[hamdani yusuf (OP)]

It can't be "refuted" because it is a sound mathematical model of what happens when λ << d. You might as well claim that newtonian mechanics can be "refuted" when v → c. Fact is that Babinet and Newton are useful mesoscopic approximations, as everyone knows.

A valid mathematical model doesn't necessarily represent physical reality.

[alancalverd]

A valid mathematical model doesn't necessarily represent physical reality.

It's only "valid" if it does adequately predict reality! Validation is what experimental physics is all about!

[hamdani yusuf (OP)]

In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.[1] It is not required for a valid argument to have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas (also called wffs or simply formulas).
https://en.m.wikipedia.org/wiki/Validity_(logic)

So, why diffraction and interference patterns of a single wire different from a single slit at distant screen?

[alancalverd]

We aren't talking about arguments but mathematical models.

So, why diffraction and interference patterns of a single wire different from a single slit at distant screen?

Because a wire is not a slit. Babinet says so - the central fields necessarily differ.

[hamdani yusuf (OP)]

It can't be "refuted" because it is a sound mathematical model of what happens when λ << d.

If it's not falsifiable, then it's not scientific.

[hamdani yusuf (OP)]

Because a wire is not a slit. Babinet says so - the central fields necessarily differ.

What would be the Babinet equivalent of a single slit?

[alancalverd]

There is no "Babinet equivalent". All Babinet does is point out the similarities between the diffraction patterns from objects and their spatial complements.

The spatial complement of a slit is the piece of material that you removed to make it. For most purposes this is very similar to a round wire but if you want to be pedantic it would be a wire of rectangular cross section. The best demonstrations of optical diffraction actually use razor edges. 

[hamdani yusuf (OP)]

The spatial complement of a slit is the piece of material that you removed to make it.

That's what I called Babinet equivalent.
There are many videos demonstrating single slit experiment made from various kind of materials: razor blades, aluminum foil, paper with various thickness, pencils, AA batteries. All show similar results.

[alancalverd]

Similar, because the laws of physics apply (as far as we know) to everything. But not indefinitely identical.

A circular cylinder like an AA cell is a pretty good approximation to a sharp edge from the point if view of a light ray (since the "edge" of a perfect circle is infinitesimal)  but as the slit or its complement gets narrower,  so the approximation becomes less valid, as you have demonstrated with your thick, semitransparent objects and microwaves.

[hamdani yusuf (OP)]

Similar, because the laws of physics apply (as far as we know) to everything. But not indefinitely identical.

A circular cylinder like an AA cell is a pretty good approximation to a sharp edge from the point if view of a light ray (since the "edge" of a perfect circle is infinitesimal)  but as the slit or its complement gets narrower,  so the approximation becomes less valid, as you have demonstrated with your thick, semitransparent objects and microwaves.

The similarity of single slit diffraction-interference pattern from various thickness of the objects that make the slit is that central bright spot is twice as wide as the other bright spots.
We can't say the same about the pattern from single wire experiment.

[alancalverd]

Pretty much as Babinet analysis (or common sense) would predict.

[hamdani yusuf (OP)]

Pretty much as Babinet analysis (or common sense) would predict.

How do you predict this pattern from a thin wire diffraction?

Investigation on Diffraction of Light 21 : Long Shot Diffraction
Experiment on diffraction of light with long distance between the obstacle and the screen to show the difference of interference pattern between single slit and thin wire diffraction.

[alancalverd]

Looks as though the incident beam was significantly wider than the wire, so the central spot is primary beam, not diffracted.

This is a problem in x-ray crystallography: most of the primary beam is not diffracted so it's difficult to find small-angle diffraction spots among the central scatter.

[hamdani yusuf (OP)]

Looks as though the incident beam was significantly wider than the wire, so the central spot is primary beam, not diffracted.

That's where the babinet principle failed.

[alancalverd]

"Principles" always apply to ideal situations. You have spotted the difference between physics and engineering. Or between economics and reality.