In this chapter we describe the case of a free electric field that does not interact with any material, so we start from Eq. (3.30) with the atomic polarization P0 set equal to zero. In addition, we focus on the stationary solutions, hence we drop also the term with the time derivative, and the transverse Laplacian plays the crucial role in determining the field configuration.

In Section 26.1 we show that the field equation admits stationary solutions in which the field has a Gaussian configuration with a beam radius (equal to the halfwidth of the Gaussian) that varies as a function of the longitudinal coordinate z, and that the variation is governed by a parameter, called the Rayleigh length, which characterizes the diffraction of the beam. The Gaussian solution corresponds to the fundamental mode of the radiation field, and the plane-wave solutions constitute a limit case of the Gaussian solutions obtained when the beam radius tends to infinity.

On the other hand, in Section 26.2 we derive the higher-order modes, in the form of Gauss-Hermite modes, given by the product of the fundamental Gaussian mode and Hermite polynomials, and Gauss–Laguerre modes, given by the product of the fundamental Gaussian mode and Laguerre polynomials. The Gauss–Hermite modes are appropriate to treat problems with square symmetry; the Gauss–Laguerre modes are appropriate to treat problems with axial symmetry. In both cases the modes depend on one independent parameter, the Rayleigh length, and in both cases the modes constitute an orthonormal and complete set of functions of the transverse variables x and y for any arbitrary value of the longitudinal variable z.

Sections 26.3–26.5 are devoted to the case of Gaussian modes in a cavity. To be precise, in Section 26.3 we focus on the case of a Fabry–Perot cavity with spherical mirrors and calculate the Rayleigh length which characterizes the set of cavity modes as a function of the parameters of the cavity, i.e. the radii of curvature of the spherical mirrors and the distance between the mirrors.

The relaxation rates κ, γ⊥ and γ∥ which appear in the single-mode model are irrelevant for the stationary state. Instead, their values become critical when we consider dynamical, i.e. time-dependent, phenomena. In particular, they control the possibility of eliminating adiabatically some of the variables in play, thus reducing the complexity of the problem. In this chapter we first illustrate, in Section 19.1, the useful laser classification introduced by Arecchi. Next, in Section 19.2, we discuss the dynamical behavior of the solutions of a model obtained from the single-mode model (derived in Chapter 12) by performing the adiabatic elimination of the atomic variables, which procedure is called the good-cavity limit in the literature. Section 19.2.1 is devoted to the case of the free-running laser (class-A laser) and discusses in particular, at the end, the cubic single-mode laser model which is the simplest model capable of describing a laser. On the other hand, in Section 19.2.2 we discuss the case of optical bistability in the good-cavity limit.

If we eliminate adiabatically the atomic polarization only, we obtain, instead, the single- mode rate equations which are discussed in Section 19.3. In particular, Section 19.3.1 concerns the case of class-B lasers and focusses especially on the phenomenon of relaxation oscillations. Special attention is paid to demonstrating the equivalence of the model which describes class-B lasers to the rate-equation laser model derived in Chapter 1 and the semiconductor laser model obtained in Chapter 16. In Section 19.3.2 we describe the generation of a giant pulse in class-B lasers, a phenomenon which is called active Q- switching.

The final section, Section 19.4, deals with the model obtained by adiabatically eliminating the field variable in the so-called bad-cavity limit.

Classification of lasers

Arecchi [243] identifies three especially important kinds of laser, which he calls class-A, class-B and class-C lasers, respectively.

(1) Class-A lasers are characterized by the condition

κ ≪ γ⊥, γ∥.

In this case the atomic variables can be adiabatically eliminated. The He–Ne laser, argon laser and quantum-cascade laser belong to this class.

In Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care.

In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that nB = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample.

By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.

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 ω0 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.

