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Journal of the Optical Society of America B

Effects of laser-beam temporal pulse shape on signal strength and spectral line shape of forward degenerate four-wave mixing

Dai‐Hyuk Yu, Jai-Hyung Lee, Joon-Sung Chang, Jae-Seok Ryu, Jae Won HAHN, Paul M. Danehy

Published Aug 1, 2001• 1 citation
Publisher

Abstract

We investigate the effects of the input laser beam’s temporal pulse shape on a forward degenerate four-wave mixing (FDFWM) signal, assuming two-level saturable absorbers. Under conditions in which the coherence time is far shorter than the pulse-duration, we can calculate the signal easily in the time and frequency domains. From this result it is shown that, when the pulse duration is longer than ten times the population relaxation time, the calculated signal intensity is almost the same as that obtained by assuming a steady state throughout duration of the laser pulse. With this assumption, we obtain the signal by summing all the steady-state signals at each time-domain grid point during the pulse. The calculation is performed with three input beam temporal pulse shapes: square, Gaussian, and typical Q-switched pulses. After reaching a maximum, the signal intensity decreases. The rates of the signal increase and decrease depend strongly on the temporal shape of the input beam. For the same average power, the square-pulse input saturates the absorber faster than the other pulses. These results indicate that the weak part of the pulse input plays an important role in generating the signal, yielding differences in saturation behavior of the signal intensity and the spectral line shape. We compare the calculations with experiments by fitting the saturation curve of the FDFWM signal of argon atoms in a radio frequency inductively coupled plasma. Also, the spectral line shape of the DFWM signal is fitted with the calculation by use of published experimental data.

DOI
10.1364/josab.18.001111

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We investigate the effects of the input laser beam’s temporal pulse shape on a forward degenerate four-wave mixing (FDFWM) signal, assuming two-level saturable absorbers. Under conditions in which the coherence time is far shorter than the pulse-duration, we can calculate the signal easily in the time and frequency domains. From this result it is shown that, when the pulse duration is longer than ten times the population relaxation time, the calculated signal intensity is almost the same as that obtained by assuming a steady state throughout duration of the laser pulse. With this assumption, we obtain the signal by summing all the steady-state signals at each time-domain grid point during the pulse. The calculation is performed with three input beam temporal pulse shapes: square, Gaussian, and typical Q-switched pulses. After reaching a maximum, the signal intensity decreases. The rates of the signal increase and decrease depend strongly on the temporal shape of the input beam. For the same average power, the square-pulse input saturates the absorber faster than the other pulses. These results indicate that the weak part of the pulse input plays an important role in generating the signal, yielding differences in saturation behavior of the signal intensity and the spectral line shape. We compare the calculations with experiments by fitting the saturation curve of the FDFWM signal of argon atoms in a radio frequency inductively coupled plasma. Also, the spectral line shape of the DFWM signal is fitted with the calculation by use of published experimental data.
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 effects of finite laser bandwidth on resonant degenerate four-wave mixing (DFWM) are calculated with use of a model in which the intense, counterpropagating pump beams are characterized by a chaotic field, the probe beam is weak and monochromatic, and the medium consists of a gas of two-level atoms. We present a steady-state solution in the limit where the pump-laser bandwidth exceeds all other atomic relaxation rates. Although the mean intensity due to the fluctuating fields is spatially independent (no steady-state standing-wave pattern is established), the analytic results indicate that, for intensities above the saturation intensity Isat{I}_{\mathrm{sat}}Isat​, spatially periodic saturation effects are important. Increasing bandwidth is shown to lead to an increase in the effective saturation intensity resulting in lower phase-conjugate reflectivity for IIsat{I}_{\mathrm{sat}}Isat​ than for coherent pump fields, in contrast to the results for narrow-bandwidth chaotic fields. The resonant DFWM line shape is also calculated and compared to the coherent result. We comment on the application of the model to other four-wave-mixing processes employing broad-bandwidth lasers.
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.
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.
The transient behavior of degenerate four-wave mixing (DFWM) in a three-level saturable absorber is analyzed and verified experimentally. A sharp peak of phase-conjugated reflectivity is predicted and observed at high pump intensities for pulsed beams. The influence of the time-dependent behavior of the conjugate wave signal on results of pulsed-beam experiments is studied. It is shown that transient effects could account for discrepancies between previous experimental results and steady-state theory.