Photon Ring Interferometric Signatures Beyond The Universal Regime

We calculate the interferometric signatures of black hole photon rings beyond the universal regime by perturbatively including the effects of finite ring width. Our approach first slices a thick ring into a series of thin rings, each of which falls within the universal regime. We thus calculate the visibility of the thick ring by aggregating the contributions from each thin ring, and then perturbatively expand the result into polynomials of the baseline length u. We show that the visibility amplitude of a thick ring depends on its "center-of-light" diameter; it also includes additional higher-order corrections due to the width of the ring, with the leading correction terms proportional to u² for the envelope and u³ for the phase. We apply our method to images ray traced from general-relativistic magnetohydrodynamic (GRMHD) simulations and demonstrate that incorporating the higher-order corrections is crucial for accurately modeling the visibility of the first photon ring around M87*.