Double Slit Interference - Santa Rosa Junior College

Double Slit Interference

A lab Report by Jimmy Layne

With Partner Joe Nolan

Abstract

The purpose of this experiment was to test the wavelength of light using the diffraction pattern of a
650nm laser passing through a double-slit apparatus. We measured the interference pattern for four
different slit configurations of varying width and separation. For the first configuration the slits were
separated by d=.5mm and were a=.08mm in width. From this configuration we calculated an average
lambda value of (623nm±2.70%) with an average deviation of ±32.8nm. the second configuration
was using d=.25mm and a=.08mm from this configuration we calculated λ to be (684nm±3.07%)
with an average deviation of ±23.9nm. for the third configuration, we reduced the width of the
slits to a=.04mm and set d=.5mm, this yielded a λ of (692nm ±2.29%) with an average deviation of
±40.2nm. our final configuration had dimensions a=.04mm and d=.5mm from which we calculated
λ to be (665nm±3.09%) with an average deviation of ±21nm. For each measurement we used the
rules for propagation of error to determine an absolute uncertainty, and divided it by the total
measurement in order to obtain our percent uncertainty. We also calculated the average
deviation using an excel spreadsheet, which gave us another method of determining our
uncertainty in λ. Our average value of λ from all four configurations was (671nm ±2.69%) which
differed from the value printed on the laser by 3.21%. This is higher than our uncertainty would
predict however there were many sources of error in this experiment. It was difficult to see the
fringes clearly enough to accurately trace their outline on the page, and the confined space in
which we were conducting the experiment introduced further uncertainty due to people
inadvertently bumping the laser apparatus.

Theory

The purpose of this experiment is to experimentally verify the wavelength of a laser using a
double slit interference pattern projected on the wall. For our experiment we will be using four different
slit configurations which will generate four different interference patterns. A diagram of our experiment
is shown here:

We assume three things about the experiment:

• That the light entering the slits is coherent, that is they maintain a constant phase relationship with one
another.

• The light entering the slits is monochromatic; there is only one wavelength of light that is being scattered.
• The distance from the slits to the screen is much greater than the distance between the slits. That is: L>>d

Our third assumption gives rise to an important consequence. Since L>>d the rays r1 and r2 are
essentially parallel for the purposes of computation. This creates a right triangle with hypotenuse d
which will give us an easy expression for the path difference:

∆r = r2 − r1 = d ⋅sin (θ )

And from our work in chapter 18, we know that these will be constructive when:

∆r = d sin (θ ) = m ⋅ λ

And Destructive, when:

∆r= d sin (θ )=  m + 1  ⋅ λ
 2 

From the diagram, we can define tan(θ) as:

tan (θ ) = ym

L

For small angles we are able to make the approximation: tan (θ ) ≈ sin (θ ) (the effectiveness of this

approximation is discussed later) which allows us to find the linear position of both bright and dark
fringes, with respect to the central maximum:

constructive: d ⋅  ybright  = m⋅λ → ybright = mLλ
 L  d


Destructive: d ⋅  ydark  =  m + 1  λ → ydark = Lλ ⋅  m + 1 
 L   2  d  2 

Given that we have now determined a relationship between the linear position of a fringe and the
wavelength of the fringe, our job is now to measure the linear position and use it to determine the
wavelength of the incident light. In our experiment we marked the position of the bright fringes on a
sheet of paper, and measured the distances of these fringes from the central maximum. After obtaining
these values we determined the wavelength of light using:

λ = d ⋅ ybright
m⋅L

This model predicts that we should see an alternating pattern of bright and dark fringes across
the wall with even, regular spacing. We soon see experimentally that this is not the case, and that the
bright and dark fringes are further spaced, by alternating areas of bright and darkness. As a
consequence of a rather long derivation, we find that the equation modeling the intensity of a two slit
apparatus is:

  π ⋅ a ⋅sin (θ )  2
 sin  
I =Imax ⋅ cos2  π d sin (θ )  ⋅   λ 
   π
λ   ⋅ a ⋅sin (θ ) 

