Integration of a high finesse cryogenic build-up cavity with an ion trap.

We report on the realization of a hemispherical optical cavity with a finesse of F = 13 000 and sustaining inter-cavity powers of 10 kW, which we operate in a closed-cycle cryostat vacuum system close to 4 K. This was designed and built with an integrated radio-frequency Paul trap in order to combine optical and radio-frequency trapping. The cavity provides a power build-up factor of 2300.