Complex Band Structure for Subwavelength Evanescent Waves

TL;DR

This paper develops a mathematical and numerical framework for analyzing evanescent waves in subwavelength band gap materials, providing explicit formulas and numerical methods for complex band structures in one and two dimensions.

Contribution

It introduces explicit formulas for complex band structures in 1D and extends the analysis to 2D using layer potential techniques and lattice-summation methods.

Findings

Explicit formulas for 1D complex band structure using capacitance matrix

Accurate prediction of decay rates of interface modes in photonic models

Numerical computation of complex band structures in various settings

Abstract

We present the mathematical and numerical theory for evanescent waves in subwavelength band gap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the SSH-model. In two dimensions, we derive the band gap Green's function and characterise the subwavelength gap functions via layer potential techniques. By generalising existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.