 λ 

This is a rather nasty equation, but each part governs a different part of the pattern we see. The
individual bright and dark fringes are given by the cosine term, if we ignore the term in square brackets,
then we see that the points of maximum intensity are given when the cosine term is equal to one. This
allows us to determine a function for the angular position of a bright fringe:

cos2 π ⋅d ⋅ sin (θmax )  =1 → π ⋅d ⋅ sin (θmax ) =m ⋅π → θmax =sin −1  λ⋅m 
   d 
 λ  λ

The second term of the intensity function is responsible for the variation in intensity from one
bright fringe to the next. This term creates the envelope under which the intensity of each bright fringe
varies. It may be that a bright fringe corresponds to a minimum of the envelope, and as such it would
appear dark in the projection. A similar calculation as above, will give us the angular position of dark
fringes which result from the intensity envelope:

π ⋅ a ⋅sin (θmin )  =0 → π ⋅ a ⋅sin (θmin ) =nπ → θmin =sin −1  nλ 
sin    a 
λ  λ


In order to find any places where an interference maximum is suppressed by an envelope
minimum, we must look for areas where θmin=θmax:

θmin = θmax → sin −1  nλ  = sin −1  λ ⋅m  → m= nd = d
 a   d  a a

So, whenever the ratio of the distance of slit separation to slit width, is an integer number, we will see a
dark fringe where we would normally expect an interference maximum. Furthermore, it is expected that
we will see some variation in the fringe pattern as we change the dimensions of the slits. As light moves
through a smaller slit, it is bent more. We predict that this will cause the fringes to become wider.

During our experiment, we were able to measure the linear position of the fringes on the wall using a
small angle approximation. We have determined that this approximation is effective to 5° or
approximately .0873 radians. We will be also noting this when taking our data, by recording also the
value of θ which we compute using:

tan (θ ) = y →θ = tan −1  y 
L  L 

By monitoring this value we will ensure that we do not record any data for which our mathematical
approximations are ineffective

In this lab, it is important for us to keep track of error and uncertainty, as there are many
opportunities for it to be introduced. We will be determining our uncertainty in lambda using two
methods; the first is using propagation of error. Recall that the rules for propagation of error state that
the uncertainty associated with a given measurement of λ is given by:

∆=λ ∂λ ⋅ ∆y + ∂λ ⋅ ∆d + ∂λ ⋅ ∆L
∂y ∂d ∂L

By differentiating our expression for λ with respect to each variable, we obtain the following expression
for the theoretical uncertainty associated with our calculation of λ.

=∆λ d ⋅ ∆y + y ⋅ ∆d + d ⋅ y ⋅ ∆L
L ⋅ m L ⋅ m L2 ⋅ m

We will also calculate the Average Deviation of our data points using a spreadsheet program. These two
methods will make up our uncertainty in our calculated lambda values.

Procedure

We begin our experiment by setting up a Pasco Basic Optics Diode Laser which will project the
interference pattern we seek. We set up the laser and measure its distance from the wall, keeping in
mind that the distance L should be sufficiently large so as to validate our initial assumption. We then
take four sheets of paper, and on each one, we write the dimensions of each slit configuration that we
intend to test. We ensure that the disc which contains our slits is perfectly aligned with the center of the
laser. After that, we tape the laser to the table so as to minimize of the pattern on the opposite wall.

After we record all dimensions of the setup, and ensure that the laser is as stable and as
perfectly aligned as possible, we tape one of our pages to the wall with the blank side facing the laser.
We rotate the double slit to the corresponding dimensions, and turn on the laser. The laser then
projects interference fringes onto the page that we’ve taped to the wall. It’s very important that the
room in which we’re conducting this experiment be dark, so that we can see as many fringes as possible.
Using a pencil (and a very steady hand) we trace each fringe onto the paper, making special note of the
central maximum fringe. After we trace as many fringes as we can see, we remove the paper and repeat
the process for each of the three other slit configurations.

Once we’ve got the fringes drawn, we use a ruler to draw straight lines down from the middle of
each fringe. Then, we very carefully measure the y-distance from the central maximum fringe, to the
center of each subsequent fringe on either side. We record these values in our spreadsheet, and repeat
the process for each of the four cases. To determine the values of m we simply count the fringes on
either side of the maximum and record those values next to the corresponding y-distance in our
spreadsheet.

