You're right, but soap bubbles in the sun also appear colorful.

You only care about those that come at a specific angle and are reflected to an observer. For most angles it doesn't matter, remember that lakes polarize their reflection (as seen by an observer).

Soap bubbles appear colourful in the sun in spite of low coherence because the soap film is thin. Soap films can be as thin as tens of nanometers meaning that light has to be only coherent for a femtosecond for interference to happen -- and this is very comparable to the cycle time of the light itself so that coherence is practically assured. But the round trip time in a 5mm window pane is 50 picoseconds, which is thousands of times larger making coherence important.

Sunlight's coherence length is on the order of 5µm. If this weren't the case, reflection holograms wouldn't work in sunlight. 5 - 10 µm is the typical thickness of the holographic layer. If you want, a reflection hologram is a somewhat sophisticated soap bubble.

In what sense does sunlight have any coherence length at all? I was under the impression that coherence length was essentially a function of how tightly controlled the wavelength is of a monochromatic source like a laser. Sunlight is completely incoherent.

I mean, the spectrum still has a width, it's just a very large width -- which is why the coherence time is very low.

In a more abstract way, you're right that we are reaching for a metaphor here: if a frequency distribution is sharply peaked around one frequency then we can roughly say that the light "is that frequency" but then modify this by saying that the bits that are off-frequency, while they are "also that frequency," become "phase-incoherent" after a characteristic time or distance. That is not actually correct; they are simply different frequencies -- but it makes the interpretation easier. That metaphor begins to break down when you've got such a wide spread of frequencies: red light can only "pretend to be" blue light for a vanishing period of time.

To put it more plainly, the phase-coherence time is the inverse of the spectral width in frequency-space; multiply it by $c$ to get a phase-coherence length. If you applied it to a truly monochromatic ideal source, you would find that the time and length are infinite -- it only makes sense for real distributions with normal spread. But that's all distributions, including blackbody distributions like sunlight -- it's just that those distributions have very large spread and therefore very low coherence lengths: but it's the same definition either way.

Coherence is defined as a correlation of the light with itself. In simpler terms "how similar it is to itself after letting it evolve for a bit". Sunlight has a lot of different wavelengths evolving in many different ways, therefore it is not too coherent compared to an ideal single wavelength laser which will be identical even after "evolving" for a while.

However, if you look at sunlight at a certain time and then look at it one femtosecond later, the second look will still be similar to the first look to some extent, therefore you would have to say that the light is somewhat coherent.

This is why sunlight is definitely NOT "completely incoherent".

In fact any thermal radiator (e.g. black body) will be coherent over short distances proportional to the temperature of the body. Effectively the light's phase doesn't (on average) change faster than the quantum of energy (h-Plank's constant) divided by the thermal energy of the body (kT) so the power of a particular wavelength averaged in your eye can still have significant interference effects for a high temperature radiator. The distance of coherence is then related to that coherence time times the speed of light (so redder-longer wavelengths are coherent for fewer wavelenghths).

That leads to an equation of L= c*h / 4kT where the Sun can be treated roughly as 6000K. This gives a coherence length measured in microns, which is conveniently several times the half wave is necessary for interference for visible light (0.4-0.7um full wave). It's pretty cool, because I think it means that if you evolved eyes to see light near the peak of your black body radiator then you will be able to see Newton Rings for internally reflective thin membranes sized close to that peak due to the coherence length (due to Wien's law).

A nice review paper: http://hank.uoregon.edu/teaching-modules/Broadband-Interfero...

Coherence ultimately has to do with photons interfering with themselves. Films and beamsplitters can “cut” a photon in two pieces in the sense that the photon takes both paths. Separately, the photon has an uncertainty in its energy which corresponds to a bandwidth and hence a coherence time. If the system conspires to bring the photon paths back together again, meaningful interference effects occur when the photon is destroyed (measured) provided the path difference is within the coherence time. This is relatively easy to do in a bulk optical instrument like a Michelson or Mach-Zender interferometer: just carefully adjust the two optical paths to be equal. Remarkably, if you sweep one path back and forth with respect to the other, while passing through the zero optical path difference point, and take the Fourier transform of the detected signal, an accurate depiction of the light source spectrum is produced.

A fascinating read from 1890, "Soap Bubbles and the Forces Which Mould Them", C. V. Boys. Talks about colors in a film of soap (and a primer on scientific deduction and reasoning) :

http://www.gutenberg.org/files/33370/33370-h/33370-h.htm

Ah I see your point. I guess a similar thing happens when you look at plastic with polarized sunglasses, then you see different colors (though there's a chemical issue with plastic as well that affects the polarization of light)

You could do a 5mm(+/-10nm) glass pane for it to reflect/absorb a specific wavelength but it would be hard.

Most plastic sheets are birefringent due to the way they are produced. In the right circumstances, that can make a very colorful appearance when viewed in polarized light. As the difference in refraction index is normally rather small this works also for thicker sheets as no long coherence length is required for the effect to work. The effect should be strongest if you hold up the sheet to the clear sky with the sun behind you (the light coming from there is polarized horizontally). Sunglasses have the polarization vertically (to block exactly this light). As the birefringence turns the polarization the patch viewed through sheet becomes brighter. If you add in dispersion, you can get colored patches.

It's the same effect mineralogists use with polarizing microscopes.

I address that point in the notebook. Thanks to Fourier you could decompose any waveform, coherent or not, into a sum (integral really) of sine waves. So the lack of coherence should not make no difference to the argument.

I would strongly disagree with "coherence should not make no difference to the argument". This is exactly what we mean when we say "a coherence length of X um" - you take all the spectral components and correlate them with their future selves and then look at something akin to average correlation.

Therefore the "coherence answer" is exactly and precisely the correct one. What you are doing is simply explaining it with different words. "The lack of coherence" of the wave as a whole is THE argument.

What your explanation is doing is applying a narrow wavelength filter and evaluating the conditions for each \Delta\lambda, which is absolutely a valid approach, but it is the same thing. Overall I like your explanation very much, but please be careful about statements like "not really the answer, at least not directly" as they only point at a misunderstanding of the coherence explanation.

> Therefore the "coherence answer" is exactly and precisely the correct one.

The correct answer to what question?

If the question is why we don't see iridiscence when "white" light (wavelength 380-740nm) is reflected on a "wide" film (several orders of magnitude larger than the wavelength), do you really need to talk about lack of coherence? The fact that for a particular width the film will supress/enhance thousands or millions of colors across the visible spectrum should be enough to make the resulting light white as far as human perception is concerned, I think.

I agree although it is interesting to consider that even very coherent light would probably not produce a diffraction effect visible to our eyes.