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.
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.
We give a general nonperturbative treatment of cooperative emission in systems of
N two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to
N2, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length
L of the active volume is much larger than a suitable threshold length
LT (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length
L is smaller or of the same order of a suitable cooperation length
Lc (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to
N). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to
N, so that superfluorescence effects disappear.
It is now known that conical emission in a two-level atomic system is a fundamental propagation effect brought about by the breakup of the exciting laser pulse into solitary waves.1 This report presents a semiclassical numerical calculation that extends this previous work by solving the optical Bloch and Maxwell equations for a three-level atomic vapor excited by laser fields of two different frequencies. The applied fields satisfy a two-photon resonance condition between the ground and final states and are slightly detuned from the intermediate state to enhance the nonlinear polarization.2 As the fields propagate, oscillations in the nonlinear polarization of the pump beam dominate the interaction and lead to the pump's breakup into solitary waves. The Stokes polarization, depending on the pump radiation, also experiences these oscillations. Because of the intensity-dependence of the refractive index, the solitary waves experience a curvature in the time-radius plane. This curvature manifests itself in the spectrum as a radial component of the wave vector at the Rabi frequency. The Stokes field, experiencing parametric breakup, displays a similar curvature. Spectral analysis reveals a radial component of the Stokes wave vector as well. This frequency-shifted off-axis radiation is identified as conical emission in the Stokes field.
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.
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