Compendium on Electromagnetic Analysis

The following sections are included:

• Foreword
• Preface from the Editor-in-Chief
• Preface from the Editor
• About the Editor-in-Chief
• About the Editor
• Contents

In this chapter, we present classical electrodynamics as a theory with two separate components. On the one hand, Maxwell’s equations are stated in a way that is completely independent of the ambient medium and relies only on the metric-independent and differential structure of space–time. On the other hand, when homogenization is possible, various media are described by constitutive relations, which can be of a very complicated form and include the metric aspects. After a general introduction, some classical examples of constitutive laws are discussed.

The scalar and dyadic electromagnetic Green’s functions in spaces of two and three dimensions are studied in this chapter. The equations they satisfy, representations for them in both momentum and coordinate space, the singularities they possess, their regularization, and their behavior in the far field are discussed. Applications of these functions are illustrated by their use in determining the scattered field when a small particle, or a randomly or periodically rough impenetrable surface, is illuminated by an electromagnetic field. The numerical solution of the integral equations that arise in obtaining these scattered fields is described, as well as how the singularities of these Green’s functions and their derivatives can be dealt with in these calculations.

This chapter is devoted to the problem of interaction of electromagnetic fields with heterogeneous media. We restrict our attention, first, to the classical phenomena by assuming that the scale of inhomogeneities is sufficiently large so that all quantum effects can be neglected and, second, to one-dimensional systems. These two assumptions mean that the object under consideration is a regular or disordered system of plane-parallel dielectric layers. These assumptions do not result in a complete disappearance of interesting phenomena but make it possible to solve many problems analytically. In addition, the ease of experimental realization has traditionally attracted many researchers to one-dimensional layered systems.

Photonic crystals are periodic structures, which inhibit the propagation of light in one or more directions in certain frequency intervals. The band spectrum of a photonic crystal is usually analyzed using the Floquet–Bloch decomposition. This spectrum is located on the real axis and is continued to the complex plane when absorption and dispersion in the dielectric permittivities of the material constituents of a photonic crystal are taken into consideration. In this chapter, we review the fundamental definitions and properties of the dispersion law and the group velocity of electromagnetic waves in photonic crystals. Theoretical considerations are illustrated with numerical examples.

We discuss some of the basic topological concepts that can be applied to describe and design photonic topological systems. For periodic systems, we used to focus on the symmetry properties of the Bloch wave functions at each point in k-space. Recently, it was discovered that the global property of the Bloch wave functions and the change in their symmetry across the Brillouin zone (BZ) manifest as topological invariants which can be used to classify periodic systems into different classes. In addition, there is an elegant bulk-edge correspondence that relates the topological property of the bulk and that at the boundary. These concepts have shown a profound influence in condensed matter physics recently, and we will see that they can be applied equally well to photonic systems. We begin by introducing topological notions such as Berry curvature and topological invariants such as Chern numbers. We then review the recent development of topological photonic materials, and we use a simple model to illustrate the principle of designing photonic crystals that carry synthetic gauge flux and topological Weyl points. We provide examples of topologically non-trivial photonic crystals, and we will see how these novel materials and their edge modes are characterized.

In this chapter, we expound the foundations of the theory of electromagnetic field interaction with gain media. We focus on the stimulated emission, while the induced emission is taken into account in the relaxation terms. The gain medium is described using the density matrix formalism from which the Maxwell–Bloch equations are derived. The cases of two-, three-, and four-level quantum systems are considered. In the limit of weak fields, the linear approach is introduced in which a gain medium is described in terms of a complex permittivity with a negative imaginary part. Using this approach, the examples of propagation through a slab and the Kretschmann illumination scheme are considered.

The radiative transport equation (RTE) is widely used to describe the propagation of multiply-scattered light in disordered media. In this chapter, we derive and investigate the RTE for electromagnetic waves in random media. The derivation is performed by means of diagrammatic perturbation theory and holds for statistically homogeneous weakly-scattering random media.

The following sections are included:

• Introduction to the Index
• Index