Wigner Distribution Moments Measured as Intensity Moments in Separable First-Order Optical Systems
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Wigner Distribution Moments Measured as Intensity Moments in Separable First-Order Optical Systems Martin J. Bastiaans Faculteit Elektrotechniek, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, The Netherlands Email: [email protected]
Tatiana Alieva Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Ciudad Universitaria s/n, Madrid 28040, Spain Email: [email protected] Received 31 March 2004; Revised 19 August 2004 It is shown how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of separable first-order optical systems (generally anamorphic ones). The minimum number of such systems that are needed for the determination of these moments is derived. Keywords and phrases: Wigner distribution moments, beam characterization, first-order optical systems.
1.
INTRODUCTION
After the introduction of the Wigner distribution (WD) [1] for the description of coherent and partially coherent optical fields [2], it became an important tool for optical signal/image analysis and beam characterization [3, 4, 5]. The WD completely describes the complex amplitude of a coherent optical field (up to a constant phase factor) or the mutual coherence function of a partially coherent field. As the WD of a two-dimensional optical field is a function of four variables, it is difficult to analyze. Therefore, the optical field is often represented not by the WD itself, but by its global moments. Beam characterization based on the second-order moments of the WD thus became the basis of an International Organization for Standardization standard [6]. Some of the WD moments can directly be determined from measurements of the intensity distributions in the image plane or the Fourier plane, but most of the moments cannot be determined in such an easy way. In order to calculate such moments, additional information is required. Since first-order optical systems [7]—also called ABCD systems— produce affine transformations of the WD in phase space, the intensity distributions measured at the output of such systems can provide such additional information. The application of ABCD systems for the measurements of the second-
order WD moments has been reported in several publications [8, 9, 10, 11, 12, 13]. It is the aim of this paper to show how all global WD moments can be measured as intensity moments only. We show that not only the second-order moments, but also all other moments of the four-dimensional WD can be obtained from measurements of only intensity distributions in an appropriate number of (generally anamorphic) separable first-order optical systems. 2.
WIGNER DISTRIBUTION
Let partially coherent light be described by a temporally stationary stochastic process f (x, y; t); as far as the time dependence is concerned, the ensemble average of the product f (x1 , y1 ; t1 ) f ∗ (x2 , y2 ; t2 ), where the asterisk denotes complex conjugation, is then only a function of the time difference t1 − t2 :
E f x1 , y1 ; t1 f
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