cosβ
1
C2 QC2 2
`
µ2 sinθ cosθ
“ C3 ´ C2
`
µ sinθ `
η
η
2
ρ2
2Qρ4
„ ´
˙
¯ 5cos2 β ` 8 ˆ 1
5
3
C3 ` C2 2 `
`
`
QC
C
cosβ
µ3 sin3 θ
`
´
2
2
2
4η 2
ρ2 2η
˙
ˆ
˙
„
ˆ
x1
QC2
1
1
1
1 cosβ 3cosα
´ 30 sinβ “ C3 `
C2 cosβ `
` 2´
´
´
µcosθ
η
2 ¸ρ
η
2ρ2 η Qρff2
2Qρ2
«˜
˙
ˆ
1
1
C2 2
`
C3 `
cosβ ` QC2 ´ ` 2 C2 cos2 β µ2 sin2 θ
2
η ρ
„
˙
ˆ
˙
ˆ
C3
QC2
1
1
1
`
C2 `
C2 cosβ
` C2 `
` 2 ´
2
2
Qρ2
ρ˜ 2η
¸
ff
ˆ
˙
1
1 cosβ 3cosα
QC2 2 cos3 β
tan2 θ µ3 sin2 θ cosθ
`
´
´
`
2
2ρ2 2η Qρ2
4Qρ2
˙
„
ˆ
1
cosβ
cosβ sin2 β
31 z
µ sinθ
“ C4 ´
C3 ` C2 ` 2
η
η
η
2η 2
21 z
x1
´ 40 sinβ “
η
1
ˆ
˙
ˆ
1
1
C3 cosβ
C4 ` QC2 ` 2 ´
ˆρ 2 η
„
˙
˙
ˆ
2
1
1
1
5
1
ρ
1
2 β ` cos β ` 3
` 4´ 2 `
´
sin
cosα
cosβ
`
´
3
2
2
2
4
η¸ ρ
ηρ
η
η
ρ
2Qρ2
ρ
˜
*
„ 2
˙
ˆ
2
2
1
QC2 Q C2
sin β
2 1
2 β ´ 3 cosα cosβ C2 µ cosθ
`
3
cos
`
`
´
QC
`
`
2
3
2
η2
ρ2 η
ρ2
ρ2
ρ4
¸
ˆ
˙˜
2
1
1
1
1
QC2
C2
` C2 `
µ2 cos2 θ
`
`
` 2 ´
2
Qρ2 Qη
ρ
2Qη 2 2Qρ4
"ˆ
˙
6QC2 ´ 3Γ
3
2QC2 C2 3
2
`
3Q2 C2 2 `
cos β ` pC2 C3 ` C4 q cosβ
`
´
´
η
2
ρ2
ρ4 ηρ2
ff
+
˙
ˆ
˙
„ˆ
1
1
1 C2 2
1
2
C3 ` QC2 ` 2 ´
` 2QC2 ` 2 ´
cos β µ2 sin2 θ
η
η
2
ρ
ρ
(48)
(49)
(50)
"
˙
„
ˆ
Qcos3 β
1
QC2
2
´
C2 cosβ ` QC2 cos β `
µ sinθ tanθ
sinβ “ pcosβ ´ cosαq ` C3 ` 1 `
η
2
¸ 2
ff 2 +
«˜
!
C2 2
cosα
cosβ ` QC2 2 cos2 β ` QC2 cos3 β tan2 θ µ2 sin2 θ
`
C3 `
` C2 ` 2cosβ ´
4
2
„ˆ
ˆ
˙
˙
1
2
1
C2
cosα cosβ
1
1
3
1
1
22 z
“ 1`
`
`
QC
`
`
`
´
`
2
2
2
2
2
4
η
η
η
2
2Q
ρ
ηρ
ρ
η
ρ
˙
ˆ
ˆ
˙
2
1
` 1 ´ 2 ´ C3 ´ C2 cosα tanθ ` 3C4 ´ C3 cosβ tan2 θ µ cosθ
η
η
12 x
(47)
(51)
1
x1
´ 31 sinβ
η
„
"
„
1 C2
2
2
“ 3C4 ` 1 ´ ` 2 ` Q pcosβ ` 3C2 q C3 cosβ ´ 1 ` pcosβ ` C2 q Q ` 2
η ρ
η
„
ρ
3 2 2
2 ` 6 pC2 ` cosβq Q ´ 3Γ
3
cosα
2
` QC2 ` Q C2 ´ p1 ` Qcosβq Γ C2 cos β `
`
`
2
4
2
η
2ρ
2ρ
˙
*
˙
ˆ
ˆ
sin2 β
3 2 2
1
5Γ
3
3
µ sinθ
` Q C2 cos β ` 1 ´ 2 cosβ ` pC2 ´ 4cosβq
´
`
ηρ
2η 2
2ρ4 2ρ2 2
(52)
(53)
(54)
Photonics 2016, 3, 3
18 of 44
where the substitute variable Q ” pµcosθq {sin2 β is of order unity at grazing incidence. While the
principal ray Equations (9) and (11) include the off-plane (z) coordinates of a tilted entrance slit
(x “ z tanψ, q, the effect of slit length on the aberrations is negligible at present (cf. the solid and dashed
line profile raytracings in Figure 6), thus Equations (41)–(54) do not include any z-components.
A summary of the accuracy provided by Equations (41)–(54) is given in Table 2, using the
ultra-high resolution soft X-ray monochromator (parameterized at the end of Section 2.6) as a test
case. The RMS deviations between these light-path equations and independent numerical raytrace
simulations (Section 7) are given in the last 5 columns. Also listed are the lateral rays, including
˘
`
aberrations, which form the reference image position ξij , ζ ij . The top-down sequence in this table
provides the terms needed for the reference image in subsequent expansions. The bracketed {terms}
and corresponding {errors} in the S.F. column refer to the standard light-path formulation {paraxial
reference image, including astigmatism per Section 3.1}.
Table 2. Reference image lateral positions and residual calculation errors.
RMS error in lateral extreme width: unit = 10´6 m at 10 m
Term Name
11
S.F.
ξij
σ
0
z1 01
z1 02 Astigmat.
x1
ζ ij
Power
Hor.Tilt
z1 11 Ver.Tilt
0
ω
0
z1 01
01
` z1
(
02
ω
0
z1
z1 03 p0, 3q
σ2
x1 11
z1 01 ` z1 02
z1 12 Ver. p1, 2q
ωσ
x1
20
Defocus
σ
z1
(
x1 21 Hor. p2, 1q
z1
21
(
x1 20 ` x1 11
(
x1 20 ` x1 11
ωσ
ω2
Ver. p2, 1q
x1
20
(
x1 30 Mer.Coma
ω2
x1 20
z1 31 Ver. p3, 1q
ω3
x1 40 Spherical
ω3
(
x1 20 ` x1 30
x1 12 Sag.Coma
σ2
z1 22 Ver. p2, 2q
ω2 σ
(
x1 20 ` x1 30
` x1 11 ` x1 21
x1 31 Hor. p3, 1q
ω2 σ
(
x1 20 ` x1 30
1
`x 11 ` x1 21
(
(
x1 20 ` x1 30
x1 11
01
(
` z1
to µ1
to µ2
to µ3
to µ4
350
1.15
0.020
0.00006
0.00003
20
0.10
0.00118
0.114
N/A
0.00004
0.124
0.00735
0.0045
0.00003
(
11
z1 01 ` z1 02
` z1 11
µ0
(
(
(
z1 01 ` z1 02
` z1 11 ` z1 12
(
z1 01 ` z111
(
z1 01 ` z1 11
1
` z 21
(
z1 01 ` z1 11
1
` z 21
(
z1 01 ` z1 11
1
` z 21 ` z1 31
(
z1 01 ` z1 02
` z1 03
(
z1 01 ` z1 02
` z1 11 ` z1 12
` z1 21
(
z1 01 ` z1 02
` z1 11 ` z1 12
` z1 21 ` z1 22
179
t93u
t10.8u
10.8
t92u
t1.4u
0.018
0.00009
0.84
0.0013
0.00022
0.008
0.0005
0.00005
1.00
0.007
0.00012
0.000025
0.0004
0.027
0.00028
0.19
0.018
0.00036
0.43
0.016
t0.6u
0.019
As listed in Table 2, the reference points for the highest-degree mixed terms (x1 31 and z1 22 ) include
numerous lower-degree aberrations and were therefore undertaken only for the linear sub-term in µ.
These terms provide the worst accuracies („ 0.02 microns laterally at the image plane), albeit of no
practical significance at present. All the other aberration terms were expanded to include the powers
in µ needed to decrease the lateral errors to the size of an atom („ 0.001 microns) or below, insuring
they will endure any conceivable future physical application.
By comparison, the standard light-path expansion causes significant formulation errors. This may
be seen by the defocus term resulting from use of only the paraxial point in forming the reference image:
´
20 x
η
1
sinβ “
ˆ
1
1
C2 `
`
Qρ2 Qη
˙
1
1
µ cosθ ` µ2 sin2 θ ` C2 µ3 sin2 θ cosθ
2
2
Photonics 2016, 3, 3
19 of 44
Due to the neglect of the z111 aberration in forming the image reference position, the µ2 sub-term in
this equation differs significantly from that in the rigorous Equation (44). This results in the calculation
error of „ 93 microns listed in Table 2. As shown, there is a similar calculation error resulting from the
standard formulation of the x1 21 term:
´
21 x
1 sinβ
η
ˆ
˙
3cos2 β ´ 1
» 2C3 `
µ sinθ
η
This is due to neglecting three relevant aberrations (z1 11 , x1 11 and z1 12 ) in forming the reference
image. While the above equation lists only the first sub-term (linear with µ), those of degree µ2 and µ3
were also generated but made little improvement due to the linear term already being flawed.
5. Manipulation and Analysis of the Ray Aberrations
5.1. Horizontal Tilt (Sagittally-Induced)
1
The lowest-degree mixed horizontal aberration created by the surface-normal rotation is x11
(Equation (42)) resulting from the nonzero value of N11 “ C2 sinθ in Equation (5). Noting the linearity
1 with the sagittal pupil coordinate σ, this is simply a tilt of the astigmatic image by the angle
of x11
1 /z1 ) about the image coordinates (0, z1 q. Employing the expansions given in
1
ψs “ ´arctanpx11
02
01
Equations (41) and (42):
˙
„ ˆ
`
˘
1
pµsinθq C2 1 ` µ2 sin2 θ ` 1 ` µC2 sinθ tanθ ` µ2 sin2 θ cosβ
2
˙
tanψs1 “ ´ „ˆ
3
1
3 2 2
3
3
` µC2 sinθ tanθ ` µ sin θ ` C2 µ sin θ tanθ sinβ
1`
η
2
2
(55)
where a negative value is counter-clockwise for an upstream observer. As in the case of the
single-rotation “pure” SNR monochromator [2,3], a sympathetic rotation of the exit slit by ψs1 about
2
p0, z101 q will cancel the rotated value of x11 and thus ∆λ11 . A stationary rotation axis p0, 0q would
require adjusting the focusing condition (not detailed here) to include the horizontally offset image
center intercepting the slit.
