**4. Polarization singularities in partially coherent light beams**

In this section we describe specific polarization singularities arising in incoherent superposition of coaxial orthogonally polarized laser beams. It is shown that in transversal cross-section of paraxial combined optical beams of this class, instead of common singularities, such as amplitude zeroes (optical vortices) inherent in scalar fields [2], and polarization singularities such as C points and L lines inherent in completely coherent vector fields [18], *phase singularities of the complex degree of polarization* (CDP) arise, whose description and investigation have been initiated by papers [61 - 65] basing on earlier studies [66 - 68] concerned to the Young's concept of the edge diffraction wave in connection with diagnostics of phase singularities of spatial correlation functions of optical fields. There are U contours along which the degree of polarization equals zero and the state of polarization is undetermined (singular), and isolated P points where the degree of polarization equals unity and the state of polarization is determined by the non-vanishing component of the combined beam. (Note, discussed here notion of CDP differs from the definition of the complex degree of mutual polarization, CDMP [6] that is two-point function of an optical beam.)

Let us briefly argue the relevance of the introduced approach.

It is known [69] that each level of description of optical phenomena possesses its own set of singularities, *i.e.* the set of elements of a field (points, lines, surfaces, depending on considered dimension) where some parameter of a field is undetermined. Importance of detecting such elements of a field is caused by the fact that such elements form peculiar skeleton of a field, so that if one knows behavior of a field at such singular elements (and at nearest vicinities of them), one just can predict, at least in qualitative manner, but with high level of validity, behavior of a field at all other areas.

Conditionally, one can classify singularities of optical fields in the following manner [69]:


Singularities of geometrical optics are caustics where the field amplitude reaches infinity. Singularities of completely coherent wave fields are divided into two sub-classes: (i) for scalar (homogeneously polarized) fields and (ii) for vector (inhomogeneously polarized) fields. In scalar fields, when polarization can be neglected, so-called wave front dislocations take place (which are also referred to as amplitude zeroes or optical vortices). Phase of the complex amplitude is undetermined at such elements and is step-like changed at crossing of them. In vector fields optical vortices are absent, though they remain in any polarization ("scalar") component. Instead of vortices, polarization singularities arise at cross-section of a field, *viz.* field elements where azimuth of polarization (C points) or handedness (L lines) is undetermined [18]. Vector skeletons of coherent inhomogeneously polarized fields were elaborated in details in papers [70 - 72]. By crossing L lines, handedness is step-like changed into opposite one; by crossing C point, azimuth of polarization is changed into orthogonal one. These types of singularities are blurred in quantum-mechanical description being "camouflaged" by so-called quantum vacuum [69], though the distance of influence of such quantum core is rather small. Its linear size is of order of magnitude <sup>3</sup> (about 6 Å for He-Ne laser).

All mentioned singularities disappear in the case of partially coherent wave fields (though they remain in each completely coherent component, mode in a set of which partially coherent radiation is decomposed. Instead (beside) of them, new singularities appear inherent just in partially coherent fields. Let us emphasize, that singularities of partially coherent fields have formed the novel topic in the field of singular optics just at the beginning of the Third Millenium [3].

For that, two situations arise again: (i) scalar case when polarization can be ignored while the state of polarization is the same at all point of a field, and (ii) vector case when the state of polarization of *partially coherent* field changes from point to point that requires explicit taking into account of vector nature of light. The first (scalar) case became the subject of intense investigations in last years [67, 73 - 76]. As a result of these investigations, new phase singularities of spatial and temporal correlation functions of quasi-monochromatic light fields have been revealed, as well as singularities of spectral components of polychromatic ("white-light") radiation [68, 77 - 82]. For that, vector singularities of partially coherent light fields have been revealed just recently [61 - 65]. Such singularities are elaborated in this section.
