  Detection of $\gamma$-ray polarization in Positron Emission Tomography (PET)
is as yet an unexploited feature that could be used as an additional handle to
improve signal-to-background ratio in this imaging modality. The $\gamma$
polarization is related to the azimuthal angle in the Compton scattering
process, so the initial correlation of polarizations of the annihilation quanta
translates to the correlation of the azimuthal angles in events where both
annihilation photons undergo Compton scattering. This results in a modulated
distribution of the azimuthal angle difference for true events, while this
modulation is lacking for the background events. We present a comprehensive
experimental study of five detector configurations based on scintillator
matrices and silicon photomultipliers, suitable for measuring the azimuthal
modulation. The modules consist of either GaGG:Ce or LYSO:Ce pixels with sizes
varying from 1.9x1.9x20 $\mathrm{mm^3}$ to 3x3x20 $\mathrm{mm^3}$. The
distinctive feature of the modules is that they can reconstruct the Compton
scattering by detecting the recoil electron and the scattered gamma in a single
detector layer, which simplifies extension to larger systems. The amplitude
modulation of the azimuthal angles' difference is clearly observable in all
configurations ranging from $0.26\pm0.01$ to $0.34\pm 0.02$ depending on the
event selection criteria. The results suggest that finer detector segmentation
plays a leading role in achieving higher modulation factors.
