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.
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.
This chapter concludes the discussion about temporal instabilities and self-oscillations, considering the instabilities that arise in the systems analyzed in Chapter 13, with the exception of optical bistability, which has been discussed separately in Chapter 11. Section 25.1 focuses on the laser with an injected signal. As already mentioned in Section 13.1, in this system self-pulsing instabilities are rather ubiquitous, and on exploring the parameter space one meets an extremely rich variety of oscillatory behaviors. For example, under extremely high-gain conditions, breathing, spiking and chaos have been identified over various ranges of the driving field strength [294]. In this section we show some results that arise under parametric conditions that are more accessible experimentally. We illustrate especially the coexistence, for the same parametric values, of different oscillatory behaviors that are reached by starting from different initial conditions. On the other hand, Section 25.2 deals with a laser with a saturable absorber. In this system even the full scenario of stationary intensity solutions in the single-mode model is very complex. For the sake of simplicity, in this section we focus on a simpler rate-equation model that is obtained from the single-mode model of Eqs. (13.13)–(13.17) by adiabatically eliminating the atomic polarization of both active and passive medium. As in Chapter 13, we assume that the atomic transition frequencies are equal and also coincide with a cavity frequency. In this way all the variables in the model can be assumed real, and the linearization around the stationary solution described in Section 13.2 leads to a cubic eigenvalue equation. The spontaneous oscillations which arise from this instability are called repetitive passive Q-switching in the literature. Section 25.3 considers the stability of the stationary solutions of the degenerate optical parametric oscillator described in Section 13.4 and outlines the scenario of period doubling and chaos that arises under appropriate parametric conditions.
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 complement we discuss several examples of optical phenomena in media where the refractive index depends nonlinearly on the intensity , known as optical Kerr media. This nonlinear effect exists in all materials, even isotropic ones, like glass or fused silica, but it is particularly marked in certain physical systems to be exemplified in Section 7B.1. After investigating the propagation of light through such media in Section 7B.2, we shall discuss three applications of the optical Kerr effect (which can be studied in any order). We begin by describing a bistable optical system , when this nonlinear medium is inserted in a Fabry–Perot cavity (Section 7B.3). We then study phase conjugate mirrors and examine their potential applications in adaptive optics (Section 7B.4). Finally, we discuss certain effects occurring during the propagation of an isolated wave, bounded either transversely or temporally, in a Kerr medium, and describe self-focusing effects (Section 7B.5) and self-phase-modulation effects (Section 7B.6). In particular, we shall show that nonlinear effects and dispersion effects can compensate to produce stable structures known as solitons , which maintain their shape during propagation. Examples of third-order nonlinearities Nonlinear response of two-level atoms We begin by studying a simple case of a nonlinear interaction, namely a two-level quantum system under the effects of a plane wave.
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.
Investigating the behavior of an atom in response to the applied electric field when the latter frequency is in resonance with the natural frequency of the atom is an interesting subject. Near resonance, some of the most interesting optical phenomena such as dispersion, and absorption are more pronounced. Moreover, considering resonant and near resonant interaction of light with two-level atoms, uncovers fascinating physical phenomena such as area theorem and self-induced transparency describing stable pulse propagation regimes in which the pulses maintain their identity. Existence of these optical structures in on-resonance optical media has not been investigated in detail. In this thesis, the on-resonance interaction of optical field with atoms is considered and the formation of several novel self-similar and kink waves in linear and nonlinear resonant media is discovered and theoretically explored. First, self-similar pulse formation in homogeneous broadened linear amplifiers in a vicinity of an optical resonance is analyzed. It is demonstrated that the self-similar pulses serve as universal asymptotics of any near resonance short pulses propagating in coherent linear amplifiers. Second, ultrashort self-similar pulse propagation in coherent linear absorbers near optical resonance is investigated. Third, existence of self-similar optical waves with kink structure in resonant optical systems is discovered. Fourth, it is found that self-induced transparency quadratic solitons are realizable in the media with quadratic optical nonlinearities, doped with resonant impurities. Finally, stable spatial similaritons supported by homogeneous conservative optical media with quintic nonlinearities are explored. To experimentally realize the presented results, physical models are presented for all systems under consideration. The stability of the proposed near resonance optical systems is demonstrated through a series of numerical case studies.
Chapter 7 initially discusses mechanical effects on light and their application to optical tweezers. This leads to a discussion of laser cooling of gases and solids, magneto-optic traps, and velocity-selective coherent population trapping. Coherent population transfer is a counterintuitive way of altering the occupation of energy levels in an exceptionally efficient manner. Coherent transverse optical magnetism explores an optical effect mediated by the Lorentz force of light. Induced magnetic scattering is discussed and a closed form solution presented. Electromagnetically induced transparency is an interference effect that is useful for rendering opaque media transparent and enhancing high-order nonlinear optical processes via intermediate steps that may be fully resonant. Slow light is discussed. Squeezed light is given as an example of a coherent state with altered noise properties. Cavity quantum electrodynamics treats the interaction of atoms with light fields inside cavities, showing that strong atom-field interactions are possible in which the energy structure of the atoms is altered. A section on quantum information processing explains how quantum bits may be realized in practice and used to perform logic operations for quantum computing.