5.2. Vertical Tilt (Meridionally-Induced) and the Rigorous Focusing Condition
` ˘
Given the above astigmatic image tilt, the lowest-order mixed vertical aberration z111 presents
1 “ 0) would be
the interesting condition that a perfect horizontal focus for meridional rays (x20
extended into a vertically extended line, projected upon the tilted slit as a defocus normal to its length
(multiplied by tanψs1 ). In effect, the vertical aberration due to the meridional rays induces a second tilt
˘
`
1 “ ´arctan x1 {z1 . Employing Equations (43) and (44):
angle ψm
20 11
1
tanψm
˙
„ ˆ
`
˘
1
1
`
µ cosθ ` pC2 cosβq µ2 sin2 θ ` µ4 sin4 θ
“ ´ C2 `
2
Qρ ˙ ˆ Qη
˙
ˆ
1 2 2
3 4 4
1
µ
sin
θ
`
µ
sin
θ
µcosθ
` C2 `
2
8
"„ˆ Qρ
˙2 ˆ
˙
*
cosβ
cosβ
3
{
C2 ´
C2 ´
`
µ2 sin2 θ pµsinθq sinβ
η
2
2η
(56)
The solution is to purposely “defocus” the grating horizontally to force equality of the two
tilts (Equations (55) and (56)). Making the following substitutions: Q “ pµcosθq {sin2 β, µ cosθ “
Photonics 2016, 3, 3
20 of 44
1
ε ´ ε2 pcosβq tan2 θ and C2 “ 2N2 cosθ constrains the single parameter θ to provide the following
2
rigorous focusing condition:
„
0“
˙
ˆ
˙s 1
˙ˆ
{
1
1
1
1
40 x pθo q
` 2
sinβ ωbal 2 cos4 θ
sin2 β ´ 1 `
1`
η ρ ˙
η „ˆ
˙ η
"ˆ
*
s η 1
{
`
˘
x pθo q
1
1
1
2
2
`
1`
` 2 sin β ´ 40
sinβ ωbal 2 sin2 θ 2N2 cos3 θ ε
cos θ `
η˙
η ρ„
η
˙
ˆ
„ˆ
1
1
1
2
3
2 β cosθ
2
sin
´ 1 pN2 cosβq cos θ `
`
` 2`
`
η 2η ρ
2ηρ2
ˆ η
˙
`
˘
`
˘
1
1
´
` 2
N2 sin2 β cosβ sin2 θ sin2 θ cosθ ε2
η
ρ
"
„
ˆ
˙
1
1
2
1
2 β pcosβq sin2 θ
`
` 2`
sin
` 3N2 cos3 θ ´
2
˙ η
ˆ2η ρ˙ 2ηρ
*
„ˆ
`
˘
1
4
3
2
2
2
` 2 sin β ` 1 ´
cos β N2 sin θ cosθ sin2 θ ε3
`
η ρ
η
„ˆ
(57)
accurate to order ε3 , where ε ” pcosβ ´ cosαq is a small quantity (of order µ) and where ωbal “ 0
at present (the terms involving 40 x1 are derived in Section 5.3). If one drops the ε2 and ε3 terms,
a quadratic emerges whose closed-form root θ ” θo :
„ ˆ
˙
˙
ˆ
1
1 ` 2 ˘
1
sin β pN2 εq cos2 θo
0 “ 2 1`
` 2
´2
η ˙ ˆ η ˙ρ
„ˆ
ˆ
˙
1
1
1
1 ` 2 ˘
1
`
sin2 β cosθo ` 2
` 2
` 2
sin β pN2 εq
1`
η ρ
η
η ρ
(58)
Note that the linear approximation to Equations (57) or (58) has the even simpler solution
“`
˘
‰
1 “ 0q independent of
cosθoo “ ´ 1{η ` 1{ρ2 sin2 β { p2N2 εq, being a pure horizontal focus (x20
the image tilt. In the small-angle approximation, this is equivalent to that previously given by
Equation (16).
The final solution for θ is obtained by numerical iteration of Equation (57) using the above “initial
guess” of θo or θoo . Even at the maximum rotation angle of θoo „ 71.000o , only a small adjustment in
1 “ ψ1 . If using θ „ 71.245o as the initial guess, the
θ is needed to provide the desired condition ψm
o
s
´
12
residual is negligible (ă 10 ) after only 2 linear interpolations, refining θ by „ 0.064˝ .
Equalizing the two image tilts of a point source is critical to providing fine spectral resolution
at high rotation angles. In the absence of such a constraint (i.e., if the horizontal defocus alone is
corrected), the net resolution of the example monochromator is calculated to degrade by a factor „ 6 at
θ „ 71o .
5.3. Balancing of Spherical Aberration and Defocus
Though typically a very small correction, a fine adjustment in θ may also be employed to partially
balance the spherical aberration term. This is analogous to the classical technique of offsetting the
1 “ 0) focus to the plane of “least confusion” which minimizes
detection plane from the Gaussian (x20
1 9 ω and x1 9 ω 3 q. Unlike the
the sum of these two lateral horizontal aberrations of odd power (x20
40
3
defocus term, the aberrations of high power (including the ω term) vary only slowly with θ (except
1 term to be evaluated at the root of Equations (57)
near their correction points). This allows the x40
1 for a refined determination of the
or (58) and to then be treated as a constant to be added to x20
1
1 at θ from Equation (58),
meridionally-induced tilt angle (ψm ) given above. For example, evaluating x40
o
𝜃
′
𝑥20
= 0 focus to the plane of “least confusion” which minimizes
′
′
𝑥20
∝ 𝜔 and 𝑥40
∝ 𝜔3 )
3
𝜔
𝜃
′
Photonics 2016, 3, 3
21 of 44
𝑥40
′
𝑥20
′
′
𝜓𝑚
𝑥40
𝜃𝑜
the full focusing condition shown in Equation (57) employs a nonzero value for ωbal𝜔and
the following
𝑏𝑎𝑙
constant approximated by the dominant lowest-order (µcosθ) term
of
Equation
(51):
𝜇 𝑐𝑜𝑠𝜃
−(
40𝑥
′ (𝜃 )
𝑜
𝜂
𝑠𝑖𝑛𝛽) ≃
⟦4𝑁4 𝑐𝑜𝑠 3 𝜃𝑜 + [4 −
+ [(
1
𝜌2 1
5
1 3𝛤 3𝛤
1
+ ( 3 + 2 − 2 ) 𝑠𝑖𝑛2 𝛽 + ( 4 − 2 + ) 𝑐𝑜𝑠 2 𝛽]
2
1 − 𝜌2
𝜂
𝜌
𝜌
𝜂
𝜌
𝜂
𝜂𝜌
1
1 1
𝑠𝑖𝑛2 𝛽
2
1
1
− 1) 𝑁2 𝑐𝑜𝑠𝜃𝑜 + 2 − ] (2𝑁3 𝑐𝑜𝑠𝛽)𝑐𝑜𝑠 2 𝜃𝑜 + { 2 + 2 − ( 2 − ) 𝑁2 𝑐𝑜𝑠𝜃𝑜
2
𝜌
𝜌
𝜂
𝜂
𝜂𝜌
𝜂
𝜂𝜌
+[
2
1
3
(𝑁2 (1 − 𝜌2 )𝑐𝑜𝑠𝜃𝑜 + 1 − 𝜌2 𝛤) + ( 2 − 1) 𝑁2 2 𝑐𝑜𝑠 2 𝜃𝑜 ] 𝑐𝑜𝑠 2 𝛽} 𝑁2 𝑐𝑜𝑠𝜃𝑜 ⟧ 𝜀
4
𝜌
𝜌
(61)
(59)
𝑏𝑎𝑙 ≈ 2/3 ω
An optimized balancing of defocus and spherical aberration 𝜔
employs
bal « 2{3 of the
𝜃
semi-meridional aperture, resulting in the root of θ changing by „ ´0.0067˝ at the scan wavelength
𝜆/𝜆o = 6.58
of λ{λo “ 6.58,
and the extremum convolution of all the aberrations decreasing by „ 30% (cf.
Figure 6g,h). More substantial improvements are expected at larger apertures, where spherical
aberration is increasingly dominant over the lower-power aberrations.
5.4. Rotation of the Entrance Slit
𝜓
𝜓′𝜓
′ (−𝑧)
Given an entrance slit at angle𝑥 ′ψ
to the𝑥 ′vertical,
its
ψ1 ψ derives simply from Equations (9)
(𝑧) 𝑧tilt
𝑧 ′10image
10 (𝑧)
10 (−𝑧)
10
𝑡𝑎𝑛 𝜓 ′ 𝜓 ≡ [
−
−
]/[
]
and (11):
𝜂
𝜂
„ 1 𝜂
„𝜂 1
1
1
pzq x𝜇𝑠𝑖𝑛𝜃
x 10 𝑡𝑎𝑛𝜓
z 10 pzq z3 10 p´zq
10 p´zq
1
tanψ ψ ” ≃ − [
´
{2 2 / (1´+ 𝜇2 𝑠𝑖𝑛2 𝜃)
(62)
η„ 𝜌 + 𝑡𝑎𝑛𝛽η (1 + 𝜇 𝑠𝑖𝑛 𝜃)]
η
˙
2ˆ η
(60)
˘
3
tanψ µsinθ `
`
1 ` µ2 sin2 θ { 1 ` µ2 sin2 θ
»´
ρ
tanβ
2
To maintain focus along the slit length, this must be set equal to the image tilt for a point source
from Equation (55). Thus, ψ1 ψ “ ψ1 s “ ψ1 m ” ψ1 , constraining ψ as a function of the scan parameters:
ˆ
˙
cosβ
cosβ
` 4N2 cosθ ´
µ2 sin2 θ
2N2 cosθ ´
η
η
ˆ
˙
tanψ »
ρ µ sinθ
1
3 2 2
2
1 ` ` 2µN2 sin θ ` µ sin θ sinβ
η
2
(61)
The required entrance and exit slit rotation angles, ψ and ψ1 , are plotted vs. scan wavelength in
Figure 4. While these rotation values are comparable in magnitude, they are opposite in direction. This
is due to the present treatment of the entrance slit as being an isotropic emitter (e.g., back-illuminated
by a diffuse source), thus each point along the length of the slit has its own principal ray whose vertical
coordinate reverses sign at the grating pole. However, if the entrance slit were illuminated by a distant
horizontal (vertically-narrow) source, there would by a mapping between the z-values of points along
the entrance slit and the sagittal pupil coordinate σ. This may be treated by a nonzero value of z02 , as
indicated in Section 3.2 and provided by the general expansion equations. The result would be no sign
reversal of the required entrance slit rotation angle, opening the possibility for the entrance and exit
slit rotations to be replaced by a third rotation of the grating (about its ω-axis), similar to the technique
first employed in a SNR monochromator in 1993 as given in Section 4.2d of the cited thesis [15].
Photonics 2016, 3, 3
22 of 44
Figure 4. The grating scan angles are δ (about the groove axis) and θ (about the surface normal axis).
The entrance slit tilt angle (ψ) is slightly smaller in magnitude and opposite in sign (negated here for
plotting convenience) than the exit slit tilt angle (ψ1 ).
= +
= −
The sequence of analytical calculations which determine
the scan operating
parameters are
and
now specified:
, and
(1) The grating first rotation angle δ, which sets α “ γ ` δ and
β “ γ ´ δ;
′
and
(2) The grating second nominal rotation angle θo from Equation
and (58), using above α and β;
(3) Exact θ obtained, numerically
from
Equation
(57),
with
or without balance of spherical
and and
aberration (Equation (59));
∆ /
(4) The dimensionless
wavelength µ from Equation (10) or (12), using α, β and θ;
∆
(5) The exit slit tilt angle ψ1 from Equation (56), using above θ and µ; and
(6) The entrance slit tilt angle ψ from Equation (61), using above θ and µ.
[ "]
]
]
=[ ]
+[
+[
Using α, β and θ and µ as specified
above (and
an
object
position
z), the principal ray terms and
aberrations are then determined from Equations (9,11) and (41–54), respectively. Transformation of the
horizontal (x1 ) and vertical (z1 ) lateral positions to spectral resolution ∆λ{λ and exit slit height ∆z2
′
requires a final rotational transformation at the image plane, as′ given in Section
6.
5.5. Image Curvature
Spectral curvature along the image length is given by
“ 2‰
“ 2 ‰
x curv “ x10 z2
term
` r12 x2 sσ2
term
` r11 x2 szσ term
(62)
which is the sum of three components (the paraxial position and two aberrations). The first term is the
paraxial image curvature of a straight entrance slit, resulting in a deviation from a straight exit slit
(albeit rotated in accordance with Equations (55), (56) or (60)) obtained by the rotational transformation
1 ) and vertical (z1 ) positions (Equations (11) and
of Equation (67) on the z2 terms of the horizontal (x10
01
(9), respectively):
“
2
x10
‰
z2 term
˘
‰ µsinθ `
µ2 sin2 θ “
µcosθ
1 ` 3tan2 β tanψ
`
1 ` pΓ ` 3q tan2 β `
2
ρtanβ {
ˆ
ˆ 2cosβ ˙ 2tan*β
˙
1 1
Γ
cosψ1
2
`
` 1`
pµsinθq sinψ1 z2
´ Γ tan ψ
2 ρ2
tanβ
2
“ ´η
s"
(63)
This component of the image curvature may be eliminated only by curving the entrance (not exit)
slit. However, such correction is both difficult (due to its dependence on θ) and unnecessary; for the
example monochromator parameters, the uncorrected curvature at the ends of the 74 mm long image
Photonics 2016, 3, 3
23 of 44
of the entrance slit is only `1.05 microns at θ “ 0 and decreases to ´0.21 microns at θ “ 71.245o . This
is confirmed (within 0.1 microns) by the numerical raytracings (1D profile of Figure 6i), revealing that
the entrance slit length causes only a slight asymmetry and horizontal shift in the spectral line.
