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Physics 263
Experiment 8
Michelson Interferometer
1 Introduction
The Michelson interferometer uses light interference to measure distances in units of the wavelength of light from a particular source. It was developed by Albert Michelson and used, in 1893, to measure the standard meter in units of the wavelength of the red line in the cadmium spectrum. It is also known for its use in demonstrating the non-existence of an electromagnetic wave-carrying “aether”. Contemporary uses include precision mechanical measurements and fourier transform spectroscopy. In this laboratory, we will use a Michelson interferometer to (a) measure the wavelength of light from a Ne-He laser, (b) measure the index of refraction of air, and (c) study the interference of linearly polarized light beams.
1.1 How the Interferometer Works
A simplified version of the Michelson interferometer is shown in Figure 1. The basic idea is to split a beam of light into two beams; delay one with respect to the other, and then recombine them to observe their interference.
Observation Screen
Light
Source
mirror
Half−silvered
Mirror 1
Mirror 2
Figure 1:
Light from a monochromatic source is directed at a “half-silvered” mirror i.e. a mirror with a very thin metallic coating. Approximately half the light intensity is reflected down
to Mirror 1 and half transmitted, so it strikes Mirror 2. The light reflected by these mirrors goes back to the half-silvered mirror, and half the intensity of each beam then goes upward to an observation device, in our case just a screen. If the light source is a point source, we can easily find its images made by the com- bination of mirrors. This is shown in Figure 2. The image of a point source is located “behind” the mirror, at the same perpendicular distance as the object.
Light
Source
Mirror 1
Mirror 2
A
B
C
D
D’
C’
d
2d
image of source
image of source
image of C’
image of A image of C
Figure 2: Interferometer schematic. The simplifications are: a point source of light, and a very thin half-silvered mirror.
In this case, A is the image of the source made by the half-silvered mirror. B is the image of A made my Mirror 1. C is the image of the source, for rays transmitted by the half-silvered mirror, made by Mirror 2, and D is the image of C made by the half-silvered mirror. If the distances between the half-silvered mirror and Mirror 1 and between the half- silvered mirror and Mirror 2 are the same, then images B and D overlap.
Figure 4:
- Next, remove the beam-splitter mirror with a screwdriver. Turn on the laser beam, and adjust its position to direct the beam straight into the center of the the “mov- able” mirror (Mirror 2). The beam should be reflected right back into the laser. A good way to find the beam is with a small piece of paper, or an index card.
- Then restore the beam-splitter mirror and look at the observation screen. You may see two sets of small spots, one from each of the mirrors. In each set there is a bright spot, and several less-bright ones, which are produced by multiple reflections.
- Use the adjusting screws on the “fixed” mirror (Mirror 1) to make the bright spots overlap each other. As you are doing that, the overlaping spots will twinkle, due to the interference of the beams.
- Finally, restore the lens. Slide it around until you can see the ring pattern on the observation screen. It is OK to fiddle a bit the the laser orientation, if necessary.
3.2 Measuring Light Wavelength
- Turn the microcrometer either clockwise or anti-clockwise for about 1 rotation. Then turn it, in the same direction, enough to see N fringes pass some point on the screen. N should be at least 20. Measure the number of ticks on the micrometer which correspond to the N fringes. Each tick is 1 micron (10−^6 m). This corresponds to d in Equation 1. The wavelength is then
λ = 2d/N
- Compare this with the known wavelength of 633 nm.
3.3 Measuring the Index of Refraction of Air
The index of refraction of a medium is
n = c/v
For a vacuum, v = c, so n is exactly 1. For gases, the index of refraction increases linearly with pressure, as shown in Figure 5.
1
0 20000 40000 60000 80000 100000
Index of Refraction
Gas Pressure, kPa
n(P)
Figure 5:
We can express this as n − 1 = P
dn dP
We then simply have to find the rate of change of the n with respect to pressure, P. To do this, mount the air cell in front of Mirror 2, as shown in Figure 6, and connect it to the hand vacuum pump with the tubing. Then slowly pump the air out of the cell, counting the number of fringe transitions as this is done. It is sufficient to reduce the pressure down to about 50kP a. Record the file pressure, and find ∆P , using 101kP a as the starting pressure. Now the light wavelength in a medium of index of refraction n is
λ =
λ 0 n
where λ 0 is the wavelength in vacuum. Recall that when a wave traverses a distance x, its phase changes by
∆φ =
2 πx λ
In this case, if the thickness of the cell is t, since the light travels through it twice, the phase shift for constant n would be
∆φ =
2 π 2 t λ 0 /n
4 πnt λ 0
So, with the cell in place, with atmospheric pressure, we have a certain fringe pattern. When the air is pumped out, there are N fringe transitions, one for each phase shift of
