Electromagnetic Waves

EM waves composed of undulating electrical and magnetic fields. Means of transporting energy or information.

Intrinsic Wave Impedance

ratio of electric and magnetic field phasors (complex amplitudes)
\[\eta = \frac{E_{zs}}{H_{xs}} = \frac{ E_o e^{-\gamma y} }{ \frac{\gamma}{j \omega \mu} E_o e^{-\gamma y}} = \frac{j \omega \mu}{\gamma}\]
$\gamma = \sqrt{j\omega\mu (\sigma + j\omega \epsilon)} = \alpha + j\beta$
where $\fbox{$ \alpha = \omega \sqrt{\frac{\mu\epsilon}{2} \left[ \sqrt{1+ \left( \frac{\sigma}{\omega\epsilon} \right)^2} - 1 \right] } $}$
where $\fbox{$ \beta = \omega \sqrt{\frac{\mu\epsilon}{2} \left[ \sqrt{1+ \left( \frac{\sigma}{\omega\epsilon} \right)^2} + 1 \right] } $}$ and $\fbox{$ v = \frac{\omega}{\beta}, \lambda = \frac{2\pi}{\beta} $}$
$\fbox{$ \eta = |\eta|e^{j\theta_\eta} = \sqrt{\frac{j\omega \mu}{\sigma + j\omega \epsilon}}$}$
where $|\eta| = \frac{ \sqrt{\mu/\epsilon} }{\left[1 + \left( \frac{\sigma}{\omega\epsilon} \right)^2 \right]^{1/4}}$
and $\tan\theta_\eta = \frac{\sigma}{\omega\epsilon}, 0 \le \theta_\eta \le 45^o$
lossy dielectric: $\sigma > 0$

Consider linear, homogeneous, isotropic media in a source-free region($\vec{J} = 0, \rho_v = 0$). Now Maxwell’s equation in phasor form: $\begin{align*} \nabla \times \vec{E}_s &= -j\omega\mu\vec{H}_s \\ \nabla \times \vec{H}_s &= \sigma \vec{E}_s + j\omega\mu\vec{H}_s\\ \nabla \cdot \vec{E}_s &= 0 \\ \nabla \cdot \vec{H}_s &= 0 \\ \end{align*}$
Taking curl on the curl equations, substituting and using the vector identity $\fbox{$ \nabla\times\nabla\times \vec{F} = \nabla(\nabla\cdot \vec{F}) - \nabla^2 \vec{F} $}$, we get: $\begin{align*} \nabla^2 \vec{E}_s - \gamma^2 \vec{E}_s &= 0 \\ \nabla^2 \vec{H}_s - \gamma^2 \vec{H}_s &= 0 \\ \end{align*}$ where $\gamma^2 = j\omega\mu (\sigma + j\omega \epsilon)$
These equations are called phasor vector wave equations or Helmnoltz equations.
Here $\nabla^2$ is the vector Laplacian operator.

The complex constant $\gamma = \sqrt{j\omega\mu (\sigma + j\omega \epsilon)} = \alpha + j\beta$ is defined as propagation constant.
The real part($\alpha$) is called attenuation constant. It defines the rate at which the fields of the wave are attenuated as the wave propagates. For lossless media $\alpha = 0$
The img part($\beta$) is called phase constant. Defines rate at which phase changes as wave propagates.

The vector laplacian operator in terms of scalar Laplacian:
$\nabla^2 \vec{F} = (\nabla^2 F_x)\hat{x} + (\nabla^2 F_y)\hat{y} + (\nabla^2 F_z)\hat{z}$
where $\nabla^2 f = \frac{\partial^2 f}{\partial x^2} +\frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2}$
Use this in the phasor vector wave equations: $\nabla^2 \vec{E}_s = \gamma^2 \vec{E}_s$ and $\nabla^2 \vec{H}_s = \gamma^2 \vec{H}_s$
We get, six partial differential equation one for each scalar components($E_{xs}, E_{ys}, E_{zs}, H_{xs}, H_{ys}, H_{zs}$). And EM wave must satisfy all six of these PDEs. A EM wave may not have all six of the components and based on this various types of waves are defined.

$\vec{E}$ and $\vec{H}$ lie in a place perpendicular to the direction of propagation
$\vec{E}$ and $\vec{H}$ are perpendicular to each other.

A plane wave is called uniform if $\vec{E}$ and $\vec{H}$ in the plane perpendicular to the direction of propagation i.e. they vary only in direction of the propagation.