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 video, can be projected for the benefit of large audiences), but often the type of idea 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 students learn far more from seeing real experiments carried out by a live lecturer, with whom they can interact personally, and from whom they 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 the 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 that we have, during the years, found particularly valuable in demonstrating the material covered in this book, and can be prepared with relatively cheap and easily available equipment. Need we say that we also enjoyed developing and performing these experiments?

Most optical systems are used to create images: eyes, cameras, microscopes, telescopes, for example. These image-forming 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 Ernst Abbe in 1873, microscopists thought that spatial resolution was only limited by their expertise in grinding and polishing lenses. Abbe showed that the basic scale is the wavelength of light, which now seems obvious. The relationship between geometrical and physical optics is like that between classical and quantum (wave) mechanics; although 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 physical optics is 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 coherence (Chapter 11) and leads naturally both to an understanding of the limits to image quality and to ways of extending them. We shall learn:

• how Abbe described optical imaging in terms of wave interference;

• that imaging can be formulated as a double process of diffraction;

• what are the basic limits to spatial resolution;

• how microscopes are constructed to achieve these limits;

• […]

Why did it take so long for the wave theory of light to be accepted, from its instigation by Huygens in about 1660 to the conclusive demonstrations by Young and Fresnel in 1803–12? In retrospect, it may be that Huygens did not take into account the wavelength; as a result the phenomenon of interference, particularly destructive interference, was missing. Only when Huygens' construction was analyzed in quantitative detail by Young and Fresnel did interference fringes and other wavelength-dependent features appear, and when these were confirmed experimentally the wave theory became generally accepted. It was because the wavelength, as measured by Young, was so much smaller than the size of everyday objects that special experiments had to be devised in order to see the effects of the waves; these are called ‘diffraction’ or ‘interference’ experiments and will be the subject of this chapter. Even so, some everyday objects, such as the drops of water that condense on a car window or the weave of an umbrella, do have dimensions commensurate with the wavelength of light, and the way they diffract light from a distant street light is clearly visible to the unaided eye (Fig. 7.1).

The distinction between the terms diffraction and interference is somewhat fuzzy. We try to use the term diffraction as a general term for all interactions between a wave and an obstacle, with interference as the case where several separable waves are superimposed.

This book is intended to explain the physical basis of classical optics and to introduce the reader to a variety of wave phenomena and their applications. However, it was discovered at the end of the nineteenth century that the description of light in terms of Maxwell's classical electromagnetic waves was incomplete, and the notion of quantization had to be added. Since then, in parallel to the development of wave optics, there has been an explosive growth of quantum optics, much of it fuelled by the invention of the laser at the end of the 1950s, which also provided a great incentive to reconsider many topics of classical optics, such as interference and coherence theory. It would be inappropriate that this book should ignore these developments; on the other hand, the subject of quantum optics is now so wide that a single chapter can do no justice to the field. In this chapter, we therefore set out modestly to explain the way in which quantum optics is different from classical optics, and give a qualitative introduction to lasers, followed by a taste of some of the new phenomena that have developed in recent years and are currently at the forefront of optics research.

In this chapter we shall discuss:

• how the electromagnetic field can be quantized, by creating an analogy between an electromagnetic wave and a simple harmonic oscillator;

• the concept of the photon, and some of its properties;

• […]

J. B. J. Fourier (1768–1830), applied mathematician and Egyptologist, was one of the great French scientists working at the time of Napoleon. Today, he is best remembered for the Fourier series method, which he invented for representation of any periodic function as a sum of discrete sinusoidal harmonics of its fundamental frequency. By extrapolation, his name is also attached to Fourier transforms or Fourier integrals, which allow almost any function to be represented in terms of an integral of sinusoidal functions over a continuous range of frequencies. Fourier methods have applications in almost every field of science and engineering. Since optics deals with wave phenomena, the use of Fourier series and transforms to analyze them has been particularly fruitful. For this reason, we shall devote this chapter to a discussion of the major points of Fourier theory, hoping to make the main ideas sufficiently clear in order to provide a ‘language’ in which many of the phenomena in the rest of the book can easily be discussed. More complete discussions, with greater mathematical rigour, can be found in many texts such as Brigham (1988), Walker (1988) and Prestini (2004).

