IN §4.9 it was mentioned that within the domain of geometrical optics the departure of the path of light from the predictions of the Gaussian theory may be studied either with the help of ray-tracing or by means of algebraic analysis. In the latter treatment, which forms the subject matter of this chapter, terms which involve off-axis distances in powers higher than the second in the expansion of the characteristic functions are retained. These terms represent geometrical aberrations.

The discovery of photography in 1839 by Daguerre (1789–1851) was chiefly responsible for early attempts to extend the Gaussian theory. Practical optics, which until then was mainly concerned with the construction of telescope objectives, was confronted with the new task of producing objectives with large apertures and large fields. J. Petzval, a Hungarian mathematician, attacked with considerable success the related problem of supplementing the Gaussian formulae by terms involving higher powers of the angles of inclination of rays with the axis. Unfortunately, Petzval's extensive manuscript on the subject was destroyed by thieves; what is known about this work comes chiefly from semipopular reports. Petzval demonstrated the practical value of his calculations by constructing in about 1840 his well-known portrait lens [shown in Fig. 6.3(b)] which proved greatly superior to any then in existence. The earliest systematic treatment of geometrical aberrations which was published in full is due to Seidel, who took into account all the terms of the third order in a general centred system of spherical surfaces. Since then, his analysis has been extended and simplified by many writers.

IN Chapter V we studied the effects of aberrations on the basis of geometrical optics. In that treatment the image was identified with the blurred figure formed by the points of intersection of the geometrical rays with the image plane. Since geometrical optics gives an approximate model valid in the limit of very short wavelengths, it is to be expected that the geometrical theory gradually loses its validity as the aberrations become small. For example, in the limiting case of a perfectly spherical convergent wave issuing from a circular aperture, geometrical optics predicts for the focal plane an infinite intensity at the focus and zero intensity elsewhere, whereas, as has been shown in §8.5.2, the real image consists of a bright central area surrounded by dark and bright rings (the Airy pattern). In the neighbourhood of the focal plane the light distribution has also been seen to be of a much more complicated nature (see Fig. 8.41) than geometrical optics suggests. We are thus led to the study of the effects of aberrations on the basis of diffraction theory.

The first investigations in this field are due to Rayleigh. His main contribution was the formulation of a criterion (discussed in §9.3) which, in an extended form, has come to be widely used for determining the maximum amounts of aberrations that may be tolerated in optical instruments. The subject was carried further by the researches of many writers who investigated the effects of various aberrations, and we may mention, in particular, the more extensive treatments by Steward, Picht, and Born.

THE physical principles underlying the optical phenomena with which we are concerned in this treatise were substantially formulated before 1900. Since that year, optics, like the rest of physics, has undergone a thorough revolution by the discovery of the quantum of energy. While this discovery has profoundly affected our views about the nature of light, it has not made the earlier theories and techniques superfluous; rather, it has brought out their limitations and defined their range of validity. The extension of the older principles and methods and their applications to very many diverse situations has continued, and is continuing with undiminished intensity.

In attempting to present in an orderly way the knowledge acquired over a period of several centuries in such a vast field it is impossible to follow the historical development, with its numerous false starts and detours. It is therefore deemed necessary to record separately, in this preliminary section, the main landmarks in the evolution of ideas concerning the nature of light.

The philosophers of antiquity speculated about the nature of light, being familiar with burning glasses, with the rectilinear propagation of light, and with refraction and reflection. The first systematic writings on optics of which we have any definite knowledge are due to the Greek philosophers and mathematicians [Empedocles (c. 490–430 BC), Euclid (c. 300 BC)].

Amongst the founders of the new philosophy, Rene Descartes (1596-1650) may be singled out for mention as having formulated views on the nature of light on the basis of his metaphysical ideas.

