Transfer-matrix method based on perturbation expansion for periodic and quasi-periodic binary long-period gratings

Abstract

A transfer-matrix method based on perturbation expansion is proposed as an alternative way of simulating the transmission spectrum of a binary long-period grating (LPG). We first generalize the concept of transfer matrices for a heterojunction waveguide. For the couplings among copropagating modes, forward transfer matrices are used to describe the evolution of mode amplitudes along the grating. We show that these elements are related to the well-known coupling coefficients. The method is then used for the study of ideal two-mode grating couplers, and analytic solutions are obtained. We also use the matrix method to study multimode couplings in a LPG and compare the results with those obtained by using the coupled-mode theory. To further demonstrate its usefulness, we apply the method to a special quasi-periodic LPG, the Fibonacci grating. The results show that each cladding mode contributes to several transmission dips and that the dips of different cladding modes are grouped according to the special resonance conditions.