Vector generalization of the discrete Time Reversal formalism brings an electromagnetic application on overcoming the diffraction limit

In a recent work we found equivalence between the Time Reversal techniques and the use of Meta-materials in order to achieve the overcoming of the diffraction limit. Now we present the vector generalization of a formalism developed in a former study of Discrete Acoustic Time Reversal. The ulterior purpose is to apply it to electromagnetic waves, not only as an alternative to the use of left-hand materials with a negative refraction index but also in a very important area like communications. To this aim we take some tools from scattering theory and obtain the Neumann series for the discrete time reversal problem, then we build an equation (vector-matrix) for the Fourier transform of the complete Green's function that expresses the inverted signals (time reversed) in terms of the actual ones. As mentioned above, among its multiple applications is the subwavelength focusing for microwaves (~2.45GHz) which actually have an experimental success. We include as an example an application of the present formalism to an academic but simple problem. Also, we show how we can complete the missing information by introducing formally a sink term in the time reversing process and how the so called evanescent waves can be recovered as travelling waves even without the presence of a negative index material.