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Digest on Nonlinear Optics: Materials, Phenomena and Devices

Nonperturbative Calculation Of Nearly Degenerate Multi-wave-mixing Spectra

Robert P. Lucht, Rick P. Trebino, Larry A. Rahn

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Trebino and Rahn observed subharmonic resonances in high-order wave-mixing spectra of the sodium atom in flames. They used two pulse-amplified, single-mode dye lasers to generate two nearly degenerate pump beams at 590 nm. The pump beams were tuned 2 cm-1away from the sodium D1line resonance. At low laser intensity resonances at the hyperfine splitting ωhfs=0.06 cm-1and at zero frequency difference were clearly visible. At higher intensities the hyperfine resonances broadened and a subharmonic resonance at 1/2 ωhfsappeared. Trebino and Rahn also used higher order wave mixing geometries to observe subharmonic resonances at 1/3, 1/4, and 1/5 ωhfs. Trebino and Rahn have used diagrammatic perturbation theory to calculate high-order wave-mixing spectra.

DOI
10.1109/nlo.1990.695900

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Trebino and Rahn observed subharmonic resonances in high-order wave-mixing spectra of the sodium atom in flames. They used two pulse-amplified, single-mode dye lasers to generate two nearly degenerate pump beams at 590 nm. The pump beams were tuned 2 cm-1away from the sodium D1line resonance. At low laser intensity resonances at the hyperfine splitting ωhfs=0.06 cm-1and at zero frequency difference were clearly visible. At higher intensities the hyperfine resonances broadened and a subharmonic resonance at 1/2 ωhfsappeared. Trebino and Rahn also used higher order wave mixing geometries to observe subharmonic resonances at 1/3, 1/4, and 1/5 ωhfs. Trebino and Rahn have used diagrammatic perturbation theory to calculate high-order wave-mixing spectra.
We perform nonperturbative calculations of resonant multiwave mixing of two high-intensity laser beams of different frequencies both for a general three-level system and for the 3 2^{2}2S1/2{\mathit{S}}_{1/2}S1/2​--3 2^{2}2P1/2{\mathit{P}}_{1/2}P1/2​ electronic resonance of the sodium atom. Our calculations proceed by direct numerical integration of the density-matrix equations and subsequent Fourier analysis of the calculated time-dependent density-matrix elements. We examine the case where two nearly degenerate input beams are each detuned many collisional linewidths from electronic resonance, and the frequency difference between the input beams is tuned through a ground-state resonance. Unlike previous high-intensity analyses, each input beam can be resonant with numerous transitions simultaneously; for a three-level system, we refer to this as an ``M-type'' system, in analogy with \ensuremath{\Lambda}- and V-type systems. Calculated four-, six-, and eight-wave-mixing spectra exhibit subharmonic resonances of the type observed experimentally by Trebino and Rahn [Opt. Lett. 12, 912 (1987)]. The three-level system is studied to gain insight into the physics of these subharmonic resonances. For more quantitative comparison with experiment, we use a model that includes the 16 Zeeman and hyperfine states in the 3 2^{2}2S1/2{\mathit{S}}_{1/2}S1/2​ and the 3 2^{2}2P1/2{\mathit{P}}_{1/2}P1/2​ levels, and accounts for arbitrary linear polarization of the input laser radiation. The relative intensities of the resonance features in the calculated multiwave-mixing spectra show good agreement with experiment. These resonance features broaden and shift as laser intensity increases, as predicted for four-wave-mixing processes in previous, less general high-intensity analyses. Both theory and experiment also show additional resonant features in the eight-wave-mixing spectra that appear at very high intensity.
Using four-, six-, and eight-wave-mixing geometries, we observe subharmonic resonances in collision-enhanced spectra in a sodium-seeded hydrogen-air flame. Appearing at high intensity, these resonances occur at +/-1/2, +/-1/3, +/-1/4, and +/-1/5 times the frequency of the ground-state hyperfine splitting omega(hfs). We argue that these resonances result from higher-order processes, specifically chi((n)), where n >/= 5. In particular, to describe the resonances at +/-omega(hfs)/5, at least chi((13)) is required. These high-order collision-enhanced effects are true perturbative nonlinearities and not sequential four-wave mixing processes.
Using diagrammatic perturbation theory, we calculate the higher-order susceptibility for the n-wave process, omega(0) = (n/2 - 1)(omega(1) - omega(2)) + omega(1), in a phase-matched n-wave-mixing geometry. We include the 16 Zeeman and hyperfine levels of the sodium ground (3S((1/2))) and excited (3P((1/2))) states, finding resonances at subharmonics [+/-(1/2), +/-?, ..., +/-1(n/2 - 1)] of the ground-level transition frequencies. The computed spectrum for eight-wave mixing is in satisfactory agreement with experiment. In addition, a theoretical twelve-wave-mixing spectrum predicts a new higher-order selection rule.
Degenerate four-wave mixing in sodium vapor is investigated using a flashlamp-pumped dye laser synchronized on a cw dye laser. At low atomic density, the reflectivity versus the pump intensity and the line shapes are in good agreement with theoretical predictions in the case of both plane waves and Gaussian beams. At high atomic density, reflectivities much higher than unity have been observed. The influence of the pump imbalance has been studied as well as the deformation of the wave front of the conjugated beam introduced by the pump beams. In the high-atomic-density case, the theoretical problem is more complex, and our experimental results exhibit only qualitative agreement with theory.
We present the results of a pump-probe laser spectroscopic investigation of the Doppler-broadened sodium D1D1D1 resonance line. We find 13 resonances in the resulting spectra. These observations are well described by the numerical predictions of a four-level atomic model of the hyperfine structure of the sodium D1D1D1 line. We also find that many, but not all, of these features can be understood in terms of processes originating in a two-level or three-level subset of the full four-level model. The processes we observed include forward near-degenerate four-wave mixing and saturation in a two-level system, difference-frequency crossing and nondegenerate four-wave mixing in a three-level V system, electromagnetically induced transparency and optical pumping in a three-level lambda system, cross-transition resonance in a four-level double-lambda system, and conventional optical pumping. Most of these processes lead to sub-Doppler or even subnatural linewidths. The dependence of these resonances on the pump intensity and pump detuning from atomic resonance are also studied.
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