Spatiotemporal energy-density distribution of time-reversed electromagnetic fields

Time reversal exploits the invariance of electromagnetic wave propagation in reciprocal and lossless media to localise radiating sources. Time-reversed measurements are back-propagated in a simulated domain and converge to the unknown source location. The focusing time (i.e., the simulation instant at which the fields converge to the source location) and the source location can be identified using field maxima, entropy, time kurtosis and space kurtosis. In this paper, we analyse the spatial energy-density distribution of time-reversed electromagnetic fields by introducing a convergence metric based on the spatial average and variance of the energy density. We analytically prove that the proposed metric identifies the focusing time and the source location, with direct links to the source frequency content. We verify the analytical results in a free-space numerical simulation and then compare the proposed metric to existing ones in a simulated inhomogeneous medium. Next, we apply and compare this metric in an experimental case study to localise electromagnetic interference sources. The proposed metric outperforms existing ones to identify the focusing time and can also be used to locate the source. Finally, because of its tensorial nature, it can handle anisotropic media, opening the door to quantitative analyses of time-reversal focusing in metamaterials.