PHYS 351
Augustana College
Winter 2010-2011
Observation of Single-slit and Double-slit Interference and Diffraction Patterns
and Determination of the Wavelengths of a Diode Laser
Derick Peterson, Mitchell Anliker, Steven Ash, Augustana College, Rock Island, Illinois, USA
ABSTRACT
This paper presents a study of the interference and diffraction patterns of light passing through
a single slit and a double slit. The angular position of the first minima in the single slit interference
pattern was measured, and this was used to calculate the wavelength of the diode laser (646.21 +/1.34% nm). The double slit intensity pattern was plotted, and the plot was used to fit a graph using
equation 3 in order to obtain values for the maximum intensity I0 ( 1.021 +/- .30%), slit width a (.00004 +/.30% m), and slit separation d (.000125 +/- .08% m).
INTRODUCTION
Light waves of a single wavelength passing through a slit do not emerge as a single spot of light,
but a linear pattern of areas of varying intensity separated by dark regions. This occurs because when
light passes through different parts of the slit it interferes with the parts passing through different parts
of the slit, producing constructive or destructive interference. In addition, if two slits are used instead of
one, the interference is compounded by the light from the other slits. The overall geometry of the
experiment can be seen below in figure 1.
PHYS 351
Augustana College
Winter 2010-2011
For a double slit experiment using light of wavelength λ and slits of width a separated by distance d we
can make some statements about the interference pattern. The minima in the interference pattern
(seen in figure 2a) are given by the equation
m = a sin
(1)
where m is any integer. This is also the interference pattern of a similarly constructed single slit
experiment. The minima in the diffraction pattern (seen in figure 2b) are similarly given by the equation
m = d sin
(2)
where m again is any integer. The overall pattern of intensity produced by the double slit experiment is
a function of both the interference pattern and the diffraction pattern. The diffraction pattern shown in
figure 2a is the function sin2α/ α2.This is dependent upon the width of the slits. The interference pattern
shown in figure 2b is due to the presence of different slits and is shown in the equation cos2 δ. This
interference pattern is dependent upon the separation of the slits. The final intensity pattern is obtained
by combining these two functions and is shown in figure 2c. The Intensity at a given angle  is given by
the equation
PHYS 351
Augustana College
Winter 2010-2011
In this experiment we will be obtaining a single slit diffraction pattern, and using equation 1 we
will calculate the wavelength of the light that is used and compare it to the standard value. We will also
be obtaining a double slit diffraction pattern and we will attempt to fit a graph of equation 3 to our data
using data analysis software, and we will use this to find values for I0, a, and d, and compare them to
their known values.
METHOD
Single-slit
A light sensor with an aperture bracket was mounted on a rotary motion sensor that moves
along a horizontal track, as seen below in figure 3. The laser was positioned 110-120cm in front of the
light sensor, and the single slit of width a = .04mm was positioned right in front of the laser. The setup
PHYS 351
Augustana College
Winter 2010-2011
was adjusted so that the laser travelled straight down the length L from the slit to the sensor and hit the
sensor directly in the middle. Initially the aperture width was set at .5mm and the gain on the light
sensor was set to 10, but in our readings we were unable to make out the 2nd or 3rd set of fringes, so we
had to increase the gain of the light sensor to 100. Once we obtained satisfactory data we found the
angular position of the minimum in the spectrum and used equation 1 to calculate the wavelength of
the laser. This value was compared to the standard value.
Figure 3. Single slit apparatus
Double-slit
The same experimental setup was used as in the last part, but the single slit module was
replaced with a double slit module. The width of the two slits, a, remained at .04mm, and they were
separated by a distance d = .125mm. The many fringes in the intensity pattern, as seen in figure 2c, were
numerous and greatly differed in intensity. Therefore we increased the aperture size to 1.0mm to allow
for additional light to enter. This increased the amount of unwanted interference from other sources,
but we decided that any error that it introduced would be insignificant. Our final graph showed middle 7
principle peaks as well as several other peaks outside them. This graph was fit to a graph of the form of
the equation 3, and I0, a, and d were set as the parameters, and a value of the Chi-squared for the fit
was obtained. Data analysis software was used to find values for the parameters, and they were
compared with their known values.
