Michelson's Interferometer — Question 7

Question 7

A plate insertion can be compensated by shortening one arm

A plane-parallel plate of thickness $t=10.0\,\mu\mathrm m$ and index $n=1.50$ is inserted normally into one initially balanced Michelson arm in air. It replaces air on both passes. Let the source wavelength be $600\,\mathrm{nm}$ and define positive mirror motion away from the beam splitter.

1. Find the additional optical path and number of phase cycles caused by the insertion.

2. Find the signed mirror motion in that arm that restores exact optical-path equality, and explain why a single-color bright fringe is not sufficient to prove equality.

Question 7 - Solution

A plate insertion can be compensated by shortening one arm

(a) Method and calculation

$$\Delta\mathcal L=2(n-1)t=\boxed{10.0\,\mu\mathrm m}.$$$$\boxed{\Delta N=\Delta\mathcal L/\lambda=50/3=16.6667\ \text{cycles}}.$$

(b) Interpretation and further calculation

$$2\Delta x+2(n-1)t=0\Longrightarrow\boxed{\Delta x=-(n-1)t=-5.00\,\mu\mathrm m}.$$$$\delta=2\pi m\Longleftrightarrow\Delta\mathcal L=m\lambda\Longrightarrow\boxed{\text{brightness alone allows integer-path ambiguity}}.$$

Physical check

Exact compensation means zero path difference, not merely an integer number of monochromatic wavelengths.