A method for calculating arbitrary-order diffraction efficiencies of thick, lossless transmission gratings with arbitrary periodic grating shapes has been developed. This represents an extension of previous work to nonsinusoidal gratings and to higher-order Bragg angles. A Fourier-series representation of the grating is employed, along with a coupled-mode theory of diffraction. For illustration, numerical values of the diffraction efficiencies at the first three Bragg angles are calculated for sinusoidal, square-wave, triangular, and saw-tooth gratings. Numerical results for the same grating shapes with the same parameters are also calculated for comparison, by extending Burckhardt’s numerical method for analyzing thick sinusoidal gratings. The comparison shows that the coupled-mode theory provides results with relative computational ease and results that are in agreement with calculations obtained by extending the more-rigorous Burckhardt theory to nonsinusoidal grating shapes and to higher-order Bragg angles.
S. F. Su and T. K. Gaylord, "Calculation of arbitrary-order diffraction efficiencies of thick gratings with arbitrary grating shape," J. Opt. Soc. Am. 65, 59-64 (1975)