# Fabry-Perot Etalon

1. Aug 20, 2014

### roam

1. The problem statement, all variables and given/known data

Design a Fabry-Perot etalon that can resolve the longitudinal modes of a 514 nm laser. The laser is 2 m long.The FWHM of the gain spectrum is 8.8 GHz. What value of d (separation) and R (reflectivity) must be chosen?

2. Relevant equations

Finesse coefficient is:

$F=\frac{4R}{(1-R)^2}$

Full width half maximum relationship to free spectral range (intermode spacing):

$FWHM = FSR/\mathcal{F}$

3. The attempt at a solution

The finesse is given by

$\mathcal{F} = \frac{2\pi}{FWHM} = \frac{2 \pi}{8.8 \times 10^{9}} = \frac{\pi \sqrt{R}}{1-R}=\frac{\pi\sqrt{F}}{2}=7.14\times 10^{-10}$

$FWHM = \frac{4}{\sqrt{F}} \implies F=2.066 \times 10^{-19}$

So why do I get such small and unrealistic numbers? (a typical value of finesse is ~100 for a reflectivity of 97%)

Furthermore, I don't know how to calculate R from:

$\frac{2}{8.8 \times 10^9}=\frac{\sqrt{R}}{1-R}$

My calculator solver gives a value of 2.78x1020, which again is an unrealistic number for reflectivity. So what is wrong with my calculations?

As for the required length of the etalon, I think we want the mode spacing (FSR) to be greater than or equal to the the separation between longitudinal modes of the laser:

$\frac{c}{2d} \geq \frac{c}{2nl_{laser}}=75 \times 10^6 \ Hz$

Is that right?

Any help is greatly appreciated.