## Optics basics: Coherence

In previous optics basics posts, the interference of waves has played a major role.  When two or more monochromatic (single-color) waves are combined, they form a pattern of light and dark regions, in which the combined light fields have constructively or destructively interfered, respectively.  The simplest of these patterns can be created by the interference of two plane or spherical waves, and would appear as shown below:

One way of producing such a pattern is by Young’s two-pinhole (or double-slit) experiment, which we will have cause to discuss in more detail below.  The actual size of the interference pattern depends on the experimental setup, including the wavelength of the light, but can be easily made visible to the naked eye.

An astute observer of nature, however, will find something fishy about this whole discussion of interference: it does not seem to manifest itself in everyday experiences with light.  Sunlight streaming through a window, for instance, doesn’t interfere with the light emanating from a lamp inside the room.  Something is missing from our basic discussion of interference which explains why some light fields, such as those produced from a single laser source, produce interference patterns and others, such as sunlight, seemingly produce no interference.  The missing ingredient is what is known as optical coherence, and we discuss the basic principles of coherence theory and its relationship to interference below the fold.

Optical coherence refers to the ability of a light wave to produce interference patterns, such as the one illustrated above.  If two light waves are brought together and they produce no interference pattern (no regions of increased and decreased brightness), they are said to be incoherent; if they produce a ‘perfect’ interference pattern (‘perfect’ in the sense that regions of complete destructive interference exist), they are said to be fully coherent.  If the two light waves produce a ‘less-than-perfect’ interference pattern, they are said to be partially coherent.

Coherence is one of those topics which is best explained by starting with experimental observations.  We first describe two interference experiments, which highlight the two broad ‘types’ of coherence, and then explain in more detail the origins of the phenomena.

We first consider the Michelson interferometer, illustrated schematically below:

Light from a source S (which could be anything: sunlight, laser light, starlight) is directed upon a half-silvered mirror M0, which sends 50% of the light towards mirror M1 and 50% of the light towards mirror M2.  The light is reflected from each of these mirrors, returns to the beamsplitter M0, and equal portions of the light reflected from M1 and M2 is combined and projected on the screen B.  The interferometer is ‘tunable’, in that we can adjust the distance of the mirror M1 from the beamsplitter.

The Michelson interferometer, in essence, interferes a light beam with a time-delayed version of itself.  The light which travels along the path to mirror M1 must travel a distance 2d further than the light which travels along the path to mirror M2.

What do we see on the screen?  When d = 0, we see a circular set of very clear interference fringes.  When d increases, however, the fringes becomes less distinct: the dark rings become brighter and the bright rings become dimmer.  Finally, for very large d, past some critical value D, the bright and dark rings have vanished completely, leaving only a diffuse spot of light:

Evidently, a light field cannot interfere with a time-delayed version of itself if the time delay is sufficiently large.  The distance 2D is referred to as the coherence length: interference effects are only noticeable when the path length difference is less than the coherence length.  This quantity can be converted into a coherence time T, by dividing by the speed of light c:

$T = 2D/c$.

The Michelson experiment measures the temporal coherence of a light wave: the ability of a light wave to interfere with a time-delayed version of itself.  Some typical values of coherence length:

• well-stabilized laser: coherence time, $T \sim 10^{-4} \mbox{ seconds}$, coherence length $2D = 30\mbox{ km}$
• filtered thermal light: coherence time, $T \sim 10^{-8} \mbox{ seconds}$, coherence length $2D = 3 \mbox{ m}$

We can use a different sort of interferometer — Young’s two-pinhole (double slit) interferometer — to measure another, distinct, type of coherence.  Young’s interferometer, discussed briefly in the basics post on interference, is illustrated below:

A light source is used to simultaneously illuminate a pair of small pinholes, separated by a distance d,  in an opaque screen.  The light emanating from the pinholes interferes on a second screen some distance away.  In a manner similar to that described for temporal coherence, we imagine increasing the separation d of the two pinholes.  For small values of d, one sees a distinct pattern of bright and dark interference fringes on the screen.  As d is increased, however, the fringes become less distinct: the dark lines become brighter and the bright lines become dimmer.  Finally, for sufficiently large d, past some critical value D’, no fringes are visible and only a diffuse spot of light remains on the screen.  This critical value D’ is referred to as the transverse coherence length of the light field.  More commonly, one refers to the coherence area of the light field as the area within which the pinholes must lie to observe interference.  This area is approximately

$A\sim \pi D'^2$.

Young’s interferometer measures the spatial coherence of a light field: the ability of a light field to interfere with a spatially-shifted version of itself.  It is clear that this cannot be just another manifestation of temporal coherence because the light arriving at the two pinholes has traveled the same distance from the source.  Typical values of the coherence area:

• Filtered sunlight: $A \sim 4 \times 10^{-3} \mbox{mm}^2$,
• Filtered starlight: $A \sim 6 \mbox{m}^2$.

The coherence area of sunlight gives us an idea of why interference effects are not observed in everyday experience: interference can only be created with sunlight which lies within a spot roughly a tenth of a millimeter across!

Why are some light fields coherent and others incoherent?  Interference requires a definite phase relationship between the interfering fields.  If we look at the interference of two monochromatic waves as a function of time, we see that the waves undergo complete constructive interference if they are both ‘up’ simultaneously and both ‘down’ simultaneously:

The dashed lines show that the two waves are both up/down at the same time.  They therefore enhance each other, producing a larger wave.  Similarly, two waves undergo complete destructive interference if they have opposing amplitudes:

Realistic waves, however, are never perfectly monochromatic.  Ordinary thermal sources of radiation, such as a lightbulb or the Sun, consist of a large number of atoms radiating light almost independently of one another.  The field emanating from such a source has random fluctuations of amplitude and phase.  An example of a wavefield which is ‘almost’, but not truly, monochromatic is pictured below:

This wave contains what seem to be regular up-and-down motions but clearly has large fluctuations in the amplitude (size) of the wave.  Less clear is that the phase of the wave (the spacing of the peaks) also changes.  We can see this by superimposing a monochromatic field on top of this one:

At the point labeled ‘in phase’, the monochromatic wave and the random wave both go up and down together at the same time.  At the point labeled ‘out of phase’, however, the two waves are oscillating in the opposite direction. The spacing of the wavefronts only gradually changes, and for a significant period of time the random wave looks almost perfectly monochromatic.

