A variational method for electromagnetic diffraction in biperiodic structures

Consider a time-harmonie electromagnetic plane wave incident on a bipenodic structure in U The periodic structure séparâtes two régions with constant dielectnc coefficients The dielectnc coefficient mside the structure is assumed to be a gênerai bounded measurable function The magnetic permeabihty is constant throughout IR. We desenbe a simple variational method for finding weak « quasipenodic » solutions to Maxwell's équations in such a structure Our formulation is simple and computationally attractive because it only involves three field components The problem is formulated by constructing a variational form over a bounded région, with « transparent » boundary conditions The boundary conditions corne from the Dinchlet-Neumann map s for the problem, which can be calculated exphcitly We show that the variational problem admits unique solutions for ail sufficiently small frequenties, and more generally for ail but a discrete set of frequencies We also show that the weak solutions satisfy a conservation of energy condition Finally, we briefly discuss an implementation of a three-dimensional numerical finite element scheme which solves the discretized variational problem, and present the results of a simple numerical experiment