Stochastic band structure for waves propagating in periodic media or along waveguides

TL;DR

The paper introduces the stochastic band structure method to determine wave dispersion relations in periodic media with losses or damping, providing a new approach that captures local propagation information.

Contribution

It presents a novel stochastic band structure technique that maps the resolvent set in dispersion space, applicable to any time-harmonic wave system with losses or radiation.

Findings

Successfully applied to lossy sonic crystals

Effective for radiating surface phononic crystals

Applicable to optical waveguides with radiation

Abstract

We introduce the stochastic band structure, a method giving the dispersion relation for waves propagating in periodic media or along waveguides, and subject to material loss or radiation damping. Instead of considering an explicit or implicit functional relation between frequency $\omega$ and wavenumber $k$, as is usually done, we consider a mapping of the resolvent set in the dispersion space $(\omega, k)$. Bands appear as as the trace of Lorentzian responses containing local information on propagation loss both in time and space domains. For illustration purposes, the method is applied to a lossy sonic crystal, a radiating surface phononic crystal, and a radiating optical waveguide. The stochastic band structure can be obtained for any system described by a time-harmonic wave equation.