Because in many of the experiments discussed in this volume the semiconductor CU2O was used, and because this material is less well known than other semiconductors such as Si and GaAs, we review here some of the basic properties of CU2O.

CU2O is actually one of the earliest semiconductors studied [1], and the proof of the existence of excitons in this crystal [2, 3], which confirmed the theories of Frenkel [4], Wannier [5], and Mott [6], led to an exciting period of expansion in the field of excitonic physics in the late 1950s and early 1960s, during which time many of the optical properties of this intriguing crystal were established [7-19]. (For general reviews, see Refs. 20 and 21.) The same property that makes it an excellent material for nonlinear optical effects, namely, a very large excitonic binding energy (150 meV), makes it a poor electrical conductor even at room temperature. Therefore CU2O has not been studied or fabricated extensively by the electronics industry, and high-quality samples are still not widely available.

Band Structure. Most of the complexities of the band structure of CU2O stem from the d orbitals of the Cu atoms in the valence band. CU2O forms into a cubic lattice with inversion symmetry, which is the Oh symmetry group [22]. In the Oh symmetry, the five 3d orbitals are split by means of the crystal field into a higher threefold degenerate Γ+5 level and a lower twofold degenerate Γ+3level. The lowest conduction band, on the other hand, is formed from the 4s orbitals that have Γ+1 symmetry in the Oh group.

What is an Exciton?

Many people seem to have trouble with the concept of an exciton. Is it “real” in the same sense that a photon or an atom is? Does the motion of an exciton correspond to the transport of anything real in a solid?

Simply put, an exciton is an electron and a hole held together by Coulomb attraction. Of course, for some people the idea of a “hole” is a difficult concept, so this may not help much. Nevertheless, a hole is a “real” particle and so is an exciton.a Modern solid-state theory [1,2] gives equal footing to both free electrons and holes as charge carriers in a solid, exactly analogous to the way that electrons and positrons are both “real” particles, even though a positron can be seen as the absence of an electron in the negative-energy Dirac sea, i.e., a backwards-in-time-moving electron.

All excitons are spatially compact. The strong Coulomb attraction between the negatively charged electron and the positively-charged hole keeps them close together in real space, unlike Cooper pairs, which can have very long correlation lengths because of the weak phonon coupling between them. The sizes of excitons vary from the size of a single atom, e.g., approximately an angstrom up to several hundred angstroms, extending across thousands of lattice sites. Excitons are roughly divided into two categories based on their size. An exciton that is localized to a single lattice site is called a “Frenkel” exciton, after the pioneering work of Frenkel [3] on excitons in molecular crystals. Frenkel excitons appear most commonly in molecular crystals, polymers, and biological molecules, in which they are extremely important for understanding energy transfer.

Nonequilibrium Theory of the Optical Stark Effect in the Excitonic Range of the Spectrum

One of the most important applications of the theory of Bose condensation of excitons is the optical Stark effect, or “AC Stark effect, ” first demonstrated for excitons in Cu2O [1] and since seen in several semiconductors and semiconductor heterostructures (e.g., Refs. 2-5). This effect is a promising tool for the field of optical communications. One of the major interests in this field is the development of all-optical-switching methods, by which light signals are switched on or off directly by other light signals, just as in electronics, electrical signals switch other electrical signals. At the present, most optical communications systems use electrical signals (e.g., electro-optic or acousto-optic devices) to switch the light signals, which means that the limiting bandwidth of the system is controlled by that of the electrical signals, not the optical bandwidth. The optical Stark effect offers this possibility. When one laser beam impinges on a sample, it can drastically alter the excitonic absorption line shape. Therefore a second beam can see either a transparent or an absorbing medium, depending on the presence of the control beam. This is the optical equivalent of a transistor.

The basic effect is shown in Fig. 6.1. When an intense laser is tuned to a photon energy that is just below the excitonic ground state, the exciton ground state is shifted in frequency and the oscillator strength is altered. This leads to a strong reduction in the optical absorption at the original wavelength of the exciton ground state.

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.