35-5 Michelson’s Interferometer#
Prompts
With a sketch, explain how Michelson’s interferometer works. What splits the light? What is the path length difference when the beams recombine?
If mirror \(M_2\) is moved by \(\lambda/2\), how many fringes does the pattern shift? Why? What about a move of \(\lambda/4\)?
A transparent material (thickness \(L\), index \(n\)) is inserted into one arm. Derive the phase change in wavelengths. How is this related to the fringe shift?
How can the interferometer be used to measure the thickness of a material or the wavelength of light? What made it historically important for the definition of the meter?
Lecture Notes#
Overview#
Michelson’s interferometer (1881) splits a light beam into two, sends them along different paths (arms), and recombines them. The interference pattern depends on the path length difference.
Moving a mirror or inserting a transparent material changes the phase difference and shifts the fringes. Counting fringe shifts allows precise measurement of distances in terms of wavelengths.
The device was historically crucial: Michelson expressed the meter in wavelengths of light (Nobel Prize 1907), leading to the modern definition of length.
Setup#
Beam splitter \(M\): transmits half the light, reflects half. One beam goes to mirror \(M_1\) (arm 1), the other to mirror \(M_2\) (arm 2). Both reflect back to \(M\), recombine, and enter telescope \(T\). The observer sees an interference pattern (curved or straight fringes).
Path length difference (when beams recombine):
where \(d_1\) and \(d_2\) are the arm lengths.
Mirror shift#
Changing the path length difference shifts the fringe pattern. Moving mirror \(M_2\) by \(\Delta d\) changes the path by \(2\,\Delta d\) (to and fro).
Mirror move |
Path change |
Fringe shift |
|---|---|---|
\(\lambda/2\) |
\(\lambda\) |
1 fringe |
\(\lambda/4\) |
\(\lambda/2\) |
\(\frac{1}{2}\) fringe |
One fringe = one complete cycle from bright to bright (or dark to dark).
By counting fringes as \(M_2\) moves, we measure the displacement in units of \(\lambda/2\).
Insertion of a transparent material#
If a material of thickness \(L\) and index \(n\) is placed in one arm (e.g., in front of \(M_1\)), the light in that arm travels \(2L\) through the material instead of air.
Number of wavelengths in the material (to-and-fro path \(2L\)):
Number of wavelengths in the same distance \(2L\) in air:
Phase change (in wavelengths):
Each wavelength of phase change shifts the pattern by one fringe. So the number of fringe shifts \(N\) when the material is inserted is
From this, we can find \(L\) (if \(n\) and \(\lambda\) are known) or \(n\) (if \(L\) and \(\lambda\) are known).
Connection to section 35-1
Same idea as phase difference from different materials: \(\lambda_n = \lambda/n\) means more wavelengths fit in the same length, so the beam through the material gains phase relative to the beam through air.
Standard of length#
Michelson used his interferometer to express the meter in terms of the wavelength of cadmium red light: 1 m = 1 553 163.5 wavelengths. This work led to:
1961: Abandonment of the physical meter bar.
1983: Redefinition of the meter via the defined speed of light \(c\).
The interferometer demonstrates that interference provides a practical link between macroscopic distances and the wavelength of light.
Summary#
Michelson’s interferometer: beam splitter \(\to\) two arms \(\to\) recombine \(\to\) interference fringes.
Mirror move \(\lambda/2\) \(\Rightarrow\) path change \(\lambda\) \(\Rightarrow\) 1 fringe shift.
Material insertion: phase change \(2(n-1)L/\lambda\) wavelengths \(\Rightarrow\) same number of fringe shifts.
Applications: measure \(L\), \(n\), or \(\lambda\); historically, define the meter in terms of wavelengths.