  We demonstrate pulsed operation of a bidirectionally pumped polarization
Sagnac interferometric down-conversion source and its generation of narrowband,
high-visibility polarization-entangled photons. Driven by a narrowband,
mode-locked pump at 390.35 nm, the phase-stable Sagnac source with a type-II
phase-matched periodically poled KTiOPO$_4$ crystal is capable of producing
0.01 entangled pair per pulse in a 0.15-nm bandwidth centered at 780.7 nm with
1 mW of average pump power at a repetition rate of 31.1 MHz. We have achieved a
mean photon-pair generation rate of as high as 0.7 pair per pulse, at which
multi-pair events dominate and significantly reduce the two-photon
quantum-interference visibility. For low generation probability $\alpha$, the
reduced visibility $V=1-\alpha$ is independent of the throughput efficiency and
of the polarization analysis basis, which can be utilized to yield an accurate
estimate of the generation rate $\alpha$. At low $\alpha$ we have characterized
the source entanglement quality in three different ways: average
quantum-interference visibility of 99%, the Clauser-Horne-Shimony-Holt $S$
parameter of $2.739 \pm 0.119$, and quantum state tomography with 98.85%
singlet-state fidelity. The narrowband pulsed Sagnac source of entangled
photons is suitable for use in quantum information processing applications such
as free-space quantum key distribution.
