Lab_5_Interference_and_Diffraction.pdf

Lab 4: Interference and Diffraction

David SirajuddinPartner: Aimee Covert

Dept. of PhysicsPhysics 341 - Waves, Heat, and Light Laboratory

Experiment Date: November 12/19, 2007Due Date: November 26, 2007

Newtons Rings

1.

Colored bands were exhibited in the Newtons rings apparatus given in class. The reasonfor this is the same for why there are any bands at all. Diffraction of light causes aninterference pattern to appear in the form of rings. The extent to which light is diffractedis evidently also a function of the wavelength. Different wavelengths are resolved by resultof the interference of light interacting with varying thicknesses of the air between the twocontacting surfaces. This finding is similar to one found in Lab 3.

Single Slit Diffraction and Babinets Theorem

2.

Diffraction of a He-Ne laser (wavelength = 632.8 nm) was investigated through singleslits, corresponding to a slit thicknesses of 10, 25, and 50 m. A diffraction pattern wasrecorded by affixing a sheet of paper to a bae situated at some distance away from the slit.Experimental diffraction angles m were calculated from direct measurements of the recordeddiffraction pattern. Eqn. (5.4) in the laboratory manual was used to compute theoreticalpredictions of the diffraction angle t of the minima, and a juxtaposition of experiment andtheory is summarized in Tables 1-3.

m [deg] t [deg] Percent Difference [%]

-7.54 -7.27 3.72-3.78 -3.62 4.380 0 0

3.46 3.62 -4.476.98 7.27 -3.91

Table 1 - Theoretically predicted and measured diffraction angles for a 10 m single slit.

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Sirajuddin, David Lab 5 - Interference and Diffraction


-2.99 -2.90 3.11-1.56 -1.45 7.70 0 0

1.43 1.45 -1.263.12 2.90 7.59



-1.37 -1.45 -5.43-0.564 -0.731 -22.10 0 0

0.806 0.725 11.21.53 1.40 5.68


The experimentally measured angles hold a close agreement to theory. It should be notedthat a center line was identified (corresponding to a diffraction angle of 0 in the abovedata). All measurements were taken with respect to this centerline, and provided for thescenario in which the measured centerline diffraction angle is zero by consequence. Also, theconvention was assigned that distances measured to the left of the centerline were negative,and distances measured to the right were positive.

3.

When the line/slit comparison target was placed in front of the He-Ne laser, no diffractionpattern was observed. The observation verifies Babinets theorem, in that the diffractionpattern from the obstructed line in the path of the laser must have produced the oppositediffraction pattern as that of the single slit of the same size. The two in combination must,therefore, cause destructive interference, yielding no pattern. Thus, the slit plus line isequivalent to a dark screen.

4.

The observations of part 3 allow for the use of Eqn. (5.4) in the laboratory manual to beused with the diffraction pattern of a human hair. A strand was placed in front of the laser,and the resulting diffraction pattern was recorded. The measured angle m was calculatedfrom the pattern and taken to be the quantity 0 in Eqn. (5.4). Furthermore, it is notedfrom part 2 that each peak represented consecutive integers in the corresponding theoreticalprediction. Thus, from the determined centerline, integers n were assigned appropriately.Finally, Eqn. (5.4) was employed to solve for the quantity a, the thickness of the human hair,for each integer n. The data and calculations are given in Table 4 (where the calculationsfrom the centerline, corresponding to n = 0, are omitted).

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Integer, n m [deg] a [m]

-3 -1.77 61.3-2 -1.21 59.9-1 -0.56 64.11 0.56 64.12 1.12 64.13 1.69 64.2

Table 4 - Data and calculations for the thickness of a human hair, a.

The data admitted an average value of 62.9 m for the thickness of a human hair.

Diffraction from a Circular Aperture

5.

Using circular apertures with diameters of 25, 50, and 100 microns, circular diffractionpatterns were created by using a He-Ne laser. The radii of the circles were measured, andexperimental diffraction angles m were calculated from these measurements. These werecompared with theoretical predictions, from Eqn. (5.6), where the first three minima of thefirst order bessel function of the first kind J1() are indicated to be = 3.83171, 7.01559,and 10.17347. Data comparison is given in Tables 5-7.


