A rigorous description of the light–matter interaction requires the use of quantized Maxwell equations to describe the light field and the Schrödinger equation to describe the material medium. This book does not attempt to deal with quantum optical problems that necessitate a field quantization. Instead we focus on properties that are generally grouped under the umbrella of nonlinear optics. Although many definitions of this expression exist, we use it in the sense that the properties we deal with do not depend in a crucial way on the field quantization. For practical purposes, this means in general that the average photon number is large and that we can neglect spontaneous emission as a dynamical process. Rather, we include the consequence of spontaneous emission, that is, the instability of the atomic levels, in a purely phenomenological way. When this is done, we call the material equations the Bloch equations instead of the Schrödinger equation. References [1] through [4] are the classical textbooks that contain a tentative justification for the transition from the Schrödinger to the Bloch equations. None of these attempts is satisfactory from a fundamental viewpoint because the real difficulty is to incorporate the finite lifetime (or natural linewidth) of the atomic energy levels in the Schrödinger equation. This remains an open problem as of now.

The Maxwell–Schrödinger equations

We consider atoms interacting with an intense electromagnetic field. By intense we mean a field for which the quantization is not necessary.

Laser theory has attracted a large number of studies centered on the stability problem [1]–[4]. There are at least three motivations for these studies. First, the laser equations are rather simple equations that can be derived from first principles with a minimal admixture of phenomenology. The complexity of their solutions was not fully appreciated until Haken showed the equivalence between the three single-mode laser equations on resonance [equations (1.58)–(1.60) with Δ = 0] and the Lorenz equations derived in hydrodynamics [5]. However, the domains of parameters that are relevant for optics and hydrodynamics are not the same. New asymptotic studies were suggested for the laser equations. Second, infrared lasers have been built that can be modeled quite accurately by the two-level equations, at least in some domains of parameters. A good analysis of this topic is found in [6]. A marked advantage of optics over hydrodynamics is that in general the time scales are much shorter. Hence experimental data can be accumulated, and averaging procedures can be used to separate the effect of noise from deterministic properties. Third, the laser stability becomes an essential question when the laser is used as a tool in scientific or industrial applications.

Stability means that a perturbation applied to the laser decreases in time. This requires the solution of an initial value problem that in general is too difficult to be solved analytically.

Introduction

Usage has reserved the expression optical bistability mostly for coherently driven passive systems. An atomic system is called passive when there is no population inversion. This is the case, for example, of a system at thermal equilibrium. On the contrary, the laser that we have described in Chapter 1 is a driven active system, because a population inversion is created. The laser equations we have studied so far describe an incoherently driven laser. However, nothing prevents driving a lasing cavity with a coherent field emitted by another laser. This is realized in a whole class of lasers that are used mainly as frequency converters. In optical bistability (OB), the driving field is usually a coherent field. Thus we have to account for two differences between the laser and the optically bistable system: (1) In OB, there is no inversion of population: 〈|A2|〉>〈|B2|〉, in the notation of Chapter 1. Hence, in the absence of interaction with a field, the population difference D relaxes toward a negative value. (2) The pumping is coherent, meaning that an external laser field is added to the cavity field.

Using the single-mode equations (1.48)–(1.50), we have to change the sign of D and Da (which amounts to keeping D defined in (1.34) as it is but changing P into –P and A into –A) and to add a source term in the equation for the complex field amplitude.

Nonlinear optics is a fairly young science, having taken off with the advent of the laser in 1960. Nonlinear optics (NLO) deals with the interaction of electromagnetic waves and matter in the infrared, visible, and ultraviolet domains. The frontiers of NLO are somewhat blurred, but microwaves and γ-rays are clearly outside its domain. This book is entirely devoted to a study of NLO in a resonant cavity and when the quantum nature of the electromagnetic field is not of prime importance. More precisely, we study those aspects of cavity NLO in which fluctuations in the number of photons and atoms are not relevant. This particular area of optics is dominated by the Maxwell–Bloch equations, which constitute its paradigm in the sense of T. S. Kuhn. The status of the Maxwell–Bloch equations is quite peculiar. From a fundamental viewpoint, they describe the laws of evolution of the first moments of a density operator, which verifies the von Neumann equation. However, to account for the finite lifetime of the atoms and of the field in the necessarily lossy cavity, some legerdemains have to be introduced to obtain the Maxwell–Bloch equations. Stated more explicitly, the von Neumann equation for a large but finite system does not explain irreversibility, whereas the Maxwell–Bloch equations fully include the irreversible decay of the atoms and of the cavity field. This problem is not specifically related to optics but reflects the general failure of statistical mechanics to explain convincingly the irreversible evolution of macroscopic systems.

