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Physical Review A

Propagation of the electromagnetic field in optical-limiting reverse-saturable absorbers

S. Kim, David W. McLaughlin, Mary J. Potasek

Published Jan 11, 2000• 13 citations

Abstract

Reverse-saturable absorbers are of considerable interest for optical limiting. Using the electric dipole perturbation, we derived the rate equation for a five-level system describing reverse-saturable absorbers. Traditional theories for the propagating laser beam in these materials are expressed in terms of the optical intensity. However, with the introduction of high-power short-pulsed lasers, the propagation of light in these materials may be subject to nonlinear phenomena such as self-focusing and self-phase modulation. Furthermore, conventional theories treat the laser light as a continuous wave or as a very broad temporal pulse in which dispersive effects are neglected. In order to incorporate these other nonlinear or dispersive effects, and therefore determine their influence in reverse-saturable absorbers, we derived an equation for the propagation of the electromagnetic field, rather than the intensity, coupled to the rate equations for a five-level system. We also coupled our theory to experimentally measurable parameters for these materials and detailed the various physical approximations used to obtain the rate equations.

DOI: 10.1103/physreva.61.025801

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Reverse-saturable absorbers are of considerable interest for optical limiting. Using the electric dipole perturbation, we derived the rate equation for a five-level system describing reverse-saturable absorbers. Traditional theories for the propagating laser beam in these materials are expressed in terms of the optical intensity. However, with the introduction of high-power short-pulsed lasers, the propagation of light in these materials may be subject to nonlinear phenomena such as self-focusing and self-phase modulation. Furthermore, conventional theories treat the laser light as a continuous wave or as a very broad temporal pulse in which dispersive effects are neglected. In order to incorporate these other nonlinear or dispersive effects, and therefore determine their influence in reverse-saturable absorbers, we derived an equation for the propagation of the electromagnetic field, rather than the intensity, coupled to the rate equations for a five-level system. We also coupled our theory to experimentally measurable parameters for these materials and detailed the various physical approximations used to obtain the rate equations.

We derived a numerical technique for the propagation of the electromagnetic field in a five-level reverse saturable absorber including the nonlinear Kerr effect and dispersion. The numerical method combines the split step beam propagation method and the Crank–Nicholson method. Using our numerical technique we observed new behavior, not previously observed nor predicted to our knowledge, including the temporal splitting caused by the dynamics of the carrier densities in a reverse saturable absorber and the enhancement of absorption due to the Kerr nonlinearity. Our numerical calculation enables the prediction of nonlinear absorption using material parameters such as the absorption cross-sections and decay rates. We can also investigate the interplay between the optical pulse properties such as the temporal pulse width, spatial radius, incident energy and the carrier dynamics and nonlinear absorption of the reverse saturable absorber.

We develop an analytical approach to the dynamics of band populations of reverse saturable absorbers modeled by the three-level approximation of the five-level rate equations. We find high-accuracy approximate solutions to these rate equations, taking into account the temporal shape of the incident laser pulse for different regimes of excitation. The results obtained are confirmed by direct numerical integration of the rate equations and are verified by solution of the full system of the rate and the propagation equations. The validity ranges of the approximations are determined. We also prove that for input pulses that are much longer than the lifetime of the first excited state the ground-state depletion obeys the same functional dependence on the input fluence as in the case of rectangular input pulses. The dynamics of the excited states, however, explicitly depends on the pulse shape. We quantitatively estimate the effect of various parameters on the nonlinear absorption coefficient and discuss implementation of the approach by a beam propagation method to reduce the computational time.

We theoretically and experimentally study absorption of picosecond laser pulses in materials described by a four-level system that exhibit reverse saturable absorption (RSA). Using an approximate solution to the rate equations, we derive, analyze, and verify, numerically and experimentally, a single dynamical equation for the spatial evolution of the pulse fluence that includes both the rate equations and the propagation equation. This analytical approach considerably simplifies the study of optical limiting with picosecond pulses and helps to predict the behavior of the nonlinear transmittance, the level of output signal clamping, and a possible turnover from RSA to saturable absorption that restricts the performance of optical limiters based on RSA. The results obtained can also be used to characterize RSA materials by the pump–probe technique.

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.