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Journal of Modern Optics

The Steady State Solution and Symmetry of Four-wave Mixing with Absorption

David A. Fish, A.K. Powell, Trevor J. Hall

Published Jan 1, 1993• 6 citations
Publisher

Abstract

The solution of the scalar four-mixing equations in the transmission geometry in the presence of material absorption has been found to represent a difficult problem when dealing with photorefractive materials. A new matrix approach is taken here which allows the identification of an underlying symmetry based upon the SU(2) group. This symmetry is used to uncouple the wave equations, and solutions are then found in terms of hypergeometric functions. This represents an analytical solution to the coupled wave equations, but a numerical solution to the boundary value problem is required. The full numerical solution is then performed in terms of the Euler angles that parameterize the SU(2) group. The symmetry introduced by the group reduces the computation to the evaluation of two real variables, instead of the evaluation of four complex beam amplitudes. Results are shown for parallel and orthogonally polarized pump four-wave mixing with shifted gratings. The novel result of absorption enhanced phase conjugate reflectivities is shown, and results are presented for the first time that show the effect of absorption on arbitrarily shifted gratings.

DOI
10.1080/09500349314550121

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Journal of the Optical Society of America B•Oct 1, 1996•2 citations
The solution of the scalar four-mixing equations in the transmission geometry in the presence of material absorption has been found to represent a difficult problem when dealing with photorefractive materials. A new matrix approach is taken here which allows the identification of an underlying symmetry based upon the SU(2) group. This symmetry is used to uncouple the wave equations, and solutions are then found in terms of hypergeometric functions. This represents an analytical solution to the coupled wave equations, but a numerical solution to the boundary value problem is required. The full numerical solution is then performed in terms of the Euler angles that parameterize the SU(2) group. The symmetry introduced by the group reduces the computation to the evaluation of two real variables, instead of the evaluation of four complex beam amplitudes. Results are shown for parallel and orthogonally polarized pump four-wave mixing with shifted gratings. The novel result of absorption enhanced phase conjugate reflectivities is shown, and results are presented for the first time that show the effect of absorption on arbitrarily shifted gratings.
We present an eight-dimensional matrix formulation of vector four-wave mixing. We study the case of a nonoptically active photorefractive for the transmission grating. The coupled equations are represented in a group formalism and irreducible forms are found. We show that when cross-polarization coupling is neglected, the system of equations is governed by two coupled SU(2) groups. We develop a theory that allows for pump depletion and electric-field-induced bulk birefringence. Analytical solutions are found when cubic 4¯3m materials are considered in two standard experimental geometries, i.e., electric fields perpendicular to the (110) crystal faces and electric fields perpendicular to the (001) crystal faces. Results are presented that show the effects of linear and circular polarized beams, and the use of these materials for vector phase conjugation is discussed. Solution routes to other geometries and noncubic photorefractive materials are also presented.
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.
The effects of linear absorption on the degenerate and nondegenerate self-oscillation of a photorefractive phase-conjugate mirror under condition of anisotropic four-wave mixing are considered. The phase conjugation results from nonlinear mixing of two strong undepleted orthogonally polarized counterpropagating pump waves with a probe wave. We consider two types of situation for orthogonally polarized pump beams corresponding to two crystal families: one such as the Bi12SiO20 crystal family (sillenite family) and the other such as the BaTiO3 and strontium barium niobate crystal family. In both of these crystal families the degenerate and nondegenerate self-oscillations are shown to occur in the presence of linear absorption, which tends to increase the value of the coupling coefficient required for self-oscillation. Numerical results are presented that exhibit various features of degenerate and nondegenerate self-oscillation in both crystal families for different values of the phase ϕb of the backward grating.
A generalized approach to the solution of paraxial four-wave-mixing problems is identified in this paper. It is shown that a Lie group symmetry SU(2, 2) exists for the multiple-grating four-wave-mixing problem. The cases of a reflection or a transmission grating are examined and shown to be irreducible subgroups of the full case. It is shown that a transmission or a reflection grating can be reduced to previous formalisms within this new framework and that a twofold degeneracy exists for these cases. These solutions provide the basis for attempting more complex problems within this framework. Results are also presented for the transmission and reflection gratings, which show stark contrasts between the solution manifolds. An approach to the solution of the mixed-transmission–reflection-grating problem is identified by use of the group formalism.
The effects of linear absorption on the degenerate and nondegenerate self-oscillation conditions of a photorefractive phase-conjugate mirror are explored. In this study, phase-conjugation results from nonlinear mixing of two strong undepleted parallel-polarized counterpropagating pump waves with a probe wave. Numerical results are presented that illustrate that the linear absorption tends to increase the value of the coupling coefficient required for self-oscillation. It is also found that the pump intensity ratio r is almost independent of the linear absorption for positive normalized detuning Δωτ between the probe and the pump frequencies and of the zero phase shift, ϕ, between the gratings and the light interference patterns when the gratings are stationary. For ϕ = ±π/2, the variation of r with respect to absorption is symmetric for positive and negative detunings.