Rigorous analysis of focusing grating couplers using a time-domain spectral collocation method

Diffractive optical elements comprising sub-wavelength aperiodic surface reliefs of finite length require the use of rigorous solvers for Maxwell's equations. We present a detailed analysis of Focusing Grating Couplers (FGC's) using a recently introduced 2D spectral collocation method. The method, solving Maxwell's equations in the time domain, is based on a high-order Chebyshev collocation scheme has the advantage over traditionally used Finite Difference methods that much fewer points per wavelength is needed to accurately resolve wave propagation in diffracting structures. At the same time, the new method exhibits no numerical dispersion in contrast to, e.g., the Finite Difference Time-Domain method. In this presentation we analyze a number of sub-wavelength FGC's with lengths of up to 1000 wavelengths. The FGC's use analog surface reliefs due to their superior diffraction properties. For structures yielding a perpendicular out-coupling, we find that typically 10-12 collocation points per wavelength is sufficient. We find that the focal length depends strongly upon the depth of the surface relief, e. g. that a significant shift of the focal plane from the value expected from geometrical optics is seen if deep surface reliefs are used