Introduction

Albert Michelson was born in Prussia in 1852 and at the age of two his family emigrated to the United States where he received an education and was appointed by President Grant to the U.S. Navel Academy. After teaching physics and chemistry at the Academy, Michelson took leave from the Navy to study in Europe and later to take positions at Universities in the United States. Michelson’s main point of study was in optics. He made careful measurements of the speed of light and with his invention, an optical interferometer, he was able to show that light travels at the same velocity in all inertial reference frames. Michelson’s interferometer also had the ability to measure distances in great detail, so great that he was requested by International Committee of Weights and Measures to measure the standard meter, which he did in terms of the wavelength of cadmium light¹.

Theory

Michelson’s Interferometer works by splitting a beam of light into two beams, allowing them to travel different distances, and then recombining the two beams so one can observe the interference of the two beams. Figure 1 is a simple representation of Michelson’s Interferometer.

When a beam of sodium light is emitted into Michelson’s Interferometer it must strike a beam splitter, a partially silvered mirror that allows half of the beam to pass through and the other half to reflect. Each beam is then rebounded by a mirror and sent back to the beam splitter where they are recombined². The two mirrors are set at different distances in order to cause an interference between the two beams of sodium light. An observer can then view the interference of the two beams of light as fringes.

When the electrons in a sodium vapor are excited they emit a pinkish-yellow glow. This can be seen commonly in street lamps². The glow is caused by the electrons in the sodium vapor being excited to the 3p energy level. When the electron decay back to the 3s energy level they emit a photon with a wavelength of 589 nm.

It turns out that there is more to the glow of sodium vapor than meets the eye. Electrons can experience a spin angular momentum called spin-up and spin-down. Sometimes the electron in the 3s energy level will be excited to the 3p energy level with a spin-up angular momentum and sometimes with will be excited to the 3p energy level with a spin-down angular momentum³. The energy of each is very close and can easily be overlooked, but with care, the photons emitted by each can be measured through Michelson’s Interferometer.

Method and Experiment Details

In the Michelson interferometer used for this experiment, mirror 1 included was equipped with two screws to ensure that both mirrors in the interferometer were parallel and mirror 2 included a micrometer adjustment for changing the distance of the mirror. As the light from the sodium vapor lamp reached the detector – in this experiment, a human eye – the screws on the adjustable mirror were turned slightly until a fringe pattern like the one in Figure 2 could be seen. When changing the distance of mirror 1 with the micrometer, the detector could observe the fringes slowly decreasing and increasing in intensity. The fringes themselves are caused by the interference of the wavelengths as they recombine at the detector, but the periodic change in the intensity of the fringes is due to something entirely different; the change in intensity is a beat frequency caused by two very close frequencies of light known as the sodium doublet.

The beat frequency was easily measured through careful inspection of the number of fringes per beat length measured on the micrometer. To find this, the width of a fringe had to be determined by counting a small number of fringes and reading on the micrometer the change in distance of the first mirror. This was accomplished by slightly turning one of the adjustment screws on the second mirror until only three fringes were visible. Then, while slowly turning the micrometer, the detector counted 20 fringe spacings. The micrometer was read and recorded before and after each observation. This procedure was repeated ten times with constant results.

The distance between beats was then measured by finding the distance between points of maximum destructive interference. The detector found the dimmest fringes and measured the distance to the next set of dimmest fringes. This procedure was repeated ten times in order to receive an average distance and to calculate a standard deviation.

Figure 3. Spectra of Sodium Gas Lamp

Using a spectrometer and computer software, the frequency of the sodium beam was measured as shown in Figure 3. The spectrometer had a resolution of one nanometer and yet was not capable of resolving the sodium doublet leading to the conclusion that the sodium double must have a spacing less than a nanometer. Using the frequency of light obtained by the spectrometer and the information gathered from the Michelson Interferometer, the difference between the frequencies in the sodium doublet was measured.

Results

The detector counted 20 fringes as mirror 1 moved a distance of 0.03 mm. The measurement was repeated 10 times, each time receiving the same result. A fringe spacing of 1.5 µm was concluded to be an accurate measurement.

The detector carefully measured the distance between beats by find the dimmest fringes and measuring the movement of the mirror until the next set of dimmest fringes could be found. This was repeated 10. The beat was measured to be 1.46 mm with an error of .02 mm or 1.42%.

At 1.5 µm per fringe there would be 973 fringes per beat. Since only one beat is being considered the difference in the frequency of the fringes would only be by 1 so the second frequency would have 974 fringes. (The second frequency could have 972 fringes, this still being a difference of 1 fringe, but as will be shown later, though the choice of one less fringe yields the same spacing in the sodium doublet, it does not give the correct wavelength of the second frequency in the doublet.)

The spectrometer measured the wavelength of light from the sodium gas lamp to be 589 nm. All of the data required to determine the difference in frequency of the light emitted by the sodium gas lamp is ready. By observation, the difference in the wavelengths should be the same as the radio of the difference in the number of fringes created by the wavelengths. So,

λknown / λunknown = fringe1 / fringe2

(1)

where the unknown wavelength in the sodium doublet can be found by,

λunknown = fringe2 / fringe1 * λknown.

(2)

Using Equation 2 and the values stated above, the second wavelength of light in the sodium doublet is 589.6 nm, a difference of only 0.6 nm. With a 1.42% error in the measurement of the length of each beat, this accounts for an error of only .008 nm. The results from this experiment agree with the actual measurements of the sodium doublet.

Discussion

Through this experiment it is impossible to tell if the decision to use 974 fringes for the second wavelength was correct without comparing the results to actual measured and verified wavelenths. What would have changed if the assumption were made that the second wavelength of light had one less fringe than the first wavelength? Evaluation of Equation 2 for the unknown wavelength would yield 588.4 nm, a difference of 0.6 nm from the first wavelength, the same difference as obtained in the previous results. Though this experiment can accurately determine the spacing of the sodium doublet, it cannot be used to determine which wavelength is correct for the second wavelength in the doublet.

Conclusion

Though the Michelson Interferometer can not visually resolve the interference patterns of the two wavelengths present in the sodium doublet, it can be used to prove the presence of the sodium doublet. Through careful observation, an observer will notice that the fringe pattern of sodium does not appear the same as the fringe pattern of, say, a mercury over changes in the distance of mirror 1. Where the fringe pattern of mercury is consistent as mirror 1 moves, the fringe pattern of sodium increases and decreases in intensity, evidence of a beat frequency caused by two close wavelengths. Through careful calculation the difference in those wavelengths can be measured.

References

1. Nobel Lectures, Physics 1901-1921, Elsevier Publishing Company, Amsterdam, 1967.
2. Knight, R. D., “Physics For Scientists and Engineers,” Pearson Education, Inc., San Francisco, CA, 2004.
3. Beiser, A., “Concepts of Modern Physics,” Von Hoffmann Press, Inc., 2003.