Fractional-order Fourier transforms are adapted to the mathematical expression of Fresnel diffraction, just as the standard Fourier transform is adapted to Fraunhofer diffraction. The continuity of fractional Fourier transforms with respect to their orders corresponds to the continuity of wave propagation, and their composition is in accordance with the Huygens principle.
The propagation of transverse spatial correlations of photon pairs through
arbitrary first-order linear optical systems is studied experimentally and
theoretically using the fractional Fourier transform. Highly-correlated photon
pairs in an EPR-like state are produced by spontaneous parametric
down-conversion and subject to optical fractional Fourier transform systems. It
is shown that the joint detection probability can display either correlation,
anti-correlation, or no correlation, depending on the sum of the orders
$\alpha$ and $\beta$ of the transforms of the down-converted photons. We
present analytical results for the propagation of the perfectly correlated EPR
state, and numerical results for the propagation of the two-photon state
produced from parametric down-conversion. We find good agreement between theory
and experiment.Comment: 9 pages, 7 figures, to appear PR
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