Introduction

List of topics investigated in this chapter

The topics covered in this chapter and summarized in Table 7 are similar to those covered in Chapter 6. Here, however, we deal with a double- rather than a singleinterface structure which consists of identical lossless dielectric (DPS-type) cover and substrate bounding a lossy metallic (ENG-type) guide, as shown in Fig. 7.1. Such a symmetric structure produces two solutions to the mode equation that are distinct from each other for a finite guide thickness. As the thickness of the guide increases, the two solutions converge to that of a single-interface DPS-ENG-type structure. The propagation constants of these modes are calculated for the freely propagating case and for the case where the modes are excited and loaded by a prism coupler. For each mode the electric and magnetic fields are evaluated, together with the local power flow, wave impedance and surface charge density at each guide interface. In Chapter 6 we dealt with a single-interface structure and used the Otto (O) or Kretschmann (K) configurations to excite a mode. Here, because we have a double-interface structure, we use the general prism coupling configuration (G). The reflectivity of an incident electromagnetic wave off the base of the prism, ℛ, is calculated for this configuration. The theory of double-interface surface plasmons was adapted from Refs. [1] to [4] and the concept of short-range (SR) and long-range (LR) SPs and recent reviews are from Refs. [5] to [11].

In 1952 Pines and Bohm discussed a quantized bulk plasma oscillation of electrons in a metallic solid to explain the energy losses of fast electrons passing through metal foils [1]. They called this excitation a “plasmon.” Today these excitations are often called “bulk plasmons” or “volume plasmons” to distinguish them from the topic of this book, namely surface plasmons. Although surface electromagnetic waves were first discussed by Zenneck and Sommerfeld [2, 3], Ritchie was the first person to use the term “surface plasmon” (SP) when in 1957 he extended the work of Pines and Bohm to include the interaction of the plasma oscillations with the surfaces of metal foils [4].

SPs are elementary excitations of solids that go by a variety of names in the technical literature. For simplicity in this book we shall always refer to them as SPs. However, the reader should be aware that the terms “surface plasmon polariton” (SPP) or alternately “plasmon surface polariton” (PSP) are used nearly as frequently as “surface plasmon” and have the advantage of emphasizing the connection of the electronic excitation in the solid to its associated electromagnetic field. SPs are also called “surface plasma waves” (SPWs), “surface plasma oscillations” (SPOs) and “surface electromagnetic waves” (SEWs) in the literature, and as in most other technical fields, the acronyms are used ubiquitously. Other terms related to SPs which we will discuss in the course of this book include “surface plasmon resonance” (SPR), “localized surface plasmons” (LSPs), “longrange surface plasmons” (LRSPs) and of course “short-range surface plasmons” (SRSPs).

Many of the physicists studying lasers in laboratories have been confronted by the appearance of erratic intensity fluctuations in the laser beam. This type of behavior was already evident in the early days of the laser (1960s) when it was found that the intensity of the light generated by the ruby laser displayed irregular spiking. Russian theoreticians showed that equations describing an active medium coupled to an electromagnetic field could display such pulsations. Laser physicists K. Shimoda and C.L. Tang tried to relate these outputs to saturable absorption and mode competition, respectively. But the discrepancy in the values for the instability frequencies, the fact that simple rate equations only predicted damped oscillations, and the development of stable lasers shifted interest towards new topics. About the same time, spontaneous instabilities were found to play key roles in fluid mechanics, chemistry, and the life sciences. Except for some isolated pioneers like L.W. Casperson, laser physicists only understood in the early 1980s that the pulsating outputs were not the result of environmental fluctuations but rather originated from the interaction between the radiation field and matter. On June 18–21, 1985, the University of Rochester organized the first International Meeting on “Instabilities and Dynamics of Lasers and Nonlinear Optical Systems”. Two special issues of the Journal of the Optical Society of America later appeared. But it took until the early 1990s before the idea became widely accepted among physicists that lasers exhibit the same type of bifurcations as oscillating mechanical, chemical, and biological systems.

In the study of the laser subject to an electrical feedback (see Section 4.1), we assumed that the response time of the feedback was instantaneous. The time to sense information and react to it was neglected because it was much smaller than any time scale of the CO2 laser. This is, however, not the case for semiconductor lasers (SLs) exhibiting a very short photon lifetime inside the cavity (∼ 10−12 s) and optical feedback response times 103 times larger.

In this chapter, we consider a variety of systems in which the dynamics are greatly affected by the response time of the feedback. We first concentrate on the so-called low frequency fluctuations or LFF observed with SLs, because they have been the topic of many investigations in the last 30 years. We first describe the LFF from an experimental point of view and then interpret the phenomenon in terms of numerical bifurcation diagrams. In the second part of this chapter, we show how optical feedback may also be used to improve the sensitivity of imaging systems. The last section is dedicated to optoelectronic feedback systems for which pulsating instabilities appear as a possible source of high frequency (microwave) electrical signals.

