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.

Introduction

IN Chapter III a geometrical model of the propagation of light was derived from the basic equations of electromagnetic theory, and it was shown that, with certain approximations, variations of intensity in a beam of light can be described in terms of changes in the cross-sectional area of a tube of rays. When two or more light beams are superposed, the distribution of intensity can no longer in general be described in such a simple manner. Thus if light from a source is divided by suitable apparatus into two beams which are then superposed, the intensity in the region of superposition is found to vary from point to point between maxima which exceed the sum of the intensities in the beams, and minima which may be zero. This phenomenon is called interference. We shall see shortly that the superposition of beams of strictly monochromatic light always gives rise to interference. However, light produced by a real physical source is never strictly monochromatic but, as we learn from atomistic theory, the amplitude and phase undergo irregular fluctuations much too rapid for the eye or an ordinary physical detector to follow. If the two beams originate in the same source, the fluctuations in the two beams are in general correlated, and the beams are said to be completely or partially coherent depending on whether the correlation is complete or partial. In beams from different sources, the fluctuations are completely uncorre-lated, and the beams are said to be mutually incoherent. When such beams from different sources are superposed, no interference is observed under ordinary experimental conditions, the total intensity being everywhere the sum of the intensities of the individual beams.

IN Chapters I and II it was shown that the propagation of electromagnetic waves may be studied either by using Maxwell's equations, supplemented by the material equations, or by means of certain integral equations which utilize the polarization properties of the medium. In particular, either of these methods may also be applied to the study of the propagation of light through a medium whose density depends on space coordinates and on time. Though the former method has been used extensively in the past, the latter has only more recently been applied to such studies. In this chapter we shall apply the integral equation method to the problem of diffraction of light by a transparent homogeneous medium, disturbed by the passage of ultrasonic waves. It will be useful, however, to give first a qualitative description of this diffraction phenomenon and a brief summary of the theoretical work on this problem based on Maxwell's differential equations.

Qualitative description of the phenomenon and summary of theories based on Maxwell's differential equations

Qualitative description of the phenomenon

Ultrasonic waves are sound waves whose frequencies are higher than those of waves normally audible to the human ear. The angular frequencies of the ultrasonic waves produced in laboratories lie from about 105 s-1 to about 3 X 109 s-1, the former value representing the limit of audibility of the human ear. The corresponding range of wavelengths A of course depends on the velocity v of these waves in the medium in which they travel.

THE idea of writing this book was a result of frequent enquiries about the possibility of publishing in the English language a book on optics written by one of us more than twenty-five years ago. A preliminary survey of the literature showed that numerous researches on almost every aspect of optics have been carried out in the intervening years, so that the book no longer gives a comprehensive and balanced picture of the field. In consequence it was felt that a translation was hardly appropriate; instead a substantially new book was prepared, which we are now placing before the reader. In planning this book it soon became apparent that even if only the most important developments which took place since the publication of Optik were incorporated, the book would become impracticably large. It was, therefore, deemed necessary to restrict its scope to a narrower field. Optik itself did not treat the whole of optics. The optics of moving media, optics of X-rays and y-rays, the theory of spectra and the full connection between optics and atomic physics were not discussed; nor did the old book consider the effects of light on our visual sense organ – the eye. These subjects can be treated more appropriately in connection with other fields such as relativity, quantum mechanics, atomic and nuclear physics, and physiology. In this book not only are these subjects excluded, but also the classical molecular optics which was the subject matter of almost half of the German book.

As mentioned in the Preface to the seventh edition of this work, a change to a new publisher has made it possible to reset the whole text. Not surprisingly such a large amount of typesetting introduced some typographical errors. Those found by now have been corrected, as has been other inaccuracies which the reviewers and readers of the book brought to my attention. A small number of additional references have also been included. For the sake of completeness the Prefaces to the third, fourth and fifth editions have also been added.

