Supermode analysis of two-dimensional coupled coherent laser arrays based on block tridiagonal matrix

Modal properties of two-dimensional (2D) coupled coherent laser array have been investigated based on coupled-mode theory. Block tridiagonal matrix formalism is employed to present the coupling matrix and calculate array supermodes for the first time. Based on the decomposition of the block tridiagonal matrix, analytical descriptions for supermodes in N x N-element array with lasers coupling their nearest and second nearest neighbors are obtained. According to the near-and far-field distributions, it can be seen that although the number of supermodes remains NxN, the variety of intensity profiles is reduced to N(N+1)/2. In addition, the maximum wavelength splitting in the modes is estimated from the coupling strength and found to be broadened with the additional couplings that take place along the two diagonals, which will contribute to supermode selectivity. Our work provides a new method for future research on 2D coherent laser arrays under a variety of couplings as well as a basis for supermode engineering.