Tolerance of phase-locked VCSEL arrays to random and systematic parameter deviations among cavities

Cavity parameter deviations from their nominal values affect phase-locking in vertical-cavity surface-emitting lasers (VCSEL) arrays. The phase-locking tolerance is studied analytically and numerically for both random as well as systematic parameter variations. Parameter-induced deviations among the cavity eigenfrequencies cause the dominant effect. For small externally imposed variations, self-regulated nonlinear frequency pulling among cavities overcomes original detuning and permits phase-locking. The root mean square (rms) tolerance threshold is derived by solving a quasi-random matrix eigenvalue under simplifying assumptions. Tolerance increases with intercavity coupling strength and decreases with the array size M. Systematic variations, such as thermal drifts, exhibit a lower tolerance decreasing as ∝1/M or faster, compared to random variations ∝1/√M. Theory results qualitatively agree with large size array simulations employing actively coupled rate equations.