Fractional Fourier transform-based description of the Talbot effect: application to analog signal processing

The Talbot effect, or self-imaging, is a fascinating feature of Fresnel diffraction, where an input periodic wavefront is periodically recovered after specific propagation distances through free space. Interestingly, the Fresnel propagator shows a great similarity to the fractional Fourier transform (FrFT). In this paper, we provide an interpretation of the Talbot effect in the frame of the FrFT and derive simple summation formulas between the FrFT of a function and the function itself. In particular, we show that both the FrFT and the Fourier transform (FT) of any input function can be generated by coherent addition of spatially shifted replicas of the function itself, multiplied by a quadratic phase term. Transposed into the temporal domain, these results may have important applications for real-time analog computation of the FrFT/FT of arbitrary signals.