The effects of the phase mismatch are analysed within the propagation model
for the degenerate optical parametric oscillator (DOPO). The transverse st
ructures generated via a modulational instability are found to be strongly
dependent on the mistuning of the signal and the phase mismatch. Islands of
squares, quasi-hexagons and zigzags are found to emerge from a sea of stri
pes, in the plane defined by the two mistunings.