# Bloch Model

Introduction

The phenomological equations proposed by Felix Bloch in 1946 have had a profound effect on the development of magnetic resonance, on the ways in which the experiments are described and on the analysis of line widths and saturation behavior. Here we will describe the phenomological model, derive the Bloch equations and solve them for steady-state conditions.

Bloch Equations

When a magnetic field is applied to a spin, the spin quantization axis is defined by the field direction. Spin magnetic moments aligned with the field are only slightly lower in energy than those aligned opposed to the field. If we consider ensemble of spins, the vector sum of all the spin magnetic moment or macroscopic magnetization:

$\displaystyle \vec{M}=\Sigma_i \vec{\mu}_i \ \ \ \ \ (1)$

At equilibrium ${\vec{M}}$ is in the direction of the field ${\vec{B}}$. If somehow ${\vec{M}}$ is tilted away from ${\vec{B}}$ there will be a torque that causes ${\vec{M}}$ to precess about ${\vec{B}}$ with the equation of motion

$\displaystyle {d\vec{M} \over dt}= \gamma \vec{B}\times \vec{M} \ \ \ \ \ (2)$

where ${\gamma=g\mu_B \hbar}$. In addition to the precessional motion , there are two relaxation effects.

If ${M_0}$ is the equilibrium magnetization along ${\vec{B}}$ and ${M_z}$ is the z-component magnetization under non-equilibrium conditions, then we assume ${M_z}$ approach ${M_0}$ with first-order kinetics:

$\displaystyle {dM_z \over dt}=-{M_z-M_0 \over T_1} \ \ \ \ \ (3)$

where ${T_1}$ is the chararcteristic time to reach the equilibrium. Since this process involve transfer of energy from the spin system to the surrounding lattices, ${T_1}$ is called the spin-relaxation time.

There is a second kind of relaxation process besides ${T_1}$ process. Suppose that ${\vec{M}}$ is somehow tilted down from the z-axis toward the xy-axis and the precessional motion is started. Each individual magnetic moment undergoes this precessional motion, but the individual spins may precess at slightly different rates that can be caused by either local shielding that makes small variations in ${\vec{B}}$ or the effective g-factor may vary slightly through the sample. Thus an ensemble of spins that all start out in phase will gradually lose phase coherence. The characteristic time for this process is called transverse relaxation time ${T_2}$, and that the transverse magnetization components decay to the equilibrium value of zero accordingly:

$\displaystyle {dM_x \over dt}=-{M_x \over T_2},\quad \quad {dM_y \over dt}=-{M_y \over T_2} \ \ \ \ \ (4)$

We have to notice that the dephasing of the transverse magnetization does not affect ${M_z}$; a ${T_2}$ process involves no energy transfer but, being a spontaneous process, does involve an increase in the entropy of the spin system.

The approach to equilirium by a ${T_1}$ process, in which ${M_z}$ approaches ${M_0}$, also causes ${M_x}$ and ${M_y}$ to approach zero. Thus, the ${T_2}$ of eq. (4) must include both of the effects of spin-lattice relaxation as well as the dephasing of the transverse magnetization. Transfer relaxation is often much faster than spin-lattice relaxation and ${T_2}$ is then determined mostly by spin dephasing. However, in general we should write:

$\displaystyle {1 \over T_2}={1 \over T_1}+{1 \over T_{2'}} \ \ \ \ \ (5)$

where ${T_{2'}}$ is the spin dephasing relaxation time, and ${T_2}$ is the observed transverse relaxation time.

Derivation of Bloch Equation

Combining eqs. (2)-(4), we get:

$\displaystyle {d\vec{M} \over dt}= \gamma \vec{B}\times \vec{M}-\hat{i}{M_x \over T_2}-\hat{j}{M_y \over T_2}-\hat{k}{M_z-M_0 \over T_1} \ \ \ \ \ (6)$

In a magnetic resonance experiment, we apply not only a static field ${B_0}$ in the ${z}$-direction but also an oscillating radiation field in ${B_1}$ in the ${xy}$-plane, so that the total field is:

$\displaystyle \vec{B}=\hat{i}B_1\text{cos} \omega t + \hat{j} B_1 \text{sin}\omega t+ \hat{k}B_0 \ \ \ \ \ (7)$

Note there are other possible ways to impose a time-dependent ${B_1}$. The one described in eq. (7) corresponds to a circularly polarized field initially aligned along the ${x}$-axis and rotating about the ${z}$-axis in a counterclockwise direction.

