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arXiv:2607.13287

Exactly Solvable Diffraction-Grating Scattering Problems for TE and TM waves

Farhang Loran, Ali Mostafazadeh

physics.opticsmath-phquant-phSubmitted Jul 14, 2026• Announced Jul 16, 2026
v1 · Jul 14, 2026v2 · Jul 20, 2026
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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, $v(x,y):=iV_0(e^{i\mathfrak{K}\,y}-1)$ for $0\leq x\leq\ell$ and $v(x,y):=0$ otherwise, where $V_0, \mathfrak{K}$, and $\ell$ 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, $\hat\varepsilon(x,y):=1-v(x,y)/k^2$, 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.

Journal reference
Prog. Theor. Exp. Phys. 2026, 093A04 (2026)
DOI
10.1093/ptep/ptag147
Comments
34 pages, 4 figures

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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)v(x,y):=iV_0(e^{i\mathfrak{K}\,y}-1)v(x,y):=iV0​(eiKy−1) for 0≤x≤ℓ0\leq x\leq\ell0≤x≤ℓ and v(x,y):=0v(x,y):=0v(x,y):=0 otherwise, where V0,KV_0, \mathfrak{K}V0​,K, and ℓ\ellℓ 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, ε^(x,y):=1−v(x,y)/k2\hat\varepsilon(x,y):=1-v(x,y)/k^2ε^(x,y):=1−v(x,y)/k2, where kkk 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.
We provide an exact solution of the scattering problem for the potentials of the form $v(x,y)={\ensuremath{\chi}}_{a}(x)[{v}_{0}(x)+{v}_{1}(x){e}^{i\ensuremath{\alpha}y}]$, where ${\ensuremath{\chi}}_{a}(x):=1$ for $x\ensuremath{\in}[0,a]$, ${\ensuremath{\chi}}_{a}(x):=0$ for $x\ensuremath{\notin}[0,a]$, vj(x){v}_{j}(x)vj​(x) are real or complex-valued functions, ${\ensuremath{\chi}}_{a}(x){v}_{0}(x)$ is an exactly solvable scattering potential in one dimension, and $\ensuremath{\alpha}$ is a positive real parameter. If $\ensuremath{\alpha}$ exceeds the wave number kkk of the incident wave, the scattered wave does not depend on the choice of v1(x){v}_{1}(x)v1​(x). In particular, v(x,y)v(x,y)v(x,y) is invisible if v0(x)=0{v}_{0}(x)=0v0​(x)=0 and $k<\ensuremath{\alpha}$. For $k>\ensuremath{\alpha}$ and ${v}_{1}(x)\ensuremath{\ne}0$, the scattered wave consists of a finite number of coherent plane-wave pairs ${\ensuremath{\psi}}_{n}^{\ifmmode\pm\else\textpm\fi{}}$ with wave vector: ${\mathbf{k}}_{n}=(\ifmmode\pm\else\textpm\fi{}\sqrt{{k}^{2}\ensuremath{-}{[n\ensuremath{\alpha}]}^{2}},n\ensuremath{\alpha})$, where $n=0,1,2,...<k/\ensuremath{\alpha}$. This generalizes to the scattering of wave packets and suggests means for generating quantum states with a quantized component of momentum and pairs of states with an entangled momentum. We examine a realization of these potentials in terms of certain optical slabs. If $k=N\ensuremath{\alpha}$ for some positive integer NNN, ${\ensuremath{\psi}}_{N}^{\ifmmode\pm\else\textpm\fi{}}$ coalesce and their amplitude diverge. If kkk exceeds $N\ensuremath{\alpha}$ slightly, ${\ensuremath{\psi}}_{N}^{\ifmmode\pm\else\textpm\fi{}}$ have a much larger amplitude than ${\ensuremath{\psi}}_{n}^{\ifmmode\pm\else\textpm\fi{}}$ with n<Nn<Nn<N. This marks a resonance effect that arises for the scattered waves whose wave vector makes a small angle with the faces of the slab.
A medium described by an imaginary potential (or imaginary (refractive index) ), that varies sinusoidally in one direction, acts as a volume grating for plane waves incident on it obliquely or normally. Two peculiar features are identified. First, if the potential is weak, so that there are only two significant diffracted beams near the Bragg angle, and three for normal incidence, diffraction is strongly affected by degeneracies of the non-Hermitian matrix generating the `Bloch waves' in the grating; the effect of these degeneracies is very different from that of the Hermitian degeneracies for transparent gratings. Second, if the potential is strong and the grating thick, the asymptotic distribution of intensities among the diffracted beams (momentum distribution) is a rather narrow Gaussian, and dominated by a single set of complex rays; this is very different from the semiclassical limit for transparent gratings, where the rays form families of caustics proliferating with thickness, and with a wider momentum distribution.
