| Summary: | The goal of this study includes two targets: to extend the region of application for the modal theory, including Classical Modal Theory [CMT] and Rigorous Coupled Wave Theory [RCWT], and to investigate the convergence characteristics of CMT and RCWT.First, the RCWT algorithm for one-dimensional isotropic gratings is reviewed along with the details of its mathematical formulation, and the advantages of applying the inverse rule in the Fourier expansion are also explained. Then the CMT formulation for dielectric lamellar gratings with multiple indices and sub-periods is developed. Several numerical examples are tested and compared with the results obtained from RCWT. The convergence properties of the present CMT formulation are demonstrated with several examples and discussed in relation to the parameters used in the formulation.Next, the convergence characteristics of RCWT for continuously index-modulated gratings are investigated. It is demonstrated that the RCWT convergence is strongly dependent on the convergence of the Fourier coefficients for the index modulation functions, and the convergence profiles of diffraction efficiencies and those of the Fourier series are closely related.Finally, the formulation of RCWT for diffraction gratings in bi-anisotropic media, which exhibit linear birefringence and/or optical activity, is developed. All of the incident, exiting and grating materials can be isotropic, uniaxial or biaxial, with or without optical activity. The principal values of the electric permittivity tensor, the magnetic permeability tensor and the gyrotropic tensor of the materials can take arbitrary values. The optical axes may be arbitrarily and independently oriented. The symmetric constitutive relations for bi-anisotropic materials are adopted. The procedures for Fourier expansion of Maxwell's equations are also described.The present RCWT formulation is implemented and applied to various problems. Diffraction efficiencies for single layer bi-isotropic gratings are calculated and compared with those obtained from scalar diffraction theory. Characteristics of multilayer gratings in gyrotropic biaxial media are also demonstrated. Distinctive polarization coupling effects due to optical activity are observed in both cases. The fast convergence of the present RCWT formulation is also demonstrated. As a limiting case, diffraction efficiencies for a multilayer grating made with non-gyrotropic uniaxial material exhibit good agreement with available data.
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