Single and Double Slit Diffraction
Steve Ash, Mitch Anlicker, Derek Peterson
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
Abstract: One of the simplest ways to observe the wavelike nature of light is by passing
a beam of monochromatic light through a single or double slit. The light passes through
the slit and interferes with itself both constructively and deconstructivly, creating patterns
of light and dark. This experiment was performed here using a single slit of width a =
.04mm and a double slit of width a=.04mm and d=0.25mm. We were able to take these
known values along with the distance from the diffraction slit to the light sensor and the
measured data to both predict the wavelength and model the curve of each patterns
intensity curves. We calculated the wavelength of the light to be ""nm with a percent
error of ""%in the case of the single slit and ""nm with ""% error for the double slit.
I. Introduction
This experiment was focused on observing and measuring the wavelike nature of light. We observed this
wavelike nature in two situations, the first being when light undergoes single slit diffraction, and the second
when it undergoes double slit diffraction. In the case of single slit diffraction a monochromatic light
source is set just behind a single slit diffraction device. The beam passes through the thin slit of width a
and is spread out in a wavelike manner. This wavelike behavior is observed by the alternating bright and
dark fringes on the viewing screen, which are caused by the differences the lights path length from the slit
to the viewing screen. The concept is shown in figure 1 below:
The central bright fringe is located at angle 0, in order for us properly calculate anything we must
determine the relationship between the change in x and the change in θ. Using trigonometry and using the
fact that our angle changes by very small amounts we can derive that:
x = Lθ [1]
Which can be rewritten as:
θ= x/L [2]
Where all the variables are as they appear in figure 1. This calculation gives us the angle in radians, if the
angle is desired in degrees theta only needs to be multiplied by (π/180deg). The minima of the intensity
versus position plot can be predicted using the following formula:
mλ = a*sin(θ) [3]
where m is the integer number of minima away from the central peak, a is the slit width, and θ is the
distance from the central maxima in degrees. You could alternatively solve this equation for λ and use your
experimental values for m,a, and θ to determine the wavelength of the incident light. Please refer to figure
3.
To further study the effects of interfering with itself we can now look at double slit interference. This is
similar to single slit interference, differing in that the light is shone through two closely spaced slits
separated by a distance d. This is represented in the figure below:
Here when we plot the light versus intensity we see the same general shape as with a single slit, but inside
of the envelope of the single slit peaks peaks there are several maxima and minima located within the
envelope. They decrease in intensity as a function of distance from the bright central peak. The angle at
which the single slit and double slits first major minima occurs is the same for both cases if the slit width is
not changed. The minima of the interference patterns within the main envelopes can be determined using
the following equation:
mλ = d*sin(θ) [4]
where m is the integer number of minima away from the bright central maximum, and d is the slit
separation Please see figure.
When the two patterns interfere the create the shape of the single slit pattern but with alternating maxima
and minima of the double slit pattern within the envelope of the single slit. This curve can be predicted by
the following equation:
𝐼 = 𝐼0
sin⁡(𝛼)2
𝛼2
cos⁡(𝛿)2 [5]
Where
𝛼 =⁡
𝜋𝑎
λ
sin⁡(𝜃) [6]
and
𝛿 =⁡
𝜋𝑑
λ
sin⁡(𝜃) [7]
Where I is the intensity, I0 is the initial intensity normalized to 1, and λ is the wavelength of the incident
light.
II. Experimental Setup
This lab was carried out using the Science workshop software along with several pieces of hardware
including a laser emitting module, a filter with adjustable slit width and seperation, and a light sensor.
These were also accompanied by a long metal track to hold all the equipment in alignment. The light
sensor was mounted to a cart on a linear track that ran perpendicular to the incident beam. The carts
position on the track was recorded in radians through the science workshop software.
The laser was mounted to the track directly behind the filter, the light passed through the slit in the filter
and was projected onto the screen in front of the sensor. The screen had a 0.5 mm thick slit to allow light
into the sensor. The diffraction pattern was spread out on to the screen and the cart was moved across the
track so that light from each portion was let through the filter into the detector in small increments. A
representation of the setup can be seen below
The light sensors gain was set to 10, and Science Workshop was set to sample light at 1000 Hz, and the
slit with was set to .04 mm. We recorded the data in tandem allowing us to correlate the light sensors
position the intensity at that point. Using the recorded data we plotted the following curve:
Figure 1 -A plot of intensity versus angle relative to the center for our single slit data. Our peak is
focused around 0 and is normalized to 1
We can see in this graph that we have a bright central maximum centered around 0 and two secondary
maximums symmetrically spaced from the central maximum. The data consists of roughly 9,300 points.
To carry out the double slit measurements the exact same procedure was followed. We only changed the
filter to a double slit with a slit width of .04mm and a slit separation of .125mm. This resulted in the
following graph:
Figure 2 - A plot of the double slit diffractions intensity verses position relative to the center. Our
data once again shows the characteristic maxima near zero and the data has been normalized to 1.
This plot also appeared as expected, for a quick check we can glance at the first major minima in both the
single and double slit cases and see that they are at least approximately the same.
III. Results
Single Slit
Our intensity reached a maximum at an angle of 0 degrees as expected, and we found our first minima at
1.063942171 degrees from the center. We then used this information to calculate λ from equation 3, which
gave us a value of 7.42729E-07 m giving us a percent difference of 14.27% .
Double Slit
The maximum intensity was also reached at an angle of 0 degrees as expected. We found the first major
minima to be at 1.059777415 degrees, giving us a difference in local minima's from single to double slit of
.00416 degrees. This number is very close to the prediction of them having the same angle. Using Origin
we calculated a best fit curve using equation 5. This produced a very close fit for our data, as seen below:
Figure 3 - Shows our double slit data plotted in Origin with our calculated best fit curve using
equation 5.
This plot resulted in a reduced chi^2 fit of 0.00157 a very low standard of error. The values of I, a, and d
were all fit and errors estimated giving us values of:
I0 = 1.021 +/- .00308
a = .00004 +/- 1.1872x10-7
d = .00015 +/- 1.0646x10-7
IV. Discussion
The lights intensity was normalized to 1and the measured linear position of the cart was converted into
degrees from the center of the intensity for all calculations. There are some fundamental errors that must
be observed when regarding our measurements. The length L is measured on a cm scale with .5cm
increments giving us a +/-.25cm on our measurement of L. Also our measurement of the cart position x,
has a minimum increment of 0.022rads, giving us an uncertainty of +/- 22rads in our x measurements. And
although not accounted for our minimum intensity measurement increments are 0.098 arbitrary units,
giving us an uncertainty of 0. 098 AU in our intensity measurements. All error was propagated through in
all calculations. The error found through our later fit calculation is used throughout the results, with values
close to our computed values. The calculated error in θ was :
a cos(𝜃)
𝜎𝜃 = ⁡ √(
𝑚
) 2 * σθ
Please refer to the lab notes for information on further calculations.
The lab itself was performed quite successfully, we were able to easily observe the wave nature of light,
and most our results matched theory. The fit of our curve was quite similar to our data and our minima
angles lined up extremely close. The lab appears to validate our theorys.
References
Taylor, John R. An Introduction to Error Analysis: the Study of Uncertainties in Physical
Measurements. Sausalito, CA: University Science, 1997. Print.