Effective schema for the rigorous modeling of grating diffraction with focused beams

Most modal diffraction methods are formulated for incident plane waves. In practical applications, the probing beam is focused. Usually, this is simulated by means of numerical integration where Gaussian quadrature formulas are most effective. These formulas require smooth integrands, which is not fulfilled for gratings due to Rayleigh singularities and physical resonances. The violation of this condition entails inaccurate integration results, such as kinks and other artifacts. In this paper, a methodology for the efficient treatment of the numerical integration with improved accuracy is presented. It is based on the subdivision of the aperture along the lines of Rayleigh singularities, mapping of these subapertures into unit squares, and separate application of the Gaussian cubature formulas for each subarea.