Double-slit with half-transparent mirror

[jk22]

Mentor's note: This thread has been split from https://www.physicsforums.com/threads/double-slit-with-semi-transparence.845980

What about if the wall were not transparent but one of the slits equiped with a half transparent mirror ?

Then we would half-know which path was taken hence we should only obtain a half interference but it would not be completely destroyed ?

 

[DrChinese]

What about if the wall were not transparent but one of the slits equiped with a half transparent mirror ?

Then we would half-know which path was taken hence we should only obtain a half interference but it would not be completely destroyed ?

That's correct, you can have 100% interference effects, 0% (none), or any value in-between if set up accordingly.

 

[jk22]

I will try to do the calculation, but from the onset I cannot tell if the maximum will be no more at the middle point between the slits, or if it will only diminish the interference.

 

[Heinera]

I will try to do the calculation, but from the onset I cannot tell if the maximum will be no more at the middle point between the slits, or if it will only diminish the interference.

The maximum will still be at the middle point. What you will see is a pattern where half of the intensity is from an interference pattern, and half from a blurred out pattern with no interference. In other words, it will look like the two are overlayed. So you basically end up with the same interference pattern, but with reduced differences in intensity between regions with constructive vs. destructive interference.

 

[jk22]

Ok. But there still is an interference and on the other hand we know which path the single photon takes since points with half intensity have gone through the half mirror?

 

[Heinera]

Ok. But there still is an interference and on the other hand we know which path the single photon takes since points with half intensity have gone through the half mirror?

Yes, half of the photons will create the usual interference pattern, and the other half will not.

 

[jk22]

But in some sense we get an interference pattern with the which path information ?

 

[Heinera]

But in some sense we get an interference pattern with the which path information ?

Yes, because the which path information is not perfect.

 

[jk22]

Yes, because the which path information is not perfect.

If we could measure the intensity of each photon then we could say which slit it passed through. I don't know what is meant by perfect.

 

[Heinera]

If we could measure the intensity of each photon then we could say which slit it passed through. I don't know what is meant by perfect.

No, the intensity of each photon would be the same. A photon with frequency $f$ can only have energy $E = hf$, where $h$ is Planck's constant.

 

[jk22]

But we agree that the beam passing through the half mirror will have less intensity ? Can we say the flux of photons is smaller ?

 

[Heinera]

But we agree that the beam passing through the half mirror will have less intensity ? Can we say the flux of photons is smaller ?

In a sense, yes. Half of the photons will pass through both slits and will generate an interference pattern, the other half pass through one slit only and do not contribute to the pattern.