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Optics Communications

solution to four-wave mixing in the reflection geometry utilising the SU (1, 1) group symmetry

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

Published Apr 1, 1992• 8 citations
Publisher
DOI
10.1016/0030-4018(92)90250-u

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

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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.

Journal of Modern Optics•Jan 1, 1993•6 citations
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.