Diffraction of plane waves, spherical waves, and beyond
Emily R. Korfanty, Jan Mazáč
TL;DR
This work extends diffraction theory beyond bounded functions by analyzing plane-wave and spherical-wave diffraction within Besicovitch spaces, including unbounded settings. It develops a radial counterpart to Besicovitch almost periodicity, introducing the Besicovitch radially almost periodic space $Brap^2(\mathbb{R}^d)$ and proving the existence of autocorrelations with explicit, radially concentrated diffraction patterns. A key result is that spherical waves $f_a(x)=e^{2\pi i a\|x\|}$ have autocorrelation $\eta_{f_a}(x)$ expressed via Bessel functions and diffraction a single sphere with measure $\theta_{|a|}$, and that superpositions yield diffraction as a sum of radially shifted spheres, i.e., $\widehat{\eta_{F}} = \sum |c_m|^2 \theta_{|a_m|}$. The paper also provides a practical criterion for detecting circles in diffraction directly from structure, bypassing explicit autocorrelation computation, with potential applications to radially symmetric tilings and aperiodic order.
Abstract
We review the diffraction theory for plane waves and establish its connection to the diffraction of Besicovitch almost periodic functions, extending the theory to an unbounded setting and providing explicit formulas. Then, we give an alternative proof that the diffraction of a spherical wave in $\mathbb{R}^d$ is a single sphere, which was recently shown in \cite{BKM25}. After developing a suitable framework for working with spherically symmetric measures, including a radial analogue of the usual Lebesgue decomposition, we introduce the notion of radial almost periodicity. In particular, we define a space of Besicovitch radially almost periodic functions and show that this space contains precisely the functions whose radial part is Besicovitch almost periodic. The paper concludes with a diffraction analysis of these functions.