Single-Slit and Double-Slit Interference
Ahmed AbuZant, Edgar Valle, Mitch Musgrove
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
Abstract: This lab is oriented towards understanding and using the concept of
interference in both configurations of single slit and double slit. The first part of this lab
is used in order to demonstrate how to utilize this concept in order to find some unknown
parameters. In this case, the unknown parameter was the source wavelength which we
calculated to be between 6.73895E-7m + 8.56199E-8m and 6.73909E-7m+8.562E-8m.
The previous part is also a good method in order to validate the instruments and assess
their error generation. The second part of this lab is directed towards confirming and
verifying the use of a theoretical interference function by comparing the results of
intensity vs angle plots from an experimental double slit diffraction data with known
parameters to the fitted function with the same parameters. While keeping in mind the
amount of error generated from the first part, our results were a bit off for the graphs (see
Figure 6) but still fell within the margin of error of the experimental data, and thus our
method of fitting can be trusted.
I. Introduction
Light Diffraction is a phenomenon which occurs when light waves encounter an obstacle with a narrow
path (aka: slit). When light beam reaches the slit, the light beam waves escaping close to the top of the slit
interfere with the light beam waves escaping from the bottom part of the slit. If it were displayed on a
screen, the resulting beam would then be what is known as “diffraction pattern”, which consists of one
central bright spot, known as the central spot, followed by couple of spots on each side that get dimmer as
they get further away from the central spot. Between every two bright spots there is a dark region; those
regions are called minima[2], while the bright spots are called maxima.
Diffraction patterns depend on the slit width a, the wavelength of the diffracted light, the angle we are
looking at as a function of horizontal distance on the screen, and m, an integer number which represents a
specific minima/maxima, for example, m=0 represents the first maxima, which is the central spot, while
m=1 represents the first dark spot on either side of the bright central spot.
The relation between all the parameters above is governed by[2]:
m = asin𝜃 (1)
𝑎 𝑖𝑠 𝑠𝑙𝑖𝑡 𝑤𝑖𝑑𝑡ℎ, 𝜃 𝑖𝑠 𝑎𝑛𝑔𝑙𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑠𝑝𝑜𝑡,  is the wavelength of the beam, m is the order of minima
In case of presence of more than just one slit, the diffraction pattern becomes an “Interference pattern”
shown in Figure1 and is governed by[2]:
m = dsin𝜃 (2)
𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑜𝑛𝑙𝑦 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑠 𝑑: 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑠𝑙𝑖𝑡𝑠
Figure 1. Interference schematic of multi-slit setup[2]
Our main objective in this experiment is to initially perform a calibration procedure by producing a single
slit interference which is then used to figure out the wavelength of our source from equation (1). Then once
that is done, we do the most important part of this experiment, which is setting up a double slit interference
pattern with the same source, and then verifying that it can be modeled using the same experimental
parameters and a theoretical interference pattern equation[2] :
sin 2 α
cos 2 δ , (3)
2
α
I = intensity at angle ,
I0 = intensity at center of pattern,
 = (a/)sin, and
 = (d/)sin.
I  I0
where
Modeling interference patterns allows us to investigate patterns of unknown parameters and then identify
their parameters using this experiment’s procedure. Or, we can equations in this lab in order to theoretically
produce desired patterns and identify the needed parameters without having to go through the trouble of
practically setting up trials.
II. Experimental Setup
Our experimental setup consists of laser beam source of a known wavelength (650nm), both a single slit
and multi slit diffraction gratings. For our purposes, we chose the slit width in both gratings to be the same,
a=.00004m[2]. While the distance between the two slits for the double slit configuration of the multi slit
grating was .000125m. a small stand was used to align the gratings at the same height as the laser source.
For an interference pattern screen, we used rotary light sensor which can be moved sideways along with the
produced interference pattern in either direction. The sensor is separated from the slit mount by a distance
L as shown in the final configuration in Figure 2. When scanned horizontally along an interference pattern,
the light sensor collects intensity data as a function of distance with the help of a computer program called
Data Studio. That being said, it is important to make sure all light sources besides the Laser beam are
turned off so they wouldn’t be picked up by the sensor.
Motion
Sensor
Light Sensor
Figure 2: Experimental set up[2]
For the first part of this experiment, we had to do a single slit procedure that would verify the validity of
our equipment and is then used later for the second part. The parameter in question in this case is the laser
source wavelength. We do this part by setting up a single slit diffraction apparatus, and then we scan slowly
through the pattern using the rotary light sensor. The collected intensity and position data shown in Figure
3 will contain information about the pattern’s parameters, specifically the angles of the produced
maxima/minima beams from the central spot beam.
Figure 3: a sample of our collected graphs for single slit graphs produced by Data Studio. Left
graph: contains time vs position data. Right graph: contains intensity vs time data. By matching
the timing between both graphs, we can figure out the intensities and their positions for all data.
