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Journal of the Optical Society of America B

Oscillatory instability of an induced absorber in a ring cavity

Markus Lindberg, S. W. Koch, H. Haug

Published May 1, 1986• 26 citations
Publisher

Abstract

The instabilities of a nonlinear optical medium with induced absorption in a ring cavity are theoretically analyzed. Stable, bistable, and unstable stationary solutions are found for various values of incident intensity and detuning. The dynamic treatment yields, in the unstable regime, oscillations of the transmitted intensity in various modes for constant incident intensity. If the medium relaxation time is short compared with the round-trip time, oscillatory solutions are obtained, which show strong locking of the oscillation frequency to multiples of the inverse round-trip time. If the round-trip time is comparable with the medium relaxation time, numerical integrations yield periodic solutions with substructures because of the mismatch of the two characteristic times.

DOI
10.1364/josab.3.000751

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The instabilities of a nonlinear optical medium with induced absorption in a ring cavity are theoretically analyzed. Stable, bistable, and unstable stationary solutions are found for various values of incident intensity and detuning. The dynamic treatment yields, in the unstable regime, oscillations of the transmitted intensity in various modes for constant incident intensity. If the medium relaxation time is short compared with the round-trip time, oscillatory solutions are obtained, which show strong locking of the oscillation frequency to multiples of the inverse round-trip time. If the round-trip time is comparable with the medium relaxation time, numerical integrations yield periodic solutions with substructures because of the mismatch of the two characteristic times.
Absorptive, dispersive, and diffractive effects are studied in a nonlinear ring-cavity configuration containing an intrinsically bistable medium modeled generically by a nonlinear oscillator. The output intensity is shown to exhibit steady-state and periodic characteristics for constant input intensity, but no chaotic solutions are found. We find that the plane-wave analysis is inconsistent with the inclusion of diffraction effects and that periodic output characteristics are affected even for Fresnel numbers of order 1000. Our results are contrasted with previously reported experimental and theoretical work.
We analyze a simple model of dispersive optical bistability in a ring cavity filled with a Kerr medium in the presence of two and three (carrier and one sideband pair) different frequencies. The steady-state characteristics are shown and the condition for selective switching is studied. We show the phenomenon of the so-called competition for resonance between different frequencies. The characteristic phenomenon of locking of the output intensities is presented. The model of two frequencies may also be used to describe bistability in the presence of two different polarizations. For three frequencies our discussion is based on the rigorous propagation equations, which admit a real Hamiltonian description and are completely integrable. We also discuss a general numerical approach, based on a relaxation method, for finding steady-state solutions and checking their stability. The relaxation dynamics is physically related to a finite response time of the medium. The effect of this finite response time on the stability of the steady states and on the dynamics is also discussed.
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.
Using a simple model of a ring cavity comprising a cubically nonlinear medium, we analyze dispersive optical bistability in the presence of more than one spectral component. We show the phenomenon of so-called competition for resonance. In addition to presenting cavity characteristics for the cases of two and three different frequencies, we also discuss the general method for finding steady-state solutions and checking their stability. A simple and efficient algorithm, based on a relaxation method, is devised to find steady-state solutions satisfying appropriate boundary conditions. The relaxation dynamics is physically related to a finite response time of the medium.
Long-time behavior of transmitted and reflected field amplitude of optical bistability by a cavity including an absorptive resonant medium from an initial to a final steady state on switching the incident field is investigated by solving numerically the semiclassical dynamical equations without the mean-field approximation. The transmitted field amplitude as a function of time is monotonic as a whole, and an oscillatory character is seen on it in a shorter period. The dynamical character such as the oscillation amplitude and the duration of a whole process depend strongly on two parameters included in the equations.
The transmission characteristics of a Fabry-Perot cavity containing a resonant absorber are described analytically and graphically for internal fields of arbitrary intensity. The multimode instability considered by Bonifacio and Lugiato [Opt. Commun. 19 , 72 (1976)] for the unidirectional ring laser is treated for the standing-wave cavity. In particular, an instability may exist when the medium is located at one of the ends of the cavity, but not if the medium fills the cavity. The results are explained in terms of induced side-mode gain and spatial hole burning.