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Optics Letters

Subharmonic resonances in higher-order collision-enhanced wave mixing in a sodium-seeded flame

Rick P. Trebino, Larry A. Rahn

Published Nov 1, 1987• 46 citations
Publisher

Abstract

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.

DOI
10.1364/ol.12.000912

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Dephasieg-induced phenomena are a novel class of effects that are ordinarily forbidden because the two or more relevant quantum-mechanical amplitudes add coherently and cancel out.1 When dephasing is present, however, these amplitudes no longer cancel, and previously unseen effects can be observed. Indeed, a number of novel dephasing-induced phenomena have been seen.2-5 Of particular interest, both for fundamental reasons and for diagnostic purposes, are (third-order) dephasing-induced resonances between equally populated levels of a ground electronic state,6,7 specifically, Zeeman and hyperfine coherences. Working with a sodium-seeded flame, we8 have observed these latter effects and, at higher intensities, subharmonics of them.

•Jul 18, 1988•1 citation
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.
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.
Resonances of the six-wave mixing susceptibility χ(5)(ω1, ω1, ω1,-ω2,-ω2) at frequency 3ω1-2ω2 of a system of two-level atoms dissipating through the spontaneous emission and collisions and driven by a weak bichromatic external field of frequencies ω1 and ω2 in a finite Q cavity are analyzed. It is shown that the atomic collisions not only induce the usual Bloembergen resonances but that they also play a major role in enhancing the transitions among the higher-order dressed states, thereby bringing out some of the quantum resonances that are suppressed in the absence of collisions. The important contribution of collisions in this higher-order nonlinear process is the generation of the subharmonic resonances at ±1/2 times the hyperfine splitting of the first and the second excited manifolds.
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.
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.
Dephasieg-induced phenomena are a novel class of effects that are ordinarily forbidden because the two or more relevant quantum-mechanical amplitudes add coherently and cancel out.1 When dephasing is present, however, these amplitudes no longer cancel, and previously unseen effects can be observed. Indeed, a number of novel dephasing-induced phenomena have been seen.2-5 Of particular interest, both for fundamental reasons and for diagnostic purposes, are (third-order) dephasing-induced resonances between equally populated levels of a ground electronic state,6,7 specifically, Zeeman and hyperfine coherences. Working with a sodium-seeded flame, we8 have observed these latter effects and, at higher intensities, subharmonics of them.