Resonant mode coupling approximation for calculation of optical spectra of photonic crystal slabs. Part II

We propose further development of the resonant mode coupling approximation for the calculation of optical spectra of stacked periodic nanostructures in terms of the scattering matrix. We previously showed that given the resonant input and output vectors as well as background scattering matrices of two subsystems, one can easily calculate those for the combined system comprising two subsystems. It allows us to write a resonant approximation for the combined system and speed up calculation significantly for typical calculation problems. The main drawback of this approach is that the background matrix in such approximation was considered constant which is not always sufficient if the energy range of interest is relatively wide. The aim of this article is to solve this problem by utilizing more complicated approximations for the background matrices. In particular, we show that consideration of energy-dependent correction terms for the background matrices remarkably reduces the resonant energies' calculation error. Here we first consider a linear approximation, and although it is not suitable for large energy ranges, it is used as a base for a piecewise-linear approximation which allows one to keep the approximation error negligibly small with only a few sample points. Moreover, interpolation of the background matrices allows one to apply resonant mode coupling approximation in almost arbitrary large energy ranges. We also consider approximation of background matrices by an arbitrary matrix function and propose a technique to derive the resonant poles in this case. The methods described here could be considered as an alternative approach for calculation of optical spectra stacked systems.