Weimann Steffen, Perez-Leija Armando, Lebugle Maxime, Keil Robert, Tichy Malte, Gräfe Markus, Heilmann René, Nolte Stefan, Moya-Cessa Hector, Weihs Gregor, Christodoulides Demetrios N, Szameit Alexander
Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria.
Nat Commun. 2016 Mar 23;7:11027. doi: 10.1038/ncomms11027.
Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.
整数阶和分数阶傅里叶变换是基础科学和应用科学中无处不在的数学工具。当然，由于普通傅里叶变换仅仅是连续分数傅里叶域集合中的一个特殊情况，普通傅里叶变换的每一个性质和应用都成为分数傅里叶变换的一个特殊情况。尽管离散傅里叶变换具有重大的实际重要性，但相应离散运算的分数阶实现却一直难以捉摸。在此，我们报告离散分数傅里叶变换的经典光学和量子光学实现。在经典光学背景下，我们实现了示例性波函数的离散分数傅里叶变换，并通过实验证明了移位定理。此外，我们将此方法应用于量子领域，对可分离和路径纠缠的双光子波函数进行傅里叶变换。所提出的方法具有通用性，可在傅里叶变换是必不可少工具的各个领域找到应用。
