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references
The literature this implementation is built from, mapped to the part of the
project each one serves. Where two sources disagree, the resolution is recorded
in conventions.md §10 and summarised in
findings.md.
Heilmann, R. K., Huenemoerder, D. P., McCoy, J. A. & McEntaffer, R. L. Diffraction Gratings for X-ray Spectroscopy. Springer ISSI Scientific Reports (2024). arXiv:2409.02297
§2.1 gives the conical framework — the generalized grating equation, the wave-vector construction, the Kirchhoff scalar form. §4 gives the blazed reflection-grating scalar efficiency and the validity guards.
This is normative where it disagrees with any other source here: it is published, peer-reviewed, and self-consistent under Parseval.
McCoy, J. A. Scalar Treatment of Gratings, PhD thesis Appendix D, Pennsylvania State University.
The primary derivation, and the most detailed treatment of the closed forms:
square wave, sinusoid and sawtooth, with the off-plane substitution
λ → λ csc γ that lets one module serve both mounts. Four transcription errors
and one substantive difference from the ISSI chapter are recorded in
conventions.md §10.
McCoy, J. A. On X-ray Reflection, PhD thesis Appendix C.
Fresnel reflectivity, the soft-X-ray index of refraction, penetration depth, and §C.1.4 surface roughness — the basis for the Névot–Croce and Debye–Waller factors once the materials layer lands.
Harvey, J. E. & Pfisterer, R. N. Understanding diffraction grating behavior. Opt. Eng. 58(8), 087105 (2019); part II, 59(1), 017103 (2020).
A modern parametric / linear-systems treatment that handles conical diffraction natively. Not yet implemented; it is the natural basis for a second scalar backend to compare against the Kirchhoff one.
Born, M. & Wolf, E. Principles of Optics.
Kirchhoff diffraction theory and the Fraunhofer limit.
Pommet, D. A., Grann, E. B. & Moharam, M. G. Effects of process errors on the diffraction characteristics of binary dielectric gratings. Appl. Opt. (1995).
Benchmark cases, and directly relevant to profile-error sensitivity — the question of how much a measured deviation from an ideal profile changes efficiency.
Li, L. — Fourier factorization rules.
Non-negotiable for the RCWA backend. Without the correct factorization, TM metallic cases converge badly and cross-method plots would misrepresent RCWA rather than reveal anything about physics.
Dusséaux, R., Faure, C. & Chandezon, J. New perturbation theory of diffraction gratings and its application to the study of ghosts. J. Opt. Soc. Am. A 12(6), 1271 (1995).
From the group that originated the coordinate-transformation method.
Breidne, M. & Maystre, D. Variational theory of diffraction gratings and its application to the study of ghosts. J. Opt. Soc. Am. 72(4), 499 (1982).
Maystre, D. & Popov, E. Integral Method for Gratings, ch. 4 in E. Popov (ed.), Gratings: Theory and Numeric Applications, Institut Fresnel / CNRS / AMU (2012). Freely available.
The implementation guide — now the implemented guide. 59 pages, including
§4.5 on conical mounting and §4.6 on the numerical tooling. The
perfect-conductivity solver follows it directly; the anchors actually used:
eqs. (4.31)–(4.34) TE, (4.37)–(4.42) TM, (4.65) the conical invariance
theorem, (4.70)–(4.74) the equal-arc rectangular rule, (4.82)–(4.87) the
Kummer kernel acceleration, (4.95)–(4.100) the log-singularity split, and
Tables 4.1/4.2 as literature benchmarks (Table 4.1 is reproduced in
tests/test_integral_core.py). Caveats worth knowing before extending:
§4.6.5 on edges (the measured first-order TM corner convergence), §4.6.6 on
graded meshes (the fix). One transcription warning: the printed sgn/sign
placement in eqs. (4.33)/(4.40)/(4.41) did not survive PDF text extraction —
the implementation derives the kernels from the spectral form and pins every
sign with the flat-mirror and Table 4.1 tests instead.
Goray, L. I. & Schmidt, G. Solving conical diffraction grating problems with integral equations. J. Opt. Soc. Am. A 27, 585 (2010).
Conical plus integral equations is exactly the off-plane X-ray case. Goray is
the author of the standard commercial implementation; Schmidt supplied the
rigorous analysis. Implemented at M20 (conductivity="tabulated";
theory/integral.md §8) with the anchors: eq. (6) the
conical jump conditions, eqs. (15)–(21) the potential operators, eq. (23) the
coupled system (re-derived in project conventions — their normal points into
the metal and their potentials carry a factor 2, so transcription is a sign
trap; the flat-interface Fresnel tests pin every sign instead), §2.C the
energy balance, eq. (26) the absorption integral, §3 the numerical-option
note that V⁺∂ₜV⁻ may be discretised by numerical differentiation (the
option taken, via the spectral D_t matrix). Their comparison Tables 3/5/6
are usable validation data; Table 4 is a publisher's duplication of
Table 3 — see findings.md. Table 2's incident state needs
Li's polarization-angle convention (δ measured so tan δ = |E_s/E_p|, ψ the
relative phase — decoded from the Table 3/4 footnotes) and its deep lamellar
profile is corner-limited on the current mesh, so it is left to the
graded-mesh milestone.
Li, L. Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity (and the conical-mount companion data). J. Opt. Soc. Am. A 10, 2581 (1993).
The independent cross-method source behind Goray & Schmidt's comparison tables — what makes their Table 3 a genuine two-code anchor rather than a self-comparison.
Maystre, D. Analytic Properties of Diffraction Gratings, ch. 2 of the same volume.
Anomalies and analytic structure — directly relevant to the Rayleigh-anomaly
instability documented in findings.md.
Kalhor, H. A. & Neureuther, A. R. Effects of conductivity, groove shape, and physical phenomena on the design of diffraction gratings. J. Opt. Soc. Am. 62 (1972).
Early integral-equation treatment; useful for the perfectly-conducting limit.
McCoy, J. A. Modeling Diffraction Efficiency, PhD thesis Chapter 2 §2.2.
The same derivation in this project's notation: boundary value problem → Helmholtz with boundary conditions → Green's function → Dirichlet enforcement → the perfectly-conducting limit, with Σℰ = 1 proved via Green's theorem. §2.1 documents the ALS beamline 6.3.2 measurement method.
§2.2.3 is outdated and should not be ported as written — it describes the commercial code as internally multiplying by Fresnel reflectivity in perfect-conductivity mode, which contradicts the Σℰ = 1 result proved immediately above it. See
conventions.md§10, item 5.
Moharam, M. G. & Gaylord, T. K. — canonical RCWA test cases.
Li, L. — crossed-grating benchmarks.
These live in the visible and near-IR, which is why that regime is the
project's validation regime even though soft X-ray is the application. See
roadmap.md.
Heuberger, G., Klepp, J., Guo, J., Tomita, Y. & Fally, M. Light diffraction from a phase grating at oblique incidence in the intermediate diffraction regime. Appl. Phys. B 127, 72 (2021).
Measured data on where scalar theory stops working — the empirical basis for a scalar validity map.
Tutt, J. et al. — diffraction efficiency testing of sinusoidal and blazed off-plane reflection gratings. J. Astron. Instrum. (2016).
Marlowe, H. et al. — polarization dependence at grazing incidence. The justification for treating soft-X-ray reflectivity as polarization-independent, which is why the reference corpus carries only TE.
McCoy, J. A. et al. — J. Vac. Sci. Technol. B (2018); ApJ 891, 114 (2020); OSA Continuum 3(11), 3141 (2020).
Villarrubia, J. S. Algorithms for scanned probe microscope image simulation, surface reconstruction, and tip estimation. J. Res. Natl. Inst. Stand. Technol. 102(4), 425–454 (1997).
The tip-deconvolution milestone (roadmap §4): morphological dilation/erosion, the certainty map that marks unrecoverable points, and blind tip estimation. Erosion bounds the true surface; it does not recover it.
PyXFocus — sequential X-ray raytracer (Allured; forks by kbuffo and this
project's author). Its grating.py holds the analytic Littrow-mount formulas
(blazeYaw, blazeAngle) slated for a native port (roadmap §3), and its GUI's
order-centroid/spot-width estimator is the instrument-level cross-check of
gratinglab.resolution. The raytracer itself is deliberately not absorbed.