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 S1) 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 N-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus T2 = S1 × S1 and two parameters running over the Fourier sphere S2. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the T2 coordinates; the eigenfunctions, only on the S2 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)