Abstract

It is well known that constitutive parameters, namely, the electrical permittivity, ε, and the magnetic permeability, μ, in a medium determine the response and reaction of such medium or material when exposed to external time-varying electromagnetic fields. Furthermore, most materials are lossy and dispersive, that is, both permittivity and permeability are complex and frequency-dependent. Interestingly, by controlling the sign of real parts of ε and μ in a medium, unique electromagnetic properties can be achieved that are not readily available in nature. Recently, subwavelength composite engineered structures, also known as metamaterials, have evolved in many engineering and optical applications, due to their unique electromagnetic properties that are not found in nature, including but not limited to negative refractive index, backward wave propagation, subwavelength focusing and super lenses, and invisibility cloaking. The main aims of this chapter are to provide an overview of electromagnetic field behavior and interaction with metamaterials and to explore such behavior in various metamaterials both analytically and numerically.

Keywords: double negative medium, electromagnetic waves, metamaterials, plane wave, single-negative medium

## 1. Introduction

Electromagnetic field is a physical behavior that is produced in a space due to time-varying electric charges and represents the interaction between electric and magnetic fields. Unlike static charges that can only produce static electric fields in space, time-varying electric charges are one of sources for the rise of magnetic fields, which in turn produce time-varying electric fields. This is summarized in the four time-varying Maxwell's equations given in differential form:

$$\nabla.\mathbf{E} = \rho\_v(\mathbf{t})/\mathbf{e} \tag{1}$$

$$\nabla.\mathbf{B} = \mathbf{0} \tag{2}$$

$$\nabla \times \mathbf{E} = -\mu \frac{\partial \mathbf{H}}{\partial t} \tag{3}$$

$$\nabla \times \mathbf{B} = \mathbf{J}(t) + \varepsilon \frac{\partial \mathbf{E}}{\partial t} \tag{4}$$

where ρ<sup>v</sup> is time-varying volume charge density, ε and μ are the electric permittivity and magnetic permeability, respectively, J is the time-varying electric current

density in a medium, D and B are time-varying electric and magnetic flux densities, respectively, and E and H are time-varying electric and magnetic field intensities, respectively.

In 1864, James Maxwell showed through Eqs. (1)–(4) that oscillating electric and magnetic fields give rise to electromagnetic waves that travel at the speed of light in free space [1], which also implies that light is electromagnetic in nature. By taking the curl of Eqs. (3) and (4), it is also straightforward to show that electromagnetic wave propagation can exist.

In a medium, there are two main quantities, also known as the constitutive parameters, namely, electric permittivity, ε, and the magnetic permeability, μ, in addition to the conductivity, σ, that determine the nature of electromagnetic wave and its behavior in such a medium. In other words, the aforementioned parameters along with the boundary conditions in a medium determine uniquely the response of such medium to an incoming electromagnetic wave. This is also summarized through two equations, given below, that describe the relationship between electric and magnetic field quantities in a simple linear and isotropic medium:

$$\mathbf{D} = e\mathbf{E} \tag{5}$$

$$\mathbf{B} = \mu \mathbf{H} \tag{6}$$

where in Eqs. (5)–(6), both ε and μ in a lossy dispersive medium are commonly complex and frequency-dependent and are real quantities in a lossless isotropic medium. From such relations, Eqs. (1)–(6), important parameters, such as the wavenumber, k, the refractive index, n, and the intrinsic wave impedance, η, in a medium can be determined, which are given respectively as:

$$\mathbf{k} = \alpha \sqrt{\mu \varepsilon} \tag{7}$$

$$
\eta = \sqrt{\mu/\varepsilon} \tag{8}
$$

$$m = \sqrt{\mu\_r \varepsilon\_r} \tag{9}$$

where ω = 2πf is the radian frequency (in rad/sec), f is the frequency (in Hz), and μ<sup>r</sup> = μ/μ<sup>0</sup> and ε<sup>r</sup> = ε/ε<sup>0</sup> are the relative permeability and permittivity, respectively, while μ<sup>0</sup> and ε<sup>0</sup> are the free-space permeability and permittivity, respectively.

