Physics 123C Waves

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Physics 123C Waves
Lecture 13Single Slit Diffraction April 29, 2005

• John G. Cramer
• Professor of Physics
• B451 PAB
• cramer_at_phys.washington.edu

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Lecture 13 Announcements

• Lecture Homework 4 has been posted on the
  Tycho system. It is due at 900 PM on
  Wednesday, May 4.

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Lecture Schedule (Weeks 1-3)
We are here
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Huygens Principle
The Dutch scientist Christian Huygens, a
contemporary of Newton, proposed Huygens
Principle, a geometrical way of understanding the
behavior of light waves.

• Huygens Principle Consider a wave front of
  light
• Each point on the wave front is a new source of a
  spherical wavelet that spreads out spherically at
  wave speed.
• At some later time, the new wave front is the
  surface that is tangent to all of the wavelets.

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Single Slit Diffraction
As we have seen in the demonstrations with
light and water waves, when light goes through a
narrow slit, it spreads out to form a diffraction
pattern. Now, we want to understand this
behavior in more detail.
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Analyzing Single Slit Diffraction
For an open slit of width a, subdivide the
opening into segments and imagine a Hyugen
wavelet originating from the center of each
segment. The wavelets going forward (q0) all
travel the same distance to the screen and
interfere constructively to produce the central
maximum. Now consider the wavelets going at
an angle such that l a sin q _at_ a q. The
wavelet pair (1, 2) has a path length difference
Dr12 l/2, and therefore will cancel. The same
is true of wavelet pairs (3,4), (5,6), etc.
Moreover, if the aperture is divided into p
sub-parts, this procedure can be applied to each
sub-part. This procedure locates all of the dark
fringes.

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Conditions for Diffraction Minima
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Pairing and Interference
Can the same technique be used to find the
maxima, by choosing pairs of wavelets with path
lengths that differ by l? No. Pair-wise
destructive interference, but pair-wise
constructive interference does not necessarily
lead to maximum constructive interference. Below
is an example demonstrating this.
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Calculating theDiffraction Pattern
We can represent the light through the
aperture as a chain of phasors that bends and
curls as the phase Db between adjacent phasors
increases. b is the angle between the first and
the last phasor.
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Calculating theDiffraction Pattern (2)
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Diffraction Patterns
The narrower the slit opening a, the broader
is the diffraction pattern.
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Angles of the Secondary Maxima
The diffraction minima are precisely at the
angles wheresin q p l/a and a pp (so that
sin a0). However, the diffraction maxima
are not quite at the angles where sin q (p½)
l/aand a (p½)p (so that sin a1).
l 633 nm a 0.2 mm
1
2
3
4
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q (radians)
To find the maxima, one must look near sin
q (p½) l/a, for places where the slope of the
diffraction pattern goes to zero, i.e.,
whered(sin a/a)2/dq 0. This is a
transcendental equation that must be solved
numerically. The table gives the qMax solutions.
Note that qMax lt (p½) l/a.
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Example Diffraction of a laser through a slit
Light from a helium-neon laser (l 633 nm)
passes through a narrow slit and is seen on a
screen 2.0 m behind the slit. The first minimum
of the diffraction pattern is observed to be
located 1.2 cm from the central maximum. How
wide is the slit?
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Width of a Single-SlitDiffraction Pattern
w
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Clicker Question 1
l1
l2
Two single slit diffraction patterns are
shown. The distance from the slit to the screen
is the same in both cases. Which of the
following could be true?
(a) The slit width a is the same for both
l1gtl2. (b) The slit width a is the same for both
l1ltl2. (c) The wavelength is the same for both
width a1lta2. (d) The slit width and wavelength is
the same for both p1ltp2. (e) The slit width and
wavelength is the same for both p1gtp2.
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Combined Diffractionand Interference
So far, we have treated diffraction and
interference independently. However, in a
two-slit system both phenomena should be present
together.
Notice that when d/a is an integer,
diffraction minima will fall on top of missing
interference maxima.
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Five Things You Should Have Learned from This
Lecture

• Huygens Principle provides a geometrical way of
  constructing the propagation of waves, using
  wavelets from each point on a wave front.
• When light passes through a small slit, is
  spreads out and produces a diffraction pattern,
  showing a principal peak with subsidiary maxima
  and minima of decreasing intensity. The primary
  diffraction maximum is twice as wide as the
  secondary maxima.
• We can use Huygens Principle to find the
  positions of the diffraction minima by
  subdividing the aperture, giving qmin p l/a, p
  1, 2, 3, ... .
• Calculating the complete diffraction pattern
  takes more algebra, and gives IqI0sin(a)/a2,
  where a p a sin(q)/l.
• To predict the interference pattern of a
  multi-slit system, we must combine interference
  and diffraction effects.

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End of Lecture 13

• Before the next lecture, read Knight, Chapters
  22.5 and 22.6
• Lecture Homework 4 has been posted on the Tycho
  system and is due at 900 PM on Wednesday, May 4.