Curvilinear coordinate method as an initial value problem: application to gratings

We propose a C-method approach for the efficient calculation of the scattering matrix of a grating. We consider two horizontal planes above and below the grating and define a coordinate system such that the grating surface and both horizontal planes correspond to surface coordinates. Inside the area A delimited by the horizontal planes, the covariant formalism of Maxwell's equations leads to an initial value problem that is solved for independent initial conditions satisfying the boundary conditions. Outside the area A, the fields are represented by Rayleigh expansions. The scattering matrix is obtained by using continuity relations between different components of fields on the horizontal planes.