Here is our explanation of coherence time: the coherence time of a wave is the duration over which the field looks approximately monochromatic.  As long as we interfere a wave with itself within the coherence time, it is effectively the same as interfering two monochromatic waves together.

The less monochromatic a wave is, the broader its spectrum.  A wavefield containing many colors, such as sunlight, therefore has a very low coherence time.  A measurement of the coherence time can be used to estimate the spectral width of a light wave, and vice versa.

What about spatial coherence?  Spatial coherence is a measure of the interference-causing capabilities of a wavefield at two different points in space.  To see what factors influence spatial coherence, let’s first imagine a single point source, producing light which is not necessarily monochromatic.  A point source produces spherical waves, as illustrated schematically below:

The wavefronts (the peaks of the wave, illustrated in blue) are not equally spaced for a non-monochromatic wave.  However, at points P1 and P2 which are equally distant from the source, the fields are identical.  If the wavefields have a definite relationship to each other — in other words, they are correlated — at those two points in space for all time, they can interfere with one another; this is illustrated schematically below:

The two examples on the left show fields which are correlated with one another: if they are combined using a Young’s interferometer, they will produce an interference pattern.  When the two fields ‘do their own thing’, however, as in the example on the right, they produce no interference pattern.  As an example of a source which produces an uncorrelated field, we imagine the field produced by two independent point sources.

In the figure on the left, the wavefronts from the red and blue sources arrive at points P1 and P2 at the same instant of time.  In the figure on the right, however, the wavefronts from the red and blue sources arrive at points P1 and P2 at different times, resulting in different fields at the two points.  In short, the field at P1 changes independent of the field at P2.  There is no definite phase relationship between the fields at the two points, and they will not produce an interference pattern when added together.

As a consequence of this argument regarding point sources, one can show that a larger (or closer) thermal source of light produces light of smaller coherence area than a smaller (or more distant) thermal source of light.  Sunlight therefore has low spatial coherence, while starlight has significantly higher spatial coherence.  This argument holds only for thermal light sources, i.e. collections of atoms/molecules radiating independently.  In lasers, which emit light by a very different physical process, light is typically spatially fully coherent.

Coherence theory can be thought of as the merging of electromagnetic theory with statistics, just as statistical mechanics is the merging of ordinary mechanics with statistics.  It is used to quantify and characterize the effects of random fluctuations on the behavior of light fields.  We have roughly divided coherence effects into two classes — spatial and temporal coherence — which are summarized by the following picture:

It is worth noting that we cannot typically measure the fluctuations of the wavefield directly.  The individual ‘ups and downs’ of visible light is not something that we can detect directly with our eyes, or even with sophisticated sensors: the frequency of visible light is on the order of $10^{15}$ cycles/second (a ‘1′ with fifteen zeros after it).  We only measure the average properties of the light field.

As a field of study, coherence has a number of applications.  It has been shown that partially coherent fields are less affected by atmospheric turbulence, which makes such fields useful in laser communications.  Partially coherent fields have been applied in studies of laser-induced fusion: the reduction of interference effects results in a ‘smoother’ beam impinging on the fusion target.  Coherence effects have been used in astronomy for a number of applications, notably to determine the size of stars and the separation of binary star systems.

Coherence is important in the study of quantum-mechanical fields as well as classical fields.  In 2005, Roy Glauber won half of the physics Nobel Prize “for his contribution to the quantum theory of optical coherence.”

There’s a lot more I could say about coherence (it is one of my specialties, actually), but we’ll leave further discussion for future posts.

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### 69 Responses to Optics basics: Coherence

1. stuwat says:

Having a good grasp of coherence is key to understanding the principles behind many powerful tools which use optical interference. Optical interferometry allows us to make highly accurate measurements of distance, temperature, pressure, strain and velocity, to name but a few applications. Your blog is fast becoming a great resource for learning these fundamentals and I recommend it to anyone wanting to learn more about optics.

2. meichenl says:

thanks for another great post!

i’ve heard it’s possible to make two different lasers coherent for short periods. how difficult is that and how long can they be coherent? is it the same order of magnitude as the temporal coherence time for a single laser?

3. meichenl: That’s a great question, and the answer is yes – it can be done! I have to go track down the original paper again for the details, which I think will turn into another ‘classic science papers’ post.

In short, the work was done by Leonard Mandel in the 1960s, and the results stirred a significant amount of debate. In a great example of quantum weirdness, he and a collaborator later showed that one could get the two beams to interfere even if the light intensity was so low that only one photon was typically present in the interference pattern!

More to come…

4. P.S. My intuition says that the fields will only interfere, as you’ve suggested, over times on the order of the coherence time.

Thanks for another excellent post! I do have a minor question about the discussion of coherence length in the Michelson interferometer, though.

I thought if you use a laser as your light source, then as you increase d, the interference fringes would disappear when 2d = 0.5 lambda * n, for n >= 1, due to destructive interference when the two beams are half a wavelength out of phase at the detector.

The explanation of the coherence length made it sound like the fringes just get gradually worse, without any sort of periodicity. Is this effect not visible under normal experimental conditions?

Thanks for any clarification you might offer — I’m probably just confused about something.

6. Wade: Thanks for the comment! I think you’re mixing up the conditions for destructive interference (2d = 0.5 lambda + 2pi n) with the conditions for any sort of interference (2d < coherence time). My picture of the rings is produced in the Michelson interferometer is a little misleading: as d is increased, one will see the rings ‘move’: the light and dark rings will move outward, and new ones will appear in the center. This is due to the change in the specific conditions for constructive/destructive interference. However, one will gradually see this pattern vanish, i.e. the visibility of the fringes will decrease: this is the coherence effect. How ‘quickly’ this pattern vanishes is dependent on the bandwidth of the light, while the shape of the interference pattern is dependent on the center frequency.

In other words, “coherence” refers to the overall ability of the light field to produce constructive or destructive interference. When I talk about coherence, I’m talking about the presence or absence of an entire fringe pattern, rather than the presence or absence of an individual bright or dark fringe.

I think I understand my problem now — I was thinking of what would happen at a single point on the screen, instead of to the entire interference pattern.

A single point on the screen should see a periodic variation in light intensity as d is adjusted, due to the rings in the interference pattern moving across it as they spread out from the center.

Then as d is made larger, the pattern as a whole will gradually fade out as the coherence length is approached.

Does this sound right to you?

8. RWS says:

The paper you’re looking for is by Magyar and Mandel from Nature in 1963. And it’s worth noting that if you’re working with sufficiently fast detectors, the coherence properties are rather meaningless (you trade the speed for time-averaging).

