The
diffraction of
electromagnetic radiation from
periodic grating profiles is determined using rigorous coupled-wave analysis, with intermediate calculations cached to reduce computation time. To implement the calculation, the
periodic grating is divided into
layers, cross-sections of the ridges of the
grating are discretized into rectangular sections, and the
permittivity, electric fields and magnetic fields are written as
harmonic expansions along the direction of periodicity of the
grating. Application of Maxwell's equations to each intermediate layer, i.e., each layer except the
atmospheric layer and the substrate layer, provides a matrix
wave equation with a wave-vector matrix A
coupling the
harmonic amplitudes of the
electric field to their partial second derivatives in the direction perpendicular to the plane of the
grating, where the wave-vector matrix A is a function of intra-layer parameters and incident-
radiation parameters. W is the eigenvector matrix obtained from wave-vector matrix A, and Q is a
diagonal matrix of square roots of the eigenvalues of the wave-vector matrix A. The requirement of continuity of the fields at boundaries between
layers provides a matrix equation in terms of Wand Q for each layer boundary, and the solution of the series of matrix equations provides the
diffraction reflectivity. Look-up of W and Q, which are precalculated and cached for a useful range of intra-layer parameters (i.e.,
permittivity harmonics, periodicity lengths,
ridge widths,
ridge offsets) and incident-
radiation parameters (i.e., wavelengths and angles of incidence), provides a substantial reduction in computation time for calculating the
diffraction reflectivity.