**1.**** ****The
single-mode vortex in a nonlinear medium**

For two
reasons we shall consider here a single-mode vortex. The first reason: this
spatial configuration has shown the greatest stability in experiments with the
equations of Maxwell-Hertz. The second
reason: all the details of evolution of electrical and magnetic fields can be
easily observed in this elementary vortex.

The single-mode vortical structure is the lowest configuration and corresponds
to the attached Legander polynomial_{}. The starting position of fields is presented schematically in Fig.20. Here the electric field** ****E** has the shape of the dipole, oriented across the spin
axis. The magnetic field** ****H** has the shape of a closed ring, the axis of which is
perpendicular to the spin axe and the electric-dipole axe. The symbol** ****P** designates Poiting
vector in the corresponding places of a vortex.

Fig. 20

All the calculations were done in conditions of a scalar approximation of dielectric
and magnetic conductivity. Parameters of the induced-by-medium nonlinearity are
chosen so that* **e* and* **m** *in a vortex zone do
not exceed 1 in the area of small amplitudes and 3,5 in the area of their
maxim. The tensor representation of the medium parameters has been difficult
mainly for technical reasons: it requires much greater on-line storage of a
computer. But modern personal computers are essentially limited with regard to
this characteristics. As result, all the described below numerical experiments
are practically illustrative.

Stationary values are imposed on variable fields. The electric field has a
radial direction (charge), and a magnetic field has an axial direction
(magnetic moment). Both have 5 % of maxima of the variable-components
amplitudes. Being set artificially, these fields remain unchanged further on,
in the course of a numerical experiment. Possibly, the process of their
relaxation up to any equilibrium natural state requires much more time, than
half the turn of a vortex.

The evolution of a single-mode vortex in case it makes a turn within the
limits of 0 - p are represented in Fig. 21 by
means of isosurfaces of squared values of E (in a red colour)
and H (dark blue).
Numbers of cycles are designated by numerals by means of which the
corresponding position is fixed. Here you can observe the way the general field
configuration rotates.

Fig. 21

Fig. 22
illustrates the situation in the middle of a vortex in a plane, which is in the
middle of it and perpendicular to the axis* **z*. The tic mark is the same as in Fig. 21. Here vectors indicate an
electric field, and their color and intensity indicate the distribution of the
amplitude* **z* – the components of
a magnetic field.

Fig. 23

Let us
note once again the fact that the approximate solution is presented here. When calculating,
the exact equations of Maxwell-Hertz in the form of (Ï3.8) from Appendix 3 have been used. But initially tensor
parameters of the medium were replaced by scalars*
**e* and *m**.* Therefore, we
could not see the picture at a turning angle, exceeding 180°.