We develop a one-dimensional theory of diffraction-induced transverse effects in nonlinear Fabry-P\'erot resonators addressed by finite-width incident beams. This is achieved in the framework of the coupled-mode analysis, which takes full advantage of the fact that nonlinear Fabry-P\'erot devices are resonant. As compared with a recently published theory dealing with the same subject [M. Haelterman, Opt. Commun. 75, 165 (1990); M. Haelterman, G. Vitrant, and R. Reinisch, J. Opt. Soc. Am. B 7, 1309 (1990); and G. Vitrant, M. Haelterman, and R. Reinisch, ibid. 7, 1319 (1990)], where the nonlinearity is introduced in an approximate way, the formalism developed here takes the nonlinearity associated with the optical Kerr effect rigorously into account. This feature has an important consequence: It leads to a theory that can be generalized to the TM case and also to anisotropic nonlinear media. The theory presented here is valid for any value of the angle of incidence. Under normal (or quasinormal) incidence, two counterpropagating modes, having the same absolute value of the wave-vector component parallel to the plane of the mirrors, are resonantly (or nearly resonantly) excited, whereas under oblique incidence, only one of these modes is excited.This allows us to point out that the feedback leading to optical bistability occurs along the direction of propagation of these resonantly excited counterpropagating modes of the nonlinear optical resonator. This explains why optical bistability disappears under oblique incidence. Although developed in the case of nonlinear Fabry-P\'erot resonators, this theory applies to a wide range of nonlinear optical resonators such as nonlinear prism couplers, nonlinear grating couplers, nonlinear interference filters, etc. The theory developed here applies whenever the excited electromagnetic field can be accurately described by a single mode or two counterpropagating modes of the nonlinear device. As an example, numerical results are given in the case of a nonlinear multiple-quantum-well Fabry-P\'erot-type structure of ${\mathrm{Al}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$Gax{\mathrm{Ga}}_{\mathit{x}}Gax​As/GaAs with ${\mathrm{Al}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$Gax{\mathrm{Ga}}_{\mathit{x}}Gax​As/AlAs mirrors.
The diffraction of a coherent, stationary, beam of atoms from a standing wave of light is considered. Particular attention is paid to long interaction times for which the motion of the atoms inside the light field must be taken into account. This is the regime of dynamical diffraction, and is treated within the framework of a set of differential difference equations due to Raman and Nath. The popular Raman-Nath approximation, in which the kinetic energy imparted to the atoms by the light wave is ignored, is no longer valid and the full equations must be solved. When the interaction between the atoms and the light is very weak then few beams are produced. To describe spontaneous emission from the atom one can invoke a complex potential as a model for dissipation. If the atom beam is obliquely incident one finds anomalously high transmission close to the Bragg angles which is the physical manifestation of degeneracies in non-Hermitian matrices. The classical limit is approached when the atom-light interaction is strong, producing many beams. The classical dynamics (geometrical rays) reveal that the farfield intensity pattern is dominated by caustics. These proliferate with increasing interaction distance. Two methods are presented for analytically treating the semiclassical limit of the wave mechanics. The first begins with W.K.B. solutions to the continuised Raman-Nath equations. Mapping onto parabolic cylinder functions removes the spurious divergences of the W.K.B. solutions, giving eigenvectors which are uniformly valid for all scattering angles and may be summed to give the wavefunction. The second method transforms the original W.K.B. eigensum, using the Poisson summation formula, giving a new sum, each term of which has classical significance. This method shows that wave theory decorates the classical caustics with an Airy function intensity profile, as predicted by catastrophe theory.