Resonances and different recording times on electromagnetic scalar and vector potentials

For a best broadcasting efficiency we give in this paper an alternative treatment with electromagnetic potentials (EP) and propose a powerful tool based on the resonance properties. Recently, we have developed some methods for studying the change on broadcasting conditions and explained theoretically as the brake of confinement conditions for the so-called evanescent waves. We also named the former as left-hand material conditions because the event that occurs inside a left-hand material can be analyzed with exactly the same tool as the broadcasting case. So we have proposed that the equivalent mathematical behavior in case of electromagnetically broadcasting is really a transformation of the confined near field into travelling waves. In addition, we know that if we make a time reversal, the quality of the signal becomes much better and can be focused in a more restricted area. In other words, the theory of the generalized Fredholm integral equations (GFE) has been used to first introduce a source for the time reversal procedure but then we transform the inhomogeneous equations (GIFE) into generalized homogeneous ones (GHFE) for describing the resonances, that is the frequencies for which we can see the brake of confinement conditions and because of their singular properties, prevent information lose. But in the past we never talked about the electromagnetic potentials neither the retarded Green functions and applying them to the mixing of signals in the same thin band but with different recording times. In this paper we built the equivalent formalism that involves both the electromagnetic potentials and the retarded (and advanced) Green functions. We must underline that, for convenience, the system considered is a discrete one, but we can recover a continuum version by including the discrete indexes as continuum variables. To convert evanescent waves into travelling waves we must also locate the sink in the near field region, restriction that we have explained as a consequence of orthogonality rules followed by the resonances. Once we have obtained the inhomogeneous vector-matrix equation we inquire ourselves about a singular condition that is a situation similar to that of scattering quantum mechanics theory when we have an outgoing wave but we don't have an incident one. That condition on scattering theory describes an unstable system known as resonance. If we demand the same condition on our vector-matrix equation, we obtain also resonances but which describes travelling waves that come not for an incident wave but for evanescent waves. In addition, in accordance with Communication Theory we have shown that it is possible to introduce the recording time as an additive contribution to the so-called power. The fact that we are using the same band but different cutting limits, also suggests that we can design an appropriate filter that can distinguish between signals according to the recording time that is we can superpose signals with the same frequency range but with different recording times. In a previous work we have sketched a filter, but now we give a better-defined device. In addition we introduce the so-called Plasma Sandwich Model (PSM) with the purpose to optimize the broadcasting, which propose that regions or zones with a fluctuating refraction index constitute the transmission media and that strongly depends on the magnetization and local values of the EP. Finally we give a simple example for obtaining the resonant frequencies for a particular form for the Fourier transform of the kernel in the GHFE.