Frequency locking of supermodes and stability of the out-of-phase-locked state in one-dimensional and two-dimensional arrays of vertical-cavity surface-emitting lasers

In large arrays with equally pumped lasers the out-of-phase-locked state is a combination of several supermodes of the array that are locked to a common frequency. The far-field pattern is four lobed because the out-of-phase supermode is largely dominant. An approximated analytical expression for the locking threshold is found, and saturable dispersion is indicated as the physical mechanism that selects the out-of-phase-locked state.