### 2. Overview of metamaterials and their realizations

Figure 1 depicts a general overview of possible materials based on their constitutive parameters: the electric permittivity and the magnetic permeability values. The aforementioned constitutive parameters are in principle complex, and their signs are based on the signs of their real parts, while their imaginary parts indicate the presence of electric or magnetic losses, respectively. While in naturally occurring materials, both real parts of the permittivity and permeability are positive (i.e., >0); it is possible that either one of the real parts of the constitutive parameters or even both have negative values. In the second quadrant, while permeability is above zero, the permittivity is below zero (negative), which can be termed as a singlenegative (SNG) or ε-negative (ENG) medium. Similarly, when a medium possesses negative permeability value, while its permittivity is positive, this is also termed as an SNG medium or μ-NG (MNG) medium, where μ < 0 in this type of material, as shown in the third quadrant in Figure 1. An interesting medium is the case when both real parts of permittivity and permeability are negative (i.e., the third quadrant in Figure 1). This is termed as double negative (DNG) medium or left-handed

Electromagnetic Field Interaction with Metamaterials DOI: http://dx.doi.org/10.5772/intechopen.84170

Figure 1.

Classification of materials, based on their constitutive parameters, ε and μ.

medium (LHM), due to its unique resultant electromagnetic features, like negative refraction and negative phase velocity, as it follows a left-handed system rule. In summary, metamaterials have three classes, depending on the signs of their constitutive parameters: ENG, MNG, and DNG.

Practically, natural SNG media are available that possess ENG response, for instance, metals at visible and near-ultraviolet regime. However, at much lower frequencies, one commonly adopted realization of SNG medium is the periodic arrangement of metallic wires, which results in possessing negative effective permittivity below the plasma frequency of metallic wires or rods [2]. It is instructive to mention here that naturally occurring materials with permeability values below zero are not yet available in nature, especially within the radio frequency/microwave regime. However, such response can be obtained through engineered arrangement of metallic inclusions printed on a dielectric medium [3], as it will be discussed further later on.

After the seminal work of Veselago in [4], where he investigated mathematically the possibility of electromagnetic wave propagation through materials with both negative permittivity and permeability values, the word "metamaterials" evolved, which refers to what is beyond naturally occurring materials. Metamaterials can be defined as artificially engineered structures that have electromagnetic properties not yet readily available in nature. Such artificial composite structures are realized in one way by periodically patterning metallic resonant inclusions in a host medium, i.e., dielectric or magneto-dielectric material, either in a symmetric or nonsymmetric fashion. When exposed to an electromagnetic field, the metamaterials alter the electromagnetic properties of the new host medium due to mainly the inclusions' response and features. Figure 2 depicts a general view of one possible realization of a metamaterial structure.

Tremendous efforts had been put forward in the past with the goal to provide efficient numerical means for the retrieval of constitutive parameters of arrays of metamaterials in order to advance the design and characterization of metamaterials [5–7]. Such numerical retrievals provided engineering and physical means in replacing local electromagnetic response details of individual metamaterials elements with averaged or homogenized values for the effective electric permittivity, εeff, and effective magnetic permeability, μeff. As a matter of fact, this retrieval approach is a direct translation of the characterization of natural media, which consist of atoms and molecules with their dimensions that are much smaller in magnitude than the wavelength. The electromagnetic wave response and propagation within the effective metamaterial medium can then be fundamentally

Figure 2. General sketch of a metamaterial composite structure.

described using constitutive parameters along with Maxwell's equations. In principle, this effective response would be permissible if the unit cell dimension is sufficiently small enough or a fraction of an operating wavelength [5], say, for example, L, as shown in Figure 2, satisfies the relation below:

$$L \ll \lambda \tag{10}$$

where L is the unit cell dimension and λ is the operating wavelength of the incoming electromagnetic field. When the condition in Eq. (10) holds, quasi-static behavior for the artificial metamaterials can be applied, in which an equivalent resonant circuit, composed of resistor-inductor-capacitor (RLC) elements, is permissible to use in order to provide qualitative description of the physical behavior of the artificial materials [8].
