Electromagnetic Anisotropy and Bianisotropy

The following sections are included:

• Frontispiece
• Prologue
• Dedication
• Preface
• Acknowledgments
• Acronyms and Principal Symbols
• Contents

The action of complex arrangements of matter on electromagnetic waves lies at the heart of this book. Herein, we generally regard matter from a macroscopic perspective, which means that electromagnetic wavelengths are assumed to be large compared with interatomic distances — the atomic (and sub-atomic) nature of matter does not directly concern us. Accordingly, ‘mediums’ and ‘materials’ are referred to in this book, rather than ‘assemblies of atoms and molecules’. The theoretical basis is provided by the Maxwell postulates for macroscopic fields combined with constitutive relations. The fundamental features of macroscopic electromagnetic theory, which underpin the remainder of the book, are presented in this chapter.

A simple understanding of the notions of anisotropy and bianisotropy is that the adjectives ‘anisotropic’ and ‘bianisotropic’ describe mediums which are not isotropic. Accordingly, prior to discussing anisotropic and bianisotropic mediums in this chapter, we first establish our terms of reference by briefly considering isotropic mediums. We then present a survey of the commonly encountered classifications of anisotropy and bianisotropy, as expressed in terms of their constitutive relations and constitutive dyadics. Our attention is restricted in this chapter to linear anisotropic and bianisotropic mediums; a description of the more complex constitutive relations associated with nonlinear anisotropy and bianisotropy is postponed to Chap. 7. Furthermore, we are chiefly concerned with nonactive mediums (cf. Sec. 1.7.2.2).

The electromagnetic properties of a linear medium are characterized in terms of its constitutive dyadics: the four 3×3 dyadics ${\underset{\_}{\underset{\_}{ϵ}}}_{\text{EH}}(\underset{\_}{r},\omega ),{\underset{\_}{\underset{\_}{\xi }}}_{\text{EH}}\left(\underset{\_}{r},\omega \right),{\underset{\_}{\underset{\_}{\zeta }}}_{\text{EH}}\left(\underset{\_}{r},\omega \right)$ and ${\underset{\_}{\underset{\_}{\mu }}}_{\text{E}\text{H}}(\underset{\_}{r},\omega )$ in the Tellegen representation (or ${\underset{\_}{\underset{\_}{ϵ}}}_{\text{EB}}(\underset{\_}{r},\omega )$, ${\underset{\_}{\underset{\_}{\xi }}}_{\text{E}\text{B}}(\underset{\_}{r},\omega ),{\underset{\_}{\underset{\_}{\zeta }}}_{\text{E}\text{B}}(\underset{\_}{r},\omega )$ and ${\underset{\_}{\underset{\_}{v}}}_{\text{E}\text{B}}(\underset{\_}{r},\omega )$ in the Boys–Post representation). The symmetries of these constitutive dyadics reflect the underlying spacetime symmetries of the medium. Indeed, the general form of the constitutive dyadics may be deduced from a consideration of macroscopic spacetime symmetries. In this chapter, we largely focus upon those spacetime symmetry operations which bring a medium into self-coincidence, leaving a point within the medium fixed in position. These point symmetry operations — which form the basis of the point group classifications — determine the general forms of the constitutive dyadics…

The passage of electromagnetic signals through a complex medium is generally a complicated issue, from a theoretical perspective at least. Often it is best broached by considering the propagation of plane waves. Although plane waves themselves are idealizations, being of limitless spatial and temporal extents and possessing infinite energy, their consideration can provide a reasonable understanding of fields far away from sources. Furthermore, realistic signals may be usefully represented as superpositions of plane waves…

Often in practical situations one wishes to find the (frequency-domain) electromagnetic field generated by a given distribution of sources immersed a specific linear, homogeneous medium. By virtue of the linearity of the Maxwell postulates, this fundamental issue may be tackled by means of dyadic Green functions (DGFs) [1, 2]. However, the explicit delineation of DGFs for complex mediums poses formidable challenges to theorists. Closed-form representations of DGFs are available for isotropic mediums and for some classes of relatively simple anisotropic and bianisotropic mediums, but not for general anisotropic and bianisotropic mediums [2]. Notwithstanding that, integral representations of DGFs can always be found. Furthermore, for certain applications — such as in the determination of the scattering response of electrically small particles, used in homogenization studies — suitable approximative solutions may be constructed using only the singular part of the DGF, which gives rise to the depolarization dyadic [2, 3]. A short survey of DGFs and depolarization dyadics for unbounded anisotropic and bianisotropic mediums is presented in this chapter. All of the mediums considered are homogeneous, with the exception of HBMs in Sec. 5.4.2.

A mixture of two (or more) different mediums may be viewed as being effectively homogeneous, provided that wavelengths are much longer than the length-scales of the mixture’s nonhomogeneities [1–4]. In fact, the process of homogenization implicitly underpins our descriptions of anisotropy and bianisotropy, since the constitutive relations relate macroscopic electromagnetic fields which represent the volume-averages of their microscopic counterparts [5, 6]. On a more practical level, homogenization is important in the interpretation of experimental measurements and in material design. This latter topic, in particular, continues to motivate much research [4]. For example, bianisotropic mediums may be readily conceptualized as homogenized composite mediums (HCMs), arising from constituent mediums which are themselves not bianisotropic (or even anisotropic in certain instances)…

In the previous six chapters, we have largely concentrated on mediums in which the induction fields are linearly proportional to the primitive fields. We now focus on the considerably more complex scenario wherein the proportionality is nonlinear. Mediums described by nonlinear constitutive relations have been studied and exploited for technological purposes since the earliest days of electromagnetics. For example, the nonlinear permeability of ferromagnetic mediums was widely harnessed in electrical machines of the nineteenth century [1]. The Luxembourg effect — which involves the transfer of amplitude modulation between radio waves of different frequencies in the ionosphere, as observed by Tellegen in 1933 [2] — is due to the nonlinearity of plasma at sufficiently high wave intensities [3]. Nonlinear properties at optical wavelengths are generally exhibited only at high light intensities; thus, the speciality of nonlinear optics grew rapidly in the 1960’s following the invention of the laser [1, 4]. The combination of nonlinearity and anisotropy or bianisotropy presents formidable challenges to theorists, but the vast scope for complex behavior also presents tantalizing opportunities.

Reflection and refraction are associated with abrupt changes in constitutive dyadics at the planar interface of two dissimilar mediums. When total internal reflection occurs, Isaac Newton realized that the incident wave, which appears not to enter the refracting medium, actually penetrates that medium with an amplitude that decays exponentially with distance from the interface on the microscopic scale [1]. The penetrating wave is nowadays being exploited for near-field spectroscopy [2]…

From a macroscopic viewpoint, a topological insulator functions as an electrical insulator within its interior but as an electrical conductor on its surface. The crucial distinction between a topological insulator and a regular electrical insulator with a conducting surface is that the surface conducting states of a topological insulator are protected by time-reversal symmetry, for quantum-mechanical reasons [1]. The macroscopic electromagnetic characteristics of topological insulators began to be investigated just a few years ago [2–4]. For example, the topologically insulating chalcogenides, such as Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃ [5–7], are possible vehicles for Lorentz nonreciprocity in the terahertz regime, without the need for an external agent such as an applied field or translational motion. This functionality could be exploited for one-way optical devices, for example…

The following sections are included:

• Appendix A: Dyadic Notation and Analysis
• Epilogue
• Index
• About the authors