Class B lasers naturally exhibit damped relaxation oscillations and, as for any nonlinear oscillator, their responses to a time-periodic modulation of a parameter are rich and varied. The study of forced oscillators itself has a long history. Systematic studies started with Edward Appleton (1922) and Balthasar van der Pol (1927) who showed that the frequency of a triode generator can be entrained by a weak external signal with a slightly different frequency. These studies were of high practical importance because such generators became basic elements of radio communication systems. The next impact on the development of the theory of forced oscillators came from the Russian school when control engineering became an emerging discipline. Alexandr Aleksandrovich Andronov (1901–1952) was a key figure in the development of mathematical techniques for driven oscillators, yet his name, and his contributions to control theory and nonlinear dynamics, are much less well known in the West than they deserve to be. As we shall demonstrate later in this chapter, these analytical techniques are totally appropriate for our laser problems.

Today, lasers and fiber optic cables have replaced the electronic amplifying tubes and cables. Light signals are modulated with the information to be sent into fiber optic cables by lasers. Telephone fiber drivers may be solid state lasers the size of a grain of sand and consume a power of only half a milliwatt. Yet they can send 50 million pulses per second into an attached telephone fiber and encode over 600 simultaneous telephone conversations.

A Hopf bifurcation marks the transition from a steady state to a time-periodic solution. We already encountered an example of a Hopf bifurcation in Section 3.4 as we analyzed the laser subject to an injected signal.

The emergence of spontaneous time-dependent regimes in lasers is not a purely academic problem because physicists have been confronted by the appearance of “noise-like” intensity fluctuations in the laser's beam since the beginning of the laser. This type of behavior was evident even during the earlier investigations of the laser in the 1960s where it was found that the intensity of the light generated by the ruby laser displayed irregular spiking, as shown in Figure 4.1. Were these spikes the result of a noisy environment or were they coming from the laser itself? A lot of experiments have been undertaken on the ruby laser under various conditions (see). It eventually appeared that the oscillatory output of the ruby laser resulted from the combined effect of several mechanisms. Research on this topic vanished because of the advent of new lasers whose parameters are much better controlled and therefore capable of delivering cw power or pulses with well-defined and reproducible properties.

For many years, attempts to understand the appearance of such oscillatory instabilities in lasers were limited (for instance, the extensive but isolated effort of Lee W. Casperson to describe the pulsations of the Xe laser), until Hermann Haken showed the equivalence of the laser equations with the Lorenz system.

From the standpoint of a solid-state physicist, nanocrystals are nothing else but a kind of low-dimensional structure complementary to quantum wells (two-dimensional structures) and quantum wires (one-dimensional structures). However, nanocrystals have a number of specific features that are not inherent in the two- and one-dimensional structures. Quantum wells and quantum wires still possess a translational symmetry in two or one dimensions, and a statistically large number of electronic excitations can be created. In nanocrystals, the translational symmetry is totally broken and only a finite number of electrons and holes can be created within the same nanocrystal. Therefore, the concepts of the electron–hole gas and quasi-momentum fail in nanocrystals.

From the viewpoint of molecular physics, nanocrystals can be considered as a kind of large molecule. Similar to molecular ensembles, nanocrystals dispersed in a transparent host environment (liquid or solid) exhibit a variety of guest–host phenomena known for molecular structures. Moreover, every nanocrystal ensemble has inhomogeneously broadened absorption and emission spectra due to distribution of sizes, defect concentration, shape fluctuations, environmental inhomogeneities and other features. Therefore, the most efficient way to examine the properties of a single nanocrystal which are smeared by inhomogeneous broadening is to use selective techniques, including single nanoparticle luminescence spectroscopy. The present chapter summarizes the principal quantum confinement effects on absorption and emission of light, recombination dynamics and many-body phenomena in semiconductor nanocrystals, as well as primary application issues.

Since every non-periodic medium is in a sense a unique structure, it seems at first glance that there is no regularity to be traced with respect to light propagation in such a medium. However this statement is not correct. First, in non-periodic media with absence of any regularity, i.e. in random media, the definite laws of light propagation resulting from random scattering can still be evaluated. There are analogies with the length-dependent resistance (and conductivity) of a conductor, coherent backscattering and Anderson localization of light. Second, there are well-identified classes of aperiodic media featuring definite geometrical regularities. These are fractal media with self-similar geometry and quasi-periodic media. Certain regularities of light propagation through aperiodic media with well-defined geometrical algorithms have been discovered to date and are still an issue of current research. Finally, it will be shown there are certain conservation relations that are valid for all structures independent of their spatial organization. These issues will all be the subject of consideration in this chapter.

The 1/L transmission law: an optical analog to Ohm's law

Consider the propagation of light waves in a medium which contains randomly distributed scatterers. It is refractive index inhomogeneity that makes the light wave scatter. Scatterers may differ in shape, size and refractive index of material.

