  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 $u^2$ for the envelope and $u^3$ 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*.
