1.5.3.3. The averaged Poynting vector of the vector field

A

Figure 1.90 Displacement of the Poynting vector singularity relatively to the position of -point.

s it has been shown above, appearance of singularity of the averaged Poynting vector in the area of the elementary polarizations cells is connected with the presence of -points with specified characteristics. Obviously, this pattern of the phenomenon is directly extended on the case of singularities of the Poynting vector at the field of general form. Namely, appearance of the Poynting vector’s singularity is always associated with -point positioned near this defect, cf. Figure 1.90.

The type of singularity (vortex or passive) depends on the relation of the signs of the topological charge of the vibration phase of -point and of the handedness factor within the plane of analysis. The vortex singularity arises, when these signs are opposite [104]:

. (1.179)

Passive singularity is formed, if the signs of and are the same. As it follows from consideration in subsection 1.4.1, the connection between the topological charge and the index of -point is of the form:

, (1.180)

T

Figure 1.91. Interconnection between singularities of the transversal component of the Poynting vector and C-points. and – negative and positive -points, respectively; and – vortex and passive singularities. Chirality of the vortex singularities is indicated by bold white arrows. Black solid lines are -contours. The numbers are the examples of specified pairs of -point – to the associated singularity.

hus, one can conclude that the vortex singularity of the transversal component of the Poynting vector corresponds to C-points with the negative index (to the negative

-points), and the passive defect of the Poynting vector appears near the positive -points.

This circumstance is illustrated by the data of computer simulation for a random vector field, see Figure 1.91. It is seen from this figure that the negative -points are located near the vortex singularity indicated by the numbers 1,1. Passive singularities gravitate towards the positive -points, numbers 2,2. Note, the vortex singularities differ in chirality ( or ), which is determined by the the sign of the handedness factor of the area containing the negative -point. The transversal component of the Poynting vector circulates clockwise around the Poynting vortex within the areas with clockwise polarization, , , and in the opposite direction within the areas with counterclockwise polarization, , .

In conclusion of the subsection, we represent two tables summarizing the main properties of the Poynting vector’s singularities and their connection with the conventional optical singularities.

Instantaneous singularities of the Poynting vector

Kind of the Poynting vector’s singularity	Edge	Vortex singularities (VSs) VSs possess be the same topological index and differ in chrality	Passive singularities (PSs)
Scalar field	Localization coincides with equiphase lines	Do not exist	Moving PSs unavoidably pass through the stationary points of phase and intensity
Vector field	Do not exist	As a rule, VSs coincide with disclinatons	PSs can appear independently from disclinations

averaged singularities of the Poynting vector

Kind of the Poynting vector’s singularity	Vortex singularities (VSs)	Passive singularities (PSs)
Scalar field	1. Localization of VSs coincides with the vortices. 2. Chirality of VSs is determined by the topological charge of the vortex phase.	1. Positions of PSs coincide with the stationary points of a phase. 2. As a rule, in far zone PSs are located in the saddle points of a phase.
Vector field	1. VSs are associated with the negative -points. 2. Generally, localizations of VSs and -points are different. 3. Chirality of PSs is determined by the handedness factor within the region where PS occur.	1. PSs аre associated with the positive -points. 2. Generally, localizations of PSs and -points are different.