Horizontal curvature of the (vertical) astigmatism from a point source is given by x1 12
(Equation (52)), often referred to as “sagittal coma” or “astigmatic coma”. This term, together with z1 03
from Equation (45) determines the curvature in the spectral direction (x2 12 ) using Equation (67):
r12 x2 sσ2 term “
˙"
˙
*
„
ˆ
sˆ
Qcos3 β
1
QC2
cosψ1
C2 cosβ ` QC2 cos2 β `
µsinθ tanθ
µcosθ ` C3 ` 1 `
´η
sinβ „ˆ2
2
˙
{ 2
˘
`
3
1
2
2
`η sinψ
` C3 tan θ µsinθ σ
2
(64)
In spatial units, this resolution corresponds to 0.36 µm at the minimum scan wavelength, zero at
the (passive) correction wavelength (λ{λo „ 1.48) and 1.16 µm at the maximum wavelength, confirmed
by the numerical raytracings within 0.07 µm. As given in Figure 5, these correspond to spectral
resolutions (∆λ{λ) of 2 ˆ 10´6 , zero and 3 ˆ 10´6 , respectively.
It is noted that there are no (i,1) aberrations when θ “ 0, where the straight grooves are
perpendicular to the meridional plane. In this case, the dominant mixed aberration is astigmatic
2
coma (x12 ), for which the residual nonzero component of Equation (64) yields simply ∆λ12 {λ “ φs2 {8,
independent of η. This is the same result previously reported [8,15] for a plane grating in a converging
(stigmatic) beam where η « ´ 1. While of importance and of historical significance in that original
VLS application to fast XUV telescope beams (φs „ 0.1), this aberration is negligible for most soft
X-ray laboratory applications (e.g., ∆λ12 {λ “ 2 ˆ 10´6 for φs “ 0.004). It is also noted that the
passive correction point seen in Figure 5 may be obtained by zeroing the coefficient of the µ term in
Equation (52), yielding θ12 » 36˝ , though this aberration remains small across the scan range.
When expanded to include an off-plane (z ‰ 0) object point, the spectral aberration of
2
sagittally-induced image tilt (x11 ), otherwise zero by proper rotation of the exit slit (Section 5.1),
also has a small curvature component. However, because nonzero values of z are not included in the
expansion of the aberration terms (i + j ą 1) at present, the required horizontal and vertical component
terms are not given in Equations (42) and (41), respectively.
5.6. The Horizontal Mixed Aberrations (2,1) and (3,1)
As shown in Figure 5, the dominant spectral aberration at long wavelengths is due to the (2,1)
mixed term, which vanishes at one scan wavelength. A good approximation to this “passive correction”
point may be obtained by zeroing the coefficient of the linear term in µ from Equation (47):
˙
ˆ
˙
ˆ
ˆ
˙
1
cosβ
1
1
2
2C3 ` 1 ´ ` QC2 ` 2 C2 cosβ ` QC2 ` 2 cos β ´ cosα `
cosβ “ 0
η
η
ρ
ρ
(65)
Simplifying this expression by treating α and β as small angles yields:
cosθ21 » r1{ pη cosθ21 q ´ p1 ´ 1{ηq N2 s {N3
(66)
which occurs at θ21 » 47.8˝ (λ{λo „ 2.08) for the example monochromator. However, this aberration
is dominant at larger rotation angles, reaching a full-width comparable to that of the pure meridional
(3,0) aberration at the long-wavelength end of the scan range (θ “ 71.25˝ ). As the magnitude of this
mixed aberration scales with ωσ, it is highest at the corners of the solid aperture. An elliptically-shaped
illumination would thereby halve the aberration with only a 22% intensity loss. However, for clarity
the present analysis employs a simple rectangular aperture. The horizontal (2,1) ray aberration was
also required to obtain the non-paraxial reference image position in the expansions of the (4,0) and
(3,1) horizontal aberrations. As this is the dominant mixed aberration in the horizontal direction, the
Photonics 2016, 3, 3
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coefficients to degree µ3 were expanded so as to maintain accurate results even if µ increases (e.g., due
to use of a higher graze angle).
As will be clear from the spectral resolution plot (Figure 5) and the raytrace diagram (Figure 6), the
higher-degree (3,1) aberration causes only a small distortion to the above (2,1) aberration. Therefore, its
laborious Fermat expansion (requiring nine lateral aberrations to compose the proper reference image,
as listed in Table 2) was performed only for the linear component in µ and is given by Equation (54).
5.7. Minor Vertical Aberrations (1,2), (2,1), (3,1) and (2,2)
Given that the horizontal (1,2) aberration is very small, the vertical (1,2) aberration is surprisingly
large (11 microns for the example design). This result would not be obtained from the standard
light-path formulation, even given the aforementioned “astigmatism exception”. The aberration is
correctly determined here by the inclusion of two additional non-paraxial reference points, namely
1
1
11 x and 11 z , as listed in Table 2. Though not being of practical significance in itself for a (highly)
astigmatic monochromator, the image plane rotation of Equation (67) transforms this vertical aberration
into a non-negligible („ 1 micron) component of the (2,1) image width in the spectral direction. The
vertical (1,2) aberration is also required as a non-paraxial reference for the correct expansion of the
(dominant) horizontal (2,1) aberration.
Similarly, the vertical (2,1), (3,1) and (2,2) aberrations are of no importance in themselves, as they
are comparatively negligible additions to the astigmatism term. However, their explicit expansions
have been given in Section 4 to provide the comprehensive set of vertical reference image coordinates
required for the expansion of the horizontal (3,0), (4,0) and (3,1) aberrations, respectively. The image
plane rotation of Equations (67) and (68) also transforms the (2,1), (3,1) and (2,2) vertical aberrations
into non-negligible components of the (3,0), (4,0) and (3,1) spectral aberrations.
6. Spectral Resolution
The spectral resolution equals the grating dispersion times the ray aberration normal to the slit
length. The latter is determined by a rotational transformation of the image plane coordinates (x1 , z1 )
to those of the ψ1 -tilted image plane (x2 , z2 ). The power series is carried over to the x2 coordinate by
linearly combining the vertical and horizontal coefficients in accordance with:
ij x
2
“ij x1 cosψ1 ´ i´1,j`1 z1 sinψ1
(67)
Division by the linear dispersion per fractional wavelength yields the corresponding wavelength
shift coefficients
ij ∆λ
ij x
2 sinβ
¯
sinβ ´ 1
1
1
x
´
z
tanψ
“
i
´
1,j
`
1
η µ cosθ cosψ1
η µ cosθ ij
(68)
ˇ
ˇ
ˇ
ˇ
∆λij {λ “ pij ˇij ∆λ{λˇ ω̆ i´1 σ̆ j “ pij ˇij ∆λ{λˇ pφm {sinαqi´1 φs j
(69)
λ
“
where tanψ1 is given by Equation (56). The wavelength variation modulus (∆) of each term over the
full rectangular grating aperture (ω “ ˘ ω̆, σ “ ˘ σ̆) is its “extremum” (full width) aberration:
where pij “ 21´pi` jq if i is odd and j is even; otherwise pij “ 22´pi` jq ; and in which the full angular
frontal apertures are φm “ 2ω̆sinα (meridionally) and φs “ 2σ̆ (sagittally). Using Equations (67)–(69)
with the constituent horizontal (x1 ) and vertical (z1 ) ray aberrations given by the light-path
Equations (41)–(54), Figure 5 plots each of the nonzero spectral aberration terms ∆λij {λ. In the
absence of spherical aberration balancing (Section 5.3), ∆λ20 and ∆λ11 are each zero relative to the
rotated exit slit normal, as devised in Sections 5.4 and 5.5. The Ni and η corrections for (3,0) and (4,0)
specified in Section 2 are verified by Figure 5 to indeed minimize their peak magnitudes over the
intended scan range of three octaves.
Δ
Δ
Photonics 2016, 3, 3
∆
/
25 of 44
Figure 5. Geometrical spectral aberrations of a soft X-ray single-element plane grating monochromator
(SEPGM) at the Gaussian image plane. The grazing angular deviation is 6˝ and the incident aperture
is 2 mrad horizontally ˆ 4 mrad vertically. The meridional and sagittal aperture dependences of the
individual terms are given in the legend, and plotted as extrema (peak-to-valley) over the full grating
aperture. The colored curves result from rigorous expansion and Fermat differentiation of light-path
functions, while the open circles are algebraic extractions from numerical raytracings at 29 wavelengths.
Discrepancies between these two independent methods of analysis are negligible (~ 10´10 in ∆λ{λ),
being four orders of magnitude smaller than the physical diffraction width at a wavelength of 1 nm.
−
~ curve
/
The black
is a∆ conservative
index of the net RMS geometrical resolution, summed from the
individual terms (see the text).
An easily-calculable index of the net (measureable) spectral resolution is the root-mean-square
(RMS) of the wavelength deviations (relative to that diffracted from the grating pole):
∆λ RMS
λ
,1{2
$
»
fi2
’
/
& 1 { ÿ´
.
¯
def
i ´1 j fl
–
“
∆λ{λ
ω
σ
dωdσ
ij
’
/
% 4ω̆ σ̆
i,j
(70)
Photonics 2016, 3, 3
26 of 44
„
$ „
1
1
1 2
1 2 4
1
2
2
2
2
4
’
’
F F ∅ ` F20 `
F ∅ `
F30 F12 g ` pF30 ` F12 g q ∅2
’
& 40 20 40
12
448 40
72
80
1 2 2 2
1 2 2 4
“∅
`
F g ∅ `
F g ∅
’
’
’
144 21
960 31
%
def
def
def
,1{2
/
/
/
.
(71)
/
/
/
-
¯
∆λ{λ
{sini´1 α and where Equation (71) includes all terms
ij
˘
`
`
˘
up to a power-sum of 3, except for the insignificant terms of 22 ∆λ{λ ω σ2 and 13 ∆λ{λ σ3 .
This RMS value is also plotted in Figure 5. The surface integral in Equation (70) corresponds to
rays distributed uniformly on the grating surface, which differs somewhat from a uniform angular
distribution originating at the object (source) point. It is also noted that the presence of asymmetrical
aberrations, particularly the dominant (3,0) term, results in a nonzero mean value for the deviation.
This offset causes Equation (71) to calculate somewhat larger values than a true “RMS width”, where
the deviations would be calculated relative to the mean.
The first [bracketed sum] in Equation (71) contains a cross-product of the F 20 (defocus) and
F 40 (spherical aberration) coefficients; by departing from the (abstract) condition of being “in-focus”
2 and F 2 terms
(F 20 “ 0), this product can be made negative and thus partially balance the positive F20
40
(as accomplished by the focus adjustment derived in Section 5.3). However, such (partial) balancing of
these two different terms is aperture-dependent. Similarly, the second [bracketed sum] shows that
combined coma may be made smaller than the sum of the positive meridionally-induced and positive
sagittally-induced components. While exploited in normal-incidence optics (having „ a 3:1 ratio
between these components), grazing angles result in the much larger (ą 10:1) ratios evident in Figure 5,
enabling little such coma balancing. The maximum value of sagittally-induced coma (1,2) is 2.9 ˆ 10´6
at θ max “ 71.25˝ (being only 1.5 times its in-plane value at θ “ 0) and is small compared to either the
(25 times larger) meridionally-induced coma (3,0) or the (50 times larger) dominant aberration (2,1).
More generally, given a design optimized in resolution over a scan range of q octaves, and given
a ratio of γ{φ between „ 10 and 100, meridional coma dominates the spectral aberration (provided the
aperture aspect ratio g À 2); fitting to several such sets of parameters yields the following approximate
relation for the nominal resolving power (R) over the designed scan range:
where ∅ “ ∅m , g “ ∅s {∅m , Fij “
´
def
R “ 1{x∆λ RMS {λy „ p7γ{φq2 23´q
(72)
This simple result shows a resolving power which scales quadratically with the graze angle and
inversely with both the solid angle of acceptance and the spectral range. For the ultra-high resolution
design plotted in Figure 5, γ{φ “ 25 and q “ 3, for which Equation (72) estimates R „ 30, 000
(in agreement with the value of 29,000 obtained by averaging Equation (71) over the scan range).
As specified in Section 5.3, optimization of the resolution for higher apertures (e.g., γ/φ ~ 8) requires
some “least-confusion” defocusing to balance the dominant spherical aberration term. At yet higher
apertures, several limitations would emerge: (1) spherical aberration would finally dominate and thus
invalidate Equation (72); (2) more than a 50% line-space variation (Section 8.4) would be required;
and (3) the large (ą 12%) variation in graze angle in the meridional direction would compromise the
average reflectivity.
