We study the behaviour of electromagnetic waves in terms of their propagation through time and space.

First, while time typically flows in only one direction for complex systems, the reciprocal nature of most wave systems allows for a form of time reversal. Indeed, for a wave f(t, r), its time-reversed counterpart f(-t, r) can also physically exist. Moreover, methods have been developed to create this time-reversed wave, enabling the application of time reversal to wave focusing and imaging. In this context, we record the direct-time wavefront f(t, r) emitted by an unknown source on a (ideally closed) surface, known as a time-reversal mirror. This recording is then time reversed and played back from the mirror, causing the resulting wave to converge back to the original source location. If the original source is absent, the wave continues propagating, slightly blurring the focus and exhibiting the diffraction patterns observed in classical optics.

A wide range of methods has been developed to enable wave imaging and focusing beyond the diffraction limit, achieving "super-resolution." We begin with a review of the theory, methods, and applications of super-resolution imaging and focusing in the radio-frequency range. We then contribute to the theoretical framework of time-reversal by, first, proposing a new convergence metric based on a combination of probability theory and electromagnetic energy density, and second, generalising the theory of the time-reversal cavity for nonreciprocal media, high-order multipole sources, and proposing a link between the dispersion relation of homogeneous media and the attainable resolution. We subsequently present three applications of time-reversal imaging for electromagnetic compatibility pre-compliance testing aimed at imaging the current from an electrostatic discharge: i) using a resonant metalens, ii) leveraging cable radiation, and iii) with a low-cost setup.

In the second part of the thesis, we study the spatial behaviour of complex electromagnetic sources. This behaviour can be described analytically by the multipole expansion, a widely used method in wave systems. Traditionally, this method projects complex field distributions onto a basis of spherical harmonics. The strong connection between this method and Green's function warrants, first, an analysis of the singular behaviour of this function in uniaxial media. We then present the novel Cartesian time-domain multipole expansion, along with a practical recursive implementation. We illustrate the method on time-reversal imaging of intricate sources. Additionally, we apply this expansion to model the time-domain fields radiated by, first, impulse-radiating antennas, and second, lightning. Finally, we discover that the Cartesian approach is essential in anisotropic media to describe novel degrees of freedom in wave propagation.