Computer-aided determination of existence and stability of optical patterns

Summary form only given. In recent years, studies into transverse effects in nonlinear optical systems have revealed a wealth of spatial complexity, ranging from periodic patterns to localised structures. We present here a computer-assisted technique, valid for arbitrary values of the system parameters, which can find the stationary solutions of a model. It uses a Newton-type method and a Fourier transform to evaluate the spatial derivatives. The Newton method automatically gives the linearisation around the solutions found and therefore can be extended to find their eigenspectrum and hence stability. We will demonstrate the method by finding exact hexagonal, roll and square solutions for a model of a saturable absorber in a cavity. We will show these pattern solutions as a function of their wavenumber and of the intensity of the input pump.