1.4.4. Interconnection of the component vortices and -points

Let us again represent the vector field , cf. Figure 1.32, as the sum of two orthogonally linearly polarized components, . Complex amplitudes of these components are connected with the parameters of the orthogonally circularly polarized components, cf. Figure 1.33, by the following relation [3,78,79]:

(1.81)

where are the amplitudes and the phases of clockwise and counterclockwise circularly polarized components. Let for determinicity that the -contour limits the area with clockwise polarization. In the points of the -contour where the axes of the decomposition basis are parallel to the direction of oscillation of the field, one of the components has a vortex.

Figure 1.32 Figure 1.33 Figure 1.34

Consider in more details one of two components, to say the -component. The first of Eqs. (1.81) can be rewritten in the form:

. (1.82)

For linear polarization ( -contour, ) one obtains:

. (1.83)

Decomposing the field into linearly polarized components, one has:

. (1.84)

Note, is a continuous and smooth along the -contour function. is undetermined in singular points and changes by at crossing vortex. Singular point corresponds to the azimuth of polarization in Eq. (1.83). Changing the sign of cosine at crossing these points just corresponds to addition or subtraction of a constant phase whose magnitude is . Let us rewrite Eq. (1.83) in the form:

. (1.85)

It follows from Eq. (1.85) that

. (1.86)

Let divide the -contour in the elements , where the function is a continuous and smooth, and the elements where the azimuth of linear polarization is , viz. vortices of the -component. The length of the intervals of the -contour containing vortices, , approaches zero. Consider the sum:

, (1.87)

where are the corresponding phases at intervals . Derivative from Eq. (1.87) is of the form:

, (1.88)

where is the number of vortices of the -component along the -contour.

Integrating Eq. (1.88) over the intervals of the -contour (direction of integration is counterclockwise) we obtain:

. (1.89)

Consider the integral over a half of a circle with radius , see Figures 1.34(a),(b), within the domain including the -component vortex. While is small, the linear approximation is valid for description of the field within such domain. Then, owing to periodicity of , this integral under circumvention of the singular point (period , see subsection 1.1.1) will be equal:

(1.90)

Let us add to the left and right sides of Eq. (1.89) the quantity:

, (1.91)

where is the total topological charge of the -component vortices occurring at the -contour. For that, integration over a half of a circle was performed following the scenario (a). The quantity corresponds to the integral from over some closed -contour, along which the function is continuous:

. (1.92)

The domain limited by such a contour contains all -points lying within the area of clockwise polarization, and the -component vortices lying at the -contour. It is assumed that magnitude of is small enough, so that -points lying within the area of counterclockwise polarization are not covered by the domain limited by the -contour.

It follows from Eq. (1.56) that the integral from over the -contour is equal to the doubled topological charge of -points occurring within the domain bounded by this contour:

. (1.93)

Besides, while the function is continuous along the -contour and the magnitude of is small, the following expressing id true:

. (1.94)

On the other hand [3,11]:

. (1.95)

Then, the following relation follows from Eqs. (1.97) - (1.95):

. (1.96)

Eq. (1.96) can be also derived in supposition that the relation kind of (1.92) is computed following the scenario (b), cf. Figure 1.34. In this case, the closed contour of integration, -contour, limits the domain, which does not contain the -component vortices. The integral from along this contour is equal to zero, and the following relation takes place:

, (1.97)

which coincides with Eq. (1.96).

It can be shown that, while the choice of the -component is arbitrary, the total topological charge of the -component vortices two times exceeds the total topological charge of -points also:

. (1.98)

L et us assume now that the considered -contour limits the area A with wit clockwise polarization, which contains the area B with counterclockwise polarization (see Figure 1.35). The boundary of the areas A и B is the -contour. The total topological charge of -points within the area B equals a half of the total topological charge of the -component vortices at the -contour:

. (1.99)

I

Figure 1.35

t can be shown that integration over the contour limited the area A results in transformation of Eq. (1.96) to the form:

. (1.100)

It follows from Eqs. (1.99) and (1.100) that

. (1.101)

Eq. (1.101) is also true in the case, when the number of the areas B inside the area A exceeds unity.

Thus, the total topological charge (index) of -points in the area A is determined by the total topological charge of all vortices occurring at -contours limiting this area. This statement is also true, if the interior areas contain the areas with another type of polarization.

