In Section 1.4 we have introduced the concept of gauge transformation for quantum electrodynamics and we have shown that Maxwell's equations are invariant under a combined transformation of the scalar and vector potentials. In Section 3.5 we have seen that the principle of gauge invariance can lead to a Lagrangian for the interacting Schrödinger and electromagnetic field. This possibility, as we will discuss in detail in this appendix, is not limited to nonrelativistic QED, but it applies also to relativistic QED as well as to other fields such as those of electroweak interactions and of quantum chromodynamics. The importance of gauge invariance stems from the fact that field theories that can be obtained by a gauge principle are “renormalizable”, in the sense that all ultraviolet divergences can be removed at all orders of perturbation theory by introducing a finite number of renormalization constants. An extensive discussion of this point would lead us beyond the scope of this book; consequently, in this appendix we will only show how the Lagrangian of relativistic QED can be derived by a gauge principle, and we will extend this to the case of quantum chromodynamics (QCD), where it leads to new and unexpected features.

The Lagrangian of the free Dirac field is given in (3.59).

Introduction. The second part of the book is dedicated to the dressed atom, and it begins with this chapter, which deals mainly with the quantum-dynamical description of an atom dressed by a real electro-magnetic field. Here the emphasis is on the adjective ‘real’, by which we mean that the field is in an excited state populated by real photons and not just by the zero-point photon background. Due to the coupling discussed in the first part of the book, atom-photon correlations are established which admix, shift and split the levels of the system atom plus radiation field. The admixed and correlated states are called dressed-atom states. In Section 5.2 we obtain the Hamiltonian for an atom in a cavity with perfectly reflecting walls. The cavity selects a discrete set of field modes, and this leads us naturally to consider the simplest possible nontrivial atom-field system: the Jaynes-Cummings model describing a two-level atom coupled to a single-mode system. In Section 5.3 we develop the theory, based on a unitary transformation, to dress a two-level atom by a mode of the cavity populated by real photons. A necessary preliminary for dealing with more complicated atom-field models is the theory of spontaneous emission in free space, which is discussed in Section 5.4 in a Wigner-Weisskopf framework.

Introduction. In the previous chapter we have obtained the form of the coupling between matter and the electromagnetic field. We are thus in a position to treat electrodynamics in the presence of charges and currents. We start by obtaining the Euler-Lagrange equations of the matter-electromagnetic field system. These turn out to be the Maxwell-Lorentz equations, which is a set of coupled equations of motion in which the matter field acts as a source for the electromagnetic field and vice versa. In Section 4.2 we derive the Hamiltonian of the complete system in the Coulomb gauge, and this leads to the so-called minimal coupling Hamiltonian, containing both the electromagnetic potential and the matter-field amplitude. Specializing the nonrelativistic matter field to the case of a neutral atom, consisting of the field of electrons in a static nuclear potential, in Section 4.3 we obtain the atom-photon Hamiltonian in the minimal coupling scheme. Some of the basic processes induced by the atom-photon interaction part of the Hamiltonian are also discussed in this section. This gives us the possibility of introducing at this stage a fundamental simplification of the interaction Hamiltonian, namely the electric dipole approximation. The minimal coupling scheme, however, is not the only possible atom-field coupling.

Introduction

Optics is the ideal subject for lecture demonstrations. Not only is the output of an optical experiment usually visible (and today, with the aid of closed circuit television, can be made visible to large audiences), but often the type of idea which is being put across can be made clear pictorially, without measurement and analysis being required. Recently, several institutes have cashed in on this, and offer for sale video films of optical experiments carried out under ideal conditions, done with equipment considerably better than that available to the average lecturer. Although such films have some place in the lecture room, we firmly believe that the student learns far more from seeing real experiments carried out by a live lecturer, with whom he can interact personally, and from whom he can sense the difficulty and limitations of what may otherwise seem to be trivial experiments. Even the lecturer's failure in a demonstration, followed by advice and help from his audience which result in ultimate success, is bound to imprint on the student's memory far more than any video film can do.

The purpose of this appendix is to transmit a few ideas which we have, ourselves, found particularly valuable in demostrating the material covered in this book, and can be prepared with relatively cheap and easily-available equipment. Many other ideas are given by Taylor (1988). Need we say that we also enjoyed developing and performing these experiments?

