What is the pattern in a double slit experiment near the slits?

[Heidi]

TL;DR Summary

Does this picture show that there is no fringe near the slits?

I am looking to this picture
https://en.wikipedia.org/wiki/De_Broglie–Bohm_theory#Double-slit_experiment
it seems that fringes only appear on a screen when it is not too close to the fringes.
and that the electron near the screen would give a pattern as when the path is known. Is it right?

 

[vanhees71]

Yes, but I wouldn't argue with Bohmian mechanics but with standard wave mechanics. Qualitatively you can understand the diffraction problem with the Huygens principle, i.e., the wave pattern behind the slits is found by taking the wave field in the slits as given and construct the field behind the slits by a superposition of spherical waves originating from any point in the slits. Close to the slits these partial waves from two different slits do not overlap and thus there's no interference of contributions from different slits. Going farther away the contributions start to overlap and interference occurs. Very far from the slits there's no way to distinguish the contributions from the individual slits, i.e., there's no more which-way information and you have the full interference pattern of in Fraunhofer conditions.

This picture also explains that you need a coherent source which "illuminates" the slits to begin with. Otherwise you don't see interference fringes at all.

 

[Heidi]

Going farther away the contributions start to overlap and interference occurs. Very far from the slits there's no way to distinguish the contributions from the individual slits, i.e., there's no more which-way information and you have the full interference pattern of in Fraunhofer conditions.

So the distance between the screen and the slits is a parameter which control the which way information. full information (1) near and null at infinity. Is this WWI(d) an exponentian?

Do you know other interferometry devices which also have a continuous parameter controlling the fringe visibility?
thanks.

 

[Heidi]

Have we also a null visibility of the fringes near the slits when their width tend to zero?

 

[vanhees71]

Very close to the slits there are no interference fringes, because the partial waves originating from the different slits are not overlapping in this region and thus there's nothing to interfere.

 

[Heidi]

Very close to the slits there are no interference fringes, because the partial waves originating from the different slits are not overlapping in this region and thus there's nothing to interfere.

e"

Very close to the slits there are no interference fringes, because the partial waves originating from the different slits are not overlapping in this region and thus there's nothing to interfere.

Yes they do not overlap at a close distance. But wouldn't they if the slits were narrower?

 

[vanhees71]

Yes, thinking further leads to answer the question, what "close" quantitatively means.

 

[Heidi]

Please let me ask the question differently.
The width of the slits is infinitesimal. the particle has a wave length lambda.
the distance between the slits is l. the screen is in a plane parallel to the slits ans the distance
between these plane is D. Let P be a point with abscissa x.
What is the exact formula for the amplitude of probability associated to P?
this formula has to be the same for every values of D,l,x and lambda.[/QUOTE]

 

[vanhees71]

Have a look at this:

https://arxiv.org/abs/1110.2346v3

 

[Heidi]

Thanks a lot!

 

[Heidi]

in this paper the source is at (0 0 0). the slits at z=D and are centered a x= -b and +b.
the screen is at D+L

I read this in the paper:
D is supposed to be very large compared to the dimensions in
the x-direction, more precisely, x, a, b ≪ D, L
I would have no problem with x,a,b very smaller than D+L but here the screen is far from the slirs (compared to the distance 2b of the slits).

This cannot answer my question.

 

[vanhees71]

I was referring to Fig. 4, where I think the point that you see through which slit the particle came under Fresnel conditions, i.e., with the screen not too far from the slits, and that then there is no double-slit interference. This is not inconsistent with the assumptions made. The criterion is the "Fresnel Number", $N_F$. The above definitions are necessary only, because the trajectories in the path integral, where the particle is moving through the slits several times are neglected.

 

[Heidi]

I was referring to Fig. 4, where I think the point that you see through which slit the particle came under Fresnel conditions, i.e., with the screen not too far from the slits, and that then there is no double-slit interference. This is not inconsistent with the assumptions made. The criterion is the "Fresnel Number", $N_F$. The above definitions are necessary only, because the trajectories in the path integral, where the particle is moving through the slits several times are neglected.

I had in mind that the presence or the abscence of fringes was only depending on L (fringes far away, no fringes near the slits). In fact the Fresnel number is important and it is a function of the
product of L (distance between the two planes) and the wave length of the particle

 

[jtbell]

In classical optics, the intensity distribution near the slits is called "Fresnel diffraction", as opposed to the far-field "Fraunhofer diffraction" which is the basis for the diffraction / interference patterns that you see in most books.

Try a Google search for something like "Fresnel diffraction double slit" and see what you get. You'll probably have to weed out stuff for single slits, which is what I mostly discussed when I taught Fresnel diffraction a long time ago as part of an intermediate optics course.

Even Fresnel diffraction involves approximations that fail when you get very close to the slits.

 

[Heidi]

thanks for this answer, Jtbell
I am still trying to understand the paper given by Vanhees71. Maybe you could help me to understand two points. The amplitudes A(x,a,b) is defined by a path depending on T + tau. Look at equation 16. But time disappears in the left notation. Why?
The second question is about the purety of the state. can this path formulation lead to mixed states?

 

[vanhees71]

In classical optics, the intensity distribution near the slits is called "Fresnel diffraction", as opposed to the far-field "Fraunhofer diffraction" which is the basis for the diffraction / interference patterns that you see in most books.

Try a Google search for something like "Fresnel diffraction double slit" and see what you get. You'll probably have to weed out stuff for single slits, which is what I mostly discussed when I taught Fresnel diffraction a long time ago as part of an intermediate optics course.

Even Fresnel diffraction involves approximations that fail when you get very close to the slits.

Yes, and it's often not so carfully discussed in textbooks. As usual, Sommerfeld gives a clear and solid explanation in Vol. IV (optics) of his Lectures on Theoretical Physics. BTW, exact (electrodynamical) refraction theory was the content of his habilitation thesis.

 

[Heidi]

thanks for this answer, Jtbell
I am still trying to understand the paper given by Vanhees71. Maybe you could help me to understand two points. The amplitudes A(x,a,b) is defined by a path depending on T + tau. Look at equation 16. But time disappears in the left notation. Why?

Could anybody tell me why the amplitude A(x,a,b) is not written A(x,a,b,T+tau)?