What is the relationship between coherence and interference in wave propagation?

[Niles]

Hi all

From http://en.wikipedia.org/wiki/Interference_%28wave_propagation%29: "Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. "

Coherent means that they are monochromatic and have a definite phase relationship. Let's say we have two waves of e.g. 16 Hz and 18 Hz. They are not monochromatic and their phase relationship is not constant, even though they start out in phase - but they will still interfere (they will beat against each other). Here the waves are not coherent, but we still have interference. How is that?

 

[Vanadium 50]

The words "usually" and "nearly" are both important.

 

[Born2bwire]

I would also mention that coherence is more of a measure than a black or white property. The waves, 16 Hz and 18 Hz, have a limited amount of coherence. They are not perfectly coherent, but in real life it would be hard to ever produce two perfectly coherent waves in the first place. So coherence is descibed by such numbers of merit like the coherence length or time. The longer the length/time, the more coherent the two waves are.

 

[Niles]

I have just checked my book, and they write that constructive/destructive interference is not possible unless the waves are coherent. That is the part i don't quite get, since in my OP it is quite clear that it is possible. But it seems like post #2 and #3 is a contradiction to this statement?

 

[Doc Al]

Certainly any two waves that overlap will 'interfere' in the sense that they will add or subtract at the point of overlap. But to get a persistent interference pattern the waves need to be coherent (or close to it). I suspect that that's what your book is talking about.

 

[Cthugha]

Coherent means that they are monochromatic and have a definite phase relationship.
Lets say we have two waves of e.g. 16 Hz and 18 Hz. They are not monochromatic and their phase relationship is not constant, even though they start out in phase - but they will still interfere (they will beat against each other). Here the waves are not coherent, but we still have interference. How is that?

No. All you need is a well defined phase relationship. The phase relationship does not need to be constant. Two monochromatic waves of 16 Hz and 18 Hz ARE mutually coherent (in the classical meaning of coherence).

 

[Andy Resnick]

Hi all

<snip>
Lets say we have two waves of e.g. 16 Hz and 18 Hz. They are not monochromatic and their phase relationship is not constant, even though they start out in phase - but they will still interfere (they will beat against each other). Here the waves are not coherent, but we still have interference. How is that?

I don't understand what you mean- your example is indeed two monochromatic waves (each wave is composed of a single frequency), and there is also a definite phase relationship between the two- I can exactly calculate what that phase difference is for any point in time. Those two waves are indeed mutually coherent (you left out polarization, but that's fine for now).

 

[Niles]

I don't understand what you mean- your example is indeed two monochromatic waves (each wave is composed of a single frequency), and there is also a definite phase relationship between the two- I can exactly calculate what that phase difference is for any point in time. Those two waves are indeed mutually coherent (you left out polarization, but that's fine for now).

So "definite phase relationsship" does not equal "constant phase relationsship"? If not, then what is meant by "definite phase relationsship"?

 

[Cthugha]

You need a well predictable phase relationship. If you have a completely monochromatic wave and know its phase at some point and time and its frequency, you will be able to predict the phase at any other time and place for all eternity. This is perfect coherence. Now if you superpose two monochromatic waves with different frequency, the superposition will look different, but you will still be able to predict the phase at every point and time - there is a fixed and determined phase relationship between both waves at every position and every time.

 

[Niles]

You need a well predictable phase relationship. If you have a completely monochromatic wave and know its phase at some point and time and its frequency, you will be able to predict the phase at any other time and place for all eternity. This is perfect coherence. Now if you superpose two monochromatic waves with different frequency, the superposition will look different, but you will still be able to predict the phase at every point and time - there is a fixed and determined phase relationship between both waves at every position and every time.

Thanks. That is a very good explanation. But in the light of your post, then how can one explain that the 16 Hz wave and the 18 Hz wave are not perfectly coherent?

The waves, 16 Hz and 18 Hz, have a limited amount of coherence.

 

[Cthugha]

But in the light of your post, then how can one explain that the 16 Hz wave and the 18 Hz wave are not perfectly coherent?

Well, I do not know. Most of all I am not sure, what exactly Born2bwire was talking about.

