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Physical Review A

Exact linear stability analysis of the plane-wave Maxwell-Bloch equations for a ring laser

L. A. Lugiato, Lorenzo M. Narducci, M. Squicciarini

Published Oct 1, 1986• 39 citations
Publisher

Abstract

We perform an exact linear stability analysis of the Maxwell-Bloch equations for a unidirectional ring laser. The model analyzed in this paper is based on the plane-wave assumption for the cavity field, but it allows arbitrary values of all the other parameters for the cavity and for the active medium. This problem had been solved previously only under exact resonance conditions, or in the uniform-field (mean-field) limit, or after adiabatic elimination of the atomic polarization.

DOI
10.1103/physreva.34.3101

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Singular perturbation analysis of the mean-field limit of semiclassical optics

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A linear stability analysis of a plane-wave model of a unidirectional ring cavity, which allows the slowly varying field amplitudes to vary along the length of the atomic medium, yields a characteristic equation having singularities coincident with estimates of the atomic-dominant eigenvalues derived under the mean-field limit. This result calls into question the validity of previous estimates of these eigenvalues. Using this general characteristic equation, we estimate the cavity- and atom-dominant eigenvalues by applying a perturbation expansion in sequential powers of the uniform-field parameter \ensuremath{\eta}=2 ln\ensuremath{\Vert}scr\ifmmode \bar{E}\else \={E}\fi{}(L)/scr\ifmmode \bar{E}\else \={E}\fi{}(0)\ensuremath{\Vert}. The resulting eigenvalue estimates appear as asymptotic expansions with limited domains of validity. The cavity-dominant eigenvalues, which are responsible for the well-known ``self-pulsing'' instability in unidirectional ring lasers and optically bistable resonators, are shown to be consistent with estimates derived in previous studies. However, the new expressions for the atom-dominant eigenvalues exhibit a more intricate distribution in the complex plane.

