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Physical Review A

Conical emission as a result of pulse breakup into solitary waves

Michael E. Crenshaw, Cyrus D. Cantrell

Published Jan 1, 1989• 38 citations
Publisher

Abstract

We report computational results that show that the conical emission observed when a nearly resonant laser pulse propagates through an atomic vapor is the result of the breakup of the pulse into solitary waves. For an incident pulse with a radial as well as a temporal profile, the crests and troughs of the two-dimensional solitary waves are curved in a time-radius plane. The temporal modulation associated with pulse breakup appears as Rabi sidebands in the spectrum, while the curvature of the solitary waves in the time-radius plane results in a transverse spatial modulation that leads to conical emission in the far field. These results were obtained by performing detailed numerical calculations of the time-dependent paraxial propagation of a cylindrically symmetric laser pulse through a vapor of two-level atoms under the rotating-wave and slowly-varying-envelope approximations in the limit of no collisional damping or Doppler broadening. Our results can be interpreted readily in terms of optical nutation on the Bloch sphere and in terms of noncollinear phase matching through the curvature of the solitary waves rather than through additional parametrically generated waves. We find that self-focusing is not required for the generation of conical emission, although it would be difficult to separate these effects experimentally.

DOI
10.1103/physreva.39.126

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We report computational results that show that the conical emission observed when a nearly resonant laser pulse propagates through an atomic vapor is the result of the breakup of the pulse into solitary waves. For an incident pulse with a radial as well as a temporal profile, the crests and troughs of the two-dimensional solitary waves are curved in a time-radius plane. The temporal modulation associated with pulse breakup appears as Rabi sidebands in the spectrum, while the curvature of the solitary waves in the time-radius plane results in a transverse spatial modulation that leads to conical emission in the far field. These results were obtained by performing detailed numerical calculations of the time-dependent paraxial propagation of a cylindrically symmetric laser pulse through a vapor of two-level atoms under the rotating-wave and slowly-varying-envelope approximations in the limit of no collisional damping or Doppler broadening. Our results can be interpreted readily in terms of optical nutation on the Bloch sphere and in terms of noncollinear phase matching through the curvature of the solitary waves rather than through additional parametrically generated waves. We find that self-focusing is not required for the generation of conical emission, although it would be difficult to separate these effects experimentally.
Computational results show that the conical emission and Rabi sidebands observed when a nearly resonant laser pulse propagates through an atomic vapor are the result of the breakup of the pulse into solitary waves. For an incident pulse with a transverse as well as a temporal profile, the crests and troughs of the 2-D solitary waves are curved in a time-radius plane. The temporal modulation associated with pulse breakup appears as Rabi sidebands in the spectrum, while the curvature of the solitary waves in the time-radius plane results in a transverse spatial modulation, resulting in conical emission in the far field. These results were obtained by performing detailed numerical calculations of the time-dependent paraxial propagation of a cylindrically symmetric laser pulse through a vapor of two-level atoms. Our results can be interpreted readily in terms of optical nutation on the Bloch sphere and in terms of noncollinear phase matching through the curvature of the solitary waves rather than through additional parametrically generated waves. The cone opening angle depends on the initial transverse profile of the pulse and the process of beam breakup as well as self-focusing.
Computational results show that the conical emission and Rabi sidebands observed when a nearly resonant laser pulse propagates through an atomic vapor are the result of pulse breakup into temporal-spatial (2-D) solitary waves. The temporal modulation associated with pulse breakup appears as Rabi sidebands in the spectrum. Because the process of pulse breakup depends on the pulse area, the 2-D solitary waves are curved in the time-radius plane due to the transverse profile of the pulse. The curvature of the solitary waves in the time-radius plane results in a transverse spatial oscillation and a corresponding transverse wave vector that leads to conical emission in the far field.
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.
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.
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.
An intense laser pulse that is nearly resonant with an atomic transition and that can initially be described as smooth and nearly adiabatic can acquire significant temporal and spatial modulation as the result of propagation through an atomic vapor. Computer calculations demonstrate that the temporal modulation is a result of enhancement, under propagation, of the initially small nonadiabatic portion of the transient nonlinear atomic response. The transient nonlinear atomic response varies with the field strength so that the transverse variation in field strength results in a spatial modulation, known as conical emission.