Eigenvectors in anisotropic media

Question

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?

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

Answer 1

The physical significance of the off-diagonal elements being zero is that your coordinate system's 3 basis vectors are aligned with the principle axes of the medium.

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.