Hexagonal spatial patterns for a Kerr slice with a feedback mirror

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Abstract
We study analytically and numerically a very simple nonlinear optical system, a thin slice of Kerr material with a single feedback mirror. Theoretical analysis shows that for both a focusing and defocusing medium the plane-wave solution is unstable above a certain input intensity and a hexagonal pattern of bright spots should form. The amplitude equations for this system are three coupled Ginzburg-Landau type equations. Numerical analysis confirms these results; moreover by further increasing the input intensity, the hexagonal solution becomes itself unstable and turbulent motion sets in.

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