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Strong organized magnetic fields have been studied in the upper main sequencechemically peculiar stars for more than half a century. However, only recentlyhave observational methods and numerical techniques become sufficiently matureto allow us to record and interpret high-resolution four Stokes parameterspectra, leading to the first assumption-free magnetic field models of thesestars. Here we present a detailed magnetic Doppler imaging analysis of thespectropolarimetric observations of the prototypical magnetic Ap star alpha^2CVn. The surface abundance distributions of Fe and Cr and a full vector map ofthe stellar magnetic field are reconstructed in a self-consistent inversionusing our state-of-the-art magnetic Doppler imaging code Invers10. We succeededin reproducing most of the details of the available spectropolarimetricobservations of alpha^2 CVn with a magnetic map which combines a globaldipolar-like field topology with localized spots of higher field intensity. Wedemonstrate that these small-scale magnetic structures are inevitably requiredto fit the linear polarization spectra; however, their presence cannot beinferred from the Stokes I and V observations alone. Our magnetic Dopplerimaging analysis of alpha^2 CVn and previous results for 53 Cam support theview that the upper main sequence stars can harbour fairly complex surfacemagnetic fields which resemble oblique dipoles only at the largest spatialscales. Spectra in all four Stokes parameters are absolutely essential tounveil and meaningfully characterize this field complexity in Ap stars. Wetherefore suggest that understanding magnetism of stars in other parts of theH-R diagram is similarly incomplete without investigation of their linearpolarization spectra.

Magnetic Doppler imaging of alpha^2 Canum Venaticorum in all four Stokes parameters. Unveiling the hidden complexity of stellar magnetic fields