History

Optical feedback (OFB) cannot be fully avoided in experiments. Any optical element placed in front of a laser, such as a detector or even an antire flection coated lens, back-scatters part of the laser beam.

Optical Parametric Oscillators (OPOs) are based on multiwave interaction in a nonlinear medium. They have been realized in a variety of configurations, giving rise to an extended range of new dynamical problems. Like lasers, OPOs admit a steady state bifurcation at threshold and, in addition, they may exhibit bistability or Hopf bifurcations. Moreover, thermal effects may be dominant in cw oscillators leading to interesting slow–fast responses where the temperature is a new dynamical variable. Second-harmonic generation (SHG) is in a sense the inverse process of degenerate parametric amplification. Devices based on SHG are described by similar evolution equations but show different phenomena.

Parametric processes

An OPO is a light source similar to a laser, but based on optical gain from parametric amplification in a nonlinear crystal rather than from stimulated emission. Like a laser, such a device exhibits a threshold for the pump power, below which there is negligible output power. A main attraction of OPOs is that the signal and idler wavelengths, which are determined by a phase-matching condition, can be varied in wide ranges. We may thus access wavelengths (e.g. in the mid-infrared, far-infrared, or terahertz spectral region) which are difficult or impossible to obtain from any laser and we may also realize wide wavelength tunability. The downside is that any OPO requires a pump source with high intensity and relatively high spatial coherence. Therefore, we always need a laser as the pump source, generally a diode-pumped solid state laser.

What is common to the self-combustion of grain dust in a storage silo and the bursting activity of neurons? In both cases a quick time evolution follows a period of quiescence during which a key parameter is slowly varying. It can be the surrounding temperature in the case of the grain storage silo or the concentration of calcium ions that turns the neuronal activity on and off. These dramatic changes are possible because a slowly varying parameter passes a limit or bifurcation point of a fast dynamical system. But because of the system's inertia close to the bifurcation point, the expected jump or bifurcation transition is delayed. This delay has raised considerable interest not only for lasers but in other areas as well, such as fluid mechanics and chemistry. In mechanical engineering, slow passage problems are referred to as “nonstationary processes”. They occur in the start-up and shut-down of engines or in high-rise building elevators when the length of the rope is slowly changing. Although most delay effects are now well understood and illustrated by simple first or second order equations, the study of slow passage problems remains a fascinating topic of research for mathematicians, biologists, and students learning bifurcation theory in the laboratory.

Quantitative comparisons between experiments and theory for slow passage problems are always delicate. The evolution equations of a real physical system cannot be reduced to a simple equation if the rate of change is gradually increased, and we often need to take into account the effect of noise present in experiments.

In Chapter 1, we introduced the standard rate equations (SRE) for a laser containing two-level atoms between which stimulated emission is possible. But real lasers exhibit much more complicated energy level schemes. Throughout this chapter, we consider several models in which the light–matter interaction is described by population equations for all the levels involved in the laser operation. Although they are mathematically more complicated, we shall investigate these equations in the same way as in Chapter 1, i.e. by formulating dimensionless equations and by analyzing the stability properties of the steady states.

The basic ingredients of the SRE model used up to here, namely pumping and relaxation processes, play key roles in the efficiency of lasers. But already during the pioneering days of the laser, it appeared important to investigate three- and four-level models in order to obtain more reliable information on quantities such as the power conversion efficiency or the response time. The common extensions of the two-level SRE typically consider an open two-level system, or three- or four-level systems, depending on the nature of the active medium. Restriction to as few as four levels is again a crude simplification of the complex population dynamics occurring in most lasers. But it is rather surprising to see how good these simple kinetic models are. As we demonstrate by studying specific examples, the solution of the three- or four-level rate equations differs only slightly from the solution of the SRE.

There is a recurrent need for high power, frequency-stable lasers in applications as diverse as laser radar, remote sensing, gravitational wave interferometry, and nonlinear optics. This need is often satisfied by using a low power, frequency-stable laser followed by a chain of amplifiers, but a preferred approach is to injection-lock a high power (slave) laser to a lower power, frequency-stable (master) laser. Other advantages of the injection-locking technique are the possibility of ensuring single-mode operation, eliminating mode partition noise, mode hopping, preventing spurious feedback effects, and synchronizing one or more free-running lasers to the same pump. As explained in Section 3.4, the main benefits of optical injection occur when the frequencies of both lasers are close together and for sufficiently large injected power. The slave laser then gets the spectral properties of the master one in terms of frequency and linewidth. Stover and Steier did the first optical injection experiment in 1966 using gas lasers. The first optical injection experiment using semiconductor lasers (SLs) came much later and was done by Kobayashi and Kimura in 1980. At that time, it was not clear that SLs would be useful lasers, but the performance of these lasers has dramatically improved during the last 30 years, providing reliable devices for a large variety of applications. Optical injection is used to reduce noise (frequency noise, mode partition noise, or intensity noise, to generate microwave signals, or to produce chaotic outputs for secure communication.

