Diffraction by crossed anisotropic gratings

Summary form only given. We consider the diffraction of time-harmonic electromagnetic waves by biperiodic structures, which are composed of nonmagnetic anisotropic optical materials and separate two homogeneous regions. By biperiodic or doubly periodic we mean that the structure is periodic in two not necessarily orthogonal directions. The scattered far field consists of a finite number of outgoing plane waves of known directions. To determine the intensity and phase of these diffracted modes one has to solve numerically the electromagnetic field equations. In the talk some analytical and numerical properties of this problem are discussed. Assuming quasiperiodic solutions the time-harmonic Maxwell equations can be reduced to a system of second order partial differential equations in a bounded periodic cell with jump conditions at material interfaces and with nonlocal boundary conditions. We formulate an variational equation which is equivalent to the boundary value problem. The variational form is strongly elliptic for all physical relevant parameters, which allows to prove quite general existence and uniqueness results for the direct diffraction problem. It can be shown that the diffraction problem is solvable for all frequencies and incident angles of the incoming plane wave, and that the solution is unique with exception of a discrete sequence of frequencies. If the structure contains absorbing materials, then the solution is always uniquely determined. The magnetic field of the solution belongs locally to the Sobolev space HI and depends smoothly on frequencies and incident angles of the incoming plane wave with exception of the Rayleigh frequencies. These properties allow to use standard 3D finite elements to solve the boundary value problem