Single and Double Slit Diffraction and Interference
Mitchell Anliker, Steve Ash, Derick Peterson
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
In this lab, single slit and double slit diffraction were used to determine locations of
minima and maxima for each diffraction grating as well as the wavelength of light.
Constructive and destructive interference play a major role when measuring the relative
intensities of both single and double slit diffraction patterns. A single slit diffraction
grating was used to determine the wavelength of light hitting the sensor with the equation
mλ=dsinθ, and found to be 742.7291nm. This pattern of minima’s was later compared
with the double slit diffraction to ensure accuracy of the graphs with the minima used for
the single slit at .018569rads and the double slit at .018496rads from the chosen center
point. In addition, a fit was made of the intensity graph for the double slit diffraction to
determine the relative error in intensity at the center point, slit width, and slit separation
for comparison with their experimental values.
I. Introduction
When waves of light collide with an obstacle, they undergo what is known as diffraction. Thomas Young
was the first to scientifically observe this phenomena in his single slit diffraction experiment, giving proof
that light travels as both waves and particles[1]. In single slit diffraction, light of a uniform wavelength
passes through a tiny slit causing constructive and destructive interference[1]. These diffraction patterns
create various minima and maxima of light intensity, seen as spots of bright light when looked at with the
naked eye. Information can be drawn from these interference patters to determine the wavelength of light,
the slit width the light is passing through, and the distance between the slit and the surface the light is
hitting. Double slit diffraction is of a similar nature to single slit as it is the use of two or more slits to
produce more complicated patterns of minima and maxima due to the increased number of slits the light
passes through. In this lab, single slit diffraction was used to determine the wavelength of light passing
through the slit. This was done by finding the minima’s angular position relative to the largest maxima
peak. The calculated wavelength of light was found to be 742.7291nm. A double slit diffraction was then
done using the same laser to compare the minima’s between the two trials. It was found that the minima
used corresponded almost exactly, with the minima used in the single slit being 018569rads and the double
slit at .018496rads from the center point. A fit was done on the graph of the double slit diffraction to find
values for I0, a, and d. These values were found to be 1.021±.00308 for intensity at the center of the
pattern, .00004±1.1872x10-7m for slit width, and .00015m±1.0646x10-7m for the separation between the
slits. These values correspond fairly well with our calculated values of 1, .00004m, and .000125m for
intensity, slit width, and separation respectively.
Theory
In this lab, single slit diffraction was used to find the wavelength of light hitting the sensor. The following
equation shows the relationship between distance and slit width;
s  r
(1
Where r corresponds to the length between the slit and the light sensor, s corresponds to the arclength or
distance from the center maxima, and θ is the measured angle. The equation;
m  d sin 
(2
corresponds to a single slit diffraction, where d corresponds to the slit width, m corresponds to a whole
integer based on the order of minima, and θ corresponds to the angle from the chosen center point. In
addition the equation;
I  I0
sin 2 
2
cos 2 
(3
was used to create a fit of the parameters I0, a, and d which correspond to intensity at the center maxima,
slit width, and slit separation respectively for the double slit portion of the lab. In equation 2, I 0
corresponds to intensity at the center point, I stands for intensity at angle θ, α stands for (πa/λ)sinθ, and δ
corresponds to (πd/λ)sinθ.
II. Experimental Setup
The basic setup for the lab can be seen in the figure below:
[Figure 1: Single and double slit apparatus setup]
AS seen in figure 1, there are three essential parts to both sections of this lab. The laser, diffraction slit, and
light sensor. The light sensor panel and track motion dial also play important roles. In the first section of
this lab, a single slit diffraction grating was chosen and placed approximately 110-120cm away from the
light sensor. The light sensor was then hooked up to science workshop so that the intensity of light could
be monitored while the sensor moved along the track. All lights were then turned off and the laser was
position so that it passed through the single slit grating and its maxima hit the light sensor. The gain on the
sensor was set at 100 and the slit used on the sensor was .5mm wide. The laser was tilted up and down to
ensure the maximum reading was being taken through the sensor before any data was taken. The light
sensor was then slowly pushed down a linear track in front of the interference patterns created by the single
slit. These interference patterns showed up as bright dots on the light sensor panel and as the whole sensor
apparatus was moved horizontally in front of the light, the respective intensities for each maxima and
minima were recorded for each point. A figure similar to the obtained data can be seen below;
[Figure 2: Single slit intensity graph (http://hyperphysics.phy-astr.gsu.edu/)]
This figure is a good representation of what the intensity data looked like relative to the center maxima in
intensity. This data appeared as a series of five peaks in the shape of a bell curve, with the most intense
spot at its center. The single slit data was then taken and normalized versus a central point where the center
maxima of light occurred in the graph. This was done by dividing each coordinates linear position with
respect to the chosen center maxima’s position then subtracting each corresponding point from it so that the
center maxima would be at a linear position of 0 when graphed. This ensured that the maxima was set at
the center and allowed the angle from that center point to be calculated for each set of data. The angle from
the center chosen maxima was calculated using equation 1 where r is the length to the center point, in this
case 101.5cm, and s is the corresponding distance from the center. Theta, θ, was solved for at every point
and converted into degrees. The first minima from the center maxima was then by observing where the
dips in intensity occurred relative to the chosen center point. The angle for this minima, .01856degrees,
was taken and plugged through equation 2 with the known values for slit width, .04mm, and the integer
order of minima, 1, to find the wavelength. The double slit portion of the lab was exactly the same as the
single slit except a double slit diffraction grating was used instead of single. The two slits used were
.04mm in width and 1.25mm apart from one another. The gain for this portion of the lab was set at 100 and
the aperture of the light sensor was changed to 1.0mm from .5mm. The light sensor was passed in front of
the interference pattern caused by the double slit diffraction in a slow consistent manner to obtain concise
data. The pattern observed from the double slit showed five central peaks, along with six smaller peaks
counting eleven in total. The data obtained was similar to figure 3 seen below:
[Figure 3: Double Slit diffraction intensity graph (http://hyperphysics.phy-astr.gsu.edu/)]
This data was then taken and normalized versus a central maxima point similar to the single slit. The angle
for the minima chosen in the single slit portion for the lab was compared with that in the double slit data to
ensure they were the same and to be sure that the data taken was consistent throughout both sections of the
lab. This intensity data was plotted versus the linear center point of the data in origin. A function was
created using equation 3 with the known values for I0, a, and d to create a fit of the double slit data.
