Fabry-Perot Etalon Design: Resolving 514 nm Laser Modes

[roam]

Homework Statement

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?

Homework Equations

Finesse coefficient is:

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

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

$FWHM = FSR/\mathcal{F}$

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.

 

[mark726]

Thank you for your interesting question. The values you obtained for the finesse and reflectivity are indeed very small and unrealistic. This is most likely due to an error in your calculations.

To calculate the finesse, you can use the following formula:

$\mathcal{F}=\frac{\pi\sqrt{R}}{1-R}$

Substituting the given values, we get:

$7.14\times 10^{-10}=\frac{\pi\sqrt{R}}{1-R}$

Solving for R, we get:

$R=0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999$

This value is much more realistic for the reflectivity of an etalon. It is also important to note that the finesse and reflectivity values are not independent of each other. In fact, they are related through the finesse coefficient, as shown in the homework equations section.

For the required length of the etalon, your calculation is correct. The mode spacing (FSR) should be greater than or equal to the separation between longitudinal modes of the laser. This ensures that all the modes are resolved by the etalon.

I hope this helps clarify your doubts. Let me know if you have any further questions. Good luck with your design!