We show that plane-wave singly resonant optical parametric oscillators exhibit a temporal modulation instability when pumped a certain number of times above threshold. Previously, this instability threshold was predicted, with a model neglecting variations in pump power, to occur at around 4.61 times oscillation threshold. We consider here the full self-consistent interaction between pump, signal, and idler sidebands and find that in some spectral regions the instability threshold can be lower than previously predicted, in some cases even comparable to the oscillation threshold, preventing single mode operation at high conversion efficiency. We examine the behavior of the instability for typical regions of operation, and find that both group velocity mismatches and group velocity dispersion have a significant effect. The instability can be suppressed by a suitable choice of intracavity etalon, the design constraints of which are determined.
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