Some further errors and misprints that were found in the earlier editions of this work have been corrected, the text in several sections has been improved and a number of references to recent publications have been added. More extensive changes have been made in §§13.1–13.3, dealing with the optical properties of metals. It is well known that a purely classical theory is inadequate to describe the interaction of an electromagnetic field with a metal in the optical range of the spectrum. Nevertheless, it is possible to indicate some of the main features of this process by means of a classical model, provided that the frequency dependence of the conductivity is properly taken into account and the role that the free, as well as the bound, electrons play in the response of the metal to an external electromagnetic field is understood, at least in qualitative terms. The changes in §§13.1–13.3 concern mainly these aspects of the theory and the revised sections are believed to be free of misleading statements and inaccuracies that were present in this connection in the earlier editions of this work and which can also be commonly found in many other optical texts.

I am grateful to some of our readers for informing me about misprints and errors. I wish to specifically acknowledge my indebtedness to Prof. A. D. Buckinham, Dr D. Canals Frau and, once again, Dr E. W. Marchand, who supplied me with detailed lists of corrections and to Dr É. Lalor and Dr G. C. Sherman for having drawn my attention to the need for making more substantial changes in Chapter XIII.

Introduction

ON the basis of Maxwell's equations, together with standard boundary conditions, the scattering of electromagnetic radiation by an obstacle becomes a well-defined mathematical boundary-value problem. In the present chapter some aspects of the theory of diffraction of monochromatic waves are developed from this point of view, and in particular the rigorous solution to the classical problem of diffraction by a perfectly conducting half-plane is given in detail.

In the early theories of Young, Fresnel, and Kirchhoff, the diffracting obstacle was supposed to be perfectly ‘black’; that is to say, all radiation falling on it was assumed to be absorbed, and none reflected. This is an inherent source of ambiguity in that such a concept of absolute ‘blackness’ cannot legitimately be defined with precision; it is, indeed, incompatible with electromagnetic theory.

Cases in which the diffracting body has a finite dielectric constant and finite conductivity have been examined theoretically, one of the earliest comprehensive treatments of such a case being Mie's discussion in 1908 of scattering by a sphere, which is described in Chapter XIV in connection with the optics of metals. In general, however, the assumption of finite conductivity tends to make the mathematics very complicated, and it is often desirable to accept the concept of a perfectly conducting (and therefore perfectly reflecting) body. This is clearly an idealization, but one which is compatible with electromagnetic theory; furthermore, since the conductivity of some metals (e.g. copper) is very large, it may represent a good approximation if the frequency is not too high, though it should be stressed that the approximation is never entirely adequate at optical frequencies.

THE three preceding chapters give an account of the geometrical theory of optical imaging, using for the main part the predictions of Gaussian optics of the Seidel theory. An outstanding instance of the invaluable service rendered by this branch of optics lies in its ability to present the working principles of optical instruments in an easily visualized form. Although the quality of optical systems cannot be estimated by means of Gaussian theory alone, the purpose served by the separate optical elements can be indicated in this way, so that a simple, though somewhat approximate, picture of the action of the system can often be obtained without entering into the full intricacy of the techniques of optical design.

The development of optical instruments in the past has proceeded just as fast as technical difficulties have been overcome. It is hardly possible to give a step-by-step account of the design of optical systems, for two reasons. Firstly, the limitations of a given arrangement are not indicated by the predictions of the simple theory; in particular cases this needs to be supplemented by a fuller analysis often involving tedious calculations. Secondly, difficulties of a practical nature may prevent an otherwise praiseworthy arrangement from being used. It is not intended in this account to discuss the theoretical and practical limitations in individual cases; only the basic principles underlying the arrangement of some of the more important optical instruments will be given, in order to provide a framework for some of the later chapters which deal with the more detailed theories of optical image formation.

Approximation for very short wavelengths

THE electromagnetic field associated with the propagation of visible light is characterized by very rapid oscillations (frequencies of the order of 1014 s-1) or, what amounts to the same thing, by the smallness of the wavelength (of order 10-5 cm). It may therefore be expected that a good first approximation to the propagation laws in such cases may be obtained by a complete neglect of the finiteness of the wavelength. It is found that for many optical problems such a procedure is entirely adequate; in fact, phenomena which can be attributed to departures from this approximate theory (so-called diffraction phenomena, studied in Chapter VIII) can only be demonstrated by means of carefully conducted experiments.

