Media, Electromagnetic Characteristics

Definition

Media, electromagnetic characteristics. Macroscopic permittivity and permeability properties of media.

Introduction

When electromagnetic wave propagates in some medium other than vacuum, since the characteristic wavelength is several orders larger than the atoms of which the medium is composed, the detailed behavior of the fields over atomic distance becomes irrelevant. What do matter are the quantities averaged over the atomic scale, including the macroscopic fields and macroscopic sources. Such treatment implies that the inhomogeneous medium is essentially replaced by a homogeneous one with macroscopic permittivity and permeability (Jackson, 1998).

Macroscopic properties

In macroscopic media, the electric displacement field $\mathbf{ D}$ and magnetic field $\mathbf{ H}$ are related to the electric field $\mathbf{ E}$ and magnetic induction $\mathbf{ B}$ through the macroscopically averaged electric dipole, magnetic dipole, electric quadrupole, and higher-order multipoles. In most materials, only the electric and magnetic polarizations are significant.

In an isotropic medium, the constitutive relations are $\mathbf{ D}=\varepsilon \mathbf{ E}$, and $\mathbf{ B}=\mu \mathbf{ H}$, where ε and μ are the permittivity and permeability, respectively. A diamagnetic medium has μ larger than μ₀, since diamagnetic substance, whose atoms or molecules have no angular momentum, creates induced magnetic moments that tend to oppose the applied magnetic field. A paramagnetic medium has μ smaller than μ₀, since paramagnetic substance has a net angular momentum which is aligned parallel to the applied magnetic field.

For anisotropic media, the constitutive relations are $\mathbf{ D}=\overline{\overline{\varepsilon}}\mathbf{ E}$ and $\mathbf{ B}=\overline{\overline{\mu}}\mathbf{ H}$, where $\overline{\overline{\varepsilon}}$ and $\overline{\overline{\mu}}$ are the permittivity and permeability tensor, respectively. Crystals are generally characterized by symmetric permittivity tensors, which can be transformed into a diagonal matrix as

$$\overline{\overline{\varepsilon}}=\left[\begin{matrix}{{\varepsilon_x}} & 0 &0 \cr 0 &{{\varepsilon_y}} & 0 \cr 0 & 0 & {{\varepsilon_z}}\end{matrix}\right] $$

When ε_x = ε_y = ε_z, the crystals are isotropic, such as the cubic crystals. Uniaxial crystals have two of the three parameters equal. Examples are tetragonal, hexagonal, and rhombohedral crystals (Kong, 2005).

For bianisotropic media, the constitutive relations are $\mathbf{ D}=\overline{\overline{\varepsilon}}\mathbf{ E}+\overline{\overline{\xi}}\mathbf{ H}$ and $\mathbf{ B}=\overline{\overline{\zeta}}\mathbf{ E}+\overline{\overline{\mu}}\mathbf{ H}$ (Kong, 2005).

Dispersion

All media are dispersive in that the phase velocity of a wave depends on the frequency. That is, waves with different frequencies travel in a dispersive medium at different speeds. The frequency dependence of ε and μ leads to new effects when an arbitrary wave train containing a range of frequencies travels.

What enter the picture of dispersion are the molecular constitution of matter and the dynamics of molecules. A simple model leads to a dispersion formula which in most cases serves as an adequate representation of the dielectric constant as a function of frequency. The success of this model is truly amazing considering the several assumptions being made, including treating the relative permeability equal to unity, neglecting magnetic force effects, and confining the oscillation to be sufficiently small. The forces exerting upon an electron include an electric force due to the electric field, a restoring force, and a damping force. The resultant formula for the dielectric constant is

$$ \varepsilon (\omega )={\varepsilon_0}+\frac{{N{e^2}}}{m}\sum\limits_k {\frac{{{v_k}}}{{\omega_k^2-{\omega^2}-i\omega {\gamma_k}}}}, $$

where – e and m are the charge and mass of an electron, respectively. Here it is assumed that there are N molecules per unit volume and v_k electrons per molecule with binding frequency ω_k and phenomenological damping constant γ_k (Jackson, 1998).

When real part of $\varepsilon (\omega )$ increases with ω, it is called normal dispersion. Anomalous dispersion refers to the reverse. Since in general the resonant frequencies ω_k are large compared to the damping factors γ_k, in the neighborhood of ω_k, the behavior of $\varepsilon (\omega )$ is expected to be rather violent, with disruption of the normal dispersion and appearance of appreciable imaginary part of $\varepsilon (\omega )$ or resonant absorption.

