All optical oscillators and amplifiers can be analyzed as an electromagnetic structure, such as a cavity or a transmission line, which contains an amplifying medium. The operating characteristics of lasers are governed by both the electromagnetic properties of the structures and the properties of the amplifying medium.

Stimulated emission and absorption of radiation from energy states are the physical basis of amplification in all laser materials. In order to study lasers, it is necessary to understand the energy levels of the amplifying medium and the stimulated transitions involving them. Quantum mechanical analysis of the energy states and the stimulated emission and absorption is presented in most books on lasers. However, quantum mechanical analyses cover only the analysis of individual atoms. We need to relate the effects of the quantum mechanical interactions to the macroscopic properties of the materials so that properties of the lasers may be analyzed. Therefore, the macroscopic susceptibility of materials related to stimulated emission and absorption is the focus of discussion in this chapter.

It is not the purpose of this book to teach quantum mechanics. There are already many excellent books on this topic. The readers are assumed to have a fundamental knowledge of quantum mechanics. However, it is necessary first to review some of the major steps in order to understand precisely the notation used and the meaning of the results. This is presented in Section 5.1.

It is well known that basic solid state and gas laser cavities consist of two end reflectors that have a certain transverse (or lateral) shape such as a flat surface or a part of a large sphere. The reflectors are separated longitudinally by distances varying from centimeters to meters. The size of the end reflectors is small compared with the separation distance. All cavities for gaseous and solid state lasers have slow lateral variations within a distance of a few wavelengths (such as the variation of refractive index and gain of the material and the variation of the shape of the reflector). Therefore these cavity modes are analyzed using the scalar wave equation. Laser cavities are also sometimes called Fabry–Perot cavities because of their similarity to Fabry–Perot interferometers. However, Fabry–Perot interferometers have distances of separation much smaller than the size of the end reflectors. The diffraction properties of the modes in Fabry–Perot interferometers are quite different from the properties of modes in laser cavities.

The analysis of the resonant modes is fundamental to the understanding of lasers. Modes of solid state and gas lasers are solutions of Eqs. (1.28a) and (1.28b), known as Gaussian modes. They are TEM modes. The analysis of laser modes and Gaussian beam optics constitutes a nice demonstration of the mathematical techniques presented in Chapter 1.

When I look back at my time as a graduate student, I realize that the most valuable knowledge that I acquired concerned fundamental concepts in physics and mathematics, quantum mechanics and electromagnetic theory, with specific emphasis on their use in electronic and electro-optical devices. Today, many students acquire such information as well as analytical techniques from studies and analysis of the laser and its light in devices, components and systems. When teaching a graduate course at the University of California San Diego on this topic, I emphasize the understanding of basic principles of the laser and the properties of its radiation.

In this book I present a unified approach to all lasers, including gas, solid state and semiconductor lasers, in terms of “classical” devices, with gain and material susceptibility derived from their quantum mechanical interactions. For example, the properties of laser oscillators are derived from optical feedback analysis of different cavities. Moreover, since applications of laser radiation often involve its well defined phase and amplitude, the analysis of such radiation in components and systems requires special care in optical procedures as well as microwave techniques. In order to demonstrate the applications of these fundamental principles, analytical techniques and specific examples are presented. I used the notes for my course because I was unable to find a textbook that provided such a compact approach, although many excellent books are already available which provide comprehensive treatments of quantum electronics, lasers and optics.

The general principles of amplification and oscillation in semiconductor lasers are the same as those in solid state and gas lasers, as discussed in Chapter 6. A negative χ″ is obtained in an active region via induced transitions of the electrons. When the gain per unit distance is larger than the propagation loss, laser amplification is obtained. In order to achieve laser oscillation, the active material is enclosed in a cavity. Laser oscillation begins when the gain exceeds the losses, including the output. However, the details are quite different. In this chapter, the discussion on semiconductor lasers will use much of the analyses already developed in Chapters 5 and 6; however, the differences will be emphasized.

In semiconductor lasers, free electrons and holes are the particles that undertake stimulated emission and absorption. How such free carriers are generated, transported and recombined has been discussed extensively in the literature. We note here, in particular, that free electrons and holes are in a periodic crystalline material. The energy levels of electrons and holes in such a material are distributed within conduction and valence bands. The distribution of energy states within each band depends on the specific semiconductor material and its confinement within a given structure. For example, it is different for a bulk material (a three-dimensional periodic structure) and for a quantum well (a two-dimensional periodic structure).

In order to understand optical fiber communication components and systems, we need to know how laser radiation functions in photonic devices. The operation of many important photonic devices is based on the interactions of several guided waves. We have already discussed the electromagnetic analysis of the individual modes in planar and channel waveguides in Chapter 3. From that discussion, it is clear that solving Maxwell's equations simultaneously for several modes or waveguides is too difficult. There are only approximate and numerical solutions. In this chapter, we will first learn special electromagnetic techniques for analyzing the interactions of guided waves. Based on these techniques, practical devices such as the grating filter, the directional coupler, the acousto-optical deflector, the Mach– Zehnder modulator and the multimode interference coupler will be discussed. The analysis techniques are very similar to those techniques used in microwaves, except we do not have metallic boundaries in optical waveguides, only open dielectric structures.

The special mathematical techniques to be presented here include the perturbation method, the coupled mode analysis and the super-mode analysis (see also ref.). In guided wave devices, the amplitude of radiation modes is usually negligible at any reasonable distance from the discontinuity. Thus, in these analyses, the radiation modes such as the substrate and air modes in waveguides and the cladding modes in fibers are neglected. They are important only when radiation loss must be accounted for in the vicinity of any dielectric discontinuity.

