  Dielectric metasurfaces, extremely thin nanostructured dielectric surfaces,
hold promise to replace conventional refractive optics, such as lenses, due to
their high performance and compactness. However, designing large field-of-view
(FOV) metalenses, which are of particular importance when imaging relatively
big objects at short distances, remains one of the most critical challenges.
Recently, metalenses implementing a quadratic phase profile have been put
forward to solve this problem with a single element, but despite their
theoretical ability to provide $180^\circ\,$FOV, imaging over very large FOV
has not been demonstrated yet. In this work, we provide an in-depth analysis of
the imaging properties of quadratic metalenses and, in particular, show that
due to their intrinsic barrel distortion or fish-eye effect, there is a
fundamental trade-off between the FOV achievable in a given imaging
configuration and the optical resolution of the metalens and/or the detector
resolution. To illustrate how to harness the full potential of quadratic
metalenses, we apply these considerations to the fingerprint detection problem,
and demonstrate experimentally the full imaging of a $5\,$mm fingerprint with
features of the order of $100\,\mu$m, with a metalens ten times smaller in size
and located at a distance of only $2.5\,$mm away from the object. This
constitutes the most compact system reported so far for the fingerprint
detection.
