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Journal of the Optical Society of America B

Investigation of the dependence of degenerate four-wave mixing line intensities on transition dipole moment

Roger L. Farrow, Thomas K. Dreier, David J. Rakestraw

Published Oct 1, 1992• 100 citations
Publisher

Abstract

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.

DOI
10.1364/josab.9.001770

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Physical Review A•Sep 1, 1990•47 citations
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.
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.
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 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 object of this study is to quantitatively elucidate the laser-power dependence in transient degenerate four-wave mixing (DFWM) with an emphasis on the high laser-pump intensity region. We discuss our investigation on the power dependence of transient DFWM by taking gas-phase iodine (I2{\mathrm{I}}_{2}I2​) molecules as a testing example. The distinct physical feature is that in the high-power laser pump, where both laser-pulse duration and the inverse of pump rate are much shorter than the molecular population relaxation time, the steady-state DFWM theory of Abrams and Lind [Optical Phase Conjugation, edited by R. A. Fisher (Academic, New York, 1983), Chap. 8, pp. 211--284; Opt. Lett. 2, 94 (1978); 3, 205 (1978)] is not appropriate. The prediction by the steady-state theory has shown the DFWM to decrease with increasing laser-pump intensity as a function of 1/Ilaser{\mathrm{I}}_{\mathrm{laser}}Ilaser​, which disagrees with the saturation behavior observed in the experiment. To elucidate the dependence of DFWM on the laser pump intensity, a non-steady-state extension of the nonperturbative theory of Abrams and Lind is required. The non-steady-state theoretical result will be shown to be in good agreement with the experimental power dependence at resonance transient DFWM especially in the high-power pump region.
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.
The steady-state theory of degenerate four-wave mixing (DFWM) with broad-bandwidth lasers is extended to treat the time dependence of the interaction allowing calculation of the intensity dependence and saturation behavior of DFWM with pulsed lasers. An experimental study of this intensity dependence as a function of laser bandwidth is reported, showing qualitative agreement with the theory. The line shape of DFWM induced by broad-bandwidth pump fields, probed by a monochromatic probe, is shown to have a Doppler-broadened profile in agreement with the theoretical predictions.