Michelson Interferometer

1- Michelson Interferometer Abstract: We can use The Michelson Interferometer to find the wavelength of light waves. It uses the interference of light and the path difference resulted from the beam splitter and the reflection to find the wavelength. Introduction: The Michelson Interferometer was introduced by Albert Michelson. It is a amplitude-splitting interferometer, meaning that it uses a beam splitter that allows part of the waves to be reflected and the other to be refracted and pass through. It plays an important role in the development of modern physics. The device is as follows: First, light is emitted from the source (S) which is usually a laser source. The beam of ray (1) then hits the beam splitter. It hits the beam splitter and half of the beam gets reflected into beam (2) and the other half gets refracted into beam (3). Beam (3) gets reflected by M1. As for beam (2), a compensator plate C made of the same material and dimensions as BS is inserted parallel to BS in its path to compensate for the refraction. Beam (2) will get reflected by M2. Both the beams will then reach the beam splitter. (2) will get transmitted and (3) will get reflected. Both rays then will interfere and leave the interferometer as beam (4). The mirrors M1 and M2 are perpendicular. This means that the image of M1 (M1’) resulting from the beam splitter is going to be parallel to M 2. Also, one of the mirrors can be moved towards or away from the beam splitter. The distance d between M2 and the M1’ determines the path difference between beam 1 and 2. If the optical path difference was an integer multiple of the wavelength, then the interference is constructive. 𝑝𝑎𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = ∆𝑝 = 2𝑑 𝑐𝑜𝑠(𝜃) where the angle 𝜃 measures the inclination of the beams relative to the optical axis. For normally incident rays 𝜃 = 0. The net optical path difference is ∆= ∆𝑝 + ∆𝑟 where a relative phase difference of 𝜋 occurs. 𝜆 1 ∆= 2𝑑 𝑐𝑜𝑠(𝜃) + = (𝑚 + )𝜆 2 2 Or we say: 2𝑑 𝑐𝑜𝑠(𝜃) = 𝑚𝜆 𝑚 = 0,1,2, . .. (dark fringes) Since we focus on the normally incident rays, 𝑐𝑜𝑠(𝜃) = 1 and we can conclude that: 2𝑑 = 𝑚𝜆 If we move the mirror so that we change m, we can get that: ∆𝑑 𝜆 = ∆𝑚 2 Or 𝜆=2 ∆𝑑 ∆𝑚 Procedure: 1- Set up your Interferometer and use the light source you want to measure. 2- Adjust the mirror so that the central fringe is bright. Take the reading on the rotating disc for do. 3- Adjust the mirror using the disc so that a number of fringes disappear. Take the reading on the disc. Repeat 10 times. 4- Graph a d – m graph. Take the best fit line and calculate the slope. 5- Use the slope to calculate the wavelength. Data: No. of fringes m d-do(𝜇m) 17 0.3 29 0.7 41 0.9 53 1.5 65 1.7 77 2.1 89 2.2 101 2.4 113 3.0 125 3.3 ∆𝑑 (3.25 − 1.75) × 10−6 𝑠𝑙𝑜𝑝𝑒 = = = 2.609 × 10−8 𝑚 ∆𝑚 125 − 67.5 𝜆 = 2 𝑠𝑙𝑜𝑝𝑒 = 2 × 2.609 × 10−8 = 52.18 𝑛𝑚 Slope Error: 3.375 × 10−6 − 0 𝑠𝑙𝑜𝑝𝑒 𝑢𝑝 = = 2.81 × 10−8 𝑚 120 − 0 1.875 × 10−6 − 0 𝑠𝑙𝑜𝑝𝑒 𝑑𝑜𝑤𝑛 = 2.39 × 10−8 𝑚 78.5 − 0 ∆𝑠𝑙𝑜𝑝𝑒 = 𝑠𝑙𝑜𝑝𝑒 𝑢𝑝 − 𝑠𝑙𝑜𝑝𝑒 𝑑𝑜𝑤𝑛 = 0.21 × 10−8 𝑚 2 𝑠𝑙𝑜𝑝𝑒 𝑢𝑝 + 𝑠𝑙𝑜𝑝𝑒 𝑑𝑜𝑤𝑛 = 2.6 × 10−8 𝑚 2 𝑠𝑙𝑜𝑝𝑒 ′ ± ∆𝑠𝑙𝑜𝑝𝑒 = 2.6 × 10−8 ± 0.21 × 10−8 𝑚 𝑠𝑙𝑜𝑝𝑒 ′ = 𝑠𝑙𝑜𝑝𝑒 ≅1 𝑠𝑙𝑜𝑝𝑒′ Error: (𝜆 − 𝜆′)2 m d(μm) 𝜆 (𝑛𝑚) 𝜆 − 𝜆′(𝑛𝑚) 17 0.3 35.3 -14.07 29 0.7 48.3 -1.07 1.14 41 0.9 43.9 -5.47 29.9 53 1.5 56.6 7.23 52.3 65 1.7 52.3 2.93 8.58 77 2.1 54.5 5.13 26.3 89 2.2 49.4 0.03 0.0009 101 2.4 47.5 -1.87 3.50 113 3.0 53.1 3.73 13.9 125 3.3 52.8 3.43 11.8 𝜆′ = 49.37 𝑛𝑚 2 ∑((𝜆 − 𝜆′ ) ) ∆𝜆 = √ = 1.96 𝑛𝑚 𝑁(𝑁 − 1) 𝜆′ ± ∆𝜆 = 49.37 ± 1.96 𝑛𝑚 Sources of error: 1- Accuracy in measurement tools. 2- Human error. Not reading the measurement accurately. 3- Lack of accuracy in counting the fringes. 4- Approximation. × 10−18 198 Discussion and Conclusion: We were able to find the wavelength of the source. The conclusion we reach: 1- Beam splitters are optical devices that splits light into two. A part gets transmitted mostly into the refractive medium and a part gets refracted. Beam splitters split light into 2 halves. 2- Since Part of the rays get reflected before passing through any refractive medium, the compensator plate is added to cover up for the resulted path difference. 3- We can change the distance d to change the pattern on the screen as fringes get smaller and later disappear. 4- This is an example on fringes of equal inclination. 5- We can use a convex lens with a small focal length to diffuse the rays before it reaches the beam splitter. 6- We can use Michelson interferometer to find the wavelength of light or the refractive index of the material the beam splitter (and thus the compensation plate) 7- The relationship between m and d is linear.