Solving the Scattering Problem for Open Wave-Guide Networks, I Fundamental Solutions and Integral Equations

TL;DR

This paper develops a layer potential method using fundamental solutions for solving transmission problems in open wave-guide networks, leading to integral equations that are shown to be Fredholm of second kind and solvable.

Contribution

It introduces a novel fundamental solution construction and integral equation formulation for open wave-guide transmission problems, demonstrating their solvability.

Findings

Integral equations are Fredholm of second kind.

Fundamental solutions are constructed via Fourier transform and ODE theory.

The method effectively represents guided modes.

Abstract

We introduce a layer potential representation for the solution of the transmission problem defined by two dielectric channels, or open wave-guides, meeting along the straight-line interface, $\{x_1=0\}.$ The main observation is that the outgoing fundamental solution for the operator $\Delta +k_1^2+q(x_2),$ acting on functions defined in ${\mathbb R}^2,$ is easily constructed using the Fourier transform in the $x_1$-variable and the elementary theory of ordinary differential equations. These fundamental solutions can then be used to represent the solution to the transmission problem in half planes. The transmission boundary conditions lead to integral equations along the intersection of the half planes, which, in our normalization, is the $x_2$-axis. We show that, in appropriate Banach spaces, these integral equations are Fredholm equations of second kind, which are therefore generically solvable. We analyze the representation of the guided modes in our formulation.