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Physical Review A

Cone emission from laser-pumped two-level atoms

Li You, Jan Mostowski, J. Cooper, R. Shuker

Published Dec 1, 1991• 26 citations
Publisher

Abstract

In spite of the fact that cone emission was first observed over a decade ago, there is still no generally accepted theory of the effect. Here we give an outline of a unified approach and point out a mechanism that supports Cherenkov-type radiation due to vacuum fluctuations as a possible source of cone emission. This can be considered as an additional mechanism to the previously discussed four-wave mixing, and initial encoding and follow-up refraction effects. We compare our treatment to previously considered models.

DOI
10.1103/physreva.44.r6998

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Cone emission from laser-pumped two-level atoms. II. Analytical model studies

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We establish a mechanism, which supports Cherenkov-type radiation, as a source of cone emission. This can be considered as an additional mechanism to the previously discussed four-wave mixing (4WM) and initial encoding and follow-up refraction effects. The radiation source is provided by spontaneous emission of the driven atoms at the frequencies of interest. Due to the medium dispersion, the pump field propagates faster than the phase velocity of the field that is generated at the Rabi sideband at lower frequency. Cherenkov radiation is then possible if the source has a long correlation length, which is indeed the case despite the random nature of spontaneous emission. Coherent superposition of the source for the lower-frequency field is in an off-axis direction given by the Cherenkov radiation condition. Differences between 4WM and Cherenkov-type models are consequently resolved.

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Conical light emission is observed when an intense quasiresonant laser beam propagates in an atomic vapor.1-4 A parametric four-wave mixing process was suggested to explain this effect.1 However, this model is inconsistent with experiments both quantitatively in the value of the predicted cone angle and qualitatively due to the absence of the obligatory blue counterpart to the red-shifted cone photon.2-4 A model of Cherenkov radiation by a laser-induced moving polarization was proposed to account for the origin and angular/spectral dependencies of the conical emission.3

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The cone emission obtained from pumped two-level atoms is related, by a perturbation theory, to time-dependent terms in the macroscopic Bloch equations. It is described as a cooperative superfluorescence radiation from collections of spatially distributed atoms, including in the theory the far-zone dipole-dipole interactions, in the presence of the strong monochromatic field. The distribution of the cone emission as a function of frequencies and cone angles is analyzed. A phase-matching condition is obtained for the wave vectors of the pump and conical beams that is similar to the Cherenkov condition. The frequency shift for the maximum of the cone light is found to be equal to the generalized Rabi frequency. The broadening of the cone emission is related to inhomogeneity in Rabi frequencies and decoherence effects. An approximate result is obtained for the ratio of intensities of the blue-shifted cone relative to the red-shifted cone so that the blue-shifted cone can appear only for extremely strong laser intensities.

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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

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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 $(m>0)$ 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.

Physical Review A•Dec 31, 2002•8 citations
In spite of the fact that cone emission was first observed over a decade ago, there is still no generally accepted theory of the effect. Here we give an outline of a unified approach and point out a mechanism that supports Cherenkov-type radiation due to vacuum fluctuations as a possible source of cone emission. This can be considered as an additional mechanism to the previously discussed four-wave mixing, and initial encoding and follow-up refraction effects. We compare our treatment to previously considered models.
We establish a mechanism, which supports Cherenkov-type radiation, as a source of cone emission. This can be considered as an additional mechanism to the previously discussed four-wave mixing (4WM) and initial encoding and follow-up refraction effects. The radiation source is provided by spontaneous emission of the driven atoms at the frequencies of interest. Due to the medium dispersion, the pump field propagates faster than the phase velocity of the field that is generated at the Rabi sideband at lower frequency. Cherenkov radiation is then possible if the source has a long correlation length, which is indeed the case despite the random nature of spontaneous emission. Coherent superposition of the source for the lower-frequency field is in an off-axis direction given by the Cherenkov radiation condition. Differences between 4WM and Cherenkov-type models are consequently resolved.
Conical light emission is observed when an intense quasiresonant laser beam propagates in an atomic vapor.1-4 A parametric four-wave mixing process was suggested to explain this effect.1 However, this model is inconsistent with experiments both quantitatively in the value of the predicted cone angle and qualitatively due to the absence of the obligatory blue counterpart to the red-shifted cone photon.2-4 A model of Cherenkov radiation by a laser-induced moving polarization was proposed to account for the origin and angular/spectral dependencies of the conical emission.3
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.
The cone emission obtained from pumped two-level atoms is related, by a perturbation theory, to time-dependent terms in the macroscopic Bloch equations. It is described as a cooperative superfluorescence radiation from collections of spatially distributed atoms, including in the theory the far-zone dipole-dipole interactions, in the presence of the strong monochromatic field. The distribution of the cone emission as a function of frequencies and cone angles is analyzed. A phase-matching condition is obtained for the wave vectors of the pump and conical beams that is similar to the Cherenkov condition. The frequency shift for the maximum of the cone light is found to be equal to the generalized Rabi frequency. The broadening of the cone emission is related to inhomogeneity in Rabi frequencies and decoherence effects. An approximate result is obtained for the ratio of intensities of the blue-shifted cone relative to the red-shifted cone so that the blue-shifted cone can appear only for extremely strong laser intensities.
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.