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Annual Meeting Optical Society of America

Stability and time-dependent behavior of the detuned ring laser with a saturable absorber in the vicinity of multiple steady-state solutions

David E. Chyba, Neal Broadus Abraham, A.M. ALBANO

Published Jan 1, 1985• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1985.fw3

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Physical Review A•Sep 1, 1985•84 citations
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.
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 provide a comprehensive analysis of the standard model of the bidirectional ring laser in which only one mode can be supported in each direction and in the limit that the polarization can be eliminated adiabatically. The interaction between the two counterpropagating modes can be derived and it is most naturally viewed as a coupling between them via scattering from a spatial grating formed in the population inversion. If the grating is a sufficiently small modulation of the spatial average of the population inversion, it can be approximated by a sinusoidal function. A systematic derivation of the model with only a sinusoidal grating for the homogeneously broadened case is presented that reveals and corrects errors in several previously published analyses. The stability of the steady-state solution is analyzed. The bidirectional steady-state solution is unstable and the unidirectional steady-state solutions may be stable or unstable depending on the parameters. The well-established result of bistability between the two modes when the cavity is tuned to resonance is recovered, a result that persists, in part, even when the losses between the two modes are different.With detuning, the unidirectional steady-state solutions can be destabilized, creating regions in the parameter space where only time-dependent solutions are found. For parameters characteristic of solid-state or molecular gas lasers, the instability can occur for very small detunings and even very close to the lasing threshold. Asymptotic limits of the instability boundaries for these parameters are presented. Additional results are derived formally for inhomogeneous broadening with a Lorentzian line shape and for arbitrary inhomogeneous linewidths. Explicit analytic results are presented in the limits of very small and very large inhomogeneous linewidths compared with the homogeneous linewidth. Time-dependent solutions in the homogeneously broadened case for relaxation rates appropriate to CO2{\mathrm{CO}}_{2}CO2​ lasers show that there are broad regions in the parameter space of gain and detuning for which the behavior is dynamically chaotic in the form of nearly ``square-wave'' alternation between the counterpropagating modes of the laser with irregular switching times. In other regions of that parameter space we find more irregular pulsations and pulse alternations. Regions of periodic pulsations also have been found.
We formulate a first-principles description of the behavior of a bidirectional ring cavity containing a saturable absorbing medium, and driven by an external coherent field. The setting of interest to this study is an extension of the ordinary optically bistable system, where a ring resonator supports only one of the two possible directions of propagation of the cavity field. In this more general case and in the uniform-field limit, we show that the dynamics of the fields-atoms system is described by an infinite set of coupled equations that can be readily solved by standard numerical means after appropriate truncation of the number of atomic variables. With the help of two different approaches, we find unexpected, long-time stationary solutions such that the forward and backward fields oscillate with different carrier frequencies: the first is based upon the time-dependent equations and the second on a set of nonlinear-algebraic equations describing the steady states. As an added confirmation, we carry out a linear-stability analysis of the unidirectional steady state for the purpose of identifying the conditions under which a backward field can grow from an initial fluctuation. By this approach we also suggest a possible strategy for the experimental observation of stationary, frequency-shifted output fields.
We study the properties of the laser field in the case where the cavity contains two cells, one which amplifies and one which absorbs the radiation. We adopt the simplest possible model, in which there is only one running mode in the cavity and all atoms are fixed and homogeneously distributed in space. We first set up a semiclassical description. In the frame of the stationary semiclassical (SC) theory the most striking feature is the occurrence of a domain in which three solutions coexist. A linear-stability analysis shows that two of these solutions are stable. This gives rise to an hysteresis cycle. Most of the results in this section are analytic. In the frame of a purely quantum description, we derive a generalized Fokker-Planck equation which is exactly solved in the stationary case. This yields a stationary quasiprobability distribution function for the field. In the domain where the SC theory predicts three solutions, the field-distribution function has two maxima and one minimum whose positions are precisely given by the SC intensities. With this distribution function we study the intensity and the intensity fluctuations: the intensity shows a rather abrupt increase when crossing the narrow transition region; in this region the intensity fluctuations are extremely high. By means of an approximate method to solve the time-dependent Fokker-Planck equation we study the linewidth and show that it drastically decreases above threshold. Finally we consider the linearized and quasilinearized Fokker-Planck equations which can be solved analytically. We show that the domain in which neither of these equations fits the results of the generalized Fokker-Planck equation is extremely small. The properties of the metastable states are analyzed in terms of the generalized and asymptotic Fokker-Planck equations.
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.