Physical Review A•Jan 1, 1990•1 citation
We perform an exact linear stability analysis of the Maxwell-Bloch equations for a unidirectional ring laser. The model analyzed in this paper is based on the plane-wave assumption for the cavity field, but it allows arbitrary values of all the other parameters for the cavity and for the active medium. This problem had been solved previously only under exact resonance conditions, or in the uniform-field (mean-field) limit, or after adiabatic elimination of the atomic polarization.
This paper contains a description of the behavior of a multimode unidirectional ring laser with a homogeneously broadened active medium. Our formulation is based on the conventional Maxwell-Bloch (MB) equations, but is distinguished from other treatments by the inclusion of a finite mirror reflectivity and an arbitrary value of the gain parameter. We review the steady-state behavior of the system and analyze the longitudinal profile of the field and of the atomic variables. With an appropriate transformation of variables, we transform the boundary conditions of the ring cavity into standard periodicity type, even in the presence of a finite reflectivity, and derive an infinite hierarchy of coupled mode equations. We analyze exactly the linear stability of the system, and investigate the dependence of the instability domain on the reflectivity and gain parameters. A numerical study of the full MB equations for a parameter range of the type explored in the recent experiments by Hillman et al. [Phys. Rev. Lett. 52, 1605 (1984)] reveals similarities, but also considerable differences between the results of the theory and the main experimental signatures of their instability. However, the injection of numerical noise shows the presence of numerous coexisting basins of attraction which are likely to play a significant role in the dynamics of a noisy laser.
In the previous chapters we considered the Maxwell–Bloch equations with the boundary condition appropriate for a unidirectional ring cavity and calculated the stationary solutions for laser and optical bistability, respectively. Next, we showed that the exact stationary solutions simplify appreciably in the low-transmission limit. In this chapter we want, first of all, to apply the low-transmission approximation on the dynamical equations directly, beyond the stationary case. We start from the Maxwell–Bloch equations (4.35)–(4.37) and the boundary condition (8.36). The latter is complex and includes the retardation Δ t and some parameters such as the transmissivity and reflectivity coefficients T and R , the input field y and the cavity detuning δ 0 . We introduce a transformation of the coordinates z and t and a transformation of the variables F and P [121–123]. As a result, the boundary condition reduces to a simple periodic boundary condition, and the electric-field equation incorporates the parameters which previously appeared in the boundary condition. The virtue of the periodic boundary condition is that it allows one to expand the electric field in terms of modal amplitudes and to recast the field equation in the form of a set of time-evolution equations for the modal amplitudes. All of this is shown in Section 12.1. In Section 12.2 we apply the low-transmission approximation and the Maxwell–Bloch equations assume a form that is a generalization of a set of equations commonly used in the literature to describe the ring laser [124]. Together with the Bloch equations for the atomic variables, the modal equations provide a model that lends itself ideally to a numerical resolution of the equations. In Section 12.3 we show that, if we assume that only the resonant cavity mode is active, this model reduces to a simple single-mode model consisting in three coupled ordinary differential equations, which will be analyzed in Part II of this book. Next, in the same section we derive the single-mode model in a more direct way, without passing through the multimodal equations.
The dynamic behavior of a coherently pumped single-mode unidirectional ring laser with a homogeneously broadened three-level active medium is studied. Our formulation is based on a set often real equations of the plane-wave, mean-field Maxwell–Bloch type. The instability domain in the main control parameters space is determined. Our numerical study of these equations for a parameter range of the type explored in the recent experiments by Weiss Brock [ Phys. Rev. Lett.57, 2804 ( 1986)] reveals some similarities, but striking differences between our theoretical predictions and their experimental observations are also noted.
We present an examination of the stability and time-dependent behavior of a semiclassical model of the ring laser with a saturable absorber, in the vicinity of multiple steady-state solutions. We consider particularly the case in which the cavity is tuned away from the resonance frequency of the gain medium. The model considers the absorbing and amplifying media to consist of two-level atoms. Only homogeneous spectral broadening and unidirectional propagation of the light are considered. We are able to treat the steady state with an approximate mean field solution and also with its exact longitudinal dependence. The latter is found numerically from an implicit function solution to the Maxwell-Bloch equations in the steady state. There may be as many as four steady-state solutions for a given set of parameter values. When there are four solutions, we find that three solutions are of one intensity class (high or low) and that one is of a second intensity class (low or high). When the absorbing and emitting atoms are in resonance and the cavity is tuned to their line-center frequency, two of the members of the three-solution class have the same intensity. This degeneracy is broken by detuning.
This paper presents new results for the steady states of a detuned ring laser with a saturable absorber. We employ a semiclassical model which assumes homogeneously broadened two-level atoms. We proceed by solving the Maxwell-Bloch equations for the longitudinal dependence of the steady states of this system, and then simplify our solution by use of the uniform-field approximation. We present uniform-field results for squared electric field versus operating frequency, and for each of these versus cavity tuning and laser excitation. Various cavity linewidths and both resonant and nonresonant amplifier and absorber line-center frequencies are considered. The most notable finding is that cavity detuning breaks the degeneracies found in the steady-state solutions of the fully tuned case. This leads to the prediction that an actual system will bifurcate from the zero-intensity solution to a steady-state solution as laser excitation increases from zero, rather than to the small-amplitude pulsations found for the model with exactly resonant tuning of the cavity and the media line centers. Other phenomena suggested by the steady-state results include tuning-dependent hysteresis and bistability, and instability in both intensity and frequency due to the appearance of one or more new steady-state solutions as tuning is varied. These effects of detuning are being tested by a linearized stability analysis whose results will be reported separately.
A linear stability analysis of a plane-wave model of a unidirectional ring cavity, which allows the slowly varying field amplitudes to vary along the length of the atomic medium, yields a characteristic equation having singularities coincident with estimates of the atomic-dominant eigenvalues derived under the mean-field limit. This result calls into question the validity of previous estimates of these eigenvalues. Using this general characteristic equation, we estimate the cavity- and atom-dominant eigenvalues by applying a perturbation expansion in sequential powers of the uniform-field parameter \ensuremath{\eta}=2 ln\ensuremath{\Vert}scr\ifmmode \bar{E}\else \={E}\fi{}(L)/scr\ifmmode \bar{E}\else \={E}\fi{}(0)\ensuremath{\Vert}. The resulting eigenvalue estimates appear as asymptotic expansions with limited domains of validity. The cavity-dominant eigenvalues, which are responsible for the well-known ``self-pulsing'' instability in unidirectional ring lasers and optically bistable resonators, are shown to be consistent with estimates derived in previous studies. However, the new expressions for the atom-dominant eigenvalues exhibit a more intricate distribution in the complex plane.