Context. Polarimetric imaging is one of the most effective techniques for high-contrast imaging and for the characterization of protoplanetary disks, and it has the potential of becoming instrumental in the characterization of exoplanets. The Spectro-Polarimetric High-contrast Exoplanet REsearch (SPHERE) instrument installed on the Very Large Telescope (VLT) contains the InfraRed Dual-band Imager and Spectrograph (IRDIS) with a dual-beam polarimetric imaging (DPI) mode, which offers the capability of obtaining linear polarization images at high contrast and resolution. Aims. We aim to provide an overview of the polarimetric imaging mode of VLT/SPHERE/IRDIS and study its optical design to improve observing strategies and data reduction. Meth...