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Physical Review A

Degenerate four-wave mixing with broad-bandwidth pulsed lasers

D. R. Meacher, A. Charlton, P. Ewart, J. Cooper, G. Alber

Published Sep 1, 1990• 47 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.42.3018

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Degenerate Four Wave Mixing With Pulsed Lasers Theory And Applications

P. Ewart

Resonant Degenerate Four Wave Mixing, DFWM, is a nonlinear optical process arising from the third order susceptibility X(3). Resonance enhancement of X(3)in absorbing media makes the process useful for spectroscopic applications[1]. In many situations pulsed, high power lasers are used which have broad spectral bandwiths. The stochastic fluctuations of amplitude and phase associated with this bandwidth complicates the theoretical description in such cases. An averaging of the atom-field interaction is required simultaneously over the temporal and spatial variations of the fields. A solution has been found in the broad bandwidth limit which uses an appropriate decorrelation approximation. Theoretical and experimental results will be presented for the effects of laser bandwidth on the power dependence and temporal behaviour of resonant DFWM[2].

Digest on Nonlinear Optics: Materials, Phenomena and Devices•Jan 1, 1990•0 citations
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.
We present a revision of the theory of finite laser bandwidth effects on degenerate four-wave mixing (DFWM) of Alber, Cooper, and Ewart (ACE) [Phys. Rev. A 31, 2344 (1985)]. The same model is used for intense, broad-bandwidth pump lasers interacting with a weak monochromatic probe in a medium composed of two-level atoms. The density-matrix equations describing the time evolution of the atomic polarization coupled to fluctuating fields are solved using an appropriate decorrelation approximation. A steady-state analytical solution is found for resonant and near-resonant DFWM, for intensities that do not saturate the medium. For more intense fields it is found to be necessary to form a more complicated set of differential equations for all products of the field and atomic density-matrix elements. The results show that as in the ACE theory, increasing the bandwidth b leads to an increased effective saturation intensity. However, we find the DFWM reflectivity scales as 1/b2{b}^{2}b2 when the probe is monochromatic and not 1/b as in the ACE theory. Furthermore, the saturation behavior of nonresonant DFWM is found to differ from the predictions of ACE. Atomic motion effects are shown to yield a Doppler-broadened line shape and also to affect the saturation behavior of the signal.
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 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.
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.
Resonant Degenerate Four Wave Mixing, DFWM, is a nonlinear optical process arising from the third order susceptibility X(3). Resonance enhancement of X(3)in absorbing media makes the process useful for spectroscopic applications[1]. In many situations pulsed, high power lasers are used which have broad spectral bandwiths. The stochastic fluctuations of amplitude and phase associated with this bandwidth complicates the theoretical description in such cases. An averaging of the atom-field interaction is required simultaneously over the temporal and spatial variations of the fields. A solution has been found in the broad bandwidth limit which uses an appropriate decorrelation approximation. Theoretical and experimental results will be presented for the effects of laser bandwidth on the power dependence and temporal behaviour of resonant DFWM[2].