Rigorous three-dimensional analysis of surface-relief gratings using a spectral collocation method

We present a general 3-D spectral collocation method for the analysis of diffractive optical elements (DOEs). The method computes a direct solution to the Maxwell's equations in the time domain. The computational domain is decomposed into a number of small subdomains in which a high-order Chebyshev spectral collocation scheme is used to approximate the spatial derivates in Maxwell's equations. The local solutions in each subdomain are integrated using a Runge-Kutta scheme, and the global solution is reconstructed by using the characteristic variables of the strongly hyperbolic set of equations. A smooth mapping technique is used to correctly model curvi- linear boundaries thus making the method a strong tool for analyzing, e.g., grating couplers with analog surface reliefs. The accuracy and efficiency of the method is verified using simple test cases and examples of the analysis of analog grating couplers of finite length are given. The examples demonstrate the superior properties of the method such as the low number of points per wavelength needed to accurately resolve wave propagation and the absence of numerical dispersion.