Use of the van Cittert–Zernike theorem for partially polarized sources

Abstract

We introduce an extension of the van Cittert -Zernike theorem to spatially incoherent sources with partial polarization. We show through a simple example that fields generated by such sources can possess correlation matrices with interesting properties. In particular, we show that by suitable modulation of the polarization state across the incoherent source, the correlation between the orthogonal components of the field as well as the degree of polarization may drastically change on propagation. © 2000 Optical Society of America OCIS codes: 030.0030, 060.2380, 230.6080, 260.5430. There is currently an interest in optical beams that are both partially coherent from the spatial standpoint and partially polarized. -4Such beams can be described by an approximate version of the general tensorial theory of the electromagnetic f ield developed by Wolf. 5In fact, a single 2 3 2 matrix, called a beam coherencepolarization (BCP) matrix, is suff icient to yield a complete account of the second-order statistical properties of the f ield. 3In scalar coherence theory, the van Cittert-Zernike theorem is a fundamental tool for studying the propagation processes of partially coherent fields. 5,6 Moreover, several partially coherent beams are generated starting from a primary spatially incoherent source. 5,7,8 In such synthesis procedures the van Cittert -Zernike theorem plays a major role. We have observed that a spatially incoherent source can exhibit partial polarization and that the polarization state can change from one point to another across the source. Thus a suitable extension of the van Cittert -Zernike theorem to these sources should be sought. In this Letter we introduce such an extended version of the theorem. We then work out a specific example. It will be shown that the correlation functions appearing in the BCP matrix can behave in a rather different way with respect to each other. This will help the reader to appreciate the significance of the vectorial case. Furthermore, our results suggest that in the vectorial case, too, the van Cittert-Zernike theorem can serve as a useful tool in synthesis processes.We use a reference frame in which the z axis coincides with the mean direction of propagation of the beam. At a typical transverse plane, the vector r is used to specify the position of a point. Let us recall that, for a quasi-monochromatic field, the BCP matrix is def ined as 3 J͑r 1 , r 2 , z͒ ∑ J xx ͑r 1 , r 2 , z͒ J xy ͑r 1 , r 2 , z͒ J yx ͑r 1 , r 2 , z͒ J yy ͑r 1 , r 2 , z͒whereThe angle brackets denote time average, and E a ͑a x, y͒ is a Cartesian component of the time-dependent electric f ield. In the framework of scalar theory a spatially incoherent source is characterized by means of a d-like mutual intensity function, which expresses the fact that the fields at any two distinct points across the source plane are uncorrelated. 5In the same way we def ine a partially polarized, spatially incoherent source as one whose BCP matrix elements J ab have the formwhere d is the two-dimensional Dirac function...

Synthesis and characterization of partially coherent beams with propagation-invariant transverse polarization pattern

“…This means that all points belonging to a circle ρ =constant has to be perfectly correlated with one another. Such a requirement prevents the use of the van Cittert-Zernike theorem for the synthesis of such beams, as it is customary for all sources of the Schell-model type [3,23,24], and a different approach must be used.…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). In particular, it is well known that a nonuniformly polarized beam generally changes its polarization pattern upon propagation, even in the case of perfectly polarized and deterministic field distributions in paraxial conditions.…”

Partially Coherent Linearly Polarized Sources with Inhomogeneous Azimuth

“…Recently there is a great interest in proposing new sources of light with special characteristics of coherence and polarization including non uniformly polarized fields with controllable distributions across the transversal section (Ambrosini et al, 1994;Gori et al, 1998;Seshadri, 1999;Gori et al, 2000;Gori, 2001;Gori et al, 2001;Piquero et al, 2002;Pääkkönen et al, 2002;Shirai et al, 2005;Zhan, 2009;Brown and Zhan, 2010;Liang et al, 2014;Rodrigo and Alieva, 2015;Borghi et al, 2015;Mei and Korotkova, 2016;Wang and Korotkova, 2016). Works can be found in the literature that deal with partially coherent and partially or totally polarized sources, from both theoretical and experimental points of view (Friberg and Sudol, 1982;Friberg and Turunen, 1988;Serna et al, 1992;Santarsiero et al, 1999;Piquero et al, 2002;Santarsiero et al, 2009;Ramírez-Sánchez et al, 2010;de Sande et al, 2012;Maluenda et al, 2013).…”

3D polarisation speckle as a demonstration of tensor version of the van Cittert-Zernike theorem for stochastic electromagnetic beams

“…The matrix form, tensor version extension of the conventional Van Cittert-Zernike theorem is first described by Gori 8,20 in the temporal domain, and then similar work in the spectral domain by Ostrovsky 19 . Further research work about the variation of the degree of coherence and degree of polarization within this field was developed both theoretically 18,21,22 and experimentally 23 soon after.…”