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I recently watched this video that illustrates a laser's coherence length: https://www.youtube.com/watch?v=LixwAXsN8vg

I've learned in class that coherence length of a laser with several wavelengths is the distance over which adjacent wavelengths of light become pi out of phase. We also showed that given this definition, the coherence length is equal to the length of the laser cavity. However I don't understand why the contrast in the interference fringes is periodic with period equal to the coherence length as shown in the video. All the coherence length seems to describe is the phase difference between wavelengths of light on one path and not the phase difference between two different paths. Can someone please explain this phenomenon?

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  • $\begingroup$ well, the whole thing is two different wavelengths travelling the same path versus one traveling the same path. Why do you think that coherence length should be defined over two paths?en.wikipedia.org/wiki/Coherence_length $\endgroup$
    – anna v
    Mar 10, 2015 at 6:58
  • $\begingroup$ Well it seems like the light that travels each path has a coherence length and both coherence lengths should be the same. The main thing I'm confused about however is how coherence length is the same as the period of the contrast in the interference fringes as shown in the video. $\endgroup$
    – user35734
    Mar 10, 2015 at 7:04

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You may be familiar with the fact that an interference pattern is really a Fourier Transform of sorts (depending on how the pattern was created) - you find out about the frequency components. When you have a range of frequencies (all the frequencies that "fit" in a laser cavity) you can think of this as a top hat function (or box) in frequency space. The FT is a sinc function which has multiple minima and maxima - but successive maxima are not as strong.

Intuitively the first minimum happens when the first and last frequency are exactly $2\pi$ out of phase (you have every phase represented equally so nothing adds up to anything), then a smaller maximum when the difference is $3\pi$ (2/3 of waves cancel, 1/3 enhances), etc.

If the coherence is due to other effects (e.g. Doppler broadening) then the frequency envelope is Gaussian (like) and once the signal becomes incoherent it "doesn't come back".

Another way to say this is that the FT of a Gaussian is another Gaussian - not a function with multiple zeros like the sinc function.

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My answer is in agreement with what the explanations in Wikipedia as indicated by Anna:

"In physics, coherence length is the propagation distance over which a coherent wave (e.g. an electromagnetic wave) maintains a specified degree of coherence. Wave interference is strong when the paths taken by all of the interfering waves differ by less than the coherence length.".

You saw in the movie that the laser waves following the two paths give a good contrast when the path-length difference is zero. That happens because the two waves are exactly in phase. But when the path-length difference approaches the value where the phase difference is $\pi/2$ the contrast disappears. When the phase difference is around $2\pi$ we get again good contrast because again maxima of one wave fall on the maxima of the other wave, and negative maxima of one wave on negative maxima of the other wave. When the path difference is so that the phase difference is around $3\pi$ we have again bad contrast, and so on. This is the periodicity of the tableau with one frequency.

With more frequencies the tableau is more complicated because the periodicity of the waves is more complicated. We have in each wave points where all the partial waves (of different frequencies) add up in phase, while on the sides of these points the waves add up with some difference in phases and these differences increase the more we go to the sides. This is why in the pattern of interference between what comes on the two paths we have a region of strong maxima, and on the sides weaker maxima. Thus a good contrast is obtained not after a phase difference of a multiple of $2\pi$ but when we get again a superposition of the points, in the two waves, where all the frequencies add up in phase.

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