# Laser Diffraction ```Laser Diffraction
Introduction and Theory
When light waves of a single wavelength pass
through a slit and hit a screen, the image is not a
single spot of light but a line of spots of varying
intensity separated by dark regions. This pattern,
the diffraction pattern, occurs because light from
one part of the slit interferes with light passing
through other parts of the slit. Depending upon
the relative phases of the light waves, this can
produce constructive or destructive interference.
If two or more slits are present, the diffraction
pattern is complicated by the interference
between light passing through the different slits.
In this experiment, you will study the diffraction
patterns produced by two, three, four, and five slit
combinations.
Figure 2 (a). This pattern is determined by the
width of the two slits. The interference pattern is
due to the interference of the light arriving at the
screen from the two different slits and is
described by the cos2 term. This function is
plotted in Figure 2(b). Figure 2(c) shows the
resulting intensity pattern when these two
functions are combined.
Figure 1. Laser Diffraction Experiment
Figure 1 shows the geometry of the experiment.
Consider a double slit experiment in which light
of wavelength  is diffracted by two slits each of
width a and separated by a distance d. In this
case, the intensity at a point on the screen
associated with angle  is given by the equation
sin 2 α
I  I0
cos 2 δ ,
α2
where I = intensity at angle ,
I0 = intensity at center of pattern,
 = (a/) sin , and
 = (d/) sin .
This is actually a diffraction pattern and an
interference pattern superimposed on one another.
The diffraction pattern is described by the
function D() = sin2/2 which is plotted in
Laser Diffraction
Figure 2. Double Slit Intensity Patterns
Of particular interest to us is the fact that the
minima in the interference pattern are given by
m = d sin , m = 0,  1,  2, ….,  integer. (1)
The minima in the diffraction pattern are given by
m = a sin , m = 0,  1,  2, ….,  integer. (2)
1
If a minimum in the diffraction pattern is located
at the position of a maximum in the interference
pattern, the interference maximum will be
eliminated. This is called a &quot;missing order&quot; in the
an aperture bracket is mounted on a rotary motion
sensor that moves along a linear translator bar.
Signals from the light sensor and rotary motion
sensor are fed into an interface box that is
connected to a computer. These signals are then
analyzed using the Science Workshop program.
The position of the light sensor can be measured
in radians or distance along the linear translator
bar in cm.
Turn on the interface box and click on the
Science Workshop icon. Open the file slitdif.sws.
Position the laser so that it is located110-120 cm
in front of the light sensor. With the light sensor
approximately at the center of the linear
translator, adjust the laser so that it is shining at
the midpoint of the aperture opening in front of
the light sensor. Now put the bracket with the
single slits just in front of the laser and position it
so that the laser is shining through the slit with
slit width a = .04 mm. Measure the distance L
between this single slit and the slit in front of the
light sensor. Try to keep L constant throughout
the data collection process.
Figure 3. Interference patterns for multiple slits
interference pattern.
If there are N-slits where N  3, it can be shown
that there will be &quot;secondary&quot; maxima between
the principal intensity maxima. These secondary
maxima are much reduced in intensity relative to
the intensity of the primary maxima. There will
be N-1 equally spaced minima separated by N-2
secondary maxima in the space between principal
maxima. Figure 3 shows the interference pattern
for two, three, and four slits. These interference
patterns will be modified by the diffraction
pattern given by D() as shown in Figure 2(c). It
should be noted that the positions of the principal
maxima do not change as N increases as long as d
does not change. However, the principal maxima
peaks do become narrower with increasing N.
Procedure
The experimental setup that will be used is shown
in Figure 4 on the next page. A light sensor with
2
Laser Diffraction
Figure 3. Interference patterns for multiple slits
Move the rotary motion detector along the linear
translator until it is in contact with the &quot;rack
clamp&quot;. Put the setting on the light sensor at 10.
Click on the record (REC) button on the
computer display. Move the rotary motion
detector by hand along the total length of the
linear translator. Try to move the motion detector
as smoothly as possible and keep the back of the
detector in contact with the upright edge of the
base of the linear translator. When the rotary
detector reaches the end of the linear translator,
stop the data collection. Expand the graph and
examine the spectrum. It should clearly show the
large central principal peak and at least two
secondary peaks on either side of the center peak.
If you don't have these 5 peaks, make additional
runs until you do. It may be necessary to move
the rotary sensor closer to the rack clamp when
initially lining up the laser.
After you have a suitable spectrum for the single
slit, replace it with the double slit for which a =
0.04 mm and d = 0.125 mm. Keep the setting on
the light sensor at 10. Repeat the process used to
Figure 4. Experimental setup to obtain light intensity measurements of slit spectrum patterns
obtain the intensity spectrum of the single slit.
side of the central peak. Then s is the difference
Examine the resulting spectrum. It should clearly
between these two values. The corresponding
show a group of 7 principal peaks at the center.
angular distance is  = s/L ( has units of
Then there will be two smaller peaks on either
radians). The displacement of the first minimum
side of the central group, separated from the
from the central peak in radians is just  = /2.
central group of peaks by a region of zero
Use Equation 2 to determine . Compare this
intensity. You should obtain at least these 11
value to the accepted value  = 632.8  0.1 nm.
peaks in your spectrum. You may well have
Figurethis
4. Experimental
Setup to Obtain
Light Intensity Measurements of Slit Spectrum Patterns
more peaks beyond
minimum number.
If you
Now look at the graph associated with the double
don't have at least these 11 peaks, make
slit. Determine the positions of the 6 minima in
additional runs until you do. It may be necessary
the center grouping of peaks. Then determine the
to change the relative position of the laser and the
average value of the angular separation between
optical bench, vertically and/or horizontally, in
adjacent minima min and the average deviation.
order for all of these peaks to be picked up by the
Use min in Equation 1 to determine .
sensor.
Compare your results with the accepted value
given above. Compare your two results for  and
Once you have an appropriate set of data for the
comment on their relative experimental precision
double slit, repeat the process for the three- and
and accuracy.
four-slit sets that are on the same card as the
double slit. Before taking data on the three- and
Using the graph of the double slit data, measure
four-slit patterns, adjust the slit position until the
and record the intensity and position of each of
secondary minima are clearly visible to the naked
the central 11 peaks. Repeat for the graphs of the
eye. If they are indistinct to you, they will be
3- and 4-slit data. (In these cases, note the
indistinct on the Science Workshop graph. Also,
secondary peaks between the principal peaks, but
you may need to reduce the setting on the light
only make measurements of intensity and
sensor to 1, for these two spectra. Once you have
position for the principal peaks.)
all four spectra, you may want to save your data
file with a new filename or on a different drive.
Plot a scatter graph of percent maximum intensity
vs. angular distance from the center peak. Since
Analysis
the central peak has the maximum intensity, it
should be correspond to the point (0, 100). Add
Consider the graph associated with the single slit.
plots of the three- and four-slit data to the plot of
Use the cross hair measuring device to find the
the double-slit data. (To do this, click on Graph
angular positions of the two minima on either
and then on Add Plot to Layer.)
Laser Diffraction
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The theoretical relativity intensity of the peaks is
given by the diffraction function D(). Graph
this function on the graph with your data. Be sure
to multiply by 100 to make it a percent instead of
a fraction. You may assume the small-angle
approximation, so that
 ≈ a/,
Observe how closely this function fits your data
and how consistent the various data sets are with
respect to each other.
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Laser Diffraction
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