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IEEE Journal of Quantum Electronics

Exact solution of the rate equations for the three energy level laser

Ezio Marchi, Leopaldo Millan, E. Crinó

Published Apr 1, 1980• 2 citations
Publisher

Abstract

The system of two nonlinear differential equations for a three-level laser is solved exactly. It relates the number of photonsq(t)in the laser cavity and the population inversionn(t). The functionsn(t)andq(t)are expanded in a power series and two recurrence relations are obtained. These are solved and the solutions are in excellent agreement with experimental results.

DOI
10.1109/jqe.1980.1070489

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The system of two nonlinear differential equations for a three-level laser is solved exactly. It relates the number of photonsq(t)in the laser cavity and the population inversionn(t). The functionsn(t)andq(t)are expanded in a power series and two recurrence relations are obtained. These are solved and the solutions are in excellent agreement with experimental results.
In Chapter 1, we introduced the standard rate equations (SRE) for a laser containing two-level atoms between which stimulated emission is possible. But real lasers exhibit much more complicated energy level schemes. Throughout this chapter, we consider several models in which the light–matter interaction is described by population equations for all the levels involved in the laser operation. Although they are mathematically more complicated, we shall investigate these equations in the same way as in Chapter 1, i.e. by formulating dimensionless equations and by analyzing the stability properties of the steady states. The basic ingredients of the SRE model used up to here, namely pumping and relaxation processes, play key roles in the efficiency of lasers. But already during the pioneering days of the laser, it appeared important to investigate three- and four-level models in order to obtain more reliable information on quantities such as the power conversion efficiency or the response time. The common extensions of the two-level SRE typically consider an open two-level system, or three- or four-level systems, depending on the nature of the active medium. Restriction to as few as four levels is again a crude simplification of the complex population dynamics occurring in most lasers. But it is rather surprising to see how good these simple kinetic models are. As we demonstrate by studying specific examples, the solution of the three- or four-level rate equations differs only slightly from the solution of the SRE.
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.
A nondegenerate three-level atomic system is considered, and the possibility of lasering without population inversion via numerical computations is investigated. The atomic system consists of one upper level and two closely spaced lower levels, which are coherently mixed by a microwave field interaction. A numerical calculation is performed, based on a variable step formulation of the classic Adams formulae for solving first-order differential equations. This method of solution has been outlined by C. W. Gear and by L. F. Shampine and M. K. Gordon. The authors adapt this algorithm for use with complex double precision arrays and apply the resulting program to the Maxwell-Schrodinger equations of this atomic system. This enables the authors to carry out a strong field analysis of the atom-field interaction, since they are not restricting their analysis to perturbation theory.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.