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IEEE Journal of Quantum Electronics

Transverse effects associated with the propagation of coherent optical pulses in resonant media

F. P. Mattar, Maurice C. Newstein

Published Jul 1, 1977• 26 citations
Publisher

Abstract

Analytical and numerical methods are used to study the transverse reshaping that occurs when ultrashort optical pulses propagate through media resonant to the pulse-carrier frequency. The field is described by the scalar quasi-optic equation, and the material system by the full Bloch equations describing a distribution of homogeneously broadened two-level molecules. Transverse derivatives in the field equation provide the coupling leading to the development of radially dependent frequency variations and phase curvature. We have studied the origin and evolution of transverse energy flow and investigated the conditions under which this leads to significant coherent self-focusing. The physical mechanisms responsible for the onset and development of the focusing phenomenon are discussed and related to other types of self-confinement of optical pulses.

DOI
10.1109/jqe.1977.1069385

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Bulletin of the Russian Academy of Sciences Physics•Mar 1, 2012•1 citation
Analytical and numerical methods are used to study the transverse reshaping that occurs when ultrashort optical pulses propagate through media resonant to the pulse-carrier frequency. The field is described by the scalar quasi-optic equation, and the material system by the full Bloch equations describing a distribution of homogeneously broadened two-level molecules. Transverse derivatives in the field equation provide the coupling leading to the development of radially dependent frequency variations and phase curvature. We have studied the origin and evolution of transverse energy flow and investigated the conditions under which this leads to significant coherent self-focusing. The physical mechanisms responsible for the onset and development of the focusing phenomenon are discussed and related to other types of self-confinement of optical pulses.
The spatiotemporal evolutions of ultrashort pulses in two dimensions are investigated numerically by solving the coupled Maxwell-Bloch equations without invoking the slowly varying envelope approximation and rotating-wave approximation. For an on-axis $2n\ensuremath{\pi}$ sech pulse, local delay makes the temporal split $2\ensuremath{\pi}$ sech pulses crescent-shaped in the transverse distribution. Due to the transverse effect, the temporal split $2\ensuremath{\pi}$ sech pulses become unstable and experience reshaping during the propagation process. Then, interference occurs between the successive crescent-shaped pulses and multiple self-focusing can form.
Analytic and numerical solutions of the Maxwell–Bloch equations, including transverse and timedependent phase variations, predict on-resonance self-focusing (SF) and elucidate its formation due to the combined effect of diffraction and inertial response of the medium. This new SF can be characterized by a simple parameter in terms of the beam and medium parameters. Recently, two independent experiments on SF in Na and Ne demonstrated convincingly this new SF effect for spatially nonuniform self-induced transparency pulses propagating in thick resonant absorbers.
Coherent pulse propagation began with McCall’s & Hahn’s discovery of self‐Induced‐Transparency (SIT)[1] theoretically and experimentally. Even for collimated beams of finite extent a diffraction‐induced phase develops. Using a perturbational treatment[2–5] the time‐ and radially‐dependent phase φ, its wavefront curvature ∂φ/∂ρ, and the associated transverse energy flux current JT=A 2∂φ/∂ρ are calculated. The energy is redistributed across the beams. The evolving pulse departs from the uniform‐plane‐wave (UPW) counterparts. The onset of the radial reshaping and dynamic self‐action is demonstrated.
This paper is devoted to understanding the temporal synchronization of the two-color superfluorescence output powers. Following McCall and Feld et al., the monochromatic superfluorescence can be described in terms of a semi-classical propagational model that shares the same equations as Self-Induced Transparency (SIT). Thus it became important to understand double SIT that was discovered by McCall and Hahn (MH) to correctly interpret co-propagating beams with coherent exchange of energy and nonlinear phase evolution. A further motivation would be to elucidate pump and Stokes synchronization as well as pump depletion effect on Stokes buildup in Raman forward amplifier. In this work, we describe how the diffraction affects the longitudinal and transverse reshaping associated affects the simultaneous resonant propagation of two different-wavelength optical pulses in a three-level medium. The pulse lengths are shorter than any of the relaxation times, thus, the interaction is coherent. The material response has an inertial characteristic, which precludes its description with two instantaneous susceptibilities, and requires instead the density matrix formulism. A semi-classical formulism, as introduced by Konopnicki and Eberly (KE), is adopted in which two paraxial Maxwell field equations are closely intertwined in the generalized three-level Bloch equations.
When cylindrical symmetry is assumed, a numerical study of coherent on-resonance self-focusing of laser pulses in atomic media reveals a well-defined dependence on the interaction's parameters for the focusing distance and the maximal on-axis energy density reached at the focus. We interpreted this focusing to be the result of a diaphragm effect at the edge of the pulse. This can be readily explained by considering the transverse dependence of local self-induced transparency phenomena first described by McCall and Hahn [Phys. Rev. Lett. 18, 908 (1967); Phys. Rev. 183, 457 (1969)]. From this interpretation and the well-known Maxwell-Bloch equations, we derived a theoretical quantitative model for coherent on-resonance large-scale self-focusing, that is to say, focusing of the beam as a whole. Our main results include the above-mentioned exact parametric dependence for the focusing distance, as well as predictions about the ratios of on-axis energy densities and pulse transverse sizes between the input and focus planes. All of these predictions appear to corroborate the results of an experiment on a Tm169{\mathrm{Tm}}^{169}Tm169 vapor.
A technique for the numerical simulation of coherent optical responses of the photon echo type formed in resonance media with strong inhomogeneous broadening under the action of femtosecond laser pulses is developed. This approach is based on solving the Maxwell-Bloch equations using the finite-difference time-domain (FDTD) method without application of a fast rotating field and slowly varying envelope approximations. The method can be used to simulate coherent responses in different resonance media with a complex structure of energy levels. The technique was validated using the example of describing experiments on narrowing a photon echo pulse and femtosecond echo processing.