Low-frequency scattering defined by the Helmholtz equation in one dimension

• URL: http://arxiv.org/abs/2105.07895v1
• Date: Fri, 14 May 2021 11:58:01 GMT
• Title: Low-frequency scattering defined by the Helmholtz equation in one dimension
• Authors: Farhang Loran and Ali Mostafazadeh
• Abstract summary: The Helmholtz equation in one dimension describes the propagation of electromagnetic waves in effectively one-dimensional systems. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schr\"odinger equation. The fact that the potential term entering the latter is energy-dependent obstructs the application of the results on low-energy quantum scattering in the study of the low-frequency waves satisfying the Helmholtz equation. We use a recently developed dynamical formulation of stationary scattering to offer a comprehensive treatment of the low-frequency scattering of these waves for a general finite-range scatterer. In particular, we give explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix of the system which in turn allow for determining the low-frequency expansions of its reflection, transmission, and absorption coefficients. Our general results reveal a number of interesting physical aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss.

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