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The electromagnetic vortex in a nonlinear medium



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. 22


Distribution of Poiting vector in the same plane is shown in Fig.23.



Fig. 23


Now we shall analyze the result of this numerical experiment.

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 (3.8) 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.

However, from our point of view the turning mechanism is shown quite clearly. The basic conclusions are the following:

-    the field configuration rotates, remaining definitely localized;

-    the elemental electromagnetic vortex possesses total energy and spin;

-         leastwise, the constant electrical charge and the constant magnetic moment do not disappear.

It is obvious that the following two steps should be made to acquire fuller information with a view of studying the problem:

-    It is necessary to execute numerical experiments with due account of tensor character of the medium parameters - e and m.

-    It is necessary to organize a real experiment in the medium that is similar to the one, in which self-focusing of laser radiation was observed.


Thus, even a macroscopic vortex possesses all those properties, which are inherent to steady fundamental particles - an electron and a proton.


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