Instability of optical tetrahedral dissipative crystals

Using a normal form analysis, we derive general conditions under which tetrahedral crystals are stable. This result is applied to two driven nonlinear optical systems: degenerate optical parametric oscillator and passive cavities with defocusing Kerr media, for which we prove that tetrahedral crystals are always unstable close to the Turing instability. This result arises from the interaction of the quasi-neutral mode with the Turing modes. Without this interaction, tetrahedral crystals may be stable.