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Physical Review A

Single-mode approximation in laser physics: A critique and a proposed improvement

Luigi A. Lugiato, Lorenzo M. Narducci, Donna Kay Bandy, Jorge R. Tredicce

Published Feb 1, 1986• 18 citations
Publisher

Abstract

We propose the use of a laser model in which the single-mode approximation is introduced for the electric field but not for the atomic variables. We show that our improved single-mode model is in good agreement with the exact steady-state solution for a reasonable range of experimentally relevant parameters well beyond the limits of validity of the mean-field approximation.

DOI
10.1103/physreva.33.1109

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We propose the use of a laser model in which the single-mode approximation is introduced for the electric field but not for the atomic variables. We show that our improved single-mode model is in good agreement with the exact steady-state solution for a reasonable range of experimentally relevant parameters well beyond the limits of validity of the mean-field approximation.
The fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function p(n,m)p(n,m)p(n,m) to find nnn photons in the light field and mmm atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.
We consider an inhomogeneously broadened, unidirectional ring laser formulated in the traditional Maxwell–Bloch configuration of a two-level atom with transverse modes. We numerically analyze the steady state for a single longitudinal mode in the paraxial approximation, using the uniform-field limit with cylindrical symmetry and Gaussian profile for inhomogeneous broadening. Several domains of steady-state solutions are found for the resonant case in the good-cavity limit. Both single- and multiple-transverse mode solutions are observed; the expected TEM00 mode is seen in the limit of single-mode operation. The laser threshold is found to depend on the mode spacing (A1) as well as on the gain (2C) and the scaled Doppler broadening (σD). This means that as A1 is increased, the laser requires a larger gain to operate, up to a critical gain; beyond this critical gain, the laser threshold is independent of A1. Hysteresis between zero- and nonzero-intensity solutions at the laser threshold is obtained under realistic operating conditions for both increasing and decreasing 2C and A1 (i.e., for changes in the initial conditions). These distinct, but overlapping, domains of parameter space where both single- and multiple-mode steady states exist are qualitatively the same for increasing values of σD ≥ 1 and may indicate the presence of competing attractors.
In an attempt to explain the instability observed by Hillman et al.,1 we have analyzed the domain of parameters in which their dye laser operates. This leads to the consideration of an asymptotic limit for which the slowly varying envelope approximation no longer holds. As a result, the electric field equation must retain its second-order space and time derivatives. This opens new possibilities for the steady states and their stability properties, (i) Even in the case of perfect matching between atomic and cavity frequencies, the laser operating frequency is displaced from the cavity resonance, (ii) The single-mode stable steady state is not necessarily the mode whose frequency is nearest to the atomic transition, (iii) Instability of the steady state can occur without restriction on the laser cavity quality.