Degenerate four-wave mixing spectroscopy with short-pulse lasers: theoretical analysis

Abstract

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.