In the previous chapter, we investigated the case of weakly modulated lasers. We found that a bistable response is possible if the modulation frequency is close to the relaxation oscillation (RO) frequency or to twice the RO frequency of the laser. In this section, we consider stronger modulation amplitudes, which is the case in most experimental studies. A strongly modulated laser may lead to chaos through successive period-doubling bifurcations as we shall see in this chapter.

In the late 1970s and early 1980s, there was a lot of excitement about “deterministic chaos” in all fields of physics, chemistry, and even biology. Deterministic refers to the idea that the future state of a system can be predicted using a mathematical model that does not include random or stochastic influences. Chaos refers to the idea that a system displays extreme sensitivity to initial conditions so that arbitrary small errors in measuring the initial state of the system grow exponentially large and hence practical, long term predictability of the future state of the system is lost. As far as optics was concerned, Kensuke Ikeda suggested in 1979 that an optical ring resonator containing a two-level medium and subject to a delayed feedback could exhibit chaos. His work triggered a lot of experimental research on optical chaos but, as explained in Section 4.2, it took several years before quantitative comparisons were possible. Following a quite different approach, Arecchi et al. modulated the losses of a CO2 laser at a frequency close to the RO frequency and obtained in 1982 a clear period-doubling cascade to chaos (see Figure 6.1).

Far-infrared (FIR) molecular lasers have a restricted domain of application because their technology in the 100 μm to 1 mm spectral range is not yet mature. This wavelength range is, however, unavoidable in radioastronomy because of the transparency windows of the Earth's atmosphere, and in semiconductor physics because of the energy domain of some lattice excitations. So far, applications of FIR lasers are limited. They have been used for checking high-voltage cable insulation and, more recently, for security-screening systems. On the other hand, FIR lasers are highly interesting for their instabilities and they have been studied in several laboratories.

The analogy found by Haken between the Lorenz equations and the laser (Maxwell–Bloch) equations for the homogeneously broadened laser triggered the search for an experimental laser system that could be well described by these equations. Haken's model of the laser is based on a semiclassical approach in which the electric polarization is explicitly considered, contrary to the standard rate equations where this variable is absent. By contrast to the laser rate equations, Haken–Lorenz equations admit sustained pulsating intensities and could be relevant for lasers that exhibit spontaneous pulsating instabilities. We have already discussed the complicated case of the ruby laser spiking. The 3.51 μm Xe laser self-pulsations were also known and investigated in detail but the mechanism responsible for this particular instability was partly masked by the difficulty in accounting for the inhomogeneous broadening, which is a dominant process in this laser.

Modeling lasers may be realized with different levels of sophistication. Rigorously it requires a full quantum treatment but many laser dynamical properties may be captured by semiclassical or even purely classical approaches. In this book we deliberately chose the simplest point of view, i.e. purely classical equations, and try to extract analytically as much information as possible. The basic framework of our approach is provided by the rate equations.

In their simplest version, they apply to an idealized active system consisting of only two energy levels coupled to a reservoir. They were introduced as soon as the laser was discovered to explain (regular or irregular, damped or undamped) intensity spikes commonly seen with solid state lasers (for a historical review, see the introduction in). These rate equations are discussed and sometimes derived from a semiclassical theory in textbooks on lasers. They capture the essential features of the response of a single-mode laser and they may be modified to account for specific effects such as the modulation of a parameter or optical feedback.

The most basic processes involved in laser operation are schematically represented in Figure 1.1. N1 and N2 denote the number of atoms in the ground and excited levels, respectively. The process of light–matter interaction is restricted to stimulated emission and absorption.

In a laser with a saturable absorber (LSA), two spatially separated cells are placed in the laser cavity as shown in Figure 8.1. The roles of the two cells are quite different: one of them is pumped so that the atoms have a positive population inversion (active or amplifying medium); the other one is left with a negative population inversion (passive or absorbing medium). As these two media are in general different, they saturate at different power levels. The most interesting case corresponds to the situation where the absorber saturates more easily than the active medium, introducing nonlinear losses inside the cavity. This new nonlinearity is responsible for two phenomena. An LSA may exhibit optical bistability, i.e. two distinct stable steady states may coexist for a range of values of a parameter. It may also produce pulsating intensity oscillations which have been called “passive Q-switching” (PQS) in contrast to “active Q-switching” experiments such as the “gain switching” experiments discussed in Section 1.3.2.

The interest in LSAs varied very much over time, with peaks in the late 1960s for their large intensity pulses, in the mid 1980s for their chaotic outputs, and in the late 1990s for the design of compact microlasers. Historically, physicists trying to explain the irregular intensity pulses delivered by the ruby laser suspected the possible destabilizing role of a saturable absorber. Shimoda proposed that the nonlinear losses generated by a saturable absorber could explain this phenomenon.