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Applied Physics Letters

Generation of time-reversed wave fronts using a resonantly enhanced electronic nonlinearity

D. Grischkowsky, N. S. Shiren, R. J. Bennett

Published Nov 1, 1978• 46 citations
Publisher

Abstract

Time-reversed optical wave fronts were generated by degenerate four-wave mixing in rubidium vapor. An extremely large and resonantly enhanced electronic nonlinearity, described by the adiabatic following (AF) model, was responsible for the mixing. The intensity dependence of the AF nonlinear susceptibility was shown to explain the observed reduction in conversion efficiency at high pumping intensities.

DOI
10.1063/1.90554

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Time-reversed optical wave fronts were generated by degenerate four-wave mixing in rubidium vapor. An extremely large and resonantly enhanced electronic nonlinearity, described by the adiabatic following (AF) model, was responsible for the mixing. The intensity dependence of the AF nonlinear susceptibility was shown to explain the observed reduction in conversion efficiency at high pumping intensities.
Two-photon resonantly enhanced parametric generation processes have generally been described using time-dependent perturbation theory. In this paper we show that a theory of two-photon coherent effects can be used to derive and explain these nonlinear mixing processes. Our technique makes use of the adiabatic following (AF) approximation to obtain solutions to a vector model describing the two-photon resonance. We show that the usual results for the nonlinear susceptibilities correspond to the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ vector of Feynman, Vernon, and Hellwarth adiabatically following the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\gamma}}}$ vector in the small-angle limit. Consequently, the theory allows a natural extension to large angles, and power-dependent nonlinear susceptibilities are obtained. We then use these AF results for the polarization to study the propagation of pulses nearly resonant with a two-photon transition, and we demonstrate that the pulse reshaping is due to the two related effects of a nonlinear pulse velocity and self-phase modulation.
The use of resonantly enhanced four-wave mixing to generate high reflectivity phase-conjugate waves is discussed. A physical understanding of the mechanisms underlying the nonlinear response for a simple two-level system is presented followed by a discussion of the complicating effects of pump absorption, pump depletion, unbalanced pumps, and thermal motion. Modifications of the theory to describe four-level behavior are presented and compared with experiments. The application of nearly degenerate four-wave mixing to narrow-band filtering complicated by the A. C. Stark effect is dis-cussed along with examples of experimental observations.
Nonlinear optical mixing of existing laser frequencies to access portions of the spectrum where lasing action is not easily obtainable is common practice today. Although nonlinear solid-state media are available [1] for frequency conversion from just under 200 nm in the ultraviolet to about a few microns in the infrared, the generation of widely-tunable laser radiation in the mid-to-far infrared and VUV regions typically requires four-frequency mixing in atomic vapors. In the past, frequency conversion in atomic vapors suffered from poor conversion efficiencies due to the small magnitude of the third-order nonlinear susceptibility and the difficulty in phasematching. We demonstrate a method [2] for performing four-frequency mixing which overcomes both of these shortcomings, allowing efficient frequency conversion with moderate pump intensities.
Short pulses of narrow-line low-intensity dye-laser light nearly resonant with the Zeeman-split 2P12^{2}P_{\frac{1}{2}}2P21​​ resonance line (7948 \AA{}) of rubidium were observed to propagate through dilute rubidium vapor as slowly as (114)c(\frac{1}{14})c(141​)c. These slow pulse velocities showed that most of the energy in the propagating wave was contained by the vapor as coherent atomic excitation. The observed pulse velocities vp{v}_{p}vp​ are in good agreement with the equation ${v}_{p}=\frac{d\ensuremath{\omega}}{\mathrm{dk}}$ for the group velocity obtained from linear-dispersion theory. Also, the experimental results are quantitatively explained by adiabatic following, in which the pseudomoments of the atoms remain aligned along the effective field of the laser light. The adiabatic-following model allows for a direct comparison of our results with the work on self-induced transparency. For high-intensity light, adiabatic following predicts a nonlinear pulse velocity and the possibility of observing self-steepening.
Quasi-degenerate four-wave mixing (FWM) of low-intensity radiation (with an intensity less than 1 mW cm -2 ) was observed for the 5 S 1/2 ( F =2)—5 P 3/2 ( F '=3) transition in an optically transparent 87 Rb vapour under conditions of electromagnetically induced absorption. The efficiency of frequency conversion was no less than 1% with respect to the intensity of pump waves. Nonlinear FWM is highly selective relative to the frequency detuning of pump waves (the FWM full width at half maximum was less than 50 kHz).