Inserting the ${\vec{B}}$ into eq. (7) and separating the results into their components, then:

$\displaystyle {dM_x \over dt}=-\gamma B_0 M_y + \gamma B_1 M_z \text{sin} \omega t-{M_x \over T_2} \ \ \ \ \ (8)$

$\displaystyle {dM_y \over dt}=\gamma B_0 M_x - \gamma B_1 M_z \text{cos}\omega t-{M_y \over T_2} \ \ \ \ \ (9)$

$\displaystyle {dM_z \over dt}=\gamma B_1[M_y \text{cos}\omega t-M_x \text{sin}\omega t]-(M_z-M_0)/T_1 \ \ \ \ \ (10)$

It is convenient to write ${M_x}$ and ${M_y}$ as:

$\displaystyle M_x=\mu \text{cos}\omega t + \nu \text{sin} \omega t \ \ \ \ \ (11)$

$\displaystyle M_y=\mu \text{sin}\omega t-\nu \text{cos} \omega t \ \ \ \ \ (12)$

or

$\displaystyle \mu=M_x \text{cos} \omega t+ M_y \text{sin} \omega t \ \ \ \ \ (13)$

$\displaystyle \nu=M_x \text{sin} \omega t-M_y \text{cos} \omega t \ \ \ \ \ (14)$

This is equivalent to transformation into a coordinate system that rotates with the oscillating field; ${\mu}$ is that part of ${M_x}$ which is in phase with ${B_1}$ and ${\nu}$ is the part which is 90 out of phase. Differentiating eq. (13) and substituting eqs. (8) and (9), we get:

$\displaystyle {d\mu \over dt}=-[\omega-\gamma B_0]\nu-{\mu \over T_2} \ \ \ \ \ (15)$

Similarly, we obtain:

$\displaystyle {d\nu \over dt}=-[\omega-\gamma B_0]\mu-{\nu \over T_2}+\gamma B_1 M_z \ \ \ \ \ (16)$

$\displaystyle {dM_z \over dt}=-\gamma B_1 \nu-{M_z-M_0 \over T_1} \ \ \ \ \ (17)$

Equations (15-17) are the Bloch equations in the rotating coordinate frame.

In a continuous wave (CW) magnetic resonance experiment, the radiation field ${B_1}$ is continuous and ${B_0}$ is changed only slowly compared with the relaxation rates (slow passage conditions). Thus a steady-state solution to eqs (15-17) is appropriate. Setting the derivatives to zero and solving the three simultaneous equations, we get:

$\displaystyle u= {\gamma B_1 M_0 (\omega_0-\omega)T_2^2 \over 1+T_2^2 (\omega_0-\omega)^2+\gamma^2 B_1^2 T_1 T_2}, \ \ \ \ \ (18)$

$\displaystyle v= {\gamma B_1 M_0 T_2 \over 1+T_2^2 (\omega_0-\omega)^2+\gamma^2 B_1^2 T_1 T_2}, \ \ \ \ \ (19)$

$\displaystyle M_z= {M_0 \left[1+(\omega_0-\omega)T_2^2\right] \over 1+T_2^2 (\omega_0-\omega)^2+\gamma^2 B_1^2 T_1 T_2}, \ \ \ \ \ (20)$

where ${\omega_0=\gamma B_0}$ is Larmor frequency and corresponds to the frequency of the energy level transition.

