IT was mentioned in §3.1.1 that the eikonal equation of geometrical optics is identical with an equation which describes the propagation of discontinuities in an electromagnetic field. More generally, the four equations §3.1 (lla)-(14a) governing the behaviour of the electromagnetic field associated with the geometrical light rays may be shown to be identical with equations which connect the field vectors on a moving discontinuity surface. It is the purpose of this appendix to demonstrate this mathematical equivalence.

Relations connecting discontinuous changes in field vectors

In §1.1.3 we considered discontinuities in field vectors which arise from abrupt changes in the material parameters £ and fi, for example at a surface of a lens. Discontinuous fields may also arise from entirely different reasons, namely because a source suddenly begins to radiate. The field then spreads into the space surrounding the source and with increasing time fills a larger and larger region. On the boundary of this region the field has a discontinuity, the field vectors being in general finite inside this region and zero outside it. We shall first establish certain general relations which hold on any surface at which the field is discontinuous. For simplicity we assume that at any instant of time t > 0 there is only one such surface; the extension to several discontinuity surfaces (which may arise, for example, from reflections at obstacles present in the medium) is straightforward.

Forty years ago this month Max Born and I dispatched to the publishers the Preface to the first edition of Principles of Optics. Since that time the book has been published in six editions and has been reprinted seventeen times (not counting unauthorized editions and several translations), usually with only a small number of corrections. A recent change to a new publisher, who expressed willingness to reset the whole text, has given me the opportunity to make more substantial changes.

The first edition was published a year before the invention of the laser, an event which triggered an explosion of activities in optics and soon led to the creation of entirely new fields, such as non-linear optics, fiber optics and opto-electronics. Numerous applications followed, in medicine, in optical data storage, in information transfer and in many other areas. On a more fundamental level, quantum optics emerged as a vibrant and rapidly expanding field, which has provided new ways of testing some basic assumptions of quantum physics relating, for example, to localization and indistinguishability. The progress made in these fields has been rapid and broad and some of the newer areas have themselves become the subjects of books.

It is clear that a fully updated new edition of Principles of Optics would require that it be expanded into several volumes. Consequently, in order to preserve a singlevolume, only a few new topics have been added; they were selected to some extent so as not to necessitate major revisions of the original text.

So far we have been concerned with the propagation of light in nonconducting, isotropic media. We now turn our attention to the optics of conducting media, more particularly to metals. An ordinary piece of metal is a crystalline aggregate, consisting of small crystals of random orientation. Single crystals of appreciable size are rare, but can be produced artificially; their optical properties will be studied in Chapter XV. A mixture of randomly oriented crystallites behaves evidently as an isotropic substance, and as the theory of light propagation in a conducting isotropic medium is much simpler than in a crystal, we shall consider it here in some detail.

According to §1.1, conductivity is connected with the appearance of Joule heat. This is an irreversible phenomenon, in which the electromagnetic energy is destroyed, or more precisely transformed into heat, and in consequence an electromagnetic wave in a conductor is attenuated. In metals, on account of their very high conductivity, this effect is so large that they are practically opaque. In spite of this, metals play an important part in optics. Strong absorption is accompanied by high reflectivity, so that metallic surfaces act as excellent mirrors. Because of the partial penetration of light into a metal, it is possible to obtain information about the absorption constants and the mechanism of absorption from observations of the reflected light, even though the depth of penetration is small.

The characteristic functions of Hamilton

IN §3.1 it was shown that, within the approximations of geometrical optics, the field may be characterized by a single scalar function S(r). Since S(r) satisfies the eikonal equation §3.1 (15), this function is fully specified by the refractive index function (r) alone, together with the appropriate boundary conditions.

Instead of the function S(r), closely related functions known as characteristic functions of the medium are often used. They were introduced into optics by W. R. Hamilton, in a series of classical papers. Although on account of algebraic complexity it is impossible to determine the characteristic functions explicitly for all but the simplest media, Hamilton's methods nevertheless form a very powerful tool for systematic analytical investigations of the general properties of optical systems.

In discussing the properties of these functions and their applications, an isotropic but generally heterogeneous medium will be assumed.

The point characteristic

Let (x0, y0, z0) and (x1, y1, z1) be respectively the coordinates of two points PQ and P\ each referred to a different set of mutually parallel, rectangular axes (Fig. 4.1). If the two points are imagined to be joined by all possible curves, there will, in general, be some amongst them, the optical rays, which satisfy Fermat's principle. Assume for the present that not more than one ray joins any two arbitrary points.

THE SOFT X-RAY AND EXTREME ULTRAVIOLET REGIONS OF THE ELECTROMAGNETIC SPECTRUM

One of the last regions of the electromagnetic spectrum to be developed is that between ultraviolet and x-ray radiation, generally shown as a dark region in charts of the spectrum. It is a region where there are a large number of atomic resonances, leading to absorption of radiation in very short distances, typically measured in nanometers (nm) or micrometers (microns, µm), in all materials. This has historically inhibited the pursuit and exploration of the region. On the other hand, these same resonances provide mechanisms for both elemental (C, N, O, etc.) and chemical (Si, SiO2, TiSi2) identification, creating opportunities for advances in both science and technology. Furthermore, because the wavelengths are relatively short, it becomes possible both to see smaller structures as in microscopy, and to write smaller patterns as in lithography. To exploit these opportunities requires advances in relevant technologies, for instance in materials science and nanofabrication. These in turn lead to new scientific understandings, perhaps through surface science, chemistry, and physics, providing feedback to the enabling technologies. Development of the extreme ultraviolet and soft x-ray spectral regions is presently in a period of rapid growth and interchange among science and technology. Figure 1.1 shows that portion of the electromagnetic spectrum extending from the infrared to the x-ray region, with wavelengths across the top and photon energies along the bottom.

In our studies of radiation from charged particles moving at velocities approaching that of light, a number of interesting phenomena are observed, such as the searchlight effect wherein radiation from the charged particle is constrained to a very narrow forward radiation cone. Furthermore, the calculation of detailed angular radiation patterns, in the frame of reference moving with the charged particle, and wavelength distributions are readily accomplished. The results can then be transformed back to the laboratory, or observer, frame of reference. For instance, the calculation of undulator radiation reduces to use of the well-known formula for so-called dipole radiation from a simple oscillating electron. With this approach we need solve Maxwell's equations for only the simplest radiating system, a small amplitude oscillating electron. This approach is not only simple to follow, but gives valuable physical insights to the radiation process and the parameters that characterize it.

In order to relate calculations in one frame of reference to those in another frame of reference when the relative speed between the two approaches that of light, we must make use of the Lorentz space-time transformations, which provide relationships between spatial and temporal scales in the two frames of reference, and are consistent both with Einstein's postulates of special relativity and with all known experiments (see Ref. 2 for a discussion of the Lorentz transformations and their reduction to Galilean transformations as v/c → 0).