I am particularly indebted to Dr E. Hecht who in a thorough review of the seventh edition noted several errors and inaccuracies that have now been corrected. I am also obliged to Dr S. H. Wiersma and Mr Damon Diehl who read much of the text very carefully and supplied me with long lists of misprints and other errors. I must also thank my friends and colleagues Professor Taco Visser, Dr Daniel F. V. James, Dr Peter Milonni, Professor Richard M. Sillitto, Mrs Winifred Sillitto and Dr Andrei Shchegrov for drawing my attention to a number of errors. Finally I wish to express my indebtedness to my colleague and former student Dr Greg Gbur for much help with the preparation of the corrected version.

ADVANTAGE has been taken in the preparation of a new edition of this work to make a number of corrections of errors and misprints, to make a few minor additions and to include some new references.

Since the appearance of the first edition almost exactly three years ago, the first optical masers (lasers) have been developed. By means of these devices very intense and highly coherent light beams may be produced. Whilst it is evident that optical masers will prove of considerable value not only for optics but also for other sciences and for technology, no account of them is given in this new edition. For the basic principles of maser action have roots outside the domain of classical electromagnetic theory on which considerations of this book are based. We have, however, included a few references to recent researches in which light generated by optical masers was utilized or which have been stimulated by the potentialities of these new optical devices.

We wish to acknowledge our gratitude to a number of readers who drew our attention to errors and misprints. We are also obliged to Dr B. Karczewski and Mr C. L. Mehta for assistance with the revisions.

IN 1831 William Rowan Hamilton discovered the analogy between the trajectory of material particles in potential fields and the path of light rays in media with continuously variable refractive index. By virtue of its great mathematical beauty, the ‘Hamiltonian Analogy’ survived in the textbooks of dynamics for almost a hundred years, but did not inspire any practical applications until 1925 when H. Busch first explained the focusing effect of electric and magnetic fields on electron beams in optical terms. Almost at the same time E. Schrodinger took the Hamiltonian Analogy a step further by passing from geometrical optics to wave optics of particles with his wave equation, in which he incorporated the wavelength of particles, first conceived by Louis de Broglie in 1923.

Practical electron optics developed rapidly from 1928 onwards. By this time the Hamiltonian Analogy was widely known and inspired the invention of electron-optical counterparts of light-optical instruments, such as the electron microscope. Though the mathematical analogy is general, the two techniques are not exactly parallel. Some electron-optical instruments such as cathode-ray tubes and systems with curved optic axes have no important counterparts in light optics. In the available space only those problems of electron optics will be considered whose light-optical analogues were developed at length in the previous chapters of this work, so that the results can be transferred almost in toto, with few modifications. It may be noted that this applies in particular to the most recondite chapter of electron optics: the wave theory of lens aberrations.

Introduction

IN carrying out the transition from the general electromagnetic field to the optical field, which is characterized by very high frequencies (short wavelengths), we found that in certain regions the simple geometrical model of energy propagation was inadequate. In particular, we saw that deviations from this model must be expected in the immediate neighbourhood of the boundaries of shadows and in regions where a large number of rays meet. These deviations are manifested by the appearance of dark and bright bands, the diffraction fringes. Diffraction theory is mainly concerned with the field in these special regions; such regions are of great practical interest as they include the part of the image space in which the optical image is situated (region of focus).

The first reference to diffraction phenomena appears in the work of Leonardo da Vinci (1452–1519). Such phenomena were, however, first accurately described by Grimaldi in a book, published in 1665, two years after his death. The corpuscular theory, which, at the time, was widely believed to describe correctly the propagation of light, could not explain diffraction. Huygens, the first proponent of the wave theory, seems to have been unaware of Grimaldi's discoveries; otherwise he would have undoubtedly quoted them in support of his views. The possibility of explaining diffraction effects on the basis of a wave theory was not noticed until about 1818. In that year there appeared the celebrated memoir of Fresnel (see Historical introduction) in which he showed that diffraction can be explained by the application of Huygens’ construction (see §3.3.3) together with the principle of interference. Fresnel's analysis was later put on a sound mathematical basis by Kirchhoff (1882), and the subject has since then been extensively discussed by many writers.