Analytical treatment of the polychromatic spatially multiplexed volume holographic grating

An alternative model to N-coupled wave theory of the spatially multiplexed finite thickness volume holographic reflection grating is developed from the parallel stacked mirrors (PSM) model in terms of N infinite arrays of parallel stacked mirrors each characterized by a different grating vector. A plane reference wave interacts with each of the N sets of stacked mirrors, producing N signal waves. First-order coupled partial differential equations describing the detailed process of Fresnel reflection within the grating are derived for the reference and N signal waves. These equations can be solved analytically at Bragg resonance where agreement with conventional N-coupled wave theory is exact. The new model is compared for the case of some simple multiplexed volume phase reflection gratings at and away from Bragg resonance with a rigorous coupled-wave solution of the Helmholtz equation. Good agreement is attained for even rather high values of index modulation. For lower modulations more characteristic of modern holographic materials, agreement appears extremely good at and around Bragg resonance, although differences inevitably appear in the higher-order diffractive sideband structure. The analytic model is extended to cover polychromatic spatially multiplexed volume phase gratings at Bragg resonance, where once again agreement with rigorous coupled-wave calculations is very good for index modulations typical for modern holographic gratings. Finally, the model is extended to cover the case of the lossless multicolor phase-reflection hologram, where analytic and graphical results are presented concerning diffractive efficiency.