We propose an original method for analytically solving the propagation equation in a resonant dense medium, and we give explicit solutions, using a parametric formulation under the thin-sample approximation. We then use the knowledge of the field transformation in a single-pass process to study the transmission of a resonant dense medium inserted into a ring cavity. Implicit analytical results are given, together with an explicit formula under the thin-film approximation. The nonlinear phase shift that is associated with propagation through the medium is shown to have a great influence on the cavity transmission. Optical bistability is obtained in the most general case. We can easily adjust the switching intensities and the intensity shift by modifying the initial cavity detuning ϕ0. For positive cavity detunings, perfect optical limiting can be obtained: Induced absorptive and dispersive losses limit the output intensity. This device can lead to the stabilization of a noisy incident continuous wave or to the realization of optical limiters.
We propose an original method for analytically solving the propagation equation in a resonant dense medium, and we give explicit solutions, using a parametric formulation under the thin-sample approximation. We then use the knowledge of the field transformation in a single-pass process to study the transmission of a resonant dense medium inserted into a ring cavity. Implicit analytical results are given, together with an explicit formula under the thin-film approximation. The nonlinear phase shift that is associated with propagation through the medium is shown to have a great influence on the cavity transmission. Optical bistability is obtained in the most general case. We can easily adjust the switching intensities and the intensity shift by modifying the initial cavity detuning ϕ0. For positive cavity detunings, perfect optical limiting can be obtained: Induced absorptive and dispersive losses limit the output intensity. This device can lead to the stabilization of a noisy incident continuous wave or to the realization of optical limiters.
We formulate a first-principles description of the behavior of a bidirectional ring cavity containing a saturable absorbing medium, and driven by an external coherent field. The setting of interest to this study is an extension of the ordinary optically bistable system, where a ring resonator supports only one of the two possible directions of propagation of the cavity field. In this more general case and in the uniform-field limit, we show that the dynamics of the fields-atoms system is described by an infinite set of coupled equations that can be readily solved by standard numerical means after appropriate truncation of the number of atomic variables. With the help of two different approaches, we find unexpected, long-time stationary solutions such that the forward and backward fields oscillate with different carrier frequencies: the first is based upon the time-dependent equations and the second on a set of nonlinear-algebraic equations describing the steady states. As an added confirmation, we carry out a linear-stability analysis of the unidirectional steady state for the purpose of identifying the conditions under which a backward field can grow from an initial fluctuation. By this approach we also suggest a possible strategy for the experimental observation of stationary, frequency-shifted output fields.
We study the propagation of electromagnetic radiation through a ring-cavity device containing a dispersive nonlinear medium. Under conditions appropriate for introducing the slowly varying envelope approximation and for adiabatically eliminating the atomic variables, we calculate analytical expressions between the steady-state-output intensity and the input-field amplitude for two models of the nonlinear susceptibility, the two-level atomic medium and the Kerr medium. The stability of these curves is determined, with special attention given to changes in the thresholds by varying the return-trip phase shift, the linear absorption, and the reflection coefficients of the mirrors. We investigate the positive-branch instabilities resulting in a pulsing output intensity and calculate the bifurcation rate for the period-doubling hierarchy and the fractal dimension of our two-dimensional maps for the Kerr medium. The convergence of the bifurcation rate to a constant is slow, and we find, as in the publications of Snapp et al. [ Opt. Commun.40, 68 ( 1981)] and Carmichael et al. [ Phys. Rev. A26, 3408 ( 1982)], that this constant is consistent with that found for the class of one-dimensional maps with a quadratic maximum. This universal constant is observed over a large range of fractal dimensions for the chaotic attractor. Discussions of pulse-output bistability, intermittency, and anomalous switching are given.
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.
We study optical bistable behavior of a Rydberg electromagnetically-induced transparency (EIT) atomic medium in a unidirectional optical ring-cavity. Due to strong van del Waal (vdW) interactions between the atoms, both optical nonlinear dispersion and nonlinear absorption coefficients are enhanced substantially. Under the condition of two-photon on resonance, we show that probe one-photon detuning can change the phase of the third order nonlinearity coefficient, which tunes the character of the optical bistability within different ratios of dispersive and absorptive types. This enables the single-photon control over photonic devices for further manipulation of light other than switches and transistors. More interestingly, we predict appearance of a scaling phenomena for optical bistabilities with the factors of coupling Rabi frequency and atomic density. Additionally, we also discuss the influence of the cavity detuning and the mirror transmission coefficient on the optical bistable behavior. The strong bistable feature provides a good ingredient for realizing all-optical logic gate devices in optical computing.
We analyze instabilities in mixed absorptive and dispersive optical bistability in the rate-equation approximation and the mean-field and good-cavity limits. Our starting point is a set of multimode equations derived from the Maxwell-Bloch equations for ring-cavity boundary conditions. We obtain analytic expressions for the instability conditions. In a plane-wave analysis, we find that a portion of the lower transmission branch can be unstable in addition to the upper-branch instability found in purely absorptive bistability. Also, a new disconnected region of instability can exist on the upper branch. Our analysis becomes particularly simple for equal and opposite cavity detunings and we explore this case in detail. We extend our treatment to include a Gaussian transverse intensity profile and show that the instabilities remain in the presence of Gaussian averaging. We also show that many of the results obtained in the rate-equation approximation hold when the atomic linewidth and atomic decay rate are of the same order.
We review and extend the theory of instabilities in absorptive optical bistability for ring cavities containing homogeneously broadened two-level media. The instabilities occur in the high-transmission branch and result in either a precipitation to the low-transmission branch or in pulsed multimode operation. Two numerical approaches are used to predict operation away from the simple mean-field limit. First, an iterative method is used to solve the general field eigenvalue equation for cavity modes with frequencies displaced from the input frequency. This work shows that the instability region decreases in size as the mirror transmission increases. These calculations are confirmed by the second numerical technique, consisting of direct integrations of the coupled Maxwell-Bloch equations. This second approach also provides time histories of the instability evolutions, and divides the instability range into a precipitation regime and a self-pulsing regime. The results are discussed in terms of first- and second-order phase transitions, and agree with the analytical results obtained within the dressed-mode description of optical bistability. They show that when the incident field is adiabatically decreased along the high-transmission branch, the spiking behavior always appears abruptly. By further decreasing the incident intensity, the self-pulsing disappears either continuously if the system remains in the high-transmission branch, or discontinuously if the system precipitates to the low-transmission branch. Connection is made with induced probe gain known in the saturation spectroscopy of absorbers (uninverted media), in which population pulsations transfer energy from a saturating wave to the probe waves. The very close relationship with multimode operation in homogeneously broadened unidirectional ring lasers is also established.