Welcome to Coherent Wave Explanation

[Niles]

Hi all

I read in a book that coherent means when two waves are monochromatic and have a definite phase relationship. What is meant by a definite phase relationship?

 

[Born2bwire]

Coherence is a bit tricky because it is related by a statistical relationship. It is much easier to think of it in extremes. A perfectly coherent light is made up of monochromatic waves that have a constant phase difference between them. This is temporal coherence, I guess you would call it the propensity for two waves to remain in the same temporal relation to each other over time. The mathematical relationship is the time correlation between the two waves, or better yet, the autocorrelation of the light.

As we drift away from coherence, we can imagine the two waves starting to drift in their phase relation. Perhaps you have a bad source for one that is drifting its phase about. Or the frequency of one of the waves shifts around in time due to changes in the source. A perfectly coherent source is a monochromatic wave, as we drift away from the monochromatic ideal, we lose coherence. There are numbers of merit to describe the coherence, for example the coherence time and length.

The coherence time would be the time it takes for the phase or coherence to drift by a certain metric. The coherence length is the distance the wave traveled over the coherence time. So these numbers, when referenced to a common metric, often come up in optics. You will often hear of coherence length with lasers. Ideally we would want a monochromatic light from a laser but in truth we get a small bandwidth of light. The coherence length would be a measure of how monochromatic the laser's output would be.

The wikipedia article seems to be rather sufficient in describing coherence. http://en.wikipedia.org/wiki/Coherence_(physics)#Temporal_coherence

 

[Niles]

Ok, a lot of your reply I did not understand - the same as with the Wikipedia-reference.

First of all, if we have two waves of different frequencies, then why will they eventually getout of phase with each other?

 

[Born2bwire]

Ok, a lot of your reply I did not understand - the same as with the Wikipedia-reference.

First of all, if we have two waves of different frequencies, then why will they eventually getout of phase with each other?

Two waves of different frequencies are incoherent. We cannot talk about phase shifts as producing incoherence with waves of different frequencies. These two waves can remain in perfect phase relationship, no drifts, over time but they still will be incoherent. That is because the waveform of one "drifts" in time with relation to the other. The peaks of one wave will drift away from the other wave's peaks as time progresses due to the difference in their frequency. If the frequency difference is large, then this will happen even before a full cycle finishes.

For example, let's say wave one is frequency f and wave two is frequency 2f. In this case, in half a cycle, 1/f, wave two will have a peak a wave one's trough. They fall out of sequence very very quickly and thus are very incoherent. But what if they are very very close in frequency. Say wave one is f and wave two is 1.01f. It now takes 50 cycles before wave two's crest falls over wave one's trough. So in this case, we do have an appreciable coherence because the time over which the temporal correlation of the two waves holds is much much longer.

Light is really a superposition of waves. So if we were to measure the coherence of the light from the two sources, we would need to measure it from a single light wave, the superposition of the two waves. We do this by finding the autocorrelation of the light. The autocorrelation will relate how similar the signal is over time. If it is a perfectly monochromatic signal with no temporal phase shifts, then we maximize the autocorrelation function since we can, with only two points of measurement at the Nyquist frequency, always predict the future behavior of the wave. If we have a wave that is the superposition of f and 2f, then we weaken the autocorrelation because now we see the waveform change over time. True, it is periodic and so that will give it some autocorrelation, but the extreme nature of its variance over time means that it will be a very weak autocorrelation.

 

[Niles]

For example, let's say wave one is frequency f and wave two is frequency 2f. In this case, in half a cycle, 1/f, wave two will have a peak a wave one's trough. They fall out of sequence very very quickly and thus are very incoherent. But what if they are very very close in frequency. Say wave one is f and wave two is 1.01f. It now takes 50 cycles before wave two's crest falls over wave one's trough. So in this case, we do have an appreciable coherence because the time over which the temporal correlation of the two waves holds is much much longer.

Thank you for this answer. I need to be able to visualize this.

If I have a wave of frequency f, and a wave of frequency 2f, then the wavelength of the 2f-wave will be half of the wavelength of the f-wave, right? And this is because they both travel with c. If they travel at the same speed, then how can they fall out of sequence? I mean, the phase relationship should be constant?

This is the only part of the post I do not understand.

 

[zero_kilo]

the path that they have/had taken are different.

 

[Doc Al]

If I have a wave of frequency f, and a wave of frequency 2f, then the wavelength of the 2f-wave will be half of the wavelength of the f-wave, right? And this is because they both travel with c. If they travel at the same speed, then how can they fall out of sequence? I mean, the phase relationship should be constant?

Say they start out exactly in phase at x = 0. Then they both travel along the x-axis at the same speed. When they've traveled a distance equal to the wavelength of the 2f-wave, the 2f wave will be back at its original phase but the 1f-wave, having only traveled half a wavelength, will be 180 degrees out of phase.

 

[Niles]

But how can they be in phase to begin with when the wavelengths aren't equal?

 

[Doc Al]

But how can they be in phase to begin with when the wavelengths aren't equal?

So? All "being in phase" means is that the two waves are at the same position along the 360 degrees of their cycle. Since they have different wavelengths they can't be in phase everywhere, only at certain points.

 

[mikeph]

Coherence is a bit tricky because it is related by a statistical relationship. It is much easier to think of it in extremes. A perfectly coherent light is made up of monochromatic waves that have a constant phase difference between them. This is temporal coherence, I guess you would call it the propensity for two waves to remain in the same temporal relation to each other over time. The mathematical relationship is the time correlation between the two waves, or better yet, the autocorrelation of the light.

If two waves have the same frequency, isn't that sufficient to say that they have constant phase difference?

I was always taught coherence is when the phase difference is zero, so two waves of equal frequency hit the max/min at the same time.

 

[Born2bwire]

If two waves have the same frequency, isn't that sufficient to say that they have constant phase difference?

I was always taught coherence is when the phase difference is zero, so two waves of equal frequency hit the max/min at the same time.

Coherence is often discussed in the context of real world systems. Like I stated earlier, coherence length is a characteristic of merit given for lasers that relate the monochromatic-ism (??) of the laser. Here, I was pointing out that we could have two sources that merge to produce the signal that are both the same frequency, but one can be less than ideal. The limitations of the source may cause its phase to vary over time. It would be no different than the frequency shifting over time, but it could be a much more complicated description in the frequency as a function of time as opposed to phase as a function of time.