9. Wade: Yep, that sounds about right. At a single point on the screen, as d is increased, one would see the intensity oscillate but the oscillations would gradually ‘flatline’ to a constant value of intensity.

RWS: Thanks for the comment! I actually got the Magyar paper via my precious ILL yesterday, and will hopefully discuss it in a future post, as it is quite interesting. One of the lessons of the paper, I would say, is an illustration that coherence properties necessarily involve averaging: fields always interfere over short time periods, but that interference is only visible over long times if the fields are coherent.

10. juana says:

I have one question about the partially coherent beam
I read some stuff about partially coherent beam and geting rid that partially coherent beams are less effected by air turbulent but my problem is that how can I show MATHEMATICALL RELATIOSHIP BETWEEN PHASE, AMPLITUDE AND REFRACTIVE INDEX OR THE DECAY OF INTENSITY that will give me evedence that there is less effect on atmospheric turbulence

11. juana wrote: “how can I show MATHEMATICALL RELATIOSHIP BETWEEN PHASE, AMPLITUDE AND REFRACTIVE INDEX OR THE DECAY OF INTENSITY that will give me evedence that there is less effect on atmospheric turbulence”

That’s a tricky question to answer; mathematically, one looks at the scintillation index (fluctuations of intensity) of the wavefield on propagation. Conceptually, I tend to explain the ‘resistance’ of partially coherent beams as follows: A partially coherent beam consists of a large number of mutually incoherent ‘modes’ which propagate independently through the turbulence. At the detector, one sees the contributions from many modes at once. Because these modes are incoherent, they don’t interfere with one another and produce a relatively uniform intensity pattern. Relatively uniform intensity low scintillation index.

12. juana says:

Thank you for your insight .
you give me one step ahead to break the queestion a little further

13. Alex says:

So I’m fuzzy on electron interference. The idea, quantum mechanically, is that a single electron when shot at two slits actually interferes with itself right? It takes all possible paths, including bouncing off the the divider containing the two slits. My question is, if detectors were set up in a 360 degree frame, shouldn’t you, after shooting one electron, also get a electron behind the cannon, or above, along with the electron detected by the plate in front of the slits. Mainly my question is how can the electron exist simultaneously in two states, interfere with itself but not show up also as one of the infinite other possibilities at the same time in another place?

• Alex: It’s a fair question. I think the resolution of your question, if I understood it correctly, is as follows:

A single electron, though it has wave properties, is still only detected at a single location effectively as a ‘point’ particle. The electron interference pattern only builds up on average as many, many electrons are sent through the system; I sketched a picture of this in my ‘basics’ post here.

The wave nature of any particular electron does include the possibilities of the electron being reflected/absorbed by the screen; if it does so, though, the electron is not detected in the interference pattern and does not contribute to it.

So, in essence, the electron acts like a wave up until the point that it ‘chooses’ to take a stand and be detected somewhere, which is the idea of the collapse of the wavefunction in quantum mechanics. You may justifiably ask: when does the electron ‘choose’ to take a stand? This is, in essence, one of the great unsolved problems in physics, so I can’t really give an answer!

14. Alex says:

So I understand it travels as a wave, but when shot one at a time, it creates an interference pattern, meaning somehow the particles are interfering with themselves correct?

• Alex wrote: “but when shot one at a time, it creates an interference pattern, meaning somehow the particles are interfering with themselves correct?”

Well, yes and no! 🙂 We never see the wave of an individual electron; as I said, the electron is detected as a point-like particle. If you were to do the experiment with a single electron, you wouldn’t see any reason to believe that an electron has wave-like properties.

What the wave function represents, roughly, is the probability of finding the electron at a particular point in space. This `probability wave’ is what creates an interference pattern, but we don’t see this pattern directly — the electron is finally detected only at one place. When we send many electrons through the same double slit interferometer, however, the screen acts as a histogram, counting how many electrons appear at a particular point on the screen. We indirectly see the probability wave by making the relationship: ‘large # of electrons = high probability’, ‘small # of electrons = low probability’.

This is actually a very subtle point (and, as I pointed out, still not perfectly understood); I may come back and write a more detailed post on this in the near future.

15. Andrew says:

Have you seen the remarkable paper by Lindner et al (quant-ph/0503165) that describes observations of quantum interference in time? Horwitz analyses the results in Phys Lett A v355 (2006) 1-6.

He write on p2 to the effect that e/m waves can interfere in time since they obey 2nd order d.e.’s, but Schrodinger particles cannot since they obey 1st order d.e.’s. (I do accept his earlier argument that the recombined electron state is a mixed state, thus no interference).

Well, the Schrodinger eq is a special case of the Pauli eq which is a low energy approx to the Dirac eq which looks 1st order but the dispersion relation is as for the Klein-Gordon eq, that is, second order. This seems quite a separate matter from choosing the retarded propagator.

Can you relate differential order and temporal coherence, please?

Thanks

16. Tom says:

Thanks! This is one of the most concise explanations of coherence I have found. Kudos on avoiding equations! (I took Wolf’s graduate modern coherence class).

• You’re welcome! I still remember taking Wolf’s first midterm — and basically leaving the correlation function completely out of one of my solutions.

17. tm says:

I liked the article but didnt find what I was looking for, the coherence time of sunlight. I found an article giving a solar radiation coherence time of 10^-12 sec, but I dont see how they got it. Also, does the solar radiation coherence time vary from the radio through the IR and visible ? thanks.

• Formally, coherence time is the inverse of the spectral bandwidth of the radiation — if you can find a number for the spectral bandwidth of sunlight, you’ll have the coherence time.

It’s a little subtle, however, I imagine, because one could also talk about an “effective” coherence time. Because sunlight is so broadband, there are hardly any detectors that are sensitive to all frequencies! This means that the coherence time might seem effectively larger than it “physically” is.

• tm says:

I have come a ways since posting this question and I believe the reply includes a common misconception about coherence time dating from a paper by Wolf in the 50’s. The treatment used to derive the 1/bandwidth coherence time was for a special case of a radiation field for which light at all frequencies is at the same phase. Then, the Fourier treatment gives a pulse of light with a 1/bandwidth time width, so they call that the coherence time. But, no source in nature is in phase at all wavelengths. This formula has been disproven by unfiltered interferometric solar observations (my field) which showed the coherence time of unfiltered sunlight is at least 1.5 10^-9 sec, not 10^-15 sec as predicted by the 1/bandwidth formula. So, while optical physicists believe this formula, solar physicists disprove it every day. I have derived the coherence time correctly using the van Cittert-Zernike theorem and am close to submitting the paper. It has nothing to do with that 1/bandwidth formula that Wolf has promoted.