Introduction

Up to now, we have described in detail many properties of steady bifurcations and limit points, that is, critical points where a stable steady state solution loses its stability and coincides with another steady state solution. At a few places, we have also met the so-called Hopf bifurcation where a steady solution loses its stability and a time-periodic solution emerges. However, we have not yet studied in any detail a Hopf bifurcation for lack of a suitable example. Even the simple-looking trio of laser equations on resonance [equations (1.58)–(1.60) with Δ = 0, E and P real] yield such complex expressions that it is hard to separate conceptual difficulties from mere computational problems. In this chapter, we make an intrusion upon a domain that has not yet been considered in this book. The motivation is both to cover an important topic of cavity nonlinear optics and to provide a pedagogical example of a Hopf bifurcation.

In the preceding chapters, this book has dealt exclusively with processes in which only one photon is either absorbed or emitted. Other phenomena, however, rely on multiphoton transitions [1, 2]. In the original Bohr formulation of atomic transitions and in much of the ensuing quantum mechanical formulation, resonance conditions on atomic transitions express conservation laws but give no constraint on the number of photons needed to achieve the transition.

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].

Introduction

Until now, we have always assumed that the cavity losses are linear, that is, field-independent. This need not always be true, and in this chapter we investigate how nonlinear losses affect the operation of a nonlinear optical device. We consider two examples. The first one is the laser with a saturable absorber, the second is parametric amplification in the presence of a saturable absorber.

The modeling of lasers with a saturable absorber (LSA) has a history that is practically as long as that of the laser. Problems related to the LSA are still a subject of debate. An LSA is a laser that contains both an active (or amplifying) medium and a passive (or absorbing) medium. For instance, if population excitation is produced in the laser cavity but inversion is not achieved in the whole cavity, some domains of the laser amplify the radiation and others absorb it. The salient feature of this situation is that both amplification and absorption result from the resonant interaction of light with atoms. Hence, both processes contribute to the nonlinear response. One way to look at an LSA is to consider it as a generalization of a laser in which the losses are as nonlinear (i.e., intensity-dependent) as the gain. From the viewpoint of dynamical systems, the LSA is a prototype of competition between nonlinear gain and nonlinear losses. As a result, there has been over the years an irresistible temptation to attribute to saturable absorption properties that are not explained by the standard laser equations, derived in Chapter 1.

In Chapters 1 to 5, we have dealt with single-mode ring cavities, either for lasers or for optical bistability. In this chapter, we come back to laser theory to consider the properties of multimode cavities. This subject is immense and our goal can only be modest.

The single-mode unidirectional ring laser is the model of choice for theoreticians who want to study fundamental aspects of laser theory. The simplicity of its evolution equations, equations (1.58)–(1.60), makes the model attractive. Its equivalence with the Lorenz equations [1], which have become the generic model to study chaos in ordinary differential equations, increases the relevance of the ring laser model. Of importance is the fact that the laser model lends itself quite naturally to a complexification of the variables. It suffices that the detuning be nonzero to have a coupling between the phase and the amplitude of the electric field and of the atomic polarization. This opens the door to an even richer phenomenology of complex behaviors.

The ring configuration for a laser is not simply an idealization intended for theoreticians. A number of lasers operate in this configuration. Dye lasers and some coherently pumped lasers are built with ring cavities. Laser gyroscopes are essentially ring lasers. If the ring cavity is perfectly symmetric with respect to the two directions of propagation, there is no preferential direction of oscillation and both directions must be taken into account. This is the simplest example of a multimode laser and we analyze some of its properties in Section 9.3.

Introduction

At the end of the previous chapter, we have seen that a characteristic of nonlinear systems is the presence of critical points, that is, values of the control parameter at which two solutions coincide. Note that at a critical point, more than two solutions can coexist and that the coexisting solutions need not be stationary: They can have any time dependence. In the next four chapters, though, we concentrate on the steady critical points at which two steady solutions coincide. If the solutions exist on both sides of the critical point, we call it a bifurcation point. Another type of critical point that will also draw much of our attention later is the limit point. However, the solutions exist on only one side of the critical point.

The main feature of critical points is that their presence is always signaled by the vanishing of the real part of at least one characteristic root in the stability analysis. For steady critical points, it is a real root that vanishes. In some cases, however, more than one root will vanish at criticality. Then one deals with degenerate critical points that may have richer properties. Physically, the absolute value of the real part of a characteristic root λ is a relaxation rate if Re(λ) < 0 and a divergence rate if Re(λ) > 0. Let λc be the root that vanishes at the critical point. The vanishing of a relaxation rate at criticality means the divergence of a relaxation time. Thus, an unconventional dynamics occurs in the domain surrounding the critical point that is characterized by critical slowing down.

