Filip Niewiński, Michael D. Johnson

We derive and showcase a novel approach to approximating Fourier transforms in higher dimensions, focusing specifically on the case of 2D radially concentrated ('ring-like') functions. We first reduce the problem to that of evaluating the Hankel transforms of each angular mode of the image and then use our focal expansion to approximate these remaining Hankel transforms. Our method provides a single approximation that remains accurate from small to large spatial frequencies, bridging regimes where moment-based or large-frequency asymptotic expansions are individually reliable. We explore a series of examples, showing that the leading focal term provides an accurate global approximation for a broad range of functions. We demonstrate the utility of this approximation by examining the interferometric response for toy models of a black hole's "photon ring," highlighting the application to experiments designed to measure this feature such as the Black Hole Explorer.