Turing patterns in nonlinear optics

Summary form only given. A well known pattern formation mechanism in nonlinear optical resonators is "off-resonant" excitation. If the central frequency of the gain line of the laser /spl omega//sub A/ is larger than the resonator resonance frequency /spl omega//sub R/, then the excess of frequency /spl Delta//spl omega/=/spl omega//sub A/-/spl omega//sub K/ causes transverse (spatial) modulation of the laser fields, with characteristic transverse wavenumber k obeying a dispersion relation ak/sup 2/=/spl Delta//spl omega/ (a is the diffraction coefficient of the resonator). Typical for off-resonance patterns is that the dominating transverse wavelength of emerging patterns scales with the square root of the diffraction constants, and strongly depends on the off-resonance detuning. We show another pattern formation mechanism in nonlinear optics, principally different from the off-resonant excitation mechanism: the patterns can occur due to the interplay between diffractions or diffusions of coupled field components. The reported mechanism is similar to that of local activation and lateral inhibition (LALI) found in reaction-diffusion systems by Turing, where the coupling between components with different diffusion coefficients cause spatial instabilities and lead to pattern formation. We generalize the Turing pattern formation mechanism to an arbitrary form of nonlocalities encountered in nonlinear optics, both diffraction and diffusion. We study two different nonlinear optical systems: 1) degenerate optical parametric oscillators; 2) lasers with saturable absorbers.