We provide a matrix formalism of optical systems based on the Wigner distribution function (WDF).The resulting Wigner matrix includes elements that characterize wavelength dependent parameters f the optical systems.By factoring the general Wigner matrix of optical fractional Fourier transformations (ERT),we discuss the different physical interpretations about the input and output scale parameters.By undrstanding what is needed to reduce the wavelength dependence.we further design the optical systems which can implement the achromatic FRT with low chromatic aberrations over the entire visible spectrum.