Complex scaling for open waveguides

TL;DR

This paper applies complex scaling to analyze wave propagation in open dielectric waveguides, demonstrating that integral equation kernels can be analytically continued and that solutions exhibit exponential decay, enabling efficient numerical discretization.

Contribution

It shows that kernels in integral equations for open waveguides admit analytic continuation and that solutions decay exponentially, improving numerical methods for these problems.

Findings

Kernels admit rapidly decaying analytic continuation.

Solutions to integral equations have analytic continuation and decay exponentially.

Numerical examples demonstrate the effectiveness of the method.

Abstract

In this work we analyze the complex scaling method applied to the problem of time-harmonic scalar wave propagation in junctions between `leaky,' or open dielectric waveguides. In [arXiv:2302.04353, arXiv:2310.05816, arXiv:2401.04674, arXiv:2411.11204], it was shown that under suitable assumptions the problem can be reduced to a system of Fredholm second-kind integral equations on an infinite interface, transverse to the waveguides. Here, we show that the kernels appearing in the integral equation admit a rapidly decaying analytic continuation on certain natural totally real submanifolds of $\mathbb{C}^2.$ We then show that for suitable, physically-meaningful, boundary data the resulting solutions to the integral equations themselves admit analytic continuation and satisfy related asymptotic estimates. By deforming the integral equation to a suitable contour, the decay in the kernels, density, and data enable straightforward discretization and truncation, with an error that decays exponentially in the truncation length. We illustrate our results with several representative numerical examples.