Numerical Solution of Large Non-Hermitian Eigenvalue Problem Arising from Model of Vertical Cavity Surface Emitting Laser Array

Abstract

Modal behavior of a 2-D (square lattice geometry) antiguided vertical cavity surface emitting laser (VCSEL) array was studied numerically. The background of the numerical model of VCSEL array is scalar diffraction theory and 3-D bidirectional beam propagation method. Resonator modes were found as eigen-functions of the socalled round-trip operator which transforms the transverse distribution of electro-magnetic field when light have a round-trip in the device. The round-trip operator after corresponding discretization becomes a linear non-hermitian operator in a complex linear large dimensional space. Using the Arnoldi algorithm, a number of array optical modes were found. In calculations, both Fourier and space variable descriptions of beam propagation were combined. Calculations were made for various spacing length between elements and a size of the array. 4x4 and 10x10 laser arrays were studied numerically. Array optical modes having different symmetry properties were found. They include in-phase mode with constant phase over the array, out-of-phase mode with alternating phase between elements and modes with mixed symmetry. Conditions are found for favorable lasing of the in-phase mode providing high laser beam quality.