Wide angle near-field diffraction and Wigner distribution.pdf

a r X i v : p h y s i c s / 0 3 0 5 0 2 6 v 1 [ p h y s i c s .o p t i c s ] 8 M a y 2 0 0 3 Wide angle near-field diffraction and Wigner distribution Jose? B. Almeida Universidade do Minho, Physics Department, 4710-057 Braga, Portugal Vasudevan Lakshminarayanan University of Missouri - St. Louis, School of Optometry and Department of Physics and Astronomy, St. Louis, MO 63121, USA Abstract Free-space propagation can be described as a shearing of the Wigner distribution func- tion in the spatial coordinate; this shearing is linear in paraxial approximation but assumes a more complex shape for wide-angle propagation. Integration in the frequency domain al- lows the determination of near-field diffraction, leading to the well known Fresnel diffrac- tion when small angles are considered and allowing exact prediction of wide-angle diffrac- tion. The authors use this technique to demonstrate evanescent wave formation and diffrac- tion elimination for very small apertures. The Wigner distribution function (WDF) provides a convenient way to describe an optical signal in space and spatial frequency [1, 2]. The propagation of an optical signal through first- order optical systems is well described by the WDF transformations [3, 4], which allows the reconstruction of the propagated signal. On the other hand, some authors have linked Fresnel diffraction and the fractional Fourier transform (FRFT) [5, 6]; both of these papers associate free-space propagation to a rotation of the WDF and the corresponding FRFT accompanied by a quadratic phase factor. 1 In this paper we show that free-space propagation is always associated with shearing of the WDF. This can be used to evaluate the near-field diffraction. In the paraxial approximation our results duplicate the well known Fresnel diffraction, with the advantage that we do not need to resort to the Cornu integrals. The same procedure can be extended to wide angles, where other phenomena are apparent, namely the presence of e