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Journal of the Optical Society of America

Theory of phase conjugation by stimulated scattering in a waveguide*

R. W. Hellwarth

Published Aug 1, 1978• 68 citations
Publisher

Abstract

We consider the backward optical wave stimulated by a multimode, monochromatic, incident optical wave in a waveguide filled with a transparent nonlinear medium, when the incident wave is negligibly perturbed by the nonlinear processes. We derive the conditions on guide length, area, mode number, and Stokes shift in order that a given high percentage of the power in the backscattered field be the “phase conjugate” of the incident field, i.e., be proportional to its complex conjugate in the entrance plane of the waveguide.

DOI
10.1364/josa.68.001050

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Journal of Experimental and Theoretical Physics•Feb 1, 1999•4 citations
We consider the backward optical wave stimulated by a multimode, monochromatic, incident optical wave in a waveguide filled with a transparent nonlinear medium, when the incident wave is negligibly perturbed by the nonlinear processes. We derive the conditions on guide length, area, mode number, and Stokes shift in order that a given high percentage of the power in the backscattered field be the “phase conjugate” of the incident field, i.e., be proportional to its complex conjugate in the entrance plane of the waveguide.
We review the history and nature of stimulated scattering of optical beams and of the large phase-conjugate component often observed in stimulated backscattering, whether it be of Raman, Brillouin, Rayleigh, or other origin. We review the theory of the magnitude of the phase-conjugate component in stimulated backscattering of waves in a multimode optical guide (the usual experimental configuration) or in free focus. Wide applicability of stimulated backscattering for optical phase conjugation is found.
Recently it was shown that in disordered media, nonlinear backscattering of phase-conjugated signal light is possible in the presence of only a single pump wave, in contrast to the case of an ordered medium where two counterpropagating pump waves are necessary. Formally, this disorder-induced nonlinear process corresponds to the cooperon-channel (back)scattering of the signal wave in the presence of diffusion propagation of the pump wave. (The cooperon is the Cooper particle-particle diffusion propagator.) In this contribution, the question is considered whether the disorder-induced backscattering of phase-conjugated light can restore the disturbed signal wave front as it does in the ordered case. Examining correlations of the outgoing waves corresponding to different incoming signal waves, it is shown that the backscattered waves possess a ``phase memory'' even for a large difference of the angles of incidence of incoming waves. This is in contrast to the case of linear backscattering, where such correlations take place only at a very small difference of the angles of incidence (of the order of backscattering peak width). Thus the phase conjugation in disordered media does possess the peculiar coherent properties of that in ordered media.
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.
Optical phase conjugation is concerned with the use of nonlinear optical effects to precisely reverse the direction of propagation of each plane wave in an arbitrary beam of light. This causes the return beam to exactly retrace the path of the incident beam, leading to many applications of automatic pointing and tracking (even in the presence of an inhomogeneous medium). The main goals of this course are familiarization with how each of the many nonlinear optical effects gives rise to a phase conjugate signal, and understanding and appreciation of the many potential applications for optical phase conjugation. Conjugation techniques involving Kerr-like degenerate four-wave mixing, resonant degenerate four-wave mixing, stimulated Brillouin scattering, photorefractive effects, surface phenomena, and others will be carefully reviewed. Applications of particular importance to this course will include distortion correction, pointing, tracking, laser-resonator design, optical information processing, etc. Throughout the course, key references will be identified for those wishing to pursue any topic more vigorously.
A quasi-collinear and partially degenerate four-wave mixing model is proposed to explain the optical phase-conjugation property of various types of stimulated backscattering. According to this model, after passing through a phase-disturbed medium or an aberration plate, the input pump beam can be resolved into two portions: a stronger undisturbed regular portion and a weaker phase-disturbed irregular portion. These two portions interfere with each other and create a volume holographic grating in the pumped region of the scattering medium. Only the stronger undisturbed portion of the pump field can excite an initial backward stimulated scattering beam with a regular wavefront. When the latter (as a reading beam) passes through the induced holographic grating, a diffracted wave will be created and then amplified together with the reading beam. A rigorous mathematical analysis shows that under certain conditions the combination of these two portions (the reading wave and the diffracted wave) of the backward stimulated scattering can be an approximate phase-conjugate field of the input pump field. The major theoretical conclusions are basically supported by the experimental results based on a specially designed two-beam interference setup.