Lossy dielectric gratings have been analyzed using a Raman-Nath formalism modified to incorporate losses. Four second-order coupled wave equations are retained for computation of the zero, first- and second-order diffracted beams for a multitude of practical cases. Significant differences are found in comparison with computations in which only two coupled waves are retained. The entire range of losses and thicknesses encountered for holograms in film emulsions has been studied using this unified approach. Graphs have been prepared to show the efficiency, i.e., power diffracted in the first-order relative to the total incident power, vs the index modulation for a wide range of thicknesses and losses. At a given thickness, optimum frequency requires a specific exposure. The efficiency for an optimum exposure is plotted vs the loss factor with thickness as a parameter. New experimental data are presented for bleached gratings in which several diffracted orders are measured and compared to our theory for a wide range of index modulation and loss factors.
Milton Chang and Nicholas George, "Holographic Dielectric Grating: Theory and Practice," Appl. Opt. 9, 713-719 (1970)