With ${\chi={\partial M \over \partial H}}$ and ${\vec{B}=\mu_0 \vec{H}}$, we get the two components of ac susceptibilities:

$\displaystyle \chi '= {\gamma \mu_0 M_0 (\omega_0-\omega)T_2^2 \over 1+T_2^2 (\omega_0-\omega)^2+\gamma^2 B_1^2 T_1 T_2}, \ \ \ \ \ (21)$

$\displaystyle \chi ''= {\gamma \mu_0 M_0 T_2 \over 1+T_2^2 (\omega_0-\omega)^2+\gamma^2 B_1^2 T_1 T_2}. \ \ \ \ \ (22)$

${\chi'}$ is the ac susceptibility component in-phase with the driving field ${B_1}$. In general a response that is exactly in phase with a driving signal does not absorb power from the signal source and in spectroscopy corresponds to dispersion. In contrast, an out of phase corresponds to absorption. In magnetic resonance, it is usually the absorption, or ${\chi''}$, that is detected.

When the microwave or radiofrequency power, proportional to ${B_1^2}$, is small so that ${\gamma^2 B_1^2 T_1 T_2<<1}$, Eqs. (21) and (22) becomes

$\displaystyle \chi '= {\gamma \mu_0 M_0 (\omega_0-\omega)T_2^2 \over 1+T_2^2 (\omega_0-\omega)^2} \ \ \ \ \ (23)$

$\displaystyle \chi ''= {\gamma \mu_0 M_0 T_2 \over 1+T_2^2 (\omega_0-\omega)^2} \ \ \ \ \ (24)$

Equation (24) of ${\chi''}$ corresponds to the classical Lorentzian line shape function. The absorption curve will be a Lorentzian line with the half-width of the half height:

$\displaystyle \Delta\omega= {1 \over T_2} \quad \text{or} \quad \Delta B={\hbar \over g_s \mu_B T_2} \ \ \ \ \ (25)$

# Cavity Perturbation Technique

Introduction

The Cavity Perturbation Technique (CPT) is one of the contact-free technique most widely used because of its high sensitivity and relative simplicity. A microwave resonant cavity is a box fabricated from high-conductivity metal (usually from oxygen-free copper) with dimensions comparable to the wavelength. The cavity has simple geometrical form (rectangle or cylinder) to make it easier to calculate the distribution of electromagnetic fields inside. At resonance, the cavity is capable of sustaining microwave oscillations, which form an interference pattern (standing wave configuration) from superposed microwaves multiply reflected from the cavity walls. Each particular cavity size and shape can sustain oscillations in a number of standing wave configurations called modes. In this technique one measures adiabatic change of the characteristics of a resonator upon the introduction of a foreign body (the sample under investigation). The foreign body must be small compared to the spatial variation of the field (quasi-static limit) and its disturbing influence must not be strong enough to force a jump from the unperturbed cavity mode. The perturbation is viewed as a serial expansion in powers of the filling factor (the volume of the sample ${V_s}$ divided by the volume of the cavity ${V_c}$) and only first order effects are accounted for in this analysis.

The principles to determine the sample properties when inserted inside a cavity are simple. In this technique one simply takes the difference of the measured quality factor Q (or a halfwidth of the resonant curve ${\Gamma =f_0/Q}$) and center frequency ${f_0=\omega_0}$ with the sample in and out. However, the relative changes of quality factor (${\Delta {1 \over 2Q}}$) and center frequency (${{\Delta \omega \over \omega_0}}$) upon introducing the sample are connected to the complex conductivity in a nontrivial way, depending on the sample “electrodynamical” geometry. Namely, the sample can either be a good conductor and microwaves penetrate only in the length scale of skin depth into it which is called skin depth regime, or it can be a bad conductor (or even a dielectric) and microwave radiation penetrates completely through it (since the skin depth exceeds the principal dimensions of a sample) which is called depolarization regime.