• The treatment used to derive the 1/bandwidth coherence time was for a special case of a radiation field for which light at all frequencies is at the same phase.

Actually, the opposite is true: traditional coherence theory deals with fields that are statistically stationary, which means that the statistical properties are independent of the origin of time. This automatically results in a field for which the different frequencies of light are completely uncorrelated — there is no definite phase relation between *any* frequencies.

The mathematics of the spectral properties of pulses and the spectral properties of partially coherent fields are structurally very similar. The difference is that pulses have correlated spectral phases and PC fields have uncorrelated spectral phases. This similarity has led to a lot of confusion.

• tomsol@comcast.net says:

No, You need to go reread the derivation of the 1/bandwidth coherence time. It starts with a spectral distribution given by e^-(nu^2/sig^2). Then the Fourier transform is taken, and the result given the 1/bandwidth formula. but, the spectral amplitude is in general complex and and the ratio between the real and imaginary parts gives the phase. So the above spectral distribution corresponds to a frequency-independent phase, which is not found in nature. Didn’t you read the part about unfiltered solar interferometric observations providing the full solar spectrum ? This could not happen if the 1/bandwidth coherence time formula were true. The interferometer used for the observations creates a path length difference of 0.5 m, or 1.5 nanoseconds.

The field of optics has been led astray by a paper by Wolf from the 50’s, but I dont expect a true believer to accept that. I will submit my paper on coherence time in the next week and then we’ll see what happens.

By the way, your reply did not address these points the first time. That is why I repeated them.

• The field of optics has been led astray by a paper by Wolf from the 50′s, but I dont expect a true believer to accept that.

“True believer”? *sigh* I can see that there’s little point in trying to discuss this with you further. Best of luck with your paper.

• J Thomas says:

Five months ago you considered Dr. Skyskull a resource who might give you helpful information. That could still be true, even if it turns out he has accepted something that is false.

I don’t fully understand your points but I think I see the hinges.

“The treatment used to derive the 1/bandwidth coherence time was for a special case of a radiation field for which light at all frequencies is at the same phase. Then, the Fourier treatment gives a pulse of light with a 1/bandwidth time width, so they call that the coherence time. But, no source in nature is in phase at all wavelengths.”

The first question is whether the results of the special case generalize. Does it really matter? What result should we expect in the general case where things are out of phase? Dr. Skyskull might know.

“This formula has been disproven by unfiltered interferometric solar observations (my field) which showed the coherence time of unfiltered sunlight is at least 1.5 10^-9 sec, not 10^-15 sec as predicted by the 1/bandwidth formula. So, while optical physicists believe this formula, solar physicists disprove it every day.”

Do optical physicists really believe a formula that predicts 10^-15 instead of 10^-9? Dr. Skyskull probably knows whether they do. If they don’t, or if some of them don’t, he may be able to steer you to the good literature instead of the bad stuff you have been looking at.

He replied, “Actually, the opposite is true: traditional coherence theory deals with fields that are statistically stationary, which means that the statistical properties are independent of the origin of time. This automatically results in a field for which the different frequencies of light are completely uncorrelated — there is no definite phase relation between *any* frequencies.”

He didn’t look at the special case situation you described. It’s possible that optical physicists actually do use that bad formula even though most of the time they use statistically stationary fields. In that case you would do them a good turn to present the correct result. But maybe they (or some of them) use a completely different derivation along the lines he describes. Does the other formula also give the wrong result? Dr. Starskull might know whether they use the formula you complain about and if not what the alternative predicts.

“The mathematics of the spectral properties of pulses and the spectral properties of partially coherent fields are structurally very similar. The difference is that pulses have correlated spectral phases and PC fields have uncorrelated spectral phases. This similarity has led to a lot of confusion.”

He might have been giving you an important clue here. Maybe Wolf thought he was using correlated phases like for pulses, but really he was not. If you ask nicely he might say what he means plainly enough that you will understand. Maybe when he says it plainly it will be obvious to both of you that he’s wrong. Or right.

When I took freshman mathematics, they made a rule for arguments about proofs. Before you say what’s wrong with the other guy’s argument, you repeat back to him what he’s saying and get it clear whether what you think he said was what he actually said. It worked very well. Dr. Skyskull has not done this, but if you want to communicate with him you could do it.

Last August he said that what the formula actually predicts is the multiplicative inverse of the spectral bandwidth? (or was it the inverse function?) Anyway, he was saying that what you measure for coherence time is not the “real” coherence time the formula predicts. That would seem to leave room for a formula that predicts the coherence time that can be measured. 😉

“No, You need to go reread the derivation of the 1/bandwidth coherence time. It starts with a spectral distribution given by e^-(nu^2/sig^2). Then the Fourier transform is taken, and the result given the 1/bandwidth formula. but, the spectral amplitude is in general complex and and the ratio between the real and imaginary parts gives the phase. So the above spectral distribution corresponds to a frequency-independent phase, which is not found in nature.”

That looks plausible when I read it. You say how the coherence time is derived. And you say the result will give a single spectral amplitude so every frequency will have the same phase?

Maybe it would be clearer if you provided a link to a more detailed explanation.

At this point, I see the following possibilities:

1. The traditional formula you use is not actually the one Dr. Skyskull uses. He could tell you his.

2. The traditional formula you use is correct and the correct interpretation of it does not give you a frequency-dependent phase. Dr. Skyskull might give you a link to an explanation. Also, however, it apparently gives you an answer which is useless in practice.

3. The traditional formula you use is in widespread use but you are correct that its derivation depends on a special case which does not actually fit the stationary fields that traditional coherence theory deals with. Dr. Skyskull might tell you that you are right.

I have asked Dr. Skyskull several questions which would probably take considerable work on his part to find coherent answers. He has not replied at all. This is appropriate — if he was more interested in answers to my questions than answers to his own questions, he would not specialize in his research areas.

You are asking him to do some work he would otherwise not do. I don’t know how hard it would be for him but my uninformed estimate is 20 minutes to 2 hours. After he does this favor for you, he can expect you to thank him. Why would you call him names when his first quick answer was unclear? He had no obligation to answer at all.

