The electric field injected into a unidirectional ring cavity with spherical mirrors is assumed to be the TEM_(10) mode of a Gaussian beam. In the framework of the single transverse mode model, the stabilities of the stationary state case and the purely absorptive case are discussed. The results show that the instabilities may appear.
In Chapters 27 and 28 we discussed the topic of optical spatial pattern formation in planar cavities. In this chapter, instead, we consider the case of ring cavities with spherical mirrors, on the basis of the discussion of transverse modes in Chapter 26. In the first section we derive, first of all, a set of modal equations to be coupled with the atomic equations, with a configuration similar to that of the modal equations of Chapter 12, with the difference that now transverse modes are included in addition to longitudinal modes. This provides an extremely complex context, and therefore in the following we focus on the case of quasi-planar cavities, in which it is possible, by introducing the low-transmission and single-longitudinal-mode approximations, to derive a single-longitudinal-mode model that represents, in the framework of cavities with spherical mirrors, the counterpart of the single-longitudinal-mode model for planar cavities discussed in Chapter 27. In Section 29.2 we consider the simplest configuration, in which only the fundamental Gaussian transverse mode is assumed to have negligible losses, so that it is possible to consider a single-mode model that includes only the Gaussian mode. This is precisely the model considered in Section 24.3.1 for the comparison with experimental data concerning the single-mode instability of optical bistability. We calculate the stationary curve, and in the laser case we demonstrate the validity of the mode-pulling formula also in this configuration. In the following section we consider, instead, the general case in which several transverse modes of a laser are involved and demonstrate that the modal equations admit multimodal stationary intensity states, a configuration that is not possible in the case of longitudinal modes, at least in the low-transmission approximation. In this phenomenon, which we call cooperative frequency locking , the modes agree a common oscillation frequency that is given by the average of the (mode-pulled) modal frequencies, with weights given by the modal intensities. Numerical and experimental evidence of cooperative frequency locking is provided.
In this paper we analyze the steady-state and stability properties of bistable optical systems, lasers with an injected signal, and ordinary free-running lasers. The starting point of our study is a unified model of a ring cavity containing a finite-size cylindrical sample of homogeneously broadened two-level atoms, and capable of supporting a single field mode with a Gaussian transverse profile. After solving the equations of motion in the steady state, we carry out a linear stability analysis of the stationary solutions, identify the domains of unstable operation of each of the three systems, and compare in detail the results of this work with those of earlier plane-wave uniform-field calculations. In the case of the laser with an injected signal, we find a significant enhancement of the instability domain relative to the plane-wave limit. A similar conclusion holds for mixed absorptive and dispersive optical bistability, although the enhancement of the domain of instability is less pronounced. Unstable behavior is predicted to occur for experimentally accessible values of the bistability parameter, and to be favored by the absence of bistability and by the selection of atomic and cavity detunings having opposite signs. This configuration ensures that the instability will be responsible for the emergence of undamped output pulsations of the type that will make the bistable system behave as an optical clock.
In this paper, the general ring cavity model of passive and active optical systems is presented. In the mean field and the Gaussian one transverse mode approximations, the steady state and stability properties of the systems are analyzed for mixed absorptive and dispersive cases. It is found that when the transmissivity ratio M (equals T1/T2) varies, the bistable loop changes its size and even the systematic behavior varies from the bistability mode into the amplification mode, and the instability domain on the steady state curve varies obviously. The mode-pulling formula and bistable threshold with varying values of M are also discussed.
Of the Lorenz-type instabilities expected in optical systems, the single-mode instability in optical bistability has been the subject of detailed theoretical and experimental work by our groups. A treatment employing the plane-wave and mean-field limits of the Maxwell-Bloch equations predicts a rich variety of higher-order bifurcations including period doublings and intermittency.1 An extension of this work to include the transverse dependence of the intracavity field reveals no significant qualitative differences between the plane-wave theory and the Gaussian beam theory as regards the stability boundaries obtained from a linearized analysis.2 The instability persists in the presence of a Gaussian transverse profile and the boundaries are not radically modified compared with those obtained from the plane-wave theory. However, numerical integration of the Maxwell-Bloch equations in the single transverse mode approximation does not show the wealth of higher-order dynamic states present in the plane-wave case. An experimental investigation of optical bistability for two-level atoms in a single transverse, single longitudinal mode cavity exhibits limit cycle oscillation with stability boundaries that agree well with the boundaries obtained from the work in Ref. 2. However, for the range of values C < 300, |Δ| < 5, |θ| < 50, X ~ 102, the experimental results do not show the diversity of higher dynamic states expected from the plane-wave theory, in agreement with our numerical results. In addition the measurements support the existence of a single transverse mode oscillation. The sum of our theoretical and experimental work provides a quantitative statement for the role of the transverse effects in optical instabilities.
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.