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.

http://www.terrestrial-planet-finder.com/status.html
http://en.wikipedia.org/wiki/Terrestrial_Planet_Finder

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.

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.

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.

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.

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

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!

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