Lasers and Laser Light — Question 1

Question 1

Temporal coherence can be inferred from unequal-arm fringe visibility

A laser has Lorentzian power linewidth $\Delta\nu$. Use $|g^{(1)}(\tau)|=e^{-\pi\Delta\nu|\tau|}$. An interferometer combines equal intensities with path difference d in vacuum; visibility is $\mathcal V=|g^{(1)}(d/c)|$. Take $c=3.00\times10^8\,\mathrm{m/s}$.

1. Derive the path difference giving visibility $1/e$.

2. For $\Delta\nu=2.00\,\mathrm{MHz}$, find that length and the visibility at $d=30.0\,\mathrm m$.

Question 1 - Solution

Temporal coherence can be inferred from unequal-arm fringe visibility

(a) Method and calculation

$$\mathcal V(d)=\exp[-\pi\Delta\nu d/c].$$$$\mathcal V=e^{-1}\Rightarrow\boxed{L_c=c/(\pi\Delta\nu)}.$$

(b) Interpretation and further calculation

$$\boxed{L_c=47.75\,\mathrm m}.$$$$\boxed{\mathcal V(30.0\,\mathrm m)=e^{-.2\pi}=.5335}.$$

Physical check

This coherence length uses a stated 1/e visibility convention and a Lorentzian spectrum. Other linewidth shapes or conventions produce different numerical factors.