1 waves

1.1 environments

1.1.1 dispersion

Dispersion occurs when electromagnetic properties, such as permittivity, permeability or conductivity, vary with frequency.

1.1.2 dielectrics

A material is called dielectric if it reacts with an external electric field E. Then the polarization density P is: \vec P = \epsilon_0 \chi_e \vec E.

We say a material is conductive if it allows electrons/ions to flow through it.

1.1.2.1 losses

Losses are quantified with a loss tangent: \epsilon_r = \epsilon_r'(1 - j\tan \delta)

However if losses have conductive behavior then: \epsilon_r = \epsilon_r'j\frac{\sigma}{\omega \epsilon_0}

1.1.3 electric susceptibility

\chi_e is electric susceptibility, it can be expressed in terms of relative permittivity: \epsilon_r = 1 + \chi_e. Therefore: \vec D = \epsilon_0(1+\chi_e)\vec E

1.2 wavefront

Surface of equal phase, for example a spherical wave

1.3 plane wave

Approximation of a real wave.

When we move far from the source of a spherical wave, the wave becomes approximately straight.

Let \vec k be the normal unit vector of the plane wave. Let \vec r be the distance from source Then \vec r \vec k - vt = const.

1.3.1 phase

\beta = \frac{\omega}{v} = \omega\sqrt{\mu \epsilon}

(in radians): \phi = \omega t - \vec \beta \vec r = - \beta (\vec r \vec k - vt) = const

1.3.2 properties

A plane wave is a transverse electromagnetic wave (TEM)

1. Velocity v = \frac{1}{\sqrt{\mu \epsilon}}
2. Orthogonal vectors \vec E = \sqrt{\frac{\mu}{\epsilon}}\vec H \times \vec k and \vec H = \sqrt{\frac{\epsilon}{\mu}}\vec k \times \vec E
3. Impedance Z_w = \frac{E_\perp}{H_\perp} = \sqrt{\frac{\mu}{\epsilon}}