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Physical Review A

Multimode instabilities in a homogeneously broadened ring laser

Luigi A. Lugiato, Lorenzo M. Narducci, Elia Eschenazi, Donna Kay Bandy, N. B. Abraham

Published Sep 1, 1985• 84 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.32.1563

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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.
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 study the dynamic behaviour of a coherently pumped single-mode unidirectional ring laser with an active medium of Doppler broadened three-level atoms. Our formulation is based on a set of 9N+1 real equations of the plane-wave mean-field Maxwell-Bloch type (N is the number of velocity groups of molecules considered). This model predicts, for a parameter range of the type explored in the experiments by Weiss and Brock (Phys. Rev. Lett. 57 (1986) 2804), a behaviour that reproduces most of their main experimental features. These include the instability pump threshold and the sequence of instabilities ocurring on resonator tuning and on increasing pump strength.
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.
We solve numerically the time-dependent equations of motion for an inhomogeneously broadened single-mode ring laser. As expected from the linear stability analysis, pulsations develop in the output of the laser for appropriate values of the gain parameter. Under resonant conditions, the tendency is for the system to slip from a periodic regime into irregular oscillations; out of resonance, however, periodic oscillations and period-doubling bifurcations are more typical over a large range of gain values. We find that, as the ratio of the population-to-polarization decay rates varies from a value of two (radiative limit) to smaller values, the periodic oscillations turn into a train of well-separated pulses whose peak intensity scales approximately as the square of the atomic density.
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.