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Applied Optics

Degenerate four-wave-mixing line shapes of hydroxyl at high pump intensities

Michael S. Brown, Larry A. Rahn, Robert P. Lucht

Published Jun 20, 1995• 18 citations
Publisher

Abstract

The degenerate four-wave-mixing spectral profile of the R(1)(9) transition in the A(2)Σ(+) ? X(2)∏(0, 0) band of OH has been measured for various combinations of saturating pump beams. With increasing pumpbeam intensity the degenerate four-wave-mixing line shape changes dramatically near line center. In phase-conjugate geometry, three distinct spectral line shapes were observed for the cases of (1) equally intense pump beams, (2) a strong forward pump and a weak backward pump, and (3) a weak forward pump and an intense backward pump. A significant saturation dip appears in the spectrum near line center for case (3). The measured spectra have been modeled by the use of nonperturbative numerical solutions of the density matrix equations, and agreement between the calculations and the experimental results is excellent. The differences in the saturated line shapes for cases (2) and (3) are explored theoretically, and the calculated results are compared with previous theoretical work [Bloch and Ducloy, J. Opt. Soc.Am. 73, 635 (1985)] in which the line shapes were calculated in the limit of infinite Doppler broadening.

DOI
10.1364/ao.34.003274

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Physical Review A•Jun 19, 2003•3 citations
The degenerate four-wave-mixing spectral profile of the R(1)(9) transition in the A(2)Σ(+) ? X(2)∏(0, 0) band of OH has been measured for various combinations of saturating pump beams. With increasing pumpbeam intensity the degenerate four-wave-mixing line shape changes dramatically near line center. In phase-conjugate geometry, three distinct spectral line shapes were observed for the cases of (1) equally intense pump beams, (2) a strong forward pump and a weak backward pump, and (3) a weak forward pump and an intense backward pump. A significant saturation dip appears in the spectrum near line center for case (3). The measured spectra have been modeled by the use of nonperturbative numerical solutions of the density matrix equations, and agreement between the calculations and the experimental results is excellent. The differences in the saturated line shapes for cases (2) and (3) are explored theoretically, and the calculated results are compared with previous theoretical work [Bloch and Ducloy, J. Opt. Soc.Am. 73, 635 (1985)] in which the line shapes were calculated in the limit of infinite Doppler broadening.
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).
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.
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)].
An analytical expression for the degenerate four-wave mixing signal produced in a medium composed of two-level atoms by pump and probe lasers of arbitrary intensities and parallel linear polarization, suffering negligible absorption and depletion, has been derived. The Abrams–Lind approach [Opt. Lett.2, 94 and 3, 205 (1978)] for deriving the phase-matched polarization has been extended to find all the other higher-order polarization terms contributing to the signal. The signal field amplitude is expressed as an infinite series whose convergence rate depends on the intensity of the input fields and their detuning from resonance. The solution is successfully verified by comparison with a full numerical nonperturbative calculation and is found to be equivalent to the Abrams–Lind result in the limit of weak probe intensity. The result is used to calculate the saturation effects on spectral line shape that are relevant to simulation of molecular spectra obtained by degenerate four-wave mixing with saturating pump and probe fields. The result is extended to treat atomic motion in the case of forward geometry of the input beams.
A theoretical description for a forward degenerate four-wave mixing process in homogeneously broadened two-level absorbers with arbitrary intensities of all four beams is presented. Assuming no pump absorption and depletion, we obtain an analytical solution of forward degenerate four-wave mixing signal intensity in the limit of the strong pumps and weak probe and signal. In the case of low absorption or large detuning, this solution is shown to become equivalent to that of the phase-conjugate degenerate four-wave mixing derived by Abrams and Lind [Opt. Lett.2, 94 (1978)]. For arbitrary beam intensities and absorption parameters, the signal intensity is numerically calculated by solution of the coupled equations of complex wave amplitudes. Comparing the results of the numerical calculation with those of the analytical solution, we discuss the validity of using the analytical solution in practical experiments and show the saturation behavior that is due to the strong probe beam. The line shapes under the various ratios among the input beams and absorption parameters are also obtained and discussed.
We study phase-conjugate degenerate four-wave mixing in an optically pumped, two-level atomic system (the 3s2S1/2,{3s}^{2}{S}_{1/2},3s2S1/2​, F=2,F=2,F=2, ${m}_{\mathrm{F}}=\stackrel{\ensuremath{\rightarrow}}{2}{3p}^{2}{P}_{3/2},$ F=3,F=3,F=3, mF=3{m}_{\mathrm{F}}=3mF​=3 transition in a diffuse, collision-free, thermal beam of sodium). In this investigation, we study the detailed line shape and strength of the four-wave mixing signal as a function of the intensity of the cw pump waves, and of the velocity of the atoms. Our technique allows us to examine the four-wave mixing interaction under conditions in which the Doppler shift, the natural linewidth of the transition, and the Rabi frequency for the interaction are each comparable with one another. From our experimental measurements and computational results, we show that the line shapes are very complex in this regime, and that they depend sensitively upon the atomic velocity, especially in the case when the intensity of the forward and backward pump beams exceeds the saturation intensity for the transition.