Data and Graphs

Since we included the distance to the center of each fringe, for each configuration, our data tables are
somewhat large, and extremely cumbersome to search through. So for convenience sake, I have
included a summary for each configuration in the following table:

Config 1 a (mm) d (mm) avg λ (nm) Average avg % % Difference

Config 2 .08 .50 2.29% Deviation Uncertainty 6.60%
Config3 .08 .25 3.07% 6.92884E-07 4.02162E-08 5.28%
Config 4 .04 .50 2.29% 6.84308E-07 2.39E-08 6.60%
.04 .25 3.09% 6.92884E-07 4.02162E-08 2.31%
6.65003E-07 2.14E-08

The remaining data tables and fringe tracings is included on the next few pages.

Configuration 1

d(m) a(m) L(m)

Config 1 0.0005 0.00008 2.642

my 0.0787 lambda theta (rad) Absolute Percentage
23 0.0748 6.47566E-07 0.0297792 Uncertainty uncertainty
22 0.071 6.43452E-07 0.0283043
21 0.0677 6.39847E-07 0.0268671 1.04754E-08 1.62%
20 0.0575 6.40613E-07 0.0256189 1.05154E-08 1.63%
17 0.0537 6.40112E-07 0.0217604 1.65%
16 0.0505 6.35172E-07 0.0203227 1.0571E-08 1.67%
15 0.0471 6.37144E-07 0.019112 1.06936E-08 1.73%
14 0.037 6.36693E-07 0.0178255 1.11046E-08 1.77%
11 0.0342 6.3657E-07 0.0140036 1.12143E-08 1.80%
10 0.0306 6.47237E-07 0.012944 1.83%
9 0.0278 6.43452E-07 0.0115816 1.1437E-08 1.98%
8 0.0157 6.57646E-07 0.0105219 1.16565E-08 2.03%
5 0.013 5.94247E-07 0.0059424 1.25766E-08 2.12%
4 0.0095 6.15064E-07 0.0049205 1.31453E-08 2.20%
3 0.0061 5.99293E-07 0.0035957 1.36218E-08 2.89%
2 0.003 5.77214E-07 0.0023089 1.44635E-08 3.22%
1 0.004 5.67752E-07 0.0011355 1.71877E-08 3.93%
1 0.008 7.57002E-07 0.001514 5.40%
2 0.0105 7.57002E-07 0.003028 1.9824E-08 9.63%
3 0.0145 6.62377E-07 0.0039742 2.35617E-08 7.55%
4 0.0171 6.86033E-07 0.0054882 3.11601E-08 4.43%
5 0.0289 6.47237E-07 0.0064723 5.46934E-08 3.68%
9 0.0311 6.07705E-07 0.0109382 5.71537E-08 3.02%
10 0.0348 5.88569E-07 0.0117708 3.34974E-08 2.76%
11 0.0382 5.9872E-07 0.0131711 2.43818E-08 2.17%
14 0.0487 5.16384E-07 0.0144577 2.07466E-08 2.10%
15 0.0526 6.14434E-07 0.0184309 1.78766E-08 2.02%
16 0.0548 6.22161E-07 0.0199065 1.31571E-08 1.95%
17 6.10055E-07 0.0207389 1.23827E-08 1.81%
20 0.06 5.67752E-07 0.0227062 1.20845E-08 1.78%
21 0.0736 6.63278E-07 0.0278505 1.00925E-08 1.76%
22 0.0761 6.54635E-07 0.028796 1.11418E-08 1.72%
23 6.58263E-07 0.0302708 1.10451E-08 1.64%
0.08 1.07138E-08 1.63%
1.61%
9.7464E-09
1.08756E-08
1.06608E-08
1.06145E-08

Configuration 2

d(m) a(m) L(m)

Config 2 0.00025 0.00008 2.642
m
y(mm) lambda theta Absolute Fractional
Uncertainty uncertainty

