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arXiv:physics/9708010 · Physical Review A

Reflection of light from a disordered medium backed by a phase-conjugating mirror

J.C.J. Paasschens, M. J. M. de Jong, Piet W. Brouwer, C. W. J. Beenakker

Published Nov 1, 1997• 16 citations

Abstract

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.

DOI: 10.1103/physreva.56.4216

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Nature of the interference pattern produced on reflection at a phase-conjugate mirror

Arturo A. Jacobs, Wayne R. Tompkin, Robert W. Boyd

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

Annual Meeting Optical Society of America•Jan 1, 1986•0 citations

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.

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.

In the case of disordered nonlinear media, it is shown that in four-wave mixing a phenomena analogous to phase conjugation may occur. In contrast to the case of transparent media in which two counterpropagating pump waves (with frequency ${\ensuremath{\omega}}_{p}$) are required, phase conjugation may occur in the presence of a single pump wave injected into the disordered nonlinear material. We find that in the angular distribution of radiation of frequency ${\ensuremath{\omega}}_{s}$ (${\ensuremath{\omega}}_{s}$=2${\ensuremath{\omega}}_{p}$-${\ensuremath{\omega}}_{i}$, where ${\ensuremath{\omega}}_{i}$ is the frequency of probe wave) there is a narrow peak in the backscattered direction associated with the probe wave. The intensity of this peak may be some orders of magnitude larger than the diffuse background at the frequency ${\ensuremath{\omega}}_{s}$.

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.

Motivated by the analogy between a phase-conjugating mirror (PCM) and a superconductor, we search for optical counterparts of the well-known DC and AC Josephson effects. We show that in a system consisting of two PCM`s separated by vacuum an ,,optical supercurrent`` arises as a function of an applied phase difference between the PCM`s, which is the optical analogue of the DC supercurrent flowing in a superconducting weak link. The corresponding AC effect occurs when the two PCM`s are pumped by light of a different frequency, causing the phase difference to oscillate in time with the frequency difference.

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