Chapter 1
Introduction

1.1 Basic Equations

Maxwell Equations

Electromagnetic problems involving fields and sources are governed by the Maxwell equations, medium equations and boundary conditions. For time-harmonic fields and sources, with time dependence , the Maxwell equations, in the form of Heaviside [19] and the formalism of Gibbs [13], can be expressed as

Here, and denote the respective electric and magnetic current density vectors while E, H, D and B represent the respective electric field, magnetic field, electric flux-density and magnetic flux-density vectors. All of these vectors may have complex components.

Medium Conditions

Conditions between the field vectors in a bianisotropic medium can be expressed in the form [29, 35]

where and are medium dyadics. In terms of a given vector basis, any dyadic can be represented in terms of a matrix involving nine scalar components (for rules of dyadic algebra, see Appendix C).

Here we mainly consider problems involving the simple-isotropic medium ('free space, vacuum'), for which the medium conditions are reduced to

The wave impedance , expressing the ratio of the electric and magnetic field magnitudes in the simple-isotropic medium, is defined by

Boundary Conditions

Boundary conditions form an essential part of formulating electromagnetic problems. By a boundary we mean a surface on which secondary electromagnetic sources are induced by the primary fields so that the fields beyond the surface vanish. A boundary surface is different from an interface of two media because the fields beyond the interface are not necessarily zero. From the mathematical point of view, boundary conditions are required to ensure existence and uniqueness of solutions for a particular problem. From the engineering point of view, two aspects of boundary conditions can be separated which may be called analytic and synthetic.

An analytic aspect of boundary conditions is encountered when a given physical problem requires mathematical analysis. Due to natural complications, a given structure must often be approximated by certain boundary conditions to find a numerical solution [21, 88]. As an example, solving radio wave propagation over ground requires that the ground be approximated by an impedance boundary or, if well-enough conducting, by the perfect electric conductor (PEC). Thus, in such a case, to analyze the effect of a given structure, we replace it by some approximate boundary conditions.

In contrast, a synthetic aspect emerges when we wish to realize given boundary conditions by some physical structure. As an example, when designing mobile phones with submerged antennas, the concept of perfect magnetic conductor (PMC) boundary has been suggested to solve the problem of efficient radiation [92]. In this case, the problem becomes a synthesis, how to find a physical structure to realize the PMC conditions.

In this book we are concerned about the synthetic aspect by studying different types of boundary conditions, their effect on electromagnetic fields, and possible realizations by interfaces of media defined by medium parameters. More practical physical realizations by (meta) materials are, however, beyond the topic of the book. Let us start with three most basic boundary conditions, properties of which will be considered in due course.

• Perfect Electric Conductor (PEC) (1.6)
• Perfect Magnetic Conductor (PMC) (1.7)
• Perfect Electromagnetic Conductor (PEMC) [45] (1.8) The PEMC, involving a parameter (the PEMC admittance), is a generalization of both the PEC and the PMC. For , the PEMC reduces to the PMC, and for , it reduces to the PEC.

1.2 Duality Transformation

A given electromagnetic problem can be transformed to another one in terms of duality transformation which does not change the geometry of the problem, but the sources, fields, medium and boundary parameters are transformed to have other values, in general. The concept of duality was evidently unknown to Maxwell, because he presented his equations in a very nonsymmetric form, in terms of 20 scalar field and potential quantities [79]. The concept was introduced by Heaviside in 1886 [19].

Duality transformation is based on the apparent symmetry of the Maxwell equations (1.1), (1.2), written more compactly as

In fact, for the simple change of symbols , and , the pair of equations (1.9) is invariant. More generally, the same property can be expressed in terms of the duality transformation , defined by [35]

The wave impedance has been included to obtain dimensionless transformation parameters . They are assumed to satisfy

whence the inverse transformation exists and has the form

Applying (1.10) to (1.1) and (1.2), the associated transformation rules can be expressed as

One can show that the PEMC boundary conditions (1.8) are transformed to

with the transformed PEMC admittance satisfying

From this it follows that both PEC and PMC boundaries are transformed to PEMC boundaries with and , respectively. Also, any given PEMC boundary can be transformed to PEC and PMC boundaries when the transformation parameters are chosen to satisfy the respective restrictions and .

In the general case, the dyadic parameters of the electromagnetic medium will be changed when the fields are subject to the duality transformation (1.10). For a bianisotropic medium defined by conditions of the form (1.3), the transformed medium dyadics can be shown to obey the relations [42]

Requiring a transformation in which the simple isotropic medium (, ) is invariant leads to a choice of the form

where is a free transformation parameter. The resulting transformation matrix

can be recognized as a rotation matrix. The corresponding transformation rule for the medium dyadics (1.17) becomes in this case

with the matrix defined by

For the simple-isotropic medium with and , from (1.20) we obtain and , as required. One can further show that the matrix satisfies

i.e., it is an orthogonal matrix. The proof is left as an exercise.

Expressing the PEMC admittance parameter in terms of another parameter as

the transformation rule (1.16) takes the simple form

Thus, for the choice of the duality parameter, the PEMC boundary is transformed to the PMC boundary , while for , it is transformed to the PEC boundary .

1.3 Plane Waves

Basic Conditions

Let us consider time-harmonic plane waves with dependence in a simple isotropic medium, in front of a boundary surface defined by . For simplicity we assume that the surface is a plane, i.e., that is a constant unit vector. The electric and magnetic field components of waves incident to the boundary are expressed by

and the reflected fields are

To satisfy the Maxwell equations, the two wave vectors must satisfy the dispersion equation as

If the boundary condition is linear, the and vectors must have the same components tangential to the boundary surface, i.e., they can be represented as

Figure 1.1: The incident and reflected plane waves have the same wave-vector components tangential to the boundary surface.

A class of boundaries with conditions deviating from linear ones is considered in Chapter 6.

Substituting the plane-wave fields in the Maxwell equations (1.1) and (1.2), equations relating the electric and magnetic fields are obtained as

and

The field components normal to the boundary can be expressed in terms of their tangential components from the orthogonality conditions as

valid for . Similar conditions are valid for the magnetic fields.

Field Relations

From (1.33) and (1.34) we obtain the following relations between the tangential components of the incident fields:

Assuming also...