# Are monochromaticity and coherence in the context of lasers two sides of the same coin?

We know that monochromatic lasers produce monochromatic light, i.e., all photons have the same wavelength $$\lambda$$ (ideally). Coherence, on the other hand, states that the phases of photons are in sync w.r.t. each other. Of course, if the photons had different wavelengths, their phases could not match up due to different wavelength or "repetition cycles". But, if I had a "double wavelength laser" that produced two wavelengths $$\lambda_1$$ and $$\lambda_2=2\cdot \lambda_1$$, wouldn't the phases still match up, at least from the point of view of $$\lambda_2$$? Such a laser could be "coherent", but would definitely not be monochromatic.

In other words: are monochromaticity and coherence two distinct qualities of laser light, or more two very similar qualities? I'm thinking of a hypotehtical "laser" that would emit photons of the same wavelength, but with non-matched photon phases. Of course this wouldn't work because of the disrupted stimulated emission process etc., but it's more of a gedankenexperiment.

For context, I am defining the qualities of laser light, and I'm having an argument with my supervisor about this issue.

• Short pulse lasers aren’t monochromatic. LEDs, while not as monochromatic as a laser diode from the same materials, are monochromatic compared to a light bulb. Mar 29 '20 at 16:02
• The glib answer is that people use the word "coherence" to mean a hundred different things, usually something like "X is coherent if it makes the equipment I'm currently using work good". So depending on the definition you're using that day, your example could be coherent or not. Mar 29 '20 at 18:18
• Certainly, the Fourier spectrum of the laser output has all the information you need, so you can just talk in terms of that. In fact, to avoid long arguments that just go in circles, it would be wise to just forbid using the word "coherent" in discussion at all. Focus on things that you actually agree on the definitions of, or else you'll basically be arguing over whether a taco is a sandwich. Mar 29 '20 at 18:19

I would say that they are distinct features. A source that is not monochromatic cannot be coherent between it's different wavelengths, but each wavelength can be coherent. At the same time a monochromatic source can be incoherent (e.g. a gas discharge lamp with all emission lines but one filtered out). This would just mean, that the different waves are out of phase, e.g. one is $$\sin(\omega t - k x)$$ and the other one $$\sin(\omega t - k x + \phi)$$.
• A monochromatic wave can't be incoherent. A sum of multiple $\sin(\omega t+\Delta\varphi_n)$ with different $\Delta\varphi_n=\operatorname{const}$ and the same $\omega$ is still proportional to $\sin(\omega t+\Delta\varphi)$ for some $\Delta\varphi=\operatorname{const}$. Mar 30 '20 at 8:05