Introduction

In Chapter 8 we discussed the theory of Fraunhofer diffraction and interference emphasizing in particular the relevance of Fourier transforms. In this chapter we shall describe the applications of interference to measurement; this is called interferometry. Some of the most accurate dimensional measurements are made by interferometric means, particularly using waves of different types – electromagnetic, matter, acoustic etc. The variety of techniques is enormous, and we shall limit ourselves in this chapter to a discussion of several distinctly different interferometric principles, without any intention of describing the variety of instruments or methods within the classes. There are several monographs on interferometry which discuss practical aspects in greater detail, for example Tolansky (1973), Steel (1983) and Hariharan (1985).

The discovery of interference effects by Young (§1.2.4) enabled him to make the first interferometric measurement, a determination of the wavelength of light. Even this primitive system, a pair of slits illuminated by a common point source, can be surprisingly accurate, as we shall see in §9.1.1. In general interference is possible between waves of any non-zero degree of mutual coherence (§11.4), including different sources (light beats), but for the purposes of this chapter we shall simply assume that waves are either completely coherent (in which case they can interfere) or incoherent (in which case no interference effects occur between them). In the case of complete coherence, there is a fixed phase relationship between the waves, and interference effects are observed that are stationary in time, and can therefore be observed with primitive instruments such as the eye or photography.

Introduction

The coherence of a wave describes the accuracy with which it can be represented by a pure sine wave. So far we have discussed optical effects in terms of waves whose wave-vector k and frequency ω can be exactly defined; in this chapter we intend to investigate the way in which uncertainties and small fluctuations in k and ω can affect the observations in optical experiments. Waves that appear to be pure sine waves only if they are observed in a limited space or for a limited period of time are called partially coherent waves, and a considerable part of this chapter will be devoted to developing measures of the deviations of such impure waves from their pure counterparts. These measures of the coherence properties of the waves are functions of both time and space, but in the interests of clarity we shall consider them as functions of each variable independently. Fig. 11.1 illustrates, in a very primitive manner, one wave which is partially coherent in time (it appears to be a perfect sine wave only when observed for a limited time) and a second wave which is partially coherent in space (it appears to be a sinusoidal plane-wave only if observed over a limited region of its wavefront).

The understanding of the coherence properties of light has had numerous practical consequences. Amongst these are the technique of Fourier-transform spectroscopy and several methods of making astronomical measurements with high resolution.

We were most encouraged by the publisher's request to revise Optical Physics for a third edition. The request involved considerably more work than we had anticipated; on the one hand we were specifically asked to enlarge our coverage of geometrical optics, fibre optics and quantum optics, and on the other hand not to increase the total length, which obviously necessitated rewriting much of the rest of the book! The requests for the two last topics in particular were very welcome. Both fibre optics and quantum optics have taken great strides forward in the last decade, and a basic understanding of them is essential for any student of physics, not only for a specialist. Since we are not mathematicians, we hope that the approach used for these subjects – an analogy with well-known elementary solved problems in quantum mechanics – will appeal to that section of our readership of a similar ilk. We have certainly learnt a lot in preparing both of these topics. We decided to present geometrical optics in a practical form, which we hope makes it attractive to today's students who have an easy familiarity with computers. We limited ourselves mainly to Gaussian optics, which is of most service to the physicist in general.

We have used the previous editions for more than a decade as the basis for two courses. One is an undergraduate course, which has as prerequisite a knowledge of high-school optics and a familiarity with elementary wave theory and quantum mechanics. This course covers most of Chapters 3, 4, 7, 8, 9, 11 and 12.

Introduction

Most optical systems are used for image formation. Apart from the pinhole camera, all image-forming optical instruments use lenses or mirrors whose properties, in terms of geometrical optics, have already been discussed in Chapter 3. But geometrical optics gives us no idea of any limitations of the capabilities of such instruments and indeed, until the work of Abbe in the middle of the nineteenth century, microscopists thought that the only limit to spatial resolution was their technical capability of grinding and polishing lenses. But (it now seems obvious) the basic scale is the wavelength of light, although recently several imaging methods have been devised which achieve resolution considerably in excess of this limit. The relationship is again like that between classical and quantum mechanics. Classical mechanics predicts no basic limitation to measurement accuracy; it arises in quantum mechanics in the form of the Heisenberg uncertainty principle.

This chapter describes the way in which wave optics are used to describe image formation by a single lens (and by extension, any optical system). The theory is based on Fraunhofer diffraction (Chapter 8) and leads naturally to an understanding of the limits to image quality and some of the ways of extending them.

The diffraction theory of image formation

In 1867 Abbe proposed a rather intuitive method of describing the image of a periodic object, which brought out clearly the limit to resolution and its relationship to the wavelength.

