Abstract: We consider a degenerate optical parametric oscillator consisting of a crystal in a ring resonator. The occurrence of spatial instabilities in such system due to diffraction are studied and it is shown that they may lead to the formation of stationary or dynamical spatial patterns. The thresholds for such instabilities are derived when either the pumped and the generated modes are detuned with respect to the cavity reference frequency; the patterns that emerge from the instability are calculated analytically by bifurcation analysis. Numerical simulations substantially confirm the onset of such phenomena giving evidence of the formation of rolls, zig-zag patterns, dislocations, filamentation and optical chaos.!15
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