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Journal of the Optical Society of America

Theory of saturated line shapes in phase-conjugate emission by resonant degenerate four-wave mixing in Doppler-broadened three-level systems

Daniel Bloch, M. Ducloy

Published May 1, 1983• 67 citations
Publisher

Abstract

We present a theoretical study of phase-conjugate (PC) emission by backward degenerate four-wave mixing (DFWM) processes in a resonant gas medium. In this study, high-intensity (pump) saturation and Doppler broadening are simultaneously taken into account. We develop a general formalism to deal with the saturation effects in the case of a three-level atom (with degenerate frequencies) and obtain analytical expressions of the PC field amplitude (after integration over the velocity distribution) under the following conditions: (1) the probe beam and one of the pump beams have intensities weak enough to be treated at the lowest perturbation order, whereas the other pump intensity is arbitrarily high; (2) the probe crosses the standing pump wave at grazing incidence. It is shown that the saturation induced by the forward pump (copropagating with the probe beam) leads to entirely different behavior of the nonlinear susceptibility from that induced by the backward pump (counterpropagating with the probe beam). This saturation anisotropy is specific to the inhomogeneous broadening of the gas medium, which is also responsible for the frequency splitting experimentally observed in the intensity line shape of the PC emission for large pump intensities. At saturation, this splitting is theoretically shown to be proportional to the Rabi frequency, with a proportionality coefficient critically dependent on the relaxation processes; in the case of a single-relaxation model, the splitting is equal to 1.5 times the Rabi frequency. The PC intensity line shape is demonstrated to be of dispersive origin in the case of backward saturation, whereas for forward saturation, narrow structures are predicted in both real and imaginary parts of the nonlinear susceptibility, also leading to the intensity line-shape splitting. One also finds that the reflectivity of PC mirrors based on resonant DFWM strongly depends on the origin of the saturation. For a saturating forward pump, the PC emission intensity tends toward zero with increasing pump intensities, and it reaches a finite (nonzero) limit in the case of backward saturation. Most of these predictions have been observed experimentally [ BlochD., Phys. Rev. Lett.49, 719 ( 1982)].

DOI
10.1364/josa.73.000635

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Theoretical investigation of the forward phase-matched geometry for degenerate four-wave mixing spectroscopy

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We examine theoretically the degenerate four-wave mixing (DFWM) signal intensities and line shapes obtained with the forward phase-matched geometry in which all beams propagate in the same direction and compare the results to those of the phase-conjugate geometry with counterpropagating pump beams. To examine the forward phase-matched geometry, we modify a theoretical approach used previously to calculate phase-conjugate DFWM signal intensities. This theoretical approach, which involves numerical integration of the time-dependent density-matrix equations, is validated for the forward phase-matched geometry by comparison of our calculated line shapes to both a perturbative solution and to experimental data. This methodology is then used to compare the signal intensities and line shapes obtained with the forward phase-matched geometry and the phase-conjugate geometry in the perturbative (low laser power) and saturated (high laser power) regimes. In the perturbative regime the forward phase-matched signal exhibits less sensitivity to the Doppler linewidth. At pump laser intensities approximately equal to the saturation intensity the signal for the forward phase-matched geometry is stronger than that for the phase-conjugate geometry for primarily Doppler-broadened resonances, assuming the same probe volume for both geometries. These advantages warrant further investigations employing the forward phase-matched configuration for DFWM measurements of gas-phase species.