In the direction along the slit length, the rotational transformation of the image plane yields
ij z
2
“
`
ij z
1
˘
´ 01 z1 cosψ1 ` i`1,j´1 x1 sinψ1
(73)
resulting in the following full-width aberrations in the direction parallel to the slit and relative to its
center (z2 “ 0):
2
2
2
(74)
∆zij “ p ji zij pω̆, σ̆q “ p ji ij z ω̆ i σ̆ j´1
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7. Numerical Raytrace Simulations
Independent confirmation of the light-path equations derived in Sections 2–6 is obtained here
by three-dimensional numerical raytraces, using the commercial code “BEAM4” developed by
M. Lampton [16]. An angular deviation of 2γ “ 6˝ is chosen, as it provides a single-bounce gold
reflectance of „ 50% at λo “ 1 nm in the soft X-ray. To display the optical aberrations, the raytraced
source is a point at the entrance slit. However, if 5 micron wide slits are to contribute a dispersive
component (Section 8.5) equal to the nominal optical resolving power of R “ 25,000, the object distance
must be r “ 3000 mm. This scale converts the design parameters (listed in Section 2.6) to the following
dimensional values:
(Image distance)
(Line density at the pole)
(VLS ruling coefficients)
r1 “ 11,100 mm
M1 “ 1{do “ 1000 mm´1
2M2 “ 0.955315252 mm´2
3M3 “ ´2.792863 ˆ 10´4 mm´3
4M4 “ 1.906 ˆ 10´7 mm´4
From Equation (4), one may easily determine the required accuracy for kMk is M1 {R{ p2rω̆qk´1 ,
thus (1000/25000)/(214 mm) „ 0.0002 mm´2 for 2M2 . This translates to a groove positioning error
of „ 0.0005 mm (1/2 groove width). This tolerance is 3 orders of magnitude less stringent than
demonstrated (0.1 „ 1 nm) by existing technologies in the fabrication of low scatter gratings [4,17].
To accept the 2 mrad (φm ) ˆ 4 mrad (φs ) frontal aperture at every angular orientation (δ, θ) across the
wavelength scan, the grating must have a physical aperture of 214 mm in diameter. At each value of ρ,
only the light-path equations given in Sections 2–5 were used to determine the fixed (“cold”) inputs
(α, θ, ψ and Mk ) for raytrace simulations run at 29 sample wavelengths across the three-octave scan
range. No numerical optimization was performed by the raytrace routine (e.g., no “auto-focus” used).
7.1. Spot Diagrams
The right-hand panels of Figure 6 are the result of uniformly illuminating the grating rectangular
pupil aperture with 10,000 randomly placed rays from the on-axis object point. Eight scan wavelengths
were selected to highlight the different characteristic aberrations, and their convolutions, as listed in
the caption. In these phase-space plots, horizontal sagittal coma (1,2) appears as a finite width at ω “ 0
(visible only in Figure 6a), meridional coma (3,0) appears as a parabola (e.g., Figure 6a,d, with this also
being a component aberration in Figure 6c,e,f,i), spherical aberration (4,0) causes a cubic (S-shaped)
asymmetry between the –ω and `ω regions (most evident in Figure 6a,f,g and i), the dominant
mixed-term aberration of horizontal (2,1) appears as a “bow-tie” shape (in Figure 6b,c,e,f,g,h,i) and the
minor mixed-term aberration of horizontal (3,1) causes the widths of the bow-tie to be different at the
2 ends (lopsided), as visible only in Figure 6g,h and i.
The bow-tie shaped aberration (2,1) is absent only for the in-plane orientation (θ “ 0q at which
λ{λo “ 1 and at the passive correction point given by θ21 using Equation (66) for which λ{λo „ 2.08.
At those two wavelengths, the phase-space spot diagrams of Figure 6a,d show the classical curves
resulting from the addition of the quadratic (coma) and cubic (spherical aberration) terms. Conversely,
in Figure 6b these pure meridional aberrations are absent or small, resulting in the near-exclusive
presence of the mixed aberration (2,1). It is also noted that Figure 6h confirms the least-confusion
balancing (Section 5.3) of the defocus and spherical aberration terms given by Equation (59).
7.2. Line Profiles
The left-most panels in Figure 6 display the simulated (raytraced) spectra of the line doublet near
three representative wavelengths within the scan range. The thick line segment shows the “RMS”
value calculated by Equation (71). Due to the dominant aberration of coma (3,0) being more highly
peaked than a normal distribution, the actual marginal optical resolution is much finer than the usual
Photonics 2016, 3, 3
28 of 44
measure of a full-width-at-half-maximum “ 2.355 RMS, except at the 2 coma-corrected wavelengths
(λ{λo “ 1.26 and 6.58) where the marginal resolution is „ 1 RMS. Though Section 7.3 will reveal
more quantitative detail for each aberration, it is evident in Figure 6 that the spectral resolution is
comparable to or better than the 1/20,000 separating the two raytraced lines, thus confirming the net
convolution of the light-path terms as plotted (black curve) in Figure 5.
~
Figure 6. Numerical raytracings of an ultra-high resolution SEPGM, displayed in phase-space 2D spot
diagrams (x2 vs. ω) and 1D spectral profiles (intensity vs. x2 ). The meridional pupil coordinate ω spans
" in angular aperture. Wavelength
„ "140 mm at λo to „ 214 mm at 8λo , corresponding to „ 2 mrad
increases to the top of ~
each panel, in which the two wavelengths (shown
~ here in green and red)
are separated by 1 part in 20,000; their dispersed separation („ 9 microns at λo to „ 18 microns at
8λo ) provides the dimensional scale for the image plane coordinate x2 . (a) λo (θ “ 0˝ ): dominant
~ o : (3,0) canceled,
coma (3,0), some spherical aberration (4,0) and slight sagittal coma (1,2); (b) 1.26λ
" 2.08λo :
dominant (2,1) and slight (4,0); (c) 1.43λo : (4,0) canceled, (1,2) nearly canceled, (2,1)–(3,0); (d)
(2,1) canceled, dominant (3,0); (e) 2.5λo : (3,1) canceled, dominant (3,0) near maximum, (2,1)–(4,0);
(f) 5.2λo : dominant bow-tie (2,1), (3,0)–(4,0); (g) 6.58λo : (3,0) canceled, dominant (2,1), some (4,0),
slight (3,1); (h) per 6(g), but with (2,0) adjusted to balance (4,0) for least-confusion; and (i) 7.99λo :
(θmax „ 71.25˝ ), (3,0)–(2,1)–(4,0), slight (3,1).
Four objects were used in the 1D simulations: a point (_____ ), as used for the 2D diagrams; a 3 µm
horizontal width (____ ), a 20 mm long slit (- - - - - -) and a 3 µm wide ˆ 20 mm long slit (- - - -).
~
The entrance slit was an isotropic source (equivalent to a passive slit backlit by a diffuse source) and
tilted in accordance with Equation (61) for its image to coincide with the exit slit. The absence of
any significant broadening in the line profiles obviates the need at present to include off-plane object
m
≠0
=0
~
=
m
Photonics 2016, 3, 3
29 of 44
(z ‰ 0) terms in the aberration expansions Equations (41)–(54) which were derived in Section 4 for
a point source (z “ 0). The 1D spectral profiles exhibit the following aberration ratios:
(b) ∆λ21 „ 5∆λ40 „ 6∆λ31 „ 59∆λ12 „ 89∆λ22 ;
(d) ∆λ30 „ 4∆λ40 „ 76∆λ31 „ 118∆λ12 „ 165∆λ22 ;
(i) ∆λ21 „ 1.6∆λ30 „ 2.7∆λ40 „ 7.5∆λ31 „ 41∆λ12 .
The small longward shifts of the peak positions (compare the solid and dashed curves) for the
2 shorter wavelengths ((b) and (d)) is due to „ 1 micron image curvature of the straight entrance slit,
as given by Equation (11). For the longest wavelength trace (i), the shift is seen to be in the opposite
(shortward) direction and very small („ 0.2 microns), as predicted by Equation (63). “Pre-emptively”
curving the entrance slit by a corresponding amount in the opposite sense (see Section 5.5) is found to
eliminate such curvature and shift at the image plane (verified by a raytracing at λ{λo „ 1.26).
7.3. Power Series Extraction
Precise raytracings provide an alternate (and independent) method of determining the numerical
value of the power series aberration terms derived from first principles in Sections 3 and 4. Image
positions (a “spot diagram”) from 21 pupil points are sufficient to extract all 20 component terms
of power-sum (i + j ´ 1) ď 3, each free of contamination by any (other) term of power-sum ď6.
A numbered set of convenient grating pupil coordinate pairs used is:
˙
ˆ
˙
ˆ
˙
ˆ
˙
1
1
1
1
ω̆, 0
2 : 0, σ̆
4 : 0, ´ σ̆
3 : ´ ω̆, 0
5 : pω̆, 0q 6 : p0, σ̆q 7 : p´ω̆, 0q
2
2
2
2
˙
ˆ
˙
ˆ
1
1
ω̆, 0
14 : 0, σ̆
8 : p0, ´σ̆q 9 : pω̆, σ̆q 10 : p´ω̆, σ̆q 11 : p´ω̆, ´σ̆q 12 : pω̆, ´σ̆q 13 :
4ˆ
ˆ
˙
ˆ
˙
ˆ
˙
ˆ
˙
˙ 4 ˆ
˙
1
1
1
1
1 1
1
1 1
1
15 : ´ ω̆, 0
16 : 0, ´ σ̆
ω̆, σ̆
ω̆, ´ σ̆
17 :
18 : ´ ω̆, σ̆
19 : ´ ω̆, ´ σ̆
20 :
4
4
2 2
2 2
2
2
2
2
0 : p0, 0q 1 :
ˆ
where ω̆ and σ̆ denote the meridional and sagittal pupil half-widths, respectively. The difference
(or sum) of two image plane positions, each relative to the principal ray, is written succinctly as (for
`
˘ def
5´ def
17` def
example) x1 7 “ px1 5 ´ x1 0q ´ˆx1 7 ´ x1˙
0 “ xˆ1 pω̆, 0q ´ x1 ˙
p´ω̆, 0q, or (as another example) z1 19 “
1 1
1
1
def
ω̆, σ̆ ` z1 ´ ω̆, ´ σ̆ ´ 2z1 p0, 0q. It is also found that 13 pupil
pz1 17 ´ z1 0q ` pz1 19 ´ z1 0q “ z1
2 2
2
2
points (nos. 0 through 12) are sufficient to extract the same 20 terms, however with a small amount of
contamination from (other) terms of power-sum ą 3.
def
def
1 “ x1 0 “ x1 p0, 0q; z1 “ z1 0 “
The first two terms are simply the principal ray positions: x10
01
1
z p0, 0q. The remaining 18 are the extrema aberrations (∆ being the P-V variation in the lateral ray image
positions over the rectangular grating aperture of ω “ ˘ω̆ and σ “ ˘σ̆). Each of these “extraction”
Equations (75)–(92) are given in both 21-ray form and (after the “–” sign) 13-ray form, followed by
an identification of the lowest power contaminant term(s) for the latter. The indicated percentage
level of contamination was determined, at the highest scanned wavelength (λ{λo “ 8) of the example
monochromator, by comparing the results at two different values of the aperture in each direction,
thus exposing its (distinct) power dependence, namely ω̆ i´1 σ̆ j for ∆xij1 and ω̆ i σ̆ j´1 for ∆z1ij .
