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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Rotational Diffusion Effect On Time Reversal In Phase Conjugation Spectroscopy

J. P. Lavoine, A. A. Villaeys

Published Dec 22, 1989• 0 citations
Publisher

Abstract

Envelope time reversal can be achieved in a phase conjugation experiment. In this work, we study the influence of the various microscopic parameters of the nonlinear medium as well as the effects of rotational diffusion of the dipole moments on the envelope reversal. To this end, we consider a backward degenerate four-wave mixing experiment with pulsed fields and assume that any characteristic times of the envelopes or of the rotational diffusion are larger than the transverse decay time. Propagation and field polarizations effects are introduced. From the propagation equations solved in the slowly varying envelope approximation, we calculate the amplitude of the conjugated wave. Its expression depends on the field polarization as well as on the various overlap integrals of the pump and probe envelopes. In order to obtain a complete time reversal between the amplitudes of the conjugated wave E4(0,t) and probe field E3(0,t), supplementary contributions to E4(0,t) which do not satisfy the proportionality relation between E4(0,t) and E3*(0,-t) must wash-out. Some of these contributions cancel by taking a convenient choice of beam polarizations. It is shown here, that if the dipole moments undergo a rapid rotational diffusion, then complete time reversal is observed.

DOI
10.1117/12.961418

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Envelope time reversal can be achieved in a phase conjugation experiment. In this work, we study the influence of the various microscopic parameters of the nonlinear medium as well as the effects of rotational diffusion of the dipole moments on the envelope reversal. To this end, we consider a backward degenerate four-wave mixing experiment with pulsed fields and assume that any characteristic times of the envelopes or of the rotational diffusion are larger than the transverse decay time. Propagation and field polarizations effects are introduced. From the propagation equations solved in the slowly varying envelope approximation, we calculate the amplitude of the conjugated wave. Its expression depends on the field polarization as well as on the various overlap integrals of the pump and probe envelopes. In order to obtain a complete time reversal between the amplitudes of the conjugated wave E4(0,t) and probe field E3(0,t), supplementary contributions to E4(0,t) which do not satisfy the proportionality relation between E4(0,t) and E3*(0,-t) must wash-out. Some of these contributions cancel by taking a convenient choice of beam polarizations. It is shown here, that if the dipole moments undergo a rapid rotational diffusion, then complete time reversal is observed.
A theoretical study of the nonlinear polarization in a molecular medium is presented. It enables us to stress the importance and to evaluate the contribution of the population and orientational gratings in four-wave mixing depending on the pump- and probe-beam polarizations. The medium is described as an ensemble of two-level systems and is subject to rotational diffusion. The model is valid far or near resonance, as well. It is only assumed that time variation of the field envelopes and rotational diffusion constant are much smaller than the transverse decay rate ${\ensuremath{\Gamma}}_{2}$. The nonlinear polarization is expressed as a function of the field polarizations. The results are applied to a four-wave-mixing experiment to discuss the influence of the rotational diffusion. Finally, by including propagation effects, we use this formalism to calculate a phase-conjugated signal. This enables us to stress the influence of rotational diffusion to achieve time reversal of the envelope of a light pulse.
The main goal of this paper is to analyze a two-photon degenerate four-wave-mixing process that occurs in an isotropic media modeled by homogeneously broadened two-level systems having unequal permanent dipole moments and subject to rotational diffusion. Our theoretical treatment is based upon a third-order perturbative solution of the density-matrix equations and does not require the rotating-wave approximation. The medium being excited by three stationary fields, we give an analytical expression of the macroscopic polarization up to the third order, including permanent dipole moments, orientational diffusion, and field polarizations. We apply these general results to the description of a degenerate four-wave-mixing process by selecting, in the expressions, the contributions in the direction ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$-${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$+${\mathbf{k}}_{\ensuremath{\gamma}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathbf{k}}_{\ensuremath{\gamma}}$ being, respectively, the wave vectors of the exciting fields. We show that the two-photon degenerate four-wave mixing is dependent on the permanent dipole moments and is not present if these dipole moments are equal or zero. We analyze the influence of rotational diffusion. Different cases of field polarizations are also considered, and we discuss a comparison of the well-known one-photon degenerate four-wave-mixing process and this two-photon process. This process is also discussed in the particular case of phase conjugation.
We present the theory of resonant backward degenerate four-wave mixing in gaseous media for Doppler-broadened degenerate two-level systems. The degeneracy of the atomic levels as well as the influence of the polarization of the incident fields are taken into account within the framework of a tensorial formalism. The amplitude, line shape, and polarization of the reemitted phase-conjugate field are calculated up to the third order in the incident fields. The velocity average is performed in the approximation of large Doppler linewidths. One considers in detail the influence of depolarizing relaxation processes and the effects of pump-probe angular separation. In particular, the importance of the residual Doppler effect (shortening of the lifetime of the optically induced gratings by the atomic motion) associated with the pump-probe angle is emphasized. We show that the polarization of the reemitted fields is governed by very simple laws as soon as the residual Doppler effect overcomes the lifetimes of the atomic levels. Phase-conjugate mirrors can be simply characterized by a linear dichroism (parallel pump polarizations) or a birefringence (cross-polarized pumps). Vectorial phase conjugation is also analyzed, in the case of counter-rotating circular polarizations of the pumps. Finally, the case of two-photon transition is studied and compared with the case of a resonant transition.
We describe an experiment in phase conjugation using holographic degenerate four-wave mixing with picosecond pulses. Spatial and temporal distortions are simultaneously introduced on the probe beam, respectively, by a ground glass and a Gires-Tournois interferometer. We experimentally show that after it recrosses the ground glass we obtain a time reversed pulse on the reconstructed beam. Its temporal shape is the same as that without the ground glass. We theoretically discuss this result and show that in our configuration the temporal effect depends only on the hologram dimension parallel to the direction of propagation of the beam, while the spatial phase conjugation is related to the dimensions in the perpendicular plane.
Degenerate four-wave-mixing phase conjugation is analyzed in the case of two-photon transitions between levels having unequal dipole moments. The theoretical treatment is based upon a nonperturbative solution of the density matrix and does not require the rotating-wave approximation. We give an analytical expression for the polarization component oscillating at the field frequency for a medium excited by stationary waves. In the weak-probe-beam limit, the contributions in the direction of the probe and conjugated beams are evaluated. For instance, by using this two-photon process, instead of the usual single-photon one which is independent of the permanent dipole moments even in the strong-field limit, it is shown that lower intensities are sufficient to reach higher reflectivities. The influences of the permanent dipole moments, relaxation rates, and field frequency are also discussed.
This paper presents an exact solution to the slowly varying envelope equations for the probe and signal fluxes in degenerate four-wave mixing in Kerr media. The analysis is performed for a noncollinear geometry, and all beams have the same linear polarization in the nonlinear medium. It is assumed that the angle separating the directions of the counterpropagating pumps and probe and signal beams is large enough so that cross-coupled waves do not contribute to the envelope equations describing the spatial evolution of the pump, probe, and signal beams. Multiple values of the transmitted probe are obtained for equal pump fluxes above a threshold determined by the nonlinear phase shift in the medium. Significant pump depletion occurs in this regime, but it can also occur without the appearance of multiple solutions. Since exact optical phase conjugation occurs only in the limit where the pumps are undepleted in Kerr media, this result implies that there are fundamental upper limits on the pump intensity that can be used to increase signal reflectivities in optical phase conjugation.