Journal of the Optical Society of America B•Oct 1, 1998•15 citations
We present a theoretical study of phase-conjugate (PC) emission by backward degenerate four-wave mixing (DFWM) processes in a resonant gas medium. In this study, high-intensity (pump) saturation and Doppler broadening are simultaneously taken into account. We develop a general formalism to deal with the saturation effects in the case of a three-level atom (with degenerate frequencies) and obtain analytical expressions of the PC field amplitude (after integration over the velocity distribution) under the following conditions: (1) the probe beam and one of the pump beams have intensities weak enough to be treated at the lowest perturbation order, whereas the other pump intensity is arbitrarily high; (2) the probe crosses the standing pump wave at grazing incidence. It is shown that the saturation induced by the forward pump (copropagating with the probe beam) leads to entirely different behavior of the nonlinear susceptibility from that induced by the backward pump (counterpropagating with the probe beam). This saturation anisotropy is specific to the inhomogeneous broadening of the gas medium, which is also responsible for the frequency splitting experimentally observed in the intensity line shape of the PC emission for large pump intensities. At saturation, this splitting is theoretically shown to be proportional to the Rabi frequency, with a proportionality coefficient critically dependent on the relaxation processes; in the case of a single-relaxation model, the splitting is equal to 1.5 times the Rabi frequency. The PC intensity line shape is demonstrated to be of dispersive origin in the case of backward saturation, whereas for forward saturation, narrow structures are predicted in both real and imaginary parts of the nonlinear susceptibility, also leading to the intensity line-shape splitting. One also finds that the reflectivity of PC mirrors based on resonant DFWM strongly depends on the origin of the saturation. For a saturating forward pump, the PC emission intensity tends toward zero with increasing pump intensities, and it reaches a finite (nonzero) limit in the case of backward saturation. Most of these predictions have been observed experimentally [ BlochD., Phys. Rev. Lett.49, 719 ( 1982)].
We analyze the properties of phase-conjugate (PC) emission via resonant, Doppler-broadened, backward four-wave mixing in the saturation regime, when both counterpropagating pump waves are arbitrarily intense. One considers two atomic situations, either a four-level system modeled as a pair of uncoupled transitions, or a three-level system. The first case leads to an analytical expression of the PC field, for arbitrary pump intensities, in the Doppler limit. The second case, in general, necessitates a numerical solution, except in the asymptotic regime when one pump is very intense compared to the other one. One finds that both cases lead to quite similar predictions: (i) two-peaked or four-peaked emission line shape, depending on the forward to backward pump intensity ratio; (ii) the maximum of PC reflectivity is obtained for infinite backward saturation, but for a fi-p nite value of the forward pump intensity; (iii) for backward saturation, the emission line shape is practically independent of the forward pump intensity. Finally, the theoretical analysis is extended to the case of a saturating probe and its effect on emission line shape and PC reflectivity, as well as its application to the properties of emission processes involving higher-order optical susceptibilities (degenerate multiwave mixing).
Nonlinear effects of sublevel coherence on resonant backward-degenerate four-wave mixing are analyzed in the saturation regime of an inhomogeneously broadened and optically thin gas medium, composed of \ensuremath{\Lambda}-type three-level atoms. No limitations are set to the pump-field intensities, and an analytical approach is presented in the Doppler limit and efficient transverse-optical-pumping regime. It is shown that coherent population trapping due to strong pump-field coupling has a negligible incidence in phase conjugation, while interference of two imbalanced coherent-population-injection channels determines the mixing process. Level-crossing emission resonances are found to exhibit particular line-shape characteristics: (1) an insensitivity to power shifts, (2) strong power-broadening accompanied by incomplete line splitting, or (3) field-induced line-broadening inhibition. The conditions under which these effects should be observable are identified, and their dependence on pump imbalance is remarked.
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.
We present in this paper a theory of the saturation effects in degenerate four-wave mixing which makes use of the dressed-atom method. We mainly study the case of a Doppler-broadened line, but several results can be applied to the case of nearly degenerate four-wave mixing for a Doppler-free medium. We show that a close relationship does exist between the two problems : the generated beam is mainly created when the detuning between two incident beams is equal to the Rabi frequency. In the degenerate case, this condition is fulfilled by two velocity groups because of the Doppler shift. We analytically calculate the lineshapes for two-level atoms in the case of forward saturation (the intense pump beam propagates in the same direction than the probe beam) and in the case of backward saturation (the intense pump beam and the probe beam propagate in opposite directions). We also study collisional effects and the influence of the intensity of the weak pump beam when perturbation theory can no longer be applied.
The effect of an intense pump beam on the intensity of the phase-conjugated emission is studied in the case of a three-level atom. We first extend to the case of cross-polarized pump beams the dressed-atom model that we have presented previously [Phys. Rev. A 34, 4008 (1986)] for the case of two-level atoms. The range of validity of our theory is \ensuremath{\Gamma} (natural width) ${\ensuremath{\Omega}}_{1}$ (Rabi frequency of the intense pump) ku (Doppler width). We have studied the line shape in degenerate four-wave mixing, the influence of a static magnetic field, and the resonances that appear in nondegenerate four-wave mixing. Experiments are done in a neon discharge using cw dye lasers. The experimental results are in good agreement with the theoretical predictions. In degenerate four-wave mixing, we have shown that the saturating dip disappears when the weak beams have orthogonal polarizations. In the presence of a static magnetic field and in the nearly degenerate case, narrow resonances corresponding to transitions between energy levels of the dressed atom are observed. The intensity and the width of these resonances are in very good agreement with the predictions of the theoretical model.
We examine theoretically the degenerate four-wave mixing (DFWM) signal intensities and line shapes obtained with the forward phase-matched geometry in which all beams propagate in the same direction and compare the results to those of the phase-conjugate geometry with counterpropagating pump beams. To examine the forward phase-matched geometry, we modify a theoretical approach used previously to calculate phase-conjugate DFWM signal intensities. This theoretical approach, which involves numerical integration of the time-dependent density-matrix equations, is validated for the forward phase-matched geometry by comparison of our calculated line shapes to both a perturbative solution and to experimental data. This methodology is then used to compare the signal intensities and line shapes obtained with the forward phase-matched geometry and the phase-conjugate geometry in the perturbative (low laser power) and saturated (high laser power) regimes. In the perturbative regime the forward phase-matched signal exhibits less sensitivity to the Doppler linewidth. At pump laser intensities approximately equal to the saturation intensity the signal for the forward phase-matched geometry is stronger than that for the phase-conjugate geometry for primarily Doppler-broadened resonances, assuming the same probe volume for both geometries. These advantages warrant further investigations employing the forward phase-matched configuration for DFWM measurements of gas-phase species.