On Rayleigh's Theory of Partially Reflecting Gratings

The numerical behaviour of the Rayleigh-Fourier method is examined for sinusoidal and sawtooth gratings overlying lossless dielectric and metallic substrates. The vector of the coefficients of the reflected spectral orders is expressed as a linear transformation of the vector of the corresponding coefficients of a grating with the same profile overlying an infinitely conducting medium (TOG). The pair of fundamental polarization mode solutions of the latter problem are shown to couple into each polarization mode solution of the partially reflecting grating (PRG) problem. These facts explain why the Rayleigh theory (RT) generally works less well for the PRG than for the TOG, particularly in the case of metallic substrates with high electrical conductivity. Nevertheless, the RT is able to account for all of the features (specular reflection reduction of dielectric gratings, surface plasmon and Wood's anomalies, quasi-Brewster absorption of shallow metallic gratings, etc.) of a large variety of experimental results and rigorous theoretical predictions.