  It is common practice to test the optical properties of photomultiplier tubes
(PMTs) by illuminating the entire photocathode region from the front at once
and measuring the average performance. However, for optimal utilisation of the
PMT performance in experiments, especially in the single-photon region, it is
essential to also know the systematic variations across the photocathode, which
requires measurements with focused light sources that illuminate only small
regions of the PMT. We present a detailed uniformity characterisation of the
gain, transit time, transit time spread, and pulse shape of the 80$\,$mm
Hamamatsu R15458-02 PMT. We find that the parameters exhibit asymmetry along
one axis, likely caused by the position and geometry of the dynode system. For
all parameters except the transit time, the observed variations are small given
the intrinsic variation of the parameters. For positions with shifted transit
time we observe on average underamplified pulses which can potentially be
exploited to improve the pulse reconstruction.