1.73 1.76 -1.883.46 3.24 7.03

Table 5 - Theoretically computed diffraction angles compared with experimental data from a circularaperture of 25 m.


1.73 0.88 96.52.42 1.61 50.33.12 2.34 33.3



0.85 0.44 94.31.14 0.81 41.51.43 1.17 21.9


The angles found in experiment are roughly the same as those computed from theory.

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6.

Given an aperture of 2 mm and 2400 mm for the human eye and Hubble Space Telescoperespectively, an estimation of the diffraction limited angular resolution of both instrumentscan be calculated from Braggs law (d sin = n), where d is the aperture, and n is aninteger. The minimum angle is obtained by considering the smallest argument for the arcsine, i.e. when n = 1. A median value is taken for the ' 550nm among the spectrum ofvisible light ( 400-700 nm) for the wavelength. Inputting these numbers into Braggs law,solved in terms of the limiting angle , yields a limiting angular resolution of 0.158 and1.313105 for the human eye and the Hubble Space Telescope respectively.

Double Slit Interference

7.

The interference pattern resulting from two sets of double slits of thickness 40 m and80 m were observed. Separation distances of 250 m and 50 m were used for both slitthicknesses. From the resulting interference pattern, several minima for each were recorded.When apparent, the fine structure was recorded in addition to the coarse structure; however,a fine structure was only apparent for the 80 m double slit with a slit separation of 250m. The data for this part is left for part 9, where the collected data is compared againsttheory.

8.

A direct comparison is demonstrated with the 50 m single slit data and the coarsestructure data of the 40 m double slit. The comparison can be seen from theory in any ofthe Tables 8-11; however, for the sake of an example in experiment, the first minimum at n= 1 for the single slit was located at a diffraction angle of 0.806 (as per part 2), while thefirst minimum found in lab for the double slit (for the 250 m separation) was 0.943. Thus,they are in the same order of magnitude, and hold a percent difference of 14.5%.

9.

Using Eqns. (5.4) and (5.8) of the laboratory manual, the experimental angles werejuxtaposed with theoretical predictions for the minima of a single and double slit. Thewavelength has been taken to be that of the He-Ne laser ( = 632.8nm), so as to matchthe Excel spreadsheet given in class. The data is compared in the following tables, wherem is the measured angle, s is the prediction for a single slit, and d is the predicted doubleslit angle.

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m [deg] s [deg] d [deg] n

-4.004 -3.628 4.137 -25-3.148 -2.720 3.120 -21-1.504 -1.813 1.523 -10-1.146 -0.906 1.088 -70.859 0.906 0.943 61.790 1.813 1.813 122.862 2.720 2.829 19

Table 8 - Measured diffraction angles compared with theoretical predictions for a single slit and a doubleslit, for a slit size of 40 m, and a separation of 250 m between the two slits.


-3.814 -3.628 -3.810 -52-2.913 -2.720 -2.938 -40-1.759 -1.813 -1.777 -24-0.955 -0.906 -0.979 -130.905 0.906 0.906 121.809 1.813 1.849 252.913 2.720 2.938 403.764 3.628 3.737 50



n -1.909 -3.628 -1.088 -8-1.353 -3.174 -0.943 -7-1.114 -2.720 -0.798 -6-0.915 -2.267 -0.653 -5-0.637 -1.813 -0.508 -4-0.477 -1.360 -0.363 -3-0.239 -0.906 -0.218 -2-0.119 -0.453 -0.073 -10.398 0.453 0.363 30.915 0.906 -0.943 71.353 1.360 1.378 91.830 1.813 1.813 12



0.477 0.453 0.471 60.955 0.906 0.979 131.353 1.360 1.342 181.830 1.813 1.849 252.267 2.267 2.285 322.783 2.720 2.793 38

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It was evident during the calculation for both the single and double slit, that Eqns. (5.4)and (5.8) predicted the all minima, not only those pertaining to the coarse and fine struc-ture. Thus, the peaks measured on the diffraction pattern recorded in experiment did notnecessarily correspond to consecutive integers. Furthermore, since the fine structure wasonly apparent for one of the spectra, the integer n was mostly guaranteed not to correspondconsecutively to each of the peaks (i.e. consecutive integers would identify consecutive peaks,which would all be part of the microscopic, or fine structure of the diffraction pattern). Forthis reason, the integer n corresponding to the double slit peaks are listed in the above tablesfor identification. These integer labels were used to aid in the labelling of the peaks in thesubsequent plots.Furthermore, the spreadsheet provided in class was used to plot the expected diffractionpattern (we expect the double slit pattern to be realized in experiment, so only these spectraare shown). These plots are shown in Figures 1-4.

