Multiscale Talbot effects in Fibonacci geometry

TL;DR

This paper explores the Talbot effects in Fibonacci geometry using a cut-and-project method, revealing self-similar multiscale patterns at specific fractional Talbot distances related to Fibonacci and Lucas numbers.

Contribution

It introduces a novel approach to analyze Talbot effects in Fibonacci structures through the cut-and-project construction, capturing infinite patterns in a finite computational cell.

Findings

Identification of Talbot foci at distances involving Fibonacci and Lucas numbers.

Demonstration of multiscale, self-similar patterns in the Talbot images.

Theoretical and numerical validation of the Talbot effect 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.