1.2.2.4. Elementary topological reactions

Proceeding from the conservation law for the topological charge and the Poincare index, birth of individual vortex into the field is impossible, while this event would result in changing (growing or decreasing) of the total topological charge by unity. Birth or annihilation of vortices are always realized by pairs, as birth or annihilation of two vortices with the opposite topological charges . This statement is known as the pair principle.

At the same time, birth of two vortices results into change of the total topological index , as the vortices, irrespective of the sign, possess positive Poincare index. So, birth of two vortices is unavoidably accompanied by appearance of two saddles with the negative index . Such elementary process can be described as the following topological reaction [11]:

, (1.7)

where are the positive and negative vortices, and the phase saddle, respectively.

Proceeding from this topological equation, one can estimate specific number of topological elements, , which may spring up into the field.

Let us fold an infinitely spatially extended plane of observation into the sphere of infinite radius. We start with the topologically feasible case, when only two vortices of the opposite sign are at this plane, and will step by step increase the number of vortices. In correspondence with Eq. (1.7), we obtain the following relations for the specific numbers of topological elements:

_. (1.8)

(When the number of vortices at the field is large enough, the last term in the second relation can be neglected.)

The same conclusion follows also from the Euler characteristics of a surface, in correspondence with the Poincare-Hopf theorem on topological indices of a surface.

1.2.3. Experimental observation and identification of vortices into scalar fields

W

Figure 1.5. Forklet produced by interference of the vortex and plane reference wave. Zone of the vortex is depicted by a white circle.

hile the wave field amplitude vanishes at the vortex core, the straightforward means for identifying the vortex is to measure vortex intensity. However, photometric measurements do not result in reliable differentiation of the true vortex (absolute amplitude zero) from close (but not equal) to zero minimum of amplitude. Really, in contrast to a phase of the field (for which the vortices manifest themselves as singular points), minima of intensity are only the stationary points. As a result, intensity distribution in the vicinity of a vortex resembles intensity distribution in the vicinity of local minimum.

At the same time, peculiarities of behavior of a phase in the vicinity of a vortex results in typical bifurcation of interference fringe (in appearance of an “interference forklet”) as such field structure interferes with a plane reference wave, see Figure 1.5, [7-10,12,14,36-38]. For that, the direction of bifurcation of an interference fringe depends on the sign of the topological charge of a vortex. Thus, interference fringes produced by the vortices of opposite signs are oppositely directed.

Note, when interaction beams have curvature, interferometric pattern may be transformed into corresponding spiral [36-38].

So, the interferomentric techniques are the only reliable ones for identification of the vortices. Interference identification of vortices is considered in more details in Refs. 36-38.