Abstract.The e ect of asymmetric apodization is analysed using the fractional Fourier transform. The focusing properties of an apodizing pupil mask are investigated by means of a simple display in a generalized phase-space (or x±p domain). Some comparative computer simulations are performed and curves displaying the Strehl ratio versus defocus are presented in order to illustrate the possibilities of our approach. IntroductionSeveral e orts to improve the quality of the point spread function (PSF) of an optical imaging system have been done by using apodizer apertures. The purpose of the apodization process is to produce the suppression of the secondary maxima, or side lobes, of a di raction pattern in order to enhance the resolving power of the optical system. It is a classical problem and it has been widely studied [1,2]. In the apodization method proposed by Cheng and Siu [3], the aperture is modi®ed in an asymmetric mode. This asymmetric aperture function produces the suppression of the secondary maxima at a side of the central peak of the di raction pattern at the cost of increasing the lobes at the opposite side [4].The fractional Fourier transform (FRT) was introduced in optical signal processing by Ozaktas and Mendlovic [5] and Lohmann [6]. Two di erent optical de®nitions of the FRT have been given. In the ®rst, the FRT was de®ned based on light propagation in quadratic graded index media. In the second de®nition, the FRT of fractional order p results from a phase-space rotation of the input Wigner distribution function by an angle of pº=2. The FRT is a generalization of the classical Fourier transform and the information content stored in the FRT changes from spatial to spectral as the fractional order varies from pˆ0 to pˆ1. The relation between a FRT of order p and the free-space di raction [7,8] allows the analysis of the tolerance of optical imaging systems to focus errors and/or aberrations. There are several criteria to analyse the performance of an optical imaging system based on the on-axis irradiance. The Strehl ratio (SR), de®ned as the intensity values on axis at the di raction focus conveniently normalized, is an important image quality parameter [9]. The purpose of this letter is to analyse the