Exploring the Limitations of Diffraction: Fresnel vs. Fraunhofer

[kent davidge]

I have only seen Fresnel and Franhoufer diffraction being discussed in textbooks.

Is it because they are the only two tratable types of diffraction?

I mean, in the sense that one can really work out the equations to get results.

 

[sophiecentaur]

I think it's just a matter of convenience to separate the two conditions. The calculations in Fraunhoffer diffraction are simple because the construction 'rays' are assumed to be parallel when the distance is great enough. The basic textbook treatment shows two or more similar triangles and that the field vectors are all parallel at any point and there is no significant 1/r factor. Relatively easy stuff.
Fresnel diffraction calculations are when path lengths are short enough for the angles to change across the pattern and that the 1/r starts to be relevant when summing the vectors.
If you want to know where one takes over from the other then I guess it would be a matter of how accurate the result of the easier Fraunhoffer method is for your particular application.
You could have the same problem with specifying Distance measurement. What's considered to be close and what's far away would depend on context.

 

[DaveE]

Another way of saying what @sophiecentaur just said (the way I learned it, many years ago):

When calculating the wave propagation, for example using Huygens principle, you would use a spherical wave-front for accuracy, which is necessary in the near field. However, the math is a bit of a pain with spheres, you end up with square roots in the exponent, etc. So Fraunhoffer approximated the sphere with a quadratic (Taylor's expansion), which works well in the far field.

I thought this Wikipedia page was good; it has some approximate rules for when you should use each.
https://en.wikipedia.org/wiki/Fresnel_diffraction

 

[DaveE]

There are other ways of doing the math, like Rayleigh-Sommerfeld diffraction. I never used this and don't know much about it. But there is a good description here: http://www.physics.usyd.edu.au/teach_res/mp/op/doc/op_diffraction_integrals_theory.pdf

 

[kent davidge]

Thanks @DaveE, @sophiecentaur

However I think you didn't understand what I'm asking.

I want to know if Franhoufer diffraction and Fresnel diffraction are the only two types of diffraction that can possible occur.

 

[DaveE]

Fraunhouffer diffraction and Fresnel diffraction are not two types of diffraction. They are two ways of calculating the result of the propagation of light. Diffraction in this context is an ever-present feature of the way light behaves. I have only ever heard of one "type" of diffraction of light. Of course, different circumstances result in different answers, but the general phenomenon of diffraction is the same.

Diffraction is also used in the same way for other stuff, like electrons. But that gets into QM. We have been in the classical physics realm in this thread.

 

[sophiecentaur]

the only two types of diffraction

There is only one Diffraction and it is the result of a wave (any form of wave) interacting with a discontinuity in its path. That can be an object or an aperture. It is calculated by following the same process as Huygen's description from years ago. You (vector) sum the waves arriving by all possible paths at any particular point of interest.

I have a pet hate of 'classification' of Science terms and this is an example that has managed to confuse you. I can sympathise but hopefully it will now be clearer for you.

 

[DaveE]

In the most general sense, I would say that diffraction occurs for any wave propagation situation with multiple sources. Combined with Huygen's principle, which requires linearity (i.e. superposition), this allows you to solve wave propagation problems with initial boundary values (slits, and such).

 

[DaveE]

Oops, said the same thing at the same time, LOL!

 

[jtbell]

The Wikipedia article about Fresnel diffraction that DaveE linked to says at the very beginning,

In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field.

Follow the link in the quote above and you'll find out about Kirchhoff's diffraction formula which is the basis for both Fraunhofer and Fresnel diffraction, using different approximations. That article shows the derivations for both Fraunhofer and Fresnel, at the end.

 

[sophiecentaur]

any wave propagation situation with multiple sources.

Our posts are in agreement, of course.
You would probably need to use the term "distributed sources" because that implies the need for Integration, rather than a simple summation of point sources.

 

[vanhees71]

Indeed Fraunhofer and Fresnel approximations are used because they are treatable and applicable in their (different!) realms of validity.

Exact solutions for diffraction problems are very difficult (you have to go beyond Kirchhoff's approximations and take into account the full vector nature of the em. field). It was solved for the half-space by Sommerfeld in his Habilitation Thesis (that's a second large work on top of the PhD thesis in Germany to get the "venia legendi" at a university).

 

[DaveE]

Our posts are in agreement, of course.
You would probably need to use the term "distributed sources" because that implies the need for Integration, rather than a simple summation of point sources.

OK. I don't really see a much difference between integration and the summation though. Huygen's implies the need for integration; I don't know how else you would use it. This is more a semantic difference than conceptual, IMO.

 

[sophiecentaur]

I was just avoiding the confusion between distributed sources and the approximated point sources like Young's Slits and diffraction gratings (to first approximation)