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Laser Applications to Chemical, Security and Environmental Analysis

Degenerate Four-Wave Mixing Spectroscopy: Multistate Resonances and Multimode Laser Effects

Thomas A. Reichardt, William C. Giancola, Robert P. Lucht

Published Jan 1, 1998• 0 citations
Publisher

Abstract

Degenerate four-wave mixing (DFWM) spectroscopy is a promising diagnostic technique [1-3], but significant questions remain regarding its application for quantitative concentration measurements in combustion environments with typical laser systems. In this paper we present a summary of two numerical investigations with the goal of quantifying the measurement technique for common experimental situations. Specifically, we investigate both (1) probing a degenerate-level (multistate) resonance with a single-mode laser and (2) probing a two-state resonance with a multimode laser. DFWM signal levels are calculated by solving the time-dependent density matrix equations using direct numerical integration (DNI) [4, 5] for a two or more level system interacting with three laser beams.

DOI
10.1364/lacea.1998.lmc.24

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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)•Jan 20, 2003•0 citations
Degenerate four-wave mixing (DFWM) spectroscopy is a promising diagnostic technique [1-3], but significant questions remain regarding its application for quantitative concentration measurements in combustion environments with typical laser systems. In this paper we present a summary of two numerical investigations with the goal of quantifying the measurement technique for common experimental situations. Specifically, we investigate both (1) probing a degenerate-level (multistate) resonance with a single-mode laser and (2) probing a two-state resonance with a multimode laser. DFWM signal levels are calculated by solving the time-dependent density matrix equations using direct numerical integration (DNI) [4, 5] for a two or more level system interacting with three laser beams.
Degenerate four-wave mixing (DFWM) spectroscopy is a potentially powerful technique for quantitative measurement of minor species in gas-phase media [1-3]. In this paper we compare the DFWM signal levels and lineshapes obtained using two different phase-matching geometries: the phase-conjugate geometry and the forward phase-matched geometry. DFWM signal levels are calculated by solving the time-dependent density matrix equations using direct numerical integration (DNI) [4, 5] for a two-level system interacting with three laser beams. This numerical analysis includes the effects of molecular motion, intermolecular collisions, and high laser intensities, all of which have been shown to affect the DFWM signal level.
The effect of Doppler broadening on degenerate four-wave mixing (DFWM) signal intensities in the regime of high pump and probe laser intensities is investigated theoretically. DFWM reflectivities are calculated by solving the time-dependent density-matrix equations for a two-level system interacting with three laser fields. The density-matrix equations are integrated directly in the time domain on a grid of spatial locations along the phase-matching axis; the DFWM signal level is then calculated by summation of the polarization contribution (with the appropriate phase factor) from each of the spatial grid points. For the case in which the Doppler and the collisional linewidths are comparable, the DFWM reflectivity is found to be inversely proportional to the factor 1 + (bΔωD/ΔωC)2, where ΔωD is the Doppler width, ΔωC is the collisional width, and b is weakly dependent on the pump and the probe laser powers. We developed an analytical expression for the reflectivity of a line that is both collision and Doppler broadened by dividing the widely used Abrams and Lind expression for homogeneous reflectivity Rhom by the factor 1 + (bΔωD/ΔωC)2. This modified reflectivity expression is found to give accurate results for the DFWM reflectivity over a wide range of values for the ratio of Doppler to collisional width. With this modified Abrams–Lind expression, strategies for quantitative DFWM concentration measurements in flames and plasmas are proposed and analyzed. We conclude that, by selection of the appropriate rotational transition, a DFWM reflectivity that is directly proportional to the square of the total species number density can be obtained over a wide range of temperature for constant-laser-intensity spatial profile mapping in flames.
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.
The use of picosecond lasers for degenerate four-wave mixing (DFWM) diagnostics of atmospheric pressure flames has been suggested as a means of overcoming the complicated collision-rate dependence of the DFWM signal in the nanosecond-laser pulse-length regime. In this paper DFWM spectroscopy in the regime in which the laser pulse length (τL) is less than the characteristic collisional time (τC) is investigated theoretically. DFWM signal levels for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integration of the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and by summing of the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision broadened and Doppler broadened are considered. DFWM signal generation for these two types of resonances in the fully transient regime (τL≪τC) is strikingly different. For a homogeneously broadened resonance, the DFWM signal persists long after the laser beam has passed through the medium. The decay rate of the DFWM signal is determined by the collision rate in the medium, and the integrated DFWM signal is still dependent on the collision rate even in this fully transient regime. However, when the Doppler broadening of the resonance is much greater than the collisional broadening, the DFWM signal decays with a fall time close to that of the laser pulse. For τL≪τC, we find that it is the laser spectral width, rather than the collisional width, that dictates the range of velocity groups with which the pulses effectively interact. The signal decays with a fall time close to that of the laser pulse because the excited velocity groups radiate out of phase with one another. In a medium in which the Doppler width is much greater than the collisional width, the DFWM reflectivity is independent of collision rate for τL≪τC, but the signal level is dependent on the Doppler width. Finally, we investigate the effect of different laser pulse lengths on reflectivity saturation curves for resonances with different levels of 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.
Nowadays degenerate four wave mixing (DFWM) with forward geometry (FDFWM) has found increasing usage, for example, in the gas phase detection. In the fields of theoretical development of DFWM, most of the studies have been concentrated on the understanding of phase-conjugated DFWM. However, few theories have dealt with FDFWM. These models calculated the signal directly from the off-diagonal element of the density matrix by assuming optically thin medium. In the paper we obtain a theoretical model for FDFWM in Doppler broadened two-level absorbers including propagation effect and discuss the Doppler effect on the signal intensity and lineshape. Assuming that probe and signal field amplitudes are small compared with those of the pump fields and pump field amplitudes are constant in the interaction volume, we obtain the following coupled equations from the Maxwell equation using the slowly varying envelope approximation and the paraxial approximation.