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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Dispersive optical bistability in unidirectional ring cavity

F. Šrobár

Published Jul 1, 2003• 0 citations
Publisher

Abstract

The mathematical models of dispersive optical bistability in unidirectional ring cavity yield the transmitted intensity It as a function of the input intensity Iin with additional parameters (mirror reflectivity R and phase shift β). There are two feedback elements in this description: the energy (and, hence, also causal) feedback quantified by R and the purely causal feedback given by dependence of the phase shift on the transmitted power, β=β0+β2It. The apparatus of causal diagrams method is used to represent and analyze phenomena in the ring cavity. The region of physically unstable states is marked by the condition tloop>1, where tloop is the transmission function of the feedback loop ItβIinIt present in the diagram. Besides the critical points defined by tloop=1 (which are linked to the existence of bistability), there are two conditions marked by tloop=0 in which the cavity state is insensitive to the values of R.

DOI
10.1117/12.498518

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The mathematical models of dispersive optical bistability in unidirectional ring cavity yield the transmitted intensity It as a function of the input intensity Iin with additional parameters (mirror reflectivity R and phase shift β). There are two feedback elements in this description: the energy (and, hence, also causal) feedback quantified by R and the purely causal feedback given by dependence of the phase shift on the transmitted power, β=β0+β2It. The apparatus of causal diagrams method is used to represent and analyze phenomena in the ring cavity. The region of physically unstable states is marked by the condition tloop>1, where tloop is the transmission function of the feedback loop ItβIinIt present in the diagram. Besides the critical points defined by tloop=1 (which are linked to the existence of bistability), there are two conditions marked by tloop=0 in which the cavity state is insensitive to the values of R.
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 report the transient nonadiabatic generalization of Bonifacio-Lugiato1 cw nonlinear ring cavity optical bistability (NRCOB) with mirrors that ensure the necessary feedback. A semiclassical formalism is adopted, consisting of a paraxial wave equation coupled to the two-level sharp-line density matrix which includes detuning and relaxation times in conjunction with boundary conditions at the mirrors. The existing super-fluorescence finite difference code2 with radial variations of the Beer length has been modified to simulate the NRCOB with the free space propagation between the mirror treated with a finite Hankel series.3
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.
It is known that two-wave coupling in photorefractive crystal may result in an intensity-dependent phase shift as well as energy transfer between the two waves without phase crosstalk. In a sense, this kind of phase shift is similar to the intensity-dependent refractive-index effects in Kerr media. Consequently, optical bistability can be established in an optical cavity with a photorefractive crystal. We formulate the model and get the expected bistable transmission with numerical simulations. Several cases, including that of the codirectional coupling and the contradirectional coupling, are considered. The effects of the photorefractive crystal’s distribution absorption and the spatial phase shift between the index grating and the light interference pattern in the crystal on the nonlinear transmission characteristics have also been studied. It is interesting to note that the output of the pumping wave is dependent on the cavity intensity and thus shows bistable behavior with respect to the cavity input as well.
The absorptive bistability in a nonlinear optical system surrounded by two feedback loops is studied. The role of the nonlinear element is played by a cell with Λ-type atom vapors placed in a unidirectional ring cavity. The feedback includes two electromagnetic fields interacting with two atomic transitions. A two-dimensional domain of stability was found in the coordinates of field intensities at the input. The dependence of its shape on the values of different parameters of the optical system is studied. The input-output curves corresponding to different trajectories in the domain of stability are obtained.
When the output parameter of an optical device could be a multivalued function of the input parameter, this device can be working in an optical bistable status. The most typical optical bistable device is a Fabry–Perot cavity containing a nonlinear optical medium, the refractive index of which depends on the light intensity. Upon the action of an intense input optical beam, the transmitted (or reflected) intensity of this device may exhibit a nonlinear response to the incident light intensity variation and, especially under appropriate working conditions, the characteristic output/input curve may manifest a single (or multiple) hysteresis loop(s). Such type of optical bistable device can be used as an optical differential amplifier, optical switch, optical limiter, optical clipper, optical discriminator, or an optical memory element, depending on the chosen nonlinear medium and operation conditions.