Dynamic Solutions and Instabilities of the Four-wave Mixing Interaction Utilising the Underlying SU(2) Group Symmetry

Optical phase conjugation is an important nonlinear process, with many applications in the areas of optical communications and optical processing. This is as a result of the wave-front correction properties that phase conjugation offers and this has generated much interest in the area. A common method of producing phase conjugated wavefronts is the four-wave mixing interaction and many publications on the steady state solution to this problem have appeared over the last decade. More recently, however, interest has been focussed upon the temporal behaviour of four-wave mixing systems, with instabilities and chaos being both demonstrated by Gauthier et al [1] and predicted by Królikowski et al [2]. To date, the analysis for the temporal behaviour of anisotropic four wave mixing, in the transmission grating regime, has involved the direct numerical integration of the equations describing four wave mixing. This analysis however, does not utilise the symmetries of the four wave mixing process, and thus results in a more complex problem. Through exploitation of these symmetries the complexity of the problem has been reduced from one containing four complex variables, to one containing three real variables. This has been accomplished through the use of the Special Unitary Group 2, a group providing a two dimensional matrix, whose elements are functions of three real variables. The multiplication of this matrix, together with one containing the boundary conditions for the problem, thus enables the four complex beam amplitudes to be reexpressed in terms of three real quantities. Using this technique, the temporal nature of anisotropic four wave mixing has been studied, and shown under certain conditions, to exhibit chaotic behaviour when an electric field is present. The effect of any material absorption on the chaotic nature is also demonstrated.