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Cambridge University Press eBooks

Noisy dynamics in optically bistable systems

E. Arimondo, Daniel Hennequin, P. Glorieux

Published Apr 6, 1989• 4 citations
Publisher

Abstract

Introduction Lasers are part of a large class of non-linear systems which covers a great deal of scientific domains, including physics, chemistry, biology, sociology, etc. All these non-linear systems have common properties and present universal features which become more and more extensive as new studies are performed. In recent years, the interest in these non-linear systems has considerably increased. In particular, it is now well established that the role of fluctuations in non-equilibrium systems is far more relevant than in systems at thermodynamical equilibrium, and the study of their influence on stationary and transient states has become very important. Among the different aspects presented by the non-linearity in lasers, two of them will be particularly studied here: (i) The first one is the optical bistability (OB). An optical system is bistable if it has two output states I for the same value of the excitation A over some range of excitation values (see Gibbs, 1985, for a recent review). A typical response curve of such a system is given in Figure 5.1. Here, the system is defined as bistable between A ↓ and A ↑ , where the effectively occupied state depends on the history of the input parameter (see the caption to Figure 5.1). If OB is a common phenomenon in different fields, the OB transient phenomena have far more general features.

DOI
10.1017/cbo9780511897832.007

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Optical bistability: Constant input

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Introduction Usage has reserved the expression optical bistability mostly for coherently driven passive systems. An atomic system is called passive when there is no population inversion. This is the case, for example, of a system at thermal equilibrium. On the contrary, the laser that we have described in Chapter 1 is a driven active system, because a population inversion is created. The laser equations we have studied so far describe an incoherently driven laser. However, nothing prevents driving a lasing cavity with a coherent field emitted by another laser. This is realized in a whole class of lasers that are used mainly as frequency converters. In optical bistability (OB), the driving field is usually a coherent field. Thus we have to account for two differences between the laser and the optically bistable system: (1) In OB, there is no inversion of population: 〈| A 2 |〉>〈| B 2 |〉, in the notation of Chapter 1. Hence, in the absence of interaction with a field, the population difference D relaxes toward a negative value. (2) The pumping is coherent, meaning that an external laser field is added to the cavity field. Using the single-mode equations (1.48)–(1.50), we have to change the sign of D and D a (which amounts to keeping D defined in (1.34) as it is but changing P into – P and A into – A ) and to add a source term in the equation for the complex field amplitude.

Cambridge University Press eBooks•Jan 13, 1997•0 citations

Optical bistability

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Transient bimodality in optically bistable systems

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We analyze the main features of transient bimodality in optically bistable systems. This phenomenon is proved to exhibit a nonmonotonic behavior in its dependence on the noise level, because a decrease of noise first enhances and subsequently damps the transient bimodality. Hence we identify a range of noise levels which characterizes the phenomenon. In this range the most probable switching time exhibits a remarkably linear dependence on the logarithm of the noise parameter. We discuss the dependence of the phenomenon on the distance from the critical point, especially in connection with the experiments on critical slowing down. Our results show a very satisfactory qualitative agreement with the recent experiments on transient bimodality performed by Lange and collaborators. We argue that our picture holds in general for a large class of systems which exhibit the phenomenon of transient bimodality.

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Effect of laser amplitude and phase fluctuations on optical bistability

C. R. Willis

A generalized Fokker-Planck equation is derived for a system undergoing optical bistability when the dominant source of fluctuations is due to fluctuations in the incident field rather than spontaneous emission. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. Both amplitude and phase fluctuations are included on the incident laser field. The effect of fluctuations of the incident laser on optical bistability are different for amplitude and phase fluctuations and are different for absorptive and dispersive optical bistability. The combination of incident laser phase fluctuations and dispersive optical bistability leads to large amplitude fluctuations near the turning points of the optical bistability curve and causes the system to make a transition from one branch of the optical bistability curve to the other before reaching the mean-field turning points. We take the high-$Q$ cavity limit of our generalized Fokker-Planck equation, solve the linearized equations, and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters. The magnitude of incident laser fluctuations are typically orders of magnitude larger than fluctuations due to spontaneous emission and are therefore important for practical applications of optical bistability devices.

