## 10 Conclusions  
We guessed the possibility of generating electromagnetic waves with extremely high directivity. This prediction is suggested by an extension of [PERSON]'s equations in a vacuum aimed to simulate compact signals, traveling unperturbed at the speed of light. The enlargement of the solutions space relies on the fact that the divergence of the electric field may assume values different from zero also in the absence of charges associated with physical particles--for instance, electrons. This requirement is indeed able to explain many electromagnetic phenomena, as documented in [15]. In addition, the revised model provides the exact link between electrodynamics equations and the rules of geometrical optics, a property that the standard Maxwellian approach is not able to reproduce, unless one accepts a sequence of rough simplifications (see, e.g., [28], Section 3.1).  
The waves belonging to the category we studied in this paper display a toroidal topology, with the magnetic field distributed along closed lines following the toroidal axis. With the help of simple devices, it is actually possible to create waves having a similar topology and developing along conical patterns. These emanations cannot be described by the classical [PERSON] equation, so that the adoption of the extended model becomes a necessity dictated by reality.  
The theory predicts that antennas with infinite directivity could possibly be built. Reality is a different issue, so reasonable quantitative confirmations will only be made as the results of the first experiments are made available. Once it is demonstrated that such a project is feasible, generating these waves becomes only matter of technical effort. The reader can of course understand the importance  
Figure 10: Antenna completed with a conical reflector. The final output signal has parallel Poynting’s vectors. The magnetic field envelopes closed circles around the \(z\)-axis. The solitonic wave is expected to travel by carrying its associated information without dissipation.  
of this achievement. In this paper, we indicated a way to fabricate antennas devoted to this purpose. These are obtained by filling the space delimited by a biconic antenna with a dielectric medium placed in contact with the cones, and such that the boundary between the medium and the external vacuum is designed following a suitable curvature. The specific signals produced are expected to reach a prescribed target without dissipating their energy along the path. Moreover, since most of the theoretical passages are based on the rules of geometrical optics, we argue that the role of frequency might not be so crucial, thereby allowing transmissions ranging within a rather broad band. All these properties should guarantee a number of positive outcomes in terms of sustainable operation, safe energy transfer and secure information beaming.  
Conceptualization, D.F.; methodology, D.F. and L.D.; writing-original draft preparation, A.C., D.F. and L.D.; writing-review and editing, A.C., D.F. and L.D. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding. The APC was funded by MDPI, that we gratefully acknowledge.  
The authors declare no conflict of interest.