• To introduce some theory of electromagnetic waves. • To introduce optical bistability and show some related devices. • To discuss possible future applications. • To apply some of the theory of nonlinear dynamical systems to model a real physical system. On completion of this chapter, the reader should be able to • understand the basic theory of Maxwell’s equations; • derive the equations to model a nonlinear simple fiber ring (SFR) resonator; • investigate some of the dynamics displayed by these devices and plot chaotic attractors; • use a linear stability analysis to predict regions of instability and bistability; • plot bifurcation diagrams using the first and second iterative methods; • compare the results from four different methods of analysis.
• To introduce some theory of electromagnetic waves. • To introduce optical bistability and show some related devices. • To discuss possible future applications. • To apply some of the theory of nonlinear dynamical systems to model a real physical system. On completion of this chapter, the reader should be able to • understand the basic theory of Maxwell’s equations; • derive the equations to model a nonlinear simple fiber ring (SFR) resonator; • investigate some of the dynamics displayed by these devices and plot chaotic attractors; • use a linear stability analysis to predict regions of instability and bistability; • plot bifurcation diagrams using the first and second iterative methods; • compare the results from four different methods of analysis.
The nonlinear phenomenon known as optical bistability, which enables optical systems to maintain two stable output states for a single input, is thoroughly examined in this review article. Through a detailed analysis of two important devices—the Simple Fiber Ring (SFR) resonator and the Nonlinear Optical Loop Mirror (NOLM)—it investigates the basic concepts of optical bistability, namely nonlinearity and feedback. Numerical simulations demonstrating dynamic dynamics and stability conditions bolster a strong theoretical framework. Recent advancements in bistable optical technologies, including developments in epsilon-near-zero materials, silicon photonics, hybrid plasmonic structures, neuromorphic photonics, and AI-driven photonic processors are also discussed. Furthermore, this review highlights emerging applications of optical bistability in optical logic circuits, high-speed communication, and quantum information processing. This review work is a useful tool for researchers and engineers creating next-generation optical systems because it connects theoretical insights with cutting-edge technological developments.
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).
This chapter shows that solitons and other special solutions of soliton equations play important roles in helping people understand the transverse structure of optical beams in optical cavities, the longitudinal structure of pulses in resonant media, superfluorescence, and the production of the Stokes wave in Raman scattering processes. It describes the fundamental ingredients required for laser operation. The chapter introduces the simplest model of a laser involving a medium of two-level atoms. It discusses the manifestation of optical bistability in ring and Fabry–Perot cavities. The chapter also discusses the stability of the "stable" solutions and find that they can undergo a series of period-doubling bifurcations leading to a temporally chaotic output. It looks at the remarkable behavior of a pulse propagating through an almost perfectly tuned resonant and initially passive medium. The resonator is tuned to support frequencies close to that of the laser transition.
In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11. Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states. In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2. In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations. A laser with an injected signal From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element. In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.
This thesis presents the theory of the driven optical ring cavity containing a nonlinear material. First, an overview of the history of bistability and spatial solitons in driven nonlinear cavities is outlined. Then, the basic theory of the optical Kerr effect is presented and two of its consequences are outlined: optical bistability and spatial solitons. Within this background, the mirror feedback model is derived from the Maxwell-Bloch equations and considerations of the classical optics of the cavity. This model is analytically shown to exhibit bistability in the plane wave case. The mean-field model is subsequently derived by applying the mean-field limit to the Maxwell-Bloch equations and the cavity optics. In addition, the mean-field model is analytically shown to exhibit bistability in the plane wave case. Numerical simulations are performed on the mean-field model demonstrating the existence of: (i) plane wave bistability, (ii) transverse modulational bistability, (iii) the ability to “switch†between bistable states in both the plane wave and modulated case, (iv) instabilities in the modulated case and (v) the possibility for optimization of the switching process. Finally, an application of bistable driven cavities is explored: an all-optical digital information processor and storage device.
In this chapter we outline the general scenario for optical temporal instabilities, starting from the standard Maxwell–Bloch equations. With the exception of Section 22.7, in Part II we will always consider the ring-cavity configuration because only in the case of the ultrathin-medium case considered in Section 22.7 is the treatment of instabilities in Fabry–Perot cavities not substantially more complicated than that in the ring-cavity configuration. Some authors have introduced special approaches in order to study temporal optical instabilities, exploiting the concepts of gain and of dispersion, which are natural in optical systems and therefore help our intuition. In particular, a hint of the mechanism that produces instabilities in optical systems is offered by the so-called “weak-sideband” approach or “gain-feedback approach” [245–249], whose validity can be justified rigorously with the help of the linear-stability analysis [250]. The central point of this method is the response of the system to a weak probe passing through a collection of two-level atoms driven by a strong field [251–253]. If the strong signal with carrier frequency ω 0 saturates the medium, we can find regions of the frequency axis, which are far removed from the carrier frequency of the signal, where the weak probe can experience gain instead of absorption even if the medium has no population inversion. In this book we follow the approach of [250], which is rigorously rooted in the linear- stability analysis and, in addition, establishes a general, useful and, in a sense, surprising connection between single-mode and multimode instabilities. We classify as single-mode a temporal instability that involves the resonant mode only, and classify as multimode a temporal instability that involves only off-resonance cavity modes. The discussion in this chapter is mainly focussed on this connection. In Section 20.1 the multimodal equations derived in Section 12.2 are reformulated by introducing a modal expansion also for the atomic variables. In this way one obtains a complex hierarchy of equations that couple the modal amplitudes for the field and for the atomic variables.