# Why don't waves with different wavelengths interfere with each other in white-light?

The book I'm reading about optics says at some point that "each color (wavelength) contained in the white light interferes only with itself". But why is this so?

Edit: I moved the rest of the question elsewhere.

• Closely related: Why don't waves erase out each other when looking onto a wall?. Jul 13, 2011 at 2:25
• There’s no reason to think that photons of any frequency interfere with each other as they are propagating. The patterns of constructive and deconstructive accumulation you see are at the detection screen as they all arrive. Aug 17, 2021 at 16:25

At any instant in time, light of different wavelengths can be said to interfere. However, because of the extremely high frequencies of visible light, any cross interference will get time-averaged away very quickly unless the two waves are very close in frequency.

• Very simple and clear. Just one thing: did you mean "At any instant"? Also, I was wondering if I should strip down the question and retain only the first sentence; the rest might be a bit messy or slightly off-topic. What do you think? Jul 12, 2011 at 17:09
• I think it might be better if I moved the rest of it to another question. I'll do so now. Jul 12, 2011 at 17:19
• Care to give to mathematical proof? Dec 10, 2021 at 13:36

Interference is not an instant phenomenon but a time-lasting one. The time should be sufficient to speak of certain frequencies. Two different frequencies can be easily distnguished with the corresponding resonators but this process takes time. It is in a long term condition that two frequencies are separated (resonators pumped). Instantly we cannot even speak of a frequency.

Imagine two light beam each of its own wavelength $L_1=720\text{ nm}$ and $L_2=719\text{ nm}$. One can show I suppose that every $720$ cycles of the $L_2$ beam the total constructive interference shall repeat.

In air the $L_1$ beam light frequency is $$F_1=\frac{C}{L_1}=417\times 10^{14}\text{ Hz}$$

and therefore the frequency of the interference effect is $$F_\text{int}=\frac{F_1}{720}=57.9*10^{12}\text{ Hz}$$

This frequency is rather high to be detectable. When two light beams are of identical wavelength however, the interference effect is time independent and therefore observed as an interference pattern. It is obvious that there is the interference between light beams of a different "color" albeit difficult to observe. Please read about the white light interferometry.