Increasing spatial complexity toward near-resonant regimes of quadratic media

Spiral patterns, spiral breakup, and zigzag spirals in an optical device

Above the first lasing threshold the degenerate optical parametric oscillator with saturable absorber displays successive Hopf and Turing instabilities. Various spiral patterns and defect turbulent patterns are numerically observed on the light intensity profiles. Close to the Hopf threshold, a normal form is derived which leads to a complex Ginzburg-Landau equation where a bi-Laplacian instead of a Laplacian drives the formation of spirals. At resonance the predictions of the normal form are compared with the numerical observations of the full equations. Above the Hopf threshold, the spirals destabilize, breaking into slowly evolving patterns with small spirals and filaments. Further above the threshold, when both the Turing and Hopf bifurcations interplay, a new spiral pattern emerges, with large notched arms.

Optical patterns with different wavelengths

The semiconductor resonator is an example of an optical system where two modulational instabilities with different wave numbers coexist. In the limit of nascent bistability, the dynamics is generically described by a nonvariational real order parameter equation, of which we give a detailed derivation. This considerably simplifies the linear and weakly nonlinear stability analyses. When the two instabilities are close together, we derive normal form equations and put special emphasis on "envelope" branches of solutions. These particular solutions may connect the two instability points or form an isola. On the basis of these rigorous results, we finally discuss the case of distant modulational instabilities, in both one and two transverse dimensions.