Diffraction of Bloch Wave Packets for Maxwell's Equations

TL;DR

This paper analyzes Maxwell's equations in periodic media, constructing three-scale WKB solutions that incorporate dispersion and stability, extending scalar wave analysis to vector systems with non-elliptic generators.

Contribution

It develops a novel three-scale WKB method for Maxwell's equations in periodic media, accounting for dispersion and non-elliptic operators, advancing wave propagation understanding.

Findings

Constructed accurate approximate solutions of three-scale WKB type.

Identified dispersion governed by a Schrödinger equation from the quadratic dispersion relation.

Proved stability under a weak ray average hypothesis.

Abstract

We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schr\"odinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations.