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The fractional Fourier transform (FRT) is known to be optically implementable with use of a medium with a perfect radial  quadratic-index profile. Using the quantum-mechanical operator formalism, we examine theeffects on the FRT action of such a medium that are due to small random inhomogeneities in the longitudinal direction, the direction of propagation, and we formulate the random fractional Fourier transform (RFRT).Applying the RFRT to a self-fractional Fourier function, a Gaussian function, we discuss both the total power and the variance. The random Fourier transform is examined as a special limiting case

Random fractional Fourier transform: stochastic perturbations along the axis of propagation