$Light refracting through a prism creating a spectrum$

The Michelson interferometer

PHYS 310 · Interference and Coherence

The Michelson interferometer splits and recombines light to measure path differences with wavelength-scale sensitivity. This lesson explains its operation, fringe shifts, and applications.

Key equations

\Delta L=2(L_2-L_1)\Delta\phi=\frac{2\pi}{\lambda}\Delta L\Delta L=m\lambda\Delta L=\left(m+\frac{1}{2}\right)\lambda2\Delta x2\Delta x=N\lambda\Delta x=\frac{N\lambda}{2}\Delta(OPL)=(n-1)L2(n-1)L

Learning objectives

Describe the layout and operation of a Michelson interferometer.
Relate arm length difference to phase difference.
Calculate mirror displacement from fringe shifts.
Explain refractive index measurement with optical path length.
Identify major applications of Michelson interferometry.

Splitting and recombining light

The Michelson interferometer is one of the most important optical instruments in physics. It uses a beam splitter to divide light into two perpendicular arms. Mirrors reflect the beams back, and the beams recombine to interfere.

Because interference depends on path difference, the Michelson interferometer can measure extremely small length changes, refractive index changes, and wavelength differences.

Basic layout

A light beam reaches a partially reflecting beam splitter. One part travels to mirror $M_1$ , and the other travels to mirror $M_2$ . After reflection, the beams return to the beam splitter and recombine at a detector or viewing screen.

If the arm lengths are $L_1$ and $L_2$ , the round-trip geometric path difference is

$Delta L=2(L_2-L_1)$

in air, ignoring small corrections.

Phase difference

The phase difference due to optical path difference is

$Deltaphi= rac{2pi}{lambda}Delta L$

Constructive interference occurs when the path difference is an integer number of wavelengths:

$Delta L=mlambda$

Destructive interference occurs when

ight)lambda$$ Reflection phase shifts at the beam splitter can affect whether the center is bright or dark, but fringe shifts depend on path changes. ## Moving a mirror If one mirror moves by distance $Delta x$, the beam traveling to that mirror changes its round-trip path by $$2Delta x$$ Each time the path difference changes by one wavelength, one fringe shifts past a reference point. If $N$ fringes pass, $$2Delta x=Nlambda$$ so $$Delta x= rac{Nlambda}{2}$$ This gives extremely precise displacement measurement. ## Measuring refractive index If a sample of length $L$ and refractive index $n$ is inserted into one arm, the optical path changes by approximately $$Delta(OPL)=(n-1)L$$ If the beam passes through the sample twice, the change is $$2(n-1)L$$ Counting fringe shifts allows measurement of $n$ or changes in $n$ due to pressure, temperature, or composition. ## White-light interferometry With broadband light, interference is visible only when optical path difference is within the coherence length. White-light Michelson interferometry can locate equal-path conditions very precisely and is used in surface profiling and optical coherence tomography. The short coherence length becomes an advantage for depth resolution. ## Michelson-Morley experiment The Michelson interferometer is historically famous for the Michelson-Morley experiment, which searched for changes in light speed due to Earth's motion through a supposed luminiferous ether. The null result helped undermine ether theories and paved the way for special relativity. The experiment showed the power of interferometry to test fundamental physics. ## Modern applications Michelson-type interferometry appears in precision metrology, spectroscopy, Fourier-transform infrared instruments, gravitational wave detection, surface measurement, optical coherence tomography, and laser stabilization. Gravitational wave detectors are enormous interferometers that measure incredibly tiny changes in arm length. ## The big idea The Michelson interferometer converts tiny optical path differences into measurable interference fringes. Because a mirror displacement changes round-trip path by twice the displacement, fringe counting provides wavelength-scale precision. The instrument is central to measurement science and fundamental physics.

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