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Annual Meeting Optical Society of America

Dynamic bistability of an optical ring circuit having photorefractive gain, loss, and a photorefractive feedback coupling element: theory

David D. Crouch, Anno Hermanns, Dana Z. Anderson

Published Jan 1, 1989• 0 citations
Publisher

Abstract

We discuss the properties of an optical circuit with gain and loss provided by two-beam coupling in a photorefractive medium and with a photorefractive element as a programmable mirror to provide feedback. As with an ordinary ring resonator having photorefractive gain and loss the circuit can exhibit reversible or irreversible bistability and self-pulsing. In the limit that the feedback element has a response that is slow compared to the gain and loss, its behavior is much like the conventional case. However, whereas a conventional ring resonator has a well-defined detuning characteristic dependent on the round trip length, in the present circuit a fast photorefractive element continually compensates for changes in the round trip path length. The bistability and self-pulsing characteristics are strongly affected by the relative time constants of the three photorefractive media and by the relative magnitudes of the small signal gain and loss. In particular, the on state of the ring has a memory time determined by the time constant of the feedback element.

DOI
10.1364/oam.1989.fl6

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IEEE Journal of Quantum Electronics•Sep 1, 1985•18 citations
We discuss the properties of an optical circuit with gain and loss provided by two-beam coupling in a photorefractive medium and with a photorefractive element as a programmable mirror to provide feedback. As with an ordinary ring resonator having photorefractive gain and loss the circuit can exhibit reversible or irreversible bistability and self-pulsing. In the limit that the feedback element has a response that is slow compared to the gain and loss, its behavior is much like the conventional case. However, whereas a conventional ring resonator has a well-defined detuning characteristic dependent on the round trip length, in the present circuit a fast photorefractive element continually compensates for changes in the round trip path length. The bistability and self-pulsing characteristics are strongly affected by the relative time constants of the three photorefractive media and by the relative magnitudes of the small signal gain and loss. In particular, the on state of the ring has a memory time determined by the time constant of the feedback element.
We report the observation of bistability in a unidirectional ring resonator that employs photorefractive barium titanate for gain and also for saturated loss. One photorefractive crystal provides gain via two-beam coupling in a conventional unidirectional ring configuration.1 A second crystal is arranged to couple light out of the resonator using two-beam coupling, thereby producing loss. The orientations of the two crystals are chosen so that the magnitude of the small signal loss is greater than the magnitude of the small signal gain. Thus the oscillation in the ring cannot build up from noise since there is no net gain. However, the pump intensity for the gain medium is made larger than that for the loss medium. This means that the loss saturates at a lower signal level. Hence, at large intensities, gain can exceed net loss so oscillation can occur.
A ring resonator with photorefractive gain and loss is an active nonlinear system that exhibits self pulsing of the resonator intensity1. The novel feature of this resonator over single crystal resonators2,3 is the addition of a second crystal in our case to couple light out of the resonator providing a saturable loss4. The nonlinearity of a single crystal resonator is manifested in the saturation of the small signal gain. The nonlinearity of the two crystal system is enriched by the interplay between the two crystal time constants and the small signal gain and loss, as well as the addition of the second saturation point The dynamic behavior of the system can be described as a competitive interaction between gain and loss; if the gain wins, the resonator turns on, and if the loss wins the resonator remains off. Self pulsing occurs when the winner is alternately the gain and then the loss, and then the gain again, and so forth. The relative time constants, coupling strengths, and saturation points of the two crystals dictate the details of the competitive interaction. In this communication we include the loss crystal in a theory for photorefractive resonators to address the experimental observations of self pulsing.
We compare weak-field theory to strong field numerical results for the bistable ring resonator with saturable photorefractive gain and loss.1,2 Our weak-field model, which only considers the time dependence of the gratings and neglects the spatial dependence,3 captures an approximate picture of the two-crystal resonator. The basic model consists of two coupled complex equations for the gain and loss gratings. An analytic equation is derived for the self-pulsing period and amplitude for the case of zero detuning and near threshold gain. In this case, the system is analogous to a Van der Pol oscillator. We solve for the stationary points of the system and discuss their stability conditions. We also present computer simulation of the nonzero detuning case.
We demonstrate an optical flip-flop that consists of two unidirectional ring resonators that are coupled competitively through two-beam interactions in a photorefractive BaTiO(3) crystal. This crystal provides an active loss for each ring that depends on the oscillation intensity of the other. The coupled rings exhibit flip-flop bistability when the net gain, given by the product of the passive cavity losses and the gain, is greater than unity but is less than the small-signal active loss. The state of the system can be switched with an injected signal.
A transverse optical bistability is defined here as a bistability that would not occur with a plane-wave input, i.e., it depends crucially upon the radial variations in the input. Such bistabilities are characterized by much more pronounced changes in transverse profiles than in total transmission. An optical cavity is not required, making such systems good candidates for continuous wave studies of delayed-feedback instabilities which require a round-trip time longer than the medium response time. Trapping bistability and diffraction-free-encoding bistability are two limiting cases of transverse optical bistabilities. Trapping bistability occurs only for a long self-focusing nonlinear medium and arises from self-trapping of a single beam with a single feedback mirror or from cross trapping of two counterpropagating beams. In the diffraction-free-encoding case, diffraction is negligible through the portion of the nonlinear medium performing the encoding, but it is significant through any remaining medium and in the free-space propagation to the mirror and back. This paper is organized as follows: The physics of transverse optical bistabilities is summarized in Section I. Theoretical models are presented in Section II. A thin-sample-encoding (TSE) rings model is defined and shown not only to predict transverse bistability but also to account for the interference rings observed to accompany TSE bistability; it also yields delayed-feedback instabilities with a rich bifurcation sequence. Transverse bistability using a short length of sodium vapor is reported under TSE and DFE conditions in Section III. Recently it was discovered that TSE bistability has been seen only when the feedback beam is displaced from the input by about one beam diameter at the sodium. It remains to be determined just how much the nonconcentric feedback assists the concentric bistability predicted by the TSE rings model. A delayed-feedback instability with a period of twice the round-trip time is also seen.