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Journal of the Optical Society of America B

Exact solution of four-wave mixing of copropagating light beams in a Kerr medium

Eyal Lichtman, Asher A. Friesem, Robert G. Waarts, Henry H. Yaffe

Published Nov 1, 1987• 26 citations
Publisher

Abstract

The underlying equivalence between four-wave mixing (FWM) and self-phase modulation was exploited to develop a general formalism for FWM among light beams of different frequencies copropagating in a Kerr medium. The effectiveness of this formalism is demonstrated by analytically solving strong FWM of two and three input waves. The results predicted are then verified experimentally.

DOI
10.1364/josab.4.001801

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The underlying equivalence between four-wave mixing (FWM) and self-phase modulation was exploited to develop a general formalism for FWM among light beams of different frequencies copropagating in a Kerr medium. The effectiveness of this formalism is demonstrated by analytically solving strong FWM of two and three input waves. The results predicted are then verified experimentally.
The theory of probe beam amplification via degenerate optical wave mixing in a Kerr medium is presented. This theory includes consideration of the self-phase modulation, two-wave mixing, and multiwave mixing terms in the coupled-wave equations. Both analytical solutions, under the assumptions of a nondepleted pump beam and a transparent Kerr medium, and numerical solutions with these assumptions removed are presented. Our solutions show that under certain conditions, phase matching can be satisfied and a high probe beam gain is possible, even for a zero phase shift between the refractive index and the intensity gratings.
Intense light beams propagating in a lossless, dispersionless, single-mode optical fiber are subject to the Kerr effect, i.e., to the intensity-dependent refractive index of the fiber’s fused-silica core. Classically, Kerr-effect-induced self-phase modulation (SPM) can be used for spectral broadening of a picosecond pulse for grating-pair pulse compression down to femtosecond duration. Quantum mechanically, Kerr-effect-induced four-wave mixing (FWM) has been used to produce squeezed-state light. We present a quantum propagation theory for a lossless, dispersionless fiber with the Kerr nonlinearity. The theory includes classical SPM and quantum FWM within their regions of validity. It introduces a material time constant for the Kerr interaction, limiting the quantum phase shifts caused by the broadband zero-point fluctuations that accompany any input field, to develop a coarse-grained time multitemporal mode field analysis. Explicit expressions are obtained for the first and the second output-field moments when the fiber’s input field is in an arbitrary Gaussian state. These results are used to obtain homodyne-detection noise spectra, which are employed, in turn, to seek experimentally accessible manifestations of the Kerr time constant.
Optical phase conjugation by four-wave mixing in a Kerr medium is considered. Two strong counterpropagating lasers pump a nonlinear medium in which they excite the third-order polarization. Mixing with a weak incident field then generates the phase-conjugated image of that field. It is shown how the polarization of the pumps and the tensorial nature of the nonlinear interaction can be accounted for by a geometrical polarization tensor. The electric field is shown to obey two coupled wave equations, which couple a positive- and negative-frequency component of the field. These wave equations allow plane-wave solutions, and we derive the dispersion relation for the wave vectors of these modes. By matching the field of such a mode across the boundaries of the medium to an external incident field, we are able to obtain analytically the Fresnel coefficients for reflection and transmission. Our expressions reduce to earlier results in the appropriate limits.
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