The analysis of experiments on bistability and instability in nonlinear optical resonators depends critically on an understanding of the properties of the corresponding empty (linear) resonators. Both real and ideal cavities exhibit behavior which can lead to a misinterpretation of experimental results, and in both cases this behavior becomes more pronounced with increasing finesse. As an example of the first case, consider a real cavity which is nonideal in that different methods of measurement of the cavity loss give different values for it and hence for the cooperativity C. A simple but realistic model for such a cavity leads to the derivation of a state equation for bistability which differs in functional form from the usual state equation derived assuming an ideal cavity. In such a nonideal cavity, one might find its loss by measuring the decay time of the transmitted light after rapidly cutting off the incident light or by scanning the length of the cavity to measure the finesse. If these two different values of loss vary by about a factor of two, as they have in certain of our investigations, then making either one or the other measurement and applying the usual state equation would cause one to overestimate or underestimate the value of C necessary for the critical onset of bistability by as much as 40%.
The analysis of experiments on bistability and instability in nonlinear optical resonators depends critically on an understanding of the properties of the corresponding empty (linear) resonators. Both real and ideal cavities exhibit behavior which can lead to a misinterpretation of experimental results, and in both cases this behavior becomes more pronounced with increasing finesse. As an example of the first case, consider a real cavity which is nonideal in that different methods of measurement of the cavity loss give different values for it and hence for the cooperativity C. A simple but realistic model for such a cavity leads to the derivation of a state equation for bistability which differs in functional form from the usual state equation derived assuming an ideal cavity. In such a nonideal cavity, one might find its loss by measuring the decay time of the transmitted light after rapidly cutting off the incident light or by scanning the length of the cavity to measure the finesse. If these two different values of loss vary by about a factor of two, as they have in certain of our investigations, then making either one or the other measurement and applying the usual state equation would cause one to overestimate or underestimate the value of C necessary for the critical onset of bistability by as much as 40%.
Optical resonators, usually in the form of spherical-mirror interferometers, are employed in a wide variety of experiments, including studies of steady-state and dynamic behavior in lasers and optical bistability. To make meaningful comparisons between experiment and theory, a resonator must be characterized in terms of its losses; that is, the behavior of the empty, or linear, resonator must be well understood if one wishes to understand the behavior of the filled, nonlinear resonator. In general, however, it is not possible to characterize an optical cavity in terms of a single loss parameter. Doing so results, for example, in a different state equation for optical bistability.
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.
In this review, we will focus on optical bistability and the condition for its occurrence in cavity optomechanics. We study different optomechanical systems (basic optomechanical systems and hybrid optomechanical systems). we derived an equation that could predict the behavior of the system either bistable, transistor or monostable for each system considered depending on the value of its parameters. This equation is derived using analytical study for functional relation between the incident intensity field as a function of the optical cavity field intensity. Investigating the relation between any physical measurable quantity in the optomechanical system against the incident field intensity and its dependency on fundamental optical cavity field against the incident field intensity. Finally, studying the effects of integrating two types of atomic media into the optomechanical system, and tunability it could provide to the optomechanical system, each medium is separately studied. The atomic medium also tends to squeeze the domain of the bistability within the system relative to the basic optomechanical system
In nonlinear optical interactions, one expects the transverse spatial intensity dependence to be modified. Nonlinear interactions within an optical resonator, in the absence of a mode-selective element such as an aperture, will thus in general couple the energy into various transverse modes of the resonator. One attractive analytical model incorporating transverse effects into optical bistability assumes, in spite of the observations above, that the transverse intensity profile remains a single-Gaussian (TEM00) mode. This model seems to be supported by experimental results for bistability with two-level sodium atoms in a confocal ring resonator. To test the generality of the single-Gaussian mode model, we have performed bistability experiments with two-level atoms in standing-wave resonators. Two resonators, differing only in length, were used. The confocal resonator is transverse-mode-degenerate and will support a combination of modes, as would the ring; the slightly longer resonator will allow only a single-transverse mode to be excited. Preliminary results indicate that there may be a difference in the two cases; although the behavior near the critical onset of bistability seems to be the same in the two cases, there seems to be a difference in the evolution of hysteresis with increasing cooperativity parameter. Further results, including a quantitative comparison with the single-Gaussian mode theory for standing-wave resonators, are presented.
Absolute quantitative comparisons between experiment and theory are reported for the steady-state characteristics of optical bistability with two-state atoms in ring and in standing-wave resonators. The case treated here is the absorptive one, with both atoms and cavity resonant with the field. The experiment is designed to realize as closely as possible a standard theoretical model of two-state atoms interacting with a single mode of the electromagnetic field in a resonator and is described in detail. A detailed explanation is also given of the extensions to the standard theory that are needed to account for unavoidable experimental limitations such as transit broadening and the nonideal nature of the resonator. The good agreement observed between experiment and theory provides a foundation for quantitative investigation of nonlinear dynamics and quantum phenomena in this nonequilibrium optical system.
We discuss a mean-field quantum-mechanical model which describes the dynamics of a homogeneously broadened system of two-level atoms contained in a pencil-shaped resonant cavity and driven by a coherent resonant field EI. The model is treated in the semiclassical approximation. The model is justified on the basis of Maxwell-Bloch equations with two coherently coupled directions of propagation, with boundary conditions for the fields taken into account. Above a suitable critical density of atoms the system exhibits a bistable behavior including both the stationary situation and the transient, both the light transmitted in the forward direction and the fluorescent light. Bistability is shown to be a consequence of atomic cooperation. We give a simple description of optical bistability leading to new predictions for the transient behavior of the transmitted light and for the spectrum of the fluorescent light. The damping constant which characterizes the rate of appraoch to the stationary situation exhibits a hysteresis cycle. In the low-transmission regime the approach is monotonic, whereas in the highly transmitting situation the approach is oscillatory. One finds a critical slowing down in correspondence with the values EI(+), ${{E}_{I}}^{(\ensuremath{-})}$ of the incident field where the transmitted field (as well as the total fluorescence intensity, the rate of approach to the stationary state, etc.) changes discontinuously. On the basis of the regression hypothesis we give a qualitative description of the spectrum of the fluorescent light. This is shown to undergo hysteresis and discontinuous changes at the same values EI(+), ${E}_{I}^{(\ensuremath{-})}$ of the incident field. Below the critical density of atoms one recovers the usual picture of resonance fluorescence, with a continuous transition from a single-line spectrum to a three-peaked structure when the Rabi frequency becomes equal to the natural linewidth. Above the critical density the triplet appears only when the Rabi frequency becomes equal to the cooperative linewidth of pure super-fluorescence (i.e., for EI=EI(+)). Moreover, it appears discontinuously: when EI crosses the value EI(+) the spectrum changes from a single narrow line to a triplet with well-separated sidebands.