In this chapter we shall learn:

• what is a Fourier series;

• about real and complex representation of the Fourier coefficients, and how they are calculated;

• how the Fourier coefficients are related to the symmetry of the function;

• how to represent the coefficients as a discrete spectrum in reciprocal, or wave-vector, space;

• […]

In this chapter we shall meet examples of electromagnetic wave propagation in systems containing fine dielectric structure on a scale of the order of the wavelength, where the scalar-wave approximation is inadequate. Clearly, in these cases we have to solve Maxwell's equations directly. On writing the equations, we shall discover that they bear a close similarity to those of quantum mechanics, where the dielectric constant in Maxwell's equations is analogous to the potential in Schrödinger's equation. This opens up a vast arsenal of methods, both analytical and numerical, which have been developed for their solution.

We first discuss the optical waveguide, already familiar in everyday life as the optical fibre, which has caused a revolution in the communications industry (Agrawal (2002)). The second topic is the dielectric multilayer system which, in its simplest form (the quarter-wave anti-reflection coating) has been with us for more than a century, but can today be used to make optical filters of any degree of complexity (MacLeod (2001)).

Following these examples, we shall briefly discuss their application to photonic crystals, structures with periodic refractive index leading to optical band gaps, whose behaviour can immediately be understood in terms of the quantum analogy (Joannopoulos et al. (2008)). Photonic crystals have always existed.

Why should a textbook on physics begin with history? Why not start with what is known now and refrain from all the distractions of out-of-date material? These questions would be justifiable if physics were a complete and finished subject; only the final state would then matter and the process of arrival at this state would be irrelevant. But physics is not such a subject, and optics in particular is very much alive and constantly changing. It is important for the student to study the past as a guide to the future. Much insight into the great minds of the era of classical physics can be found in books by Magie (1935) and Segré (1984).

By studying the past we can sometimes gain some insight – however slight – into the minds and methods of the great physicists. No textbook can, of course, reconstruct completely the workings of these minds, but even to glimpse some of the difficulties that they overcame is worthwhile. What seemed great problems to them may seem trivial to us merely because we now have generations of experience to guide us; or, more likely, we have hidden them by cloaking them with words. For example, to the end of his life Newton found the idea of ‘action at a distance’ repugnant in spite of the great use that he made of it; we now accept it as natural, but have we come any nearer than Newton to understanding it?

Many aspects of the interaction between radiation and matter can be described quite accurately by a classical theory in which the medium is represented by model atoms consisting of positive and negative parts bound by an attraction that depends linearly on their separation. Although quantum theory is necessary to calculate from first principles the magnitude of the parameters involved, in this chapter we shall show that many optical effects can be interpreted physically in terms of this model by the use of classical mechanics. Some of the quantum-mechanical ideas behind dispersion will be discussed later in Chapter 14, but most are outside the scope of this book.

In this chapter we shall learn:

• about the way in which a classical dipole atom responds to an oscillating electromagnetic field;

• about Rayleigh scattering, and why sky light is blue and polarized;

• how refractive index, absorption and scattering are related;

• that dispersion, the dependence of refractive properties on frequency, results from atomic resonances;

• about anomalous dispersion near to absorption lines;

• analytical relationships between refractive index and absorption;

• about plasma absorption and magneto-optical effects;

• whether signals can be propagated faster than the speed of light in anomalous-dispersion regions;

• a little about non-linear optical properties, which arise when the wavefields are very intense;

• about harmonic generation, the photo-refractive effect and soliton propagation;

• about optics at interfaces between conventional dielectrics and materials with negative permittivity;

• about surface plasmon resonance.

