Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach

TLDR

The study proposes a unified approach to determine the characteristics of almost‑periodic grating slab waveguides, including gain. The approach divides the waveguide into short segments with locally periodic gratings, then obtains overall characteristics by multiplying the segment F matrices while applying appropriate phase conditions at each interface. The method accurately predicts behavior for tapered, chirped, and phase‑shifted gratings, agrees with existing techniques, and enables calculation of distributed‑feedback laser threshold conditions including edge reflections.

Abstract

A unified approach to obtain the characteristics of almost-periodic grating slab waveguides including gain in the waveguide is reported. In this approach the waveguides are divided into short segments, and in each segment the gratings are assumed to be periodic, that is, parameters such as coupling coefficient, grating phase, deviations from the Bragg frequency, and gain in the waveguide are independent of a propagation direction z. Then characteristics of almost-periodic grating slab waveguides can be obtained by multiplying each F matrix of a short segment with the proper grating phase conditions at the interface between two adjacent segments. The appropriateness of this approach is shown for typical aperiodic grating waveguides such as tapered, chirped, and phase-shifted gratings. The results obtained by this method are compared with others and prove to be in good agreement with the results obtained by other methods. In addition to these characteristics, it is shown that the F matrix can be used to obtain the threshold conditions for distributed feedback laser oscillations including reflections from cleaved edges.