Propagation of Electromagnetic Waves in The Medium

Introduction

My research mostly deal with the response of materials when electromagnetic waves sent to the materials. My work usually use microwave and millimeter wave spectral range.

In this post, I want to explain the response of materials in electromagnetic fields, particularly the propagation of electromagnetic waves in the materials.

Maxwell’s Equations in The Presence of Matter

The equations are:

\displaystyle \nabla \times \mathbf{E} + {1 \over c} \frac{\partial \mathbf{B}}{\partial{t}} = 0, \ \ \ \ \ (1)

\displaystyle \nabla \cdot \mathbf{B}=0, \ \ \ \ \ (2)

\displaystyle \nabla \times \mathbf{H}-{1 \over c}{\partial \mathbf{D}\over{\partial t}}={4 \pi \over c}\mathbf{J}_{cond}, \ \ \ \ \ (3)

\displaystyle \nabla \cdot \mathbf{D}=4\pi \rho_{ext}, \ \ \ \ \ (4)

The presence of a medium in electromagnetic fields may lead to electric dipoles and magnetic moments, polarization charges, and induced current. Clearly the electromagnetic fields will not be uniform within the material but fluctuate from point to point reflecting the periodicity of the atomic lattice. For wavelengths appreciably larger than the atomic spacing, nevertheless we can consider an average value of the electric and magnetic fields. These fields, however, are different compared with the fields in vacuum; consequently {\bf{D}} and {\bf{H}} are introduced to account for the modifications by the medium.

The total charge density {\rho=\rho_{total}} has two components

\displaystyle \rho_{total}=\rho_{ext}+\rho_{pol}, \ \ \ \ \ (5)

an external charge {\rho_{ext}} added from outside and a contribution due to the spatially varying polarization

\displaystyle \rho_{pol}=-\nabla \cdot \mathbf{P}. \ \ \ \ \ (6)

In a homogeneous neutral material (not in electromagnetic fields) the positive and negative charges cancel everywhere inside the material, leading to no net charge {\rho_{pol}} .

Let’s assume there is no external current present: {\mathbf{J}_{ext}=0}. The total current density {\mathbf{J}=\mathbf{J}_{total}} consists of a a contribution {\mathbf{J}_{cond}} arising from the motion of electrons in the presence of an electric field and of a contribution {\mathbf{J}_{bound}} arising from the redistribution of bound charges:

\displaystyle \mathbf{J}_{total}=\mathbf{J}_{cond}+\mathbf{J}_{bound}. \ \ \ \ \ (7)

Ohm’s law is assumed to apply to this conduction current

\displaystyle \mathbf{J}_{cond}=\sigma_1 \mathbf{E}. \ \ \ \ \ (8)

{\sigma_1} is the conductivity of the material.

The electric field strength {\mathbf{E}} and the electric displacement {\mathbf{D}} are connected by the dielectric constant (or permittivity) {\epsilon_1}:

\displaystyle \mathbf{D}=\epsilon_1\mathbf{E}=(1+4\pi\chi_e)\mathbf{E}=\mathbf{E}+4\pi\mathbf{P} \ \ \ \ \ (9)

where {\chi} is the dielectric susceptibility and {\mathbf{P}=\chi_e\mathbf{E}} is the dipole moment density or polarization density. The dielectric constant {\epsilon_1} can be either positive or negative. Similarly, the magnetic field strength {\mathbf{H}} is connected to the magnetic induction {\mathbf{B}} by the permeability {\mu_1}:

\displaystyle \mathbf{B}=\mu_1\mathbf{H}=(1+4\pi\chi_m)\mathbf{H}=\mathbf{H}+4\pi\mathbf{M}, \ \ \ \ \ (10)

where {\chi_m} is the magnetic susceptibility and {\mathbf{M}=\chi_m\mathbf{H}} is the magnetic moment density, or magnetization. The quantities {\epsilon_1, \chi_e, \mu_1}, and {\chi_m} which connect the fields are unitless. The magnetic susceptibility {\chi_m} is typically four to five orders of magnitude smaller (except in the case of ferromagnetism) than the dielectric susceptibility {\chi_e}, which is the order of unity. For this reason the dia- and para-magnetic properties can in general be neglected compared to the dielectric properties when electromagnetic waves pass through a medium. Frequently, we do not discuss the properties of magnetic materials and therefore we assume that {\mu_1=1}.