∆z102 “
¯ 8
1 ´ 1 6´
1 6´
2´
14´
2´
– z1 4 ´ z1 8 ; ∆z106 „ 0.00001%
z 8 ´ 40 z1 4 ` 256 z1 16
45
3
3
¯ 8
1 ´ 1 6´
1 6´
2´
14´
2´
1
x 8 ´ 40 x1 4 ` 256 x1 16
– x1 4 ´ x1 8 ; ∆x15
„ 0.00003%
45
3
3
¯ 8
1 5´
1 ´ 1 5´
1´
13´
1´
– z1 3 ´ z1 7 ; ∆z151 „ 0.000006%
z 7 ´ 40 z1 3 ` 256 z1 15
∆z111 “
45
3
3
¯ 8
1 ´ 1 5´
1 5´
1´
13´
1´
1
1
∆x20
“
x 7 ´ 40 x1 3 ` 256 x1 15
„ 0.0020%
– x1 3 ´ x1 7 ; ∆x60
45
3
3
1
∆x11
“
(75)
(76)
(77)
(78)
Photonics 2016, 3, 3
∆z103 “
30 of 44
¯ 8
1 ´ 1 6`
1 6`
2`
14`
2`
z 8 ´ 80 z1 4 ` 1024 z1 16
– z1 4 ´ z1 8 ; ∆z107 „ 0.00013%
90
3
6
¯ 1´
¯ 1´
¯
8 ´ 1 17´
19´
11´
11´
9´
9´
–
z 18 ` z1 20
z1 10 ` z1 12
z1 10 ` z1 12 ; ∆z114 „ 0.00005%; ∆z132 „ 0.096%
´
3
6
2
¯
¯
¯
´
´
´
1 “ 8 x1 17´ ` x1 19´ ´ 1 x1 9´ ` x1 11´ – 1 x1 9´ ` x1 11´ ; ∆x1 „ 0.03%; ∆x1 „ 0.0001%
∆x21
20
18
10
12
10
12
23
41
3
6
2
∆z112 “
¯ 8
1 ´ 1 5`
1 5`
1`
1`
13`
z 7 ´ 80 z1 3 ` 1024 z1 15
– z1 3 ´ z1 7 ; ∆z161 „ 0.00003%
90
3
6
¯ 8
1 5`
1 ´ 1 5`
1`
13`
1`
1
1
∆x30
“
x 7 ´ 80 x1 3 ` 1024 x1 15
„ 0.0016%
– x1 3 ´ x1 7 ; ∆x70
90
3
6
¯ 4
8 1´
4´
1´
13´
5´
5´
– z1 7 ´ z1 3 ; ∆z151 „ 0.006%
34 z1 3 ´ 64 z1 15 ´ z1 7
∆z131 “
9
3
3
¯ 4
8 1´
4´
1´
13´
5´
5´
1
1
34 x1 3 ´ 64 x1 15 ´ x1 7
∆x40
„ 0.30%
– x1 7 ´ x1 3 ; ∆x60
“
9
3
3
¯ 8
1 6`
1 ´ 1 6`
2`
2`
14`
1
1
x 8 ´ 80 x1 4 ` 1024 x1 16
– x1 4 ´ x1 8 ; ∆x16
ă 0.0001%
∆x12
“
90
3
6
¯
¯ 1´
16 ´ 1 17´
19´
2´
9´
10´
6´
z 20 ´ z1 18 ´ 2z1 4
z1 12 ` z1 11 ´ 2z1 8
´
∆z122 “
3
¯ 6
1 ´ 1 9´
10´
6´
1
z 12 ` z 11
–
´ z1 8 ; ∆z124 „ 0.002%; ∆z142 „ 0.10%
2
´
´
¯
¯
1 “ 16 x1 17´ ` x1 18´ ´ 2x1 2´ ´ 1 x1 9´ ` x1 10´ ´ 2x1 6´
∆x31
20
8
19
4
12
11
3
¯6
1 ´ 1 9´
10´
6´
1
1
1
1 „ 0.0012%
x 12 ` x 11
´ x 8 ; ∆x51 „ 0.18%; ∆x33
–
2
´
´
¯
¯
1 “ 16 x1 17´ ´ x1 19´ ´ 2x1 1´ ´ 1 x1 9´ ` x1 12´ ´ 2x1 5´
∆x22
20
3
7
18
10
11
3
¯6
1 ´ 1 9´
12
´
5
´
1 „ 15.9%; ∆x1 „ 0.73%
x 10 ` x1 11
´ x1 7 ; ∆x42
–
24
2
¯ 4
4´
8 2´
2´
14´
6´
6´
1
1
– x1 8 ´ x1 4 ; ∆x15
34 x1 4 ´ 64 x1 16 ´ x1 8
∆x13
“
„ 17.5%
9
3
3
¯ 4
8 2´
4´
2´
12´
6´
6´
– z1 8 ´ z1 4 ; ∆z106 „ 36%
34 z1 4 ´ 64 z1 14 ´ z1 8
∆z104 “
9
3
3
¯ 1´
¯
16 ´ 1 17´
20´
1´
9´
12´
5´
´
z 18 ` z1 19 ´ 2z1 3
z1 10 ` z1 11 ´ 2z1 7
∆z113 “
3
¯ 6
1 ´ 1 9´
12´
5´
1
–
z 10 ` z 11
´ z1 7 ; ∆z115 „ 1.34%; ∆z133 „ 0.12%
2
∆z121 “
(79)
(80)
(81)
(82)
(83)
(84)
(85)
(86)
(87)
(88)
(89)
(90)
(91)
(92)
1 „ 0.03% times the magnitude
The largest absolute contamination in the 13-ray extractions is ∆x41
1
of the ∆x21 geometric aberration. As shown in Figure 5, the latter results in a maximum spectral
aberration (at the longest wavelength) of ∆λ{λ „ 10´4 , thus the 13-ray extraction is in error by only
the negligible magnitude of „ 3 ˆ 10´8 . However, this contamination error was eliminated by use
of the 21-ray extraction when calculating the deviations (given in Table 2) between the numerical
(raytrace) extractions of Equations (75)–(88) and those calculated from first principles by the new
(rigorous) light-path expansions of Equations (41)–(54). There are no adjustable parameters in either
method, and thus no “fitting” of one to the other. The extremely small deviations shown in this
table can therefore only be the result of both accurate light-path equations and accurate raytracings.
For example, the raytracings illuminate the grating surface by the exact and centered rectangular
1
aperture used in the analytical equations (ω “ ˘ ω̆, σ “ ˘ σ̆), whereby ω̆ « φm {sinα (though
2
the exact non-linear equation is used) is a strong function of α as the grating is rotated. In addition,
the formulations take equal care in the horizontal and vertical directions, as both spatial aberrations
contribute to the spectral aberration. This is due both (directly) to the image tilt which includes the
Photonics 2016, 3, 3
31 of 44
vertical aberration term i´1,j`1 z1 in the rotational transformation of Equation (68), and (indirectly) to
the need for inclusion of the vertical reference image position ζ ij in Equation (37).
Equations (75)–(92) may be converted from extrema widths to power series coefficients by use of
`
˘
`
˘
the transformations: ij x1 “ ∆xij1 { pij ω̆ i´1 σ̆ j and ij z1 “ ∆z1ij { p ji ω̆ i σ̆ j´1 . Using Equations (68)–(70),
these terms were then converted to spectral aberrations ∆λij {λ and plotted in Figure 5 as the open
circles at 29 raytraced wavelengths. Note that, as formulated in Section 5, the (tilted) spectral
direction and the corresponding adjustment to the focusing condition cancels the effect of the first four
aberrations (Equations (41)–(44) or Equations (75)–(78)), therefore Figure 5 shows no power terms of
∆λ20 {λ or ∆λ11 {λ, as they vanish to within the accuracy of the calculations. The last four extraction
Equations (89)–(92) result in the 58 raytraced points for ∆λ22 {λ and ∆λ13 {λ each contributing less than
10´6 . Due to these exceedingly small magnitudes (below the physical diffraction limit) and the large
number of non-paraxial reference image coordinates required for their proper analytical expansion,
the explicit equations for these have not been derived.
8. Practical Considerations and Enhancements
8.1. Vertical Deflection of the Principal Ray (z101 )
The functional dependences of this off-plane deflection (Equation (9)) are most easily seen by
applying the small-angle (grazing incidence) approximation to the grating Equation (10):
µ cosθ » cosβ ´ cosα » 2γ2 p1 ´ ρq { p1 ` ρq
(93)
thus revealing that z101 {η in Equation (9) scales with „ γ2 tanθ. At very shallow graze angles (γ „ ½˝ )
for use in the X-ray, the deflection is less than ¼ mrad even at the largest rotation angles considered
here (θ “ 71.25˝ ), largely eliminating the need for vertical translation of the exit slit. However, z101 {η
grows to 5 mrad at γ „ 3˝ (soft X-rays) and to 60 mrad at γ „ 10˝ (extreme UV), being comparable to
or exceeding the image length due to astigmatism.
This deflection may be compensated by the addition of a mirror which adjusts the vertical direction
of the principal ray incident to the grating [15]. Absent such a mirror in the present “one-bounce”
configuration, a translating iris may be used to provide off-center illumination of the grating, thereby
maintaining a centered astigmatic image at the exit slit plane. In the case of the horizontal and vertical
object conjugates being coincident (a point-like source), the iris center must align with an off-plane
pupil coordinate σo determined by balancing Equations (9) and (41):
σo » ´ pµsinθq { p1 ` 1{ηq
(94)
Unfortunately, this would not also cancel the off-plane direction of the diffracted rays, which
would continue to cause an off-center illumination of any target significantly downstream of the
exit slit.
The iris translation requires an incident beam of vertical angular extent Φs ą (φs ` σo ), being
4 mrad ` 6 mrad for the example monochromator. In addition, the lateral aberrations which scale to
powers of σ2 and higher will increase substantially as a result of the off-center grating illumination.
Fortunately, these terms are generally insignificant, the largest one affecting the spectral resolution
being horizontal (1,2) which is „ 2 orders of magnitude smaller than the dominant terms (see Figure 5).
8.2. Image Tilt
˘
˘
1` 2
1` 2
Using the grazing incidence approximation pµcosθq {sinβ »
α ´ β2 {β »
1{ρ ´ 1 β »
2
2
` 2
˘
1{ρ ´ 1 γρ{ pρ ` 1q “ ´ p1 ´ 1{ρq γ results in a simplified form of Equation (55):
ψ1 {γ » rpC2 ` 1q p1 ´ 1{ρq { p1 ` 1{ηqs tanθ
(95)
Photonics 2016, 3, 3
32 of 44
revealing that the tilt is nearly linear with the graze angle. Consequently, rotations of the exit and
entrance slits become less critical as the graze angle is reduced. For an X-ray version (γ „ ½˝ ), the
maximum value of ψ from Equation (61) is „ 1.3˝ and represents only „ 0.1 mm deviation at either
end of a 10 mm long entrance slit. Such a small tilt may be accomplished by simpler and more accurate
means than required for larger rotations.
8.3. Zero Order Overlap
The reflected zero order beam from a plane grating is unfocused, thus one must insure the exit slit
does not lie within the specularly reflected diverging beam of undispersed light [10]. Such overlap
is avoided if the meridional aperture φm ă 4γ pρ ´ 1q { pρ ` 1q, assuming ρ ą 1; in the case of ρ ă 1,
substitute its reciprocal “ 1/ ρ. To provide useable dispersion, ρ is typically ą 3/2 (or ă 2/3), for which
the aperture limitation becomes φm ă 0.8 γ. A tolerable (e.g., ă 40%) variation in graze angle across
the optic independently constrains φm ă „ 0.4 γ, thereby placing the exit slit far into the shadow of
the specular zero order light. The unfocused zero order is advantageous when using high peak-power
sources, as it avoids a destructively intense focus of the integrated spectrum.
8.4. Required Variation in Spacing
Inserting Equation (15) for N2 into Equation (6) determines the magnitude of this (dominant)
linear component to the fractional variation of spacing across the ruled width:
∆d
»
d
ˆ
φm
γ
˙
1 ` ρ2o {η
1 ´ ρo
(96)
For values of φm {γ “ 0.04–0.12 (e.g., 2–6 mrad at γ “ 3˝ ), ρo „ 1.5 and η „ 3–4, the required
variation is 12–36%. Surprisingly, this is not significantly larger than the 10–29% variations required for
a converging-beam geometry (η » ´1) and is well within the factor 2 variations previously employed
for 200 mm ruled width plane gratings in an astronomical spectrometer [15] (page 37). The sub-50%
line density variation required of the present grating design is comparable to those successfully
manufactured and provided in numerous („ 50) laboratory VLS spectrometer and monochromator
systems designed and constructed by the author over the past 30 years, and thus are commercially
available using existing fabrication methods.
8.5. Dispersive Resolution
Simple differentiation of the grating equation yields the contribution from (entrance; exit) slits of
width ∆x to the fractional FWHM resolution of any single-grating monochromator as
p∆λ{λqdispersive
1
»
γ
ˆ
∆x{r
|ρ ´ 1|
˙
p1; ρ{ηq
(97)
where η{ρ is the grating magnification. For the present typical values of ρ “ 2 and η “ 3.7, or their
reciprocals, p∆λ{λqdispersive „ 1.27(∆x{rq{γ. Thus, attainment of the average geometrical resolution
shown in Figure 5 (∆λ{λ „ 4 ˆ 10´5 or ∆λ „ 10´4 nm) requires ∆x/r „ 1.65 µrad at γ “ 3˝
(e.g., ∆x “ 5 µm, r “ 3 m and a monochromator length of p1 ` ηq r “ 14.1 m). The extremely
low grating geometrical aberration of ∆λ „ 10´5 nm near the first coma-corrected wavelength of
λ “ 1.26 nm matches only exceedingly narrow (∆x „ 1 µm) slits and a grating slope error tolerance of
„ 0.05 µrad; however the latter is attainable for plane surfaces.
