Fractional-order Fourier transformations as pseudo-differential operators. Applications to diffraction and imaging by optical systems

Abstract: Circular and hyperbolic fractional-order Fourier transformations are actually Weyl pseudo-differential operators. Their associated kernels and symbols are written explicitly. Products of fractional-order Fourier transformations are obtained by composing their kernels or symbols, in accordance with the Weyl calculus. On the other hand, optical field transfers by diffraction, from spherical emitters to receivers, are mathematically expressed using fractional-order Fourier transforms and are classified into three categories, for which we provide geometrical characterizations. Respecting the Huygens-Fresnel principle requires that the associated transformations be adequately composed, which is achieved through the composition rules of pseudo-differential operators. By applying the Weyl calculus, we prove that, in general, imaging through a refracting spherical cap can be described as the composition of two fractional-order Fourier transformations, but only if those transformations are of the same kind. Basic laws of coherent geometrical imaging are thus deduced from the composition of two circular fractional-order Fourier transformations. We also prove that the arrangement of a sequence of points along the axis of a centered optical system is preserved by imaging, up to a circular permutation. This recovers a classical result from geometrical optics, but here it is derived within the framework of diffraction theory.