Dielectric tensor is a very good physical value to describe the waves in plasma with kinetic theory. This method is systematically used by T.H. Stix in his famous book “Waves in Plasmas”.

Conductivity tensor

Following the Ohm’s law, the current density in plasma is described as: $\vec{j} = \sigma\vec{E}$. If the media (eg. plasma) is anisotropy, $\sigma$ shall be a tensor, else it will only be a scalar value.

Dielectric tensor

In plasma, the Ampere’s law is modified with plasma current as: $\nabla\times\vec{B} = \mu_0\vec{j} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$. If we consider the Ohm’s law, we can get: $\nabla\times\vec{B} = \mu_0\sigma\vec{E} + \mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t} = \frac{-i\omega}{-i\omega}\mu_0\sigma\vec{E} - i\omega\mu_0\epsilon_0\vec{E}$, then we get:
$-i\omega\mu_0\epsilon_0(\frac{1}{-i\omega\epsilon_0}\sigma\vec{E} + \vec{E}) = \mu_0\epsilon_0\frac{\partial}{\partial t}((\frac{i\sigma}{\omega\epsilon_0} + I)\vec{E})$
Where, $\epsilon = \frac{i\sigma}{\omega\epsilon_0} + I$ is the dielectric tensor, $I$ is the unit tensor.

Dielectric function