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? Why?
3. Which grating will yield a wider diffraction pattern: 300 lines/mm or 600 lines/mm? Why?
4. Sketch the actual diffraction patterns that correspond to the single and double-slit intensity.
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.
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.
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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)
where
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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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:
 sin 
I  4 I o 
 
2

 d 
 cos 2  , where     sin 
  

(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
Procedure
Be sure to avoid shining the laser light into anyone’s eyes! In this lab you will use a
series of laser pointers (red, green, and blue) monochromatic light sources. Be sure to record the
wavelength for each laser. We will also use two different single slits (A and B on the first card), two
different double slit patterns (labeled A and B on the second card) and at least two different
diffraction gratings (100, 300, and 600 lines/mm available).
1. Set up the clamp and pole at your table on the opposite side of the center podium from the
nearest wall or window sill. Use 3 right angle clamps to mount the three colored lasers as
close to each other on the pole as feasible. Do not tighten down on the laser pointers just yet
– doing so will turn them on. Wait until you are ready to collect data.wait until you are ready
to collect data to turn them on.
2. Tautly tape tracing paper over the white boards (this reduces reflections and simplifies the
data collection). Drawn a vertical line down the approximate center of the whiteboard so you
have a common target for all your interference patterns.
3. Use the table podium to hold the targets (either diffraction grating, single-, or double-slit
cards) in a lens holder. You can use the blocks (0, 1, or 2) at the table to raise and lower the
metal lens holders for the different lasers. Adjust the heights of your lasers as necessary so
that you will be able to send the laser through the targets and hit your white board.
4. Record the distance to your whiteboard as well as the wavelengths of light used.
5. CAREFULLY tighten the clamps over the laser pointer buttons to turn on the laser - do not
over-tighten or you risk breaking the laser.
6. Record the interference patterns from 2 single slits, 2 double slits and at least 2 diffraction
gratings for each of the 3 colors of light (for a total of 18 patterns). Label your trace paper to
keep all the different patterns straight.
a. You can use multiple sheets of paper if it is difficult to keep track of all of your lines.
You can also just shift the white board vertically to record all the interference patterns
together. Remember to always aim your central maximum at the center line on the
white board so that your distance to the paper is constant.
b. When recording the patterns, you can just mark the maxima (easier to see – rather
than the minima) for each pattern directly on the sheet. Remember to be careful and
not look directly at the laser beams while you are recording your data.
c. Qualitatively describe the differences between the patterns based on changes in the
slits as well as wavelength of light used.
Analysis
The following analysis only needs to be completed for one single slit pattern, one double slit
interference pattern and one diffraction grating pattern. You may choose whichever laser color and
slit parameters from among the ones you measured.
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1. Measure on your paper the 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. Plot sin θ vs m, where sin θ is
approximately xm/(distance to target), by the small angle approximation. Be sure to attach
your graph – what is the slope of your line equal to? From the known lines per mm,
determine the wavelength used to create the pattern and compare this to the actual
wavelength used.
2. On the single-slit diffraction pattern, measure several xm, the distance from the central image
to the mth minimum. Use your data (plotting sin θ vs m) to determine the slit width, a, and
compare your calculated value to the nominal slit width of the card (if your card isn’t labeled,
make sure to get the slit widths from the instructor).
3. On the double-slit diffraction pattern measure several xm, the distance from the central image
to the mth minimum of the interference fine structure. Also record the diffraction envelope
minima. Determine which plots are necessary to use your data to determine both d and a for
the double-slit. Compare these values to the ones from the card you used.
Discussion
What are some applications of diffraction both in science labs as well as in your home? What
are the largest sources of error in this lab? How would you improve this lab?
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