1.5.2.2. Averaged singularities of the poynting vector of scalar field

Averaged Eqs. (1.156)-(1.158) are of the form:

, (1.160)

where is the amplitude, and are the derivatives from a phase.

Similarly to the case of the instantaneous Poynting vector, two kinds of singularities can arise.

Figure 1.62. Possible behavior of the Poynting vector in the vicinity of the averaged singularities of a scalar field. a,d,e – passive singularities; b,c – vortex singularities.

1. All components of the averaged Poynting vector vanish, cf. Figure 1.62(b),(c). This case corresponds to the averaged vortex singularity localized at the vortex center. The amplitude is zero. Conventional circulation of the Poynting vector is observed in the vicinity of the vortex center. Such a singularity of the azimuth of the Poynting vector is characterized by the positive Poincare index.

Singularities with different chirality correspond to the vortices different in sign of the topological charge.

2. Only transversal component vanishes, cf. Figure 1.62(a),(d),(e). The averaged passive singularities take place. It follows from Eqs. (1.160), their coordinates coincide with the coordinates of the stationary points of a phase. The direction of the energy flow coincides in these points with -axis.

I n other words, these points just determine the prevailing direction of the energy flow of a scalar wave.

Possible behavior of the Poynting vector within the nearest vicinity of such singularities is reduced to the situations illustrated in Figure 1.63.

T

Figure 1.63

he negative (saddle) passive singularities provide topological connection between the vortex singularities with the same chirality, while the adjacent vortices with the opposite directions of circulation of the Poynting vector are directly connected by the current lines of the transversal component of the Poynting vector.

1.5.3. Singularities of the Poynting vector at vector fields

1.5.3.1. Instantaneous singularities of vector field

Similarly to the case of scalar fields and in accordance with Eqs. (1.153)-(1.155), the instantaneous singularities of the Poynting vector arise at the points of a field, where the disclination or amplitude zero of the transversal component of this vector occur.

It is known [3,45], disclinations are the point-like defects of a vector field. It has been shown above, cf. subsection 1.4.1, that disclinations move along -contours, are born and annihilate. The number of disclinations at -contour can change by even number alone, i.e. similarly to all topological defects, disclinations are born and annihilate in-pairs [3,63,76]. Motion of disclinations, interconnection of them and connection of them with other structures of a field obey the topological regularities. That is why the events associated with the singularities of the Poinging vector resulting from discinations must obey similar regularities.

a b c

Figure 1.64. Singularities of the Poynting vector associated with disclinations. Random vector field. (a) – instantaneous distribution of the modulus of the transversal component of the Poynting vector; (b) – distribution of the instantaneous azimuth of the transversal component (the levels of grey correspond to different orientations of the vector); (c) – distribution of the modulus and the azimuth of the transversal component (orientation of the Poynting vector is indicated by white arrows).

There are no any limitations on the sign of singularity associated with disclination, which would follow from Eqs. (1.153)-(1.155). Moreover, the positive instantaneous defects of the Poynting vector can be both vortex ones and passive singularities. This circumstance is illustrated by the results of computer simulation represented in Figure 1.64.

Note, such newly arising defects can be both vortex ones, i.e. both singularities are characterized by the same Poincare indices being different in chirality. Difference in chirality is sufficient for providing interconnection between the born Poynting vortices, but insufficient for forming of the topological connection with other structures of the field.

So, two following scenarios of the birth event and annihilation of singularities of the Poynting vector associated with disclinations are possible.

1. Assume that two vortex singularities were born at -contour. Their chiralities are different, but their Poincare indices are the same (both positive). In agreement with the conservation law for the total topological index, two

Figure 1.65. Behavior of the singularities of the Poynting vector associated with the disclinations for different moments. Temporal step between the figures is 1/8 of oscillation period. a–d – distribution of the azimuth of the Poynting vector’s transversal component (the levels of grey correspond to various orientation of the vector); e–h – distributions of the modulus and the azimuth of the transversal component. Orientation of the component is illustrated by thin white arrows. Bold white arrows in fragments (a) to (d) indicate the direction of motion of singularities. and – instantaneous singularities of the Poynting vector with positive and negative indices respectively. Solid lines are -contours.

singularities with the negative index must be born simultaneously with the birth event of these singularities at the same point, viz. at -contour. It is clear, these are the passive singularities, which just after arising leave -contour and walk out to the region with elliptical polarization. There is the topological reaction corresponding to this event:

, (1.161)

i.e. four singularities of the transversal component of the Poynting vector appear and disappear.

2. One of the singularities associated with disclination has positive index (it doesn't matter vortex or passive singularity), and another has negative one. In this case, the topological reaction of appearance-disappearance of singularities is transformed to the form:

, (1.162)

Only two singularities of the Poynting vector take part in this reaction.

S ingularities of the Poynting vector associated with disclinations, similarly to the disclinations themselves, move along -contour, disappear and are born again, see Figure 1.65. Their positions are repeated twice per period of oscillation.

T

a b

Figure 1.66. Fig. 12. Instantaneous singularities associated with zero magnitude of the transversal component of the Poynting vector. a – distribution of component azimuth indicated by shades of gray; b – azimuth (indicated by white arrows) and modulus of component (indicated by shades of gray).

he birth event and annihilation of singularities can be accompanied by appearance and disappearance of additional singularities, when only transversal component of the Poynting vector vanishes.

In general case, such singularities do not belong to -contour. Similar singularities can appear independently on the birth events of disclinations, cf. Figure 1.66. For that, such non-connected with -contour singularities can be both passive and vortex ones.

Temporal behavior of the instantaneous vortex singularity, which was born within the area with elliptical polarization, is illustrated in Figure 1.67. One can see that this singularity passes through the area with non-uniform polarization.

Figure 1.67. Motion of the instantaneous vortex singularity, which was born at the area with non-uniform polarization.

Note in conclusion that the “instantaneous” angular momentum averaged over the spatial coordinates and short temporal interval , being observed in the vicinity of the vortex singularity, is of the maximal magnitude exceeding the momentum magnitude at other areas of the field with the same energy per unite area, irrespectively from origin of such singularity (from disclination or in other way).