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 nm±3.07%)
with an average deviation of ±23.9nm. for the third configuration, we reduced the width of the from this configuration we calculated λ to be (684 mm and set d=.5mm nm ±2.29%) with an average deviation of
nm mm and d=.5mm from which we calculated slits to a=.04 , this yielded a λ of (692 nm±3.09%) with an average deviation of ±21nm. For each measurement we used the ±40.2 . our final configuration had dimensions a=.04 λ to be (665 lated the average deviationrules for propagation using an excel of errorspreadsheet, to determine which an gave absolute us another uncertainty, method and of determining divided it by our the total measurement in order to obtain our percent uncertainty. We also calcu nm is is higher than our uncertainty would predictuncertainty however in λ. Ourthere average were many value sources of λ from of allerror four in configurations this experiment. was It (671was difficult±2.69%) to see which the fringesdiffered clearly from the enough value to printed accurately on the trace laser their by 3.21%.outline Th 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:
∆=rrr21 − = d ⋅sin (θ )
And from our work in chapter 18, we know that these will be constructive when:
∆=rdsin (θλ) = m ⋅
And Destructive, when:
1 ∆=rdsin (θλ) = m + ⋅ 2
From the diagram, we can define tan( ) as:
θ y tan (θ ) = m L
For small angles we are able to make the approximation: tan(θθ) ≈ sin ( ) (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:
ybright mLλ constructive: d⋅ = my ⋅→λ bright = Ld
ydark 11Lλ Destructive: d⋅ =+ mymλ →dark = ⋅+ Ld22
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:
dy⋅ λ = bright mL⋅
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:
2 πθ⋅⋅a sin ( ) sin πθ λ 2 d sin ( ) II=max ⋅⋅cos λ πθ⋅⋅a sin ( ) 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:
21πθ⋅⋅ddsin ( max ) πθ⋅⋅sin ( max ) − λ ⋅m cos =→1 =⋅→m πθmax =sin λλ d
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:
πθ⋅⋅aasin ( min ) πθ⋅⋅sin ( min ) −1 nλ sin =→0 =→=nπθmin sin λλ a
In order to find any places where an interference maximum is suppressed by an envelope
minimum, we must look for areas where min= max:
θ θ −−11nλλ ⋅m nd d θθmin= max →sin = sin →=m = a d aa
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 which we compute using:
θ yy−1 tan (θθ) = →=tan LL
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 is given by:
∂∂∂λλλλ ∆λ = ⋅∆ydL + ⋅∆ + ⋅∆ ∂∂yd ∂ L
By differentiating our expression for with respect to each variable, we obtain the following expression for the theoretical uncertainty associated with our calculation of . λ λ d y dy⋅ ∆λ = ⋅∆yd + ⋅∆ + ⋅∆ L L⋅m Lm⋅⋅ 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:
a (mm) d (mm) (nm) Average avg % % Difference Deviation Uncertainty Config 1 .08 .50 avg λ 2.29% 6.92884E-07 4.02162E-08 6.60% Config 2 .08 .25 3.07% 6.84308E-07 2.39E-08 5.28% Config3 .04 .50 2.29% 6.92884E-07 4.02162E-08 6.60% Config 4 .04 .25 3.09% 6.65003E-07 2.14E-08 2.31%
