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Michelson Interferometer
Level 1 - Physics topic page in Interference.
Principle
A Michelson interferometer splits one beam into two perpendicular arms and recombines them. Moving a mirror changes the optical path difference and shifts the interference fringes.
Notation
\(L_1,L_2\)
arm lengths
\(\mathrm{m}\)
\(\Delta L\)
change in one mirror position
\(\mathrm{m}\)
\(\Delta\)
optical path difference
\(\mathrm{m}\)
\(N\)
number of fringes shifted
1
\(\lambda\)
wavelength of light
\(\mathrm{m}\)
Method
Derivation 1: Round-trip path
Each beam travels to a mirror and back, so a mirror displacement changes the path twice.
Path difference
\[\Delta=2(L_2-L_1)\]
Mirror displacement
\[\Delta(\mathrm{path})=2\Delta L\]
Derivation 2: Fringe shift
One fringe passes when the path difference changes by one wavelength.
Fringe count
\[N=\frac{2\Delta L}{\lambda}\]
Mirror displacement
\[\Delta L=\frac{N\lambda}{2}\]
Derivation 3: Index changes
Putting a medium in one arm changes the optical path length.
Single-pass optical path change
\[\Delta\ell=(n-1)L\]
Double-pass fringe shift
\[N=\frac{2L(n-1)}{\lambda}\]
Rules
Michelson path difference
\[\Delta=2(L_2-L_1)\]
Mirror-fringe relation
\[N=\frac{2\Delta L}{\lambda}\]
Mirror displacement
\[\Delta L=\frac{N\lambda}{2}\]
Index fringe shift
\[N=\frac{2L(n-1)}{\lambda}\]
Examples
Question
A Michelson mirror is moved
\[250\,\mathrm{nm}\]
using \[500\,\mathrm{nm}\]
light. How many fringes shift?Answer
\[N=\frac{2\Delta L}{\lambda}=\frac{2(250\,\mathrm{nm})}{500\,\mathrm{nm}}=1\]
Checks
- The factor of 2 appears because the light travels to the mirror and back.
- A full fringe shift corresponds to one wavelength of optical path difference.
- Michelson measurements compare optical path lengths, not just physical distances.
- The same instrument can measure tiny displacements, wavelengths, and refractive-index changes.