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Physical Review Letters

Coherent On-Resonance Self-Focusing of Optical Pulses in Absorbers

H. M. Gibbs, Bouwe Bolger, F. P. Mattar, Maurice C. Newstein, G. Forster, P. E. Toschek

Published Dec 27, 1976• 50 citations
Publisher

Abstract

Self-focusing of spatially nonuniform coherent pulses has been seen in the center of Na and Ne transitions. Self-focusing and temporal reshaping were studied as a function of laser detuning in Na. The dependence of on-resonance self-focusing upon input pulse area was observed in Ne. Numerical solutions of the Maxwell-Bloch equations including transverse variations predict coherent self-focusing and yield a single parameter for estimating its importance for a given beam diameter and sample absorption.

DOI
10.1103/physrevlett.37.1743

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Self-focusing of spatially nonuniform coherent pulses has been seen in the center of Na and Ne transitions. Self-focusing and temporal reshaping were studied as a function of laser detuning in Na. The dependence of on-resonance self-focusing upon input pulse area was observed in Ne. Numerical solutions of the Maxwell-Bloch equations including transverse variations predict coherent self-focusing and yield a single parameter for estimating its importance for a given beam diameter and sample absorption.
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.
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.
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.
A comprehensive model of intense laser light propagation in a fully saturable medium is presented. This model employs the Maxwell-Bloch equations for a two-level atomic system. Light propagating through a medium creates intensity-dependent index of refraction gradients that reduce to a Kerr nonlinearity only for a weak field. When the laser detuning is positive, self-induced index gradients cause the beam to self-focus. Stationary filaments occur when self-focusing exactly balances diffraction. There exists a unique set of two-dimensional fundamental (and higher mode) solutions for which the filament transverse profile (not Gaussian in general) depends on the following quantities: density of saturable absorbers, laser detuning from resonance, laser power, and natural linewidth. (In contrast the Kerr solution does not have a unique filament diameter.) Emphasis will be placed on propagation of Gaussian incident beams; deviations from the stationary solution in either shape or power generally give rise to oscillating spatial patterns, including donutlike profiles, or beam breakup.
The coherent reshaping of short-duration (2–40 nsec) N2O laser pulses in a resonant NH3 absorber is studied for a variable number of absorption lengths (αl ≤ 14). For small-area square pulses, the reshaping can produce subnanosecond pulses at the leading and trailing edges. A rapid phase reversal gives rise to amplification for times comparable to the transverse relaxation time. Zero-degree pulses [∫−∞∞E(z,t) d t=0] are observed to propagate with enhanced transmission for both short-duration low-intensity pulses and longer pulses of intensity sufficient to enable observation of optical nutation effects.
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.