Diffraction of Bloch Wave Packets for Maxwell's Equations

Grégoire Allaire (1, 2) , Mariapia Palombaro (3) , Jeffrey Rauch (4)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 DeFI - Shape reconstruction and identification
3 DISIM - Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila]
4 Department of Mathematics - University of Michigan

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 Schrödinger 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.

Mots clés: Geometric optics; diffractive geometric optics; Bloch waves; diffraction; electromagnetism; Maxwell's equatons

Domaines: Equations aux dérivées partielles [math.AP]

Grégoire Allaire, Mariapia Palombaro, Jeffrey Rauch. Diffraction of Bloch Wave Packets for Maxwell's Equations. Communications in Contemporary Mathematics, 2013, 15 (6), pp.1350040. ⟨10.1142/S0219199713500405⟩. ⟨hal-00932946⟩