Exactly Solvable Diffraction-Grating Scattering Problems for TE and TM waves
Abstract
In J.~Phys.~A {\bf 31}, 3493 (1998), Berry provides an analytic solution of the problem of determining the diffracted beam intensities for atoms incident upon a grating given by the potential, v(x,y):=iV0(eiK y−1) for 0≤ x≤ℓ and v(x,y):=0 otherwise, where V0, K, and ℓ are positive real parameters. This result, which relies on the paraxial approximation, readily applies to the diffraction of transverse electric (TE) waves by the nonmagnetic optical grating given by the relative permittivity, ε̂(x,y):=1−v(x,y)/k2, where k is the incident wavenumber. We show that Berry's grating belongs to a larger class of possibly magnetic diffraction gratings whose scattering problem for both TE and transverse magnetic (TM) waves are exactly solvable. Our analysis makes use of a recently developed dynamical formulation of stationary scattering that is based on the idea of mapping the scattering problem to the quantum dynamics generated by an effective non-Hermitian Hamiltonian operator. {We use this formulation to derive explicit analytic expressions for the diffracted beam amplitudes that are valid beyond paraxial approximation.} As a concrete example, we offer the exact solution of the scattering problem for TE and TM waves incident upon a generalization of Berry's grating.
Versions
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BibTeX
@misc{loran2026exactly,
title = {{Exactly Solvable Diffraction-Grating Scattering Problems for TE and TM waves}},
author = {Farhang Loran and Ali Mostafazadeh},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.13287},
eprint = {2607.13287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13287},
}
RIS
TY - UNPB
TI - Exactly Solvable Diffraction-Grating Scattering Problems for TE and TM waves
AU - Farhang Loran
AU - Ali Mostafazadeh
PY - 2026
DA - 2026/07/20
PB - arXiv
DO - 10.48550/arxiv.2607.13287
UR - https://arxiv.org/abs/2607.13287
AB - In J.~Phys.~A {\bf 31}, 3493 (1998), Berry provides an analytic solution of the problem of determining the diffracted beam intensities for atoms incident upon a grating given by the potential, v(x,y):=iV_0(e^iK y−1) for 0≤ x≤ℓ and v(x,y):=0 otherwise, where V_0, K, and ℓ are positive real parameters. This result, which relies on the paraxial approximation, readily applies to the diffraction of transverse electric (TE) waves by the nonmagnetic optical grating given by the relative permittivity, ε̂(x,y):=1−v(x,y)/k^2, where k is the incident wavenumber. We show that Berry's grating belongs to a larger class of possibly magnetic diffraction gratings whose scattering problem for both TE and transverse magnetic (TM) waves are exactly solvable. Our analysis makes use of a recently developed dynamical formulation of stationary scattering that is based on the idea of mapping the scattering problem to the quantum dynamics generated by an effective non-Hermitian Hamiltonian operator. {We use this formulation to derive explicit analytic expressions for the diffracted beam amplitudes that are valid beyond paraxial approximation.} As a concrete example, we offer the exact solution of the scattering problem for TE and TM waves incident upon a generalization of Berry's grating.
ER -