Hexagonal optical structures in photorefractive crystals with a feedback mirror

A nonlinear theory is presented for the formation of hexagonal optical structures in a photorefractive medium equipped with a feedback mirror. Oppositely directed beams in photorefractive crystals are unstable against the excitation of sideband waves. It is shown here that as this instability evolves to its nonlinear stage, the three-wave interaction between weak sideband beams does not stabilize it, but rather leads to explosive growth of the amplitudes of beams whose transverse wave vectors form angles that are multiples of p/3. As a result, sideband beams at these angles are found to be correlated. A range of parameters is found in which four-wave interactions saturate the explosive instability, which explains the appearance of stable hexagons in the experiment. Outside this region, nonlinearities of higher order saturate the explosive instability, and the process of hexagon generation must be studied numerically. Matrix elements are obtained for the threeand four-wave interactions as functions of the distance to the feedback mirror, and an equation for the time evolution of the sideband wave amplitudes is derived that describes the hexagon generation. A comparison is made with experimental results for the photorefractive crystals KNbO3 and BaTiO3. © 1998 American Institute of Physics. @S1063-7761~98!02903-5#