Ultra-peripheral heavy-ion collisions (UPCs) occur when the impact parameter of the collision is greater than the sum of the radii of the colliding nuclei. UPCs allow one to study photon-induced reactions, such as the photoproduction of a vector meson, which is a well-established tool to probe the gluon structure of the colliding nuclei.

We will focus on the measurement of the impact-parameter dependence of the azimuthal anisotropy in the $\rho^0$ meson photoproduction. The interference originates from the linear polarization of the exchanged virtual photons and from quantum interference between amplitudes contributing to the photoproduction process.

We present the results of the first measurement of this kind at the LHC, using Pb--Pb UPCs and the ALICE detectors. The anisotropy is studied via the distribution of a variable, $\phi$, that is, approximately, the angle between the transverse momentum of the $\rho^0$ and that of one of its two decay pions. Models predict a $\cos(2\phi)$ modulation of the $\rho^0$ yield. The amplitude of such a modulation was measured in three impact parameter ranges, defined by event classes with different neutron emission requirements. The amplitude of the modulation is found to increase by about one order of magnitude from large to small impact parameters, in agreement with the available theoretical predictions.