1.4.7. “Stokes-formalism” for polarization singularites. “stokes-vortices”

As it has been shown above (see subsection 1.4.2.3), the polarization singularities can be unambiguously identified using the interference technique whose practicalility is most pronouncing at zones where intensity of the field is low, in part in the case of weakly depolarized fields where zones with large changes of polarization are drew to the amplitude zeroes of the orthogonal components. Analysis of the structure of a vector field based on measurements of intensity (Stokes parameters etc.) is rather difficult in this case. At the same time, for the fields with large integral depolarization (more than 40 to 50%), localization of the polarization singularities is not connected directly with zones of low intensity, and such singularities can be identified through analysis of conventional local polarization parameters of a field [93,94]. On this reason, it would be useful to find out the interconnection of the polarization singularities with behavior of such characteristics of a vector field.

Assume that the complete set of the normalized Stokes parameters at the each point of the field has been obtained from the experimental data. The normalized Stokes parameters of a coherent field can be written in the form [93]:

. (1.148)

For that, the following relation is valid:

. (1.149)

Consider so-called Stokes-fields, , determined by the relations [93]:

. (1.150)

The fields of such kind are characterized by the system of singularities. Hereinafter, we will refer to such singularities as the Stokes-vortices [93]. The coordinates of the Stokes-vortices of the field can be found, similarly to common scalar vortices, as the solutions of the system:

. (1.151)

It follows from Eqs. (1.148) and (1.149) that only one Stokes parameter is not vanishing at the points of the Stokes-vortices. Moreover, the magnitude of this non-vanishing parameter equals unity.

So, for example, for the vortices of the field one has , and . Following, the vortices of the field coincide with -points. Let us introduce for description of vortices of the field additional parameter, , determining the sign of the non-zero Stokes parameter. Then, the vortex of the field with ( ) corresponds to -point localized within the area with a clockwise (counterclockwise) polarization.

Similarly, for the field , i.e. the vortices of this field coincide with the component vortices (the phase difference vortices), while either or vanishes.

Let us again consider Eqs. (1.150). The solutions of the first and the second equations determine the systems of some closed contours. So, one can see from Eq. (1.148) that the first of Eqs. (1.151) determine for the field the contours, along which the components have the same intensity, and the solutions of the second of Eqs. (1.151) form the system of -contours, cf. subsection 1.4.2.2.

The solutions of similar equations for the field correspond to the systems of - and -contours. In other words, the vortices of this field, similarly to the phase difference vortices (vortices of the non-diagonal element of a coherence matrix ), occur at the cross-sections of such contours. Indeed, one can shown that

. (1.152)

At last, the vortices of the field , viz. , arise at the cross-sections of -contours and the lines, along which intensities of the components are equal to each other. It is evident, the vortices of the field do not correspond to any conventional polarization singularities, being determining the coordinates of the “mark” points at -contours, where the azimuth of linear polarization is or .

It is clear, using the Stokes-formalism, one can formulate various sign principles concerning to the phase difference vortices of vortices or -points. Various topological invariants kind of the ones described by Eq. (1.95), can be also derived.