Using Eq. (9) and Ohm’s law (8) and recalling that there is no external current, Eq. (3) can be written as

\displaystyle c\nabla\times\mathbf{H}=-i\omega\epsilon_1\mathbf{E}+4\pi\sigma_1\mathbf{E}=-i\omega\hat{\epsilon}\mathbf{E}, \ \ \ \ \ (11)

where we have assumed a harmonic time dependence of the displacement term {\partial \mathbf{D}/\partial t = -i\omega \mathbf{D}}, and we have defined the complex dielectric quantity

\displaystyle \hat{\epsilon}=\epsilon_1+i{4\pi \sigma_1 \over \omega}=\epsilon_1+i\epsilon_2. \ \ \ \ \ (12)

By writing {\mathbf{D}=\hat{\epsilon}\mathbf{E}}, the change in magnitude and the phase shift between displacement {\mathbf{D}} and the electric field {\mathbf{E}} are conveniently expressed. {\epsilon_1} and {\epsilon_2} span a phase angle of {\pi /2}. Here {\epsilon_1} is the in-phase and {\epsilon_2} is the out-of-phase component. The notation accounts for the general fact that the response of the medium can have a time delay with respect to the applied perturbation. Similarly the conductivity can be assumed to be complex

\displaystyle \hat{\sigma}=\sigma_1+i\sigma_2 \ \ \ \ \ (13)

to include the phase shift of the conduction and the bound current, leading to more general Ohm’s law

\displaystyle \mathbf{J}_{tot}=\hat{\sigma}\mathbf{E} \ \ \ \ \ (14)

and we define the relation between the complex conductivity and the complex dielectric constant as

\displaystyle \hat{\epsilon}=1+{4\pi i \over \omega}\hat{\sigma}. \ \ \ \ \ (15)

Besides the interchange of real and imaginary parts in the conductivity and dielectric constant, the division by {\omega} becomes important in the limits {\omega \rightarrow 0} and {\omega \rightarrow \infty} to avoid diverging functions.

Wave equations in the medium

To find a solution of Maxwell’s equations we consider an infinite medium to avoid boundary and edge effects. Furthermore we assume the absence of free charges ({\rho_{ext}=0}) and external currents ({\mathbf{J}_{ext}=0}). We use a sinusoidal periodic time and spatial dependence for the electric and magnetic waaves. Thus,

\displaystyle \mathbf{E}(\mathbf{r}, t)=\mathbf{E}_0 exp (i(\mathbf{q\cdot r}-\omega t)) \ \ \ \ \ (16)

and

\displaystyle \mathbf{H}(\mathbf{r}, t)=\mathbf{H}_0 exp (i(\mathbf{q \cdot r}-\omega t- \phi)) \ \ \ \ \ (17)

describe the electric and magnetic fields with wavevector {\mathbf{q}} and frequency {\omega}. We have included a phase factor {\phi} to indicate that electric and magnetic fields may be shifted in phase with respect to each other. The wavevector {\mathbf{q}} has to be a complex quantity: to describe the spatial dependence of the wave it has to include a propagation as well as attenuation part. Using the vector identity:

\displaystyle \nabla \times (\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^2\mathbf{A} \ \ \ \ \ (18)

with Maxwell’s eqs (1) and (4), we can separate the magnetic and electric components to obtain

\displaystyle {1 \over c} {\partial \over \partial t}(\nabla \times \mathbf{B})= \nabla^2\mathbf{E}-\nabla({4\pi \rho_{ext} \over \epsilon_1}). \ \ \ \ \ (19)

By substituting the three material eqs (8)-(10) into Ampere’s Law (3), we arrive at {\nabla \times \mathbf{B}=(\epsilon_1 \mu_1 / c)(\partial \mathbf{E}/\partial t)+(4 \pi \mu_1 \sigma_1 / c)\mathbf{E} }. Combining this with Eq. (20) eventually leads to the wave equation for the electric field

\displaystyle \nabla^2\mathbf{E}-{\epsilon_1\mu_1\over c^2}{\partial^2 \mathbf{E} \over \partial t^2}-{4 \pi \mu_1 \sigma_1 \over c^2}{\partial \mathbf{E} \over \partial t}=0, \ \ \ \ \ (20)

if {\rho_{ext}=0} is assumed. In Eq. (21) the second term represents Maxwell’s displacement current; the last term is due to the conduction current. Similarly we obtain the equation

\displaystyle \nabla^2\mathbf{H}-{\epsilon_1\mu_1\over c^2}{\partial^2 \mathbf{H} \over \partial t^2}-{4 \pi \mu_1 \sigma_1 \over c^2}{\partial \mathbf{H} \over \partial t}=0, \ \ \ \ \ (21)

describing the propagation of the magnetic field. From Eq. (2) we can immediately conclude that {\mathbf{H}} always has only transverse components. The electric field may have longitudinal certain cases, for from Eq. (4) we find {\nabla \cdot \mathbf{E}= 0} only in the absence of a net charge density.

