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Topical Meeting on Optical Bistability (OB)

Subharmonic Phase Memory: Symmetric Optical Multistability

A. E. Kaplan

Published Jan 1, 1985• 0 citations
Publisher

Abstract

It is well known that n-th order subharmonic oscillations (where n is integer) possess a fundamental property of having n possible and equally probable oscillation phases, each of which corresponding to the same oscillation amplitude.1 It was demonstrated both theoretically and experimentally by this author2 that very high order subharmonics can be excited in passive lump nonlinear systems driven by a sufficiently strong pumping signal. It was also shown that subharmonics can be generated by the optical parametric oscillator.3 In this paper we show that optical subharmonic oscillators may provide a novel kind of optical multistability - specifically, phase multistability of these oscillations, with the same amplitude for each steady state of the phase. This differs drastically from all presently known types of optical bistability, all of which are based on amplitude multistability. The new principle results in a unique property of the system which we call symmetry of the states: all the steady states of the system possess the same stability, noise, etc. characteristics and may be switched from one steady state to another by an external signal of the same small amplitude.

DOI
10.1364/obi.1985.wb6

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This paper presents a new model of optical bistability (OB). In view of the ideas of the mean field approximation, the slow envelope approximation and the adiabatic approximation, the OB system can be regarded as a box. Comparing with the theory of nonlinear oscillation, we find that the courses in different OBS's-includingdifferent substances and different cavities (E-P. C. or E. C. )-can be describedconsistently, in both steady state and temporal respect, with an appropriate nonlinear forced oscillation equation, in which the coefficients are measurable macroscopical parameters. Using the method of slowly changing phase and amplitude in oscillation theory, we develop the equation into a coupling and autonomous dynamic equation set of phase and amplitude. The general solutions of the equation set for OB steady state and transient behavior are obtained conveniently from this form. Several concrete results in literature are compared as the examples for the verification of the widely applicability o del.

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Cambridge University Press eBooks•Feb 28, 2015•0 citations
It is well known that n-th order subharmonic oscillations (where n is integer) possess a fundamental property of having n possible and equally probable oscillation phases, each of which corresponding to the same oscillation amplitude.1 It was demonstrated both theoretically and experimentally by this author2 that very high order subharmonics can be excited in passive lump nonlinear systems driven by a sufficiently strong pumping signal. It was also shown that subharmonics can be generated by the optical parametric oscillator.3 In this paper we show that optical subharmonic oscillators may provide a novel kind of optical multistability - specifically, phase multistability of these oscillations, with the same amplitude for each steady state of the phase. This differs drastically from all presently known types of optical bistability, all of which are based on amplitude multistability. The new principle results in a unique property of the system which we call symmetry of the states: all the steady states of the system possess the same stability, noise, etc. characteristics and may be switched from one steady state to another by an external signal of the same small amplitude.
In this paper we present a theory of optical subharmonic generation in a nonlinear Fabry-Perot resonator. Our approach, based on classical electrodynamics, leads to a Mathieu-Hill-type equation. This enables us to describe the generation of optical subharmonic oscillation in terms of a general formalism used to account for parametric phenomena, and to outline the possibility of using a nonlinear Fabry-Perot resonator as a memory device able to store the phase of a signal. The oscillation threshold condition is discussed and the rate of amplitude rise at start-up is evaluated.
Most treatments of noise characteristics of optical parametric oscillators (OPOs) have been based on linearization of the equations of motion. This approach, valid far below and far above threshold, breaks down in the threshold region as it ignores the nonlinearity of interaction. We present a treatment that takes into account both the nonlinearity and the quantum mechanical nature of noise in these oscillators. Analytical expressions for the steady-state amplitude distribution functions are derived. Both the degenerate and the nondegenerate OPOs are considered. These steady-state distributions are used to study the coherence properties of the OPOs in the threshold region. These studies bridge the noise characteristics of the OPOs below and above threshold. Results are compared with classical oscillators and differences and similarities are discussed. Special attention is given to quantum mechanical features that survive even as the oscillator amplitude grows and reaches macroscopic proportions.
Bichromatically pumped Kerr microresonators exhibit phase bistability and multistability arising from four-wave mixing between pump and generated modes. Here we analyze the symmetry structure of stationary solutions in the coupled-mode description of such systems. We show that the two-pump coupled-mode equations are a particular case of a more general model with pumps placed at equally spaced modes. For this model, any stationary solution generates a finite family of stationary solutions through a discrete phase transformation. This transformation leaves the equations invariant and preserves the stability type of the corresponding stationary states. As a consequence, the possible stationary values of each individual mode form a regular polygon in the complex plane. The number of vertices is determined by the order of the mode index μμμ in the group Zn\mathbb{Z}_nZn​, where nnn is the separation between the pumped modes. Thus, the phase multistability structure depends not only on nonlinear dynamics but also on the arithmetic relation between the mode index and the pump separation. These results provide a simple symmetry-based explanation for the discrete phase structure observed in multimode Kerr resonators under bichromatic pumping.
This paper presents a new model of optical bistability (OB). In view of the ideas of the mean field approximation, the slow envelope approximation and the adiabatic approximation, the OB system can be regarded as a box. Comparing with the theory of nonlinear oscillation, we find that the courses in different OBS's-includingdifferent substances and different cavities (E-P. C. or E. C. )-can be describedconsistently, in both steady state and temporal respect, with an appropriate nonlinear forced oscillation equation, in which the coefficients are measurable macroscopical parameters. Using the method of slowly changing phase and amplitude in oscillation theory, we develop the equation into a coupling and autonomous dynamic equation set of phase and amplitude. The general solutions of the equation set for OB steady state and transient behavior are obtained conveniently from this form. Several concrete results in literature are compared as the examples for the verification of the widely applicability o del.
Optical bistability is a phenomenon that arises in the transmission of light by an optical cavity filled with a medium which presents saturable absorption or nonlinear dispersion. From the theoretical viewpoint, optical bistability is a remarkable example of cooperative behaviour in an open system driven far from thermal equilibrium. From the practical viewpoint, optical bistability offers the possibility of realizing memory elements that may perhaps be used to construct an optical computer. We illustrate the basic physical principles of optical bistability at steady state, and the main features of the transient behaviour. The role and the effects of fluctuations (noise) are described in general. Particular attention is devoted to the situations in which the output of the system is not stationary in time, but oscillatory (self-pulsing behaviour), so that the system works as a converter of continuous wave light into pulsed. It is shown that, according to the values of the external parameters, the oscillations can be either periodic in time or completely aperiodic (optical turbulence). The state of the art in the problem of optical bistability is briefly discussed.
In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11. Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states. In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2. In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations. A laser with an injected signal From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element. In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.