LASER SPUTTERING

Direct removal of semiconductor material by laser light without a reactive intermediary has been reported for a number of solids (see reviews by Bäuerle 1986 and Ashby 1991). Direct ablation of small quantities of semiconductor material using UV laser radiation has specific applications in link breaking for circuit restructuring (Smith et al. 1981, Raffel et al. 1985). However, the majority of ablation studies have been carried out in a reactive atmosphere or medium.

A summary of work carried out on the direct ablation or photosublimation of semiconductor materials is given in Table 8.1. In general ablative effects are limited to excitation with high intensity pulses whereas photosublimation occurs by exposure of a semiconductor to CW, visible laser radiation at somewhat lower intensity. Figure 8.1 shows the intensity threshold for ablation in several semiconductors.

Early experiments on the ablation of Si at 193 and 248 nm (Shinn et al. 1986) showed that neither the ablation rate nor the ablation threshold fluence were strongly dependent on laser wavelength, for excitation in this wavelength range. Above the threshold fluence the etching rate was found to be 0.2–0.5 μm per pulse at intensities of 107–108 W cm–2. The threshold fluence was about 1.3Jcm–2 both at 193 and at 248 nm and was found to be independent of ambient gas pressure for pressures between 0–1000 Torr. A strong plasma emission showing a wide variety of Si, Si+ and Si2+ spectral lines was observed at the highest intensities used (about 108 W cm–2).

The development of lasers operating at ultraviolet wavelengths has provided mankind with a new set of unique tools. With characteristics which combine the precision to remove micrometer-thick layers of corneal tissue for the correction of refractive errors in the human eye and the ability to vaporize even the most refractory of materials, UV lasers have immediately developed into indispensable tools in many areas of materials science. The remarkable ability of high power pulsed excimer laser radiation to vaporize complex materials such as high temperature superconductors, while maintaining stoichiometry in thin films deposited from this vaporized material, offers many exciting opportunities in the creation of superconducting thin films and thin film devices. Similar unique capabilities are available in the deposition, doping and modification of semiconductors using UV laser radiation.

As a result of these and other applications, many of which can be immediately adopted by industry, UV lasers have a secure future in the field of materials science. Their implementation is limited only by our creativity in finding new applications and ways to use these new tools.

A fascinating aspect of the development of these applications involves the many fundamental questions that arise concerning the manner in which intense UV laser radiation interacts with matter. This is an area of great scientific interest and is truly interdisciplinary in nature so that answers to these questions will only come from both theoretical and experimental studies extending over a diverse range of disciplines.

INTRODUCTION

The response of inorganic insulating materials to intense UV radiative fluxes is complex and involves both photophysical and photochemical interactions. The first effect, noticeable at low exposure, involves the production of defect centers or radiative interactions with existing centers or impurities. Such changes often result in an increase in optical absorption at the laser wavelength as well as at other wavelengths. This can have a profound effect on the quality of transmissive optical components such as windows and lenses. With optical fibers, defect formation limits the fluence that can be transmitted.

Higher exposure to radiative fluxes results in changes in composition and density as the sputtering threshold is approached. Defects have been found to play a significant rôle in the initiation of ablation under irradiation with photons of energies less than that of the optical bandgap. The generation of electron–hole pairs via two-photon absorption is also an important process at high laser intensities and would appear to be the initiator of ablative decomposition of transparent insulators.

This chapter begins with a review of the formation and properties of dominant defects in several wide bandgap insulators. The relation between these defects and the coupling of UV laser radiation leading to ablation is then discussed. The chapter concludes with a summary of dry and laser-assisted etching processes and rates in a variety of insulating solids.

INTRODUCTION

Ultraviolet laser sources can initiate both photochemical and photothermal effects in condensed media. The relative importance of these two effects depends on a variety of factors including laser wavelength, pulse duration, intensity and the photochemical/photothermal response of the irradiated material. In addition, exposure to UV laser radiation can result in radiation conditioning or hardening, such that the response of the medium to subsequent irradiation may be quite different from its initial response.

This chapter explores some of the fundamental limitations of materials processing with lasers as they relate to the physical and chemical response of the irradiated medium. Some general constraints on the relative rate of ablation in photochemical and photothermal regimes are also discussed. The question of radiation resistance is shown to exhibit both geometrical and physico-chemical characteristics.

FUNDAMENTAL LIMITATIONS IN LASER MATERIALS PROCESSING

At the intensities customarily used in laser processing of materials, the irradiated sample is exposed to an intense radiative environment that is generally far from the equilibrium state of the ambient medium. The thermal or physical change in the irradiated medium is then driven by an attempt to approach a new equilibrium in the applied radiation field. In general, even at intensities that may be as large as 108W cm–2, the response function of an irradiated medium is usually described using classical heat transfer theory. There are, however, implicit limitations to the validity of this theory as well as assumptions implied by the adoption of this description of the thermal response that may be relevant at high incident laser intensities or short pulse durations (Harrington 1967, Duley 1976).