Michelson Interferometer

Michelson Interferometer Leanne Garrity 200836248 Second Year Physics Laboratory, University of Strathclyde, Glasgow Abstract: The purpose of the Michelson Interferometer is firstly to find the wavelength of the mercury green filtered line. In the second part of the experiment it is to determine the fringe visibility and to measure the distance between the maximum and minimum fringe visibility, this can then be used to calculate the difference in wavelength between the two yellow spectral lines. The wavelength of the mercury green line was found to be: λ= (4±100) ×10¯⁷m This doesn’t compare favourably to the accepted value of: λ=546nm [1]

The difference in wavelength of the yellow spectral lines was found to be: Δλ= (0.02±120) ×10¯⁷m This value compares favourable to the accepted value of: Δλ= 0.02×10¯⁷m [1] The uncertainty in both the experiments is very high; this was due to how the experiment was carried out, as the fringes were counted by the human eye instead of a counting device. However the second experiment was a success as apart from the high uncertainty, the calculated value was close to the accepted value. Introduction: In 1881, Albert Abraham Michelson invented a device called the interferometer. The interferometer uses interference fringes to measure distances in terms of wavelength with great precision. [2] The interferometer is still an important optical instrument to this day, as it is used in physical optics, spectroscopy and laser physics. [3] In this experiment, the Michelson Interferometer is used to measure the wavelength of a mercury green filtered line and the difference in wavelength between two yellow spectral lines. The Michelson Interferometer causes interference by splitting a beam of light into two parts. These two beams will appear to be reflected at the surfaces of both mirrors, these mirrors are

separated by a distance d. [3] By finding this distance the wavelength can then be calculated. The experiment clearly demonstrates how using the Michelson Interferometer, the wavelength of the light can be calculated by finding out the distance between two mirrors. Theory: The wavelength of the mercury green line was calculated when the two light beams interfere to give a maximum intensity, this equation shows this: λ= 2dcosθ/n Where

(1)

λ= wavelength (m) n= number order θ= angle between mirrors d= separation between fringes

When measuring the distance between mirror positions d1 and d2 and when only two wavelengths are present, Δλ can be calculated using this equation: Δλ= λ²/2(d2-d1) Where mirror

(2)

λ= accepted wavelength value d2-d1= the distance between the

Experimental Method:

Figure 1: shows the arrangement of the interferometer [4] Movable mirror is M1 and Fixed mirror is M2. The experiment was set up as shown in Figure 1. Firstly in experiment one, the mercury lamp was switched on and the green filter removed. The mirrors had to be parallel with one another, so when looking into the beam splitter there where two images of a small cross, the mirror M2 was adjusted so that there was only one image of the small cross, therefore they were parallel. A green filter was placed into the holder until interference fringes were seen, M1 was adjusted again so that concentric circular fringes were seen. From an experiment before this, part (a) of the Michelson Interferometer, the zero order position was found. Mirror M1 was moved away from this zero order position until a clear set of concentric fringes were found. From this M1 was moved so that the centre of the fringe

pattern would go through a maximum one hundred times, the distance that M1 moved was then recorded. This was repeated several times. Experiment two was set up again as shown in Figure 1. This time however the green filter was removed and replaced by a yellow filter. From the mercury lamp, the yellow filter passes a strong pair of spectral lines of mean wavelength 578nm. [5] M1 was the moved so that the fringe visibility went from a minimum until the next minimum position, this then was (d2-d1) and a value was recorded. This was repeated several times. Mirror M1 is pushed forward by a lever driven by a micrometer; therefore a movement of 5dmm of the micrometer causes the mirror to move dmm. [6] Experimental Results: Results for experiment 1: Original reading for ‘zero order position’ = 8.49×10¯³m Table 1: Distance M1 moved through 100 maxima Zero Order Reading (d1) (×10¯³m)

Reading on Micrometer (d2) (×10¯³m)

Difference (d1d2) (×10¯³m)

