This chapter will discuss the electromagnetic wave as a most important example of the general treatment of wave propagation presented in Chapter 2. We shall start at the point where the elementary features of classical electricity and magnetism have been summarized in the form of Maxwell's equations, and the reader's familiarity with the steps leading to this formulation will be assumed (see, for example, Grant and Phillips (1990), Jackson (1999), Franklin (2005)). It is well known that Maxwell's formulation included for the first time the displacement current ∂D/∂t, the time derivative of the fictitious displacement field D = ∈0E+P, which is a combination of the applied electric field E and the electric polarization density P. This field will turn out to be of prime importance when we come to extend the treatment in this chapter to wave propagation in anisotropic media in Chapter 6.

In this chapter we shall learn:

• about the properties of electromagnetic waves in isotropic linear media;

• about simple-harmonic waves with planar wavefronts;

• about radiation of electromagnetic waves;

• the way in which these waves behave when they meet the boundaries between media: the Fresnel coefficients for reflection and transmission;

• about optical tunnelling and frustrated total internal reflection;

• about electromagnetic waves in conducting media;

• some consequences of the time-reversal symmetry of Maxwell's equations;

• about electromagnetic momentum, radiation pressure and optical tweezers;

• about angular momentum of waves that have spiral wavefronts, instead of the usual plane wavefronts;

• […]

The helix antenna discussed in the previous chapter used a new type of element to model surfaces. The theory underlying this is described in this chapter. The basic theory is quite complex, and general implementations are especially challenging. However, by choosing a suitable problem, it proves possible to undertake a limited implementation of a three-dimensional scattering problem, using a basis function defined on a triangular patch known as the RWG element. This is named after Rao, Wilton and Glisson, who introduced the element in their classic 1982 paper [1]. It represented a new type of element, the vector or edge-based element, and a closely related class of element was also under development for finite element applications at that time, although it would be some years before the connection was fully appreciated. (This will be pursued in more detail in the later coverage of the FEM.) The RWG element underlies the surface treatment of modern codes such FEKO (although not NEC), and some examples of using existing codes (in particular FEKO) to compute scattering from more general surfaces will further illustrate this.

We will also see that not only can perfectly (or highly) conducting structures be efficiently modelled using surface currents, but also homogeneous dielectric and/or magnetic regions, using fictitious equivalent currents. (We will even briefly describe how inhomogeneous bodies can be modelled using volumetric currents, but note at the outset that this is not one of the strong points of the MoM.)

We use optics overwhelmingly in our everyday life: in art and sciences, in modern communications and medical technology, to name just a few fields. This is because 90% of the information we receive is visual. The main purpose of this book is to communicate our enthusiasm for optics, as a subject both practical and aesthetic, and standing on a solid theoretical basis.

We were very pleased to be invited by the publishers to update Optical Physics for a fourth edition. The first edition appeared in 1969, a decade after the construction of the first lasers, which created a renaissance in optics that is still continuing. That edition was strongly influenced by the work of Henry Lipson (1910–1991), based on the analogy between X-ray crystallography and optical Fraunhofer diffraction in the Fourier transform relationship realized by Max von Laue in the 1930s. The text was illustrated with many photographs taken with the optical diffractometers that Henry and his colleagues built as ‘analogue computers’ for solving crystallographic problems. Henry wrote much of the first and second editions, and was involved in planning the third edition, but did not live to see its publication. In the later editions, we have continued the tradition of illustrating the principles of physical optics with photographs taken in the laboratory, both by ourselves and by our students, and hope that readers will be encouraged to carry out and further develop these experiments themselves.

Six years after the first edition was prepared, it was clear that a revised edition was in order. Continued advances in computational electromagnetics, new capabilities in commercial codes, the continual increase in computational resources, challenging new problems and a new generation of research students and engineers required new material.