8.6. Table-Top Version
At the other extreme from the ultra-high resolution performance displayed in Figure 5,
a high-throughput version is parameterized by choosing φ{γ “ 1/8, providing „ 10 times larger
solid aperture. For the same graze angle (3˝ ), the frontal aperture becomes 6.5 ˆ 13 mrad and
Photonics 2016, 3, 3
33 of 44
the resolving power is approximated by Equation (72) as R „3000. Given this lower resolution,
Equation (97) indicates a required physical length of only „ 2 m using the same slit widths („ 5 µm).
An extreme UV version (γ “ 8˝ ) of this “table-top” length monochromator would have a frontal
aperture of 17.5 ˆ 35 mrad and yield the same resolving power using slit widths of „ 15 µm.
8.7. X-Ray Version
For a given meridional aperture (∅m ) and ruled width („ r ∅m {sinγ), the object distance r 9 γ,
resulting in λ{∆λdispersive 9 γ2 from Equation (97). Therefore, at γ “ 1˝ (providing usable broadband
reflectance to „ 5 keV in photon energy), an aperture of ∅m “ 2 mrad and a grating width of 214 mm
requires r „ 1 m (a monochromator length „ 4.7 m) and yields λ{∆λdispersive „ 2800. This figure
matches the geometrical resolving power given by Equation (72).
Further reduction of the graze angle allows extension of the monochromator (same length and
grating size) into the X-ray region (λ À 0.25 nm). For example, γ „ ½˝ results in φm “ 1 mrad,
λ{∆λdispersive „ 1400 and a scanning range of λ „ 0.125–1 nm (ℏν „ 10–1.25 keV). It is noted that
the first use of a pure (single rotation axis) SNR monochromator was at high energies, with a concave
grating at γ “ 1˝ providing strong line spectra to 3 keV and first-order Bremsstrahlung continuum
to its scan limit („ 4 keV) as shown in “Figure 4-4” on page 133 of the cited thesis [15]. The present
design additionally benefits from recent advances in grating fabrication on smooth plane surfaces [4],
and thus should extend efficient broadband grating spectroscopy to photon energies ą 5 keV.
8.8. Further Refinements and Additions
The detailed equations and raytracing results presented in this initial report demonstrate the
fundamental spectral characteristics of the new geometry. Outlined here are a few possible variations
to the basic single-element configuration, which may offer advantages dependent upon the graze
angle, aperture, source size, performance priorities or desired versatility of the monochromator.
8.8.1. Linear Upstream Source
Inclusion of a nonzero value for z02 (an object point whose z-coordinate has a linear dependence
on σ), corresponding to an entrance slit illuminated by a distant horizontal line. Such a case is
allowed in the general light-path formulation (Section 3) and in Equations (32)–(34) of Section 4.1, but
not included in the current explicit lateral aberration expansions (Section 4.2). Per Section 5.4, this
longitudinal separation of the effective object plane in the vertical and horizontal directions may allow
the required rotations of entrance and exit slits to be replaced by a (third-axis) rotation of the grating.
8.8.2. A Two-Element Monochromator with Astigmatism Control
A mirror may be added to focus (anastigmatically) or collimate in the vertical direction, the latter
being previously used to maintain high spectral resolution without resort to a rotating slit in the case
of a pure surface-normal rotation monochromator, as given in Section 4.3 of the cited thesis [15].
8.8.3. Choice of Magnification
In Section 2.6, the conjugate distance ratio (η) and the VLS coefficient N3 were chosen to eliminate
coma (3,0) at two scan wavelengths. As the separation between these wavelengths vanishes, this
optimization eliminates coma and its first derivative, resulting in a maximally-broad region nearly free
of geometrical aberrations, centered on one wavelength. Alternatively, η may be chosen independently
(e.g. to provide a desired magnification = η{ρ), resulting in a narrower high-resolution region at the
one scan wavelength where coma is eliminated by choice of N3 .
Photonics 2016, 3, 3
34 of 44
8.8.4. Curved Grating Surface
A curvature of the grating surface, as already provided in the general light-path formulation
(Section 3). Generalizing the graze-angle light-path approximations of Equations (14) and (15) to
include a finite curvature radius (R) yields:
Fi0 »
#
r1 ´ pη{ρq { pRsinαqs ρ2 {η i´1 ` r1 ´ 1{ pRsinαqs p´1qi
Ni cosθ ´
ρ2 ´ 1
Ni »
r1 ´ pη{ρo q { pRsinαo qs ρ2o {η i´1 ` r1 ´ 1{ pRsinαo qs p´1qi
ρ2o ´ 1
+
µ cosθ
(98)
(99)
Combining the above two equations, the focusing condition (F20 “ 0) becomes:
ρ2 {η ` 1 ´ pρ ` 1q { pRsinαq
cosθ » 2
ρo {η ` 1 ´ pρo ` 1q { pRsinαo q
„
ˆ
ρ2o ´ 1
ρ2 ´ 1
˙
(100)
For a given in-plane scan parameter ρ, a radius which is positive (concave), but larger in magnitude
than that required for a constant line-space (N2 “ 0) grating to focus, requires an increase in θ, while
a negative (convex) radius requires a decrease in θ. From the approximate grating Equation (93),
these changes in θ result in a more extended scan range (µ{µo ) for a concave grating and a (slightly)
reduced scan range for a convex grating. An interesting case is when the radius is positive and smaller
in magnitude than the constant line-space self-focusing radius, the latter of which completes the
equations of this paper on a classical note:
RCLS
ρo ` 1
{sinαo
» 2
ρo {η ` 1
„
(101)
In the case of 0 ă R ă RCLS , the surface curvature provides too strong of a focusing power,
requiring the VLS to weaken this by a change in sign of N2 . The result is that the required value
of θ decreases sharply (for a given ρ). For the parameters of the example ultra-high resolution
monochromator, choosing R „ 0.85 RCLS is numerically found to minimize θ; at ρmax „ 2.73, θ „ 38o
rather than the plane grating requirement of θ „ 71o . However, this reduces the maximum scan
wavelength from „ 8 λo to „ 3.2 λo . Nonetheless, the advantage of this curved grating design is that
this θ is still significantly smaller than the value of „ 58˝ required by the plane grating design at the
same wavelength. Thus, for this (more limited) scan range, the curved grating results in a significant
decrease (factor „ 2) in the required slit tilts and „ 30% decrease in the vertical image deflection.
Equation (98) with i “ 3 provides three free parameters (η, N3 and R), potentially enabling the
2 “ 0) at an additional (third) scan wavelength or the cancelation of spherical
cancelation of coma (x30
2
aberration (x40 “ 0) at an additional (second) scan wavelength.
8.8.5. Extended-Range Configurations
Dimensionless Equations (14)–(16) reveal that the horizontal focusing (first, second and third
degree) and the VLS ruling parameters (Ni ) are nearly independent of graze angles. Therefore, the
same physical grating, object and image distances provide nearly the same imaging performance at
any small angular deviation 2γ. As with other self-focusing plane grating geometries, this requires
the in-plane (initial) value of the scan parameter (ρo ) to be unchanged, which is arranged by simply
re-adjusting the initial grating normal angle δo “ arctanrptanγq pρ ´ 1q { pρ ` 1q] per Figure 1.
For example several stationary exit slits at various horizontal locations could provide different
values of λo scaling with γ2 per Equation (93). Though a linear scaling of wavelength with graze
angle would maintain the blaze of phase-diffraction grooves and provide roughly equal reflection
coefficients from a gold surface for λ ă „ 10 nm, the γ2 scaling is near optimal for obtaining reasonable
Photonics 2016, 3, 3
35 of 44
(„ 0.2–0.9) reflectances over the entire 10-octave region (λ „ 0.125–120 nm) where grazing incidence
is of advantage. This enables construction of an ultra-wide range monochromator employing multiple
(5) exit slits, each providing a band spanning three octaves in wavelength:
2γ “ 1˝ (X-ray band, λ “ 0.0625–0.5 nm); reflectance ă 10% below 0.125 nm (θ “ 60˝ )
2γ “ 2˝ (Deep SXR band, λ “ 0.25–2 nm)
2γ “ 4˝ (SXR band, λ “1–8 nm)
2γ “ 8˝ (XUV band, λ “ 4–32 nm)
2γ “ 16˝ (EUV/FUV band, λ “ 16–128 nm)
A single amplitude reflection grating could provide a constant relative diffraction efficiency
and good suppression of higher spectral orders (|m| ą 1), or multiple blazed gratings could be
interchanged to provide a higher (peak) efficiency in each band. The slit rotations would scale with γ
(per Equations (55) or (95)), as would the dispersive limit to the resolving power (per Equation (97)).
If used at high spectral resolution, then the exact (γ-dependent) focusing condition of Equation (57)
would be enforced by adjustments to the angle θ for each band.
As an alternative to the use of multiple slits, a conventional plane pick-off mirror
(rotating-translating or rotating off-axis) could be inserted into the diffracted beam to direct each
angular deviation along the same exit axis. Though such a plane mirror does not alter the basic
focusing characteristics of the optical system, while further suppressing the higher spectral orders by
preferential absorption, such a configuration requires at least two reflections (reducing its efficiency).
8.8.6. A Two-Element Time-Compensated Monochromator
A tandem grating configuration can substantially cancel the groove-to-groove phase differences
defining wavefront division for an individual grating, thereby providing a common path-length
within the aperture of the emerging beam. The new rotation scheme enables the construction of such
a monochromator using only two plane grating reflections, eliminating the need for the additional
two concave mirrors employed with current state-of-the-art time-compensated monochromators [18].
9. Conclusions
At fixed conjugate distances and horizontal deviation angle (2γ), self-focused grazing-incidence
gratings defocus rapidly upon (groove-axis) in-plane rotation. This is due either to the use of a curved
(concave) surface or to the change in ρ (ratio of diffracted graze angle to incident graze angle) which
determines the focusing condition of plane gratings. However, if varied line-spacing (VLS) provides
the focusing power rather than surface curvature, strong defocusing also results from a surface-normal
(off-plane) rotation, which can therefore be used to cancel that from the in-plane rotation. These
concerted rotations also reinforce the change in wavelength, extending the scan range beyond that
available from a single rotation. A new scanning geometry and class of monochromator has thereby
been invented which requires only one plane surface, representing a terminal point in the progression
of grazing incidence monochromator designs towards fewer and simpler surfaces.
Small-angle approximations to the light-path equations for a plane grating provide a clear
understanding and classification of the essential spectral aberration characteristics for different
optical geometries. This simplifies the equations, clearly revealing the imaging properties which
are (nearly) independent of the graze angle. A more rigorous analysis of the imaging properties
exhibited by the new monochromator geometry has precipitated the (incidental) introduction of
two mathematical tools of more general applicability in aberration analysis; being an expansion
formulation based on non-paraxial reference points and an extraction of aberration series coefficients
from raytrace simulations:
(1) The small spectral aberrations and dominance of off-plane terms in the present monochromator
geometry has unveiled a flaw in standard light-path formulations. A mathematically rigorous general
light-path expansion theory has been introduced, which systematically employs reference wavefronts
Photonics 2016, 3, 3
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centered on non-paraxial image points. This procedure, while more complex than the standard
approach, correctly isolates each power-term of the path-length series. Interestingly, comparison of the
two formulations reveals that the standard approach does not in general provide the correct expansion
(or consequently the total aberration), being inexact even in the simple case of a spherical mirror.
(2) An accurate method has been introduced for extracting the individual geometrical terms from
the image positions of a small number (13–21) of numerically traced rays. For 14 of the 18 lateral
aberrations of power-sum ď 3, a detailed comparison between these extractions and the analytical
equations (derived from the new light-path formulation) shows essentially exact („ 10´11 radians)
agreement, being several orders of magnitude finer than the physical diffraction width. This precision
suggests that such extractions may be used in the future to infer (rather than derive) the Fermat
equations, by composing them from an algebraic template of geometric parameters fit to the raytrace
extractions. However, such numerical fitting has not been used in the present work, rather all equations
have been derived from first principles.
