Michelson Interferometer Farooq Hassan Roll no: 2012-10-0101 LUMS School of Science and Engineering November 13, 2010 1 Abstract Michelson Interferometer was first used in the classic experiment in 1887 to detect the presence of aether which was thought to permeate vacuum and act as the medium in which all EM waves travel. The experiment hoped to detect aether wind caused by motion of Earth through the medium. Light that traveled parallel to direction of aether would take longer to reach the screen than light moving perpendicular to it and this difference would be measurable by the resulting fringe shift. The experiment did not detect any fringe shift which led to the abandoning of concept of aether. In this experiment, we will use Michelson Interferometer to calculate the wavelength of HeNe laser. We will also use Michelson Interferometer to measure the refractive index of glass. 2 Introduction Michelson Interferometer makes use of superposition phenomenon in waves to produce circular fringes. These fringes exit or enter the screen when the optical path of one of the beams is increased or decreased. By counting the no. of fringes that pass a reference point, distances of the order of nanometers can be measured accurately. 1
Target Screen Beam Splitter He Ne Laser M2 M1 Figure 1: Experimental setup. 3 Theoretical background Light from a monochromatic laser is split into two halves using a beam splitter. The two components are reflected by mirrors M1 and M2 and then superimposed. In this way, the beams that reach the detector are in phase(both undergo phase difference of pi upon reflection from beam splitter) and coherent. As a result, interference pattern is observed on the screen. To an observer on the screen the set up would look like: A Conceptual Rearrangement of the Michelson Interferonmeter S2 θ S1 2dcosθ 'O θ Detector S So M1 d M2 Si S2 2d S1 2dcos(θ) = Nλ N = 0,1,2... (1) 2
4 Measuring Wavelength of laser using Michelson Interferometer 4.1 Apparatus used HeNe laser, Beam Splitter, post holders, steel posts, laser and lens mounts, kinematic mounts, convex lens, rotation platform and DC servo motor controller. 4.2 Procedure The first task in the experiment was optical alignment. The HeNe laser was mounted using its clamp and M2 mirror was positioned in front of it such that the laser beam fell in the centre of M2 and reflected back on to the laser. Then M1 was placed at approx. same height as M2. Finally the beam splitter was inserted and rotated such that the reflected beam fell on M1. Once this was done, three dots were visible on the beam splitter, one from each mirror and one from laser. The three dots were aligned using the screws on the mirrors. This completed the alignment. In order to observe the fringes, we need to diverge the beam (enlargement) so two plano-convex lenses were place, one right in front of laser and another before the screen. This required realignment of the beams to keep the centre of fringes in focus. Once realignment was done, circular fringes were observed. These concentric fringes are characteristic of the plano convex lens used. Inorder to measure wavelength, the M2 mirror was moved using computer software. The settings were as follows: max vel = 0.003 mm/s, acc = 0.03 mm/s 2, step distance = 0.001 mm. The rotor was connected to M2 which was placed between parallel aluminium sheets to ensure movement only takes place parallel to the optical path of the beam reflected by M2. The software was then used to move the mirror a distance of 10µm. 4.3 Observation As the mirror M2 moved towards beam splitter, fringes were seen to disappear from the centre. The no of fringes that disappeared were counted using the centre as reference point (counting no of fringes was found to be least error prone at the centre). This procedure was repeated 5 times. The results are given below. 3
