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Appl Opt. 1989 Jun 15;28(12):2401-21. doi: 10.1364/AO.28.002401.

Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media.


The diffraction by two planar slanted fringe gratings superposed in the same volume of an anisotropic medium is treated using rigorous 3-D vector coupled wave analysis. Arbitrary angle of incidence and polarization are included. Both phase and/or amplitude slanted gratings in anisotropic media are treated in the analysis. The external boundary regions can be either isotropic (for bulk applications) or uniaxial anisotropic (for integrated applications). Both forward- and backward-diffracted orders are characterized by a number pair (i(l),i(2)), where i(l) and i(2) are integers. The Floquet condition is discussed for the case of two superposed gratings. When the external regions are anisotropic, each diffracted order has an ordinary (O), and an extraordinary (E) component. The analysis is also generalized for an arbitrary number of superposed gratings. The numerical complexity is discussed. In the case of equal grating periodicities along the boundaries, the diffracted orders become degenerate in the external regions. In this case, an alternative analysis that utilizes a cascaded stack of unslanted gratings can be used. Limiting cases are also presented. The various Bragg conditions are identified and quantified. Sample calculations presented include the quantification of the crosstalk between two superposed gratings, the evaluation of the effects of coupled Bragg conditions in beam combining applications, design and analysis of a beam splitter and a beam combiner, demonstration of the use of a cascaded stack of unslanted gratings of constant modulation to represent two superposed gratings that have the same periodicity along the boundaries, and finally evaluation of the effect of the phase difference between two gratings. The same analysis applies in the limiting cases of isotropic materials, single slanted gratings, etc. Applications of this analysis include optical storage, optical digital truth table look-up processing, neural nets, optical interconnects, beam splitting, and beam combining.

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