Experiments with single photons

Central to the entire discipline of quantum optics, as should be evident from the preceding chapters, is the concept of the photon. Yet it is perhaps worthwhile to pause and ask: what is the evidence for the existence of photons? Most of us first encounter the photon concept in the context of the photo-electric effect. As we showed in Chapter 5, the photo-electric effect is, in fact, used to indirectly detect the presence of photons – the photo-electrons being the entities counted. But it turns out that some aspects of the photo-electric effect can be explained without introducing the concept of the photon. In fact, one can go quite far with a semiclassical theory in which only the atoms are quantized with the field treated classically. But we hasten to say that, for a satisfactory explanation of all aspects of the photo-electric effect, the field must be quantized. As it happens, the other venerable “proof” of the existence of photons, the Compton effect, can also be explained without quantized fields.

In an attempt to obtain quantum effects with light, Taylor, in 1909, obtained interference fringes in an experiment with an extremely weak source of light. His source was a gas flame and the emitted light was attenuated by means of screens made of smoked glass.

Scope and aims of this book

Quantum optics is one of the liveliest fields in physics at present. While it has been a dominant research field for at least two decades, with much graduate activity, in the past few years it has started to impact the undergraduate curriculum. This book developed from courses we have taught to final year undergraduates and beginning graduate students at Imperial College London and City University of New York. There are plenty of good research monographs in this field, but we felt that there was a genuine need for a straightforward account for senior undergraduates and beginning postgraduates, which stresses basic concepts. This is a field which attracts the brightest students at present, in part because of the extraordinary progress in the field (e.g. the implementation of teleportation, quantum cryptography, Schrödinger cat states, Bell violations of local realism and the like). We hope that this book provides an accessible introduction to this exciting subject.

Our aim was to write an elementary book on the essentials of quantum optics directed to an audience of upper-level undergraduates, assumed to have suffered through a course in quantum mechanics, and for first-or second-year graduate students interested in eventually pursuing research in this area. The material we introduce is not simple, and will be a challenge for undergraduates and beginning graduate students, but we have tried to use the most straightforward approaches. Nevertheless, there are parts of the text that the reader will find more challenging than others.

“All information is physical”, the slogan advocated over many years by Rolf Landauer of IBM, has recently led to some remarkable changes in the way we view communications, computing and cryptography. By employing quantum physics, several objectives that were thought impossible in a classical world have now proven to be possible. Quantum communications links, for example, become impossible to eavesdrop without detection. Quantum computers (were they to be realized) could turn some algorithms that are labelled “difficult” for a classical machine, no matter how powerful, into ones that become “simple”. The details of what constitutes “difficult” and what “easy” are the subject of mathematical complexity theory, but an example here will illustrate the point and the impact that quantum information processors will have on all of us. The security of many forms of encryption is predicated on the difficulty of factoring large numbers. Finding the factors of a 1024-digit number would take longer than the age of the universe on a computer designed according to the laws of classical physics, and yet can be done in the blink of an eye on a quantum computer were it to have a comparable clock speed. But only if we can build one, and that's the challenge! No one has yet realized a quantum register of the necessary size, or quantum gates with the prerequisite accuracy. Yet it is worth the chase, as a quantum computer with a modest-sized register could out-perform any classical machine.

Over the past three decades or so, experiments of the type called Gedanken have become real. Recall the Schrödinger quote from Chapter 8: “… we never experiment with just one electron or atom or (small) molecule.” This is no longer true. We can do experiments involving single atoms or molecules and even on single photons, and thus it becomes possible to demonstrate that the “ridiculous consequences” alluded to by Schrödinger are, in fact, quite real. We have already discussed some examples of single-photon experiments in Chapter 6, and in Chapter 10 we shall discuss experiments performed with single atoms and single trapped ions. In the present chapter, we shall elaborate further on experimental tests of fundamentals of quantum mechanics involving a small number of photons. By fundamental tests we mean tests of quantum mechanics against the predictions of local realistic theories (i.e. hidden variable theories). Specifically, we discuss optical experiments demonstrating violations of Bell's inequalities, violations originally discussed by Bell in the context of two spin-one-half particles. Such violations, if observed experimentally, falsify local realistic hidden-variable theories. Locality refers to the notion, familiar in classical physics, that there cannot be a causal relationship between events with space-like separations. That is, the events cannot be connected by any signal moving at, or less than, the speed of light; i.e. the events are outside the light-cone. But in quantum mechanics, it appears that nonlocal effects, effects seemly violating the classical notion of locality in a certain restricted sense, are possible.

We have previously emphasized the fact that all states of light are quantum mechanical and are thus nonclassical, deriving some quantum features from the discreteness of the photons. Of course, in practice, the nonclassical features of light are difficult to observe. (We shall use “quantum mechanical” and “nonclassical” more or less interchangeably here.) Already we have discussed what must certainly be the most nonclassical of all nonclassical states of light – the single-photon state. Yet, as we shall see, it is possible to have nonclassical states involving a very large number of photons. But we need a criterion for nonclassicality. Recall that in Chapter 5 we discussed such a criterion in terms of the quasi-probability distribution known as the P function, P(α). States for which P(α) is positive everywhere or no more singular than a delta function, are classical whereas those for which P(α) is negative or more singular than a delta function are nonclassical. We have shown, in fact, that P(α) for a coherent state is a delta function, and Hillery has shown that all other pure states of the field will have functions P(α) that are negative in some regions of phase space and are more singular than a delta. It is evident that the variety of possible nonclassical states of the field is quite large.