The experimental study [79] verifying the above consideration is performed in the arrangement shown in Figure 1.36. A linearly polarized laser beam comes into Mach-Zehnder interferometer formed by the beam-splitters 2 and 12 and the mirrors 6 and 7. A plate 1 transforms a linearly polarized beam into circularly polarized one. In the object channel, the beam is focused by the microbjective 8 and impinges on the scattering object (thin polymer plate) 9. The object is chosen in such a manner that its scattered field occur to be “integrally polarized” with the degree of polarization none exceeding 50%.

A

Figure 1.36. Arrangement for analysis of a vector field. 1 and 11 – -plates, 2 and 12 – beam-splitters, 3-5 – collimator, 6 and 7 – mirrors, 8 – microobjective, 9 – scattering object, 10 – object, 13 - analyzer, 14 – polarization pair (tandem of the -plate and the linear analyzer), 15 – photodetector. Elements 14-15 forms a Stokes-polarimeter.

s it will be shown below (see subsection 1.4.6), when such an object is illuminated by the circularly polarized beam, -contours are small and include, as a rule, the areas with one type of polarization, viz. clockwise or counterclockwise. In other words, such areas do not contain the inserted -contours. Analysis of the field at far zone is provided by placing the object at the focal plane of the objective 10. In this case, the field just behind the objective obeys the far zone approximation. The reference beam is formed by a collimator 3-5. The analyzed 13 placed at the interferometer output provides selection of any polarization projection of the analyzed field visual analysis of interference of the scattered field and a plane reference wave. Such optical arrangement enables to carry out the comprehensive interference analysis of the polarization singularities on interferograms of the polarization projections of the vector field, as it has been shown above (see subsection 1.4.2.3).

For reconstruction of the -contour and determination of the loci of vortices associated with the orthogonally linearly polarized components, one looks after the corresponding projection of the vector field by turning the polarizer 13. Angular spacing of turning of the analyzer was 20°.

To determine the location of -points, one introduce a plate 11 with known orientation of its axes in respect to the axis of the analyzer, which enables to obtain the clockwise and counterclockwise circularly polarized projections of the field, just behind the objective 10. The loci of vortices of such field components coincide with the loci of -points.

Figure 1.37. Characteristics of С-points and vortices of the linearly polarized components. The degree of integral depolarization of the scattered object field is 46%. The left-hand column illustrates the polarization projections corresponding to the orientation of the main axis of the polarizer 80°, 100°, and 160° (fragments (a),(b),(c), respectively). The right-hand column shows the position of s-contours, C-points and vortices of linearly polarized projections of the vector field. - vortices of the orthogonally linearly polarized components, - the point of birth (annihilation) of them, - counterclockwise polarized C-points, - clockwise polarized C-points. The signs “+” and “–“ correspond, respectively, to positive and negative topological charges of vortices and the topological indices of C-points.

The coordinates of vortices of the orthogonal components (linearly or circularly polarized) are determined from the positions in interference forklets observed in the interferograms for the corresponding polarization projections. A Stokes-polarimeter 14-15 is placed at the second output of the interferometer, which enables to measure the average polarization characteristics of a field (including the degree of integral depolarization), when the reference beam is blocked.

The results of the experimental study of a scattered field are shown in Figure 1.37. Some interferograms of the linearly polarized projection of the object field are shown in the left-hand column. Orientation of the main axis of the analyzer 13 correspond to the angles 80°, 100°, and 160°. Fragments of the right-hand column illustrate the positions of the reconstructed -contours and -points. The vortices associated with the given polarization projection are depicted at -contours. The topological charges of them are indicated by the circles with the signs (+) or (–). Downwards and upwards interference forklets correspond to the topological charges +1 and –1, respectively. -points are depicted in figure by gray squares and rhombus with the corresponding signs.

One can see from Figure 1.37 that one -point with the topological index +1/2 belongs to the -contour with one vortex with the topological charge +1, see the upper -contour. Two -points with indices +1/2 and –1/2 are observed in the area limited by the lowest (and the largest) -contour with two vortices of opposite signs, so that the total topological charge is zero. Thus, the total topological index of -points occurring in the area limited by the contour equals zero. -points are absent in the area limited by the left -contour. Accordingly, the sum of the topological charges of the vortices of linearly polarized field components equals zero.

It is of interest, some polarization projections can be free of vortices located at -contour, if such contour limits the area where the total topological index of -points is zero. The polarization projection of a field corresponding to such situation is shown in Figure 1.37(a). For the left contour, one observes the point of birth (annihilation) of two vortices, rather than two vortices per se.