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Use of an electronic model as a guideline in experiments on transient optical bistability

Wulfhard K. Lange

Introduction The transient behavior of nonlinear optical resonators which display bistability and its reaction to fluctuations of system parameters are of extreme importance in all possible applications of these devices. Moreover, however, studies on these processes may give insight into some more general aspects of the behavior of nonlinear dynamical systems. In fact it has been emphasized in the early stages of the development of the field that optically bistable systems, in a suitable model, may exhibit a first-order phase transition (Bonifacio and Lugiato, 1978). Since the threshold behavior of a laser can be described as an example of a second-order phase transition (Haken, 1964,1975; see also Haken, 1977), optical bistability in (passive) nonlinear resonators complements nicely lasers as nonlinear dissipative systems in optics. Both types of optical systems are easily accessible to measurements, and the time scales coming into play are convenient for detailed experimental studies of the dynamics. It is worth mentioning that there is a close connection between laser operation and optical bistability in nonlinear (passive) resonators (see Lugiato, 1984). The standard single-mode laser model with distributed losses (Haken, 1970; Sargent, Scully and Lamb, 1974) can be generalized to include the effect of an external field that continuously drives the atoms within the optical resonator. The standard model of absorptive bistability, the Bonifacio–Lugiato model, is a special case of this generalized model. It represents the situation of a pumping mechanism being absent.

Cambridge University Press eBooks•Apr 6, 1989•0 citations

Dynamical aspects in optical bistability

Luigi A. Lugiato, Franco Prati, Massimo Brambilla

Although the discovery of oscillatory instabilities (both single-mode and multimode) in free-running lasers was achieved in the 1960s, investigations on oscillatory instabilities in optical bistability did not take place until the 1970s and 1980s. The first section of this chapter does not deal with oscillatory instabilities, but instead concerns an interesting dynamical phenomenon in optical bistability that arises when the input field approaches a boundary of the bistable domain, and is called critical slowing down . This has been investigated both theoretically and experimentally. The remainder of this chapter is entirely devoted to oscillatory instabilities, following a historical order. With an inverse approach with respect to the laser case, we discuss multimode instabilities first. In Section 24.2.1 we start with the description of the instability which arises under exactly resonant conditions. While this is the counterpart in optical bistability of the resonant multimode laser instability discussed in Section 22.5, a striking difference from the laser case is that in the passive configuration there is no corresponding single-mode instability. This difference disappears, however, when we consider the generalization of the multimode instability of optical bistability to the detuned configuration. Finally we show that in the detuned case it has been possible to achieve a convincing experimental observation of the multimode instability in a long microwave cavity. After some relevant miscellaneous considerations in Section 24.2.2, in Section 24.2.3 we address the subject of the multimode Ikeda instability which leads to period doubling and chaos and aroused a lot of interest in optical chaos. We emphasize also the intrinsic connections with the standard multimode instability discussed in Section 24.2.1, showing, in particular, that the Ikeda instability requires conditions far from the low-transmission limit, i.e. that ∝L is not small. In Section 24.3 we discuss single-mode instabilities in optical bistability. In Section 24.3.1 we illustrate a configuration for the single-mode instability that includes also the Gaussian shape of the electric field in the beam section, and leads to periodic, close-to-sinusoidal self-pulsations.

