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.
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.
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.
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.
In Chapter 9 we discussed the case of a nonlinear cavity containing an active medium. In this framework, the outstanding phenomenon which arises is laser emission, and we calculated the stationary solutions for the ring laser. In this chapter, instead, we turn our attention to the passive nonlinear cavity and in this framework the outstanding phenomenon is optical bistability , i.e. the existence of two distinct stable stationary states that coexist for the same fixed values of the parameters in play. This phenomenon can arise also in the case of an active system contained in a cavity, as we have already noted in Section 9.1 and as we will discuss further in connection with the laser with injected signal, but it became popular from the studies of nonlinear passive optical cavities in the 1970s and 1980s. In contrast to the laser case, in the passive configuration we use an input coherent field, which is injected into the cavity. We discussed this case in Chapter 8 and derived a general equation for the transmission of the cavity (Eq. (8.23)), which we used for an empty cavity and for a linear cavity in Sections 8.4 and 8.5, respectively. In this chapter we will discuss the full nonlinear configuration. It is customary to identify two extreme cases of optical bistability called absorptive optical bistability and dispersive or refractive optical bistability . In both cases an essential ingredient to obtain the bistable behavior is the feedback action exerted by the mirrors of the cavity. In the purely absorptive configuration, in which there is exact resonance among the frequencies of the input field, of the atoms and of the cavity, the other essential ingredient is the saturable nonlinearity of the absorption. In the dispersive case, instead, the main feature is the nonlinearity of the refractive index. These two cases will be illustrated in Sections 11.1 and 11.2, respectively, both “exactly” and in the low-transmission limit.
After reviewing the interpretation of laser operation as a nonequilibrium Bose-Einstein condensation phase transition, we illustrate the novel features arising from the nonequilibrium nature of photon and polariton Bose-Einstein condensates recently observed in experiments. We then propose a quantitative criterion to experimentally assess the equilibrium versus nonequilibrium nature of a specific condensation process, based on fluctuation-dissipation relations. The power of this criterion is illustrated on two models which show very different behaviors. Historical and Conceptual Introduction The first introduction of concepts of nonequilibrium statistical mechanics into the realm of optics dates back to the early 1970s with pioneering works by Graham and Haken [1] and by DeGiorgio and Scully [2], who proposed a very insightful interpretation of the laser threshold in terms of a spontaneous breaking of the U( 1 ) symmetry associated with the phase of the emitted light. Similar to what happens to the order parameter at a second-order phase transition, such an optical phase is randomly chosen every time the device is switched on and remains constant for macroscopic times. Moreover, a long-range spatial order is established, as light emitted by a laser device above threshold is phase-coherent on macroscopic distances. While textbooks typically discuss this interpretation of laser operation in terms of a phase transition for the simplest case of a single-mode laser cavity, rigorously speaking this is valid only in spatially infinite systems. In fact, only in this case one can observe nonanalytic behaviors of the physical quantities at the transition point. In particular, the long-range order is typically assessed by looking at the long-distance behavior of the correlation function of the order parameter, which, for a laser, corresponds to the first-order spatial coherence of the emitted electric field, the spontaneous symmetry breaking is signaled by this quantity becoming nonzero (see Chapter 5). The average is taken on the stationary density matrix of the system. In order to be able to probe this long-distance behavior, experimental studies need devices with a spatially extended active region.
The influence of noise on an optical bistable system in the vicinity of one of the bifurcation points is investigated by studying the transient response to a step input of light. The experiment is performed under absorptive conditions in a sodium-filled nonlinear Fabry-Perot in the bad cavity limit. Alternatively, a bistable electronic circuit (Schmitt trigger) is shown to exhibit the same type of behavior while admitting more detailed studies. On the monostable side of the bifurcation point, we observe a transient bimodality as well as a shift of the average switching delay to shorter times with increasing noise. Both results are in accord with a recent prediction by Broggi and Lugiato. We also present indications that the average switching time is smoothly connected with the lifetime of the metastable state on the bistable side of the bifurcation point.
The subject of spontaneous pulsations in Optical Bistability has stimulated considerable interest following its prediction by Bonifacio, and Lugiato1 in the framework of the plane-wave, ring cavity model and the realization by McCall2 of an electro-optical converter of cw coherent light into pulsed radiation. In a subsequent important development Ikeda3 showed that, in the dispersive case, the Bonifacio-Lugiato instability leads to chaotic behavior (optical turbulence). This was later observed in a hybrid device by Gibbs, et al4.