A VARIATIONAL METHOD FOR ELECTROMAGNETIC DIFFRACTION IN BIPERIODIC STRUCTURES

Consider a time-harmonic electromagnetic plane wave incident on a bipe riodic structure in R3. The periodic structure separates two regions w ith constant dielectric coefficients. The dielectric coefficient insid e the structure is assumed to be a general bounded measurable function . The magnetic permeability is constant throughout R3. We describe a s imple variational method for finding weak <<quasiperiodic>> solutions to Maxwell's equations in such a structure. Our formulation is simple and computationally attractive because it only involves three field co mponents. The problem is formulated by constructing a variational form over a bounded region, with <<transparent>> boundary conditions. The boundary conditions come from the Dirichlet-Neumann maps for the probl em, which can be calculated explicitly. We show that the variational p roblem admits unique solutions for all sufficiently small frequencies, and more generally for all but a discrete set of frequencies. We also show that the weak solutions satisfy a conservation of energy conditi on. Finally, we briefly, discuss an implementation of a three-dimensio nal numerical finite element scheme which solves the discretized varia tional problem, and present the results of a simple numerical experime nt.