Context: Polarimetry is a very powerful tool to uncover various properties of astronomical objects that remain otherwise hidden in standard imaging or spectroscopic observations. However, the reliable measurement of the low polarization signal from astronomical sources requires a good control of spurious instrumental polarization induced by the various components of the optical system and the detector. Aims: We perform a detailed multi-wavelength calibration study of the FORS2 instrument at the VLT operating in imaging polarimetric mode (IPOL) to characterize the spatial instrumental polarization that may affect the study of extended sources. Methods: We use imaging polarimetry of a) high signal-to-noise blank fields BVRI observations during full-moon, when the polarization is expected to be constant across the field-of-view and deviations originate from the instrument and b) a crowded star cluster in broad-band RI and narrow-band H{alpha} filters, where individual polarization values of each star across the field can be measured. Results: We find an instrumental polarization pattern that increases radially outwards from the optical axis of the instrument reaching up to 1.4% at the edges, depending on the filter. Our results are well approximated by an elliptical paraboloid down to less than {sim0.05%} accuracy,and {sim0.02%} when using non-analytic fits. We present 2D maps to correct for this spurious instrumental polarization. We also give several tips and tricks to analyze polarimetric measurements of extended sources. Conclusions: FORS2 is a powerful instrument allowing to map the linear polarimetry of extended sources. We present and discuss a methodology to measure the polarization of such sources, and to correct for the spatial polarization induced in the optical system. This methodology could be applied to polarimetric measurements using other dual-beam polarimeters.
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