Since the first edition appeared, several trends can be noted in the field. Firstly, in terms of commercial companies, there has been a significant shake-out in the market. Whilst not pretending to offer encyclopedic coverage of the large number of commercial codes available, the three codes whose application is discussed in this book, viz. CST, FEKO and HFSS, have further established themselves as amongst the market leaders in their regions of application during this period. These codes have evolved continuously, and this evolution is reflected in places in this revised edition. Secondly, whilst no fundamentally new techniques have been introduced (either in the field in general or in commercial codes in particular), a large number of additional features, improvements and enhancements have continued to extend the utility of these packages. Thirdly, after more than two decades of continual increase in CPU clock speeds in personal computers (which now dominate computational engineering), the last few years have seen clock speeds not only stagnate, but in some cases actually decrease. However, Moore's law has continued to hold sway, but in terms of multi-core and multi-processor systems. Exploiting parallelism has become essential to benefit from new hardware.

Throughout this book, checking convergence numerically has been continually emphasized. However, we have not discussed the more theoretical issues of whether the underlying numerical formulations are indeed convergent, in the sense that the approximate numerical solution fN of the continuous operator equation Lf = g has the property fN → f as N → ∞. The aim of this appendix is to give a brief summary of the current status of this – which readers may be surprised to learn is far from a closed subject.

With the FDTD, the Lax equivalence theorem (discussed in Chapter 2) provides us with confidence that refining the FDTD mesh will indeed result in a convergent solution. With the FEM, work in applied mechanics has provided a rich set of convergence results — although we should note that convergence for high-frequency electromagnetics problems is often in terms of the energy norm, as discussed in Chapter 12. This is a slightly weaker statement of convergence, since the energy norm does not satisfy all the properties of the norm. Also, these proofs are usually in terms of interpolation error; as has been noted, dispersion (or pollution) error is a different problem specific to the differential equation based solvers, but can usually be controlled by adequate meshing.

Optics is the study of wave propagation and its quantum implications, the latter now being generally called ‘photonics’. Traditionally, optics has centred around visible light waves, but the concepts that have developed over the years have been found increasingly useful when applied to many other types of wave, both inside and outside the electromagnetic spectrum. This chapter will first introduce the general concepts of classical wave propagation, and describe how waves are treated mathematically.

However, since there are many examples of wave propagation that are difficult to analyze exactly, several concepts have evolved that allow wave propagation problems to be solved at a more intuitive level. The latter half of the chapter will be devoted to describing these methods, due to Huygens and Fermat, and will be illustrated by examples of their application to wave propagation in scenarios where analytical solutions are very hard to come by. One example, the propagation of light waves passing near a heavy massive body, called ‘gravitational lensing’ is shown in Fig. 2.1; the figure shows two images of distant sources distorted by such gravitational lenses, taken by the Hubble Space Telescope, compared with experimental laboratory simulations. Although analytical methods do exist for these situations, Huygens' construction makes their solution much easier (§2.8).

A wave is essentially a temporary disturbance in a medium in stable equilibrium. Following the disturbance, the medium returns to equilibrium, and the energy of the disturbance is dissipated in a dynamic manner.

There are two sorts of textbooks. On the one hand, there are works of reference to which students can turn for the clarification of some obscure point or for the intimate details of some important experiment. On the other hand, there are explanatory books which deal mainly with principles and which help in the understanding of the first type.

We have tried to produce a textbook of the second sort. It deals essentially with the principles of optics, but wherever possible we have emphasized the relevance of these principles to other branches of physics – hence the rather unusual title. We have omitted descriptions of many of the classical experiments in optics – such as Foucault's determination of the velocity of light – because they are now dealt with excellently in most school textbooks. In addition, we have tried not to duplicate approaches, and since we think that the graphical approach to Fraunhofer interference and diffraction problems is entirely covered by the complex-wave approach, we have not introduced the former.

For these reasons, it will be seen that the book will not serve as an introductory textbook, but we hope that it will be useful to university students at all levels. The earlier chapters are reasonably elementary, and it is hoped that by the time those chapters which involve a knowledge of vector calculus and complex-number theory are reached, the student will have acquired the necessary mathematics.

