What is a monochromatic EM wave? As far as I know, monochromatic waves are waves in the form: $$\vec E(\vec r, t) = \vec E_o \cos(ωt − \vec k \cdot \vec r)$$ where $\vec E_o$ is a constant. Note that this waves are a subset of the plane waves family.
However, in this question: Is a plane wave necessarily monochromatic?, I read that there are monochromatic waves that aren't plane wave, for example Gaussian beams and spherical waves.
Then, I noticed that plane wave, spherical wave and Gaussian beams can all be written in the form:
$$E(x,y,z,t)=u(x,y,z)e^{i(\omega t-kz)}$$
So if I have to guess, I suppose this is the generic expression for a monochromatic wave.
My question is, what is a monochromatic wave? What is the general expression of a monochromatic wave?
 A: Mono$\textit{chromatic}$ waves have a constant frequency, meaning one single frequency $\omega$, while $\textit{plane}$ waves point to a constant direction in space (indicated by the constant vector $\vec{E}_0$), so that the surfaces of constant field are planes. You can have non-monochromatic plane waves, such as
$$
\vec{E}_0\left[\cos(\omega_1 t-\vec{k}_1\cdot\vec{r})+\cos(\omega_2 t-\vec{k}_2\cdot\vec{r})\right],
$$
as well as monochromatic non-plane waves, such as
$$
E_0\hat{r}\cos(\omega t-\vec{k}\cdot\vec{r}).
$$
A: 
Then, I noticed that plane wave, spherical wave and Gaussian beams can all be written in the form:
$$E(x,y,z,t)=u(x,y,z)e^{i(\omega t-kz)}$$

This is not quite correct. A plane wave in the $x$ direction would not have an $e^{-ikz}$ term, for example.
Simply
$$E(x,y,z,t)=u(x,y,z)e^{i\omega t}$$
should cover all of the waveforms you mentioned, though.

My question is, what is a monochromatic wave?

A monochromatic wave is one that only has a single frequency component.

What is the general expression of a monochromatic wave?

Any wave that has only $e^{i\omega t}$ time dependence (As opposed to $e^{i\omega_1 t} + e^{i\omega_2 t}$, for example) is monochromatic.
I don't think you can write a general expression for the spacial dependence, since you should consider waves traveling in all kinds of guided wave structures, waves affected by diffraction, or by refraction, etc.
A: 
As far as I know, monochromatic waves are waves in the form: $$\vec E(\vec r, t) = \vec E_o \cos(ωt − \vec k \cdot \vec r)$$
where $\vec E_o$ is a constant. Note that this waves are a subset of the plane waves family.

What makes this wave monochromatic is the single frequency term $ω$. A polychromatic wave will be a sum of such waves with different values of $ω$.
It is quite possible for all these waves to be aligned in the same plane.