Initial use of a meridional-only approach (based on the standard theory and the
graze-angle-invariant approximation) and final use of the rigorously-developed (non-paraxial image
reference) light-path formulation, have provided equations for various focusing conditions (horizontal,
spectral and least-confusion spectral) by correlation of the two grating rotations. The theoretical
analysis has also derived the required exit slit tilt (ψ1 ) and entrance slit tilt (ψ) as functions of the
scan parameters, the VLS ruling coefficients (Ni ) for self-focusing and correction of the higher-degree
meridional aberrations, an optimized value for the mount parameter (η ” r1 {r) which provides
cancelation of the higher-order (3,0) meridional aberration at two chosen wavelengths and explicit
expansion equations of ultra-high accuracy for the 14 leading lateral aberration terms.
The spectral resolving power may be approximated by p7 γ{φq2 23´q over a scan range of q
octaves for a collection aperture of φ (meridional) ˆ 2φ (sagittal) radians. This large product of spectral
resolution, solid aperture and scan range is due to the high level of aberration-correction provided by
this geometry (particularly the cancelation of coma at 2 wavelengths) combined with the multiplicative
change in wavelength provided by the two rotation motions. Such a high figure of merit may be
parameterized between these 3 components (resolution, aperture and scan range) as desired, with
examples given for an ultra-high resolution (λ{∆λ „ 25,000) soft X-ray “beamline” version (14 m
length), a high collection aperture “table-top” version (2 m length) and an X-ray version (γ „ ½˝ ).
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A. A Simple Illustration
Figure A1 compares the two light-path calculations (“standard” and “rigorous”) of spherical
aberration (4,0) vs. magnification for the simplest and most common focusing optic, namely
a spherically concave mirror.
Photonics 2016, 3, 3
37 of 44
Figure A1.
Calculated aberrations of a spherical concave mirror at the Gaussian focus.
The second-degree (“coma”) lateral ray aberration is correctly calculated by either the standard or
the rigorous light-path formulations; the open circles are independent extractions of this term from
numerical ray tracings. Light-path calculations of the third-degree (“spherical aberration”) lateral
ray aberration are compared for the standard (dashed), semi-standard (dot-dashed) and rigorous
(solid) formulations, with the open circles being independent extractions of this term from numerical
raytracings. In this example, the object distance is 1000 mm, the graze angle is 10˝ and the acceptance
aperture is 20 mrad.
2/ =
(1 + 1/ )
The first-degree lateral aberration (“defocusing”) vanishes for a mirror curvature 2{R “
p1 ` 1{ηq sin2 γ, where γ is the graze angle. Given this constraint, there is no aberration (non-paraxial
position) to include as the reference image for calculating the next higher aberration (second-degree
3 the standard
1
“coma”), so the latter result is the −
same as given
formulation:
= −by 1
−
3
sinγ
“´
´ 30 x1
η
4
4
ˆ
˙
1
1 ´ 2 cosγ sin2 γ
η
(A1)
Given the same in-focus constraint, the expansion term for spherical aberration simplifies to:
´ 40 x1
where,
sinγ
“ e1 sin4 γ ` e2 cos2 γ sin2 γ
η
1
e1 “ ´
8
and,
e2 “
1
8
˙
ˆ
1
1
1
1´ ´ 2 ` 3
η η
η
˙
ˆ
9
5
5
9´ ´ 2 ` 3
η η
η
(A2)
(A3)
(A4)
in the standard formulation (paraxial image reference).
However, to formulate 40 x1 rigorously, one must use Equations (19), (22), (24), (26) and (29) with
xij “ zij “ 0 (on-axis point source), α “ β “ γ (mirror), σ “ 0 (no sagittal rays), S “ 1 (spherical surface)
and the inclusion therein of the existing lateral ray aberration from Equation (A1) as a non-paraxial
`
˘
image reference point ξ40 “ 30 x1 ω 2 . The resulting value of e1 is unchanged compared to the standard
Photonics 2016, 3, 3
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result, due to only the first term of Equation (A2) surviving when γ “ π{2 (normal incidence), and
in this case there is no 30 x1 aberration and thus only the standard paraxial image point. However,
the rigorous formulation yields a different value for e2 . In addition to using 30 x1 as the (non-paraxial)
reference image point and employing the Fermat differential of Equation (26), the grating curvature
˘
`
˘
3`
1 ´ 1{η 2 p1 ` 1{ηq cos2 γ sin2 γ to the
term of Equation (29) is significant, adding 30 x1 {R cot γ “
4
value of ´p40 x1 {ηq sinγ. The result is:
1
e2 “
8
ˆ
˙
7
5
3
9` ´ 2 ´ 3
η η
η
(A5)
which equals Equation (A4) of the standard expansion only at η “ 1, where 30 x1 vanishes for all
γ. In the more general case (η ‰1 and γ ‰ π{2q, 30 x1 is nonzero and Equation (A5) differs from
Equation (A4). As plotted in Figure A1, this discrepancy is clear and seen to be independently
confirmed by the overlayed data points from numerical raytrace extractions. Due to the high angular
aperture (20 mrad), the “uncontaminated” (21-ray) version of Equation (85), employing seven rays
across the pupil meridian at each magnification, was used to insure an accurate extraction of 40 x1 .
This results in essentially exact agreement („ 0.000005 mm at all but the two lowest magnification
points) with the rigorous expansion.
Salient specific results of potential interest for optical designers are apparent from Figure A1:
(1) The standard formulation under-calculates spherical aberration for magnifications ą1, while
it significantly over-calculates this term for magnifications ă 1; (2) Spherical aberration vanishes
at η „ 0.7, which for grazing angles is very near the root of the cubic equation given by e2 “ 0.
Including the e1 term, the exact value of this “sweet spot” is at η “ 0.697, 0.697, 0.704, 0.718 and 0.740
at respective graze angles of 3˝ , 10˝ , 45˝ , 60˝ and 70˝ , the latter two representing a “near-normal
incidence” condition; and (3) The standard formulation (dashed curve) incorrectly identifies 40 x1
(“spherical aberration”) as comparable to 30 x1 (“coma”) at low magnifications. Figure A1 shows
that the extremum (P-V) of 40 x1 remains significantly smaller than that of 30 x1 even at the very low
magnification of η “ 0.125. Figure A1 also shows the result of spherical aberration calculations
using a semi-standard formulation (dashed-dot curve), whereby the trailing term of Equation (29)
is included, but the paraxial reference image is (erroneously) maintained. The equation for e2 then
˘
`
becomes 12 ´ 2{η ´ 8{η 2 ` 6{η 3 {8.
Compared to the classic horizontal focusing mirror example given above, the errors inherent
in the standard light-path formulation are more significant for a surface-normal rotated diffraction
grating, and increase further in the presence of varied line-spacing. Firstly, such a geometry gives rise
to numerous mixed terms, especially large being those of (i,1), and thus a proliferating 2D power matrix
of non-paraxial reference image points. Secondly, the image rotation (1,1) terms cause a transfer of
(otherwise insignificant) vertical aberrations into rotated slit-normal (spectral) components. Thirdly, the
inclusion of vertical aberrations as image reference coordinates in the rigorous formulation of horizontal
light-path terms (and vice versa) requires that even non-dominant aberrations be initially calculated
accurately to avoid significant errors in the subsequently expanded higher-degree terms. For the above
reasons, the standard formulation does not provide an accurate prediction of the aberrations for the
new design, or for other off-plane diffraction mountings in non-stigmatic geometries. As specific
examples, refer to the discussion in Section 4 and the Table 2 entries for the horizontal aberrations x1 20
and x1 21 , which are dominant terms of the new optical design and thus require accurate formulation.
Appendix B. A Rigorous Algebraic Expansion in Detail: The (2,1) Horizontal Aberration
From Equation (29):
´
21 x
+
#
„
pλq
BA21
BB21
sinβ “ h 2µN21 `
`
η
B p∆ωq B p∆ωq ωσ coe f f icient
1
(B1)
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where, from Equation (5)
N21 “ 3N3 cos2 θsinθ
(B2)
BA21
“0
B p∆ωq
(B3)
from Equation (38):
and,
BB21
“
B p∆ωq
ˆ
cosβ ´
ξ21
ω
sinβ ´
η
η
˙
˙ˆ
38 2
58 3
1
´1 ` 8
t
´
t
t
`
2 B 21 8 B 21 16 B 21
(B4)
from Equation (35) in which, from Equation (37):
8
B t21
“ ´2
ˆ
ωcosβ
η
˙
`2
ˆ
ξ21 sinβ
η
˙ˆ
ω
η
˙
ˆ
´2
ζ 21
η
ˆ
˙ˆ ˙ ˆ
˙
˙
ζ 21 2
σ
ξ21 2
`
`
η
η
η
(B5)
where the reference image terms (ξ21 and ζ 21 ) and their products are derived from Equation (23) as:
ξ21 sinβ
sinβ
“
η
η
“´
„
ÿ
IJx
1
ω
p I,J q‰p2,1q
I ´1 J
σ “
ˆ
20 x
1 sinβ
η
˙
ω`
ˆ
11 x
1 sinβ
η
˙
σ
(B6)
ˆ
˙
sin2 β
1 2
2
2
2
3
2
2
` 2µN2 cos θ ` pµ sin θq 2N2 cosβ cosθ ` sin α ` µ N2 sin θ cos θ ω
η „
2
ˆ
˙
`
˘
1 2 2
2
2
2
´ pµsinθq 2N2 cosθ 1 ` µ sin θ ` 1 ` 2µN2 sin θ ` µ sin θ cosβ σ
2
(B7)
„
ˆ
˙
2β
ξ21
ω
cos
1
´
sinβ ´
“ sin2 α ´
` µC2 cosθ ` pµ2 sin2 θq C2 cosβ ` sin2 α ω
η
η„ ˆ
η ˙
2
(B8)
`
˘
1 2 2
` pµsinθq C2 1 ` µ sin θ ` 1 ` µC2 sinθ tanθ ` µ2 sin2 θ cosβ σ
2
ˆ 1˙
ˆ 1˙
ˆ 1˙ ˆ 1˙
ÿ
ζ 21
z
1
02 z
11 z
01 z
1 I J ´1
`
σ`
ω ` 12
ωσ
(B9)
“
“
IJz ω σ
η
η
η
η
η
η
sin2 α `
p I,J q‰p2,1q
˙ „ˆ
ˆ
˙
1
3
3
1
“ pµsinθq 1 ` µ2 sin2 θ ` 1 `
` C2 µsinθ tanθ ` µ2 sin2 θ ` C2 µ3 sin3 θ tanθ σ
2 ˙
n
2
ˆ
˙
2
„ˆ
3
1 cosβ
cosβ
3
3
µsinθ `
µ sin θ ω
C2 ´
` C2 ´
η
2
2 ˆη
" ˆ
˙
˙
cosβ
1
` pµsinθ tanθq 2C3 ´ C2 cosβ
` ´ cosα `
ηˆ
η
˙
*
„
´
¯ ˆ1˙
` 2 2 ˘
cosβ
3
2
pC
cosα `
` Q C2 ` C2 cosβ `
`
cosβq
ωσ
` µ sin θ 3C2 ´
2
2
η
ρ2
ˆ
ζ 21
η
˙2
“
ˆ
˙
˙
„ˆ
`
˘
˘
1 ` 3 3 ˘
2
pµsinθq ` p2C2 q µ2 sin2 θ ptanθq ` 4 `
µ2 sin2 θ ` 2 `
µ sin θ σ
n
„ˆ
˙n
cosβ ` 2 2 ˘
` 2C2 ´ 2
µ sin θ ω
˙η
ˆ
˙
"„ˆ
4
2
cosα
2
C2 ` ´ ´ 2 ´ 2
cosβ pµsinθq
`
2`
η
η ˙η
cosβ
ˆ
` 2 2 ˘
4
2
` 4C3 ` 2C2 ´ C2 cosβ µ sin θ ptanθq
ηˆ
˙
˙
„ˆ
´
¯
1
8
3
C2 ´
` 2 cosβ ´ 4cosα ` 2Q C2 2 ` C2 cosβ
` 12 `
ˆ ˙ η
η η
*
` 3 3 ˘
2
`
pC
`
cosβq
ωσ
µ
sin
θ
2
ρ2
(B10)
`
(B11)
Using the substitution
µcosθ
“
sin2 β
ˆ
f β cosβ ´ cosα
sin2 β
˙
”Q
(B12)
Photonics 2016, 3, 3
and expanding
«ˆ
ξ21
η
˙2 ff
40 of 44
ˆ
ξ21
η
˙2
to include all the terms in µ4 {sin2 β, which then become terms in µ3 :
˙
´
¯
2
2
2
pC
`
cosβq
`
2C
`
2C
cosβ
Q pµsinθq
`
2
2
2
ρ2 η
„ˆ
˙
´
¯ `
˘
2
2
2
2
`
`
C2 cosβ ` 4C2 cosβ ` 2C2 cos β Q µ2 sin2 θ ptanθq
2
ρ „ˆ η
˙
ˆ
˙
´
¯
3
2
1
2
2
C
`
cosβ
`
2C
`
3C
cosβ
Q
`
`
`
2
2
2
ρ2 η ´
ρ2 ¯η
ı`
`
˘
˘
` 2QC2 2 cos2 β tan2 θ
µ3 sin3 θ
“
ωσcoe f f .