17 0.1241 6.91E-07 0.046937491 1.72791E-08 2.50%

16 0.118 6.98E-07 0.044633472 1.75293E-08 2.51%

14 0.1026 6.93E-07 0.038814712 1.76395E-08 2.54%

13 0.0971 7.07E-07 0.036735926 1.80756E-08 2.56%

11 0.0821 7.06E-07 0.031064946 1.83943E-08 2.60%

10 0.0763 7.22E-07 0.028871612 1.89714E-08 2.63%

8 0.056 6.62E-07 0.02119289 1.81917E-08 2.75%

7 0.0501 6.77E-07 0.018960634 1.89561E-08 2.80%

5 0.0345 6.53E-07 0.013057547 1.97483E-08 3.02%

4 0.0289 6.84E-07 0.010938247 2.16384E-08 3.17%

2 0.0151 7.14E-07 0.005715305 2.82598E-08 3.96%

1 0.0073 6.91E-07 0.002763051 3.95439E-08 5.72%

1 0.0068 6.43E-07 0.002573802 3.84557E-08 5.98%

2 0.014 6.62E-07 0.005298966 2.70628E-08 4.09%

4 0.0276 6.53E-07 0.010446251 2.09311E-08 3.21%

5 0.0397 7.51E-07 0.015025364 2.20117E-08 2.93%

7 0.0477 6.45E-07 0.018052543 1.821E-08 2.82%

8 0.055 6.51E-07 0.020814556 1.79197E-08 2.75%

10 0.0696 6.59E-07 0.026337587 1.75132E-08 2.66%

11 0.0757 6.51E-07 0.028644699 1.7128E-08 2.63%

13 0.097 7.06E-07 0.036698127 1.80589E-08 2.56%

14 0.1048 7.08E-07 0.039646134 1.79815E-08 2.54%

16 0.118 6.98E-07 0.044633472 1.75293E-08 2.51%

17 0.1253 6.97E-07 0.047390683 1.74328E-08 2.50%

Configuration 3:

Case 3 d(m) 0.0005 a(m) L(m) Absolute Fractional
m 0.1184 0.00004 2.642 Uncertainty uncertainty
y(mm)
33 lambda theta (rad) 1.02608E-08 1.51%
6.79008E-07 0.0447846

