Interference and Diffraction
Interference and Diffraction
Pre-lab Questions and Exercises
1. What are some modern technology applications of interference and diffraction?
2. Which size slit will produce a wider diffraction pattern: 0.01 mm, or 0.05 mm?
3. Which grating will yield a wider diffraction pattern: 300 lines/mm or 600 lines/mm?
4. Sketch the actual diffraction patterns that correspond to the single and double-slit intensity
graphs in your lab manual.
5. How could a laser be used to measure the thickness of a human hair?
Introduction
In geometrical optics we are concerned with the ray concept of light. In this experiment, we
will observe how wave optics manifests itself in the phenomenon of diffraction. If a is the
characteristic width of any slit or aperture and that if the wavelength is larger than or of the order
of a, then wave optics will be required to describe what is happening. This condition is satisfied for
diffraction patterns. The aim of this experiment will be to study the interference patterns that result
from the wave nature of light.
The unifying principle of elementary wave optics is Huygens’ principle: All points on a wave
front can be considered as point sources for the production of spherical secondary wavelets. After a
time t, the position of the wave front will be the surface of tangency to these secondary wavelets.
This principle, together with the principle of superposition, can be used to explain the
phenomenon of diffraction. The principle of superposition states that the resultant waveform is
found by adding algebraically, or super-imposing, the separate wavelets that make up the total
waveform.
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The Diffraction Grating
A diffraction grating is equivalent to an array of closely spaced sources. A transmission
grating, for example, is just closely spaced parallel lines scratched onto a glass plate. We will
consider the geometry of a row of point sources, remembering that each point source on the grating
will act as a source of secondary wavelets that will interfere with each other (see Fig. 1). Here d is
the distance between sources and N is the total number of sources.
d
Nd





d
m
Figure 1
Figure 2
Consider any two adjacent sources, as shown in Figure 2. There will be constructive
interference, or maxima in intensity, when the phase of all wavelets is the same. This requires that
the optical path length difference, d sin  m , be an integer number of wavelengths. Therefore,
d sin  m  m
(m  0, 1, 2, )
(1)
This is commonly called the grating equation.
Intensity Variation fora Diffraction Grating
m=0
m=1
m=2
m=3
InterferencePeak
DiffractionFringes
θ=0

Figure 3
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Interference and Diffraction
Single-Slit Diffraction
For a single-slit, we again imagine each point on the wave front as being a new point source of
radiation. Infinitesimally small distances separate these point sources and all oscillate with the same
phase. So then we have an infinite array of point sources spread over the finite width of the slit.
From this geometry the intensity of the diffraction pattern can be derived. The intensity as a
function of angle is given (without proof) by:
2
 sin  
a
 , where  
I  I o 
sin 

  
(2)
and
Io = maximum intensity (at  = 0)
a = width of the slit
 = wavelength of the light
Minima occur when sin   0 . For this condition to be true,  must equal an integer multiple of :
  m
Since  
(m  1, 2, 3, )
a
sin  , it follows that

a sin   m
(m  1, 2, 3, )
(3)
Minima

a
Central
Maxima
m=1
m=2
m=3
Figure 4
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Interference and Diffraction
Double-Slit Diffraction
Suppose now we have two parallel slits, each of width a, with their centers separated by a
distance d. If we cover one of the slits, then the result is a single-slit diffraction pattern, as discussed
before. If we open both slits, we obtain the same diffraction pattern as with a single-slit in addition
to rapid variations in intensity caused by the double-slit interference. The slit separation d can be
found from equation (1) when the distances from the center of the diffraction pattern to maxima of
the fine structure are measured.
The intensity of the double-slit diffraction pattern is given (without proof) by:
2
 sin  
 d 
 cos 2  , where     sin 
I  4 I o 
  
  
(4)
and  is defined the same as before. Minima occur when sin   0 or when cos   0 . We already
know that when sin   0 , then a sin   m (equation 3). Note in figure 5 which minima occur
when sin   0 .
d
sin  (m  0, 1, 2, ) . When this equation

is rearranged, we obtain the grating equation in slightly different form:
When cos   0 , it must be that   m  12  
d sin  

2
 m
(m  0, 1, 2, )
(5)
This may seem to contradict what was stated above, that the slit separation can be found from
equation (1), but remember that equation (5) results from minima, while the equation (1) results from
maxima.
The interference pattern is “modulated” by the diffraction pattern just as with AM radio the
carrier wave amplitude is “modulated” by the audio signal.
I()
Double Slit
Diffraction
Envelope
d
Interference
"Fine" Structure
a
a

d
a

Figure 5
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Interference and Diffraction
LASER
m th
Diffraction
Grating
xm
Intensity
s
Figure 6
Procedure
Part 1. Gratings
You will use a Helium-Neon (He-Ne) laser for your monochromatic light source ( = 632.8
nm) and several different transmission gratings (100, 300, and 600 lines/mm). Project the
interference pattern away from other students and as far away from the grating as possible (for
example, across the room on the wall). Be sure to avoid shining the laser light into anyone’s
eyes! Measure s, the distance from the grating to the image plane, and xm, the distance from the
central image to the mth interference peak (maximum). Record measurements for several interference
peaks, and use your data to compute a best estimate of the line spacing for the grating. Record the
line density of the grating you used. Qualitatively observe and describe the pattern for different
gratings.
A diffraction pattern can also be created by shining laser light onto the spiral tracks of a CD or
DVD. If the wavelength of the laser and the angular spacing are known, then you can determine the
spacing between the tracks on the disc. Devise a way to accomplish this objective.
Part 2. Single-slit Diffraction
Observe a single-slit diffraction pattern and measure several xm, the distance from the central
image to the mth minimum. Record the value of the slit width a. Qualitatively observe and describe
the pattern for different slits. If possible, make a recording of the diffraction patterns by projecting
on a sheet of paper and tracing the bright spots with a pen or pencil. This procedure also facilitates
making measurements with a ruler at a later time when the lights are on.
A similar pattern can be created by shining laser light onto a thin, narrow object, like a human
hair. Try this and use the resulting diffraction pattern to determine the width of a hair. What
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Interference and Diffraction
differences (if any) do you observe from the diffraction pattern created by a slit compared with a hair
of the same width?
Part 3. Double-slit Interference/Diffraction
Observe a double-slit diffraction pattern and measure several xm, the distance from the central
image to the mth minimum of the interference fine structure. Also record the diffraction envelope
minima. Again, these measurements can be obtained and recorded most easily by tracing the pattern
on a sheet of paper. Record the value of the slit spacing d and slit width a. Finally, qualitatively
observe and describe the pattern for another slit configuration with different d and a.
Analysis
Part 1. Gratings
Use the diffraction grating equation to find the grating line spacing. Compare with the value
advertised on the diffraction grating. For the reflection grating (CD), compare your results with the
industry standard of 1.6 microns between adjacent tracks. Do your results agree within the
experimental uncertainty of your measurements?
Part 2. Single-slit Diffraction
Use the diffraction equation to find the width of the slit you used, and compare with the
nominal value on the slit holder. Repeat for the width of a hair (typically ~70 microns).
Part 3. Double-slit Interference/ Diffraction
Use your data to determine the slit width a and slit spacing d, and compare with the nominal
values.
Discussion
Why can m = 0 for diffraction grating maxima, but m = 0 is not defined for the first minima in
single slit diffraction?
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