I have been reading a libertarian blog, and that encourages me to suggest another way to get Dr. Skyskull’s attention. Tell him what you want him to do for you, and offer him money to do it. If he will do it for a price you can afford, you may have a deal. 😉 Or he might do it for an apology and a nice thank you.

• I appreciate your attempts to clarify the confusion, though I doubt tomsol is interested in the discussion at this point. The real problem with his argument is simply that fifty+ years of experiments have confirmed the bandwidth formula for pretty much every source of light in existence. In fact, I believe the coherence/bandwidth relationship was experimentally determined before it was mathematically formulated. As I noted earlier, people don’t apply such formulas to things such as sunlight because sunlight is so broadband that issues of detector sensitivity come into play. If your detector can’t measure ultraviolet->infrared and beyond with equal sensitivity, one’s measurement of coherence time isn’t going to be accurate.

The mathematical arguments put forth by tomsol are simply incorrect, and would be considered nonsense by any researcher in optical coherence. The spectral distribution he refers to is a spectral *intensity* distribution, which possesses no phase pretty much by definition. The Fourier transform gives the correlation function, from which the coherence time is derived. I could go on further, but I run this blog to try and enlighten people about the ideas of optics, not to argue mathematical details.

I’ve been meaning to write at least a partial response to your email, BTW, but I’ve been quite busy lately — and you wrote a long email!

18. bean sagof says:

could you speculate on what the coherence between two rarely monochromatic sources would contribute to the “observers” at p1 and p2 like you described for the last few figures?

would similar assumptions hold for e.g. neurons??? i.e. coherence would theoretically allow for interference effects and then constructive or destructive interference would be obvious in the local voltage also reflected in oscillatory spiking activity timing.

as far as measuring it, would you be able to hear coherence against a background of noise in voltage recordings? perhaps with each point in one of two stereo channels?

19. Ammar says:

Hi, Thanks for writing such articles .
I am in a trouble that i don’t have enough knowledge about waves , so i am trying to learn more by self help , though i bought lots of books regarding optics and modern physics but still i am looking for a book which will provide me sufficient knowledge about coherence theory of waves. Its my request please if possible suggest me a book and also some sort of notes on coherence theory.
I am looking towards you. Hope you’ll do reply soon.

• Thanks for commenting!

My recommendations depend very much on the level of math and optics you’re currently at; in general, coherence theory is a rather tough subject and there aren’t that many easy books on the subject. For upper-level undergrads with a physics/math background, Wolf’s “Introduction to the Theory of Coherence and the Polarization of Light” is probably the best choice (and I provided the cover figure and did many of the line drawings in the book!).

Also at that level, G.R. Fowles’ “Introduction to Modern Optics” is a relatively cheap and easy to follow optics book that discusses coherence effects early on. Another inexpensive, but more advanced, book is E.L. O’Neill’s “Introduction to Statistical Optics”, though it is a little older.

I hope this helps!

• Ammar says:

Thank you Gentleman !

Hope to be in touch .

20. J Thomas says:

I feel a need to think some things out, and I hope it’s OK if I do it here.

We think of electrons as being in one place at a time, in discrete units. oil drops each have some set number of electric charges. Well, but couldn’t there be a continuous amount of charge in space, but atoms only accept a set amount? Well but electrons and positrons traveling through cloud chambers go in lines, like they’re at one place at a time.

But electrons can be diffracted like waves. And the diffraction patterns happen even when it looks like just one electron at a time.

Explanations for this all sound like double-talk. They make no sense. Quantum mechanics is supposed to describe what happens in a way that gives the right answers, but people say that quantum mechanics also does not make sense.

How is it that electrons travel in straight or curved lines in cloud chambers but they diffract like waves through slits?

Probability waves? More double-talk.

If electrons move in lines, and their directions are independent (which I’d expect if it’s one electron at a time), and they have the chance to travel through two different holes and the chance they travel through each hole is also independent, then the probability that an electron is detected at a particular place is the probability that it travels through the first hole times the probability that it then lands where the detector is, times the probability that it travels through the second hole times the probability that it then lands where the detector is. But diffraction patterns are not the sum of those two probabilities.

Something else has to be going on.

Could it have something to do with the detectors? If a particular kind of electron comes through one slit, and soon enough afterward another kind of electron comes through the other slit and lands in the same place, then the detector doesn’t detect either one? That doesn’t seem real plausible.

Could it have something to do with the slits? The presence of one slit somehow affects the probabilities at the other slit? I could vaguely imagine that. At that level a chunk of mass with two slits in it isn’t just like a stone wall at the seashore with a couple of openings that water waves can surge through. It’s a whole bunch of charges that average out at large scales. But an electron that passes close to the edge of a slit could interact with local charges. It could bounce. It could even bounce first off one wall and then the other before it gets through, if the slit is thick enough. It could just be shifted a bit. And say the electrons are spinning, that could put some english on it. And a second slit that was close enough could somehow affect the electric fields? That doesn’t seem immediately plausible either. But it’s easier for me to think of a bunch of atoms affecting each other so that a somewhat-distant bunch of missing atoms changes the way they affect an electron, than to imagine the electron itself being everywhere and knowing everything sometimes, and other times making a single path through a cloud chamber.

Does the material the slits are made in make a difference? Can it be a conductor or a nonconductor or a semiconductor etc? What if it’s made of stripes of different materials?

Does the thickness of the material the slits are cut through matter? If it was just slits in a barrier and water waves then it wouldn’t matter except that less energy gets through.

If an electron is in one place, it doesn’t make sense for it to be affected by a hole it doesn’t go through. But it could be affected by an entire electric etc field including a hole in the mass that generates the field.

Is there any way to make the electron do the work? What if an electron can travel as a pair of particles that are somehow linked? Then one of them could go through one slit and the other through the other slit. Going through the slits changes their momentum. It can only change the momentum in particular ways, because if they are the wrong distance apart then at least one of them will fail to get through a slit — and that somehow stops the other, because they are somehow linked?

Then some of them would go through just one hole and wouldn’t show self-interference, unless they *have to* be some distance apart. They would go through a slit and all be “polarized” the same or opposite.

If there are more than 2 holes, could you tell the difference between electrons that can only go through 2 at a time versus electrons that can go through more than 2 at a time? I’d guess the interference pattern would be different without the contribution of all-at-once.

If you first send your electrons through a slit that’s oriented the same way as your later interfering slits, versus through a slit that’s oriented normal to those, do you get a different result? Does it mean anything for electrons to be polarized?