This book is designed to serve as an introduction to computational electromagnetics for radio-frequency applications. It assumes the reader has completed typical undergraduate courses in electromagnetic field theory, and has some basic knowledge of antenna design and microwave systems.

For readers in a hurry, who already know which of the techniques discussed they would like to learn more about, it is possible to go directly to the relevant chapters, but it would nonetheless be useful first to read the introductory chapter. For those in a hurry, but who need first to find out which method (or methods) to use, this chapter is essential reading.

For readers who intend working through most of the book, it would be best to work through it in the sequence presented, although the chapters on the Sommerfeld formulation and practical applications thereof could be omitted without interrupting the sequence of presentation. A more detailed outline of the book may be found in Section 1.12; this will also assist readers to locate rapidly the parts of the book of interest to them.

At the end of each chapter, a list of references linked to the chapter topic is presented, for further reading and study.

Bessel functions come into wave optics because many optical elements – lenses, apertures, mirrors – are circular. We have met Bessel functions in several places (§8.3.4, §8.7, §12.2, §12.6.4 for example), although since most students are not very familiar with them (and probably becoming less so with the ubiquity of computers) we have restricted our use of them as far as possible. The one unavoidable meeting is the Fraunhofer diffraction pattern of a circular aperture, the Airy pattern, which is the diffraction-limited point spread function of an aberration-free optical system (§12.2). Another topic that involves the use of Bessel functions is the Fourier analysis of phase functions, in which the function being transformed contains the phase in an exponent. We met such a situation when we studied the acousto-optic effect, where a sinusoidal pressure wave affects directly the phase of the optical transmission function.

In this appendix we simply intend to acquaint the reader with the results that are necessary for elementary wave optics. The proofs can be found in the treatise by Watson (1958) and other places.

Introduction

With the theoretical background now established, one is in a position to start using commercial and public domain MoM programs intelligently. In this chapter, we will discuss primarily the application of the commercial code FEKO for antenna modelling, but will also discuss the use of the public domain code NEC-2 in this regard. Other than FEKO, few commercial programs (other than some proprietary NEC-2 extensions) provide good support for modelling thin-wire antennas, the topic of this chapter; such antennas are still very widely used indeed. For commercial programs, material is usually available to assist novice users to get started with the codes. Hence we will not describe the basic concepts of entering the geometry of the problem, including the source, and specifying parameters such as operating frequency and radiation patterns, since these vary from program to program, indeed quite often from release to release, and are usually quite well documented by the suppliers. However, in the case of NEC-2, some comments are in order.

NEC-2 is a “card driven” program, dating back to the days of “decks” of punched cards. A NEC model is described by a geometry file, usually with a .nec extension. An example is given in Fig. 5.1. If using NEC in this form, one must obtain a copy of the user manual [1].

The coherence of a wave describes the accuracy with which it can be represented by a pure sine wave. So far we have discussed optical effects in terms of coherent waves whose wave-vector k and frequency ω are known exactly; in this chapter we intend to investigate the way in which uncertainties and small fluctuations in k and ω can affect the observations in optical experiments. Waves that appear to be pure sine waves only if they are observed in a limited space or for a limited period of time are called partially coherent waves, and we shall see in this chapter how we can measure deviations of such waves from their pure counterparts, and what these measurements tell us about the source of the waves.

The classical measure of coherence was formulated by Zernike in 1938 but had its roots in much earlier work by Fizeau and Michelson in the late nineteenth century. Both of these scientists realized that the contrast of interference fringes between waves travelling by two different paths from a source to an observer would be affected by the size, shape and spectrum of the source. Fizeau suggested, and Michelson carried out, experiments which showed that the angular diameter of a star could indeed be measured by observing the degradation of the contrast of interference fringes seen when using the star as a source of light (§11.8.1).