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Physical Review A

Purely absorptive bistability in double-ring cavities

Pedro Martín Mejías, Rosario Martínez-Herrero, Eusebio Bernabéu

Published Mar 1, 1986• 4 citations
Publisher

Abstract

A novel type of optical bistable configuration, namely the so-called double-ring (DR) device, is presented. The stationary solutions in the purely absorptive case for a homogeneously broadened atomic system are derived. In the mean-field limit it has been pointed out that use of a DR arrangement enables us to reduce both the length of the sample of the nonlinear medium and the response time of the device. Linear stability analyses have shown that the position and the length of the unstable regions of a given hysteresis cycle can be controlled, to a great extent, by means of the two new parameters of our DR scheme. Finally, it has been remarked that, in order to observe self-pulsing, the purely absorptive DR system may present a similar behavior to that offered in the dispersive single-ring case.

DOI
10.1103/physreva.33.1836

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Nonlinear Optics•Dec 4, 2003•0 citations
A novel type of optical bistable configuration, namely the so-called double-ring (DR) device, is presented. The stationary solutions in the purely absorptive case for a homogeneously broadened atomic system are derived. In the mean-field limit it has been pointed out that use of a DR arrangement enables us to reduce both the length of the sample of the nonlinear medium and the response time of the device. Linear stability analyses have shown that the position and the length of the unstable regions of a given hysteresis cycle can be controlled, to a great extent, by means of the two new parameters of our DR scheme. Finally, it has been remarked that, in order to observe self-pulsing, the purely absorptive DR system may present a similar behavior to that offered in the dispersive single-ring case.
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 analyze the absorptive and hybrid absorptive–dispersive double-cavity optical bistability behavior in a four-level N-type atomic system driven by two optical fields circulating inside two independently unidirectional ring cavities and a trigger field outside the cavities. In this system, the two-photon absorption and cross Kerr nonlinearity play significant roles in the formation of bistability and can be controlled efficiently due to the quantum interference effect, and then the resulting nonlinear behavior exhibits different features from the two/three-level schemes. It is shown that the threshold values and hysteresis cycle width of the bistable curve can be manipulated via the adjustment of the system parameters. Meanwhile, we also obtain double-beam optical multistability. When adding positive and negative pulses to the cavity input, all-optical steady-state switching for the two cavity outputs can be achieved simultaneously. Therefore, the proposed scheme may have potential applications in all-optical communication and computing.
Beams of homogeneously broadened two-state sodium atoms in a confocal ring resonator have been employed to investigate the steady-state characteristics of absorptive [1] and dispersive optical bistability. Also studied in the same system is the dynamical behavior due to the intrinsic single-longitudinal-mode instability [2] originating in the alteration of the atomic absorption profile by a strong field as in the Lorenz-type instability in the homogeneously broadened laser. All experimental parameters are carefully measured and corrected for transit broadening, finite absorption, and residual Doppler broadening in order to be able to make an absolute comparison with the single-Gaussian-transverse-mode theory [3-6]. Despite the frequency degeneracy of the transverse modes of the resonator, the agreement with this theory for absorptive bistability is remarkably good. The dispersive bistability data are more difficult to interpret, but they seem to be in reasonable agreement with the theory; the same can be said of the results for the domain and oscillation frequency of the instability, at least within a certain region of parameter space. Outside this region, multiple unstable states appear for various intracavity intensities; recent observations suggest that these may be due in part to instabilities involving higher-order transverse modes of the resonator.
We study the semiclassical mean-field theory of single-mode dispersive optical bistability in the limit of a fully developed hysteresis cycle (i.e., in the limit of large values of the bistability parameter C) and when all decay rates have comparable magnitude. We locate (i) on the upper branch instabilities leading to time periodic regimes, (ii) on the lower branch instabilities leading to a switching before the limit point is reached. None of these instabilities are observed in the case of purely absorptive optical bistability.
In the last chapter, we examined the effect of cubic nonlinearity on selfrefraction of an optical beam. We have also seen that the presence of a quadratic nonlinearity can also contribute to change in the profile of a Gaussian beam in a crystal. In this chapter we will see how the effect of nonlinearity, alongwith the proper feedback, can give rise to optical bistability and hysteresis. This is not unexpected as hysteresis and bistability are also observed in nonlinear electronic circuits with feedback, such as in the Schmitt trigger, as well as in hybrid optical devices, such as an acoustooptic device with feedback (Banerjee and Poon, 1991). In what follows, we will examine two different types of optically bistable devices based on optical nonlinearities. The first is a nonlinear ring resonator comprising a two-level gain medium as the nonlinear element. We will treat both absorptive and dispersive bistability. Absorptive bistability is the case when the incident optical frequency is close to or equal to the transition frequency of the atoms from one level to another. In this case, the absorption coefficient becomes a nonlinear function of the incident intensity. On the other hand, if the frequencies are far apart, the medium behaves like a Kerr-type material and the system exhibits what is called dispersive bistability. In this case the material can be modeled by an effective n2 or v (3), meaning that the refractive index becomes a nonlinear function of the optical intensity. The second configurationwewill examine is a linear-nonlinear interface. In this case, the interface provides the feedback due to refractive index mismatch. In fact, the reflection coefficient (as well as the transmission coefficient) is modified due to the nonlinearity, which, as we shall see later, is responsible for demonstrating optical switching and bistability.