While the stationary behavior of the laser was described in Part I, now we focus on its dynamical aspects. Some of them have been illustrated already in Chapter 19, but only in the single-mode regime and for class-A and class-B lasers, in which the only instability which arises concerns the trivial stationary solution and leads to the transition from the non lasing to the lasing state. Chapter 20, on the other hand, provided a general picture of single-mode and multimode instabilities in active as well as in passive systems, and of the structural relations which link single-mode and multimode instabilities. In this chapter we focus on the instabilities which arise in the laser and lead to spontaneous temporal oscillations and chaos.

We start in Section 22.1 with the linear-stability analysis of the trivial stationary solution in the general multimode case for the standard laser. The results of this analysis have been anticipated in previous chapters, but here you will find their derivation. The same problem is considered in Section 22.2 for the laser without inversion, but limited to the single-mode case under fully resonant conditions as in Section 17.5.

Of fundamental importance is the analogy between the single-mode laser model and the Lorenz model, which is prototypical for the general field of chaos. This matter is discussed in Section 22.3, and is immediately followed by the treatment of the resonant single-mode laser instability in Section 22.4.

On the other hand, multimodal instabilities in the ring laser are discussed in Sections 22.5 and 22.6. The first is devoted to the amplitude instability, completing the picture given in Section 20.2, and the second to the phase instability. The multimode amplitude instability is related to a classic dynamical phenomenon in lasers, mode-locking, which topic is illustrated in Chapter 23.

Finally, Section 22.7 is devoted to the matter of amplitude instabilities again, but in the case of a Fabry–Perot laser containing an ultrathin medium discussed in Section 14.5.

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 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 Chapters 27 and 28 we discussed the topic of optical spatial pattern formation in planar cavities. In this chapter, instead, we consider the case of ring cavities with spherical mirrors, on the basis of the discussion of transverse modes in Chapter 26.

In the first section we derive, first of all, a set of modal equations to be coupled with the atomic equations, with a configuration similar to that of the modal equations of Chapter 12, with the difference that now transverse modes are included in addition to longitudinal modes. This provides an extremely complex context, and therefore in the following we focus on the case of quasi-planar cavities, in which it is possible, by introducing the low-transmission and single-longitudinal-mode approximations, to derive a single-longitudinal-mode model that represents, in the framework of cavities with spherical mirrors, the counterpart of the single-longitudinal-mode model for planar cavities discussed in Chapter 27.

In Section 29.2 we consider the simplest configuration, in which only the fundamental Gaussian transverse mode is assumed to have negligible losses, so that it is possible to consider a single-mode model that includes only the Gaussian mode. This is precisely the model considered in Section 24.3.1 for the comparison with experimental data concerning the single-mode instability of optical bistability. We calculate the stationary curve, and in the laser case we demonstrate the validity of the mode-pulling formula also in this configuration.

In the following section we consider, instead, the general case in which several transverse modes of a laser are involved and demonstrate that the modal equations admit multimodal stationary intensity states, a configuration that is not possible in the case of longitudinal modes, at least in the low-transmission approximation. In this phenomenon, which we call cooperative frequency locking, the modes agree a common oscillation frequency that is given by the average of the (mode-pulled) modal frequencies, with weights given by the modal intensities. Numerical and experimental evidence of cooperative frequency locking is provided.

Three elements, introduced in Part I of the book, play a key role in the book itself.

The first is the derivation of the Maxwell–Bloch equations, which is carried out in Chapters 2–4. In particular the derivation of the field-envelope equation, discussed in Section 3.1, is explained with special care. The derivation is based, of course, on the slowly varying envelope approximation and on the paraxial approximation. In textbooks these two approximations are usually discussed separately and rather briefly. We show that the two approximations are intimately connected, and only in this way it is possible to demonstrate that, while the second-order derivatives with respect to time and to the longitudinal coordinate z can be neglected, the second-order derivatives with respect to the transverse coordinates x and y must be kept. It is also possible to demonstrate that the term ∇(∇ · E) in the Maxwell wave equation for the electric field can indeed be neglected. An additional bonus of this approach is that, once one has shown in detail the derivation of the envelope equation in the simplest context, it becomes straightforward to generalize it to the more complex cases of frequency dispersion in the background refractive index (Section 5.4) and of quadratic and cubic nonlinearities (Chapters 6 and 7).

The second key element is the use of a multiple limit in the parameters, which in the literature is called the mean-field limit or uniform-field limit, whereas in this book we prefer to name it more precisely the low-transmission limit. This plays a central role when we derive modal equations and single-mode models from sets of equations with propagation. In this way one avoids, for example, steps like the phenomenological introduction of a damping term in the field equation(s) to describe the escape of photons from the cavity, and derives the damping from the boundary conditions of the cavity in a transparent way (Chapters 12 and 14).

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.

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.