Since the writing of the first edition the subject of Optics, as studied in universities, has grown greatly both in popularity and scope, and both we and the publishers thought that the time had arrived for a new edition of Optical Physics.

In preparing the new edition we have made substantial changes in several directions. First, we have attempted to correct all the mistakes and misconceptions that have been pointed out to us during the nine years the book has been in use. Secondly, we have made one important change in the subject matter: we have absorbed the chapter on Quantum Optics into the rest of the book. During the years, there have appeared many books devoted to laser physics, and it now seems impracticable for a book on physical optics to cover the subject at all satisfactorily in one chapter. However, since some knowledge of the principles of the laser is necessary for the understanding of physical optics today, particularly when coherence is being discussed, we have covered what we feel to be the necessary minimum as parts of Chapters 7 and 8.

In addition to the above changes in the subject matter, we have increased the number of exercises offered to the reader, organized them according to chapter, and provided solutions. We have also included a few suggestions, based on our experience, for student projects illustrating the material in the book.

We are, of course, most grateful to all those who have pointed out to us errors and room for improvement.

Electromagnetism and the wave equation

This chapter will discuss the electromagnetic wave as a specific and most important example of the general treatment of wave propagation presented in Chapter 2. We shall start at the point where the elementary features of classical electricity and magnetism have been summarized in the form of Maxwell's equations, and the reader's familiarity of the steps leading to this formulation will be assumed (see, for example Grant and Phillips, 1975; Jackson, 1975). It is well-known that Maxwell's formulation included for the first time the displacement current δD/δt, the time-derivative of the ficticious displacement field D = ε0E + P, which is a combination of the applied electric field E and the electric polarization density P. This field will turn out to be of prime importance when we come to extend the treatment in this chapter to wave propagation in anisotropic media in Chapter 6.

The development presented in this chapter emphasizes the properties of simple harmonic waves in isotropic linear media, and the way in which waves behave when they meet the boundaries between media. An isotropic medium is one in which all directions in space are equivalent, and there is no difference between right-handed and left-handed rotation. An example would be a monatomic liquid; in contrast, crystals are generally anisotropic. A linear medium is one in which the polarization produced by an applied electric or magnetic field is proportional to that field.

Introduction

Optics is the study of wave propagation and its quantum implications. Traditionally, it has centred around visible light waves, but in the modern era the concepts which have developed over the years have been found increasingly useful when applied to many other types of wave, both within and without the electromagnetic spectrum. Wave propagation in a medium is described mathematically in terms of a wave equation; this is a differential equation relating the dynamics and statics of small displacements of the medium, and whose solution may be a propagating disturbance. This chapter will be concerned with such equations and their solutions.

The term ‘displacements of the medium’ is not, of course, restricted to mechanical displacement but can be taken to include any field quantity (continuous function of r and t) which can be used to measure a departure from equilibrium, and the equilibrium state itself may be nothing more than the vacuum.

Although it is convenient, from an elementary point of view, to study wave equations arising from the mechanical relationships between displacement and velocity, we quickly learn that almost any relationships between derivatives of a field in space and time can replace them. Then the distinction, which is clear in the mechanical sense, between ‘static’ and ‘dynamic’ properties may become blurred. For example, in the electromagnetic wave, the variables are electric and magnetic fields.

Electromagnetic waves in restricted systems

This chapter will deal with two examples of electromagnetic wave propagation in systems where the scalar-wave approximation is inadequate, essentially because of the small dimensions of the constituent parts. The first is the optical waveguide, already familiar in everyday life as the optical fibre, which has caused a revolution in the communications industry. The second example is the dielectric multilayer system which, in its simplest form (the quarter-wave anti-reflexion coating) has been with us for more than a century, and can today be used to make optical filters of any degree of complexity that are common elements in the laboratory.

Optical waveguides

Transmission of light along a rod of transparent material by means of repeated total internal reflexion at its walls must have been observed countless times before it was put to practical use. In this section we shall describe the geometrical and physical optical approaches to this phenomenon, and derive some of the basic results for planar and cylindrical guides, the latter of which is a model for the optical fibre. Optical fibres have many uses, two of which will be described briefly at the end of the section; the first is for transmitting images, either faithfully or in coded form, without the use of lenses; the second is for optical communication.

Geometrical theory of wave guiding

The principle of the optical fibre can be illustrated by a two-dimensional model (corresponding really to a wide strip rather than a fibre) shown in Fig. 10.1.