June 2001 Phys. Chem. News 2 (2001) 29-34 PCN 29 THE ABCD-HANKEL TRANSFORMATION IN TWO-DIMENSIONAL FREQUENCY-DOMAIN WITH POLAR COORDINATES M. Ibnchaikh, A. Belafhal * Laboratoire de Physique Moléculaire, Département de Physique, B.P 20, Faculté des Sciences, Université Chouaïb Doukkali, 24000 El Jadida, Morocco * Corresponding author. E-mail : Belafhal@ucd.ac.ma Received 01 November 2000; revised version accepted 06 April 2001 Abstract This work is devoted to a theoretical study of the ABCD-Hankel transformation. As the frequency- domain is very important in optics, we investigate a novel general Collins formula in two dimensions with polar coordinates. We also investigate the direct relationship between input and output spatial frequency spectra of a light field in this frame system in the case of a Fourier, Fresnel and fractional Fourier (FRT) transformations for two families of beams: Bessel-Gaussian and Bessel-modulated Gaussian beams. Keywords: ABCD-Hankel; Transformation; Collins Formula; Fourier; Fresnel; Fractional Fourier. 1. Introduction One of the most basic problems in optics is the determination of the propagation characteristics of beam waves in a paraxial optical system. In 1970, Collins [1] has found an important relationship between input and output spatial frequency spectra of a light field through an optical system characterized by A, B, C and D elements of the ray transfer matrix of the system, which is called Collins formula. We know that the ABCD law [2- 5] governs the propagation and transformation of Gaussian, Hermite-Gaussian, Laguerre-Gaussian, non-Gaussian and nonspherical light beams through paraxial systems. Earlier, Zalevsky et al [6] have provided some analytical tools on the ABCD-Bessel transformation by extending the 1-D transformation to a 2-D one, by studying some special cases. In the same year, Liu et al [7] have proposed a Collins formula in frequency-domain in two dimensions with Cartesian coordinates. This work still available for a class of beams that can be expressed in terms of Cartesian coordinates. However, in our knowledge the ABCD-Hankel transformation, which is a Bessel transformation, in two-dimensional frequency- domain with polar coordinates has not been treated before. This study is very important when the propagation of rotational symmetric beams is in need, which is the case for Bessel-Gaussian and Bessel-modulated Gaussian beams. As a follow up of previous research, we will attempt to deduce the amplitude distribution at the output plane of a paraxial optical system, characterized by ABCD matrix, illuminated by a light field, in frequency-domain with polar coordinates. On the other hand, we know that the Hankel transformation embraces several other optical transformations, so we determine the corresponding Collins formula for several cases as Fourier, Fresnel, FRT and RSOS transformations. 2. Collins formula in a two dimensional space domain with the polar coordinates We present in this section the basic theory of the ABCD-Hankel transformation derived from the Collins formula. For this, we consider an ABCD optical system illuminated by a light field represented by a transverse amplitude distribution ) y , x ( u 1 1 1 at the input plane and by ) y , x ( u 2 2 2 at the output plane. In this work, we consider that the input and output planes are embedded into the same optical medium. So, the determinant BC AD − of the corresponding ABCD matrix should be unity. In a space domain, the generalized Huygens-Fresnel integral or the Collins formula for one transversal direction of an orthogonal system relates the output field to the input one, and the ABCD elements can be written as [1] ⎩ ⎨ ⎧ λ π ∫∫ = B i exp ) y , x ( u ' C ) y , x ( u 1 1 1 2 2 2 [ ] } 1 1 2 1 2 1 2 2 2 2 2 1 2 1 dy dx ) y y x x ( 2 ) y x ( D ) y x ( A + − + + + × , (1) where k is the wave number and B i ' C λ − = . With polar coordinates, one finds that the relationship between ) , ( u 1 ψ ρ and ) , r ( u 2 ϕ is given by [7]