How do you measure a distance much smaller than a wavelength of light? Split a beam in two, send the halves along different paths, and bring them back together. The phase difference between the halves turns tiny changes of path into visible changes of brightness. Albert Michelson built this instrument in the 1880s; a giant version, LIGO, with arms 4 km long, detected gravitational waves in 2015 by sensing arm-length changes of about $10^{-18}$ m.
🔑 Key idea
A half-silvered mirror splits the light into two arms, each ending in a mirror. Moving one mirror a distance $d$ changes that arm's round trip by $2d$, and the pattern shifts by $$N = \frac{2d}{\lambda}\ \text{fringes}.$$ One fringe passes for every $\lambda/2$ of mirror motion. A thin layer of index $n$ and thickness $t$ placed in one arm shifts the pattern by $N = \frac{2t(n - 1)}{\lambda}$.
💡 Why it works
The light in the moved arm goes out to the mirror and back, so its path grows by twice the mirror's motion. Each extra wavelength of path changes the phase difference by $2\pi$, which returns the pattern to the same look: one fringe. So $N = \frac{\text{extra path}}{\lambda} = \frac{2d}{\lambda}$. In a material of index $n$, a thickness $t$ holds $\frac{nt}{\lambda}$ wavelengths instead of $\frac{t}{\lambda}$; going through it twice adds $\frac{2t(n - 1)}{\lambda}$ wavelengths.
Move the mirror and count the fringes that pass the center. At 550 nm, one fringe passes for every 0.275 μm of motion.
Worked examples
✏️ Worked example: Count the fringes
A Michelson interferometer uses a 633 nm laser. One mirror is moved 0.100 mm. How many fringes pass?
Extra round-trip path: $2d = 0.200$ mm $= 2.00 \times 10^{-4}$ m.
Michelson and Edward Morley used this instrument in 1887 to look for Earth's motion through the “ether” that light was then thought to need. Turning the instrument should have shifted the fringes by a measurable amount as the arms turned with and across Earth's motion. It did not. That null result is one of the clues that led to special relativity.
⚠️ Common mistake
Counting one fringe per wavelength of mirror motion. The light travels the extra distance twice, out and back, so one fringe passes for each half wavelength the mirror moves. Check: if a mirror moves $\lambda/4$, the round trip grows by $\lambda/2$, enough to turn a bright center dark.
⚠️ Common mistake
Forgetting the $n - 1$ for a layer in one arm. Without the layer, that distance was already filled with air (or vacuum), so only the extra wavelengths count.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.