If this book were to follow historical order, the present chapter should have preceded the previous one, since lenses and mirrors were known and studied long before wave theory was understood. However, once we have grasped the elements of wave theory, it is much easier to appreciate the strengths and limitations of geometrical optics, so logically it is quite appropriate to put this chapter here. Essentially, geometrical optics, which considers light waves as rays that propagate along straight lines in uniform media and are related by Snell's law (§2.6.2 and §5.4) at interfaces, has a relationship to wave optics similar to that of classical mechanics to quantum mechanics. For geometrical optics to be strictly true, it is important that the sizes of the elements we are dealing with be large compared with the wavelength λ. Under these conditions we can neglect diffraction, which otherwise prevents the exact simultaneous specification of the positions and directions of rays on which geometrical optics is based.

Analytical solutions of problems in geometrical optics are rare, but fortunately there are approximations, in particular the Gaussian or paraxial approximation, which work quite well under most conditions and will be the basis of the discussion in this chapter. Exact solutions can be found using specialized computer programs, which will not be discussed here. However, from the practical point of view, geometrical optics answers most questions about optical instruments extremely well and in a much simpler way than wave theory could do.

The difference between Fresnel and Fraunhofer diffraction has been discussed in Chapter 7, where we showed that Fraunhofer diffraction is characterized by a linear change of phase over the diffracting obstacle, contrasting the quadratic phase change responsible for Fresnel diffraction. Basically, Fraunhofer diffraction is the limit of Fresnel diffraction when the source and the observer are infinitely distant from the obstacle. When the wavelength is very short and the obstacles are very small, such conditions can be achieved in the laboratory; for this reason Fraunhofer diffraction is naturally observed with X-rays, electrons, neutrons, etc., which generally have wavelengths less than 1Å. The study of Fraunhofer diffraction has been fuelled by its importance in understanding the diffraction of these waves, particularly by crystals. This has led to our present-day knowledge of the crystalline structures of materials and also of many molecular structures. Figure 8.1 shows a famous X-ray diffraction pattern of a crystal of haemoglobin, from about 1958, whose interpretation was a milestone in visualizing and understanding biological macromolecules. The techniques used in interpreting such pictures will be discussed in the later parts of the chapter.

In optics, using macroscopic objects in a finite laboratory, the linear phase change can be achieved by illuminating the object with a beam of parallel light. It is therefore necessary to use lenses, both for the production of the parallel beam and for the observation of the resultant diffraction pattern.

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 using waves of different types, electromagnetic, matter, neutron, acoustic etc. One current highlight of optical interferometry is the development of detectors that can measure dimensional changes as small as 10−19 m, which should be induced by gravitational waves emitted by cataclysmic events in the distant Universe. A picture of one such interferometer, which has two orthogonal arms each 4 km in length, is shown in Fig. 9.1 and the design of this instrument will be discussed in more detail in §9.7.

An enormous variety of interferometric techniques has been developed during the years, and we shall limit ourselves in this chapter to a discussion of examples representing distinctly different principles. There are several monographs on interferometry that discuss practical aspects in greater detail, for example Tolansky (1973), Steel (1983), Hariharan (2003) and Hariharan (2007).

In this chapter we shall learn about:

• Young's basic two-slit interferometer and its capabilities;

• interference in a reflecting thin film;

• diffraction gratings: how they work and how they are made, their resolving power and their efficiency;

• two-beam interferometers of several types;

• […]

As we saw in Chapter 5, electromagnetic waves in isotropic materials are transverse, their electric and magnetic field vectors E and H being normal to the direction of propagation k. The direction of E or rather, as we shall see later, the electric displacement field D, is called the polarization direction, and for any given direction of propagation there are two independent polarization vectors, which can be in any two mutually orthogonal directions normal to k. However, when the medium through which the wave travels is anisotropic, which means that its properties depend on orientation, the choice of the polarization vectors is not arbitrary, and the velocities of the two waves may be different. A material that supports two distinct propagation vectors is called birefringent.

In this chapter, we shall learn:

• about the various types of polarized plane waves that can propagate – linear, circular and elliptical – and how they are produced;

• how an anisotropic optical material can be described by a dielectric tensor ∈, which relates the fields D and E within the material;

• a simple geometrical representation of wave propagation in an anisotropic material, the n-surface, which allows the wave propagation properties to be easily visualized;

• how Maxwell's equations are written in an anisotropic material, and how they lead to two particular orthogonally polarized plane-wave solutions;

• […]