8.49

8.36

0.13

8.49

8.39

0.10

8.49

8.37

0.12

8.49

8.40

0.09

8.49

8.41

0.08

Average d2= 8.39×10¯³m Average difference= 0.104×10¯³m Therefore taking the ×5 lever on the mirror translation into account: d1-d2= (8.49-8.39)×10¯³m =0.10×10¯³m d= 0.10×10¯³m/5 = 0.02×10¯³m Results for experiment 2: Table 2: Distance M1 was moved so that the fringe visibility went from a minimum position to the next. d1 (×10¯³m)

d2 (×10¯³m)

Difference (d2d1) (×10¯³m)

8.30

8.72

0.42

8.30

8.68

0.38

8.30

8.72

0.42

8.30

8.73

0.43

8.30

8.73

0.43

Average d2= 8.72×10¯³m Therefore taking the ×5 lever on the mirror translation into account: d1= 8.30×10¯³m/5 = 1.66×10¯³m d2= 8.72×10¯³m/5 = 1.74×10¯³m d2-d1= (1.74-1.66)×10¯³m = 0.08×10¯³m Analysis: Experiment one: By using equation (1): λ= 2×(0.02×10¯³m)×1/100 = 400nm This was where d= 0.02×0¯³m Θ= 0° n= 100 fringes

λ=? Experiment two: By using equation (2): Δλ= (578nm)²/2(1.74-1.66)×10¯³m = 2nm This was where λ= 578nm d2= 1.74×10¯³m d1= 1.66×10¯³m For experiment one, the wavelength of the mercury green line was found to be: λ= (4±100) ×10¯⁷m For experiment two, the wavelength between the two yellow spectral lines was found to be: Δλ= (0.02±120) ×10¯⁷m For both experiments the uncertainties are really high this was due to the way the experiment was carried out, as the counting was done by the human eye. Discussion: In experiment one, a value was obtained for the wavelength of the mercury green line to be: λ= (4±100) ×10¯⁷m

whereas the accepted value is λ= 546nm [1] The value that was calculated was too far off from the accepted value. The large uncertainty is due to the way that the experiment was carried out; by counting the fringes with the human eye and that the room was crowded, made it difficult to concentrate on the fringes. To decrease the uncertainty and make the experiment more accurate, a more larger number of fringes could have been counted. From experiment two, a value for the difference in wavelength between two yellow spectral lines was obtained to be: Δλ= (0.02±120) ×10¯⁷m While the accepted value is 0.02×10¯⁷m [1] This was a more successful experiment but the uncertainty again was very high but as the calculated value was close to the accepted value, the experiment is deemed successful. This experiment could have been improved by using a more suitable counting device to count the fringes. Conclusion: The wavelength of the mercury green line was found to be (4±100) ×10¯⁷m which didn’t

compare favourably with the accepted value of 546nm. The difference in wavelength of the two yellow spectral lines was found to be (0.02±120) ×10¯⁷m which from aside the large uncertainty compared favourably with the accepted value of 0.02×10¯⁷m. Acknowledgements: I would just like to thank both my lab partners Lisa Toms and Gordon Hitchell for there insight into this experiment. References: [1] H. G. Kuhn, Longmans, Atomic Spectra, Second Edition, 1969, 2nd Year Lab Scripts, (Michelson Interferometer (b) Pg 5) [2] Harris Benson, University Physics, Revised Edition, 1995 37(771) [3] TPJH, Second Year Lab Scripts, 2008, Michelson Interferometer (b) Pg 2 [4] http://hyperphysics.phyastr.gsu.edu/HBASE/PHYOPT/michel.html [5] TPJH, Second Year Lab Scripts, 2008, Michelson Interferometer (b) Pg 4

[6] TPJH, Second Year Lab Scripts, 2008, Michelson Interferometer (a) Pg 3 Appendix: To work out the uncertainties in the values the equation for the best estimate of the standard deviation of the mean was used.