Nonlinear vectorial two-beam coupling and forward four-wave mixing in photorefractive materials.

In this Letter we present an exact solution with linear absorption for a system of nonlinear differential equations that model two separate photorefractive processes. The first is the coupling of two beams with general polarizations, restricted to the case when the magnitude of the coupling for one polarization is the same as that for the other polarization.' The second is scalar forward four-wave mixing, 2 which has practical importance when noncounterpropagating phase conjugation is required, as in the case of image transmission through two lengths of equivalent optical fiber.3 When a pair of beams writes a grating in a photorefractive medium, that grating in its turn affects the propagation of the writing beams, which are thus coupled nonlinearly to each other.4' 5 The nonlinear coupled-wave equations that describe the interaction in the scalar case-when the two beams have the same polarizations-have been solved analytically. 4' 5 This solution describes the energy exchange between the beams when a phase shift exists between the light interference fringe pattern and the refractive-index grating and is the basic cornerstone to an understanding of the broader effects of four-wave mixing in photorefractive materials.6 The situation becomes more complicated when one considers the interaction of two beams with general polarizations. Four coupled nonlinear differential equations then arise, describing the interactions among the four components of the beams' optical electric fields. The nonlinear source term from the selfinduced grating causes the overall coupling of the equations. Our solution for these equations is valid in the case when direct cross coupling between vertical and horizontal polarization is negligible. This will often be the case in practical situations, where birefringence does not allow phase matching for such couplings. We also require that the magnitude of the coupling constants be the same for both polarizations. Mapping this problem to the known solution of fourwave mixing in photorefractive materials 6' 7 will let us follow similar mathematical procedures toward an explicit analytical solution including the effects of linear absorption. The vectorial interaction geometry that we consider is shown in Fig. 1. Two beams with frequency w propagate in the crystal from z = 0 to z = 1. The vectorial field amplitude of each beam is represented by two orthogonal components. Al and A2 are the components of the first beam, and A4 and A3 are components of the second one. Note that the first components of