Low-Frequency Scattering of TE and TM Waves by an Inhomogeneous Medium with Planar Symmetry

TL;DR

This work develops a Dyson-series transfer-matrix framework for Bergmann's equation governing low-frequency scattering of TE and TM waves in planar-symmetric, inhomogeneous media. By mapping the problem to a non-Hermitian two-level Hamiltonian, the authors derive explicit low-frequency expansions for the transfer matrix and the reflection/transmission amplitudes, and they identify a generalized Brewster angle, conditions for transparency and reflectionlessness, and PT-symmetry implications. The analysis further extends to half-space geometries with Robin boundary conditions, yielding analytic expressions for the reflection amplitude and absorption in the low-frequency regime. The results provide robust, analytic tools for predicting and designing low-frequency wave behavior in both electromagnetic and acoustic planar structures, with potential extensions to more complex Bergmann-type problems.

Abstract

Stationary scattering of TE and TM waves propagating in an isotropic medium with planar symmetry is described by Bergmann's equation in one dimension. This is a generalization of Helmholtz equation which allows for developing transfer matrix methods to deal with the corresponding scattering problems. We use a dynamical formulation of stationary scattering to study the low-frequency scattering of these waves when the inhomogeneities of the medium causing the scattering are confined to a planar slab. This formulation relies on the construction of an effective two-level non-Hermitian quantum system whose time-evolution operator determines the transfer matrix. We use it to construct the low-frequency expansions of the transfer matrix and the reflection and transmission coefficients of the medium, introduce a generalization of Brewster's angle for inhomogeneous slabs at low frequencies, and derive analytic conditions for transparency and reflectionlessness of PT-symmetric and non-PT-symmetric slabs at these frequencies. We also discuss the application of this method to deal with the low-frequency scattering of TE and TM waves when the carrier medium occupies a half-space and the waves satisfy boundary conditions with planar symmetry at the boundary of the half-space. Because acoustic waves propagating in a compressible fluid with planar symmetry are also described by Bergmann's equation, our results apply to the low-frequency scattering of these waves.

Figures (2)

• Figure 2: Plots of the relative permittivity $\hat{\varepsilon}$ of a slab of thickness 0.1 $\mu$m made of fused Silica as a function of the wavelength $\lambda$ and plots of its reflection coefficient $|R^{l}|^2$ as functions of $\lambda$ and $k\ell$ for incidence angles $\theta=0$ and $\theta=45^\circ$. The solid curves in the middle and right panels correspond to the exact values of $|R^{l}|^2$ given by \ref{['RR-homogen']} while the dashed curves represent the outcome of the second-order low-frequency approximation. For $\lambda \gtrapprox3~\mu{\rm m}$ which corresponds to $k\ell\lessapprox0.21$, they agree with the exact results.
• Figure 3: Schematic view of an inhomogeneous slab placed in the half-space $S^+$ given by $x>0$. The region painted in different shades of blue represents the material filling the half-space $S^-$. The source of the incident wave is placed on the plane $x=+\infty$. The incident wave vector $\mathbf{k}_0$ is shown as a red arrow. The incidence angle $\theta$ ranges over the interval $(90^\circ,270^\circ)$.