Low-frequency scattering of TE and TM waves by an inhomogeneous medium with planar symmetry

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">T</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -symmetric and non- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">T</mml:mi> </mml:mrow> </mml:mrow> </mml:math> -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.