1.4.5. Elementary polarization structures and elementary polarization singularities of vector fields

To understand the general topological properties of a vector field, namely, to understand interconnection between the behavior of polarization parameters of such field and its topology, one must consider some elementary polarization structures associated with -contours and -points.

Elementary polarization structures occurring in a vector field were comprehensively analyzed early [3,75] for the areas adjacent to the polarization singularities. However, the analysis in the cited papers is of local character and concerns to the only type of polarization singularities. At the same time, it follows from the consideration in subsection 1.4.1 that all types of vector singularities are mutually predetermining. That is why, it is expedient to consider the fields with relatively simple polarization structure, which contain both types of polarization singularities, viz. both -contour and -point.

It is obvious that such elementary vector fields can be formed as a result of superposition of the proper uniformly polarized simple waves, whose characteristics can be determined from the decomposition of a whole field into orthogonal components. Elementary polarization singularities can be obtained as a result of superposition of both two orthogonally linearly polarized vortex beams and two orthogonally circularly polarized relatively simple beams.

1.4.5.1. Polarization structures resulting from interference of orthogonally linearly polarized beams

The simplest vector fields contain -contours with the minimal number (two) phase difference vortices. The area limited by -contour bears non-zero total topological charge ( ), if these vortices belong to the orthogonal components of the field [80,81,79], i.e. the total field can be regarded as a superposition of two orthogonally polarized vortex beams.

Let us assume for simplicity that both vortex beams are formed by isotropic vortices [12]. In polar coordinates with the origin at the centre of vortex, complex amplitude of such a beam can be represented by the following relation:

, (1.102)

where is the topological charge, is the initial phase of the vortex beam, and is the amplitude vanishing at the centre of a vortex.

Let the propagating vortex beams be collinear, and the centres of vortices are distanced by . One can introduce Cartesian coordinates ( ) with the centre equally distanced from the vortex centers, and the axis being connecting the vortices, while the direction of the axis being coinciding with the direction of propagation of the partial vortex beams.

It can be shown that the contour corresponding to arbitrary phase difference, , being determined within , is described by the relation:

, (1.103)

where is the initial phase difference of the vortex beams, and are the corresponding topological charges.

If and are of the same sign, Eq. (1.103) is reduced to the form:

, (1.104)

and if and are of opposite signs, Eq. (1.103) is reduced to the form:

. (1.105)

Eq. (1.105) determines the contours of hyperbolic type, which are disclosed. Now we analyze in more details Eq. (1.104), which is the equation of a circle with the radius and the center determined by the distance between partial vortices and by the phase differences, . Note, Eq. (1.104) is transformed to the equation of elliptic type, if the resulting field results from superposition of anisotropic vortices.

Similar equations describe the contours of a whole field resulting from superposition of the vortices of any structure, if the distance between the vortices is less than the radius of the vortex core.

Let us turn to the analysis of Eq. (1.104). Assume the radii of the vortex beam be much exceeding the distance between their centers, and , where . One can conclude in this case that the relation

(1.106)

is fulfilled for arbitrary point of the field, excluding the nearest vicinity of the vortex centers, where and vanish.

Interconnection between the amplitudes of the component, the phase difference and the lengths of the ellipse axes is determined by the following relation [28]:

, (1.107)

where , are small and large half-axes of the polarization ellipse, respectively, and .

It follows from Eq. (1.107) that the difference of polarization characteristics determined by the difference of ellipticities of the ellipses in different points of the field is unambiguously determined by the phase difference . We assume that the difference of polarization characteristics in these points is small if , what corresponds to the changing by 0.2. It follows from Eq. (1.104) that the area where considerable difference of the polarization characteristics take place is comparable in size and localization with the area limited by the phase difference contours obeying the inequality . There if the equation of such contours:

, (1.108)

irrespective of the phase difference in the vortex beams, i.e. it is determined by the distance of the vortex centers alone. For that, the phase difference contour corresponding to within accuracy of is the straight line containing the centers of two vortices.

a b c

Figure 1.38. Characteristics of the phase difference contours and the loci of -points in respect of a phase difference of the vortex beams.

Figure 1.38 illustrates this consideration. One can see from Figure 1.38(a) that for , -contour is a circle whose diameter equals the distance between the vortices. Both -points are equally distanced from the vortices. For , cf. Figure 1.38(с), one of -points tends to the straight line containing the vortices, and another -point is at infinity. Figure 1.38(b) corresponds to the phase difference .

Figures 1.39 to 1.42 show the results of computer simulation of the elementary polarization singularities resulting from interference of the orthogonally circularly polarized vortex beams. We controlled not only the phase difference between the vortex beams, but the parameters of the vortices, such as the sign of the topological charge, curvature of equiphase lines in the nearest vicinity, also. All parameters are represented at the right lower corners of figures. Namely, we specify the wave

a b c

Figure 1.39. Interference of isotropic vortices of the same sign with a phase difference between the vortex beams : (a),(b) – phase maps of vortices, (с) – behavior of a phase difference of the resulting beam

a b c

Figure 1.40. Interference of isotropic vortices of the same sign with small difference of curvature of the equiphase lines within the vortex core

a b c

Figure 1.41. Interference of isotropic vortices of the same sign with large difference of curvature of the equiphase lines within the vortex core

a b c

Figure 1.42. Interference of isotropic vortices of the opposite signs

front curvature along axes and , the distance and the phase difference between the vortices, as well as coincidence or non-coincidence of the topological charges in sign. Fragments (a),(b) are the phase maps of the interfering vortices. Fragments c are the phase difference of the resulting field within to (the phase differences different by designated by the same levels of grey). The positions of - and -contours are indicated by the arrows. The correspondence between the level of grey and the magnitude of a phase or a phase difference is argued in fragments (d).

Figure 1.39 corresponds to interference of isotropic vortices with a phase difference between the vortex beams . As it follows from the above consideration, all phase difference contours are the circles. Figure 1.40 illustrates the fact that, as soon as the distance between the vortices is such so the equiphase lines at the core zone are not too curved, the phase difference contour corresponding (within to the modulus of the phase difference between the vortex beams, here ) is closed, and its size is comparable with the distance between the vortex centers. It is natural, the contour form is distorted. Under further increasing of curvature of the equiphase lines of a field of phases of the interfering vortices (or under increasing the distance between them), such contours are disclosed and transformed into contours of complicate form, for example, take the form of a spiral, cf. Figure 1.41.

For comparison, the results of simulation of interference of the vortices with the opposite signs of the topological charge are shown in Figure 1.42. One can see that all phase difference contours are disclosed even for isotropic vortices.

Let us consider once more Figure 1.40. One can see from this figure that the interior -contour containing the centers of the vortex beams is surrounded the additional -contours. In other words, the inserted -contours, like as “Matryoshka” (see comments in subsection 1.3.1) are formed. Within the basis of decomposition corresponding to the figure, the phase difference vortices are absent at the external contours. It follows that the changing azimuth of linear polarization under circumference of any exterior contour does not approach . Thus, the total topological charge of -points inside the each exterior contour equals zero.