Two counterpropagating laser beams interacting in a photorefractive medium display self-organized generation of spatial patterns. The photorefractive two-wave mixing nonlinearity is mediated by the self-induced generation of a longitudinal refractive index grating - a Bragg mirror - as a result of the pump beam interference. The second pump beam is obtained by a reflection off an external feedback mirror. As is usual in pattern forming systems, one can identifiy two antagonistic processes: on one hand the generation of the self-induced Bragg mirror, on the other hand the pump beam depletion caused by the Bragg mirror which thereby depletes its own source. The solution of this antagonism is the spatial structuring of the system: Regions with low Bragg mirror amplitude provide high transmission of the incident pump beam. The transmitted parts of the pump beam propagate to the external mirror and back and created regions with a high Bragg mirror amplitude.

The fundamental process observable with a modulation instability is the initially explonential growth of a transverse modulation of the homogeneous pump beam from noise. The transverse spatial frequency of the growing mode is primarily determined by the spatial scale imposed through free space propagation to the feedback mirror and back. The azimutal orientation of the growing modes is initially undetermined, only nonlinear mode interaction leads to certain symmetries. In case that only modes with a single transverse wave number can grow, one always observed hexagonal patterns consisting of three modes with angles of 120 degrees in between [3]. If there are active modes with different wave numbers, mode interaction results in a quite different symmetries [4]. For the experimental system presented here such non-hexagonal modes are observed over a wide parameter region (multiplle pattern region[5]). Very often, more than a single pattern symmetry is a possible solution of the system, leading to effects such as coexistence and competition of patterns.

These properties mark the photorefractive single-feedback experiment as an attractive model system for the investigation of two-dimensional mode interaction in pattern formation. A high complexity of the observed patterns suggests research dealing with the phenomena of multistability and pattern competition. To this end, the investigation of static and dynamic control methods, primarily by means of amplitude and phase modulation in real and Fourier space, but also exploiting access to other parameters, is among the objectives of our research. The goal of the development of control methods is threefold: very often, simple suppression of undesired pattern formation is required. However, beyond mere suppression, control methods are sucessfully employed as tools for the analysis of pattern forming systems and, finally, self organized pattern formation is increasingly considered as a tool for the fabrication of structured materials [6],[7].

We investigate self-organized pattern formation in a single-mirror feedback experiment [8]. The first pump beam is directly obtained from the laser source and focused into the photorefractive crystal. The second beam is created by reflecting the original pump beam off the external mirror. Diffraction of the beam propagating to the mirror and back is responsible for the spatial coupling neccessary for the generation of transverse patterns.

In most of the available parameter space, the system spontaneously generates hexagonal patterns, that consist of three modes with degenerate transverse wave numbers. For virtual mirror positions approximately in the center of the photorefractive crystal however, a variety of non-hexagonal patterns is observed in the multiple pattern region. The existence of several symmetries can be traced to the nonlinear interaction of modes with different wave numbers but roughly equal growth rates [9]. The source for the individual modes and the theoretical determination of their growth rates is a topic of current research.

Starting from the paraxial wave equation and the Kukhtarev model of the photorefractive effect, the system is modelled by three coupled partial differential equations[10],[11]. The first two equations describe the two counterpropagating wave envelope amplitudes (A₁, A₂), which are coupled via the induced Bragg mirror whose amplitude is Q.

The third equation captures the temporal development of the material change, i.e. the refractive index modulation, in dependence on the local intensity of the pump beams interference pattern. Boundary conditions providing one input beam on one side and the external feedback mirror on the other side of the crystal complete the model. This system of equations can now be treated with the tools of nonlinear physics, foremost a linear stability analysis, in order to compare predictions obtained by the model with experimental data. In addition, computer experiments can be pursued by numerically solving the equations, providing a bridge between experimental reality an simplifications of the model and checking for the model's limitations.