1.2.2. Topological charge and topological index of singular points. Elementary topological reactions

Here we consider the topological characteristics of the phase singularities and of the stationary points of a phase, as well as the elementary topological reactions of their birth and annihilation.

1.2.2.1. Topological charge

Singular points of any quantity can be characterized by the topological indices of two types [3,11]. First of them, the topological charge, is introduced for the singular point (for example, for singular point of a phase field) proceeding from the following definition:

, (1.6)

where integration is taken over small circle around the singular point in anticlockwise direction.

Figure 1.4

It is easily to see that for the phase changes kind of the ones shown in Figures 1.1 to 1.3, topological charge is +1 or –1. Signs “+” or “–” correspond, respectively, to the case when a phase increases or decreases under anticlockwise circumference of the vortex. In general case, magnitude of the topological charge may exceed unity (may be equal to ). However, vortices with the charges exceeding unity occur to be topologically instable, they decay into the set of simple singly-charged vortices even under small disturbances. For this reason, the only singly-charged optical vortices, with , are relevant, at least as regard as random speckle field [3,11-13].

It is easily to show that for areas of the field that do not contain a singular point, including the stationary points.

Equiphase lines of the field within the vortex (point ) and the saddle point are sketched in Figure 1.4. While a topological charge is inherent in the area of the field possessing a vortex alone, then the topological charge of the area of the field shown in figure equals +1 or –1 .

1.2.2.2. Topological index

Topological index of the second type is the so-called Poincare index, . Poincare index is calculated in the following way [11]. Under circumference (clockwise or counerclockwise) of the singular point, one determines the direction of rotation of the lines associated with the quantity of interest (to say, equiphase lines). If the direction of rotation of these lines coincides with the circumference direction, then one prescribes the sign “+” to the index. If the direction of the lines is opposite to the circumference direction, then one prescribes the sign “–“ to the index. The magnitude (modulus) of the Poincare index equals the number of full rotations of the lines that is calculated for the closed loop. So, Figure 1.4 illustrates the equiphase lines for the field fragment, which includes the vortex and the saddle point . One can conclude from this figure that both positive and negative vortices are characterized by the same Poincare index, , while a saddle point is characterized by the Poincare index . Both the phase extrema and the vortices support Poincare index .

1.2.2.3. Conservation law for topological charge

For any area containing singular points, one can count the resulting (total) topological charge, , and the total Poincare index [3,11]. Due to the properties of universe space, the conservation law for the topological charge takes place. This law may be formulated in the following way. Any disturbance of an optical field does not lead to changing the total topological charge and index. If electromagnetic wave is freely propagating through a linear media, and if it does not meet any sources and perfect absorbers, then the magnitudes of and are constant at any cross-section of the field [3,11].