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IEEE Journal of Quantum Electronics

Criteria for multiphoton optical bistability in a high-Q ring cavity

Shenping Li, Yizhou Zhang

Published Jan 1, 1993• 0 citations
Publisher

Abstract

The static criterion for multiphoton optical bistability in a high-Q ring cavity is analytically derived for a homogeneously broadened atomic system when the input coherent field is in perfect resonance with the atoms and with the cavity. It is shown how the static threshold condition, the bistable regime, the critical point, and the threshold input field amplitude for the optical bistability, change with the variables of the photon number involved in the transition and the system parameter C/sub n/. Criteria for the stability of the stationary solutions are derived and discussed. The results show that in the input-output curves, all parts of the branches with negative slope are unstable, and some parts of the branches with positive slope may be unstable. Considering the stability of the stationary solutions, complete criteria for optical bistability are discussed.>

DOI
10.1109/3.199264

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The static criterion for multiphoton optical bistability in a high-Q ring cavity is analytically derived for a homogeneously broadened atomic system when the input coherent field is in perfect resonance with the atoms and with the cavity. It is shown how the static threshold condition, the bistable regime, the critical point, and the threshold input field amplitude for the optical bistability, change with the variables of the photon number involved in the transition and the system parameter C/sub n/. Criteria for the stability of the stationary solutions are derived and discussed. The results show that in the input-output curves, all parts of the branches with negative slope are unstable, and some parts of the branches with positive slope may be unstable. Considering the stability of the stationary solutions, complete criteria for optical bistability are discussed.>
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.
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 paper a quantum mechanics model for describing multiphoton optical bistability is given. Based on this molel, the state equation of multi-photon option1 bistability is obtained. The method which for calculating the threshold of multi-photon optical bistability is also given. After analyzing the potential function, we obtain the statics stability criterion and the explanation of switching characteristics. The stable region and the inferior stable region in the bistability characteristic curve are separated. In fact, multiphoton optical stable states are actually multi-photon optical stable-inferior stable states. We also discuss the phase transition problem of multi-photon optical bistability and multi-photon lasers.
By means of the semiclassical method, we discuss the bistability in a two-photon laser with injected signal for the homogeneously broadened single mode ring cavity. It is demonstrated that in the resonant case one can find bistable states in a two-photon laser with injected signal. In the limit of "good cavity" and "bad cavity", we give a simple description of bistability leading to a predictions for the transient behavior of the output light and the hysteresis cycle by adiabatically eliminating the fast variables. It is found that the approach to the stationary state is monotonic in the low-branch and is oscillatory in the high-branch, and a critical slowing down occurs where the output field changes discontinuousiy.
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.
This paper presents a unified theory of nonequilibrium transitions which occur in radiation interacting via an nnn-photon transition with atomic systems inside optical cavities. Incoherent pumping of the atoms and coherent driving of the cavity are included. Examples of such systems are the nnn-photon laser, the nnn-photon laser with injected signal, and nnn-photon optical bistability. The state equation and stability conditions for these phenomena are derived. Fluctuations are included via a Fokker-Planck equation in the constant-diffusion approximaton for which potential solutions may be obtained in the steady state. When these solutions are used, moments of the photon distribution can be calculated for the above systems in the specific cases of one- and two-photon transitions.