Transfer of concepts and ideas from quantum theory of solids to nanophotonics

In Chapter 3 we discussed that optics played an important role in the development of quantum mechanics at its very early stages. Wave mechanics with respect to classical mechanics has been developed by analogy to wave optics with respect to geometrical optics. A number of similarities were outlined in that chapter between quantum mechanical and electromagnetic phenomena. Many decades later the reverse process happened. The advances in single particle quantum theory of solids that dealt exclusively with analysis of the Schrödinger equation in complex potentials with no collective phenomena and spin effects included, were systematically transferred to electromagnetism, and first of all to wave optics. We have shown the bulk of these effects and phenomena in wave optics of complex structures in Chapters 7–9. The transfer of concepts and phenomena is presented in Table 12.1 with the principal dates indicated. This transfer is a remarkable event in modern science. It is indicative of the useful exchange of ideas between two large fields of physics. In a sense, quantum theory did pay back to optics with high “interest” for originally borrowing optical ideas in the 1920s. It is owing to this transfer that the writing of this very book has become topical.

Among the quantum phenomena listed in Table 12.1, the band theory of solids in terms of electron Bloch functions, conduction band and valence band concepts, Brillouin zones and electron and hole effective mass have been overviewed in detail in Chapter 4.

It is an extraordinary paradox of Nature that, being seemingly the only creatures capable of understanding its harmony, we naively attempt to chase its very essence through our daily experience based on mass-point mechanics and ray optics, while its elusive structure is mainly contained in wave phenomena. It may be nanophotonics where many pathways happily merge that promises not only mental satisfaction in our scientific quest but also an extra bonus in the form of new technologies and devices.

In this book I have tried to give a consistent description of the basic physical phenomena, principles, experimental advances and potential impact of light propagation, emission, absorption, and scattering in complex nanostructures. Introductory quantum theory of solids and quantum confinement effects are considered to give a parallel discussion of wave optics and wave mechanics of complex structures as well as to outline the beneficial result of combined electron wave and light wave confinements in a single device. Properties of metal nanostructures with unprecedented capability to concentrate light and enhance its emission and scattering are discussed in detail.

Keeping mathematics to a reasonable minimum and reducing theoretical issues to a conceptual level, the book is aimed at assisting diploma and senior students in physics, optical and electronic engineering and material science. The contents include a vast diversity of phenomena from guiding and localization of light in complex dielectrics to single molecule detection by surface enhanced spectroscopy.

The photonic crystal concept

Since the time when de Broglie published his hypothesis on the wave properties of matter particles in 1923, the wave mechanics of matter has become a well-developed field of science. It has provided an explanation for the properties of atoms, molecules and solids. Furthermore, it predicted novel properties of artificial solids like quantum wells and quantum wires. As we have seen in Chapters 2 and 3, there are many common features and phenomena in wave mechanics and wave optics. At the very dawn of wave mechanics, it essentially borrowed much from wave optics.

Nowadays, the reverse process manifests itself in science. Results of quantum mechanics which are direct consequences of the wave properties of electrons and other quantum particles are transferred to classical electromagnetism, and to wave optics. These are results that are not related directly to spin and charge. Such transfers have formed a new emerging field in modern optics of inhomogeneous media with the concept of a photonic crystal at the heart of the field.

This chapter provides a brief introduction to optical properties of metal nanoparticles in terms of plasma-based optical response, interband transitions and size-dependent properties. The chapter is important for understanding contemporary research in nanoplasmonics including surface-enhanced emission and scattering of light near metal surfaces and nanobodies which will be the subjects of Chapter 16. Amazingly, nanoplasmonics is actually an ancient field of science and technology in spite of the fact that the notation for this trend in science only emerged just a decade or so ago. The first systematic studies of brilliant colors of dispersed metal colloids date back to Michael Faraday (1857). Purposeful applications of optical properties of metal nanoparticles are well known for example, to get colors in stained glass, which dates back to ancient Roman times. Gold and copper nanoparticles have been used routinely for decades in the glass industry in red glass production.

Prior to going through this chapter, it is advisable to recall the description of the dielectric function of a gas of non-interacting charged particles (Section 3.3) and the introduction to the electron theory of solids given in Sections 4.1–4.3. For a comprehensive description of optical properties of metal nanoparticles the books by Kreibig and Vollmer and Maier are recommended.

In this chapter we shall see that electromagnetic waves and electrons feature a number of common properties under conditions of spatial confinement. Simple and familiar problems from introductory quantum mechanics and textbook wave optics are recalled in this chapter to emphasize the basic features of waves in spatially inhomogeneous media. Herewith we make a first step towards understanding the properties of electrons and electromagnetic waves in nanostructures and notice that these properties in many instances are counterparts. Different formulas and statements of this chapter can be found in handbooks on quantum mechanics and wave optics. A few textbooks on quantum mechanics do consider analogies of propagation and reflection phenomena in wave optics with those in wave mechanics.

Isomorphism of the Schrödinger and Helmholtz equations

In Chapter 2 we discussed that an electron in quantum mechanics is described by the wave function, the square of its absolute value giving the probability of finding an electron at a specific point in space. This function satisfies the Schrödinger equation (2.56) which is the second-order differential equation with respect to space and the first-order differential equation with respect to time.