8.3 The Index of Refraction of Air

In the Michelson interferometer, the characteristics of the fringe pattern depend on the phase relationship between the two interfering beams. There are two ways to change this phase relationship: one way is to change the distance travelled by one or both beams (by moving the movable mirror, for example), another is to change the medium through which one or both of the beams pass. In this part of the experiment, you will use the second method to measure the index of refraction of air.

For light of a specific frequency, the wavelength varies according to the formula:

where is the wavelength of the light in a vacuum and n is the index of refraction for the material in which the light is propogating. For reasonably low pressures, the index of refraction for a gas varies linearly with the gas pressure. Of course, for a vacuum, where the pressure is zero, the index of refractin is exactly 1. By experimentally determing the slope, the index of refraction of air can be determined at various pressures.

1. Align the laser and interferometer in Michelson mode.
2. Place the rotational pointer between the movable mirror and the beam splitter. Attach the vacuum cell to its magnetic backing and push the air hose of the vacuum pump over the air outlet hole of the cell. Adjust the alignment of the fixed mirror as needed so the center of the interference patternis clearly visible on the viewing screen. The fringe pattern will be somewhat distorted by irregularities in the glass end-plates of the vacuum cell, but this is not a problem.
3. For accurate measurements, the end-plates of the vacuum cell must be perpendicular to the laser beam. Rotate the cell and observe the fringes. Based on your observations, how can you be sure that the vacuum cell is properly aligned?
4. Evacuate the vacuum cell and record the initial pressure in the cell .
5. Slowly let air back into the cell by flipping the vacuum release toggle switch. Record m, the number of fringe transitions that occur, and P , the final cell pressure.

As the laser beam passes back and forth between the beam splitter and the movable mirror, it passes twice through the vacuum cell. Outside the cell the optical path lengths of the two interferometer beams do not change throughout the experiment. Inside the cell, however, the wavelength of the light gets longer as the pressure is reduced.

Suppose that originally the cell length d was 10 wavelengths long (of course, it's much longer). As you pump out the cell, the wavelength increases until, at some point, the cell is only 9.5 wavelengths long. Since the laser beam passes twice through the cell, the light now goes though one less oscillation within the cell. This has the same effect on the interference pattern as when the movable mirror is moved toward the beam splitter by 1/2 wavelength. A single fringe transition will have occured.

Originally there are wavelengths of light within the cell (counting both passes of the laser beam). At the final pressure there are wavelengths within the cell. The difference between these values: is just m, the number of fringes you counted as you evacuated the cell. Therefore ; so that . The slope of the n vs. pressure graph is therefore:

where is the initial (final) air pressure, is the index of refraction of air at pressure , m is the number of fringe transitions counted, is the wavelength of the laser light in vacuum (see the lab T.A. or use the value measured in the first part of the experiment for the laser wavelength in air.), and d is the length of the vacuum cell (3.0 cm).

1. Graph the index of refraction n vs. pressure for air and calculate the slope of the graph.
2. Find the index of refraction of air at a pressure of 1 atmosphere.