Time dependent optical path length vs coherence

[Anton Alice]

Hello,

I have a simple question.

Suppose a perfect point source in front of a mirror. The virtual image of the point source acts like a second point source.

Now let's look at the interference of the direct point source light and the virtual point source, at some position.
Since the path length difference between direct light and mirrored light is constant, the interference is going to be stationary.

Now, if the mirror is moved such that the path difference changes in time, what can I say about the interference or spatial coherence? How does the degree of coherence / quality of interference depend on the speed of the movement?

I assume an incoherenct interference due to doppler shift. But if the speed of the path-difference-change is constant, then the doppler shift would give rise to a constant frequency difference.
Therefore the interference would be a particular beat, right? Is it then possible to observe a stationary interference pattern?

 

[sophiecentaur]

Is it then possible to obverse a stationary interference pattern?

What does the term "obverse" mean, here?

 

[Anton Alice]

Obverse is another way of saying "observe", if you are typing on a nerve-racking notebook keyboard.

 

[sophiecentaur]

Obverse is another way of saying "observe", if you are typing on a nerve-racking notebook keyboard.

Sorry. It's obvious to me now.

 

[sophiecentaur]

The interference pattern will only be stationary if you arrange for the motion of the mirror to keep the path differences between direct and reflected paths unchanged. I think that means the mirror can only move within its initial plane. If the mirror is moving in any other direction, the path difference will always be changing and so would the pattern. The rate that the maxima and minima pass across a point on the 'projection screen' will depend on the rate that the path difference changes by λ/2.

 

[sophiecentaur]

Actually, the pattern would be expanding and contracting about the line of symmetry as the d between object and image changes. You could assume a central maximum along that line.

 

[Anton Alice]

So you are saying there is a pattern.

I mean why should it not be possible to observe an interference structure when interfering two different frequencies which add up to a beat?
Obviously for a beat the intensity is not constant. But for a simple sine wave the (unaveraged) is not constant as well. Know what I mean?

 

[sophiecentaur]

It may be easier to think in terms of the interference with a model using two radio transmitters, each with a separate antenna. If you synchronise the two transmitters at the same frequency, there will be a stationary interference pattern. If you offset the frequency of one of the transmitters by 1Hz (a much more realistic procedure than with light sources), the interference pattern will change as each second passes. In anyone spot, the received signal strength will beat at 1Hz as the relative phases sweep throug 0-360 degrees and the pattern sweeps across the Earth's surface. The issue of coherence won't arise if the waves are continuous sinusoids, whatever the separation of the antennae or the bearing of your receiver. However, if there is (identical and synchronised) amplitude modulation of the carriers, the phase of the modulating signal will start to be different when the receiver is off the mid line. This effect gets worse as the modulating frequency gets higher for any given path difference. The modulation has changed the coherence length of the signals; bursts of high level carrier will arrive at different times from the two transmitters and cancellation / enhancement will not longer be complete as the phasors are no longer of equal amplitudes all the time. Nulls will be filled in, in the 'mush areas'.

 

[Anton Alice]

In anyone spot, the received signal strength will beat at 1Hz as the relative phases sweep throug 0-360 degrees and the pattern sweeps across the Earth's surface.

Why is it that way? With "received signal" you mean the intensity, which is the averaged poynting vector at that point. The poynting vector is time-dependent. The intensity averages over one period of oscillation, and then it follows, that the intensity is one half of the maximum amplitude. But why should I necessarily average over one oscillation?
In case of a beat I could also average over one beat-period. Or in other words: if the beat period is quick enough, then the observed interference will look stationary again.
Oh... wait a minute. You said the pattern sweeps across the space, in other words: constructive and destructive interference interchange, as the relative phases sweep through 0-360 deg. If I would average over a beat period, then the intensity would be homogeneously distributed, therefore no interference pattern observable, right?

 

[sophiecentaur]

If I would average over a beat period, then the intensity would be homogeneously distributed, therefore no interference pattern observable, right?

Exactly. The pattern just averages to a continuous value.
The 'coherence' issue comes in when the time period of the modulation becomes comparable with the time difference of arrival. In this case, the variations in phase / amplitude follow a regular pattern whereas what you get from a light source will have a random element with a statistical spread in lengths of wavetrains. However, in both cases, the limit measuring / observing the pattern will be reached when the paths become too unequal.
My background is in RF and broadcasting and I often find that using RF sources as a model can give more approachable answers to diffraction questions than using an optical model. It pays to be open minded in these things so you can pick the easiest model for each case.