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Frontiers of Physics

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

Hong-Yi Fan, Junhua Chen

Published Sep 13, 2014• 14 citations
Publisher

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.

DOI
10.1007/s11467-014-0445-x

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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.
Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between [Formula: see text] and [Formula: see text], two induced entangled state representations, and working with them, we derive fractional Fourier–Hankel transformation (FrFHT) caused by the operator [Formula: see text], where [Formula: see text] is named the core operator and is essential to the fractional transformation. The fractional property (additive rule) of the FrFHT can be explicitly proved.
We employ the recently established basis (the two-variable Hermite-Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A303, 311 (2002)] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.