Progress in the science of optical materials has made the problem of modelling novel structures more complex; this is particularly true for uniaxial or biaxial materials, materials presenting optical activity as well as absorption, and multi-layered gratings. The article by Onishi et al. shows not only how all of the above can be modelled together, but also considers bi-anisotropic materials with a gryotropic tensor. All this while taking great care of the accuracy and convergence of their method.
The authors tackle this problem by applying Rigorous Coupled Wave Theory (RCWT) using a Fourier expansion. Unlike standard Coupled Mode Theory (CMT) where a single or a few terms can be used to model shallow gratings, RCWT uses a larger set of terms and waves to model deep gratings. One problem that often arose in the past when using RCWM was its slow convergence when sharp corners, deep gratings, metallic materials or TM polarization were used. A large set of waves and Fourier terms were needed to achieve accurate estimates for the reflected amplitudes. The problem at hand is even more complex and computationally intense, since when using multi-axial or bi-anisotropic materials both polarizations must be taken into account. To deal with this problem the authors follow work by L. Li that was also published in the Journal of the Optical Society of America A in 1996, and who took special care when factorizing the Fourier terms. In his article Li showed that by factorizing correctly, the appropriate field components are preserved across the discontinuities of the permittivity tensor, something that is required at an interface.
To model a bi-anisotropic material, the authors chose to consider the symmetric constitutive relations between the magnetic and electric fields. Generally, a bi-anisotropic material is only modelled with a permittivity tensor, and yields an asymmetric constitutive relation. The article presents for the first time a method to model a symmetric constitutive relation in gratings, which is more general, and provides the explicit equations to do so. Thus, the asymmetric case can be seen as a special case of the symmetric case.
To demonstrate the efficiency of their algorithm the authors made heroic calculations. For example, they present a multi layered grating in bi-axial gyrotropic material, with a four-step saw-tooth like structure (Fig. 5), a structure exhibiting many sharp corners. Around twenty Fourier terms are needed for the reflections to reach their accurate values. Using only twenty terms makes this problem tractable, and optimized designs of grating are expected to emerge from using this method.
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