Interaction between saddle node and spatial modulational instabilities in quadratic media

The aim of this work is to investigate analytically in two dimensional system the role of the quasi neutral mode in the dynamics associated with the saddle node bifurcation on the pattern forming process in the degenerated optical parametric oscillator with a saturable absorber. The study shows that the presence of the saturable absorber allows the dynamics of the degenerated optical parametric oscillator to exhibit spatial pattern for a positive detuning parameter. More importantly, the study reveals that the interaction between the quasi neutral and Turing modes stabilizes the two phase-locked hexagonal structures. This behavior can occur below the static lasing threshold and arises even in the absence of the saturable absorber. Numerical simulations confirm the analytical predictions.