Cavity Characteristics

A resonant cavity can sustain many modes above a lower cut-off frequency. Near each resonant frequency, the power absorption spectrum has a Lorentzian shape

$\displaystyle A(\omega)={1 \over 4(\omega -\omega_0)^2+(2\pi \Gamma)^2}, \ \ \ \ \ (1)$

where ${f_0=\omega_0/2\pi}$ is the center frequency and ${\Gamma}$ is the bandwidth or full frequency width at half maximum. ${f_o \text{ and }\Gamma}$ are the two characteristics of the resonator and their ratio gives the quality factor ${Q}$, defined as

$\displaystyle Q\equiv {f_0 \over \Gamma}={\omega_0 \over L}, \ \ \ \ \ (2)$

where ${\textless\emph{W}\textgreater}$ is the time-averaged energy stored in the cavity and ${L}$ the energy loss per cycle. The simplest formulation of this problem can be made with the use of a complex frequency notation

$\displaystyle \hat{\omega} \equiv \omega - i{\omega_0 \over 2Q} \ \ \ \ \ (3)$

The Polarizability

The principle of the cavity perturbation technique is to measure separately the cavity characteristics both before (o) and after (s) a small sample has been inserted. The change in the complex frequency is

$\displaystyle \Delta \hat{\omega}=\hat{\omega}_s-\hat{\omega}_o \ \ \ \ \ (4)$

If the change ${\Delta \hat{\omega}}$ is adiabatic, then the product and the time-averaged energy stored is invariant (Boltzmann-Ehrenfest theorem)

$\displaystyle {\textless W\textgreater \over \hat{\omega}}=\text{constant} \ \ \ \ \ (5)$

This implies that

$\displaystyle {\Delta \textless W\textgreater\over \textless W\textgreater}= {\Delta \hat{\omega} \over \hat{\omega}}\sim {f_s-f_o \over f_o}-{i \over 2}\left({1 \over Q_s}-{1 \over Q_o}\right), \ \ \ \ \ (6)$

$\displaystyle \textless W\textgreater={1 \over 16\pi}\int_{V_c}\left(|\mathcal{E} (\mathbf{r})|^2+|\mathcal{H} (\mathbf{r})|^2 \right) d\nu, \ \ \ \ \ (7)$

$\displaystyle \Delta \textless W\textgreater= -{1 \over 4}\int_{V_s} (\mathbf{P}\cdot \mathcal{E}^*+\mathbf{M}\cdot \mathcal{H}^*)d\nu. \ \ \ \ \ (8)$

In our definition, ${\Delta}$ is the variation caused by the introduction of a foreign body into the resonating structure, ${\Delta f=f_s-f_o}$ is the frequency shift and ${\Delta \Gamma = 1/Q_s-1/Q_o}$ the change in the width of the resonance.

For ellipsoidal samples, the spatial dependence of the depolarization field can be omitted and the quantity should be evaluated at the center of the sample. If we put the sample in the antinode of the magnetic field (${\mathcal{E}=0}$), then

$\displaystyle \mathbf{M}=\hat{\chi}_m\mathcal{H} \ \ \ \ \ (9)$

$\displaystyle {\Delta \hat{\omega} \over \omega}=-{\alpha_m \over 4} V_s {|\mathcal{H}|^2 \over \textless W\textgreater}=-4\pi \gamma \hat{\alpha}_m \ \ \ \ \ (10)$

where ${\gamma = \gamma_0 V_s/V_c}$ and ${\gamma_0}$ is a constant that depends only on the resonance mode of the cavity.

$\displaystyle \gamma_0={|\mathcal{H}|^2\over16\pi \textless W\textgreater}V_c = {|\mathcal{H}|^2\over 2 \textless |H|^2 \textgreater} \ \ \ \ \ (11)$

$\displaystyle ={1 \over V_c}\int_{V_c}|\mathcal{H}(\mathbf{r})|^2 d\nu \ \ \ \ \ (12)$

If the sample is in the antinode of the electric field, Eqs. (9)-(12) will be equivalent but with ${\mathcal{H}}$ repalced by ${\mathcal{E}}$ and M by P.

In conclusion, the absorption of electromagnetic waves by small particles is proportional to the polarizability of the sample

$\displaystyle {\Delta \hat{\omega} \over \omega}=-4\pi \gamma \hat{\alpha}. \ \ \ \ \ (13)$

# 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.