How to determine wave vector given the polarization of a plane wave

Question

This is very similar to the question asked in the past, but I need further clarification.

Say I'm given the polarization of an electromagnetic wave in free space (in phasor form), something like this:

$$\vec{E}=[2\hat{e}_{x}+3\hat{e}_{x}+(5-j)\hat{e}_{z}] e^{-j\vec{k}\cdot \vec{r}}$$

I don't particularly care about these specific coefficients (I just made them up) but lets say that $E_x$,$E_y$,$E_z$ are all nonzero, meaning the polarization of the $\vec{E}$ field isn't just in the $xy,xz,$ or $yz$ planes (as in the last post I referenced). Given this information, and knowing that $\vec{k}$ points in some arbitrary direction, how can I determine the direction of $\vec{k}$?

Here is what I have: I've been back and forth on this one for a while now, and it has been suggested to use Gauss' law in free space $\nabla\cdot\vec{E} =0$ which, when applied to $\vec{E}=\vec{E}_0e^{-j\vec{k}\cdot{\vec{r}}}$ yields:

$$-j\vec{E}\cdot\vec{k}=0$$

It is at this point I balk. Sure I can write,

$$E_xk_x+E_yk_y+E_zk_z=0$$

but even knowing this the values of $E_x$, $E_y$, and $E_z$ it doesn't seem like I know enough to find $k_x$, $k_y$, and $k_z$. What am I missing here?

I know this seems like homework, but I'm genuinely interested to know to extract $\vec{k}$ as it has direct application to some measurements I might make in the near future.

Answer 1

You cannot extract $\vec{k}$ from the polarization, just that its direction must be perpendicular to the polarization. There is no one-to-one matching between polarization and $\vec{k}$.

Answer 2

If this were possible it would mean that a wave with a given polarization can propagate in one direction only. Think about plane polarized wave, along z axis, for example. Is the wave vector of such a wave uniquely determined? Of course not. I can have waves with E along z propagating along x axis or y axis or any axis in between.