# Dielectric Tensor of an Anisotropic Medium

As we have discussed in the classical optics portion of this website, polarization refers to the wave propagation of light in a medium. In other words, in passing from free space through a medium, the free-space electromagnetic wave “becomes” a polarization wave, and then “becomes” a phase- and amplitude-modulated free space electromagnetic wave again upon emergence from the medium. In isotropic media, the polarization wave is always parallel to the electric field, and is amplitude modulated by a scalar susceptibility as described in \eqref{linpol}.

\begin{equation} P = \varepsilon_o\chi{E} \label{linpol} \end{equation}

In the case of anisotropic media, we no longer have a scalar relationship between the applied electromagnetic field and the polarization wave. The susceptibility becomes a tensor, and the right-handed triad of the polarization wave is described by Equations \eqref{pol1} – \eqref{pol3}.

\begin{equation} P_x = \varepsilon_o(\chi_{11}E_x + \chi_{12}E_y + \chi_{13}E_z) \label{pol1} \end{equation}

\begin{equation} P_y = \varepsilon_o(\chi_{21}E_x + \chi_{22}E_y + \chi_{23}E_z) \label{pol2} \end{equation}

\begin{equation} P_z = \varepsilon_o(\chi_{31}E_x + \chi_{32}E_y + \chi_{33}E_z) \label{pol3} \end{equation}

Fortunately for us, it is always possible to choose the x, y, and z axes such that the off-diagonal components vanish. This can be done by making a rotation to a coordinate system in which the light can propagate undisturbed i.e. is a mode of the medium. Once making this rotation, the remaining components would be:

\begin{equation} P_x = \epsilon_0\chi_{11}E_x \end{equation}

\begin{equation} P_y = \epsilon_0\chi_{22}E_y \end{equation}

\begin{equation} P_z = \epsilon_0\chi_{33}E_z \end{equation}

These directions are known as the principle dielectric axes of the crystal.

We can also describe the response of the crystal to an incident electric field using a dielectric tensor as opposed to a susceptibility tensor. We can see this from the subsidiary Maxwell equation relation:

\begin{equation} D = \varepsilon_0E + P \end{equation}

where, if we now solve for the dielectric tensor, we would have:

\begin{equation} \varepsilon_{i,j} = \varepsilon_0(1+\chi_{i,j}) \end{equation}

Thus, the equations for $D$ in each spatial direction are given as:

\begin{equation} D_x = \varepsilon_{11}E_x + \varepsilon_{12}E_y + \varepsilon_{13}E_z \end{equation}

\begin{equation} D_y = \varepsilon_{21}E_x + \varepsilon_{22}E_y + \varepsilon_{23}E_z \end{equation}

\begin{equation} D_z = \varepsilon_{31}E_x + \varepsilon_{32}E_y + \varepsilon_{33}E_z \label{pol3} \end{equation}