Abstract

Superimposed holographic phase gratings are used for many important bulk and integrated optics applications. Some of these applications include holographic optical content-addressable memories, optical interconnects, multiple-beam combining and generation, filtering, and wavelength multiplexing. In the present work, the rigorous coupled-wave approach has been extended to two superimposed, anisotropic, phase, and/or amplitude gratings of arbitrary slant angle and periodicity. The analysis treats the 3-D diffraction problem with the grating vectors of the superimposed gratings being coplanar. The external boundary regions can be either isotropic (bulk applications) or uniaxial anisotropic (integrated applications). The forward and backward diffracted orders are characterized by a pair of integers (i₁, i₂), where i₁ and i₂ can take any integer value. In the case of anisotropic external regions each diffracted order has two components, an ordinary and an extraordinary one. If i₁, and i₂ are restricted to be between −M and +M, where M is an arbitrary but finite positive integer, the number of the diffracted orders in the analysis is (2M + 1)². Using Maxwell’s equations and the boundary conditions a system of 4(2M + 1)² equations with 4(2M + 1)² unknowns is formed and solved. The solution of the system provides all the diffraction characteristics of all the diffracted orders. The analysis can be extended to an arbitrary number of superimposed gratings N. Then the number of diffracted orders in the analysis becomes (2M + 1)^N, and the dimensionality of the problem increases exponentially with N. In the latter case, alternative approximate techniques are discussed.

© 1988 Optical Society of America