What if you could build a sort of wall over the bridge between the two slits, so that an electron would have decided which slit to go through before it got to them. I have the idea that shouldn’t matter much to a water wave, which mostly goes through a slit as a pressure and elevation wave and then expands in all directions. Would it matter to electrons?

It doesn’t have to make sense. But suppose somehow it did, wouldn’t that be good?

• J Thomas says:

What if you could build a sort of wall over the bridge between the two slits, so that an electron would have decided which slit to go through before it got to them. I have the idea that shouldn’t matter much to a water wave, which mostly goes through a slit as a pressure and elevation wave and then expands in all directions. Would it matter to electrons?

Here’s a picture that might be visible in a nonproportional font.

——— —– ————
|
|
|

I have the idea that the wall in front of the slits and ahead of them should have no effect on water waves. Without a wall, each piece of a wave crest depends on the pressure of the water to the side to keep it going straight rather than expand in all directions. It goes about the same when it has a wall instead, the difference is only that the wall is not moving with the wave and so friction will delay the wave near the wall a little. When the wave reaches a slit part of it goes through, and then it expands in all forward directions on the far side because it can.

I don’t know how a wall like this would affect light or electrons.

/ \
/ \
/ \
/ \

Here the other side of the slit has more than 180 degrees available. Does the wave that spreads in all directions spread backward too, say at a 270 degree angle? I think a water wave would, though I haven’t tested it. What about light? Electrons? Maybe the light spreads out too slow to actually get a measurable amount going backward even if it does do that.

• J Thomas says:

That ascii art didn’t work because it removed multiple spaces.

_….._____…..___
……..|……….
……..|……….
……..|……….

………….
…../..\….
…./….\…
…/……\..
../……..\.

• so that an electron would have decided which slit to go through before it got to them.

The curious thing is that, provided there is no attempt at “measurement” of the electron, it doesn’t *have* to decide which slit to go through before it gets to them! Another way to say it: provided your barriers don’t influence the likelihood of the electron arriving at one slit or another, it shouldn’t have a significant effect on the interference pattern.

The thorny issue in this rough explanation is the meaning of the word “measurement”; which has plagued quantum mechanics since it’s inception. I’ve always loosely imagined measurement to entail those inelastic scattering processes that involve a transfer of energy between particles, or absorption of particles. This encompasses all detectors, AFAIK.

21. J Thomas says:

The real problem with his argument is simply that fifty+ years of experiments have confirmed the bandwidth formula for pretty much every source of light in existence. In fact, I believe the coherence/bandwidth relationship was experimentally determined before it was mathematically formulated.

So the formula he looked at was the correct one, and it does not imply the phase things he thought it did, and it does not apply to his special case.

The interesting thing I think I get from this is that if for any reason you want to depend on coherence of sunlight, it comes out in practice about a million times as long as the theory would say because you can ignore the wavelengths that you don’t detect and mostly ignore the wavelengths you don’t detect well.

I’ve been meaning to write at least a partial response to your email, BTW, but I’ve been quite busy lately — and you wrote a long email!

Thank you in advance. No hurry, no obligation. I’m good at learning math and I’ve had a lot more trouble with physics — it’s supposed to be descriptive rather than proceed from axioms, and sometimes the descriptions are hard for me to follow.

22. Ahasan says:

Thanks for great post we are expecting more from you !

23. Brett says:

Cool website!
I really learned something about coherence today. The figure with the random wave and monochromatic wave cleared some doubts I had…

So the coherence time of a pure harmonic signal is infinite. In my signals & systems class I learned that the autocorrelation function R(tau) of f(t)= sin(t) is a periodic function too. The amplitude of R(tau) oscillate up and down (it’s sinusoidal).
by looking at the graph of R(tau) I would not automatically infer that sin(t) has infinite coherence time…….unless I decide to look at the “envelope” of R(tau).
The envelope of R(tau) is constant for any time shift tau. Now we are talking 🙂
I would say that the coherence time is that interval over which the envelope of R(tau) remains unity or close to it. In the case of a pure harmonic that interval of time is infinite….
Any objection/correction?
thanks!!!1
Brett

• Hi,

I think your analysis is more or less correct! The product of a sinusoidal field sin(t) with a time-shifted version sin(t+tau) will give you a function that oscillates in both t and tau. If we have the ability to instantaneously calculate this product, we would measure individual “ups and downs” as time goes on. The autocorrelation is usually defined with an averaging process over t, however! In optics, for instance, the field fluctuations are too fast for us to measure the rapid “ups and downs” and we end up measuring a long time average of the field. For the example of a sin function, we calculate an autocorrelation by taking the time average of that product, which will leave us with something proportional to cos(tau), independent of time origin t. The envelope of this is constant in time, as you noted.

This envelope calculation is made explicit by using complex wavefields, and using for instance e^{it} as the field instead of sin(t). For a harmonic field, the average over the oscillations is built-in, and the autocorrelation gives you the “average envelope”. That is, E[ e^{it} e^{-it}]=1.

I hope this makes sense; there’s a bit of subtlety in the use of complex wavefields and their relation to the physically observable wavefields.

• bean sagof says:

can the y-axis of the auto-correlation you’re talking about be read as an axis of correlation coefficients? or is there an ambiguity in that nomenclature that causes those not to be the same “auto-correlation”? it seems that the lack of an average is what’s causing a “spuriously low” auto-correlation between a sin wave with itself. and you’re saying that the complex wavefield does not run into a similar problem?

is there a way to use phase synchronization to study coherence? or are the two so close as to be the same?

24. bean sagof says:

i would think most autocorrelation functions (i.e. not infinitely coherent) would only be unity at exactly 0 and start to deviate from the amplitude expected of an infinitely coherent function pretty quickly as you went ahead or backward in time lags. another way to say that might be how close to unity are you talking about for the envelope? and of course, how does that scale fit with the phenomenon you’re considering?

25. Brett says:

hello bean sagof.
Thank for your comment. I agree with you that for realist signals the function R(tau) is 1 only at tau=0 and decays for tau_>infinity…..
I am trying to reconcile the the fact that a pure sine (infinitely coherent) does not have a constant R(tau) function for all tau. Instead, it is a sinusoidal function: for certain tau’s it is zero, nonzero, etc. But we know that a sine is coherent for any tau.
The only way to is fix things is not to focus on the values of R(tau) at each tau but to analyze the envelope of R(tau) which which for a sine is constant for any tau….

thanks!
Brett

26. brett says:

Hello Everyone!
this coherence topic is become my favourite one 🙂
I have recently learned about the angular spectrum and wonder how we can relate it to the coherence. Fourier theory works for 1D, 2-D and 3-D wavefields.
In free space we can synthesize a general and monochromatic wavefield by superposition of plane waves of the same angular frequency w , all traveling in different directions ( same magnitude wavevector). All good so far. I guess the resulting wavefield would be perfectly temporally coherent but not necessarily endowed with perfect spatial coherent….

