Approximating the Fourier Transform of Ring-Like Images: the Focal Expansion

TL;DR

This paper introduces a novel method called focal expansion for approximating Fourier transforms of 2D ring-like images, effectively bridging low and high-frequency regimes with broad applicability.

Contribution

The authors develop a new approximation technique for Fourier transforms of radially concentrated functions, combining Hankel transforms and focal expansion for improved accuracy across frequencies.

Findings

The method accurately approximates Fourier transforms from small to large frequencies.

The leading focal term provides a reliable global approximation for various functions.

Application to black hole photon rings demonstrates practical utility.

Abstract

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.