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Physical Review Letters

cw conical emission: First comparison and agreement between theory and experiment

John F. Valley, G. Khitrova, H. M. Gibbs, Jeffrey W. Grantham, Xu Jiajin

Published May 14, 1990• 66 citations
Publisher

Abstract

Experimentally observed near-field and far-field spatial profiles, frequency spectra, and cone angles are in good agreement with numerical computations of a theory that includes Doppler-broadened Raman-gain amplification of Doppler-broadened resonance fluorescence, four-wave mixing, propagational coupling, self-trapping of the pump beam, and diffraction and pump-induced refraction of the new frequencies. The use of a cw laser beam detuned from resonance by only a few gigahertz makes the computational problem tractable, leading to a simple physical model of this spatiotemporal instability.

DOI
10.1103/physrevlett.64.2362

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Experimentally observed near-field and far-field spatial profiles, frequency spectra, and cone angles are in good agreement with numerical computations of a theory that includes Doppler-broadened Raman-gain amplification of Doppler-broadened resonance fluorescence, four-wave mixing, propagational coupling, self-trapping of the pump beam, and diffraction and pump-induced refraction of the new frequencies. The use of a cw laser beam detuned from resonance by only a few gigahertz makes the computational problem tractable, leading to a simple physical model of this spatiotemporal instability.
The use of a cw laser beam detuned from resonance by only a few gigahertz to observe conical emission1 makes the computational problem tractable (2 CRAY CPU hours per spectrum), leading to a simple physical model of this spatio-temporal instability. From the excellent qualitative agreement between computations and experimental observations, one understands the physics of conical emission as follows: self-trapping2 of A1, probe generation by resonance fluorescence, Raman-gain amplification3,4 of this probe which produces A3(ω3), four-wave mixing3 between A1(ω1) and A3(ω3) which generates A4(ω4 = 2ω1 - ω3) and transfers energy back and forth between A3 and A4, division of A4 into A4L and A4R by optical-Stark-shifted absorption, division of A3 into A3L and A3R by four-wave-mixing coupling, and formation of the cone from A4L (and sometimes from the low-energy shoulder of A4R) by propagation through the spatially-dependent index of refraction prepared by A1(x,y,z). We both observe and compute that light A4(ω4) of a particular frequency may appear both in the cone and in the center, ruling out phase matching as the determinant of the cone angle, unlike many other cones, e.g., two-wavelength conical emission.5
A time-dependent theory for conical emission during near-resonant propagation of laser light in an atomic vapor, which includes full propagation for the laser and frequency sidebands in a nonlinear two-level medium is presented. The density-matrix equations for the dipole moment and population are solved in the dressed atomic frame. The polarization source terms are accurate to order \ensuremath{\gamma}/R, where \ensuremath{\gamma} is a damping constant and R is the generalized Rabi frequency. Analytical plane-wave solutions and numerical, cylindrically symmetric propagation simulations including diffraction are presented. It is shown that the calculations with cylindrically symmetric fields and atomic excitation profiles are incapable of accounting for the high levels of optical gain that are responsible for the intense conical emission observed in experiments. This result is at first surprising, since the model accounts rigorously for all of the physical phenomena that have been previously proposed as being responsible for generating large gains, and the calculation matches the symmetry of the observations. The lack of large calculated gain seems to imply the existence of higher-order (m>0)(m>0)(m>0) radial modes in the field for the experimental conditions that give rise to cone emission. In the simulations, however, the cylindrically symmetric fields do produce weak red-detuned cones with angular-frequency distributions similar to those seen in experiments.
A nonlinear medium interacting with two counterpropagating pump beams is potentially unstable. This instability is related to a mirrorless four-wave mixing oscillation.1 Recent experiments done in vapors2 show that the behavior is strongly different according to the sign of the frequency detuning from resonance. When the laser frequency ω is larger than the atomic resonance frequency ω0, the oscillation appears as a sharp conical emission, the oscillation frequency being ω. In some cases the cone is fragmented into a hexagon. When ω < ω0, the oscillation is more difficult to achieve and appears as a very broad cone.
Observations of a frequency-shifted coherent conical emission from systems pumped by an intense optical field that is nearly resonant with a one-photon transition1 have made it clear that propagation effects can play a major role in determining the spectrum as well as the spatial characteristics of the transmitted pulse. Proposed theories of conical emission stress the importance of transverse effects leading to four-wave mixing1 or spatial self-phase modulation.2 We report computer results on laser propagation in a two-level system and demonstrate that spatial structure, including conical emission, can result from temporal propagation effects.
This is the first of a series of papers on the theory of cone emission from a laser-pumped two-level atomic medium. Starting from microscopic quantum theory for both the active medium and the field, equations for the slowly varying macroscopic quantities are derived. In the steady-state limit, which experimentally corresponds to cw excitation, we find that the sources for the paraxial fields of interest have a long coherence length despite the random nature of spontaneous emission. These radiation sources enter into the equations for the fields in the same form as the source generated by four-wave mixing and therefore may be considered as a ``spontaneous four-wave mixing,'' since the photons emitted in the Rabi sidebands are correlated. This has far-reaching consequences on the physics of cone emission and also sheds some new light on the fluoresence from strongly driven atomic systems.
A detailed study on yellow conical emission observed in dense sodium vapor following near-resonant one-photon excitation of the Na 42P levels was performed. The origins of conical emission are suggested to be due to Raman resonant four-wave parametric mixing processes. In order to describe spectral and spatial features of off-axis emission, we introduce self-phase modulation and induced-phase modulation as important additional nonlinear processes, explaining the nonmonotonic detuning of the wavelength of the conical radiation with the exiting laser wavelength in the vicinity of resonance transitions. The spatial evolution of emission generated via four-wave mixing is modeled by solving the two-dimensional nonlinear Schrödinger equation describing the spatial evolution of light beams in a nonlinear media. Experimental and theoretical results are presented and discussed.