# 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?

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.

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

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.