  Recent tests of a single module of the Jagiellonian Positron Emission
Tomography system (J-PET) consisting of 30 cm long plastic scintillator strips
have proven its applicability for the detection of annihilation quanta (0.511
MeV) with a coincidence resolving time (CRT) of 0.266 ns. The achieved
resolution is almost by a factor of two better with respect to the current
TOF-PET detectors and it can still be improved since, as it is shown in this
article, the intrinsic limit of time resolution for the determination of time
of the interaction of 0.511 MeV gamma quanta in plastic scintillators is much
lower. As the major point of the article, a method allowing to record
timestamps of several photons, at two ends of the scintillator strip, by means
of matrix of silicon photomultipliers (SiPM) is introduced. As a result of
simulations, conducted with the number of SiPM varying from 4 to 42, it is
shown that the improvement of timing resolution saturates with the growing
number of photomultipliers, and that the 2 x 5 configuration at two ends
allowing to read twenty timestamps, constitutes an optimal solution. The
conducted simulations accounted for the emission time distribution, photon
transport and absorption inside the scintillator, as well as quantum efficiency
and transit time spread of photosensors, and were checked based on the
experimental results. Application of the 2 x 5 matrix of SiPM allows for
achieving the coincidence resolving time in positron emission tomography of
$\approx$ 0.170 ns for 15 cm axial field-of-view (AFOV) and $\approx$ 0.365 ns
for 100 cm AFOV. The results open perspectives for construction of a
cost-effective TOF-PET scanner with significantly better TOF resolution and
larger AFOV with respect to the current TOF-PET modalities.