32 0.1184 7.00227E-07 0.0447846 1.05815E-08 1.51%

31 0.1105 6.74587E-07 0.0418 1.02958E-08 1.53%

30 0.1703 1.07431E-06 0.0643697 1.55431E-08 1.45%

29 0.1403 9.15581E-07 0.0530539 1.3534E-08 1.48%

28 0.0982 6.63729E-07 0.0371517 1.03182E-08 1.55%

23 0.0802 6.59909E-07 0.0303465 1.06359E-08 1.61%

22 0.0756 6.50334E-07 0.0286069 1.06049E-08 1.63%

21 0.0723 6.51563E-07 0.0273588 1.07233E-08 1.65%

20 0.0685 6.48183E-07 0.0259215 1.0792E-08 1.66%

19 0.0648 6.45444E-07 0.024522 1.08809E-08 1.69%

18 0.0619 6.50812E-07 0.0234249 1.1089E-08 1.70%

17 0.059 6.56811E-07 0.0223279 1.13216E-08 1.72%

16 0.0549 6.49366E-07 0.0207767 1.13988E-08 1.76%

15 0.052 6.56069E-07 0.0196795 1.16831E-08 1.78%

10 0.0371 7.0212E-07 0.0140415 1.38588E-08 1.97%

9 0.0331 6.96022E-07 0.0125277 1.43052E-08 2.06%

8 0.03 7.0969E-07 0.0113545 1.514E-08 2.13%

7 0.0266 7.19152E-07 0.0100678 1.61079E-08 2.24%

6 0.0228 7.19152E-07 0.0086296 1.72344E-08 2.40%

5 0.019 7.19152E-07 0.0071914 1.88115E-08 2.62%

4 0.0147 6.95496E-07 0.0055639 2.08696E-08 3.00%

3 0.0111 7.00227E-07 0.0042013 2.48738E-08 3.55%

2 0.0081 7.66465E-07 0.0030658 3.36204E-08 4.39%

1 0.004 7.57002E-07 0.001514 5.71537E-08 7.55%

1 0.0034 6.43452E-07 0.0012869 5.56775E-08 8.65%

2 0.0066 6.24527E-07 0.0024981 3.17752E-08 5.09%

3 0.0112 7.06535E-07 0.0042392 2.49558E-08 3.53%

4 0.0148 7.00227E-07 0.0056018 2.09311E-08 2.99%

5 0.0189 7.15367E-07 0.0071535 1.87623E-08 2.62%

6 0.0225 7.0969E-07 0.0085161 1.71114E-08 2.41%

7 0.0268 7.24559E-07 0.0101435 1.61782E-08 2.23%

8 0.0298 7.04958E-07 0.0112789 1.50785E-08 2.14%

9 0.0335 7.04433E-07 0.0126791 1.44146E-08 2.05%

10 0.0376 7.11582E-07 0.0142307 1.39818E-08 1.96%

15 0.0508 6.40929E-07 0.0192255 1.14862E-08 1.79%

16 0.0544 6.43452E-07 0.0205876 1.13219E-08 1.76%

17 0.0576 6.41225E-07 0.0217982 1.1119E-08 1.73%

18 0.061 6.41349E-07 0.0230845 1.0966E-08 1.71%

19 0.0647 6.44448E-07 0.0244841 1.0868E-08 1.69%

20 0.068 6.43452E-07 0.0257324 1.07305E-08 1.67%

21 0.0716 6.45254E-07 0.027094 1.06413E-08 1.65%

22 0.075 6.45172E-07 0.02838 1.05378E-08 1.63%

23 0.079 6.50035E-07 0.0298927 1.05075E-08 1.62%

28 0.1027 6.94144E-07 0.0388525 1.07136E-08 1.54%

29 0.106 6.91743E-07 0.0400996 1.06241E-08 1.54%

30 0.1092 6.88872E-07 0.0413088 1.05324E-08 1.53%

31 0.112 6.83744E-07 0.0423668 1.04149E-08 1.52%

32 0.1165 6.8899E-07 0.0440668 1.04354E-08 1.51%

33 0.1213 6.95639E-07 0.04588 1.0477E-08 1.51%

Configuration 4:

d(m) a(m) L(m)

Case 3 0.00025 0.00004 2.642
m
y(mm) lambda theta Absolute Fractional
Uncertainty uncertainty

18 0.1226 6.45E-07 0.046370974 1.61378E-08 2.50%

17 0.1162 6.47E-07 0.043953505 1.62678E-08 2.52%

16 0.1091 6.45E-07 0.041271026 1.63187E-08 2.53%

15 0.1032 6.51E-07 0.039041469 1.65506E-08 2.54%

12 0.0819 6.46E-07 0.030989319 1.68252E-08 2.61%

11 0.0765 6.58E-07 0.028947249 1.72863E-08 2.63%

10 0.0688 6.51E-07 0.026034994 1.73391E-08 2.66%

9 0.0623 6.55E-07 0.023576252 1.76939E-08 2.70%

8 0.0547 6.47E-07 0.020701055 1.7838E-08 2.76%

5 0.037 7E-07 0.014003627 2.08365E-08 2.98%

4 0.0299 7.07E-07 0.011316701 2.21825E-08 3.14%

3 0.0229 7.22E-07 0.008667459 2.44985E-08 3.39%

2 0.0149 7.05E-07 0.005639607 2.80422E-08 3.98%

1 0.0071 6.72E-07 0.002687352 3.91086E-08 5.82%

1 0.007 6.62E-07 0.002649502 3.8891E-08 5.87%

2 0.0147 6.95E-07 0.005563909 2.78246E-08 4.00%
3.45%
3 0.0217 6.84E-07 0.00821329 2.36279E-08 3.15%
2.99%
4 0.0295 6.98E-07 0.01116532 2.19649E-08 2.76%
2.70%
5 0.0364 6.89E-07 0.01377657 2.05753E-08 2.67%
2.63%
8 0.0547 6.47E-07 0.020701055 1.7838E-08 2.53%
2.52%
9 0.0618 6.5E-07 0.023387105 1.7573E-08 2.50%
2.49%
10 0.068 6.43E-07 0.025732396 1.7165E-08

11 0.0759 6.53E-07 0.028720337 1.71676E-08

16 0.109 6.45E-07 0.04123324 1.63051E-08

17 0.1161 6.46E-07 0.043915728 1.6255E-08

18 0.1224 6.43E-07 0.046295436 1.61136E-08

19 0.13 6.47E-07 0.049165494 1.61361E-08

Also included are the fringe patterns which I traced from the laser apparatus and the original distances
which I measured in millimeters.

Analysis and Sample Calculations.