PHYS 351
Augustana College
Winter 2010-2011
Figure 4. Double slit apparatus
RESULTS
Table 1 below shows the experimentally determined value for the wavelength of the light from
the diode laser in comparison with its standard. This value was computed using equation (1) for the first
minimum, giving  = a sin, and the uncertainty was computed using the standard propagation of error.
The sources of error in this measurement were in the measurement of L (+/-.05cm) and linear position
(+/- .022cm), which were used to calculate . The smallest reading we were able to make out on the
meter stick for measuring L was to .1cm, so we took the error to be .05cm. We would have assumed
zero error in linear position because it was recorded digitally, but the values did not increase at a
smooth rate, but instead seemed to only be able to jump by values of .022cm, so we took that to be our
error in linear position. Graph 1 below shows the diffraction pattern produced by the single-slit
apparatus.
Diode laser (λD)
λ (nm)
646.21 +/- 1.34%
Standard val. (nm)
650
% diff
0.58%
Table 1. Experimental and Standard wavelength for Diode laser
PHYS 351
Augustana College
Winter 2010-2011
Graph 1. Single slit diffraction pattern for diode laser. Intensity is normalized to 1
and the angular position of 0 was defined to be the center on the horizontal axis
Table 2 below shows the values for each parameter (I0, a, and d) that were used to create a
function fit of the double-slit data using equation 3. The software calculated the errors in each
parameter.  in the fit equation was set to a constant at 650nm, since it was not to be a
parameter. Graph 2 below shows the diffraction pattern produced by the double-slit apparatus.
The fit graph is shown as a red line.
I0
a (m)
d (m)
Experimental val.
1.021 +/- .30%
.00004 +/- .30%
.000125 +/- .08%
Standard val.
1.000
.00004
.0125
% diff
2.10%
0.00%
0.00%
Table 2. Values of the parameters of equation 3, as computed by the fit of the data
PHYS 351
Augustana College
Winter 2010-2011
Angular position (degrees)
Graph 2. Double slit diffraction pattern for diode laser. Intensity is normalized to 1
and the angular position of 0 was defined to be the center on the horizontal axis.
The experimental data is seen in red, and the fit using equation 3 is the red line.
The Chi-squared value for this fit is .000343
DISCUSSION
The percent difference between our experimentally determined value of the wavelength of the
diode laser and the standard value was .58%, which is well within the propagated error of 1.34%. Thus,
this datum falls within computed error of the standard. Additionally, the percent difference between the
experimental values of I0, a, and d and their standards were 2.1%, 0.0%, and 0.0% compared to the
computed error of .30%, .30%, and 0.0%, respectively. Therefore, I0 does not fall within experimental
error of the standard value, but a and d do. Even so, the value for I0 is very close to the standard value,
and visually the graph of the function fit (as seen in the red line on Graph 2) is very close to the
PHYS 351
Augustana College
Winter 2010-2011
experimental data, so I conclude that the data is accurate within reasonable ranges. The method of
computation in the experimental error in the values of the parameters I0, a, and d is not known to me,
so I am unable to guess as to how these values could be improved.
Initially our attempts at getting a fit of our data proved fruitless, and it was only through
changing the values of the parameters many times and running through more iterations at different
points that we were able to get a the graph to properly fit our data. Because of this difficulty and the
closeness of the fit (the chi-squared for the fit was .000343) we decided to stop here. It is possible that
the fit could have become even closer to our data if we kept iterating the function. The fit we ended
with seems to follow our data very closely, but deviates slightly in the center 4 minima. Ideally each
peak should end in a minimum at 0 intensity, as is seen in the red function-fit line, but in reality our data
showed that the intensity in the 5 central peaks did not drop off to 0 completely before climbing again
to the next peak. This is likely due to the width of our aperture, which in effect caused each point to be
the sum the intensities of a small area surrounding the point. Thus the data is not quite the ideal
representation of many exact points of intensity. Ideally we would have a very small aperture at the light
sensor in order to get the most exact intensity reading for each individual point. We could also improve
the accuracy of this experiment by using a digital position device which more accurately measured the
position, rather than jumping at .022cm steps.
PHYS 351
Augustana College
References:
1. An Introduction to Error Analysis, John R. Taylor, 2nd Ed. (1997).
Winter 2010-2011