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Journal of the Optical Society of America B

Simple analytic approximation to continuous-wave on-resonance beam reshaping

David E. McClelland, R. J. Ballagh, Weston J. Sandle

Published Feb 1, 1986• 8 citations
Publisher

Abstract

We present a simple, analytically tractable, generalized aperture model for treating the stripping of a laser beam in a medium with saturable absorption. The model departs from previous treatments in allowing a soft transition from transparency to opacity, which yields a physically realistic diffraction-free amplitude encoding for the laser beam and thereby reduces unphysical fringes in subsequent beam evolution. The model is applied to the problem of cw exactly resonant self-focusing in a vapor of two-state atoms. It is confirmed that for strong beam saturation and absorption that is not too strong, beam propagation can be well modeled as a single stripping–diffraction sequence. A simple interpretation of the on-axis enhancement features is given.

DOI
10.1364/josab.3.000212

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 1, 1992•0 citations
We present a simple, analytically tractable, generalized aperture model for treating the stripping of a laser beam in a medium with saturable absorption. The model departs from previous treatments in allowing a soft transition from transparency to opacity, which yields a physically realistic diffraction-free amplitude encoding for the laser beam and thereby reduces unphysical fringes in subsequent beam evolution. The model is applied to the problem of cw exactly resonant self-focusing in a vapor of two-state atoms. It is confirmed that for strong beam saturation and absorption that is not too strong, beam propagation can be well modeled as a single stripping–diffraction sequence. A simple interpretation of the on-axis enhancement features is given.
A single pass of an intense cw light beam through a strongly absorbing medium can be characterized by two parameters, the input on-axis intensity normalized to the saturation value I0,0{I}_{0,0}I0,0​, and the Fresnel number FFF for an absorption length ${\ensuremath{\alpha}}^{\ensuremath{-}1}$. For both large I0,0{I}_{0,0}I0,0​ and FFF, propagation can be divided in two parts: At the very beginning, the transverse profile of the beam experiences an encoding. For a beam tuned exactly on resonance, encoding reduces to a stripping of the profile from the wings to the center, progressively, as the light beam goes further into the cell. For F>>1F>>1F>>1 and I0,0>>1{I}_{0,0}>>1I0,0​>>1, and both of the same order of magnitude, this stripping can be modeled by a circular aperture placed at the end of the encoding region. Afterwards, the beam is assumed to propagate in free space, its profile exhibits Fresnel interferences and on-axis enhancement. Using a perturbative treatment, the width and the location of the aperture are analytically estimated, and the model is successfully compared with numerical predictions of on-resonance self-focusing. For ${I}_{0,0}\ensuremath{\lesssim}F$, the stripping and free-space propagation cannot be separated because near-axis stripping coexists with large diffraction effects on the wings. This case, the best for observing cw on-resonance enhancement, would gain in a more sophisticated model, taking into account diffraction by successive apertures.
Recently, Boshier and Sandle1 predicted an unexpected and interesting phenomenon of the propagation of intense cw light through a strongly absorptive two-level medium: an enhancement of the on-axis intensity occurs with the laser frequency tuned to exact resonance with the atomic transition. Previously, cw self-focusing was thought to occur only on the high-frequency side of the resonance where nonlinear dispersion gives rise to an effective positive lens that focuses the entire beam,2,3 And indeed this cw on-resonance enhancement (CORE) arises from a different mechanism: the nonlinear absorption strips away the outer portions of the beam while diffraction of the light in the outer edge of the remaining core directs some of the beam toward the axis, resulting in near-field Fresnel fringes and CORE.
An analytic model based on a single encoding–diffraction sequence is developed to describe the first main region of focusing that occurs when a cw off-resonant monochromatic Gaussian beam enters a medium of two-state atoms. A quantitative definition of the dispersive regime for self-focusing is given, and an explicit form is obtained for the field encoding produced by the initial propagation. The phase profile of this encoded field incorporates the effects of both the medium and diffraction, and we find a simple expression for the fundamental length scale, which determines where diffractive effects become dominant. An analytic expression is obtained for the on-axis field, which results once the encoded field begins to diffract freely, and simple expressions for the position and size of the main enhancement are given. We give a simple interpretation of the focusing and the two parameters required for its characterization. The model gives excellent agreement to the full numeric solution over a wide parameter range and, in particular, explains the rapid on-axis oscillations at the outset of focusing.
There has been recent interest in the phenomenon of on-axis enhancement of an intense cw laser beam in a strongly absorbing, nonrefractive, nonlinear medium.1,2 The purpose of this paper is to present experimental observations of this cw on-resonant enhancement (CORE) and the associated Fresnel fringe structure and narrowing (self-focusing) of the spatial profile. We use the collisionally broadened D1 line of atomic sodium. We also illustrate the relatively close agreement of predictions1 based on a simple model of homogeneously broadened two-state absorbers.
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.
Only for optically thin layers and plane waves may the intensity of a light wave be unambiguously expressed in terms of the nonlinear susceptibility of the medium. In aperture limited light beams the saturation parameter and the Stark shifts are local variables, while the output intensity of the pump and probe beams is an integral characteristic of the transmission process. The output spectrum is affected by diffraction, nonlinear refraction, and saturated absorption. The nonlinear refraction is a cause of a spectral line asymmetry which is well- known in saturation spectroscopy. In addition to usual line broadening, the saturated absorption causes specific beam effects, related to the formation of a self-induced transparency channel. The diffraction of the beam due to this self-induced aperture may result in the enhancement of the on-axis intensity, which is not a result of the self-induced lens effects. All these factors are taken into account by the mathematical model suggested in the present work. The propagation of the near-resonance saturating pump and non-saturating probe beams of various initial profiles through a homogeneously broadened two-level gas is simulated numerically. The calculations carried out make it possible to analyze the contribution of the propagation phenomena, mentioned above, to the output spectra of the beams.