Physics 228 Today: Diffraction, diffraction grating
Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228
Diffraction is a further expansion of the idea of interference.
We expand from two sources to a large number of sources.
Here is an example from around home:
CDs have grooves every ≈1.6 μm, or about 25000 grooves around the
disk. When light strikes the disk, each groove acts as a source,
generating an interference pattern.
A diffraction grating
has similar effects.
Monday, February 11, 2013
Are Shadows Infinitely Sharp?
IClicker
Can shadows be infinitely sharp?
a) Always!
b) Yes - for a point source and
sharp edge!
c) Yes - for an edge. But not for
a slit.
d) Never - the world just
doesn’t work that way.
e) No - aren’t we studying
diffraction today?
Monday, February 11, 2013
Are Shadows Infinitely Sharp?
No.
Many examples:
PhET demo, real physical in-class demo
With a thin slit, you can see the light expand
to both sides and form light and dark bands.
Monday, February 11, 2013
Why Does Light “Expand” Past the Edges You
Expect from Geometrical Optics?
Huygen’s Principle
Introduced back in chapter 33,
but we did not cover it then.
Each point on the wave front
can be viewed as a source for
spherical waves propagating out
from that point. The wave front
is the envelope of all the
secondary waves.
The sum of all these
wavefronts will bend around
corners, as compared to
straight geometrical optics.
Monday, February 11, 2013
Why Does Light “Expand” Past the Edges You
Expect from Geometrical Optics?
Huygen’s Principle
Introduced back in chapter 33,
but we did not cover it then.
Each point on the wave front
can be viewed as a source for
spherical waves propagating out
from that point. The wave front
is the envelope of all the
secondary waves.
The sum of all these
wavefronts will bend around
corners, as compared to
Diffraction = interference with many sources, straight geometrical optics.
not just a couple
Monday, February 11, 2013
Near-field vs far-field
There are two situations we can consider, one where the image is
close to the slit, the other where the image is far from the slit.
We will stick to the 2nd, simpler, Fraunhofer, case.
Here we are far enough away that the rays that come to a point
in the image can be treated as going out essentially parallel from
the source, the slit.
Monday, February 11, 2013
Destructive
interference
An example of a simple case: divide the slit into
two halves. Require the top point of each to
interfere destructively. Require the middle point
of each to interfere destructively. Etc., etc.
The destructive interference is when the path
length difference is λ/2.
Condition for this destructive interference:
λ/2 = (a/2)sinθ ➭ λ = a sinθ
Monday, February 11, 2013
Where the
Minima Are
Monday, February 11, 2013
How do we get all the minima and maxima?
Divide the slit into 2m
sections.
Require section i to
interfere destructively
with section i+1. The
condition for
destructive
interference now is:
λ/2 = (a/2m)sinθ ➭
λ = (a/m) sinθ
sinθ = mλ/a
The height of a dark
band a distance x
away is
y = xtanθ ≈ mxλ/a
Even More
Minima?
Isn’t there a second set of
minima, with 3λ/2, 5λ2, ... path
length differences for the slit
in 2 a/2-wide halves?
IClicker 2
a) No! You only get interference for
path length differences of λ/2.
b) Yes - there are minima not included
in the formula sinθ = mλ/a.
c) No - those minima are already
included in the formula.
d) No - each way you divide the slit
only leads to one set of minima. You
can do it one way or the other but not
both.
e) Yes - but the maxima are not so
bright, so you do not really see them.
Monday, February 11, 2013
Standard Numerical Problem
sinθ = mλ/a
or
y = mRλ/a
Give you 4 variables, ask for the 5th.
A diffraction pattern with 500 nm light
has bright bands separated by 1 mm a
distance 5 m from the slit. What is the
width of the slit?
a = Rλ/Δy = (5)(500e-9)/(1e-3) = 25 μm.
Monday, February 11, 2013
Intensity
There is a very bright central
peak surrounded by much
dimmer side peaks.
Why is this?
Monday, February 11, 2013
One slit vs two slit
Today: one slit, width
a, has minima at:
sinθ = mλ/a
m = ±1, ±2, ...
Maxima at m=0
and m half-integral.
Last Monday: two slits,
separation d, have
maxima at:
sinθ = mλ/d
m = 0, ±1, ±2, ...
Minima at m half-integral.
Two slits are actually composed of two single slits
with separation greater than their width.
The intensity patterns of the two will combine /
convolute for the total intensity pattern.
Since d > a, the maxima and minima of the two-slit
pattern will vary more quickly with angle than the
pattern from the width of the slits.
We will return to the intensity Thursday.
Monday, February 11, 2013
Diffraction grating
In a diffraction grating, we go from 2 slits
to many slits.
Diffraction gratings are usually labelled by
the number of slits per unit length.
It does not matter whether there are
multiple slits with light from the left, or
multiple grooves with light from the right.
The path length difference between
adjacent slits is: dsinθ
There will be constructive interference
between ALL slits when there is
constructive interference between
adjacent slits:
mλ = dsinθ
so
sinθ = mλ/d
This is the same algebra as for two-slit
interference.
With the larger number of slits, the
pattern gets sharper.
Let's see a demo.
Monday, February 11, 2013
Reflection grating
Diffraction gratings allow
you to nicely separate
the colors of the
incoming light.
Monday, February 11, 2013
Diffraction
Grating
I am going to change to a diffraction
grating with more lines per inch. What
happens to the diffraction pattern?
IClicker 3
a) The spots move out.
b) The spots move in.
c) The spots stay in the
same place, but get brighter.
d) The spots stay in the
same place, but get dimmer.
e) This iClicker is too easy.
Now: what happens when I go to a
different wavelength laser?
Monday, February 11, 2013