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 Absolute Percentage m y lambda theta (rad) Uncertainty uncertainty 23 0.0787 6.47566E-07 0.0297792 1.04754E-08 1.62% 22 0.0748 6.43452E-07 0.0283043 1.05154E-08 1.63% 21 0.071 6.39847E-07 0.0268671 1.0571E-08 1.65% 20 0.0677 6.40613E-07 0.0256189 1.06936E-08 1.67% 17 0.0575 6.40112E-07 0.0217604 1.11046E-08 1.73% 16 0.0537 6.35172E-07 0.0203227 1.12143E-08 1.77% 15 0.0505 6.37144E-07 0.019112 1.1437E-08 1.80% 14 0.0471 6.36693E-07 0.0178255 1.16565E-08 1.83% 11 0.037 6.3657E-07 0.0140036 1.25766E-08 1.98% 10 0.0342 6.47237E-07 0.012944 1.31453E-08 2.03% 9 0.0306 6.43452E-07 0.0115816 1.36218E-08 2.12% 8 0.0278 6.57646E-07 0.0105219 1.44635E-08 2.20% 5 0.0157 5.94247E-07 0.0059424 1.71877E-08 2.89% 4 0.013 6.15064E-07 0.0049205 1.9824E-08 3.22% 3 0.0095 5.99293E-07 0.0035957 2.35617E-08 3.93% 2 0.0061 5.77214E-07 0.0023089 3.11601E-08 5.40% 1 0.003 5.67752E-07 0.0011355 5.46934E-08 9.63% 1 0.004 7.57002E-07 0.001514 5.71537E-08 7.55% 2 0.008 7.57002E-07 0.003028 3.34974E-08 4.43% 3 0.0105 6.62377E-07 0.0039742 2.43818E-08 3.68% 4 0.0145 6.86033E-07 0.0054882 2.07466E-08 3.02% 5 0.0171 6.47237E-07 0.0064723 1.78766E-08 2.76% 9 0.0289 6.07705E-07 0.0109382 1.31571E-08 2.17% 10 0.0311 5.88569E-07 0.0117708 1.23827E-08 2.10% 11 0.0348 5.9872E-07 0.0131711 1.20845E-08 2.02% 14 0.0382 5.16384E-07 0.0144577 1.00925E-08 1.95% 15 0.0487 6.14434E-07 0.0184309 1.11418E-08 1.81% 16 0.0526 6.22161E-07 0.0199065 1.10451E-08 1.78% 17 0.0548 6.10055E-07 0.0207389 1.07138E-08 1.76% 20 0.06 5.67752E-07 0.0227062 9.7464E-09 1.72% 21 0.0736 6.63278E-07 0.0278505 1.08756E-08 1.64% 22 0.0761 6.54635E-07 0.028796 1.06608E-08 1.63% 23 0.08 6.58263E-07 0.0302708 1.06145E-08 1.61% Configuration 2
d(m) a(m) L(m)
Config 2 0.00025 0.00008 2.642 Absolute Fractional m y(mm) lambda theta 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:
d(m) a(m) L(m) Case 3 0.0005 0.00004 2.642 Absolute Fractional m y(mm) lambda theta (rad) Uncertainty uncertainty 33 0.1184 6.79008E-07 0.0447846 1.02608E-08 1.51%
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 Absolute Fractional m y(mm) lambda theta 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 0.0217 6.84E-07 0.00821329 2.36279E-08 3.45% 4 0.0295 6.98E-07 0.01116532 2.19649E-08 3.15% 5 0.0364 6.89E-07 0.01377657 2.05753E-08 2.99% 8 0.0547 6.47E-07 0.020701055 1.7838E-08 2.76% 9 0.0618 6.5E-07 0.023387105 1.7573E-08 2.70% 10 0.068 6.43E-07 0.025732396 1.7165E-08 2.67% 11 0.0759 6.53E-07 0.028720337 1.71676E-08 2.63% 16 0.109 6.45E-07 0.04123324 1.63051E-08 2.53% 17 0.1161 6.46E-07 0.043915728 1.6255E-08 2.52% 18 0.1224 6.43E-07 0.046295436 1.61136E-08 2.50% 19 0.13 6.47E-07 0.049165494 1.61361E-08 2.49%
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:
L(m) y(m) d(m) a(m) Measurement 2.642 varies varies varies Uncertainty ±.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: m⋅⋅λ L yd ⋅ .003mm⋅ .0005 −7 y = →=λ = =5.68 × 10 = 568nm d mL⋅⋅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:
d Predicted Dark Fringes when: mn=⋅= ; n 1,2,3... a d .0005 Configuration1: mn=⋅=⋅ n ≈6 n a .00008 d .00025 Configuration2: mn=⋅=⋅ n ≈3 n a .00008 d .0005 Configuration3: mn=⋅=⋅ n ≈12 n a .00004 d .00025 Configuration4: mn=⋅=⋅ n ≈ 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 for the first configuration at m=1 is shown: θ
yy−−11 .003 tan (θθ) = →=tan = tan =.00114rad LL 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 for configuration 1 at m=1:
dλ y dy⋅⋅.0005 .003 .0005 .003 ∆λ = ⋅∆yd + ⋅∆ + ⋅∆ L = ⋅.00025 + ⋅ .000005 + ⋅.003 =± 54.7 L⋅m Lm⋅⋅ L22 m 2.642 ⋅ 1 2.642 ⋅ 1 2.642 ⋅ 1 nm
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 . 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 λ calculated value. After running the experiment for all four configurations, our results are as shown:
λ Average Absolute Average % Average Average Uncertainty(m) uncertainty Deviation(m) % Discrepancy 6.70846E-07 ±1.80207E-08 ±2.69% ±3.73E-08 3.21% 훌
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.