1.5.3.2. Behavior of the Poynting vector in areas of elementary polarization singularities

Let us consider the elementary polarization cell, viz. the field area containing the minimal number (one or two) of -points. Such field structures with one -point and the methods of formation of them have been considered in details in subsection 1.3.5.

Figure 1.68. Example of formation of the elementary non-uniformly polarized area using orthogonally circularly polarized beams.

T he structure of such cell is following. -points of the same or the opposite signs are in the region with one type of elliptical polarization (clockwise or counterclockwise) bounded by -contour of relatively simple form. Similarly to the field with one -point, the field with two -points can be formed by superposition of the vortex circularly polarized beam and orthogonally polarized beam with a smooth wave front. It can be shown that superposition of circularly polarized beams results in closed by -contour of limited size. The example of formation of the elementary non-uniform in polarization cell of the field is illustrated in Figure 1.68.

Further consideration presumes that the indices of -points, as well as the phase distribution and the intensity distribution into interfering beams can change arbitrarily.

1.5.3.2.1. Symmetric distributions of amplitude and phase of the interfering beams

а. Single -point

Assume the vortex beam to be circularly polarized isotropic vortex:

, (1.163)

and the plane wave to be orthogonally circularly polarized:

, (1.164)

where is the handedness factor, and are the polar coordinates, are the Cartesian coordinates with the origin at the center of the vortex, and is the topological charge of the vortex. Correspondingly, the complex amplitudes of the resulting field are of the form:

, (1.165)

- and -components of the field are described in terms of strength by the following relations:

, (1.166)

where:

, (1.167)

and

. (1.168)

In accordance with Eqs. (1.153)-(1.155) one finds:

. (1.169)

Thus, one obtains for the - and -components of the Poynting vector:

. (1.170)

As a consequence, the modulus of the transversal component of the Poynting vector takes the form:

, (1.171)

where is the topological charge of the vibration phase in the vicinity of point.

One can see from Eq. (1.171) that the modulus of the transversal component of the Poynting vector identically equals zero, , over the whole analyzed area, if the signs of and are the same. It means physically that the angular momentum in the vicinity of such -point equals zero also. In other words, the orbital momentum is compensated by the spin momentum. Hereinafter, we assume that and are of opposite signs:

_. (1.172)

In this case, the modulus of the transversal component behaves similarly to the longitudinal one being rotating in time around -point with the double frequency of oscillation [102-104]. The direction of rotation of the minimum (maximum) is determined by the sign of the topological charge of the vibration phase. Minimum is reached at -contour and corresponds to the position of the Nye’s disclination. For the specified distribution of complex amplitudes, cf. Eqs. (1.163) and (1.164), -contour is a circle with the center at -point. Figure 1.69 illustrates temporal behavior of the modulus of the transversal component.

Figure 1.69. Rotation of the minimum of the transversal component of the Poynting vector (disclination) around -point. Position of disclination at -contour is indicated by a white spot.

Naturally, if the structure of the interference beams is more complicated, i.e. the vector field in the vicinity of -point can not be represented as a superposition of an isotropic vortex with a plane wave, the field in the vicinity of -point is modified, and Eq. (1.171) is not valid. Nevertheless, the character of changes of is similar.

b. Angular momentum of the field into vicinity of С-point

By definition, the density of the angular momentum is represented by the relation:

. (1.173)

Then, taking into account Eq. (1.170), one obtains:

. (1.174)

The averaged in time (over one period of oscillation ) angular momentum of the field within the area is of the form:

. (1.175)

In the end, one obtains:

. (1.176)

where is a power of the vortex beam.

To compare the obtained relation with the momentum of the vortex ( [95]), note that the power of circularly polarized vortex as the beam forming a non-uniform in polarization cell with -point is twice larger than . Taking into account this circumstance, one obtains for the normalized powers the following relation:

. (1.177)

Thus, the angular momentum in the vicinity of -point is twice larger than for the vortex beam, if and are of the opposite signs (spin and orbital angular momenta of the field are added), and equals zero, if and are of the same signs (spin and orbital momenta of the field compensate to each other).

a b c d

Figure 1.70. Temporal behavior of the transversal component of the Poynting vector in the vicinity of -point for non-coinciding signs of the topological charge and the handedness factor. Azimuth of the component corresponds to orientation of the arrows. Arrow length corresponds to the magnitude of the modulus of the component.

F igure 1.70 illustrates the distribution both the modulus and the azimuth of the instantaneous transversal component of the Poynting vector. The azimuth of the component corresponds to the orientation of arrows, and its magnitude is characterized by the arrow length. One can see from this figure that the instantaneous singularity of the transversal component of the Poynting vector corresponding to disclination is the defect of vortex kind with positive chirality ( , the direction of vector circulation is clockwise).

