Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique

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Abstract
A method of solving problems of diffraction and dispersion in electromagnetic theory is presented. A modal expansion technique is used with a recursive R-matrix propagation scheme. This method retains the inherent R-matrix numerical stability and yet, contrary to some recent studies, is quite easy to implement for periodic structures (both two and three dimensional), including gratings and photonic crystal media. Grating structures may be multilayered structure, linear or crossed. Photonic media may be latticelike structures of finite or infinite depth. The eigenvalues of the modes are obtained by diagonalizing a matrix rather than searching for zeros of characteristic equations. Diffraction from dielectric and metallic sinusoidal gratings is calculated, and the results are compared with other published results. Transmission is calculated through a seven-layer-deep square arrangement of dielectric cylinders. Also, with the Floquet theorem, the bulk dispersion of the same cylinder geometry is calculated, and t e results are compared with other published results. Of particular interest as a computational tool is a description of how a complex structure can be recursively added, whole structures at a time, after the initial structure has been calculated. This is very significant in terms of time savings, since most of the numerical work is done with the initial structure.