Implementation of quantum and classical discrete fractional Fourier transforms

Steffen Weimann, Armando Perez-Leija, Maxime Lebugle, Robert Keil, Malte Tichy, Markus Gräfe, René Heilmann, Stefan Nolte, Hector Moya-Cessa, Gregor Weihs, Demetrios N. Christodoulides and Alexander Szameit ()
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Steffen Weimann: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena
Armando Perez-Leija: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena
Maxime Lebugle: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena
Robert Keil: Institut für Experimentalphysik, Universität Innsbruck
Malte Tichy: University of Aarhus
Markus Gräfe: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena
René Heilmann: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena
Stefan Nolte: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena
Hector Moya-Cessa: INAOE, Coordinacion de Optica
Gregor Weihs: Institut für Experimentalphysik, Universität Innsbruck
Demetrios N. Christodoulides: CREOL, The College of Optics & Photonics, University of Central Florida
Alexander Szameit: Institute of Applied Physics, Abbe School of Photonics, Friedrich-Schiller-Universität Jena

Nature Communications, 2016, vol. 7, issue 1, 1-8

Abstract: 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.

Date: 2016
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DOI: 10.1038/ncomms11027

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