Diffraction by gratings: from the C-method to the stochastic C-method

The Chandezon, or C-method, is an efficient and versatile numerical method for modeling diffraction problems involving smooth surface relief gratings. For some grating profiles, the C-method is limited when the height-to-period ratio exceeds a factor of three. This is due to the formation of ill-conditioned matrices for inversion. Here, the stochastic C-method (SCM) is introduced as a solution that leverages stochastic differential equations to overcome these numerical difficulties. The SCM is developed by altering the physical model of the grating profile function to include an additive Brownian noise component. The inclusion of noise dramatically expands the applicability of the C-method and enriches the physical model. Numerical experiments show that the SCM achieves a precision on the order of 10^-5 for diffracted/transmitted amplitudes on sinusoidal profiles with height-to-period ratios as high as 72. These results are in agreement with those obtained using multi-precision and the rigorous coupled wave analysis (RCWA).