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Physical Review A

Self-focused light propagation in a fully saturable medium: Theory

Marla L. Dowell, B. D. Paul, Alan C. Gallagher, J. Cooper

Published Oct 1, 1995• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.52.3244

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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.
On-resonance self-action of a beam with periodically modulated amplitude in a two-level saturable absorber is studied numerically. When the period of modulation is comparable with relaxation times, one cannot describe the medium by means of susceptibility, and the response of the medium to the amplitude variation is not immediate. We solve the Bloch equations in each point of the medium to find this response correctly. Transient behavior followed by stable oscillations of population difference and polarization vector is shown to depend strongly on the frequency and amplitude of modulation. Beam propagation is considered using a simplified model of a thin absorber followed by free space, the propagation distance being small compared with the modulation wavelength. Manifestations of non-stationary response of the medium in the output intensity behavior are analyzed.
Full space-time numerical modeling of near-resonance self-action of a beam, whose frequency is harmonically modulated in time, is presented. Maxwell-Bloch equations describing the propagation of the beam through a two-level medium with saturable absorption and refraction are solved using an original stable numerical scheme for multidimensional nonlinear Schrodinger equations. Delayed redistribution of intensity over the beam cross-section due to nonlinear self-focusing and self-phase-modulation effects is shown to affect the phase and the amplitude dynamics of the output signal in a rather nontrivial way, depending on the carrier laser frequency.
Numerical modeling of self-action of a beam, whose amplitude is periodically modulated in time, in a two-level saturable absorbing medium is reported. For describe a beam self-action we solve numerically the full set of Maxwell-Bloch equations. We study the influence of the non-stationary self-action of the beam on the transmission of modulation signal through the medium. It is shown that in the time dependence of the polarization and population difference the nonlinear distortions of the modulation signal can be much stronger than in the output field intensity.
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.
Special features of generating a self-induced transparency in a medium with both resonant and non-resonant (Kerr-type) nonlinearities are considered. Soliton solutions of the Maxwell-Bloch equations are found, taking into account the diffraction divergence of the light beam. It is shown that solitons emerge in such a medium under conditions of two-dimensional optical beam scanning along the medium surface, and have non-coinciding propagating directions of the amplitude and phase fronts. As a necessary condition for the existence of solitons, there is a definite relationship between the scanning velocity and the parameters of the medium and radiation.
Accurate modeling of optical nonlinearities is crucial to describe macroscopic laser propagation in a medium, including sum frequency generation, spectral broadening due to self-phase modulation, various ionization processes and soliton formation. For incident laser light the response of the medium is given by the induced polarization of the microscopic system. The polarization is usually expanded in a Taylor series for the electric field amplitude, which is truncated after the first non-linear term being of third order for isotropic media. A third-order nonlinearity leads to the well-known optical Kerr effect, where the refractive index of the medium becomes intensity dependent via n = n0+ n2I. This leads to an inherent problem when modeling laser propagation in two or more spatial dimensions, linked to the formal divergence (n2> 0) of the refractive index for increasing intensity.