Continuous vs. discrete fractional Fourier transforms

摘要

We compare the finite Fourier (-exponential) and Fourier–Kravchuk transforms; both are discrete, finite versions of the Fourier integral transform. The latter is a canonical transform whose fractionalization is well defined. We examine the harmonic oscillator wavefunctions and their finite counterparts: Mehta's basis functions and the Kravchuk functions. The fractionalized Fourier–Kravchuk transform was proposed in J. Opt. Soc. Amer. A (14 (1997) 1467–1477) and is here subject of numerical analysis. In particular, we follow the harmonic motions of coherent states within a finite, discrete optical model of a shallow multimodal waveguide.