The ESO planet finder instrument SPHERE will search for the polarimetric signature of the reflected light from extrasolar planets, using a VLT telescope, an extreme AO system (SAXO), a stellar coronagraph, and an imaging polarimeter (ZIMPOL). We present the design concept of the ZIMPOL instrument, a single-beam polarimeter that achieves very high polarimetric accuracy using fast polarization modulation and demodulating CCD detectors. Furthermore, we describe comprehensive performance simulations made with the CAOS problem-solving environment. We conclude that direct detection of Jupiter-sized planets in close orbit around the brightest nearby stars is achievable with imaging polarimetry, signal-switching calibration, and angular differential imaging.