  For the advancement of the understanding of timing systematics in pixelized
readout chips and for the benefit of future fast timing detectors to aid 4D
tracking technology in the HL-LHC, we have performed detailed studies of the
timing properties of the Timepix3, a hybrid pixel detector developed by the
Medipix collaboration. These studies use three different measurement techniques
to investigate the timing systematics of this detector: test pulses, testbeam,
and a laser setup that we build for this work. The average delay over the pixel
matrix is determined with the testbeam and the laser and shows the same
structure for different Timepix3 chips. The difference in delay results in a
maximum difference over the pixel matrix of around 4 ns, which is large
compared to the time bins of 1.56 ns. By correcting for this difference on a
per-pixel level, the time resolution is on average improved with 159 ps
depending on the sensor. The best time resolution that is achieved after a
timewalk correction and a correction for the difference in the average delay
per pixel is 686.4$\pm$0.2 ps for a single Timepix3 chip with a 200
$\mu\text{m}$ planar silicon sensor, compared to a time resolution of
892.9$\pm$0.3 ps without any correction. However, this improved time resolution
is not yet the naively expected time resolution of 451 ps. The origin of this
delay is determined with this laser setup, and is due to a combination of
signal propagation and a difference of the start-up time of the fast oscillator
within the pixels. This work indicates that it is vital for next generation
pixel detectors to correct for these systematic timing effects in order to
reach a better time resolution to aid 4D tracking technology in the HL-LHC.