F

Figure 1.71.Behavior of the transversal component for various instants in points of on one of diameters of -contour.

or clearness, behavior of the transversal component is shown in Figure 1.71 for various points of the line crossing the area with non-uniform polarization and passing through -point (position of the line coincides with one of the diameters of -contour) for various instants. One can see from this figure that the averaged transversal component of the Poynting vector vanishes only at the point of the field coinciding with -point, and its magnitude grows as the point of observation moves from -point.

B

Figure 1.72. Distribution of the modulus and the azimuth of the averaged transversal component of the Poynting vector

ehavior of the averaged component of the Poynting vector is shown in Figure 1.72. It follows from this figure that distribution of the modulus and the azimuth of the transversal component of the Poynting vector is similar to such distribution intrinsic to the conventional vortex. However, in the case of scalar vortex all three components vanish, while in the case of interest -component is not vanishing. Thus, a power flow at the center of linearly or circularly polarized vortex is absent, reaching its maximal amplitude, as a rule, at -point. For that, the direction of energy propagation coincides with -axis.

c. Elementary polarization cells with two -points of the same signs

The phase map of a circularly polarized beam with two vortices of the same signs is represented in Figure 1.73. The position of -contour resulting from a superposition of this beam with the orthogonally polarized plane wave is also shown in this figure.

Figure 1.73. Phase map of the vortex beam. The levels of grey correspond to different magnitudes of a phase. Position of -contour of the resulting field is indicated by a white line.

In correspondence with Ref. 80,81 two disclinations appear at -contour. One can show, for the case of symmetric distributions of the parameters of the vortex and smooth beams, that the structures of these disclinations are identical. As a consequence, the structure of the Poynting singularities must be also identical.

Two vortex singularities move along -contour in one direction, cf. Figure 1.74. Additional passive singularity kind of a saddle with the negative Poincare index arises at the geometrical center of this area. This singularity provides the topological connection between the same positive vortex singularities.

a b c d

Figure 1.74. Behavior of the instantaneous Poynting vector for the field containing two -points of the same signs.

a b

Figure 1.75. Behavior of the averaged Poynting vector for the area of the field containing two C-points: (a) two C-points are spatially separated; (b) distance between C-points if negligible

B ehavior of the averaged Poynting vector is illustrated in Figure 1.75a, from which one can see that the character of distribution of the transversal component does not differ considerably from the case of one -point. Difference between behaviors of the normalized parameters becomes negligible, when the distance between -points is very small, cf. Figure 1.75b. One must remember that the magnitude of the angular momentum of the field within the area with two close -points is twice larger than in the case of the area containing only one -point. Besides, behavior of the instantaneous Poynting vector

a b c d

Figure 1.76. Distribution of the parameters of the instantaneous Poynting vector for the field containing two closely positioned -points of the same signs.

is absolutely different from the behavior of this characteristic of the field in the area with one -point, cf. 1.76.

d. Elementary polarization cells with two -points of opposite signs

Figure 1.77

The phase map of a circularly polarized beam with two vortices of the opposite signs is shown in Figure 1.77. The vortices are located at the same distance as for the field containing two identical -points. The position of -contour resulting from a superposition of this beam with the orthogonally polarized plane wave is also shown in this figure. One can see that the positions of -contour is the same as for the area containing -points of the same signs.

Similarly to the previous case, two singularities of the Poynting vector associated with disclinations appear at -contour (Figure 1.78). However, one of them is the vortex one (it is shown on the left of Figure 1.78(b)), and another is passive, i.e. negative (it is shown on the right of Figure 1.78(b)). These singularities move along -contour in opposite directions. It follows from Figure 1.78 that moving of these singularities results in their annihilation, cf. Figure 1.78(c), and appear again, cf. Figure 1.78(a). Behavior of the azimuth of the transversal component of the Poynting vector for both situations is characterized by the absence of any defects inside -contour.

a b c d

Figure 1.78. Behavior of the instantaneous Poynting vector for the field containing two C-points of the opposite signs.

Changing the azimuth as a function of the parameter of -contour does not undergo discontinuity along the contour, similarly to the case when disclinations at -contour are absent, cf. 1.78(d). At the same time, the modulus of the transversal component of the Poynting

Figure 1.79. Distributions of the parameters of the averaged Poynting vector for the field containing two C-points of the opposite signs. The length of arrows corresponds to the magnitude of the transversal component of the vector.

vector vanishes in the vicinity of -point (right -point), for which the sign of the topological charge is the same as the sign of handedness factor within the area bounded by -contour. At the point of the field coinciding with -point, one observes a passive singularity of the Poynting vector’s component.

Behavior of the parameters of the averaged Poynting vector is represented in Figure 1.79. It follows from this figure that the vortex singularity is localized in the position of -point, for which (left -point), and the modulus of the transversal component of the Poynting vector vanishes at -point, for which the sign of the topological charge is the same as the sign of the handedness factor (right -point).