Metaxial correction of fractional Fourier transformers

We study fractional Fourier transformation in the metaxial regime of geomet ric optics. Two commonly used optical arrangements that perform fractional Fourier transformation are a symmetric thick lens and a length of graded-in dex waveguide. By means of Lie methods in phase space, we can correct some of their aberrations: for the first, through deforming the lens surfaces to a polynomial shape, and for the second, by warping the output screen at th e end of the waveguide. We correct the planar cases to third, fifth, and se venth aberration orders; checks are provided on the convergence of aberrati on series in phase space. We add some comments on the usefulness of these c orrected devices for fractional transformers in scalar wave optics. (C) 199 9 Optical Society of America [S0740-3232(99)00904-7].