  We report the fabrication of a 1.2 cm long cavity directly on a nanofiber
using femtosecond laser ablation. The cavity modes with finesse value in the
range 200-400 can still maintain the transmission between 40-60%, which can
enable "strong-coupling" regime of cavity QED for a single atom trapped 200 nm
away from the fiber surface. For such cavity modes, we estimate the one-pass
intra-cavity transmission to be 99.53%. Other cavity modes, which can enable
high cooperativity in the range 3-10, show transmission over 60-85% and are
suitable for fiber-based single photon sources and quantum nonlinear optics in
the "Purcell" regime.