The branch of optics which is characterized by the neglect of the wavelength, i.e. that corresponding to the limiting case λ0 ͢ 0, is known as geometrical optics since in this approximation the optical laws may be formulated in the language of geometry. The energy may then be regarded as being transported along certain curves (light rays). A physical model of a pencil of rays may be obtained by allowing the light from a source of negligible extension to pass through a very small opening in an opaque screen. The light which reaches the space behind the screen will fill a region the boundary of which (the edge of the pencil) will, at first sight, appear to be sharp.

IT is a general feature of equations of classical physics that they can be derived from variational principles. Two early examples are Fermat's principle in optics (1657) and Maupertuis’ principle in mechanics (1744). The equations of elasticity, hydrodynamics and electrodynamics can also be represented in this way.

However, when one deals with field equations, involving as a rule four or more independent variables x, y, z, t, …, one makes little use, owing to the great complexity of partial differential equations, of the property that the solution expresses stationary values of certain integrals. The only essential advantage of the variational approach in such cases is connected with the derivation of conservation laws — e.g. for energy. The situation is quite different in problems involving one independent variable (time in mechanics, or length of a ray in geometrical optics). Then one deals with a set of ordinary differential equations and it turns out that a study of the behaviour of the solution is greatly facilitated by a variational approach. This approach is in fact a straightforward generalization of ordinary geometrical optics in every detail. Its modern representation owes much to David Hilbert, on whose unpublished lectures, given at Gottingen in about 1903, we base the considerations of the following sections. The theory is presented here for a three-dimensional space (x, y, z) only, but can easily be extended to more dimensions.

Introduction

So far we have been mainly concerned with monochromatic light produced by a point source. Light from a real physical source is never strictly monochromatic, since even the sharpest spectral line has a finite width. Moreover, a physical source is not a point source, but has a finite extension, consisting of very many elementary radiators (atoms). The disturbance produced by such a source may be expressed, according to Fourier's theorem, as the sum of strictly monochromatic and therefore infinitely long wave trains. The elementary monochromatic theory is essentially concerned with a single component of this Fourier representation.

In a monochromatic wave field the amplitude of the vibrations at any point P is constant, while the phase varies linearly with time. This is no longer the case in a wave field produced by a real source: the amplitude and phase undergo irregular fluctuations, the rapidity of which depends essentially on the effective width Δv of the spectrum. The complex amplitude remains substantially constant only during a time interval Δt which is small compared to the reciprocal of the effective spectral width Δv; in such a time interval the change of the relative phase of any two Fourier components is much less than 2π and the addition of such components represents a disturbance which in this time interval behaves like a monochromatic wave with the mean frequency; however, this is not true for a longer time interval. The characteristic time Δt = 1/Δv is of the order of the coherence time introduced in §7.5.8.

IN the preceding chapter the effect of matter on an electromagnetic field was expressed in terms of a number of macroscopic constants. These have only a limited range of validity and are in fact inadequate to describe certain processes, such as the emission, absorption and dispersion of light. A full account of these phenomena would involve an extensive study of the atomistic theory and lies therefore outside the scope of this book.

It is possible, however, to describe the interaction of field and matter by means of a simple model which is entirely adequate for most branches of optics. For this purpose each of the vectors D and B is expressed as the sum of two terms. Of these one is taken to be the vacuum field and the other is regarded as arising from the influence of matter. Thus one is led to the introduction of two new vectors for describing the effects of matter: the electric polarization (P) and the magnetic polarization or magnetization (M). Instead of the material equations (10) and (11) in §1.1 connecting D and B with E and H, we now have equations connecting P and M with E and H. These new equations have a more direct physical meaning and lead to the following conception of the propagation of an electromagnetic field in matter:

An electromagnetic field produces at a given volume element certain amounts of polarization P and M which, in the first approximation, are proportional to the field, the constant of proportionality being a measure of the reaction of the field.