Wigner function and ambiguity function for non-paraxial wavefields Cohn JR Sheppard*, Kieran G Larkin, c a School of Physics, University of Sydney; b Australian Key Centre for Microscopy and Microanalysis, University of Sydney; C Canon Information Systems Research Australia ABSTRACT The connection between the Wigner function and the generalized OTF, and between the ambiguity function and the generalized OTF is investigated for non-paraxial scalar wavefields. The treatment is based on two-dimensional (2-D) wavefields for simplicity, but can be extended to the three-dimensional case. Keywords: diffraction, propagation, phase-space representation, Wigner function, ambiguity function, OTF, transforms 1. INTRODUCTION In the paraxial regime, the Wigner function1 and ambiguity function2 are two phase-space representations that can be used to describe propagation of waves. For example Brenner, Lohmann and Ojeda-Castañeda3 described how the ambiguity function can be used as a polar display of the defocused OTF. However, it is well known that, although these phase-space reprentations are useful in the paraxial regime, for highly-convergent fields their form changes upon propagation, an effect analogous to the introduction of aberrations. Wolf et al.4 introduced a form of Wigner function for 2-D non-paraxial wavefields, called the angle-impact marginal. This has the properties that it is real and covariant under translation or rotation. An alternative representation for wave propagation is based on the concepts of the generalized pupil function,5 and the generalized OTF,612 which are two- or three-dimensional functions for two- or three- dimensional wavefields, respectively. These generalized functions have been investigated in the paraxial regime, and for highly-convergent scalar and vector wavefields. We have found that the concept of the generalized OTF is useful in visualizing the derivation of the Wigner function.13 These different representations have also found use in the phase retrieval problem, where knowledge of the intensity in the focal region can be used to reconstruct the phase variations. 14-18 2. DERIVATION OF THE ANGLE-IMPACT MARGINAL We consider the two-dimensional problem of a scalar wave propagating in a plane. The amplitude in the focal region can be written as the Fourier transform of the generalized pupil: U(r) = _f $H(m)exp(ik m r)d2m, (1) *colin@physicsusydeduau fax +61 2 9351 7727, Physical Optics Department, School of Physics, University of Sydney, NSW 2006 Australia; kieran@research.canon.com.au, fax +61 2 9805 2929, Canon Information Systems Research Australia, 1 Thomas Holt Drive, North Ryde, NSW 2113, Australia Optical Processing and Computing: A Tribute to Adolf Lohmann, David P. Casasent, H. John Caulfield, William J. Dallas, Harold H. Szu, Editors, Proceedings of SPIE Vol. 4392 (2001) © 2001 SPIE · 0277-786X/01/$15.00 99