Vector waves

Up to now, we have worked with one-dimensional sinusoidal traveling waves $f(z,t) = \tilde{A}e^{i(kz-\omega t)}.$ that satisfy the 1D wave equation,

$$\dfrac{\partial^2 f}{\partial z^2} = \dfrac{1}{v^2}\dfrac{\partial^2 f}{\partial t^2}.$$

We have also shown that sinusoidal traveling waves that propagate in 3D, $f(\mathbf{r},t) = \tilde{A}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$, satisfy the wave equation,

$$\nabla^2 f = \dfrac{1}{v^2}\dfrac{\partial^2 f}{\partial t^2}.$$

Electromagnetic radiation consist of vector waves. What kind of wave equation would a vector traveling wave solution satisfy,

$$\mathbf{f} = \tilde{A_x}e^{i(kz-\omega t)}\hat{x} + \tilde{A_y}e^{i(kz-\omega t)}\hat{y}$$

and how would you describe this wave? Is it 1D? 3D? Something else? How can we sketch a wave like this?

Polarization is a concept that describes the direction of oscillation of a wave. For example, in the 1D examples we have had they were “linearly” polarized in the direction perpendicular to their motion. How can we descirbe the polarization of the vector traveling wave solution? What would make it linearly polarized?