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

We propose the 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 the calculation significantly for typical wave scattering problems. This approach is best suitable for isolated resonances when the approximation of a constant background scattering matrix is reasonable. In this article, we aim at expanding the applicability of the resonant mode coupling approximation 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 calculation error of the resonant parameters. Here we first consider a linear approximation of the background scattering matrices which is subsequently used as a base for a piecewise-linear approximation. We show that the latter allows one to keep the approximation error negligibly small with only a few sample points and to apply the resonant mode coupling approximation within almost arbitrary large energy ranges. Finally, we consider the approximation of background matrices by arbitrary matrix functions and propose a technique to derive resonant poles in this case. The methods described here could be considered an alternative approach for calculating the optical spectra of stacked systems.