# Eigenvectors in anisotropic media

I have several questions:

1) First, In the susceptibility tensor, when it's diagonalized, i don't understand the physical significance when the off diagonal terms are zero.

$$P_x=\epsilon_0\chi_{11}E_x, P_y=\epsilon_0\chi_{22}E_y, P_z=\epsilon_0\chi_{33}E_z$$

2) As i understand D has a different direction from E because the components of E are multiplied by different refractive indices, so to find out the eigenvectors of D we use basis vectors, two of which lie in the phase plane and the third is perpendicular to it. Is this basis same as that in the first question or does it depend on K?

$$[A_\vec u]=P_\perp^\vec u[\eta]=\begin{bmatrix} 1&&0&&0\\0&&1&&0\\0&&0&&0\end{bmatrix}\begin{bmatrix}\eta_{11}&&\eta_{12}&&\eta_{13}\\\eta_{12}&&\eta_{22}&&\eta_{23}\\\eta_{13}&&\eta_{23}&&\eta_{33}\end{bmatrix}$$

• Hi and welcome to Physics.SE! Your question would sensibly improve if you could typeset your equations with Mathjax though. Mar 14, 2020 at 18:09

In the second case, you appear to be using a coordinate system in which $$\hat z$$ is aligned with $$\vec k$$ of the propagating wave. If those 2 coordinate systems are different, then $${\bf{\chi}}$$ need not be diagonal in the latter.
• @Ne123 The vanishing of $\chi_{21}$ means that the x and y axes are aligned w/ 2 principle axes of the medium. They are one in the same. If $\chi_{11}=\chi_{22}=1$ and $\chi_{21}=\chi_{12} = 1/\sqrt 2$, and $\chi_{zz}$ is the only nonzero "z", then the coordinate axes are rotated 45 degrees relative to the principle axes.