Blazars are astrophysical sources whose emission is dominated by non-thermal processes, typically interpreted as synchrotron and inverse Compton emission. Although the general picture is rather robust and consistent with observations, many aspects are still unexplored. Polarimetric monitoring can offer a wealth of information about the physical processes in blazars. Models with largely different physical ingredients can often provide almost indistinguishable predictions for the total flux, but usually are characterized by markedly different polarization properties. We explore, with a pilot study, the possibility to derive structural information about the emitting regions of blazars by means of a joint analysis of rapid variability of the total and polarized flux at optical wavelengths. Short timescale (from tens of seconds to a couple of minutes) optical linear polarimetry and photometry for two blazars, BL Lacertae and PKS 1424+240, was carried out with the PAOLO polarimeter at the 3.6m Telescopio Nazionale Galileo. Several hours of almost continuous observations were obtained for both sources. Our intense monitoring allowed us to draw strongly different scenarios for BL Lacertae and PKS 1424+240, with the former characterized by intense variability on time-scales from hours to a few minutes and the latter practically constant in total flux. Essentially the same behavior is observed for the polarized flux and the position angle. The variability time-scales turned out to be as short as a few minutes, although involving only a few percent variation of the flux. The polarization variability time-scale is generally consistent with the total flux variability. Total and polarized flux appear to be essentially uncorrelated. However, even during our relatively short monitoring, different regimes can be singled out. (abridged)
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