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Journal of Physics B Atomic and Molecular Physics

Polarisation selection rules and disorienting collision effects in resonant degenerate four-wave mixing

Daniel Bloch, M. Ducloy

Published Jul 28, 1981• 23 citations
Publisher

Abstract

Various polarisation selection rules guiding phase conjugate emission via degenerate four-wave mixing are discussed. It is shown that, in resonant gas media, depolarising collisions may induce phase conjugation in cases when this non-linear process in otherwise forbidden. However, Doppler broadening often limits the efficiency of non-collinear collision-induced four-wave mixing and introduces a very rapid decrease of the reflectivity with increasing angular separations between pump and probe beams.

DOI
10.1088/0022-3700/14/14/003

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Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: Saturation and polarization effects

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The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.

The Journal of Chemical Physics•Dec 8, 1999•22 citations
Various polarisation selection rules guiding phase conjugate emission via degenerate four-wave mixing are discussed. It is shown that, in resonant gas media, depolarising collisions may induce phase conjugation in cases when this non-linear process in otherwise forbidden. However, Doppler broadening often limits the efficiency of non-collinear collision-induced four-wave mixing and introduces a very rapid decrease of the reflectivity with increasing angular separations between pump and probe beams.
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 present in this paper a theoretical study of the angular dependence of phase-conjugate emission via resonant degenerate four-wave mixing (FWM) in Doppler-broadened gas media. The nonlinear optical polarization is calculated through an expansion of the atomic density matrix up to the third order in the incident fields. One finds that, when the incident frequency is resonant for a single-photon transition, both strength and lineshape of the phase-conjugate emission strongly depend on the angle 0 between standing pump wave and probe wave. The emission intensity, maximum at grazing incidences, decreases rapidly by several orders of magnitude when 0 increases. Simultaneously the emission linewidth increases from homogeneous to Doppler width. Similar effects are predicted for nearly-degenerate FWM (different pump and probe frequencies). In particular, one shows that the bandwidth of the equivalent optical reflection filter undergoes a Doppler-broadening linearly increasing with sin θ. On the opposite, none of the above effects is predicted in two-photon-resonant FWM which does not exhibit any angular dependence.
The theory of polarised light phase conjugation (PC) in non-degenerate four-wave mixing in a resonant medium has been developed in the irreducible tensorial formalism. The theory is applied to a wave with any polarisation in a resonant medium with levels of arbitrary momentum in the absence of pump-induced saturation effects. The formula for the non-linear susceptibility of the medium has been found and it is shown that the change of the polarisation at PC is essentially connected with the relaxation of the atomic multipole momenta (population, orientation and alignment). Wave equations were investigated for different pump-wave polarisations and three different physical mechanisms of the PC are discussed. The influence of the different relaxation rates of the atomic multipole momenta on the PC reflectivity has been investigated. The spectral dependence of the reflectivity and the spectral ranges which depend on the wave polarisation are found. For motionless atoms with small momenta (J 1 = 1 / 2 to J 2 = 1 / 2 , J 1 =1 to J 2 =0), exact formulae for the non-linear susceptibility of the medium as a function of pump intensity were found. The spontaneous decay of higher level to the lower one was taken into account. Because of optical pumping, saturation effects can be shown at considerably smaller intensities than in the usual two-level system. The exact expressions for the PC reflectivity at different polarisations of the pump waves are given.
Polarization properties of phase-conjugate (PC) mirrors obtained by degenerate four-wave mixing in resonant gas media are experimentally analyzed on various neon transitions and theoretically interpreted. We show that, because of the Doppler broadening of the decay rate of the gratings, these properties are rather insensitive to the relaxation processes in the gas phase. We demonstrate simple ways of characterizing PC mirrors with either lambda/2 birefringence (for cross-polarized pumps) or J-dependent linear dichroism (for parallel-pump polarization).
In this paper, we present a detailed analysis of saturation effects in resonant degenerate four-wave mixing (phase conjugation) or multiwave mixing (high-order diffraction on an optically induced grating). The experiments are performed in a low-pressure Ne discharge in the Doppler-broadened regime, both on an "ideal" two- or three-level system (1s_{4}(J = 1) - 2p_{3}(J = 0)) or on a degenerate multi-level system (1s_{5}(J = 2) - 2p_{9}(J = 3)). The saturation effects appearing when both pump beams are intense are studied, and a very good agreement is found with the previously developed theory [8]. It is demonstrated that there is an absolute maximum to the phase-conjugate reflectivity, and various effects on the lineshape (splitting, linewidth, etc.) are investigated. The effects produced by the Gaussian nature of the incident beams are analyzed in order to interpret some of the discrepancies between theory and experiment. Several new theoretical results for high-order diffraction are also presented, and compared with the experiments. The possibility of dealing with the saturation effects in four-wave mixing when all three incident fields are saturating appears as a consequence of these studies on multiwave mixing. Notably discussed are the case of an intense diffracting field, and of intense grating fields, whose intensity may be unbalanced. Finally, the polarization selection rules for high-order diffraction experiments are established.
The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.