Contents Introduction; 1. Reduction of the Maxwell–Schrödinger equations; 2. Parameter swept across a steady bifurcation I; 3. Parameter swept across a steady bifurcation II; 4. Optical bistability: constant input; 5. Optical bistability: variable input; 6. Multimode optical bistability; 7. Free running multimode lasers; 8. Antiphase dynamics; 9. Laser stability; 10. Second harmonic generation; 11. Saturable absorbers; 12. Transverse effects in optical bistability.
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.
In this chapter, we remove one assumption that has been implicit since the beginning of this book, namely the unidimensional aspect of the cavity. We now take into account the transverse variation of the field in the resonant cavity. In dealing with transverse effects, there are two possible approaches, very much like in the multimode optical bistability (OB) studied in Chapter 6. One possibility is to project the Maxwell–Bloch equations on a suitable basis. In the introduction of Chapter 6, the difficulty of selecting this suitable basis was explained for 1-D cavities. This difficulty is amplified by the transverse dimensions, especially because of the lateral boundaries. The other approach is to derive global equations (that are still partial differential equations) for slowly varying amplitudes. They are generally variants of well-known nonlinear partial differential equations of mathematical physics, and therefore a large number of results are directly available. However, when transferring results from another domain to optics, some care must be exercised because the relevant domains of parameters are not always compatible. The classic example is the canonical set of parameters for the Lorenz equations that includes b = γ ║ /γ ⊥ = 8/3 whereas for atomic transitions the upper bound of b is 2. The reader will find a wealth of results, mostly for the modal expansions, in the special issues and reviews [1]–[4]. A review more specifically oriented toward the global amplitude equations is found in [5] and a mathematical study of these equations is presented in [6].
Abstract In the first part we give a brief review of some theoretical problems that still hinder the practical use of bistable optical devices in logic circuits. In the second part we study the influence of a time-dependent control parameter on the solution of differential equations. We prove that even for small sweeping rates large delays can be expected, resulting in the dynamical stabilization of unstable solutions. Examples are given for dynamically induced optical bistability in a laser and dynamical stabilization in a laser with saturable absorber.
Although the discovery of oscillatory instabilities (both single-mode and multimode) in free-running lasers was achieved in the 1960s, investigations on oscillatory instabilities in optical bistability did not take place until the 1970s and 1980s. The first section of this chapter does not deal with oscillatory instabilities, but instead concerns an interesting dynamical phenomenon in optical bistability that arises when the input field approaches a boundary of the bistable domain, and is called critical slowing down . This has been investigated both theoretically and experimentally. The remainder of this chapter is entirely devoted to oscillatory instabilities, following a historical order. With an inverse approach with respect to the laser case, we discuss multimode instabilities first. In Section 24.2.1 we start with the description of the instability which arises under exactly resonant conditions. While this is the counterpart in optical bistability of the resonant multimode laser instability discussed in Section 22.5, a striking difference from the laser case is that in the passive configuration there is no corresponding single-mode instability. This difference disappears, however, when we consider the generalization of the multimode instability of optical bistability to the detuned configuration. Finally we show that in the detuned case it has been possible to achieve a convincing experimental observation of the multimode instability in a long microwave cavity. After some relevant miscellaneous considerations in Section 24.2.2, in Section 24.2.3 we address the subject of the multimode Ikeda instability which leads to period doubling and chaos and aroused a lot of interest in optical chaos. We emphasize also the intrinsic connections with the standard multimode instability discussed in Section 24.2.1, showing, in particular, that the Ikeda instability requires conditions far from the low-transmission limit, i.e. that ∝L is not small. In Section 24.3 we discuss single-mode instabilities in optical bistability. In Section 24.3.1 we illustrate a configuration for the single-mode instability that includes also the Gaussian shape of the electric field in the beam section, and leads to periodic, close-to-sinusoidal self-pulsations.
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.
The theory of optical bistability can be developed at various levels of sophistication.(1) In this overview I have chosen an approach which I hope will make the simplicity and generality of the phenomenon clear, and be suited to the emphasis of this meeting. I will describe a deterministic, semiclassical theory of optical bistability, based on coupled Maxwell-Bloch equations. Statistical and quantum statistical theories will not be included. I also limit my attention to questions concerning steady states and their stability, leaving the theory of higher bifurcations to other speakers.