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IEEE Journal of Quantum Electronics

Multilevel paraxial Maxwell-Bloch equation description of short pulse amplification in the atomic iodine laser

M. E. Riley, Thomas D. Padrick, R. Palmer

Published Mar 1, 1979• 12 citations
Publisher

Abstract

Equations appropriate for amplification of short pulses in the multilevel atomic iodine amplifier are developed from the Maxwell-Bloch equations with homogeneous line broadening. The equations are implemented in the form of a two-dimensional time-dependent finite-difference computer code with six population densities, six medium polarization densities, and adaptive propagation capabilities. The Sandia atomic iodine laser was used to examine the adequacy of the theoretical description for amplification conditions covering the full range of gain saturation. Whole-beam self-focusing is observed with a strongly saturating small-waist beam in agreement with predictions. Calculations predict that this self-focusing instability may be controlled by tuning the multiline amplifier bandwidth.

DOI
10.1109/jqe.1979.1069980

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Equations appropriate for amplification of short pulses in the multilevel atomic iodine amplifier are developed from the Maxwell-Bloch equations with homogeneous line broadening. The equations are implemented in the form of a two-dimensional time-dependent finite-difference computer code with six population densities, six medium polarization densities, and adaptive propagation capabilities. The Sandia atomic iodine laser was used to examine the adequacy of the theoretical description for amplification conditions covering the full range of gain saturation. Whole-beam self-focusing is observed with a strongly saturating small-waist beam in agreement with predictions. Calculations predict that this self-focusing instability may be controlled by tuning the multiline amplifier bandwidth.
Coherent amplification of short-pulse XeCl lasers is studied theoretically by using multilevel Maxwell–Bloch equations in which the vibrational–rotational structures of a XeCl gain spectrum are included. The model used considers 100 transitions each in the P and R branches for six different vibrational transitions of XeCl(B, υ = 0) → XeCl(X, υ′ = 0–5). Coherence components between sublevels in the B and X states are also properly calculated. The model can successfully predict coherent effects such as a quantum beat caused by the spectrum overlap of the several vibrational–rotational transitions involved in a short-pulse laser spectrum. During amplification, laser pulses experience some nonlinear effects caused by the complex gain spectrum structure and by the coherent interactions; thus a considerable change in the laser pulse shape and a substantial reduction in the duration of the amplified laser pulse are predicted. The Frantz–Nodvik equation in the rate-equation limit and even single-level Maxwell–Bloch equations are not applicable for short-pulse propagation analyses in inhomogeneously broadened gain media.
Coherent amplifications of short pulse XeCl lasers are theoretically studied using multi-level Maxwell-Bloch equations including real vibrational-rotational structures of a gain spectrum of XeCl. The model can successfully predict coherent effects such as the quantum beat caused by spectrum overlaptions of several vibrational-rotational transitions involved in the short pulse laser spectrum. Saturation of amplified energy caused by formation of a 2 π-pulse-like pulse. Since a pulse area necessary for the production of a 2π-pulse-like pulse which depletes all the upper state populations accessible to the laser spectrum depends on the laser pulse width, an effective saturation energy is a function of the laser pulse width. When absorption in the amplifier media is not negligible to the small-signal gain, an effective u-pulse-like pulse is generated instead of a 2π-pulse-like pulse. The laser pulse width is shortened by the π or 2π-pulse-like pulse formation, and even shorter pulses than the linear limitation defined by the gain spectrum width can be generated if the amplified pulse intensity becomes very strong.
Pulse propagations in an amplifier in the case of the photodissociation iodine laser are investigated with the use of semiclassical treatments, especially when the input pulse fed into the amplifier has a width comparable to or shorter than the transverse relaxation time of the amplifier medium. In our semiclassical equations for pulse propagation in an iodine amplifier, the populations of all hyperfine levels [2P1/2(F= 3, 2),2P3/2(F = 4, 3, 2, 1)], the polarizations of optically allowed transitions (\DeltaF = 0, \pm 1), and the Doppler detuning effects were taken into consideration. Based on the numerical integrations of the semiclassical equations, the propagated pulse shape and the population distributions of the hyperfine levels are discussed. From our calculations, it has been made clear that bandwidth limitations of pulses in the range of pulsewidths of 0.1 ns and below can be avoided by making use of the nonlinear amplification that results from the coherent interaction between the propagated pulse field and the amplifier medium.
Paraxial Maxwell-Bloch theory is used to describe amplified spontaneous emission (ASE) in strongly inverted media. Counterpropagation of waves coupled through gain saturation and diffraction, and pumping by a laser beam which propagates transverse to the amplifier are included in the treatment. The reactive part of the medium polarization is calculated by population-rate equations in the limit of rapid dephasing of the atomic dipoles. The spontaneous emission noise source needed for the numerical simulation of Maxwell's equations is derived from microscopic theory by coarse-graining, thereby establishing the compatibility of the spherically symmetric microscopic source with paraxial propagation theory. The three coupled Maxwell equations for the counterpropagating ASE waves and the transversely propagating pump wave are solved numerically in time and two spatial dimensions. This simulation is used to calculate the longitudinal profile of the transverse fluorescence, which measures the saturation of the gain, the longitudinal profile of the transmitted pump wave, which measures the saturation of pump absorption, and the transverse coherence function of the radiation emitted from the ends of the amplifier.
This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.
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.