With the exception of Chapter 1, up to this point we have considered cavityless optical systems. With this chapter, instead, we start focussing on the case that the atomic medium is located in an optical cavity. We describe in detail the case of a unidirectional ring resonator, the boundary condition which characterizes it and the relations between its output fields and the intracavity field (Sections 8.1–8.3). We discuss the transmission of an empty cavity (Section 8.4) and of a cavity that contains a linear medium (Section 8.5).

Optical cavities

An optical cavity or optical resonator is formed by at least two mirrors, appropriately arranged in such a way that the radiation beam follows a closed path. The beam splitters allow the radiation to escape from the cavity (output) and allow one to inject radiation into the cavity (input). In order to confine the radiation transversally with respect to the direction of propagation, the mirrors are usually spherical (see Fig. 1.2). The case of cavities with spherical mirrors will be described in Part III of this volume. On the other hand, an analytical treatment of the problems is possible only within the plane-wave approximation which we will use in the remainder of Part I and in Part II. In accord with this, we assume here that the mirrors are planar.

The two most common types of optical resonator are the Fabry–Perot cavity and the ring cavity. The planar Fabry–Perot cavity consists in two beam splitters in a parallel configuration as shown in Fig. 8.1. Within the cavity there are two counterpropagating fields, EF propagating in the forward direction and EB propagating in the backward direction with respect to the input field EI. Because we have derived the Maxwell–Bloch equations for the unidirectional configuration, in this and the following chapters we focus on the case of a ring cavity, which allows unidirectionality. The Fabry–Perot case will be considered in Chapter 14.

In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω0 can generate harmonic components with central frequencies 2ω0, 3ω0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω1 and ω2, with ω1 > ω2, can generate components with central frequency ω1 + ω2 (sum-frequency generation) and ω1 − ω2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities.

The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.

Up to this point we have always assumed that all the atoms which interact with the radiation field have the same atomic transition frequency, with an atomic linewidth given by γ⊥. This configuration is called homogeneous broadening. However, in the great majority of lasers the linewidth is inhomogeneously broadened, because the atomic frequency follows a distribution. For example, in solid-state lasers such as the ruby laser and fiber lasers the inhomogeneity is due to the imperfections in the crystalline structure of the material. In gas lasers, where the atoms or molecules are free to move, the inhomogeneous broadening is due to the Doppler effect, for which a molecule moving with velocity v in a field of frequency ω and wave vector k experiences an effective frequency ω′ = ω − k · v [8]. Since what matters is the difference between the atomic transition frequency and the field frequency, we can keep the field frequency fixed and imagine that the field interacts with different groups of atoms that move with different velocities and therefore have different transition frequencies. The realistic velocity distribution is a Gaussian, for which the corresponding distribution for the modulus of the velocity is Maxwellian. However, in the following we will mainly analyze the case of a Lorentzian distribution for the atomic transition frequency, because this allows an analytical treatment.

Again for the sake of simplicity, we will limit our analysis to the case of a ring cavity. In Section 15.1 we will generalize to the inhomogeneous-broadening configuration the modal equations for the case of homogeneous broadening, which were formulated in Chapter 12. In Section 15.2 we will specialize to the single-mode model and discuss its stationary solutions for the laser case.

Multimode dynamical equations

The atomic transition frequency of the generic group of atoms is given by ωa + δD, where ωa is the central atomic frequency and δD takes positive and negative values. We call ωc the frequency of the cavity mode nearest to ωa and denote by δac the difference ωa − ωc.

In Part I of this book we studied almost exclusively stationary solutions of the models; we also mentioned in which parametric ranges they are stable or unstable, but without proving it. In this part we not only describe the technique used to assess the stability of stationary solutions but also, especially, study the temporal phenomena which originate from the instabilities themselves. As a matter of fact, the appearance of spontaneous pulsations in the form of undamped trains of pulses was first observed in masers even before the advent of the laser in 1960. However, a strong interest in temporal optical instabilities arose much later, in the early 1980s, in resonance, as it were, with the general interest in the field of chaos in nonlinear dynamical systems.

In Part II of the book we focus on the dynamical phenomena in nonlinear optical systems, of course taking the general viewpoint of nonlinear dynamical systems. In Chapter 18 we describe first the technique of linear-stability analysis and then we discuss the most relevant and general instability-related dynamical aspects in nonlinear dissipative systems such as, for example, attractors, bifurcations and routes to chaos. This provides the language to describe the results discussed in the following chapters of Part II. For reasons of space, we do not introduce technicalities such as, for instance, Lyapunov exponents.

In order to identify the instabilities, in general we prefer to use a procedure that is more direct than the Routh–Hurwitz stability criterion described in Appendix A. A more straight-forward technique is to calculate the stability boundaries in the space of the parameters of the system, as described in Section 18.1 and Appendix B.

In Chapter 19 we discuss the prominent transient dynamical features in class-A and class-B lasers, as is usual in textbooks on laser physics, whereas in the following chapters of Part II we discuss temporal instabilities and their consequences. In Chapter 20 we outline a general scenario that encompasses and structures all temporal instabilities that arise in the framework of two-level systems in ring cavities.