The Michelson Interferometer

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1 Experiment #33 The Michelson Interferometer References 1. Your first year physics textbook. 2. Hecht, Optics, Addison Wesley - Chapter 9 in the 4th Ed. (2001). 3. Jenkins and White, Fundamentals of Optics 3rd Ed. (1957), McGraw-Hill - Chapter 13. (QC355 J4) 4. Instruction Manual, Beck Interferometer M300/6407, R.&J. Beck Ltd. (pdf available on course website). Introduction The Michelson interferometer is the best known example of a class of interferometers that are known as amplitude-splitting interferometers, that is they produce interference by means of division of the amplitude of incident light by means of arrangements of mirrors and beamsplitters. Michelson developed this type of interferometer in order to investigate questions about the wave nature of light, specifically whether the existence of a supporting medium, known as the aether, was required for light propagation. The Michelson interferometer has also been used to provide evidence for the special theory of relativity, to detect and measure hyperfine structure in line spectra, to measure the tidal effect of the moon on the earth and to provide a substitute standard for the meter in terms of wavelengths of light. In this experiment, you will use the Michelson interferometer to measure the wavelengths of the two main lines of the sodium spectrum and the refractive index of air. The basic elements of a Michelson interferometer are shown in Fig. 1. In this diagram, refraction is ignored and anti-parallel rays are slightly displaced for the sake of clarity. Light from the source S is incident at 45 on a half-silvered rear-surface M 3 of the beam splitter P 1 at point A. The transmitted ray x passes through the compensator P 2, reflects from the stationary mirror M 1 and returns through the compensator to point A where it reflects to the observer O. Meanwhile, the reflected ray y traverses P 1, reflects from movable mirror M 2, passes through P 1 and recombines with ray x as it goes to observer O. The optical compensator P 2 ensures that both beams travel the same distance in glass. Both rays then interfere at the observation point with a path difference d defined by the equation 2 d = 2 AB AC, (1) which results in a phase difference φ o given by φ o = k r = 2π λ 2 d, (2)

2 M 2 C y M 3 S A x B P 1 P 2 M 1 O Figure 1: The Michelson interferometer. where k is the wavevector of the light. Ray x undergoes two external reflections (at M 3 and M 1 ) whereas y undergoes only one (at M 2 ) so an additional phase difference φ r = π is introduced for a total φ = φ o + φ r (3) = 2π λ 2 d π (4) The condition for constructive interference is that the total phase difference φ be an integral multiple of 2 π i.e. ( 2 d = m + 1 ) λ (5) 2 where m = 0, 1, 2,... If the mirror M 2 is moved a distance d then the intensity of light detected at O will increase and decrease as successive interference maxima and minima are experienced. Counting the number m of consecutive bright or dark patterns allows the change in path length d to be calculated as 2 d = λ m. (6) It is important to note that the field of view will only alternate between completely bright and completely dark if the incident light is collimated, i.e all rays are parallel. It is more likely that you will be using an uncollimated or extended source. In this case, concentric rings or fringes will be observed, as shown in Fig. 2. In this figure, the real mirror M 1 has been replaced by its virtual image M 1 formed by reflection in P 1. M 1 is parallel to M 2. The extended source S, now behind the observer, forms two virtual sources S 1 and S 2 by

3 P 2d 2d cos P P P P M 1 M 2 d 2d Figure 2: Michelson interferometer using an extended source. reflection in the mirrors M 1 and M 2. Note that these virtual sources are coherent since the phase difference between corresponding points in the two is the same at all times. When the separation of the mirrors M 1 and M 2 is d, the optical path differences between the parallel rays coming to the observer from the corresponding points P and P is 2 d cos θ. Taking into account the phase changes occurring upon reflection, the condition for constructive interferences becomes 2 d cos θ = m λ. (7) In this experiment, you will use a Michelson interferometer made by R.&J. Beck Ltd. In the Beck apparatus, M 2 of the Michelson optics is mounted on a carriage that is driven by a micrometer. This controls the length of the AC arm of the interferometer and hence the path difference d. The instrument is calibrated by measuring the relationship between d and the change in micrometer reading Prelab Questions K = Change in micrometer reading Distance of carriage movement 1. Show schematically how an extended source implies circular fringes. 2. Show how Eq. 7 implies circular fringes. 3. It is possible to view interference fringes from a white light source if the path length difference is very small. When viewing white light fringes, which colour should appear on the outside of the circular fringes and which colour on the inside? 4. In Part 3 of the experiment, you are asked to determine the mean wavelength and the separation of the sodium doublet. There are now two sets of fringes, one for each line (8)

4 of the doublet. You will see fringes due to both when they constructively interfere, and the fringe pattern will disappear when they destructively interfere. Once you have determined K it should be obvious how you determine λ. (How will you do this?) Determination of λ is a little more subtle. Show that the mirror distance between two positions where the fringes disappear is given by Caution 2 (d 1 d 2 ) = λ 1λ 2 λ 1 λ 2. (9) [Hint: consider the conditions for interference for λ 1 and λ 2 at the two positions.] Two points of caution are worth noting: ˆ The optical surfaces of the interferometer must never be touched with fingers or any other object. (With the possible exception of camel s or squirrel s hair brushes.) ˆ It is possible to damage your eye by looking directly at the UV emission of the mercury vapour lamp. Always use a glass shield or filter and keep the metal housing around the bulb whenever it is on. Apparatus ˆ Beck Model No. M300/6407 Michelson interferometer ˆ sodium, mercury and incandescent light sources ˆ ground glass plate ˆ vacuum cell and vacuum pump Experiment 1. Investigate the operation of the apparatus. Obtain an interference pattern using the sodium light source, which consists mainly of two yellow wavelengths (known as the sodium doublet). Make the distance AM 1 and AM 2 roughly equal and use a sighting pin during the initial mirror tilt adjustments. The fringes should be circular and centred, rather thick and of good contrast. (Please look at the figure in the Beck Manual.) 2. Determine K. Using the mercury light source (green), calibrate the carriage movement. Assume the wavelength of the green mercury light to be nm. 3. Determine the mean wavelength of the sodium doublet and the separation of the two lines.

5 4. Determine the refractive index of air. Measure the difference in optical pathlength that occurs as air is removed from the vacuum cell using the vacuum pump provided. 5. (Optional) White light fringes can be observed with the Michelson interferometer when the optical path difference of the interfering beams is nearly zero. Observe the striking colour changes in the pattern as the path difference is slowly varied from zero. (Ref. 4, and Appendix to this handout.) UG2/2007

INTERFEROMETERS. There are 4 principal types of measurements that can be made with this type of interferometer.

INTERFEROMETERS. There are 4 principal types of measurements that can be made with this type of interferometer. INTERFEROMETERS NOTE: Most mirrors in the apparatus are front surface aluminized. Do not touch the surfaces nor wipe them. they can be easily permanently damaged. Introduction This experiment provides