How about synthesizing a partially coherent (time-wise) wavefield? What type of plane waves would we use in the superposition? Plane waves of different frequency?

thank!!!
Brett

• One can actually formulate the angular spectrum representation for the cross-spectral density (correlation function in the frequency domain); this is discussed, for instance, in Mandel & Wolf’s Optical Coherence and Quantum Optics, section 5.6. In the time domain, a plane wave decomposition of the temporal correlation function is in essence the Wiener-Khintchine theorem, that states that the spectrum of the signal is the temporal Fourier transform of the correlation function.

On the other hand, you may be asking how one generates a realization of a partially coherent field that has a given set of statistical properties; I would recommend looking at this paper for one method for generating a realization of a PC field!

27. brett says:

thanks again!
great paper. I will try to do a simulation today

Brett

28. brett says:

I know that a laser beam strong directionality is due to its high spatial coherence and not to its temporal coherence… I am not sure why…does anyone have an idea? A simple explanation like the apple/orange one used in the van Cittert-Zernike theorem explanation?

For example, will two different wavefields with the same spatial coherence and lateral extent but different temporal coherence diverge, expand the same way during free space propagation?
thanks
brett

• If you’re familiar with the angular spectrum representation of wavefields, it becomes somewhat clearer! A coherent wavefield will be represented by a certain collection of plane waves spreading in different directions; the angular distribution of these plane waves is a rough indicator of the beam’s spreading, and for a laser it is quite small. If one makes the same beam partially coherent, one can use the angular spectrum representation for PC fields to show that the decrease in coherence requires a larger angular spread of plane waves.

It is possible, however, to make partially coherent fields that have the same directionality as a laser. A 1978 paper by Wolf explains that a beam of low coherence but very broad intensity profile can be constructed that has the same angular spread as a narrow fully coherent laser beam.

29. brett says:

Coherence and Polarization
two wavefields can interfere only if they are coherent and their polarization is not orthogonal.
That makes me believe that coherence and polarization are not that independent of each other. However, I feel that we can mathematically come up with a field that is perfectly polarized everywhere but completely incoherent (temporally).
Or we could have a perfectly temporally coherent field (infinite coherence time) that is fully unpolarized…
In substance, it seems that coherence and polarization are actually independent, are they?
Brett

• Not quite independent! We can in principle do the perfect polarization/temporally incoherent, but in order to have an unpolarized field we need *some* random fluctuation, and therefore the field cannot be perfectly monochromatic (though, as far as the math is concerned, it can be arbitrarily close).

30. Steve Fowler says:

I have a question regarding the coherence of sunlight. Do I ask it here or elsewhere?

• You could drop a comment here, or email me at skullsinthestars*that at thing*skullsinthestars.com!

31. Brett says:

Hello everyone.
Coherence and the correlation functions used to quantify it, as Dr. Skullsinthestars says, are based on averaging and they strictly belong to random function and random processes.
Why averaging? Well, even an incoherent field can instantaneously interfere with itself and is coherent with itself for a instant of time. But what we care about is a stable, visible, measurable manifestation of coherence that can only be achieved over a long enough interval of time…
If the function shows coherence over that long period, then interference patterns (patterns based on averaging) reflect that…
If our eyes were able to follow and record the instantaneous electric field, the interference patter would look quite different and not static as the appear. Everything would be dynamic and oscillating: a point of constructive interference would continuously blink. We only see the averaged effect….

We could surely calculate the autocorrelation function of a deterministic function and get some zero value for the autocorrelation function at certain time lags. No surprise.
But a deterministic function is always perfectly predictable: if we know the function we know it all instants in time.
So does coherence and its synonymous correlation, mean similarity or predictability?
Two thing could be very predictable, one based on the other, but also be very different. Of course, if they are similar they probably behave the same way and we can infer the behavior of one from the other….
But the bottom line is that the correlations functions make full sense only when we apply them to random processes. That is why the idea of spectral components having zero phase does not make sense to me either. IF the process is stationary, the phase terms of the various spectral components are uncorrelated random variables….
Any correction?
Brett

When we refer to a field as being “statistically stationary”, we are saying that the underlying statistics that generate it are independent of the origin of time. This has built into it the result that a “stationary” field is of infinite extent in time.

It is possible to have a deterministic field be stationary, but only one type: a monochromatic field! The correlation function of such a field (in a complex representation) is unity, and its second-order correlation function is independent of the origin of time (this is another way to define a field as stationary: its averages are independent of the origin).

The phase terms of various frequencies of a wave are in fact uncorrelated random variables, which is another consequence of stationarity. However, the second-order (field time complex conjugate of field) frequency properties, or what we properly call the power spectrum of the field, is a real-valued quantity with no phase associated with it. The power spectrum of the field is, in essence, a measure of the fractional amount of *energy* in each frequency.

Hope this helps!

32. Brett says:

Thanks Dr. Skullsinthestars,
So the phase terms for each spectral component are uncorrelated, which implies time stationarity. Can the phase terms be uncorrelated but dependent? I learned that correlation and dependence are slightly different concepts.
What do we get in that case, what type of field? The uncorrelatedness should still guarantee time stationarity….

Also, are we assuming a wavefield composed of spectral components having the same frequency w but random phase shifts, of spectral components with different frequencies and random phase shifts?

Thanks,
Brett

33. J Thomas says:

I wonder whether it would make sense to delay email notification of new comments until after Dr. Skulls has decided not to delete them. Also I wonder whether it would be easy to arrange that.

It would slow down the communication when various commenters are involved with a hot topic. But it would also reduce the amount of comment spam that gets emailed to people who have notifications turned on.

I would suggest noticing how many people have notifications turned on — it might be too few to matter. Then notice whether it looks like a good thing to delay notification. If so, then notice how much time it’s worth, and quickly estimate how much time it would take. If it’s likely to take too long, perhaps send an email to somebody else who could do it, asking for that new feature.