The field of optical pattern formation concerns the spatial and spatio-temporal phenomena that arise in the structure of the electromagnetic field in the planes orthogonal with respect to the direction of propagation. Most theoretical treatments of the interaction between matter and radiation introduce the plane-wave approximation, i.e. assume that the electric field is uniform in each transverse plane. In this way, the time-evolution equations depend only on one spatial variable, i.e. the longitudinal variable z which corresponds to the direction of propagation. By dropping the plane-wave approximation, one opens the door to the fascinating world of pattern formation. In the paraxial approximation this corresponds to keeping, in the time-evolution equation of the electric field, the term with the transverse Laplacian which describes diffraction of radiation; this term couples the different points of the transverse (x, y) plane, as is necessary for pattern formation. With the exception of the discussion of spatial Kerr solitons in Section 7.3, in the analysis of Maxwell-Bloch equations, or of other nonlinear optical models, in Parts I and II of this book we have always adopted the plane-wave approximation, whereas now in Part III we study the transverse effects.

The interaction of light with linear inhomogeneous media can give rise to spatial structures of interesting and remarkable complexity. However, the field of optical pattern formation concerns mainly the interaction with nonlinear media, where the phenomena emerge spontaneously as a consequence of an instability; another name that is commonly used to designate optical pattern formation is “transverse nonlinear optics” or “optical morphogenesis”. Historically, the interest in optical pattern formation emerged as a natural evolution of the previous development of the field of optical instabilities and chaos, when the main attention shifted gradually from purely temporal effects to spatio-temporal phenomena. The evolution was made possible also by the spectacular increase of the computational capabilities of available computers.

This chapter plays a special role in the economy of this book. In the treatment of nonlinear optical systems such as the laser, we have adopted an idealized theoretical description such as the two-level picture. For any kind of real laser the physics behind it is complex. In order to let the reader appreciate this point and evaluate to what extent the models used in this book can describe such a complex reality, we have chosen the case of one of the lasers which is most commonly utilized in applications, namely the semiconductor laser [153–163], for which the physics is especially rich and interesting. A detailed description of this type of laser is not simple, because a semiconductor material cannot be modeled as a collection of identical atoms with homogeneous broadening. Even if the interaction with the electric field involves electron–hole pairs, each of which can be assimilated to a two-level atom, the picture is complicated by various elements. Owing to the Pauli exclusion principle, there may exist at most two electrons, and two holes, having the same energy, with the factor of 2 arising from the spin degeneracy. As a consequence, different electron–hole pairs correspond to different transition frequencies, and the gain line is inhomogeneously broadened. In addition, the carriers (electrons and holes) interact with the crystalline lattice, and this circumstance leads to a complex band structure of the energy levels. Finally the carriers interact with one another via the Coulomb force, i.e. there is a many-body interaction.

We will focus on a simple model that takes the inhomogeneous broadening into account, considers the simplest possible structure for the bands and, among the effects of the many-body interaction, includes only, in a phenomenological way, the so-called band-gap renormalization, i.e. the progressive reduction of the transition frequency when the carrier number increases.

The aim of our book is to provide a unified and compact vision of the tree of nonlinear optical models and of the wealth of phenomena that can be described by them. In doing that, we adopt the viewpoint of the general field of nonlinear dynamical systems, even if we keep the treatment at a certain level of simplicity, performing an in-depth analysis but avoiding all of the technicalities which are not strictly necessary.

The discussion encompasses static aspects, temporal phenomena and spatial effects, including both those arising in the longitudinal direction in which the light beam propagates and those which occur in the transverse directions. The selected material gathers and organizes a wealth of knowledge scattered in a vast literature from the sixties of the past century to our days.

The volume is subdivided into three parts of decreasing extent. The first seventeen chapters derive from the fundamental laws which govern electromagnetic radiation and matter, a variety of models that describe the radiation-matter interaction both in free propagation and in optical cavities, and discuss mainly the stationary solutions of such models. Most space is devoted to two-level systems, but attention is paid also to parametric systems and to the effects of atomic coherence in multilevel systems. Part II (Chapters 18–25) illustrates the dynamical aspects of lasers and other amplifying or absorbing systems and, in particular, the onset of instabilities that lead to phenomena of spontaneous pulsations and chaos. Part III (Chapters 26–30) deals with the phenomena which arise in the transverse section of light beams, such as Gaussian modes, spontaneous spatial pattern formation and cavity solitons.

