# mode-locking a laser on prime-powers' frequencies

Being rather a theoretician than an experimental physicist, I have a question to the community:

Is it experimentally possible to mode-lock a laser (fixed phase relationships between the modes of the laser's resonant cavity) in a way that the longitudinal modes of the cavity would be exclusively from a discrete set of frequency $\{p^m\}$, where $p$ a prime number and $m$ a positive integer? If yes, how? If not, why?

For instance: $\{2^m\}$ or $\{2^m,3^m\}$ or ...

Thanks

• This may be a mathematically deep question because the amplitudes describing transitions in such a filter would probably be proportional to the Riemann zeta function of a variable. – Luboš Motl Jun 7 '13 at 14:15
• @LubošMotl - would you be so kind to have a look on math.stackexchange.com/questions/417590/… I wonder whether you may have an idea. – al-Hwarizmi Jun 11 '13 at 17:06

## 1 Answer

The frequency range of a mode locked laser is generally less than an octave (factor of 2). So your question would entail only 0 or 1 or 2 modes locked together.

In a typical situation, a mode-locked laser might lock the 100,000th to 120,000th cavity modes or something like that (don't quote me on that). The closest I can think of to your proposal is a tiny laser cavity where the 3rd and 4th cavity modes are lasing. But that doesn't really work either because these frequencies are not in a perfect 3:4 ratio (because of material dispersion). If you could exactly correct that dispersion, I suppose you could make the two frequencies lock together in theory. In practice, I don't know, maybe with great effort.