title: Implementation of quantum and classical discrete fractional Fourier transforms.
abstract: 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.
authors.affiliation: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
authors.forename: Steffen
authors.initials: S
authors.lastname: Weimann
authors.affiliation: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
authors.forename: Armando
authors.initials: A
authors.lastname: Perez-Leija
authors.affiliation: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
authors.forename: Maxime
authors.initials: M
authors.lastname: Lebugle
authors.affiliation: Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria.
authors.forename: Robert
authors.initials: R
authors.lastname: Keil
authors.affiliation: Department of Physics and Astronomy, University of Aarhus, 8000 Aarhus, Denmark.
authors.forename: Malte
authors.initials: M
authors.lastname: Tichy
authors.affiliation: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
authors.forename: Markus
authors.initials: M
authors.lastname: Gräfe
authors.affiliation: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
authors.forename: René
authors.initials: R
authors.lastname: Heilmann
authors.affiliation: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
authors.forename: Stefan
authors.initials: S
authors.lastname: Nolte
authors.affiliation: INAOE, Coordinacion de Optica, Luis Enrique Erro No. 1, Tonantzintla, Puebla 72840, Mexico.
authors.forename: Hector
authors.identifier: http://orcid.org/0000-0003-1444-0261
authors.initials: H
authors.lastname: Moya-Cessa
authors.affiliation: Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria.
authors.forename: Gregor
authors.identifier: http://orcid.org/0000-0003-2260-3008
authors.initials: G
authors.lastname: Weihs
authors.affiliation: CREOL, The College of Optics &Photonics, University of Central Florida, Orlando, Florida 32816, USA.
authors.forename: Demetrios N
authors.initials: DN
authors.lastname: Christodoulides
authors.affiliation: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien Platz 1, 07743 Jena, Germany.
authors.forename: Alexander
authors.initials: A
authors.lastname: Szameit