Fractional Fourier transforms, symmetrical lens systems, and their cardinal planes.

We study the relation between optical lens systems that perform a fractional Fourier transform (FRFT) with the geometrical cardinal planes. We demonstrate that lens systems symmetrical with respect to the central plane provide an exact FRFT link between the input and output planes. Moreover, we show that the fractional order of the transform has real values between 0 and 2 when light propagation is produced between principal planes and antiprincipal planes, respectively. Finally, we use this new point of view to design an optical lens system that provides FRFTs with variable fractional order in the range (0,2) without moving the input and output planes.