Steady state analysis of self-organization of light into a scattering ring due to induced reflection gratings in photorefractive materials

We present a steady state theory and derive the eigenmodes for the nonlinear self-organization of light into a scattering ring in photorefractive materials such as potassium niobate, due to the formation of reflection gratings. The configuration we consider comprises solely of a mismatched sample of potassium niobate in air, with no external feedback mirror. This is the first part of the two-step process involved in the self-organization of light into a hexagonal pattern observed in this and other materials. Complete steady state solutions are derived and amplitude and phase variations of the counterpropagating main beam and its spatial sidebands (responsible for ring formation) with propagation are shown. Our computations show the bifurcation diagrams of the amplitudes and phases as a function of the photorefractive gain parameter. Dependence of the results on the amount of linear scattering is also demonstrated. Connections with available experimental results are made, and improvements on the model used are proposed.