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Physical Review A

Analysis of self-pulsing in absorptive optical bistability

M. Gronchi, Vincenzo Gino Benza, L. A. Lugiato, Pierre Meystre, M. Sargent

Published Sep 1, 1981• 73 citations
Publisher

Abstract

We review and extend the theory of instabilities in absorptive optical bistability for ring cavities containing homogeneously broadened two-level media. The instabilities occur in the high-transmission branch and result in either a precipitation to the low-transmission branch or in pulsed multimode operation. Two numerical approaches are used to predict operation away from the simple mean-field limit. First, an iterative method is used to solve the general field eigenvalue equation for cavity modes with frequencies displaced from the input frequency. This work shows that the instability region decreases in size as the mirror transmission increases. These calculations are confirmed by the second numerical technique, consisting of direct integrations of the coupled Maxwell-Bloch equations. This second approach also provides time histories of the instability evolutions, and divides the instability range into a precipitation regime and a self-pulsing regime. The results are discussed in terms of first- and second-order phase transitions, and agree with the analytical results obtained within the dressed-mode description of optical bistability. They show that when the incident field is adiabatically decreased along the high-transmission branch, the spiking behavior always appears abruptly. By further decreasing the incident intensity, the self-pulsing disappears either continuously if the system remains in the high-transmission branch, or discontinuously if the system precipitates to the low-transmission branch. Connection is made with induced probe gain known in the saturation spectroscopy of absorbers (uninverted media), in which population pulsations transfer energy from a saturating wave to the probe waves. The very close relationship with multimode operation in homogeneously broadened unidirectional ring lasers is also established.

DOI
10.1103/physreva.24.1419

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Physical Review A•Dec 1, 1989•8 citations
We review and extend the theory of instabilities in absorptive optical bistability for ring cavities containing homogeneously broadened two-level media. The instabilities occur in the high-transmission branch and result in either a precipitation to the low-transmission branch or in pulsed multimode operation. Two numerical approaches are used to predict operation away from the simple mean-field limit. First, an iterative method is used to solve the general field eigenvalue equation for cavity modes with frequencies displaced from the input frequency. This work shows that the instability region decreases in size as the mirror transmission increases. These calculations are confirmed by the second numerical technique, consisting of direct integrations of the coupled Maxwell-Bloch equations. This second approach also provides time histories of the instability evolutions, and divides the instability range into a precipitation regime and a self-pulsing regime. The results are discussed in terms of first- and second-order phase transitions, and agree with the analytical results obtained within the dressed-mode description of optical bistability. They show that when the incident field is adiabatically decreased along the high-transmission branch, the spiking behavior always appears abruptly. By further decreasing the incident intensity, the self-pulsing disappears either continuously if the system remains in the high-transmission branch, or discontinuously if the system precipitates to the low-transmission branch. Connection is made with induced probe gain known in the saturation spectroscopy of absorbers (uninverted media), in which population pulsations transfer energy from a saturating wave to the probe waves. The very close relationship with multimode operation in homogeneously broadened unidirectional ring lasers is also established.
We analyze instabilities in mixed absorptive and dispersive optical bistability in the rate-equation approximation and the mean-field and good-cavity limits. Our starting point is a set of multimode equations derived from the Maxwell-Bloch equations for ring-cavity boundary conditions. We obtain analytic expressions for the instability conditions. In a plane-wave analysis, we find that a portion of the lower transmission branch can be unstable in addition to the upper-branch instability found in purely absorptive bistability. Also, a new disconnected region of instability can exist on the upper branch. Our analysis becomes particularly simple for equal and opposite cavity detunings and we explore this case in detail. We extend our treatment to include a Gaussian transverse intensity profile and show that the instabilities remain in the presence of Gaussian averaging. We also show that many of the results obtained in the rate-equation approximation hold when the atomic linewidth and atomic decay rate are of the same order.
We compare theoretical predictions and experimental data for the multimode instability of optical bistability, which arises from the coherent dynamics of a system of two-level molecules contained in an optical cavity and driven by an external stationary laser field. Considerable insight into this phenomenon is provided by two ``rules of thumb'' that govern the behavior of the self-pulsing frequency, and establish its relation with the Rabi frequency of the intracavity field. We describe in detail the experimental apparatus and the main features of the experimental findings. The theoretical results are based on the Maxwell-Bloch equations for a ring cavity in the plane-wave approximation. Despite the crudeness of this model, the numerical data display a satisfactory qualitative agreement with the experimental results. Additional period-doubling and chaotic phenomena, predicted by the theory and not observed in the experiment, are presumably washed out by the longitudinal and radial variation of the electric field in the cavity.
We have recently observed a sinewave self-pulsing in the beam transmitted by a passive cavity containing 2-level molecules, driven by a continuous wave. In this experiment made at a wavelength λ ≅ 3.5 mm, the intracavity Rabi frequency is comparable to the free spectral range ( 830 kHz ) of the very long cavity used (l =182 m) {1} and much larger than the collisional linewidth (≅ 25 kHz) .The medium is then strongly saturated by the driving field ( saturation parameter > 1000) so that its absorption , heavy in the linear regime (αl > 100), is in fact quite moderate. We claim that the self-pulsing observed in these conditions is the first experimental evidence of the multimode instability, predicted a long time ago by Bonifacio and Lugiato {2} but not yet observed. This instability is the multimode counterpart of the single mode instability, recently observed by Orozco et al {3}.
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.
We present an analysis of absorptive optical bistability in a system of cold atoms enclosed in a bidirectional ring cavity. This analysis is carried out using a system of equations that extends the so-called Maxwell-Bloch model to self-consistently include atomic center-of-mass motion. The analysis predicts an instability resulting in the simultaneous generation of an atomic density grating and a coherently backscattered radiation field. The formation of the density grating arises from a modulation of the radiation pressure forces exerted on the atoms. A consequence of this instability is that the lower transmission branch of the usual bistability curve, where atomic motion is neglected, can become unstable and the system switches to the upper transmission branch destroying the bistability.
Instabilities are predicted for absorptive optical bistability in a nonideal Fabry-P\'erot cavity on the basis of Maxwell-Bloch theory. We show that the stability problem of the Maxwell-Bloch equations can be formulated in terms of a Riccati differential equation with boundary conditions. We have integrated this equation numerically. Our results crucially depend on the value of the transmission coefficient of the mirrors. For finite values of this quantity we find that the nearest side mode is responsible for two disconnected instability domains in the plane spanned by the output intensity and the medium response time. One of these domains can generate instabilities in the upper and the lower branch of the steady-state curve if the cooperation parameter is sufficiently large. Higher side modes can give rise to positive-slope instabilities as well. From our findings it can be understood why a recent experimental search for instabilities in absorptive optical bistability has led to a negative result. We finally demonstrate that the instability spectra of a Fabry-P\'erot cavity and a so-called equivalent ring cavity differ considerably.