We consider a class of beams that are both partially polarized and partially coherent from the spatial standpoint. They are characterized by a correlation matrix whose elements have the same form as the mutual intensity of a Gaussian Schell-model beam. We focus our attention on those beams that would appear identical to ordinary Gaussian Schell-model beams in a scalar treatment. After establishing some inequalities that limit the choice of the matrix parameters, we study the main effects of propagation. Starting from the source plane, in which the beam is assumed to be uniformly polarized, we find that in the course of propagation the degree of polarization generally becomes non-uniform across a typical section of the beam. Furthermore, we find that the intensity distribution at the output of an arbitrarily oriented linear polarizer is Gaussian shaped at the source plane whereas it can be quite different at other planes.

No 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...

An experimental procedure for the synthesis of a class of partially polarized Gaussian Schell-model sources is proposed. The experimental set-up is based on Mach-Zenhder interferometer. This kind of sources give rise to not only a coherence degree and intensity profile that are both Gaussian but also a Gaussian local degree of polarization in each plane upon free propagation. Measurements of the local degree of polarization and the intensities in the far field are presented. It is shown that from these measurements some characteristic parameters of the source can be obtained. (C) 2002 Elsevier Science B.V. All rights reserved

It is shown that any partially polarized, partially coherent source can be expressed in terms of a suitable superposition of transverse coherent modes with orthogonal polarization states. Such modes are determined through the solution of a system of two coupled integral equations. An example, for which the modal decomposition is obtained in closed form in terms of fully linearly polarized Hermite Gaussian modes, is given.