# Common interpretation of coherence

1. Dec 3, 2015

### Anton Alice

I dont like the common interpretation of coherence of two waves.
Please tell me if something is wrong in my argumentation:

1.

It is often said, that, if two waves are coherent, then the interference pattern is stationary, which means, that the amplitudes are the same. And often, this statement is used as a definition of coherent waves.
For example, two sine waves with same frequency interfering at a point, having a certain fixed phase difference at that point, would create an oscillation with a constant amplitude, i.e. constant intensity.

Now, if I take two waves with slightly different frequencies, then the interference (i.e. the superposition) at any point would be a beat. And now if I would measure the intensity, it would vary with the beat frequency, and therefore not be constant. According to the above definition of coherence, these two waves would be incoherent.

But from a mathematical point of view (using auto correlation function) the beat signal would have an infinite coherence length:
Those two waves are coherent, if the superposition (which is the beat) is to some extend auto-correlated, i.e. self-similar. And indeed, the beat is self-similar, because its periodic. This is why I would treat them formally as coherent (with infinite coherence-length), although the interference pattern is not stationary.

2.

Now instead of taking two waves with slightly different frequencies, one could also take a continuous spectrum of waves, for example created by a laser with a certain linewidth. If for example the spectrum of the laser looks like a gaussian, then the superposition of all waves is also a gauss-shaped wave packet. This gauss-shaped wave packet has a certain width (which is inversely proportional to the line-width of the spectrum). And the (normalized) autocorrelation of that gauss-shaped wave packet would tell me something about the coherence length. The wave-packet would be nicely correlated to itself for small phase shifts, because it would act approximately like a sine.

Contrary to the above example with two waves of slightly different frequency, the laser would only have a finite coherence length, because the signal is not a periodic beat, but a gaussian, which has a finite width.

Am I right, that the coherence length of the two-wave example is infinite, and the coherence length of the laser is finite?

2. Dec 3, 2015

### Staff: Mentor

Only if the periods are a rational multiple of each other.

3. Dec 3, 2015

### Anton Alice

Oh yes. f1=f2*Pi would not work.

Is this the only constraint? Am I correct with the rest?

EDIT:
Wait... why do the have to be rational multiples? The beat is still periodic, no matter what the ratio of f1 and f2 is.

Last edited: Dec 3, 2015
4. Dec 3, 2015

### jbriggs444

Are you sure? What is the definition of "periodic"? What would be the period of the combined wave?

Edit: How does "least common multiple" figure in?

5. Dec 3, 2015

### Anton Alice

Oh yes, f1=f2*Pi would actually not work, because each single beat would then envelop a different oscillating shape.

What about the rest of post#1?