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Journal of the Optical Society of America B

Comparison of degenerate four-wave mixing line shapes from population- and thermal-grating scattering

Paul M. Danehy, Roger L. Farrow

Published Jul 1, 1996• 10 citations
Publisher

Abstract

We report detailed measurements of transition line shapes of degenerate four-wave mixing in the A2Σ+(υ′ = 0) ← X2Π(υ″ = 0) band of nitric oxide. We compared spectra obtained under conditions in which either population or thermal gratings were significant. Under conditions in which population gratings were significant, spectral lines were narrower and had intensities that varied more rapidly with the absorption B coefficient than in spectra dominated by thermal-grating scattering. In both cases we found excellent agreement with calculations that used perturbation theory to describe the population-grating contributions and the squared absorption spectra to describe the thermal-grating contributions. Our results show that quantitative prediction of degenerate four-wave mixing spectra requires relatively detailed knowledge of the ratio of contributions from the two mechanisms.

DOI
10.1364/josab.13.001412

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The Journal of Chemical Physics•Dec 8, 1999•22 citations
We report detailed measurements of transition line shapes of degenerate four-wave mixing in the A2Σ+(υ′ = 0) ← X2Π(υ″ = 0) band of nitric oxide. We compared spectra obtained under conditions in which either population or thermal gratings were significant. Under conditions in which population gratings were significant, spectral lines were narrower and had intensities that varied more rapidly with the absorption B coefficient than in spectra dominated by thermal-grating scattering. In both cases we found excellent agreement with calculations that used perturbation theory to describe the population-grating contributions and the squared absorption spectra to describe the thermal-grating contributions. Our results show that quantitative prediction of degenerate four-wave mixing spectra requires relatively detailed knowledge of the ratio of contributions from the two mechanisms.
We report an investigation of the dependence on transition dipole moment of isolated line intensities in degenerate four-wave mixing in the A2Σ+ (υ′ = 0) ← X2Π (υ″ = 0) band for NO. For spectrally integrated intensities we found that the dependence was well described by a power law μx, where μ is the one-photon transition dipole moment. As a result of saturation, the exponent x depends on laser intensity. The observations are in reasonable agreement with a simple two-level model [ Opt. Lett.294 ( 1978);Opt. Lett.3, 205 ( 1978)], which predicts limiting dependences of μ8 and μ3 for low and high intensities, respectively. At intermediate laser intensities the model is consistent with an observed rapid decrease in the exponent with increasing laser intensity, followed by a plateau extending to high intensities.
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.
The polarization properties of backward-box, degenerate four-wave mixing from OH molecules are measured in a flame and compared with theory. Ratios of signal intensities for different laser polarizations are shown to exceed perturbation-theory predictions for a closed two-level system when optical intensity gratings are formed in the interaction volume. These results are shown to be inconsistent with population-nonconserving effects and are attributed to scattering from quenching-induced thermal density gratings.
Saturation effects in gas-phase degenerate four-wave mixing (DFWM) are investigated for conditions of interest for diagnostic applications in flames and plasmas. In particular, the case in which Doppler and collisional broadening are comparable, as is often the case for flame species such as NO and OH, is investigated. DFWM line shapes and signal intensities are calculated nonperturbatively and compared with high-resolution laser measurements. In the nonperturbative calculations the time-dependent density-matrix equations for a two-level system interacting with three laser fields are integrated directly on a grid of spatial locations along the phase-matching axis. The electric-field amplitude for the DFWM signal is determined by multiplying the time-varying, laser-induced polarization at each grid point by the appropriate phase factor and then by summing over all grid points. The calculations are in excellent agreement with measurements of DFWM line shapes and signal intensities for NO in a buffer gas of He over a wide range of He pressure. Under saturation conditions the pressure dependence of the DFWM signal is reduced greatly compared with the unsaturated case. The signal level is still dependent on the ratio of pure dephasing to quenching collisions, even for saturation conditions. From the standpoint of minimal dependence on collisional processes it appears that operation with pump-laser intensities approximately equal to the saturation intensity is optimal. The dependence of the DFWM line shape and signal intensity on the ratio of probe to pump-laser intensity is also investigated.
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.
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.