Trial No. Distance moved ( d/µ m) No of fringes (N) 1 10 31 2 10 32 3 10 32 4 10 31 5 10 32 Table 1: No. of fringes counted for distance of 10µm 4.4 Analysis As the mirror M2 is moved by d, the optical path difference changed by 2 d cos(θ). For every fringe that disappears, this change in optical path must equal λ. For N fringes, 2 d cos(θ) = Nλ. Now for the centre most fringe, θ = 0, cos(θ) = 1, 2 d = Nλ therefore λ = 2 d/n. From our data, d = 10µm, N(mean) = 31.6. λ = 2 10 5 /31.6 = 632.9nm. This is very close to the true value of 632nm. 4.4.1 Uncertainty There was an uncertainty of ±1 in N at start and end of each trial giving total uncertainty of ±2 in N. Uncertainty in d is same as minimum resolution of DC motor (29 nm). Uncertainity in λ: λ 2 = (2/N) 2 u( d) 2 + ( 2 d/n 2 ) 2 u(n) 2 = (2/31.6) 2 (29 10 9 ) 2 + (2 10 5 /31.6 2 ) 2 (2) 2 = 3.37 10 18 + 1.605 10 15 = 4.01 10 8 λ = (632 ± 40) 10 9 m. 5 Measuring Refractive Index of glass 5.1 Theoretical Background 5.1.1 Optical path difference The optical path traveled in presence of glass sheet is n x where n is the refractive index of glass and x is the length of path travelled inside glass. From figure 2, xcos(r) = t where r is the angle of refraction. x = t/cos(r) nx = nt/cos(r). 4
i ٩o-i d x r t Figure 2: Optical path difference. The optical path traveled without glass sheet is d. From figure 2, sin(90 i) = t/d. d = t/sin(90 i). Hence optical path difference = nx d = nt/cos(r) t/sin(90 i). optical path difference = nt/cos(r) t/cos(i). 5.1.2 Derivation of formula for refractive index. O i r a t f d e b c P Figure 3: Change in path due to rotation. The glass sheet is rotated as shown, Before rotation, Optical path between a and b = ab + bc = nt + bc cos(i) = t/ac, 5
ac = t/ cos(i). bc = ac t = t/ cos(i) t. optical path = nt + t/ cos(i) t. After rotation, Optical path = nad + de cos(r) = t/ad, ad = t/ cos(r) sin(i) = de/dc, de = dc sin(i). = (fc fd) sin(i) = (t tan(i) t tan(r))sin(i). Change in optical path = 2[(adn + de) (nt + bc)] = Nλ nt/cos(i) + t tan(i)sin(i) t tan(r)sin(i) nt t/cos(i) + t = Nλ /2. Using Snell s Law, n(2t(1 cos(i)) Nλ) = (2t Nλ)(1 cosλ) + n 2 λ 2 /4t n = (2t Nλ)(1 cos(i))/(2t(1 cos(i)) Nλ) 5.1.3 Theoretical relation between N and θ n = (2t Nλ)(1 cosθ/(2t(1 cosθ) Nλ). Taking t = 1mm, n = 1.5, 3 t(1 cosθ) 1.5Nλ = (2t Nλ)(1 cosθ) t(1 cos(θ)) = N(1.5λ λ(1 cosθ)) N = t(1 cosθ)/(1.5λ λ(1 cosθ)) When we plot this in matlab, we get a quadratic curve. 5.2 Procedure First, we place the glass slide horizontally in front on M1. Then the rotation platform on which glass slide was mounted was rotated by hand. Care was taken to ensure no weight was placed on the desk as it caused fringe pattern to be destroyed. The zero degree mark in front of rotation platform was used as reference point. We counted the no. of fringes that disappeared once again using centre as reference point. After every 20 fringes that disappeared, the degree of rotation was noted and the counting procedure continued upto 160 fringes. This experiment was repeated 5 times using the same mark on the dial as starting point. The results are given below. Next, the micrometer screw gauge was used to calculate the thickness of glass, t, at the point where the laser beam passed the glass. The calculated value was 0.99µm. t = 1.15mm 0.16mm = 0.99mm. 0.16mm is the zero error. 6