Multiple iterations and trials were done to get the graph to fit the double slit data almost perfectly and to
obtain theoretical values for I0, a, and d.
III. Results
[Figure 4: Normalized single slit data graph(3)]
[Figure 5: Normalized Double slit data(3)]
I
n
t
e
n
s
i
t
y
Position
[Figure 6: Fit of double slit data]
Figure 4 shows the normalized data from the single slit graph. Figure 5 shows the data from the
normalized double slit graph. Figure 6 shows the double slit data with the fit overlaid on top of it. The
calculated wavelength using the first order data was found to be 742.7291nm with a 14.27% difference
from the true value. Angles for the first order minima chosen for this lab were at .018569rads and the
double slit at .018496rads for single and double slit respectively. The calculated values for intensity at the
center point, slit width, and slit separation were found to be 1, .00004m, and .000125m respectively. The
theoretical values for I0, a, and d were found to be 1.021±.00308au, .00004±1.1872x10-7m, and
.00015m±1.0646x10-7m respectively. In addition, the chi-squared value found for this fit was .00157
IV. Discussion
Major sources of error in this lab include error in measurement on many fronts. The smallest measurement
on a ruler is .5cm, so the error in measurement was found to be .25cm. The measurement in the cart
position alternated within .022rads, giving an error of ±.022rads. The error in intensity reading was found
to differ by a factor of .046au, yielding an error of ±.046au in all intensity readings. Error was propagated
for θ[2] and found to be ±.0322rads for all angles calculated. Figure 4 shows the normalized plot of the
single slit diffraction data. It can be observed that there are five peaks present, as there should be with the
largest maxima. Equation 2 was used to calculate the wavelength of the laser and found to be 742.7291nm
with a 14.27% between it and the true value of 650nm. In addition, the minima used for this calculation
was found at .018569rads and was a first order minima. This minima was compared with that for the
double slit in the same position at .018496rads with a .394% difference between the two showing that data
taken between the two halves of the lab was consistent. Four the double slit graph eleven peaks were
present with five major central peaks, which is common for most double slit intensity graphs. In addition,
when the calculated values for I0, a, and d(1, .00004m, and .000125m respectively) were plugged into a fit
using equation 3, an almost identical fit was created to the double slit graph seen in figure 3. It can be seen
that the fit is much more accurate for the five principal peaks then for the lesser maxima. This is most
likely due to the relatively low intensity given off at these points, which results in a larger amount of error
based on the finite parameters used to create the graph. A chi-squared value of .00157 with a total of five
iterations was obtained for the fit, showing that it was fairly close to the graph of the double slit intensity.
The theoretical values obtained from the fit for I0, a, and d were found to be 1.021±.00308au,
.00004±1.1872x10-7m, and .00015m±1.0646x10-7m respectively. The only parameter that fell within the
theoretical value found by the fit was slit width, but the other values for intensity at the central point and
slit separation were fairly close to their theoretical values. In conclusion, it can be said results for the single
and double slit data were accurate based on the graphs obtained seen in figures 4 and 5, the small percent
difference between the minima used for each, the relatively low percent difference in calculated
wavelength, the small chi squared value, and how the calculated parameters for the fit were fairly close to
their experimental values showing results were reputable and followed the known theory for single and
double slit diffraction and interference.
References
[1] “Diffraction and Interference.” Hyperphysics. Georgia State University, GA. February 2011,
<http://hyperphysics.phy-astr.gsu.edu>.
[2] J. R. Taylor, An Introduction to Error Analysis: The Study of uncertainties in Physical Measurements.
Sausalito, CA: University Science Books. 1982, pp. 93-161.
[3] Steven Ash, Lab Report