Vectorial field reconstruction without detecting the field
Vector beams, whose polarization varies across the transverse profile, are a central resource in structured-light optics and quantum photonics. Their characterization, however, becomes challenging when the field lies in a spectral region for which efficient spatially resolving detectors are unavailable. Here we demonstrate the spatially resolved reconstruction of an undetected vector beam by exploiting induced coherence in a nonlinear interferometer. In this effect, indistinguishability between two down-conversion pathways allows information encoded in an undetected field to be read out through interference of its detected partner. A telecom-wavelength idler field acquires a spatially varying polarization transformation but is never directly detected. Instead, its local polarization information is inferred from single-photon interference in the visible signal field, enabled by momentum correlations of the photon pair. Using phase-shifting and off-axis quantum holography with two polarization projections, we reconstruct the horizontal and vertical amplitudes and their relative phase across the beam profile, thereby recovering the full vectorial structure of the undetected field. We experimentally retrieve the polarization texture of an $m=2$ vector beam and compare multi-shot and single-shot reconstruction strategies. Our results extend imaging with undetected light from scalar objects to vectorial optical fields and open a route to polarization-sensitive sensing and state reconstruction in spectral regions that are difficult to access directly.