Figure 1 - Theoretical diffraction pattern for double slit of slit size 40 m, with a spacing of 250 m.

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The experimentally found peaks are labelled in the above plots.

Multiple Slit Interference

Varying the number of slits from 2 to 5, the width of the bands were estimated (Table12).

Number of Slits, N Band width [cm]

2 0.23 0.14 0.15 0.1

Table 12 - Width of the bright bands with respect to the number of slits

It is expected that increasing the number of slits will decrease the bandwidth size due to theincreased interference between light penetrating through more slits. The data above exhibitsthis trend.

Interference Effects with CD-ROM Disks

11.

The diffraction pattern off of a CD-ROM was recorded. The distance between peaks weremeasured, and were used to calculate the distance d between grooves on the CD for eachmeasured distance. The calculation is performed via Braggs law (See part 6), an

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Distance between peaks [cm] d [m]

3.4 5.613.1 6.152.6 7.322.5 7.61

Table 13 - The distance between the peaks is shown alongside a calculation of the separation distancebetween grooves on the CD.

The data yielded an average value of 6.68 m for a separation distance between grooves.

12.

The limiting angular resolution of the blue laser would be greater than that of a red laser.The increased resolution would allow the grooves on the CD to be spaced closer together sothat more storage could be allotted per CD.

Two-dimensional Diffraction

13.

The pattern that resulted was a 2-D array of dots. The spacing horizontally was 1.3 cm,while the spacing vertically was 0.7 cm. The result follows suit with the trend exhibitedthroughout the rest of the lab. The horizontal direction has smaller distances between slits,while the vertical direction has wider spacing. This difference in spacing will yield the trendthat was shown on our diffraction pattern.

Diffraction Gratings and Atomic Spectra

14.

The 120-line diffraction grating acted as a prism when light was allowed to shine throughit. The grating caused chromatic dispersion, wherein red light was diffracted the least. Theextent of diffraction increased for decreasing wavelengths.

15.

A spectrum of colors was shown with the diffraction grating with finely spaced lines. Thisis similar to the prism from Lab 3, as described in the laboratory manual.

Fresnel Zone Plate

16.

An object was placed at varying positions for both red and blue light. Using the lensformula (Eqn. (3.5)), and Eqn. (5.12) in the laboratory manual, focal lengths were calculated

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for the data collected in the lab notebook. The lens formula yielded focal lengths of 14.44cm and 16.70 cm for red and blue light respectively. Eqn. (5.12) admitted focal lengths of14.9 cm and 18.7 cm. These values reasonably agree.

Using Light to Measure the Spacing of Lines on a Ruler;

X-ray Diffraction for Structure Determination - The

Double Helix

17.

The He-Ne laser was reflected off the surface of a ruler, and the diffraction pattern wasrecorded. The spot due to the reflected beam pertaining to the angle was identified.Furthermore, other spots were recorded in order to resolve the angles m. These anglescorrespond directly to Eqn. (5.13), and these angular values alongside distance measurementstaken from the diffraction pattern were used to calculate the spacing between the indentationsof the ruler. Spacings were calculated for each of angles m, yielding an average value of1.37103 m. This value is in close agreement to the actual value of 1/16 in (1.5875103 m).

18.

From the measurements taken from the tungsten coil itself, and the diffraction pattern,the pitch angle of the helix was able to determined. Following the notes provided by ProfessorAkerlof, the pitch P was taken to the vertical spacing measured between the helix turns.The pitch angle was then calculated directly from the relation:

= 2 tan1(

P

2pid

)Inputting the values obtained in lab (denoted in the lab notebook) yielded a pitch angle of = 12.6.

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Lab_5_Interference_and_Diffraction.pdf