Multiscale Talbot effects in Fibonacci geometry

Abstract

This article investigates the Talbot effects in Fibonacci geometry by introducing the cut-and-project construction, which allows for capturing the entire infinite Fibonacci structure into a single computational cell. Theoretical and numerical calculations demonstrate the Talbot foci of Fibonacci geometry at distances that are multiples $(\tau+2)(F_{\mu}+\tau F_{\mu+1} )^{-1}p/(2q)$ or $(\tau+2)(L_{\mu}+\tau L_{\mu+1} )^{-1}p/(2q)$ of the Talbot distance. Here, ($p$, $q$) are coprime integers, $\mu$ is an integer, $\tau$ is the golden mean, and $F_{\mu}$ and $L_{\mu}$ are Fibonacci and Lucas numbers, respectively. The image of a single Talbot focus exhibits a multiscale pattern due to the self-similarity of the scaling Fourier spectrum.