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Applied Spectroscopy

Extended Model for Saturation in a Two-Level System

Cornelius A. van Dijk, Nicolo’ Omenetto, James D. Winefordner

Published Jul 1, 1981• 20 citations
Publisher

Abstract

A model is presented, based on rate equations for a two-level system, which describes both the temporal and time-integrated dependence of the fluorescence signal due to saturation by a broad band and spatially homogeneous laser beam in an optically thin medium. In particular, we discuss the processing of the fluorescence signal by a gated integrator and relations between gate width, laser pulse length, and decay rate constant of the two-level system. It is shown that, even in the relatively simple case of a fully saturating, rectangular laser pulse, the integrated signal is still dependent on atomic and collisional parameters. On the other hand, this dependency can be used to measure these parameters. The only case in which the integrated signal becomes independent of these parameters is the case of a perfectly rectangular portion being sliced out of the time dependent fluorescence signal. An experimental method, which is expected to result in a better approximation of the saturation parameter in the case of the integrated signals, is suggested.

DOI
10.1366/0003702814732517

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A model is presented, based on rate equations for a two-level system, which describes both the temporal and time-integrated dependence of the fluorescence signal due to saturation by a broad band and spatially homogeneous laser beam in an optically thin medium. In particular, we discuss the processing of the fluorescence signal by a gated integrator and relations between gate width, laser pulse length, and decay rate constant of the two-level system. It is shown that, even in the relatively simple case of a fully saturating, rectangular laser pulse, the integrated signal is still dependent on atomic and collisional parameters. On the other hand, this dependency can be used to measure these parameters. The only case in which the integrated signal becomes independent of these parameters is the case of a perfectly rectangular portion being sliced out of the time dependent fluorescence signal. An experimental method, which is expected to result in a better approximation of the saturation parameter in the case of the integrated signals, is suggested.
On the basis of Bloch's equations an analytical expression is obtained for the pulsed saturation signal of a two-level quantum system with allowance for inhomogeneous line broadening. It is shown that in this case the signal drop is exponential over the initial portion and nonexponential for long times. It is found that starting from a certain time, which depends on the field magnitude, relaxation parameters, and inhomogeneous width of the line, a through appears in the line contour, the depth and width of which increases with an increase in pulse duration. At smaller durations of the pulse, when the line is bell-shaped, an exponential decay of the signal indicates saturation of homogeneously broadened line. Comparison of the results obtained in an experiment on pulsed saturation in ruby shows that at large amplitudes of the field and considerable inhomogeneous broadening, one observes a deviation from the Bloch theory manifesting itself as a biexponential drop in the pulsed saturation signal A conclusion is drawn that the Bloch theory satisfactorily describes the experiment until the effective time of energy transfer to the lattice is smaller than the time of the spectral propagation of excitation.
Exact numerical solution of a time-space (one dimensional) three-level lasing system (homogeneously broadened) is used to study the relevance of steady-state models to such a system. This study revealed: 1) the existence of two disconnected (quasi) steady-state regions-the small-signal gain region (slow pumping rate) and the high saturation region (large pumping rate), and 2) the spatially averaged equations for the high saturation region have a form similar to those usually employed in the small-signal gain region, except that a constant which in the latter case is unity becomes smaller than 1 in the high saturation region.
Recent theoretical treatments of the laser with a saturable absorber have utilized multilevel rate equation models with considerable success in simulating experimental results.1 Such treatments do not incorporate the detuning of the cavity or of the absorber atom line center from the amplifying atom line center. This additional complication may have a significant effect on the dynamics of the system, as has already been strongly suggested by an examination of the stability of the steady states of a semi-classical model which assumes atoms with only two levels.2 This paper extends this work on the two-level model into the timedependent regime. The results are obtained from a particularly convenient transformation of the semiclassical time-dependent equations for a homogeneously broadened, two-level atom model of the laser with a 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.
A new method is developed for the description of transfer and scattering of a time-dependent laser pulse, which is based on a hierarchy of Bogoliubov-Born-Green-Kirkwood-Yvone (BBGKY) equations for reduced density matrices. The systems of equations are obtained for the atomic and photon density matrices and the correlation operator in the second-order Born approximation. The system of equations describes stimulated and spontaneous scattering of laser radiation in an optically thin medium, where reabsorption of spontaneous radiation can be neglected. The method proposed in the paper yields for the first time the correct expressions for nonstationary resonance fluorescence spectra, which do not give the negative values for fluorescence intensity. The expressions obtained for the resonance emission spectra coincide in limiting cases with those obtained within the framework of the quantum regression theorem.
Rate equations for the population dynamics of multilevel systems in laser-induced fluorescence spectroscopy are solved by the aid of perturbation theory. This leads to a general expression for the population of the level pumped by the laser radiation. Arbitrary laser pulse shapes as well as collisional processes are admitted. In the first-order approximation three functions depending on time and the properties of the laser pulse remain. These functions have been tabulated for a bell-shaped laser pulse. The results are used in several characteristic situations and are compared with the numerical solutions of the rate equations.