Modes of a two-dimensional phase-locked array of vertical-cavity surface-emitting lasers

A simple formalism is reported for studying the modes of an infinite two-dimensional (2-D) phase-locked array of vertical-cavity surface-emitting diode lasers. In this approach, the Helmholtz equation is approximately separated into products of solutions of the associated one-dimensional (linear) arrays. This formalism allows the knowledge gained from years of research into the modal properties of linear diode arrays to be applied directly to 2-D arrays. As a result, it is expected that both 2-D gain-guided and index-guided arrays (except those with excess gain in the low-index channels) will lase in the high-order mode characterized by emission into four distinct off-axis far-field lobes. In contrast, it is expected that properly constructed 2-D leaky-mode arrays would lase preferentially in the fundamental mode well above threshold.