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arXiv:cond-mat/9608069 · Europhysics Letters (EPL)

Brightness of a phase-conjugating mirror behind a random medium

J.C.J. Paasschens, Piet W. Brouwer, C. W. J. Beenakker

Published Jun 20, 1997• 2 citations
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Abstract

A random-matrix theory is presented for the reflection of light by a disordered medium backed by a phase-conjugating mirror. Two regimes are distinguished, depending on the relative magnitude of the inverse dwell time of a photon in the disordered medium and the frequency shift acquired at the mirror. The qualitatively different dependence of the reflectance on the degree of disorder in the two regimes suggests a distinctive experimental test for cancellation of phase shifts in a random medium.

DOI
10.1209/epl/i1997-00299-3

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A random-matrix theory is presented for the reflection of light by a disordered medium backed by a phase-conjugating mirror. Two regimes are distinguished, depending on the relative magnitude of the inverse dwell time of a photon in the disordered medium and the frequency shift acquired at the mirror. The qualitatively different dependence of the reflectance on the degree of disorder in the two regimes suggests a distinctive experimental test for cancellation of phase shifts in a random medium.
This is a theoretical study of the interplay of optical phase conjugation and multiple scattering. We calculate the intensity of light reflected by a phase-conjugating mirror when it is placed behind a disordered medium. We compare the results of a fully phase-coherent theory with those from the theory of radiative transfer. Both methods are equivalent if the dwell time ${\ensuremath{\tau}}_{\mathrm{dwell}}$ of a photon in the disordered medium is much larger than the inverse of the frequency shift $2\ensuremath{\Delta}\ensuremath{\omega}$ acquired at the phase-conjugating mirror. When ${\ensuremath{\tau}}_{\mathrm{dwell}}\ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\lesssim}1$, in contrast, phase coherence drastically affects the reflected intensity. In particular, a minimum in the dependence of the reflectance on the disorder strength disappears when $\ensuremath{\Delta}\ensuremath{\omega}$ is reduced below $1/{\ensuremath{\tau}}_{\mathrm{dwell}}.$ The analogies and differences with Andreev reflection of electrons at the interface between a normal metal and a superconductor are discussed.
We theoretically study reflection and transmission of light in a one-dimensional disordered phase-conjugating medium. Using an invariant imbedding approach a Fokker-Planck equation for the distribution of the probe light reflectance and expressions for the average probabilities of reflection and transmission are derived. A new crossover length scale for localization of light is found, which depends on the competition between phase conjugation and disorder. For weak disorder, our analytical results are in good agreement with numerical simulations.
We present experimental results showing the partial cancellation of the specular reflection off a lossless partially transmitting ordinary plane mirror in the presence of a phase conjugate mirror. This effect was first theoretically predicted by Drummond and Friberg.1 In the experimental situation a single domain photorefractive crystal of barium titanate operating via degenerate four-wave mixing acted as the phase conjugate mirror. It was placed behind a lossless partially transmitting ordinary plane mirror so that an optical resonant activity was created. Collimated light was directed onto the ordinary mirror at non-normal incidence. The intensity of the specular reflection was monitored, the extent of its cancellation for the measured phase conjugate reflectivities agreed well with the proposed theory.2 The experimental results lay marginally above the theoretical curve. The reason for such a systematic error is considered to lie with the fidelity of the phase conjugate generated.
We have repeated the classic experiment of Wiener,1 which established the nature of the standing-wave pattern produced by reflection at the surface of a metallic mirror, with the metallic mirror used in his experiment replaced by a phase-conjugate mirror. We find that the positions of the fringes produced by interference between the wave incident on the phase-conjugate mirror and the wave leaving the mirror depends on the phase of the incident wave. This result is in contrast to that obtained through the use of a metallic mirror, in which case the interference pattern is found to be independent of the phase of the incident wave. The origin of the different behavior in these two cases lies in the fact that the phases of the waves that pump the PCM provide a reference through which phase information carried by the signal wave can be obtained. The experimental results are in excellent quantitative agreement with recent theoretical predictions.2
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.