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Chapter 23
SingleSlit Diffraction
211
23
SingleSlit Diffraction
Singleslit diffraction is another interference phenomenon.
If, instead of creating a mask with
two slits, we create a mask with one slit, and then illuminate it, we find, under certain conditions,
that we again get a pattern of light and dark bands.
It is not the same pattern that you get for
twoslit interference, but, it’s quite different from the single bright line in the straightahead
direction that you might expect.
Here’s how it comes about.
Firstly, here’s the setup:
Again, we get a bright fringe in the straightahead position on the screen.
From there, working
out to either side, we get bands that alternate between dark and bright.
The first maximum to the
right or left of the central maximum is not nearly as bright as the central maximum.
And each
maximum after that is less bright than the maximum preceding it.
As far as the analysis goes, I
want to start with the minima.
Consider an imaginary line extending out from the midpoint of
the slit all the way to the screen.
Incoming Plane Waves
Screen
Mask with single slit
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SingleSlit Diffraction
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So now the question is, “Under what conditions will there be completely destructive interference
along a line such as the one depicted to be at angle
θ
above?”
To get at the answer, we first
divide the slit in half.
I’m going to enlarge the mask so that you can see what I mean.
Now I imagine dividing side A up into an infinite number of pieces and side B up the same way.
When the slit is illuminated by the light, each piece becomes a point source.
Consider the first
point source (counting from the left) on side A and the first point source (again counting from
the left) on side B.
These two point sources are a distance
w
/2 apart, where
w
is the width of the
slit.
If the light from these two point sources (which are in phase with each other because they
are really both part of the same incoming plane wave), interferes completely destructively, at
some angle
with respect to the straightahead direction, then the light from the second point
source on side A and the second point source on side B will also interfere with each other
completely destructively because these two point sources are also
w
/2 apart.
The same goes for
the thirdfromtheleft point sources on both sides, the fourth, the fifth, and so on, ad infinitum.
So, all we need is to establish the condition that makes the light from the leftmost point source
on side A (overall, the leftmost point of the slit) interfere completely destructively with the
leftmost point source on side B (overall, essentially the midpoint of the slit).
So, consider any
point
P
on a proposed line of minima.
1
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 Fall '08
 RABE
 Physics, Diffraction

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