At frequencies far beyond the highest resonant frequency, the dielectric constant is simply expressed in terms of the plasma frequency ω_p of the medium as $\varepsilon (\omega )\approx {\varepsilon_0}-\frac{{\omega_p^2}}{{{\omega^2}}}{\varepsilon_0}$.

The Kramers-Kronig relations or dispersion relations are simple integral formula relating the real part to imaginary part of the complex permittivity $\varepsilon (\omega )$ or a dispersive process to an absorption process. The validity of these relations is very general due to the fact that very few assumptions are made, among which are the monochromatic components of the displacement $\mathbf{ D}(\mathbf{ r},\omega )$ and the electric field $\mathbf{ E}(\mathbf{ r},\omega )$ at position $\mathbf{ r}$ related by $\mathbf{ D}(\mathbf{ r},\omega )=\varepsilon (\omega )\mathbf{ E}(\mathbf{ r},\omega )$, as well as a causality requirement on the kernel relating $\mathbf{ D}(\mathbf{ r},t)$ and $\mathbf{ E}(\mathbf{ r},t)$. The wide generality of the dispersion relations makes them very useful in all areas of physics (Toll, 1956). For instance, in the scattering of nuclear particles, these relations connect real and imaginary parts of the diagonal elements of the scattering matrix (Jost et al., 1950).

Inhomogeneous composite

A macroscopically inhomogeneous medium is a typical representation of many materials. For instance, a snow pack is a composite of ice particles and water; a porous rock is a composite of the rock matrix and salt water if the latter is present. In each example, the medium possesses spatially varying quantities such as permittivity and conductivity.

Electromagnetic characterization of inhomogeneous composite materials is important in a wide variety of applications, such as in astronomy and atmospheric physics and selective absorbers of solar and infrared radiation.

In the effective medium approximation of Bruggeman and Landauer, a composite of two components is considered. To obtain the effective conductivity of this composite, each particle is imagined to be immersed in a homogeneous effective medium of conductivity σ_e instead of in its actual inhomogeneous medium. The self-consistency condition is invoked such that the average electric field within a particle shall equal the electric field far from the particle. In essence, in the effective-medium approximation, only electron-dipole scattering contributes. The characteristic particle dimension in the composite is required to be small compared to the characteristic wavelength. If a two-component composite has one component dominant in volume fraction, then both approaches work well. However, it is found that EMA produces significant error in far-infrared absorption by a composite composed of dielectric and small metal particles with concentrations below the percolation threshold (Stroud, 1998).

A self-consistency condition is to choose an effective dielectric constant such that for particles embedded in the medium, on the average, their forward scattering amplitude should vanish (Stroud and Pan, 1978). Generalization to continuous size distribution of particles as well as arbitrary number of components is provided in Chylek and Srivastava (1983), where both the contribution of electric and magnetic dipole terms are considered. Special attention is paid to a composite material where small metallic particles are among the components. The form of the size distribution of metallic particles is found to determine the critical volume fraction at which there is a drastic absorption increase.

Metamaterial

Metamaterial is a new class of artificially constructed electromagnetic material that can exhibit electromagnetic properties difficult or impossible to find in conventional materials (Pendry, 2004; Smith et al., 2004). For example, the material property of negative index of refraction has been realized at both GHz and optical frequencies using metamaterial concepts.

Recent advances in construction of metamaterial with independent and arbitrary varying capability of permittivity and permeability values pave the way for totally new electromagnetic phenomena. For instance, the electric displacement $\mathbf{ D}$, the magnetic field intensity $\mathbf{ B}$, and the Poynting vector $\mathbf{ S}$, by controlling the material electromagnetic properties, can be directed at will, made focused, or avoid objects. In particular is the design of cloaking of objects from electromagnetic fields (Pendry et al., 2006).

Summary

In macroscopic media, the electric displacement field and magnetic induction can be related to the electric intensity and magnetic intensity in ways quite different from that in vacuum. With increasing complexity, the medium can be characterized as isotropic, anisotropic, and bianisotropic. All media are dispersive in that the phase velocity of a wave depends on the frequency. The real and imaginary parts of the frequency-dependent complex permittivity are related by the Kramers-Kronig relations, which are very useful in all areas of physics. In characterizing the electromagnetic properties of an inhomogeneous medium, the effective-medium approximation is useful within its region of validity. Beyond that, the method base on the self-consistency condition can be employed. In terms of artificial materials, metamaterial is a new class of artificially constructed electromagnetic material that can exhibit electromagnetic properties difficult or impossible to find in conventional materials, which leads to totally new electromagnetic phenomena.

Cross-references

Electromagnetic Theory and Wave Propagation