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Journal of the Optical Society of America B

Irreducible forms of the SU(2, 2) group in four-wave mixing

David A. Fish, A. K. Powell

Published Feb 1, 1997• 1 citation
Publisher

Abstract

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.

DOI
10.1364/josab.14.000285

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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 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.
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.
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.
We report on the experimental realization and theoretical analysis of four-level systems produced in passive optical ring cavities by introducing various intracavity elements. Birefringent elements couple waves with different polarizations and thus lift the polarization degeneracy of the cavity modes. Reflecting elements couple waves propagating in different directions and lift the propagation degeneracy. A combination of these elements leads, in general, to a splitting of every longitudinal cavity mode into a four-mode system. Under the restriction that the optical components are loss-free, the description of these ring cavities in a transmission-matrix formalism leads in a natural way to the study of the Lie group U(2,2) and its Lie algebra u(2,2). We associate each of the 16 generators of this algebra with a specific type of optical element, some of which are standard components, others not. From the commutation relations of the generators, we derive a recipe for the construction of the nonstandard components as a sequence of standard ones. It follows that the entire U(2,2) group can actually be realized. The number of independent parameters---16 for a general U(2,2) element---is shown to be reduced substantially if the optical components are selected out of a restricted number of types, provided that the corresponding generators define a subalgebra of u(2,2). In such cases, a subgroup of U(2,2) is realized and the number of independent parameters of the optical system is given by the number of generators of the subalgebra. A connection is established between the subalgebras and symmetry properties of the optical components in the cavity. We consider the influence of symmetries on the mode structure and consider our experimental results from this viewpoint. In particular, we discuss time-reversal invariance, leading to the symplectic group USp(2,2), and we identify antiunitary symmetries leading to Kramers's degeneracy in the mode spectrum. We propose the application of the group U(2,2) in optics as a tool yielding direct practical results.