Understanding the Susceptibility Tensor

Question

When an electric field passes through a dielectric medium, it causes polarization for the medium, and we define the electric susceptibility $\chi_e$ at some point in the dielectric as:$$\vec{P}=\varepsilon_0.\chi_e.\vec{E}$$Where $\vec{P}$ is the electric dipole moment per unit volume and $\vec{E}$ is the total electric field at that point.

Well, if the dielectric is "isotropic", meaning $\vec{P}$ is independent of the orientation of the E-field, $\chi_e$ will be a scalar. However, if the dielectric is "an-isotropic", $\chi_e$ will be a rank-2 tensor and $\vec{P}$ and $\vec{E}$ will not necessarily be collinear.

The question is: Why shall $\vec{P}$ and $\vec{E}$ be non-collinear ? How does it happen (the physical process) ?

The expression of $P_x$, for example, will be $$P_x=\varepsilon_0\chi_{xx}E_x+\varepsilon_0\chi_{xy}E_y+\varepsilon_0\chi_{xz}E_z$$ How can the x-component of $\vec{P}$ depend on the y- and z-components of $\vec{E}$ ? Is it safe to say that the motion of polarization charges is limited to specific constraints that build these relations ? or what ?

Any help is appreciated.

P.S: I know nothing on crystallography.

Answer 1

As you correctly pointed out, in an anisotropic medium the susceptibility is represented by a rank-2 tensor, in other words, by a matrix. One can express such a $3\times 3$ matrix in terms of a so-called spectral decomposition, which basically means that we diagonalize the matrix. So let's redefine our coordinate system so that this matrix is a pure diagonal matrix in this new coordinate system. Then we would have $$ \mathbf{P} = \epsilon_0 \chi_{xx} E_x \hat{x} + \epsilon_0 \chi_{yy} E_y \hat{y} +\epsilon_0 \chi_{zz} E_z \hat{z} , $$ where the diagonal elements $\chi_{xx}$, $\chi_{yy}$ and $\chi_{zz}$ are the eigenvalues of the original matrix. The fact that the medium is anisotropic is indicated by the fact that the 3 eigenvalues are different. However two of them may be the same (say $\chi_{xx}=\chi_{zz}\neq\chi_{yy}$). In this case one has a unixial crystal or a so-called birefringent medium. (In the more general case with three different eigenvalues we have a bi-axial crystal, but for the sake of this discussion we need not consider that case.) These eigenvalues represent two different refractive indices of the medium. The odd one out ($\chi_{yy}$ in this case) is called the extra-ordinary index while the other two represent the ordinary index. The orientation of the extra-ordinary index indicates the orientation of the optic axis of the crystal. The effect of the crystal on the light propagagting through it now depends on the relative orientations of the optic axis, the propagation direction and the state of polarization.

Now that we have the background in place we can look at the question: why would $\mathbf{P}$ not be parallel to $\mathbf{E}$ when it propagates through such a medium?

Consider the case where the light propagates in the $z$-direction through a birefringent crystal with its optic axis pointing in the $y$-direction. Let's assume the light is linearly polarized at 45 degrees $$ \mathbf{E} = \frac{E_0}{\sqrt{2}} (\hat{x}+\hat{y}) . $$ Due to the difference in the refractive indices along the $x$- and $y$-directions (differences in the values of $\chi_{xx}$ and $\chi_{yy}$), the $x$- and $y$-components of $\mathbf{P}$ will not be equal to each other $$ \mathbf{P} = \frac{\epsilon_0 E_0}{\sqrt{2}} (\chi_{xx}\hat{x}+\chi_{yy}\hat{y}) . $$ Therefore, the vector $\mathbf{P}$ would not be parallel (would be non-collinear) to $\mathbf{E}$.