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On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

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Abstract

By a quantum mechanical analysis of the additive rule F α[F β[f]]=F α+β[f], which the fractional Fourier transformation (FrFT) F α[f] should satisfy, we reveal that the position-momentum mutual-transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.

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Authors and Affiliations

  1. Department of Physics, Ningbo University, Ningbo, 315211, China

    Hong-Yi Fan

  2. Department of Material Science and Engineering, University of Science and Technology of China, Hefei, 230026, China

    Hong-Yi Fan & Jun-Hua Chen

Authors
  1. Hong-Yi Fan
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  2. Jun-Hua Chen
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Correspondence to Hong-Yi Fan.

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Fan, HY., Chen, JH. On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations. Front. Phys. 10, 1–6 (2015). https://doi.org/10.1007/s11467-014-0445-x

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  • DOI: https://doi.org/10.1007/s11467-014-0445-x

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