If Faraday’s Law is expressed in {\mathbf{q}} space using Fourier transformation:

\displaystyle \mathbf{q}\times \mathbf{E}(\mathbf{q}, \omega)-{\omega \over c}\mathbf{B}(\mathbf{q}, \omega)=0, \ \ \ \ \ (22)

we immediately see that for a plane wave both the electric field {\mathbf{E}} and the direction of the propagation vector {\mathbf{q}} are perpendicular to the magnetic field {\mathbf{H}}, which can be written as

\displaystyle \mathbf{H}(\mathbf{q}, \omega)= {c \over \mu_1 \omega}\mathbf{q}\times \mathbf{E}(\mathbf{q}, \omega). \ \ \ \ \ (23)

{\mathbf{E}} is not necessarily perpendicular to {\mathbf{q}}. Without expliciyly solving the wave equations (21) and (22), we already see from eq. (24) that if matter present with finite dissipation {\sigma_1\neq 0} – where the wavevector is complex – there is a phase shift between the electric and magnetic field. Substituting Eq. (17) into (21), we obtain the following dispersion relation between the wavevector {\mathbf{q}} and the frequency {\omega}:

\displaystyle \mathbf{q}={\omega \over c} [\epsilon_1\mu_1 + i{4 \pi \mu_1 \sigma_1 \over \omega}]^{1/2}\mathbf{n_q}, \ \ \ \ \ (24)

where {\mathbf{n_q}=\mathbf{q}/|\mathbf{q}|} is the unit vector along the {\mathbf{q}} direction. Note that we have made the assumption that no net charge {\rho_{ext}} is present: i.e. {\nabla \cdot \mathbf{E}=0}. A complex wavevector {\mathbf{q}} is a compact way of expressing the fact that a wave propagating in the {\mathbf{n_q}} direction expriences a change in wavelength and an attenuation in the medium compared to when it is in free space. The propagation of the electric and magnetic field (Eqs (21 and (22)) can now be written in Hemholtz’s compact form of the wave equation:

\displaystyle (\nabla^2 + \mathbf{q}^2)\mathbf{E}=0 \quad \quad \text{ and } \quad \quad (\nabla^2 + \mathbf{q}^2)\mathbf{H}=0. \ \ \ \ \ (25)

It should be pointed out that the propagation of the electric and magnetic fields is described by the same wavevector {\mathbf{q}}, however there may be a phase shift with respect to each other ({\phi \neq 0}).

In the case of a medium with neglible electric loses ({\sigma_1 =0}), Eqs (21) and (22) are reduced to:

\displaystyle \nabla^2\mathbf{E}-{\epsilon_1\mu_1\over c^2}{\partial^2 \mathbf{E} \over \partial t^2}=0 \quad \quad \text{ and } \quad \quad \nabla^2\mathbf{H}-{\epsilon_1\mu_1\over c^2}{\partial^2 \mathbf{H} \over \partial t^2}=0, \ \ \ \ \ (26)

There are no variations of the magnitude of {\mathbf{E}} and {\mathbf{H}} inside the material, however the velocity of propagation has changed by {(\epsilon_1\mu_1)^{1/2}} compared to when it is in a vacuum. we immediately see from Eq.(25) that the wavevector {\mathbf{q}} is real for non-conducting materials. Eq. (24) and the corresponding equation for the electric field then become

\displaystyle \mathbf{H}=\left ( {\epsilon_1 \over \mu_1}\right )^{1/2} \mathbf{n_q \times E} \quad \quad \text{ and } \quad \quad \mathbf{E}=\left ( {\mu_1 \over \epsilon_1}\right )^{1/2} \mathbf{n_q \times H}, \ \ \ \ \ (27)
indicating that both quantities are zero at the same time and at the same location and thus {\phi=0}. The solutions of Eqs (27) are restricted to transferse wave. In the case {\sigma_1=0}, both E and H are perpendicular to the direction of propagation {\mathbf{n_q}}, hence these waves called transverse electric and magnetic (TEM) waves.

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