Analytic solutions for spectral properties of superstructure, Gaussian-apodized and phase shift gratings with short- or long-period

Abstract

The analytic solutions (AS) for the spectral properties of short- and long-period waveguide gratings with the effects of discrete phase shift (PS), Gaussian-apodization (GA) and superstructure are presented in this paper, which are derived from the Fourier mode coupling (FMC) theory proposed recently. The spectral properties include the reflectivity of short-period gratings, and the transmission of long-period gratings. The calculated spectra based on the analytic solutions are achieved and compared with measured cases and that on the transfer matrix (TM) method, in the case of changing grating parameters. The AS-based calculation requires the average time of several milliseconds at common PC, and the AS-based efficiency is improved up to ~6700 times the TM-based one. The comparisons have confirmed that the FMC-based analytic solutions are suitable for the real-time and accurate analyses of some non-uniform waveguide gratings.