Cambridge University Press eBooks•Feb 28, 2015•0 citations
Introduction Lasers are part of a large class of non-linear systems which covers a great deal of scientific domains, including physics, chemistry, biology, sociology, etc. All these non-linear systems have common properties and present universal features which become more and more extensive as new studies are performed. In recent years, the interest in these non-linear systems has considerably increased. In particular, it is now well established that the role of fluctuations in non-equilibrium systems is far more relevant than in systems at thermodynamical equilibrium, and the study of their influence on stationary and transient states has become very important. Among the different aspects presented by the non-linearity in lasers, two of them will be particularly studied here: (i) The first one is the optical bistability (OB). An optical system is bistable if it has two output states I for the same value of the excitation A over some range of excitation values (see Gibbs, 1985, for a recent review). A typical response curve of such a system is given in Figure 5.1. Here, the system is defined as bistable between A ↓ and A ↑ , where the effectively occupied state depends on the history of the input parameter (see the caption to Figure 5.1). If OB is a common phenomenon in different fields, the OB transient phenomena have far more general features.
Introduction Usage has reserved the expression optical bistability mostly for coherently driven passive systems. An atomic system is called passive when there is no population inversion. This is the case, for example, of a system at thermal equilibrium. On the contrary, the laser that we have described in Chapter 1 is a driven active system, because a population inversion is created. The laser equations we have studied so far describe an incoherently driven laser. However, nothing prevents driving a lasing cavity with a coherent field emitted by another laser. This is realized in a whole class of lasers that are used mainly as frequency converters. In optical bistability (OB), the driving field is usually a coherent field. Thus we have to account for two differences between the laser and the optically bistable system: (1) In OB, there is no inversion of population: 〈| A 2 |〉>〈| B 2 |〉, in the notation of Chapter 1. Hence, in the absence of interaction with a field, the population difference D relaxes toward a negative value. (2) The pumping is coherent, meaning that an external laser field is added to the cavity field. Using the single-mode equations (1.48)–(1.50), we have to change the sign of D and D a (which amounts to keeping D defined in (1.34) as it is but changing P into – P and A into – A ) and to add a source term in the equation for the complex field amplitude.
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.
We analyze the main features of transient bimodality in optically bistable systems. This phenomenon is proved to exhibit a nonmonotonic behavior in its dependence on the noise level, because a decrease of noise first enhances and subsequently damps the transient bimodality. Hence we identify a range of noise levels which characterizes the phenomenon. In this range the most probable switching time exhibits a remarkably linear dependence on the logarithm of the noise parameter. We discuss the dependence of the phenomenon on the distance from the critical point, especially in connection with the experiments on critical slowing down. Our results show a very satisfactory qualitative agreement with the recent experiments on transient bimodality performed by Lange and collaborators. We argue that our picture holds in general for a large class of systems which exhibit the phenomenon of transient bimodality.
A generalized Fokker-Planck equation is derived for a system undergoing optical bistability when the dominant source of fluctuations is due to fluctuations in the incident field rather than spontaneous emission. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. Both amplitude and phase fluctuations are included on the incident laser field. The effect of fluctuations of the incident laser on optical bistability are different for amplitude and phase fluctuations and are different for absorptive and dispersive optical bistability. The combination of incident laser phase fluctuations and dispersive optical bistability leads to large amplitude fluctuations near the turning points of the optical bistability curve and causes the system to make a transition from one branch of the optical bistability curve to the other before reaching the mean-field turning points. We take the high-QQQ cavity limit of our generalized Fokker-Planck equation, solve the linearized equations, and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters. The magnitude of incident laser fluctuations are typically orders of magnitude larger than fluctuations due to spontaneous emission and are therefore important for practical applications of optical bistability devices.
Introduction The transient behavior of nonlinear optical resonators which display bistability and its reaction to fluctuations of system parameters are of extreme importance in all possible applications of these devices. Moreover, however, studies on these processes may give insight into some more general aspects of the behavior of nonlinear dynamical systems. In fact it has been emphasized in the early stages of the development of the field that optically bistable systems, in a suitable model, may exhibit a first-order phase transition (Bonifacio and Lugiato, 1978). Since the threshold behavior of a laser can be described as an example of a second-order phase transition (Haken, 1964,1975; see also Haken, 1977), optical bistability in (passive) nonlinear resonators complements nicely lasers as nonlinear dissipative systems in optics. Both types of optical systems are easily accessible to measurements, and the time scales coming into play are convenient for detailed experimental studies of the dynamics. It is worth mentioning that there is a close connection between laser operation and optical bistability in nonlinear (passive) resonators (see Lugiato, 1984). The standard single-mode laser model with distributed losses (Haken, 1970; Sargent, Scully and Lamb, 1974) can be generalized to include the effect of an external field that continuously drives the atoms within the optical resonator. The standard model of absorptive bistability, the Bonifacio–Lugiato model, is a special case of this generalized model. It represents the situation of a pumping mechanism being absent.
Although the discovery of oscillatory instabilities (both single-mode and multimode) in free-running lasers was achieved in the 1960s, investigations on oscillatory instabilities in optical bistability did not take place until the 1970s and 1980s. The first section of this chapter does not deal with oscillatory instabilities, but instead concerns an interesting dynamical phenomenon in optical bistability that arises when the input field approaches a boundary of the bistable domain, and is called critical slowing down . This has been investigated both theoretically and experimentally. The remainder of this chapter is entirely devoted to oscillatory instabilities, following a historical order. With an inverse approach with respect to the laser case, we discuss multimode instabilities first. In Section 24.2.1 we start with the description of the instability which arises under exactly resonant conditions. While this is the counterpart in optical bistability of the resonant multimode laser instability discussed in Section 22.5, a striking difference from the laser case is that in the passive configuration there is no corresponding single-mode instability. This difference disappears, however, when we consider the generalization of the multimode instability of optical bistability to the detuned configuration. Finally we show that in the detuned case it has been possible to achieve a convincing experimental observation of the multimode instability in a long microwave cavity. After some relevant miscellaneous considerations in Section 24.2.2, in Section 24.2.3 we address the subject of the multimode Ikeda instability which leads to period doubling and chaos and aroused a lot of interest in optical chaos. We emphasize also the intrinsic connections with the standard multimode instability discussed in Section 24.2.1, showing, in particular, that the Ikeda instability requires conditions far from the low-transmission limit, i.e. that ∝L is not small. In Section 24.3 we discuss single-mode instabilities in optical bistability. In Section 24.3.1 we illustrate a configuration for the single-mode instability that includes also the Gaussian shape of the electric field in the beam section, and leads to periodic, close-to-sinusoidal self-pulsations.