Comments on: Optics basics: Coherence

By: Brett

Brett — Wed, 11 Jan 2012 15:42:15 +0000

Thanks Dr. Skullsinthestars,
your explanation surely helps.
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

By: skullsinthestars

skullsinthestars — Mon, 09 Jan 2012 17:29:52 +0000

Not entirely sure about your question, but a few thoughts:

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!

By: Brett

Brett — Fri, 06 Jan 2012 20:56:20 +0000

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

By: skullsinthestars

skullsinthestars — Tue, 01 Nov 2011 16:25:08 +0000

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

By: Steve Fowler

Steve Fowler — Tue, 01 Nov 2011 16:01:41 +0000

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

By: bean sagof

bean sagof — Thu, 25 Aug 2011 02:22:17 +0000

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?

By: skullsinthestars

skullsinthestars — Sat, 04 Jun 2011 00:56:32 +0000

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).

By: brett

brett — Fri, 03 Jun 2011 02:50:38 +0000

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

By: skullsinthestars

skullsinthestars — Tue, 31 May 2011 20:05:11 +0000

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.

By: brett

brett — Mon, 30 May 2011 14:51:33 +0000

Still about coherence…..

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