We find the angles of the maxima/minima by finding their position and then subtracting that position from
the central peak position in order to get their distance from the central spot x. Next, we calculate the angles
of the fringes by doing a simple trigonometry identity that would relate L and x :
𝑥
𝜃 = 𝑇𝑎𝑛−1 ( )
𝐿
(4)
By knowing the order and the angle of the desired maxima/minima as well as the slit width a, we use
equation (1) in order to figure out the wavelength of the laser source and then compare it to the known laser
value of 650nm. The error percentage will tell us the minimum Error produced from our set up and
equipment. Once error has been identified, we move on to the second part of our experiment.
The second part of this experiment revolves around the double slit concept. We begin this part by first
leaving all parameters, a, L, , and the entire setup to be the same. Instead of the single slit grating
however, we load a double slit grating into the grating mount.
Next, we run the same procedure we used for the single slit part. However we don’t calculate wavelength,
instead it is known to us from first part of this experiment. Once we determine the angles, we use origin to
plot the data points as intensity vs angle graph. Then by using equation (3) and having all the parameters
determined, we overlay a theoretical curve as a function of angle 𝜃 on top of the plotted data to determine
accuracy and therefore verify the equation’s validity and practicality.
III. Results
We’ve acquired two sets of data for each part in this experiment so that we can decrease uncertainty.
First, running the single slit procedure, we ended up with Figure3 above. For trials of the same procedure,
plots are in the same graph but are labeled according to their Run #.
We needed to get at least 5 peaks on our intensity vs time graphs in order to trust that certain run. Figure3
shows that we’ve gotten 5 peaks for both runs. However, looking at the right side of Figure3, we see that
plot and peaks of run #2 are much clearer than the first run. That is because the quality of the plot depends
on the sensor scanning speed, shown on the left side of Figure3 which is hand controlled. Fast scanning
speeds, represented by run #1in Figure3 correspond to steep velocity slope shown on the left side and
produce narrow and skewed peaks shown on the right side. Slow scanning speeds produce a flatter slope
and broader peaks, which are better because slower speed allows for the sensor to measure intensities and
their positions more accurately over a longer period of time. That being said, results from run #2 are more
trusted than run #1. The five peaks from the second run are:
Run#2 Peaks data.
Time(s)
11.2
12.5
14.9
17.4
19.1
Intensity
0.3
0.7
15.4
0.8
0.3
Intensity Normalized
0.019480519
0.045454545
1
0.051948052
0.019480519
Position(m)
-0.057
-0.072
-0.094
-0.116
-0.13
Error(m)
0.0005
0.0005
0.0005
0.0005
0.0005
Angleθ(⁰)
0.042503
0.025282
0
0.025282
0.041356
Errorθ(⁰)
0.000574
0.000575
0.000575
0.000575
0.000574
Although these are the important peak values, they are not the values used to calculate the wavelength,
minima are much accurate at using to calculate the wavelength than maxima because we know for a fact
that their value is completely zero at that angle whereas maxima’s center is hard to determine due to the
light. Run#2 data (see excel) produced a wavelength of 6.73895E-07m + 8.56199E-08 while run #1
produced an average wavelength of 6.73909E-07m+8.562E-08m which is virtually the same as values from
run#2. Although the values are off the known value of 650nm, they still fall within the margin of the
calculated error and thus can be used.
Our calculations of the wavelength were more confirmed when we took the data from the runs, plotted the
points, and then reproduced them through a theoretical graph using that same wavelength and all of the
other parameters. The graphs and their theoretical curves are shown in Figure4.
Single Slit Diffraction (Run #1)
1.0
1.0
0.8
0.8
Intensity (Max %)
Intensity (Max %)
Single Slit Diffraction (Run #2)
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.00
Angle From Central Peak (Degrees)
0.01
0.02
0.03
0.04
0.05
0.06
Angle From Central Peak (Degrees)
Figure 4: Graphs of the acquired data from the double slit runs (points) with their error bars, the
straight line represents the theoretical curve produced using the diffraction function[2]
As can be seen in Figure4, the data points and the theoretical curves match dead on. The theoretical
function is diffraction function [2]:
D() = sin2/ (5)
Now that we have the wavelength that falls within the margin of error, and thus confirmed the validity of
this method, we move on to the second part of the experiment.
For this part, which deals with double slit interference, our intensity vs times graphs had to have at least 11
peaks. We ran 2 trials and got the graphs shown in Figure 5.
Figure 5: Produced graphs of the collected data by Data Studio for double slit procedure. Top graph:
contains time vs position data. Bottom graph: contains intensity vs time data. By matching the timing
between both graphs, we can figure out the intensities and their positions for all data.
The scanning speeds for both runs of this part are coherent and consistent with each other; therefore they
produce less uncertainty and error. As can be seen from the bottom graph in Figure 5, we got exactly 11
peaks for each run. The scanning direction for these two runs is opposite, which can be figured out from
noticing that each graph has its own reflection image that is located on the opposite side in the other graph.
