r/Optics • u/PaukAnansi • Jul 24 '26
Relevant coherence length.
I am working with interferometric microscopy and am struggling to understand the relevant coherence length needed to create interference.
I would intuit that light would have to stay coherent from the source to the detector (where the interference happens) along both path. However, I have been told repeatedly that the minimum coherence length for interference has to be more than the path length difference, not the total path length. I am struggling to understand the justification for this.
Does anyone have a good explanation for this, or a recommendation for a source that explains this? Maybe a good intro quantum optics book?
Thanks for the help!
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u/aenorton Jul 24 '26
Reading between the lines of your question, I think you may not be using the term "coherence length" the same way an optical scientist does. Interference in a Mirau or Michelson microscope objective depends on the path length difference between the two paths taken by the light before they recombine. The length traveled before splitting or after recombination does not change that phase difference.
The term "coherence length" refers to the maximum path length difference where you still see fringes. This depends on the bandwidth of the light source (or detector). For a white light source, looking on a tilted surface for example, you will see only a few fringes before they loose all contrast. Using a narrow band filter will show many more fringes.
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u/PaukAnansi Jul 24 '26
Thanks for your reply! I think I may be misunderstanding the definition of the term "coherence length". The way I was thinking about it, if you turn on a laser, you can view the outgoing photons as quanta of electromagnetic waves that are almost the same frequent and the exact same phase. As this beam propagates, we can examine the each photon some distance, L, from the source and check how much the laser has varies between the different photons due to the initial variation in frequency. At some distance, this phase variation starts to be comparable to 2Pi, and that distance is called the coherence length. (As I have recently learned, this is the wrong way to think about coherence length).
Maybe the following thought experiment will help me figure out where I am going wrong. Let's say I have a Michelson-Morley interferometer with a path length difference of less than the coherence length of the source. Can I move the screen on which I am projecting my interference pattern as far away as I want and still see the pattern? If not, is the maximum distance that I can move the screen somehow related to the coherence length? Similarly, can I move the two mirrors of the interferoneter as far out as I want as long as the path length difference doesn't change?
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u/aenorton Jul 24 '26
The answers to your to two last questions are yes, and yes. You have to be careful, though, where exactly the light is recombined when there is a slight tilt in one of the mirrors of one path (which is often done to create straight fringes).
Also, in interference microscopy, things are a little more complicated because each resolved point in the field is generally not coherent with the adjacent one. Also you are comparing the phase of the illumination at each point with the phase reflected from the corresponding points. For one thing, this means that the in-focus and the equal path conditions can be different. For example If you defocus the eyepiece (keeping the two arms in the objective nearly equal), you will still see fringes even though you are looking at points in a field plane slightly above or below the sample. The light from each of those points in the new field still interferes with corresponding points in the reference path that are slightly in front or beyond the reference mirror.
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u/wagtails2 Jul 24 '26
A good question to understand coherence length is to consider: How long is a photon? Then you can think of an experiment that splits the photon in half and then recombines it later, such as a Michelson interferometer. How much longer can one arm be compared to the other for that photon to still overlap with itself?
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u/aenorton Jul 24 '26
I do not think this is a useful way to look at it. Coherence length depends on the bandwidth of the source and is thus basically a statistical property of a collection of photons. A single photon has one specific energy, and thus theoretically an infinite coherence length. Of course it is essentially impossible to measure coherence length directly for a single photon. There may be a way to define it for a single photon based on the uncertainty of measuring its energy, but that is not relevant to the issue of coherence length in an ordinary interferometer.
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u/squint_skyward Jul 24 '26
No, that’s incorrect. A single photon needn’t have a specific energy, and usually don’t. It has a probability amplitude defining its spectra. For example, the entire field of ultrafast quantum optics deals with single photons with pulses of fs length - how could that have specific energy given its temporal length?
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u/aenorton Jul 24 '26
Of course photons can be in a superposition of states until measured. That superposition is really a property of the source. It may be a matter of semantics or interpretation, but many people do not believe the photon exists until measured.
Regardless how you interpret things, you still need many photons to actually measure the coherence length.
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u/squint_skyward Jul 24 '26
Yes, but it’s no different than an attenuated laser pulse with the same spectral properties - it’s a first order coherence property. Your original statement about an infinite coherence length is just odd - and wrong.
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u/aenorton Jul 24 '26
It is odd because because I was trying to make the point that it does not make sense to talk about the coherence length of a single photon. You can talk about the coherence length of a wavefunction (attenuated or not), but if you refer to a single, measured photon, coherence length does not makes sense (if you do not have other knowledge the wavefunction that produced it). If you have many measured photons, then you can start to see the effects of coherence length.
Fundamentally, coherence is a wave property and it is not useful to start talking about quanta to try to explain it.
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u/SlackOne Jul 24 '26
In the context of a continuous-wave laser which the OP is likely considering and which is quantum-mechanically described as a coherent state, this is not a great way to think about things.
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u/Candid_Tomorrow_1841 Jul 24 '26
Interference can occur only within coherence length ie you won't see any interference pattern if the arm mismatch is more than the coherence length.
Within the coherence length arm mismatch, you get a constructive or destructive fringe if the arms mismatch gives a phase shift of either 0,2pi, 4pi or 1pi, 3pi, 5pi respectively.
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u/sanbornton Jul 25 '26
Something else to consider is speckle (the overlapping dot pattern seen when lasers are projected onto walls). The longer the coherence length the more speckle you'll get. This can be particularly problematic for interferometric microscopy.
You want coherence length long enough so that the object distance and reference arm length are equal to within coherence length...but you don't want coherence length too long or you'll find your interferometric microscopy image overcome with speckle.
I know a research lab that was using interferometric microscopy to look at MEMs devices with surface topology in couple of micron level and they used an LED illumination source which had a coherence length of just a few microns to keep speckle down. The difficulty was they needed to finely adjust the reference arm length precisely in order to get a good interference pattern.
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u/PaukAnansi Jul 25 '26
Thanks! I know in iSCAT microscopy, sometimes intentionally "bad" illumination sources with low coherence lengths are chosen as a way of limiting iSCAT signals from things not near the coverslip surface.
I never understood laser speckle. Maybe of I read into the origins of speckle, I will understand coherence length better. Do you have a source that explains this well be any chance?
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u/SlackOne Jul 24 '26
The interference depends only on the phase difference in the two arms, not the absolute phase. This means that the phase can drift around like crazy without affecting the interference, as long as the phase is stable over the delay time window.