In most cases having a spectral width indeed means that coherence is limited. In fact the coherence time is the decay time of the Fourier transform of the spectral power density. Therefore smaller spectral width USUALLY means longer coherence times. However this treatment is based on treating the emission processes as ergodic stochastic processes, where the spectral width is assumed to be a result from the phase of a light beam being not constant over time and varying randomly. If you have spectral broadening as a consequence of such processes, coherence goes down. If you just superpose monochromatic light beams this does not apply. Maybe he was talking about the former situation, but this is just a guess.

 

[Bob S]

Well, I do not know. Most of all I am not sure, what exactly Born2bwire was talking about.

In most cases having a spectral width indeed means that coherence is limited. In fact the coherence time is the decay time of the Fourier transform of the spectral power density. Therefore smaller spectral width USUALLY means longer coherence times..

An FM receiver has a discriminator circuit that rejects AM-modulated signals that have the same spectral power density as the FM-modulated signals. I seem to recall that the amplitudes of the positive and negative frequency FM sidebands have opposite sign, while the AM sidebands have the same sign. So although both signals have the same spectral power (i.e., amplitude squared) density, they do not interfere. So it is possible to have coherent non-interfering signals with the same spectral power density.
Bob S

[added] See eq 1-4 on page 49 and 2-9 on page 54 of

http://contact.tm.agilent.com/data/static/downloads/eng/Notes/interactive/an-150-1/hp-am-fm.pdf

These two expressions for AM and FM modulation amplitudes are identical except for the sign of the third term (lower sideband). This sign is the major difference between FM and AM modulation. The sign difference does not show up in power spectral density measurements.
(Higher modulation of FM carriers leads to Bessel functions).

 

[Born2bwire]

Thanks. That is a very good explanation. But in the light of your post, then how can one explain that the 16 Hz wave and the 18 Hz wave are not perfectly coherent?

There are different kinds of coherence. Typically, we only talk about temporal coherence as opposed to spatial or even spectral coherence. Temporal coherence is a measure of how monochromatic a wave is. Thus, a wave of mixed frequencies will not have a high temporal coherence. This correlates to the fact that the resulting wave has beat frequencies and would give rise to an inconsistent interference pattern in say a double slit experiment because the 16 Hz component has interference at different angles than the 18 Hz component. However the closer they are in frequency, the more limited this discrepancy becomes (and also the higher the coherence is as well).

So if you have a laser source, ideally it would be monochromatic. In reality it is actually a small bandwidth of frequencies. The coherence length/time is a metric of temporal coherence that we attach to such things like lasers to describe how monochromatic the source is. The wave can perfectly maintain its phase relationship as it travels but if it is composed of a bandwidth of frequencies then it will suffer a penalty to its temporal coherence regardless.

Of course temporal coherence only describes the behavior of the wave to be consistent, or correlated would be the better statistical term, over time. Spatial coherence is how correlated the wave is over space. You would want a spatially coherent wave as well to exhibit good interference because you need to have the phase fronts overlap consistently in space. If they do not, then you would only have limited hot spots of interference. The interference would fade in and out over space instead of propagating along with the wave.

 

[Cthugha]

There are different kinds of coherence. Typically, we only talk about temporal coherence as opposed to spatial or even spectral coherence. Temporal coherence is a measure of how monochromatic a wave is. Thus, a wave of mixed frequencies will not have a high temporal coherence. This correlates to the fact that the resulting wave has beat frequencies and would give rise to an inconsistent interference pattern in say a double slit experiment because the 16 Hz component has interference at different angles than the 18 Hz component. However the closer they are in frequency, the more limited this discrepancy becomes (and also the higher the coherence is as well).

I see your point, but this is not really limiting coherence. Although you will get a beating pattern, the visibility of this pattern will not decay over time if you superpose two monochromatic light sources. Accordingly strictly speaking, the coherence time will still be infinite.

So if you have a laser source, ideally it would be monochromatic. In reality it is actually a small bandwidth of frequencies. The coherence length/time is a metric of temporal coherence that we attach to such things like lasers to describe how monochromatic the source is. The wave can perfectly maintain its phase relationship as it travels but if it is composed of a bandwidth of frequencies then it will suffer a penalty to its temporal coherence regardless.

Yes, this is the realistic version, where you have one mode, which has a bandwidth due to broadening processes caused by the phase varying randomly on a specific timescale. Here the coherence time will go down. If you instead (theoretically) take a lot of monochromatic light sources and superpose them to get the same small bandwidth of frequencies, the coherence time will not go down and you will see a strange beating pattern for all eternity. However - of course - this does not happen in real experimental situations.