During our setup we carefully measured the dimensions of the setup and recorded our uncertainties
into our spreadsheet. The measurements and associated uncertainties are as follows:

Measurement L(m) y(m) d(m) a(m)
Uncertainty 2.642 varies varies varies
±.003 ±.00025 ±.000005 ±.000005

Using these values we calculated λ from the position of each fringe on the wall. A calculation of λ for
the m=1 fringe of configuration 1 is shown below:

y= m ⋅ λ ⋅ L → λ= y ⋅d= .003m ⋅.0005m= 5.68×10−7= 568nm
d m⋅L 1⋅ 2.642m

In addition to calculating the wavelength, we also needed to calculate the orders for which interference
fringes were suppressed by the single slit diffraction envelope. We did this for each of the four
configurations as shown:

Predicted Dark Fringes when: m =n ⋅ d ; n =1, 2,3...
a

Configuration1: m =n ⋅ d =n ⋅ .0005 ≈ 6n
a .00008

Configuration2: m =n ⋅ d =n ⋅ .00025 ≈ 3n
a .00008

Configuration3: m =n ⋅ d =n ⋅ .0005 ≈ 12n
a .00004

Configuration4: m =n ⋅ d =n ⋅ .00025 ≈ 6n
a .00004

Also, we calculated the angular position, associated with each linear position along the wall. We did this
using the trigonometric relationship that we defined earlier. A calculation of the angular position θ for
the first configuration at m=1 is shown:

tan (θ ) = y →θ = tan −1  y  = tan −1  .003  = .00114rad
L  L   2.642 

This is well within our small angle approximation.

Without an idea of our uncertainty however, these predictions are meaningless. We calculate
our uncertainty two ways. First, through the average deviation, which we calculate using our
spreadsheet program. The average deviation gives us an idea of how much our individual measurements
deviate from their average value. The second method is propagation of error, which accounts for the

uncertainty associated with measurement. This method requires some computation which we described
earlier, but here we will show the calculation of the absolute uncertainty associated with our
measurement of λ for configuration 1 at m=1:

∆λ = d ⋅ ∆y + y ⋅ ∆d + d ⋅ y ⋅ ∆L = .0005 ⋅.00025 + .003 ⋅ .000005 + .0005 ⋅ .003 ⋅ .003 = ±54.7nm
L⋅m L⋅m L2 ⋅ m 2.642 ⋅1 2.642 ⋅1 2.6422 ⋅1

Now that we have obtained our Data we can make the calculations for each fringe. In total our

experiment yielded 133 data points which each yielded a value for λ. The Laser that we used to project
the fringes had a wavelength of 650nm. And we used this value to determine the effectiveness of our
calculated λ value. After running the experiment for all four configurations, our results are as shown:

Average Average Average % Average % Discrepancy
6.70846E-07 Absolute uncertainty Deviation(m) 3.21%
Uncertainty(m)
±1.80207E-08 ±2.69% ±3.73E-08

Summary

In conclusion, our results differed from the accepted value of 650nm by 3.21% which is outside
our predicted uncertainty of ±2.69%. There were a number of error sources which could have caused
this, foremost being the inaccuracy of my drawing of the fringe pattern. In many places where the
intensity of the fringes was suppressed by the single slit envelope the fringes became too dim to mark
with any accuracy. This was exacerbated by rays of light which came in through a window next to the
setup, despite our best efforts to tape the curtain shut. Another great source of our error was the
confined space in which we were working. Our Apparatus was in the corner station and moving around
with two other lab partners was difficult without bumping the laser apparatus.

We did see a drastic change in the behavior of the fringes as we changed the dimensions of the
slits. As we decreased a we saw a decrease in the spacing of fringes, but not in the size of each fringe.
However as we decreased d we saw that width of each fringe increased. These results are not
conclusive, but it does show us that there is a relationship between the fringe pattern and the
dimensions of the slits causing the diffraction.

Upon Retrospect there are quite a few things that I would have done differently to reduce my
error. Foremost is my omission of fringes whose intensity was suppressed by the single slit envelope. If I
were to repeat the experiment, I would mark every single fringe and perhaps shade the fringes which
were too dark to see clearly. Or perhaps I would measure the distance of dark areas while the fringe
pattern is projected, to get the clearest picture of how many orders are omitted.