Fractional Fourier transforms in two dimensions

We, analyze the fractionalization of the Fourier transform (FT), starting f rom the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the There i s a qualitative increase in the richness of the solution manifold, from U(1 ) (the circle S') in the one-dimensional case to U(2) (the four-parameter g roup of 2 x 2 unitary matrices) in the two-dimensional case [rather than si mply U(1) x U(1)]. Our treatment clarifies the situation in the N-dimension al case. The parameterization of this manifold (a fiber bundle) is accompli shed through two powers running over the torus T-2 = S-1 x S-1 and two para meters running over the Fourier sphere S-2. We detail the spectral represen tation of the FrFT: The eigenvalues are shown to depend only on the T-2 coo rdinates; the eigenfunctions, only on the S2 coordinates. FrFTs correspondi ng to special points on the Fourier sphere have for eigenfunctions the Herm ite-Gaussian beams and the Laguerre-Gaussian beams, while those correspondi ng to generic points are SU(2)-coherent states of these beams. Thus the int egral transform produced by every Sp(4, R) first-order system is essentiall y a FrFT. (C) 2000 Optical Society of America [S0740-3232(00)00512-3] OCIS codes: 070.2590, 080.2730.