Imaging polarimetry is an important tool for the study of cosmic magnetic fields. In our Galaxy, polarization levels of a few up to $∼$10\% are measured in the submillimeter dust emission from molecular clouds and in the synchrotron emission from supernova remnants. Only few techniques exist to image the distribution of polarization angles, as a means of tracing the plane-of-sky projection of the magnetic field orientation. At submillimeter wavelengths, polarization is either measured as the differential total power of polarization-sensitive bolometer elements, or by modulating the polarization of the signal. Bolometer arrays such as LABOCA at the APEX telescope are used to observe the continuum emission from fields as large as $∼0\fdg2$ in diameter. %Here we present the results from the commissioning of PolKa, a polarimeter for Here we present PolKa, a polarimeter for LABOCA with a reflection-type waveplate of at least 90\% efficiency. The modulation efficiency depends mainly on the sampling and on the angular velocity of the waveplate. For the data analysis the concept of generalized synchronous demodulation is introduced. The instrumental polarization towards a point source is at the level of $∼0.1$\%, increasing to a few percent at the $-10$db contour of the main beam. A method to correct for its effect in observations of extended sources is presented. Our map of the polarized synchrotron emission from the Crab nebula is in agreement with structures observed at radio and optical wavelengths. The linear polarization measured in OMC1 agrees with results from previous studies, while the high sensitivity of LABOCA enables us to also map the polarized emission of the Orion Bar, a prototypical photon-dominated region.