Introduction. The principal aim of this chapter is to familiarize the reader with the notation adopted in the text, as well as to introduce some concepts, such as the energy-momentum tensor of the electromagnetic field, the partition of its total angular momentum into an orbital and a spin contribution and its expansion in vector spherical harmonics, which are not usually included in an undergraduate course on electrodynamics. The chapter is entirely dedicated to the classical electromagnetic field in the absence of charges and currents. In the first two sections we present Maxwell's equations, the vector potential and different forms of the Lagrangian density of the free field from which Maxwell's equations can be obtained as Euler-Lagrange equations. In Section 1.3 we discuss briefly the properties of the field under pure Lorentz transformation and tensor notation. Then we introduce the concept of local gauge invariance and of gauge transformation, and we define the constraints leading to the Lorentz and to the Coulomb gauge. Using a canonical formalism, in Section 1.5 we obtain the Hamiltonian density of the field in the Coulomb gauge. The energy-momentum tensor of the field, the momentum and the angular momentum, along with their important conservation properties, are discussed in Section 1.6.

Introduction. The purpose of Chapter 6 is to discuss from a general point of view the dressing of a source by the vacuum fluctuations of the field coupled to the source. In Section 6.1 we show that in quantum optics, as well as in different branches of physics, virtual quanta of the field are present in the ground state of the source-field system. Three examples are considered: a two-level atom coupled to the vacuum electromagnetic field, a static model of a nucleon coupled to the vacuum meson field and an electron coupled to the optical phonon modes of a semiconductor (Fröhlich polaron). Section 6.2 is dedicated to a qualitative discussion of the physical nature of these virtual quanta and of their spatial distribution around the source. The dressed source is then defined as the bare source together with the virtual quanta surrounding it. This virtual cloud is shown is Section 6.3 to lead to a change of the energy levels of a nonrelativistic free electron interacting with the vacuum electromagnetic field. This kind of self-energy effect can be represented by a mass renormalization of the free electron. Self-energy effects due to the virtual cloud are discussed in Section 6.4 for each of the three examples of dressed sources considered in Section 6.1.

Introduction. The main theme of this chapter is the explicit calculation of the shape of the virtual cloud surrounding different kinds of ground-state sources. In Section 7.1 two of the three examples considered in Section 6.1 are taken up again, and it is argued that a convenient description of the shape of the virtual cloud is given by the energy density of the field around the source. This energy density is evaluated in detail for the static source of mesons and for the Fröhlich polaron. Section 7.2 is dedicated to an analogous calculation of the electric energy density around a two-level atom within a perturbation scheme. The virtual cloud around a two-level atom is again the subject of Section 7.3, where we evaluate the magnetic energy density as well as the coarse-grained energy density. From the results of the first three sections we conclude that the qualitative picture of the virtual cloud proposed in Section 6.2 is well founded. Moreover, the discussion of the two-level atom leads us to an important conclusion: the space surrounding an atom can be divided into a near zone and a far zone, where the energy density of the virtual cloud behaves differently as a function of the distance from the atom. In Section 7.4 we evaluate the energy density of the virtual cloud surrounding a slow free electron, separating classical and quantum contributions.

Introduction. In this final chapter we review two topics of the literature concerning dressed atoms which are of conceptual relevance and capable of shedding some light on the physical meaning and significance of atomic dressing. The first section is devoted to some recent work in connection with the quantum theory of measurement. We include the measuring apparatus in the Hamiltonian along with an appropriate apparatus-atom coupling. We argue that the theory of measurement of finite duration provides us with a tool for detecting the spectral composition of the virtual cloud surrounding an atom. In fact, we show that in a measurement of duration T on a two-level atom fully dressed by the vacuum fluctuations, as discussed in Chapters 6 and 7, the apparatus perceives the atom as dressed only by photons of frequency larger than T-1. In the case of a two-level atom dressed by a single-mode field populated by real photons, discussed in Chapter 5, we show that if T is smaller than the inverse Rabi frequency ħΔ-1, the atom is perceived by the apparatus as bare; on the contrary, if T is larger than ħΔ-1, the atom is perceived as dressed. The time scales for the two cases of dressing, by vacuum fluctuations or by a real single-mode field, are very different, but the similarity of the effects indicates a common physical aspect of the two kinds of dressing.

The interaction with the transverse modes of the vacuum electromagnetic field has been shown to yield a radiative (or self-energy) shift of the ground state of a two-level atom in Section 6.4. In this appendix we will show that the same effect occurs also in the energy of the states of a multilevel atom of the hydrogenic kind, which can be modelled by an electron bound to a fixed nucleus of charge Ze. This self-energy shift will turn out to depend on the form of the wavefunction of the state of the electron, leading to the possibility of lifting some of the accidental degeneracies which occur in hydrogenic atoms. Indeed the first experimental observation of this effect is related to the lifting of the well-known 2s-2p degeneracy in atomic hydrogen, and it is due to Lamb and Retherford (1947). Its nonrelativistic QED explanation, on the other hand, is due to Bethe (1947), and this appendix is simply a short account of his theory.

Consider a bare one-electron atom, whose energy levels and corresponding wavefunctions are denoted by En and un(x) = 〈x | n〉. Here n indicates the triplet of quantum numbers N, L, M and the electron spin is disregarded.

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.