The Michelson Interferometer
The Michelson Interferometer
Introduction
An interferometer is a device that can be used to measure lengths or changes in length with great accuracy by means of interference fringes. In this experiment, it will be used to measure the wavelength of a HeNe laser and of sodium light and to measure the difference in wavelength between the two components of the D line.
The Michelson interferometer is a versatile instrument of great historical importance. One of these instruments, built with the greatest care and having very long 11-meter paths, was used by Michelson and Morley in 1887 to test for the presence of the luminiferous ether. It was thought that electromagnetic waves required a medium (ether) in order to propagate. The results of their experiment were negative, corroborating the second postulate of Einstein's special theory of relativity.
The interferometer was later used by Michelson to measure the length of the standard meter - the distance between two fine scratches on a certain metal bar preserved at Sevres, France. He showed that the standard meter was equivalent to 1,553,163.5 wavelengths of a certain monochromatic red light emitted from a cadmium light source. For this careful measurement, Michelson received the 1907 Nobel Prize in physics.
Theory
Figure 1 shows the basic design of the interferometer. The beam splitter is a glass plate that is half silvered so that light from the source splits at the first surface. Half of the incoming beam is transmitted to the mirror M1 (passing through the glass compensator plate on the way) and the other half is reflected toward the mirror M2. Mirrors M1 and M2 reflect the light back to the beam splitter, and half of each beam reaches viewing screen, the remainder being directed back to the source and lost. Mirror M2, mounted on a carriage which slides on a track, can be translated toward or away from the observer by means of a precision micrometer. So the displacement of mirror M2 can be measured accurately by reading the micrometer.
Figure 1. The Michelson Interferometer
The original beam of light has now been split and portions of the resulting beams brought back together. Since the beams are from the same source, they are highly correlated. When a lens is placed between the laser source and the beam splitter, the light ray spreads out, and an interference pattern of dark and bright rings or fringes is seen on the viewing screen.
The interference pattern occurs because of the phase difference between the two beams when they reach the viewing screen. When they were initially split, they were in phase. The phase difference arises from the fact that the beams traveled different optical paths before reaching the screen.
Moving M2 varies the path length of one of the beams. Since the beam traverses the path between M2 and the beam splitter twice, moving M2/4 nearer the beam splitter reduces the optical path of that beam by /2. This will change the interference pattern; the radii of the maxima will be reduced so that they now occupy the position of the former minima. If M2 is moved an additional distance of /4 closer to the beam splitter, the maxima and minima will again trade positions with the new arrangement indistinguishable from the original pattern.
By slowly moving the mirror a measured distance dm and counting m, the number of times the fringe pattern is restored to its original state, the wavelength
of the light can be calculated as
[The compensator plate is needed if white light fringes are to be obtained. Note that the light rays going to mirror M2 traverse the beam-splitter three times before reaching the observer, whereas the rays going to mirror M1 traverse it only once. In order to achieve exact equality of path through glass, therefore, the compensator plate of exactly the same thickness as the beam splitter is added. Generally, the compensator plate is not needed when the source is a laser.]
Procedure
A. Measurement of Laser Wavelength.
Adjust the laser beam so that it is approximately parallel with the top of the base. The beam should strike the center of the movable mirror (M1) and be reflected back into the laser aperture. Attach the viewing screen to its magnetic backing.
Position the beam-splitter at a 45-degree angle to the laser beam, within the crop marks, so that the beam is reflected to the adjustable mirror (M2). Adjust the angle of the beam-splitter so that the reflected beam hits M2 near its center.
There should now be two sets of bright dots on the viewing screen; one set comes from M1 and the other from M2 . Each set of dots should include a bright dot with two or more dots of lesser brightness due to multiple reflections. Adjust the angle of the beam-splitter again until the two sets of dots are as close together as possible, then tighten the thumbscrew to secure the beam-splitter. Using the thumbscrews on the back of M2, adjust the mirror’s tilt until the two sets of dots on the viewing screen coincide.
Attach the 18-mm FL lens to the magnetic backing of the component holder in front of the laser and adjust its position until the diverging beam is centered on the beam-splitter. You should now see circular fringes on the viewing screen. If not, carefully adjust the tilt of M2 until the fringes appear. (One can also remove the viewing screen and project the fringe pattern onto a wall if the setup is in a location at which this would be convenient.)