„ˆ
ˆ
„ ˆ
˙
˙
`
˘
2C2
2C2 2cosβ
pµsinθq ´
`
“ ´
cosβ µ2 sin2 θ ptanθq
η
η
ˆ η
˙
2cosβ C2 ` 3 3 ˘
´
µ sin θ ωσ
`
η
η
ˆ
˙ˆ ˙
„
µ3 sin3 θ
ζ 21
σ
2
´2
“ ´ pµsinθq ´
σ
η
η
˙η
ˆ
˙
„ˆ η
2C2
cosβ
3C2 cosβ ` 3 3 ˘
`2 2
` 2
µ sin θ ωσ
pµsinθq ` ´
` ´
η
η
η
η
2
ˆ
ˆ
ζ 21
η
ξ21 sinβ
η
˙2
˙ˆ
ω
η
˙
˙
ˆ
˙
„ˆ
`
˘
`
˘
1 ` 3 3 ˘
2
pµsinθq ` p2C2 q µ2 sin2 θ ptanθq ` 4 `
µ sin θ σ
“ µ2 sin2 θ ` 2 `
n
˙ n
„ˆ
cosβ ` 2 2 ˘
µ sin θ ω
` 2C2 ´ 2
"„ˆ
˙η
ˆ
˙
2
4
2
cosα
`
2`
C2 ` ´ ´ 2 ´ 2
cosβ pµsinθq
η
η ˙η
cosβ
ˆ
`
˘
4
` 4C3 ` 2C2 2 ´ C2 cosβ µ2 sin2 θ ptanθq
ηˆ
˙
˙
„ˆ
´
¯
8
1
3
C2 ´
` 2 cosβ ´ 4cosα ` 2 C2 2 ` C2 cosβ Q
` 12 `
ˆ ˙ η
η η
*
` 3 3 ˘
2
pC
`
`
cosβq
ωσ
µ
sin
θ
2
ρ2
(B13)
(B14)
(B15)
(B16)
Grouping like power terms together:
t21 “ t21;002 µ2 ` t21;100 ω ` t21;102 ω µ2 ` t21;011 σ µ ` t21;012 σ µ2 ` t21;013 σ µ3
`t21;111 ω σ µ ` t21;112 ω σ µ2 ` t21;113 ω σ µ3
where,
t21;002 “ sin2 θ
˙
ˆ
cosβ
t21;100 “ ´2
η
ˆ
˙
cosβ
t21;102 “ 2 C2 ´
sin2 θ
η
t21;011 “ 2sinθ
t21;012 “ 2C2 tanθ sin2 θ
t21;013 “ 4sin3 θ
˙
ˆ
˙
ˆ
„´
¯
cosα
2
4
2
´
2
´
cosβ
sinθ
t21;111 “ 2C2 2 ` 2C2 cosβ Q ` 2 ` 2 C2 `
cosβ
ρ
ρ2 η
„
ˆ
˙
´
¯
2
4
t21;112 “ 4C3 ` 2C2 2 ` 4C2 2 cosβ ` 2C2 cos2 β Q `
C
cosβ
tanθ sin2 θ
´
2
ρ2 η
(B17)
(B18)
(B19)
(B20)
(B21)
(B22)
(B23)
(B24)
(B25)
Photonics 2016, 3, 3
41 of 44
t21;113
From Equation (B4):
q
„ˆ
˙
ˆ
˙
5
8
4
´
12 ` 2 C2 `
cosβ ´ 4cosα
ρ
ρ2 η
´
¯
` 4C2 2 ` 5C2 cosβ ` 2C2 2 cos2 β tan2 θ Q sin3 θ
“
(B26)
BB21
“
B p∆ωq ωσ term
y
ω µ ` s21;102 ω µ2 ` s21;011 σ µ ` s21;012 σ µ2 ` s21;013 σ µ3 ˆ
„
s s21;000 ` s21;100 ω ` s21;101
1
t
µ2 ` t21;100 ω ` t21;102 ω µ2 ` t21;011 σ µ ` t21;012 σ µ2 ` t21;013 σ µ3 ` t21;111 ω σ µ
2 21;002
(
`t21;112 ω σ µ2 ` t21;113 ω σ µ3
3
´ t21;002 µ2 ` t21;100 ω ` t21;102 ω µ2 ` t21;011 σ µ ` t21;012 σ µ2 `t21;013 σ µ3
8
(2
`t21;111 ω σ µ ` t21;112 ω σ µ2 ` t21;113 ω σ µ3
5
t
µ2 ` t21;100 ω ` t21;102 ω µ2 ` t21;011 σ µ ` t21;012 σ µ2 ` t21;013 σ µ3
`
16 21;002
{
(3
`t21;111 ω σ µ ` t21;112 ω σ µ2 ` t21;113 ω σ µ3 ` . . .
ω,σ and ωσ terms
(B27)
where,
s21;100
s21;000 “ cosβ
˙
ˆ
cos2 β
2
“ sin α ´
η
s21;101 “ C2 cosθ
(B28)
(B29)
(B30)
1
s21;102 “ pC2 cosβ ` sin2 αqsin2 θ
2
(B31)
s21;011 “ pC2 ` cosβq sinθ
(B32)
s21;012 “ C2 cosβ tanθ sin2 θ
˙
ˆ
1
C2 ` cosβ sin3 θ
s21;013 “
2
BB21
“
B p∆ωq ωσcoe f f icient
ˆ
˙
1
3
1
1
s21;000 t21;111 ´ s21;000 t21;100 t21;011 ` s21;100 t21;011 ` s21;011 t21;100 µ
2ˆ
4
2
2
˙
1
1
1
3
1
s21;100 t21;012 ` s21;101 t21;011 ` s21;000 t21;112 ` s21;012 t21;100 ´ s21;000 t21;100 t21;012 µ2
`
2
2
2
4
ˆ2
1
3
3
3
`
s
t
´ s
t
t
´ s
t
t
´ s
t
t
2 21;000 21;113 4 21;000 21;002 21;111 4 21;000 21;100 21;013 4 21;000 21;102 21;011
1
1
1
15
` s21;000 t21;002 t21;100 t21;011 ` s21;100 t21;013 ` s21;101 t21;012 ` s21;102 t21;011
8
2
2
2
˙
1
3
1
3
´ s21;100 t21;002 t21;011 ` s21;011 t21;102 ´ s21;011 t21;002 t21;100 ` s21;013 t21;100 µ3
4
2
4
2
(B33)
(B34)
„
(B35)
Photonics 2016, 3, 3
´ 21
x1 pλq
sinβ
η
42 of 44
„
1
3
“ 6N3 cos2 θ sinθ ` s21;000 t21;111 ´ s21;000 t21;100 t21;011
2
4
1
1
` s21;100 t21;011 ` s21;011 t21;100 µ
2
„2
1
1
1
1
` s21;100 t21;012 ` s21;101 t21;011 ` s21;000 t21;112 ` s21;012 t21;100
2
2
2
2
3
2
´ s21;000 t21;100 t21;012 µ
„4
3
1
` s21;000 t21;113 ´ s21;000 t21;002 t21;111
2
4
3
3
´ s21;000 t21;100 t21;013 ´ s21;000 t21;102 t21;011
4
4
1
1
15
` s21;000 t21;002 t21;100 t21;011 ` s21;100 t21;013 ` s21;101 t21;012
8
2
2
1
3
1
` s21;102 t21;011 ´ s21;100 t21;002 t21;011 ` s21;011 t21;102
2
4
2
1
3
´ s21;011 t21;002 t21;100 ` s21;013 t21;100
4ˆ
2
˙ˆ
1
3
1 2
2
sin θ
6N3 cos θ sinθ ` s21;000 t21;111 ´ s21;000 t21;100 t21;011
`
2
2˙
4
1
1
` s21;100 t21;011 ` s21;011 t21;100
µ3
2
2
(B36)
yielding Equation (47) as it appears in the main text:
sinβ
“
´21 x1
η
„
ˆ
˙
˙
ˆ
˙
ˆ
1
1
1
cosβ
2C3 ` 1 ´ ` QC2 ` 2 C2 cosβ ` QC2 ` 2 cos2 β ´ cosα `
cosβ µ sinθ
η
ρ
ρ
"
ˆ
˙
η
*
”´
¯
1
1
1
2
2
2
` 2C3 ` C2 cosβ `
´ ` 2QC2 ` Qcosβ C2 cos β tan θ µ2 sinθ cosθ
` C2 `
Qρ2
ρ2 η ˙
ˆ
˙
„
ˆ
1
1
3
7
QC2 ` 2 cos2 β
` QC2 ` 2 C2 cosβ `
` C3 ` 2C2 2 `
2 ˙
2
ρ
ρ
ˆ
´
¯
cosβ
cosβ ` QC2 2 cos3 β tan2 θ µ3 sin3 θ
´ cosα `
η
(B37)
Appendix C. Cancelation of Vertical Deflection: 3-Axis Grating Rotation
Additionally rotate the grating about its third axis, namely that defined in Figures 1 and 2b by the
meridional coordinate ω. From Equation (9) it is approximated that such rotation by an angle
Ω » arcsin
ˆ
´µ sinθ
sinα ` sinβ
˙
(C1)
will correct for the principal ray vertical deflection (z1 01 ). This new scan angle is plotted in Figure C1
for the ultra-high resolution monochromator parameterized in the paper. Also plotted are the scan
angles required of the exit slit (ψ1 ) and entrance slit (ψ) re-oriented by subtracting Ω from Equations
(55) and (61), respectively. Using the longest scan wavelength (λmax /λo = 7.9945343) as the most
sensitive test, these approximations yield Ω » 4.1488˝ , ψ » ´12.158˝ and ψ1 » 4.3898˝ .
Photonics 2016, 3, 3
43 of 44
!
Ω ≃ 4.1488 , 𝜓 ≃ −12.158
!
. Scan angles of the grating (𝜹, 𝜽, 𝛀) and slits (−𝝍, 𝝍! ) for a 3-axi
1
Figure C1. Scan angles
𝑧 ! of the grating (δ, θ, Ω) and slits (´ ψ, ψ ) for a 3-axis grating rotation where
z101 = 0. Compare to the 2-axis grating rotation (Figure 4) where z101 ‰ 0.
Numerical raytracings have also been performed to independently, and more precisely, determine
the optimum 3-axis grating orientation at the scanned wavelength. This was conveniently provided by
the “auto-adjust” routine of BEAM4 [16] in a least-squares minimization of the spectral image width.
At λmax this numerical procedure yielded δ = 1.3951844˝ , θ = 71.40978˝ , Ω = 4.1525˝ , ψ = ´12.158˝ and
ψ1 = 4.4083˝ . For reference, the scan angles for the 2-axis grating rotation (Ω = 0) are δ = 1.3915105˝ ,
θ = 71.24507˝ , ψ = ´8.009˝ and ψ1 = +8.5386˝ , resulting in z1 01 = ´84.03 mm. In the 3-axis scan proposed
here, the fixed trajectory of the diffracted principal ray simplifies the exit slit rotation to be that about
a stationary axis (located at x110 = 0, z101 = 0). Full-aperture numerical raytracings also reveal spot
diagrams and spectral resolution of the undeflected images (z1 01 = 0) which are nearly identical to
those shown in Figure 6 (for which the vertical deflection at λmax is ~ 84 mm). Thus, the aberration
correction is maintained.
From a mechanical engineering standpoint, upgrading from a 2-axis to a 3-axis goniometer for
precise rotation of the grating is more compact and convenient than providing vertical translations of
the exit slit rotation axis and the downstream target. Note that both these translations would need to
be accurately coordinated with the basic 2-axis grating rotation. Furthermore, in the case of complex
or large target stations, it is often impractical to even consider such translation.
The proposed grating rotation about its meridional axis will also remove the vertical deflection
inherent in the (simpler) pure surface normal rotation monochromator [2]. Given rotation about both
its surface normal axis and its meridional axis, a constant line-space concave grating can focus on its
Rowland circle and its principal ray remains fixed in the horizontal and vertical directions.
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© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons by Attribution
(CC-BY) license (http://creativecommons.org/licenses/by/4.0/).