The book combines topics that are usually considered in courses on laser physics/quantum electronics and nonlinear optics. The natural attention to the standard laser is extended to other kinds of laser, such as lasers with saturable absorber or injected signal, to other light sources as the optical parametric oscillator and to passive systems that exhibit optical bistability.

The adiabatic elimination principle [91, 92] is universal in the area of nonlinear dissipative dynamical systems, and its importance arises from the fact that it allows one to reduce the number of the equations which govern the dynamics of the system. We introduce it at this stage in the book in order to be able to describe the optical pumping mechanisms which can be exerted to attain population inversion. However, this principle will be applied repeatedly in the following chapters for other purposes.

In the first subsection we describe the principle in general, while the following subsections concern the issue of optical pumping. We show first that it is not possible to obtain population inversion between the two levels of the lasing transition if the pump involves only these two levels. Next we illustrate the three-level pumping scheme and the four-level pumping scheme. We demonstrate that, by adiabatically eliminating the probabilities of the additional energy levels, one arrives at an equation identical to Eq. (4.12), which, for appropriate ranges of values of the parameters involved in the pumping, allows one to obtain population inversion.

General formulation of the principle

In a dissipative dynamical system the dynamical variables undergo relaxation processes associated with suitable rate constants such as, for example, the rate γ⊥ for the normalized atomic polarization P and the rate γ∥ for the normalized population difference D. The inverses of these rates provide the time scales which characterize the evolution of each variable. Therefore we can distinguish a set of fast variables, which evolve over short time scales and are therefore characterized by large relaxation rates, and a set of slow variables, which display small relaxation rates. A remarkable simplification in the analysis of dynamical systems is obtained by using the technique of adiabatic elimination of fast variables, which Haken considered as a basis for the discipline which he formulated and called Synergetics [91, 92].

Introduction

Unlike a conventional laser which utilizes mirrors or periodic structures to trap light, a random laser relies on the multiple scattering of light in a disordered gain medium for optical feedback and light confinement (see Figure 6.1) [67, 70, 528, 529]. Coherent laser emission has been generated from various random structures ranging from semiconductor nanoparticles and nanorods to polymers and organic materials. Over the past decade, random lasers have generated significant interest among researchers because of their unique applications [67, 529]. For example, specifically designed random lasers combine high radiance with low spatial coherence [357, 403], ideally suited for speckle-free, parallel, high-speed imaging, laser projection and ranging [404]. The sensitivity of the random lasing threshold to the amount of scattering has been explored for cancerous tissue mapping [387, 389]. Moreover, the spectral fingerprint of the random structure [70, 71], the micron size [70], the low fabrication cost, and the robustness against surface roughness and shape deformation point to potential applications in optical tagging and identification [528]. Since random laser cavities are formed by scattering and are much easier to fabricate compared with the mirror-based cavities, they may be applied to lasers at wavelengths where high-reflectivity mirrors are either not available or very expensive, e.g. UV laser, X-ray laser, γ-ray laser. A detailed description of random lasers can be found in Chapter 3.

However, a major limitation to device applications of random lasers is the lack of control and reproducibility of lasing modes. Namely, the frequencies and spatial locations of lasing modes are unpredictable, varying randomly from sample to sample. One way to solve this problem is to utilize pseudo-random structures, or more generally, deterministic aperiodic nanostructures (DANS) [98]. Unlike structures generated by a random process, DANS are generated by the iteration of simple mathematical rules with a controllable degree of spatial complexity that interpolates between periodic and random systems. DANS lasers can combine the advantages of random lasers with the fabrication and design reproducibility required for optoelectronics integration.