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Journal of Modern Optics

Instability Thresholds in General Nonlinear Ring Resonators

W. J. Firth, Scott Sinclair

Published Mar 1, 1988• 3 citations
Publisher

Abstract

By considering a passive ring cavity with linear absorption in the dispersive limit, a unified analysis is presented for tuning-optimized minimum instability thresholds; the treatment is general, allowing consideration of arbitrary material response times, cavity tuning and finesse. In the adiabatic limit (τ → 0) the results reduce to the usual Ikeda map. The case of long medium-response time (τ → ∞) is treated in some detail in the high-finesse limit; these conditions being of particular relevance to semiconductor laser amplifiers. It is found analytically that in the high-finesse limit, both optical bistability and self-pulsing instabilities should exhibit the same low-threshold behaviour. An analytical expression is derived for the ‘instability threshold curve’, A 2 B against B for minimum A 2, in the mapping limit.

DOI
10.1080/09500348814550461

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By considering a passive ring cavity with linear absorption in the dispersive limit, a unified analysis is presented for tuning-optimized minimum instability thresholds; the treatment is general, allowing consideration of arbitrary material response times, cavity tuning and finesse. In the adiabatic limit (τ → 0) the results reduce to the usual Ikeda map. The case of long medium-response time (τ → ∞) is treated in some detail in the high-finesse limit; these conditions being of particular relevance to semiconductor laser amplifiers. It is found analytically that in the high-finesse limit, both optical bistability and self-pulsing instabilities should exhibit the same low-threshold behaviour. An analytical expression is derived for the ‘instability threshold curve’, A 2 B against B for minimum A 2, in the mapping limit.
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.
In this paper we demonstrate the dynamic simulation of light traveling in a nonlinear fiber optic ring resonator (FORR) with a single coupler. The mathematical relations describing the nonlinear characteristics of the FORR are derived from light transmission theory in linear optical systems. Optical nonlinear phenomena such as bifurcation and chaos are investigated for optical bistability and the Ikeda instability. It is found that for the Ikeda instability of light inside a FORR, the ring-cavity lifetime is much longer than the Kerr-effect response time of the fiber ring. Transmission curves of bistability and instability are shown as a function of the coupler insertion losses (γ) and the coupling coefficients (κ).
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.