  We study the possibility to use the Cherenkov light for the efficient
detection of 511 keV photons with the goal to use it in TOF-PET. We designed
and tested two detection modules consisting of PbF$_2$ crystals attached to
Planacon MCP-PMT XP85012. Amplified PMT signals are digitized by the SAMPIC
module with high readout rate, up to $10^5$ events/s, and a negligible
contribution to the time resolution, below 20 ps (FWHM). We developed a fast 2D
scanning system to calibrate the PMT time response and studied in details the
timing characteristics of the Planacon PMT. Using a radioactive $^{22}$Na
source we measured a detection efficiency of 24% for 511 keV photons in a 10 mm
thick crystal and a coincidence resolving time of 280 ps. We analyzed the main
factors limiting the time resolution of the large-surface detection module and
proposed solutions to improve it, which will be tested in our future project.
