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

TL;DR

This paper develops a comprehensive low-frequency scattering theory for the Helmholtz equation in one dimension, providing explicit formulas for scattering coefficients and revealing physical insights, especially for systems with balanced gain and loss.

Contribution

It introduces a dynamical formulation for low-frequency scattering in 1D Helmholtz systems, deriving explicit low-frequency expansion formulas for transfer, reflection, and transmission coefficients.

Findings

Explicit formulas for low-frequency transfer matrix coefficients

Insights into scattering behavior with balanced gain and loss

Enhanced understanding of electromagnetic wave propagation in 1D systems

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.