Wave, propagation and interference
Michelson built the most sensitive instrument of his time to measure the motion of the Earth through the ether, and found nothing. That null result is the starting point of all twentieth-century physics.
01No coherence, no interference
Two waves always superpose, but for the interference pattern to be visible the phase relationship between them has to hold steady during the observation time. That property is coherence, and it is what distinguishes a laser from a lamp.
| Source | Coherence length | What it allows |
|---|---|---|
| Incandescent lamp | A few microns | Almost nothing: the beam has to be split |
| LED | Tens of microns | Very short-range interferometry |
| Ordinary diode laser | Centimeters to meters | Laboratory interferometry |
| Stabilized laser | Hundreds of meters | Dimensional metrology |
The coherence length is related to the spectral width: the “purer” the color, the longer it is. That is why white-light interferometry only works with paths that are almost exactly equal, and that limitation, which looks like a defect, is precisely what makes it useful for measuring absolute distances.
02The Michelson interferometer
A beam splitter divides the light in two; each half travels along an arm, bounces off a mirror and comes back. The two recombine and interfere. Since the path difference is twice the difference in arm length, moving a mirror by half a wavelength is enough to go from bright to dark.
You move the movable mirror and the fringes shift. The fringe at the center goes from bright to dark every quarter wavelength, because the path is traveled twice.
Between 1960 and 1983 the meter was defined as 1,650,763.73 wavelengths of a krypton-86 spectral line, and it was realized with an interferometer. Today the meter is defined from the speed of light, but the interferometer is still the practical standard: a coordinate measuring machine or a lithography system locates its axes by counting fringes, with a resolution of fractions of a nanometer.
03The experiment that found nothing
Every known wave needed a medium: sound needed air, water waves needed water. Light was therefore assumed to propagate in a medium that filled the universe, the ether. If it existed, the Earth had to be moving through it at no less than 30 km/s, its orbital speed.
In 1887 Michelson and Morley used the interferometer to detect that motion. The idea was simple: if there is an ether wind, light should take a different time depending on whether it travels along the wind or perpendicular to it, and when the apparatus was rotated the fringes should shift.
Nothing moved. They repeated the experiment at different times of the year, in case the Earth was momentarily at rest with respect to the ether, and still nothing moved. The most important null result in the history of physics: for almost twenty years people tried to explain it with auxiliary hypotheses —ether drag, the FitzGerald contraction— until in 1905 Einstein took the short road and postulated that the speed of light is the same for everyone, and that what needed revising were space and time.
It is worth pausing on this, because it is a lesson in method. Michelson considered his experiment a failure: he had built the most sensitive instrument in the world and had measured nothing. He received the Nobel Prize in 1907 for the instrument, not for the result. Today that “failure” is studied as one of the most important experiments ever performed, and the LIGO interferometer, which detected gravitational waves in 2015, is the same apparatus with arms four kilometers long.
04In the lab
With a Michelson interferometer and a micrometer screw, count the fringes that pass when the mirror is displaced by a known distance. Solve for the laser wavelength and compare it with the stated one.
Repeat the fringe alignment with a laser, an LED and white light, starting from equal arms. Measure how far each source can be unbalanced before the fringes disappear, and estimate the coherence length of each one.
Insert a vacuum cell in one arm and count the fringes that pass as it is slowly filled with air. Calculate the refractive index of air from that count.
05Common mistakes
- Forgetting the factor of two from the round-trip path. Moving the mirror by \( \lambda/4 \) changes the path by \( \lambda/2 \).
- Expecting fringes from two different sources. Without mutual coherence there is no stable pattern.
- Believing that white light does not interfere. It does, but only with almost equal paths.
- Taking the null result as a failure. A well-measured null result is a conclusive result.
- Confusing temporal coherence with spatial coherence. One depends on the spectral width; the other, on the size of the source.
- Assuming the ether was foolishness. It was the reasonable hypothesis given what was known; what is remarkable is that it was abandoned.
06Self-assessment
How far does the mirror have to move to change by one fringe?
Half a wavelength to go from one maximum to the next, because the path is traveled twice. A quarter of \( \lambda \) takes you from bright to dark.
What is the coherence length?
The maximum path difference that still produces fringes. It is approximately \( \lambda^2/\Delta\lambda \).
What was the Michelson-Morley experiment looking for?
The Earth's motion relative to the ether, which should have produced a fringe shift when the apparatus was rotated.
What did it find?
Nothing. The null result forced, twenty years later, a revision of the notions of space and time.
Why does white light give so few fringes?
Because its enormous spectral width gives it a coherence length of only a few microns.
Where is a Michelson interferometer used today?
In dimensional metrology, in lithography equipment and, with arms kilometers long, in gravitational-wave detectors.
07Further reading
- Eugene Hecht. Optics. 5th ed., Pearson, 2017 (Spanish edition, “Óptica”). Coherence, amplitude-division interferometers and the full treatment of the Michelson.
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica: átomos, moléculas, sólidos, núcleos y partículas”, Limusa). The book that organizes this whole subject. Its first chapter tells the story of the crisis of classical physics.
- Albert A. Michelson and Edward W. Morley. On the Relative Motion of the Earth and the Luminiferous Ether. American Journal of Science, vol. 34, 1887. The original paper, surprisingly accessible. It is worth seeing how they report a null result.