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Implementation of quantum and classical discrete fractional Fourier transforms

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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.

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DFrFT of classical light.(a) Transformation of a Gaussian input into a Gaussian profile of larger width along the evolution in the Jx-array. The FT is obtained at Z=π/2. The experimental data (blue crosses) is compared with the numeric FrFT (red curves). (b) A shifted input Gaussian profile evolves towards the centre of the array and acquires the same width as in a.
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f2: DFrFT of classical light.(a) Transformation of a Gaussian input into a Gaussian profile of larger width along the evolution in the Jx-array. The FT is obtained at Z=π/2. The experimental data (blue crosses) is compared with the numeric FrFT (red curves). (b) A shifted input Gaussian profile evolves towards the centre of the array and acquires the same width as in a.

Mentions: To experimentally demonstrate the functionality of the suggested waveguide system, we use N=21 waveguides to perform FTs of simple wave packets. We first consider a Gaussian wave packet with a full-width at half-maximum (FWHM) covering the five central sites (Fig. 2a). The input signal is prepared by focusing a Gaussian beam from a HeNe laser onto the front facet of the sample. By exploiting the fluorescence from colour centres within the waveguides21, we monitor the full intensity evolution from the input to the output plane. The fluorescence image, Fig. 2a, shows a gradual transition from an initially narrow Gaussian distribution at the input to a broader one at the Fourier plane (left and right panels Fig. 2a), demonstrating that narrow signals in space correspond to broad signals in Fourier space. For intermediate propagation distances (Z∈[0, π/2]) we extract other orders of the DFrFT, simultaneously. For comparison, we plot the continuous FrFT produced by the corresponding continuous Gaussian profile (red curves Fig. 2a). The agreement between the computed FrFT and the experimental DFrFT proves that for the considered Gaussian input signal, N=21 is sufficient to achieve the continuous limit. We now shift the input Gaussian beam by six channels towards the edge. Since the separations between adjacent waveguides at the edges are bigger than the separations between adjacent waveguides in the centre, the discretization grid is not perfectly homogeneous. Strictly speaking, the discretized shifted Gaussian just at the input plane covers slightly less than five waveguides FWHM. We observe that the well-approximated off-centre Gaussian travels to the centre at Z=π/2 (Fig. 2b), hereby showing the famous shift theorem. In additional experiments, extended signals, for example, a shifted top-hat function, are found to be well transformed according to equation (2) as well. However, we find that for this type of excitation N>21 would be required to discuss the continuous limit (see Supplementary Note 1 with Supplementary Fig. 1).


Implementation of quantum and classical discrete fractional Fourier transforms
DFrFT of classical light.(a) Transformation of a Gaussian input into a Gaussian profile of larger width along the evolution in the Jx-array. The FT is obtained at Z=π/2. The experimental data (blue crosses) is compared with the numeric FrFT (red curves). (b) A shifted input Gaussian profile evolves towards the centre of the array and acquires the same width as in a.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4814576&req=5

f2: DFrFT of classical light.(a) Transformation of a Gaussian input into a Gaussian profile of larger width along the evolution in the Jx-array. The FT is obtained at Z=π/2. The experimental data (blue crosses) is compared with the numeric FrFT (red curves). (b) A shifted input Gaussian profile evolves towards the centre of the array and acquires the same width as in a.
Mentions: To experimentally demonstrate the functionality of the suggested waveguide system, we use N=21 waveguides to perform FTs of simple wave packets. We first consider a Gaussian wave packet with a full-width at half-maximum (FWHM) covering the five central sites (Fig. 2a). The input signal is prepared by focusing a Gaussian beam from a HeNe laser onto the front facet of the sample. By exploiting the fluorescence from colour centres within the waveguides21, we monitor the full intensity evolution from the input to the output plane. The fluorescence image, Fig. 2a, shows a gradual transition from an initially narrow Gaussian distribution at the input to a broader one at the Fourier plane (left and right panels Fig. 2a), demonstrating that narrow signals in space correspond to broad signals in Fourier space. For intermediate propagation distances (Z∈[0, π/2]) we extract other orders of the DFrFT, simultaneously. For comparison, we plot the continuous FrFT produced by the corresponding continuous Gaussian profile (red curves Fig. 2a). The agreement between the computed FrFT and the experimental DFrFT proves that for the considered Gaussian input signal, N=21 is sufficient to achieve the continuous limit. We now shift the input Gaussian beam by six channels towards the edge. Since the separations between adjacent waveguides at the edges are bigger than the separations between adjacent waveguides in the centre, the discretization grid is not perfectly homogeneous. Strictly speaking, the discretized shifted Gaussian just at the input plane covers slightly less than five waveguides FWHM. We observe that the well-approximated off-centre Gaussian travels to the centre at Z=π/2 (Fig. 2b), hereby showing the famous shift theorem. In additional experiments, extended signals, for example, a shifted top-hat function, are found to be well transformed according to equation (2) as well. However, we find that for this type of excitation N>21 would be required to discuss the continuous limit (see Supplementary Note 1 with Supplementary Fig. 1).

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Related in: MedlinePlus