We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the FT. There is a qualitative increase in the richness of the solution manifold, from U(1) (the circle <i>S</i><sup>1</sup>) in the one-dimensional case to U(2) (the four-parameter group of 2 × 2 unitary matrices) in the two-dimensional case [rather than simply U(1) × U(1)]. Our treatment clarifies the situation in the <i>N</i>-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus <i>T</i><sup>2</sup> = <i>S</i><sup>1</sup> × <i>S</i><sup>1</sup> and two parameters running over the Fourier sphere <i>S</i><sup>2</sup>. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the <i>T</i><sup>2</sup> coordinates; the eigenfunctions, only on the <i>S</i><sup>2</sup> coordinates. FrFT’s corresponding to special points on the Fourier sphere have for eigenfunctions the Hermite–Gaussian beams and the Laguerre–Gaussian beams, while those corresponding to generic points are SU(2)-coherent states of these beams. Thus the integral transform produced by every Sp(4, ℜ) first-order system is essentially a FrFT.
© 2000 Optical Society of America
R. Simon and Kurt Bernardo Wolf, "Fractional Fourier transforms in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000)