Using the parallel stacked mirror model to analytically describe diffraction in the planar volume reflection grating with finite absorption

The parallel stacked mirror (PSM) model of grating diffraction is extended to include a complex refractive index for the case of an unslanted reflection geometry at oblique incidence and for the [sigma]-polarization. The well-known theoretical upper diffraction efficiency limit applicable to reflective absorption gratings described by Kogelnik's coupled-wave theory, is shown to be directly derivable from the PSM model. Analytical formulae for the diffraction efficiency of phase, absorption, and mixed harmonic-index gratings are compared with numerical computations using a rigorous coupled-wave description. A conventional truncated coupled-wave description, similar to Kogelnik's approach, is also derived from the harmonic-index rigorous coupled-wave equations by limiting the propagating modes to a signal and reference wave and by ignoring second-order derivatives. At low to moderate average loss the PSM theory is observed to provide a somewhat better fit to the rigorous coupled-wave calculations, and this is particularly evident for gratings that are dominated by phase modulation. As the average absorption coefficient is increased, all three models show the diffractive response sharpening around Bragg resonance and the characteristic sideband structure attenuating and then disappearing altogether, giving rise to a broader spectral behavior away from resonance. At high average loss, the truncated coupled-wave model is observed to very marginally outperform PSM. However, overall the PSM model is found to provide an exceedingly good description of the general mixed-phase-absorption unslanted reflection grating with finite average loss.