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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?

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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.

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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 $$ E_0\hat{r}\cos(\omega t-\vec{k}\cdot\vec{r}). $$

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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.

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