Help with dispersion relations for EM waves in anisotropic dielectric materials

Question

I am really struggling to understand the following dispersion relations which we derived in class.

For an electric field in the z-direction, we have:

$$k^2_x + k^2_y = \frac{\omega^2}{c^2}n_z^2\tag{1}$$

And for an electric field in the x-y plane:

$$\frac{k_x^2}{\frac{\omega^2n_y^2}{c^2}}+\frac{k_y^2}{\frac{\omega^2n_x^2}{c^2}}=1\tag{2}$$

Equation (1) is the equation of a circle while (2) is the equation of an ellipse.

As already said, we derived these relationships in class although we were told that the derivation was not necessary to memorise. However I am struggling to understand a few things surrounding the relationships in (1) and (2).

1. Firstly, what is special about the z-axis, in this example, that it gets a different relationship?

2. When we were told that this is the relationship 'for the electric field in the z-direction', I am assuming this is referring to the z-component of the electric field E, or is it referring to an electric field polarised in the z-direction?

3. Based off of the direction of propagation and polarisation of an EM wave, how would we choose the appropriate subscripts for the relationships in (1) and (2).

Any help would really be appreciated! I find this subject very interesting and solving the problems is very satisfying but this is a real roadblock in my understanding.

PS. My lecturer warned us that this would be tricky!

Answer 1

In a crystal, the relationship between the $E$ field and the displacement $D$ or polarization $P$ fields is a linear one for sufficiently small $E$. The relationship can be written as, say $\mathbf D = \epsilon_0 \mathbf \epsilon \mathbf E$ where $\mathbf \epsilon$ is the so called permittivity tensor that can be represented as a 3x3 symmetric matrix in an arbitrary Cartesian coordinate system. In a particular frame, say, $x_1,x_2,x_3$ the components of the fields are related such as $D_1=\epsilon_0 (\epsilon_{11} E_1+\epsilon_{12} E_2+\epsilon_{13} E_3)$, etc.

A consequence of the symmetry of the $\epsilon$ matrix is that there always is a special set of 3 orthogonal directions, the so-called principal optical axes, that when taken those as Coordinate reference, say $x,y,z,$ then the linear relationship between the $E$ and $D$ fields simplify to$$D_x=\epsilon_0 \epsilon_x E_x\\D_y=\epsilon_0 \epsilon_y E_y\\D_z=\epsilon_0 \epsilon_z E_z \tag{1}\label{1}$$

These are the special coordinate axes implied in your question.

In the formulas there is the assumption that the harmonic wave, say, $$\mathbf E(\mathbf r, t) = \tilde {\mathbf E} e^{\mathfrak j (\omega t - \mathbf {k \cdot r}} \tag{2}$$ is propagating in the $\hat k$ direction, that is $\mathbf k = k\hat k$.

When you solve Maxwell's equation with this assumption, then you get the explicit relationship between the amplitude vectors as $$-\mu_0 \omega^2 \tilde {\mathbf D} = [\mathbf k \times (\mathbf k \times \tilde {\mathbf E})] \tag{3}$$

Let $u=\frac{\omega}{k}$ then (3) is equivalent to

$$\mu_0 u^2 \tilde{\mathbf D} = \tilde{\mathbf E}-\hat k (\hat k \cdot \tilde{\mathbf E})] \tag{4}.$$

Even if $E_z\ne 0$, when the propagation is along the $z$ axis, $\hat z = \hat k$ and $\hat k_z = 1$, then the longitudinal, that is along $\hat k$, component of $\tilde{\mathbf D}$ is zero.

Now letting $u_{x,y,z}=\frac{c}{\sqrt{\epsilon_{x,y,z}}},$ you get the relationship $$\frac{k_x^2}{u^2-u_x^2}+\frac{k_y^2}{u^2-u_y^2}+\frac{k_z^2}{u^2-u_z^2}=0 \tag{5}.$$ The eq. (5) is quadratic in $u^2$ and in general it has two real solutions, say $u'^2$ and $u''^2$. Denote the associated displacement vectors by $\mathbf D'$ and $\mathbf D''$. These two vectors are mutually orthogonal, $\mathbf D' \cdot \mathbf D''=0$ and are also orthogonal to the direction of the propagation vector $\hat k$. These are the "other" two axes. For further details, please, see Sommerfeld.