Simple reformulation of the coordinate transformation method for gratings with a vertical facet or overhanging profile.

Abstract

We reformulate the coordinate transformation method (C method) for gratings with a vertical facet or overhanging profile (overhanging gratings), in which no tensor concept is involved, only the knowledge of elementary mathematics and Maxwell's equations in the rectangular coordinate system is used, and we provide a detailed recipe for programming. This formulation is easy to understand and implement. It adopts the strategy of a rotating coordinate system from Plumey et al. [J. Opt. Soc. Am. A14, 610 (1997)JOAOD60740-323210.1364/JOSAA.14.000610] and expresses it with the method of changing variables from Li et al. [Appl. Opt.38, 304 (1999)APOPAI0003-693510.1364/AO.38.000304]. We investigate several typical overhanging gratings by the reformulated C method, and we validate and compare the results with the Fourier modal method, which shows that it is superior, especially for metal deep smooth gratings. This reformulation can facilitate the research in light couplers for optical engineers.