Why does laser on a hair/wire creat a single slit?

[Gabrielgabbe]

I was wondering why laser on a hair makes the hair act like a single slit. I already know about Huygen's Principle behind it and all that. But what is it that can convince you that it actually is a SINGLE slit and not a double, other than just seeing it by measuring min. and using mλ = d * sinα + other trigonometrics.

It would be great if anyone could answer this :)

 

[collinsmark]

Hello Gabrielgabbe,

Welcome to Physics Forums!

I was wondering why laser on a hair makes the hair act like a single slit. I already know about Huygen's Principle behind it and all that. But what is it that can convince you that it actually is a SINGLE slit and not a double, other than just seeing it by measuring min. and using mλ = d * sinα + other trigonometrics.

It would be great if anyone could answer this :)

The phenomenon can be explained via Babinet's Principle.

You might wish to research Babinet's Principle yourself for more details. But I'll give you a brief rundown.

Suppose you have slide with a single slit in it (the normal, single slit, diffraction experiment), such that the slit is exactly the width of a hair. Suppose you also have a hair that matches the slit.

Suppose you take the slide with the slit and do the normal, single slit diffraction experiment. Note the diffraction pattern observed on the wall. The diffraction patter is caused by the electromagnetic field. The electric field component of is field is E_slit. The intensity of the light (the fringe pattern) is the square of the electric field, I_slit = (E_slit)².

Replacing the slit with the hair will also produce an electric field and an intensity too, E_hair and I_hair respectively. But suppose we don't know what those are yet (suppose we haven't measured or calculated them yet).

Now set up the experiment with both the slit and the hair, such that the hair completely covers up the slit (now there is neither a slit, nor a hair, but just big obstacle completely blocking the laser). The resulting pattern on the wall is no pattern at all. The intensity is zero, implying that the electric field is zero too. In other words, with just the slit alone, the electric field was E_slit, and by adding the hair (and the corresponding electric field E_hair), we end up with zero.

So with both the slit and the hair in place, we have:

E_slit + E_hair = 0​

Rearranging gives,

E_hair = -E_slit​

And the diffraction pattern on the wall shows:

I_hair = (E_hair)² = (-E_slit)² = I_slit​

meaning that both the slit and the hair produce identical patterns.

My explanation is probably over-simplistic. I'm only trying to convey a loosely general idea without going into the details.

[Edit: Elaborating on that last sentence, there is a flaw in my above logic. The I_hair will contain a central localized bright spot, sort of as if the laser was shining directly at the wall with neither a slit nor a hair in the way. This bright spot is in addition to the diffraction pattern. But if you remove or ignore the central bright spot, the diffraction patterns are more-or-less the same.]