From first glance, we can only see about 7 peaks for each plot, however, if we were to zoom in, then we
can see those really small peaks at the ends, all in total adding up to 11 peak per plot. The generated data
was then converted from position to angle using equation (4), below is a table of the generated data for
11peaks of run #1 of double slit.
Time(s)
34.1
36
41.7
43.7
46.2
48
50.3
52.2
54.1
58.1
59.4
Intensity
1.8
2.1
3.9
20.2
43.9
53.9
44.1
20.8
3.8
2.3
2
Normalized Intensity
0.033395176
0.038961039
0.072356215
0.374768089
0.814471243
1
0.818181818
0.385899814
0.070500928
0.042671614
0.037105751
Position(m)
-0.079
-0.084
-0.094
-0.099
-0.104
-0.109
-0.114
-0.119
-0.124
-0.134
-0.139
Error(m)
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
Angleθ(⁰)
0.0344691
0.0287277
0.0172397
0.0114937
0.0057471
0
0.0057471
0.0114937
0.0172397
0.0287277
0.0344691
Errorθ(⁰)
0.0005744
0.0005745
0.0005746
0.0005747
0.0005747
0.0005747
0.0005747
0.0005747
0.0005746
0.0005745
0.0005744
This table is for the 11peaks generated data from run #2 (labled as #6)
Time
30.1
32.8
36.5
38
40
42.1
43.4
45.8
47.2
51
52.4
52.5
Intensity
1.8
2.1
3.9
20.2
44.3
55.6
44.5
20.3
3.9
2.2
2.1
2
Normalized Intensity
0.032374101
0.037769784
0.070143885
0.363309353
0.79676259
1
0.800359712
0.365107914
0.070143885
0.039568345
0.037769784
0.035971223
Position(m)
-0.079
-0.083
-0.094
-0.099
-0.104
-0.109
-0.113
-0.119
-0.124
-0.135
-0.139
-0.14
Error(m)
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
0.0005
Angleθ(⁰)
0.034469
0.029876
0.01724
0.011494
0.005747
0
0.004598
0.011494
0.01724
0.029876
0.034469
0.035617
Errorθ(⁰)
0.000574
0.000574
0.000575
0.000575
0.000575
0.000575
0.000575
0.000575
0.000575
0.000574
0.000574
0.000574
From Both tables above, we can see that they both fairly agree on the position and the intensity value of the
peaks. For the rest of the data, please refer to excel sheet.
The generated data was then plotted in origin as intensity vs angle graphs shown in Figure 6.
Double Slit Diffraction (Run #2)
Double Slit Diffraction (Run #1)
1.2
1.0
1.0
0.8
Intensity (Max %)
Intensity (Max %)
0.8
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.0
-0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
Angle From Central Peak (Degrees)
-0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
Angle From Central Peak (Degrees)
Figure 6: The graphs created from data generated in data studio (points). The line in each graph represents
the theoretical curve (3) plotted with the same parameters used and found for each run.
The line in each graph is the result of the fitting equation (3) with the same parameters used for each run.
Although the red line in run #2 is slightly off the plotted points, it still falls within error bars which were
generated using error propagation equation[1].
Due to that it can be said that the results in this experiment and thus the calculated wavelength and the
theoretical curve can be used and trusted.
IV. Discussion
Based on the generated data in the first part which lead us to find our wavelength of 6.73895E-07m +
8.56199E-8m and 6.73909E-7m+8.562E-8m, the main reason of error is our apparatus, which creates room
for a reasonable amount of error. However, this error is then followed by error analysis which produces
error values that would then allow us to trust our calculated but off values.
Another reason for that is probably due to error uncertainties which we estimated for most of our
measurements, for example, error in length L we recorded using the smallest measurement that we could
see which half of a millimeter or 0.0005m. Same error was found in the measurement of the position of the
light sensor even though it was measured through the computer rather than by a meter stick. The error in
our slit width was estimated at 0.000005m.
The error in our graphs is not noticeable at all when looking at graphs of single slit theoretical values vs
experimental data plot. However, this error is visible in our double slit origin plots but is then maintained
by the error bars. That being said, and while using the same value of parameters for both theoretical curves,
the error is slight in the first part of this experiment, but is then amplified when doing the double slit, which
leads me to think that there might be something wrong with the grating, or that our apparatus has been
moved since we measured the single slit data.
Although the theoretical fit doesn’t exactly match the experimental plot, the error generated by different
components of this experiment allowed for a reasonable margin of error, which in turn allowed our
measurements of wavelength and procedure along with the fitting curve to be accepted.
References
[1] Taylor, John R. An Introduction to Error Analysis: The Study of Uncertainties in Physical
Measurements. Sausalito, CA: University Science, 1997. Print.
[2] Vogal, Cecilia. Diffraction and Interference. Augustana College Moodle. 22 Mar. 2012. Web. 22 Mar.