34. DG says:

Regarding fringe visibility as in this image https://skullsinthestars.files.wordpress.com/2008/09/michfringecoh.jpg am I correct in assuming that the reduction in fringe visibility can be regarded as the result of superposition of the fringe patterns of many individual monochromatic frequencies which when viewed with a photon-counting broadband imaging array (or black and white film) gives the appearance of the summed fringe pattern being washed out (assuming for simplicity that the detector/film has a flat spectral sensitivity). With this interpretation, by switching for a detector with a narrow spectral sensitivity a fringe pattern would again become visible, is this correct?

If the above is true then there are a quite a few things I don’t grasp about the assertion that coherence time depends solely on bandwidth but perhaps two further questions might help to illustrate where I am having difficulties.

What happens for very short pulses? Even if spectrally filtered (or dispersed using a prism) then doesn’t there reach a point where the path difference in the interferometer is longer than the duration of the pulse. Would this allow you to have two sources with the same spectral characteristics but with different fringe patterns for certain path differences?

This goes back to what tomsol was saying (if a bit rudely) but if you do the calculation for white light then the coherence length comes out at around 900nm (Hecht). If amplitude-splitting solar interferometers with path length differences of >0.5m are used to analyse the solar spectrum doesn’t this suggest that the wave can interfere with itself over a distance much larger than the coherence length?

I think part of my difficulty in understanding this is in not knowing whether breaking a wave down into independent Fourier components is always valid (as I’ve been taught). If the individual components are monochromatic then they are infinite in time and space but this seems hard to reconcile with the idea of a finite pulse which must have a start and end point. Can you explain?

Thank you.

• With this interpretation, by switching for a detector with a narrow spectral sensitivity a fringe pattern would again become visible, is this correct?

That is in fact correct: every frequency, almost by definition, produces its own interference pattern.

What happens for very short pulses? Even if spectrally filtered (or dispersed using a prism) then doesn’t there reach a point where the path difference in the interferometer is longer than the duration of the pulse.

There’s a bit of a mathematical difference between a pulse of short duration and a continuous wave beam of short coherence time, though both are usually interpreted as having a “spectrum” related to the Fourier transform. First, let’s talk about the unfiltered case — obviously, when the path difference d is larger than the pulse width, there can’t be any interference — the pulses don’t overlap! Loosely speaking, the pulse width works a little like the coherence time.

What about filtering it? I think the key here is to think about what a filter actually *does* to a light signal! The most obvious example to think of is a Fabry-Perot interferometer, which is a pair of highly reflective, parallel mirrors. The Fabry-Perot filters by bouncing light between the mirrors for many, many passes: the better the filter, the more bounces the light has made!

In essence, the Fabry-Perot “stretches” the length of the light signal by forcing part of it to bounce around many times. The more you filter the signal (i.e. the more narrow-band the filter), the more the signal is stretched within the Fabry-Perot, and the longer the path difference can be made and still produce interference. At least from a classical optics point of view, similar arguments should apply to other filtering mechanisms.

I think part of my difficulty in understanding this is in not knowing whether breaking a wave down into independent Fourier components is always valid (as I’ve been taught).

In principle, the Fourier decomposition is always formally valid. You will find that problems with the infiniteness of the monochromatic components tends to sort itself out in pretty much every case on careful inspection, like it (more or less) does for the filter mentioned above. There are a lot of subtleties to its interpretation, however, and it probably is best to think of the Fourier decomposition as a mathematically convenient tool that may not perfectly represent “reality” but is good enough for most analyses.

• DG says:

Thank you for your reply, it is most helpful. In particular your explanation of how an etalon/interference filter can “stretch” the signal seems intuitive. I suppose for a prism the stretching of a pulse would be explained by resonances and damping in the atoms of the prism material. I need to give some thought to how a Lyot or Solc filter acts to “stretch” a pulse as it’s still not clear to me in those situations.

I’m going a little off topic here, but on the topic of Fourier decomposition if you took a stable continuous monochromatic laser and passed it though a infinitesimally short gate would you end up with a broad distribution of frequencies? Intuitively I can’t see the mechanism for this to happen but from a mathematical point of view if you create an approximate delta function in the time domain by gating a constant source then you would expect a broad distribution in the frequency domain. Is my reasoning at fault here?

Thanks again.

35. KMartin says:

Thanks for the website.

If the coherence area of a star is about 6m^2, how would planet finder interferometers work? They would interfere light from many tens of meters apart.

• KMartin says:

To answer my own question, the planet finder would use infrared light around 10µm, as at that wave length the planets are at their brightest relative to their stars. I guess the longer wave length would have a bigger coherence area.

36. Wolfgang G. says:

“Filtered thermal light: coherence time = 10^-8 sec, coherence length = 3 m”

Where does this value of 3 m come from? In the case of thermal light we have spontaneous emission, and the phase of a photon should be independent of all other photons appearing within 10^-8 seconds.

• “Coherence length” is simply defined as the coherence time X the speed of light, which immediately gives us 3 m as the coherence length.

Using a photon picture to discuss coherence is usually a bit misleading, as coherence is a wave property. In terms of photons, we can roughly say that the temporal *width* of the photon is roughly 10^-8 seconds. After that period of time has passed, the original photons have gone by/ been detected and a new set of photons is being measured. (This is a loose way of looking at it, but good enough for a non-technical blog post.)

• Wolfgang G. says:

Isn’t a temporal photon width of roughly 10^-8 seconds in contradiction with Einstein’s derivation of Planck’s formula of black body radiation in “On the Quantum Theory of Radiation” of 1917?
Einstein’s derivation is based on the premise that the energy distribution of atoms or molecules “should follow purely from emission and absorption of radiation”, especially taking into account momentum transfer.

If I understand correctly, this means: emission (and absorption) of photons on the sun’s surface lead to the same momentum changes of gas particles as thermal collisions between the gas particles themselves. And if this is true, then the emission time of a single photon should be much shorter than 10^-8 s.

37. J Thomas says:

It’s something about your system. I don’t see that with Firefox or Chromium.

38. Shabir Kabiri says:

Thank you so much for such an awesome, and enjoyable article. Wishing you more success and achievements in academics and life.

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40. yeahia says:

what is the difference between short and long coherence?

41. Kameron Hansen says:

Thank you so much for this clear & helpful article.

My question concerns the figure showing spatial coherence between 2 point sources. I’m unsure the difference between left and right sides.

Are you saying the left does have spatial coherence, because the sources are monochromatic, and therefore a fixed phase relation is established? However, in the right, the source is not monochromatic, therefore no fixed phase relation– and spatially incoherent?

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