When the interference pattern is clearly visible on your viewing screen, or wall, adjust the micrometer knob to a medium reading (approximately 50 m). Turn the micrometer knob one full turn counterclockwise. Continue turning counter-clockwise until the zero on the knob is aligned with the index mark. Record the micrometer reading.
Adjust the position of the viewing screen so that one of the marks on the millimeter scale is aligned with one of the fringes in the interference pattern. It will be easier to count the fringes if the reference mark is one or two fringes from the center of the pattern.
Rotate the micrometer knob slowly counter-clockwise, counting the fringes as they pass the reference mark. Record the micrometer reading for every tenth fringe up to the 190th. Tabulate your results as indicated in Table 1.
Using Equation (1), calculate the wavelength of the laser light and its error . Notice that Table 1 is set up so that for each row m = 100. Remember that each small division on the micrometer knob corresponds to one m (106 m) of mirror movement. Determine your percentage error relative to the accepted value = 632.80.1 nm.
- Wavelength of Sodium D Lines.
When the sodium source is used, it is important that the optical paths of the two interfering beams should be nearly equal. To ensure that this is the case, set up the interferometer with the laser, mount the compensator in the appropriate position, and adjust the movable mirror until the fewest possible fringes appear on the screen. Then remove the viewing screen and replace the laser with the Na light source. At this point, view the interference pattern by looking back through the beam-splitter toward M2.
Using the same procedure followed in Part A to determine the wavelength of the laser, determine the average wavelength of the Sodium D line and the error . It will be observed that there will appear to be just one yellow line but that it will change in intensity as the pattern fringes pass the reference line. This change in intensity will be treated in Part C. Since the observed line is actually the sum of two closely spaced spectral lines, the value for wavelength will be the average value of the two sodium D lines.
C. Measurement of Wavelength Difference
The wavelength difference between two close lines such as the components of the sodium D line is determined from their average wavelength and the visibility of the fringes. At certain positions of mirror M1, it is found that the fringes are clear and sharp whereas at intermediate positions, they are very indistinct. The reason is that there are two sets of fringes corresponding to two slightly different wavelengths ( and ) and at some positions, the two sets are in step so the overall fringe pattern is sharp (max in Figure 2). At the intermediate position
Figure 2. Fringe Visibility with Two Wavelengths.
the two sets overlap and wash out the overall pattern (min in Figure 2). The separation of the positions of maximum (or minimum) visibility of the fringe pattern determines the wavelength difference. Let da represent a point where the two sets of fringes are in step (max). Then
1(2)
Let the mirror M1 be translated through a fringe
visibility minimum to the next fringe visibility maximum and call this position db. Then the longer wavelength will have given rise to (ma - mb) fringes and the shorter wavelength to one more fringe than the longer one. That is,
2(3)
The wavelength difference d is given by
3(4)
Since the shorter wavelength gives rise to one
more fringe than the longer one, it is true that
4 (5)
By combining equations 25, it can be shown that
5 (6)
Equation 6 gives the wavelength difference of two nearly equal wavelengths in terms of the distance between two successive visibility maxima (or minima). The exact positions of the maxima or minima are, however, difficult to determine. The visibility of the fringes does not go to zero between the maxima unless the two spectral lines are of equal intensity. If they are, then the overlapping fringe systems completely wash each other out so that the field of view is uniformly illuminated. In the case of sodium, one of the D lines is about twice the brightness of the other so that the minima, though reasonably apparent, are not zero.
In order to obtain a good value of the wavelength difference, you should make 10 measurements of the distance between adjacent minima. Begin at a position where the path difference is rather small and move the mirror through a minima merely observing the field without recording data. Return the
moveable mirror back beyond the starting point. Now go back and forth between two adjacent minima 10 times, recording your results.
From the average value, calculate d and compare it to the standard value. In Equation 6, use the known values for the sodium D line: = 588.995 nm, =
589.592 nm. Also, calculate the error .
The Michelson Interferometer
ITable I. Data and Results for Parts A and B.
Fringe number / Interferometer reading / Fringe number / Interferometer reading / Difference / Deviation0 / 100
10 / 110
20 / 120
30 / 130
40 / 140
50 / 150
60 / 160
70 / 170
80 / 180
90 / 190
Sample for Data Table in Report
The Michelson Interferometer