No of Fringes N θ rotated( o ) θ rotated( o ) θ rotated( o ) θ rotated( o ) θ rotated( o ) 020 10 11 11 11 11 040 15 16 16 15.5 15.5 060 18 19.5 18.5 19 19 080 21 22.5 22 22.5 22 100 24 25 25 24.5 24.5 120 26 27 27.5 26.5 26.5 140 28 29 29.5 28.5 28.5 160 30 31 31 30.5 30.5 Table 2: degrees of rotation for every 20 fringes moved. 250 data 1 quadratic data 2 200 150 100 50 0 0 5 10 15 20 25 30 35 No of Degrees Figure 4: Experimental result(blue) and theoretical prediction(red). 7
5.3 Analysis The result obtained is a quadratic curve of N(y-axis) vs θ(x-axis). Theoretical relation between N and θ also predicts a quadratic curve. Hence the result verifies the relationship. Inserting the average values of N and theta for the 5 trials in n = (2t Nλ)(1 cosθ)/(2t(1 cosθ) Nλ). N (no.of fringes) n (Refractive Index) 020 1.52 040 1.52 060 1.51 080 1.50 100 1.50 120 1.51 140 1.51 160 1.51 Table 3: Refractive index for different values of N Finally we average n to calculate mean refractive index which comes out to be n = 1.51 which is very close to true value of 1.50 Note: We found n and then averaged. If we had found average N and λ and then inserted to find n, our answer would have been incorrect because N and λ are not linearly related. 5.3.1 Uncertainty in n due to N and θ n 2 = ( θ) 2 ( (2t(1 cosθ) Nλ)(2t Nλ)sinθ (2t Nλ)(1 cosθ)2tsinθ (2t(1 cosθ) Nλ) 2 ) 2 + ( N) 2 ( (2t(1 cosθ) Nλ)(1 cosθ)( λ) (2t Nλ)(1 cosθ)( λ) (2t(1 cosθ) Nλ) 2 ) 2 n 2 = (1) 2 ((2t Nλ)((2t(1 cos θ) Nλ)sinλ (1 cos θ)2tsinθ)) ( ) 2 + (2t(1 cos θ) Nλ) 2 (2) 2 ( (λ(1 cos θ)(nλ 2t(1 cos θ)+(2t Nλ))) (2t(1 cos θ) Nλ) 2 ) 2 n 2 = ( ((2t Nλ)( 2tsinθNλ)) ) 2 (λ(1 cos θ)(2t cos θ)) + 4( ) 2 (2t(1 cos θ) Nλ) 2 (2t(1 cos θ) Nλ) 2 8
Plugging in the values of N in the above equation, N (no of fringes) n(uncertainty in n) 020 0.08 040 0.07 060 0.03 080 0.08 100 0.03 120 0.02 140 0.01 160 0.01 Taking average, n = 0.04. n = 1.51 ± 0.04 6 Conclusion Precautions: The Michelson Interferometer is an extremely sensitive experiment. Several precautions were taken while performing the experiment: 1) While taking readings, no vibrations or noise in nearby surrounding had to be ensured. The fan had to be switched off. 2) The fringes could appear or disappear as a result of movement of steel post holding the glass as well. It had to be ensured that steel post was firmly clamped to the rotation platform. 3) The starting point of all readings was take to be the same for each trial. 4) We had to make sure that all lens, glasses were clean. 5) Direct eye contact with the laser was avoided for personal safety. Care was taken while performing optical alignment to ensure that ray did not deflect in arbitrary direction. Suggested Improvements: The experiment can be further improved by: 1) Use computer software to generate rotation as well so as to minimize uncertainty in its value. 2) Increase optical path by causing beam to move multiple times between beam splitter and mirror before falling on the screen. 3) Use compensation mirror. This would reduce the error that occurs because 1 beam passes beam splitter once while the other beam passes beam splitter thrice. 4) Conduct experiment in dark environment in mercury bath( as the original experiment in 1887). 9
Other Uses of Michelson Interferometer: 1) To measure wavelength components in non-monochromatic light eg white light 2) Measure the affect of pressure of a gas on its refractive index 3) Search for gravitational waves (LIGO) 4) Find refractive index of any material previously unknown. 5) To measure thermal expansion co-efficient of a substance.. 7 References http://physlab.lums.edu.pk/images/c/c5/michelson final1.pdf. http://physlab.lums.edu.pk/images/1/1d/errorsv3.pdf. http://physlab.lums.edu.pk/images/5/50/optics Hecht.pdf. http://physlab.lums.edu.pk/images/e/ed/refractive index reference Book.pdf. http://physlab.lums.edu.pk/images/d/dc/lab Michelson.pdf. http://physlab.lums.edu.pk/images/c/c7/michelsonpaper.pdf. 10