The time-domain Cartesian multipole expansion of electromagnetic fields.

Time-domain solutions of Maxwell's equations in homogeneous and isotropic media are paramount to studying transient or broadband phenomena. However, analytical solutions are generally unavailable for practical applications, while numerical solutions are computationally intensive and require significant memory. Semi-analytical solutions (e.g., series expansion), such as those provided by the current theoretical framework of the multipole expansion, can be discouraging for practical case studies. This paper shows how sophisticated mathematical tools standard in modern physics can be leveraged to find semi-analytical solutions for arbitrary localized time-varying current distributions thanks to the novel time-domain Cartesian multipole expansion. We present the theory, apply it to a concrete application involving the imaging of an intricate current distribution, verify our results with an existing analytical approach, and compare the proposed method to a finite-difference time-domain numerical simulation. Thanks to the concept of current "pixels" introduced in this paper, we derive time-domain semi-analytical solutions of Maxwell's equations for arbitrary planar geometries.

Time-reversed electromagnetic fields in anisotropic media.

Electromagnetic time reversal is commonly used for field imaging and focusing. This Letter builds upon the concept of the time-reversal cavity, which constitutes the main theoretical framework of time reversal theory. We study the behavior of the fields using modern methods of mathematical physics involving Colombeau generalized functions. This approach allows for a direct expression of time-reversed electric and magnetic fields in anisotropic time-reversal-invariant and nonreciprocal media. Moreover, the results hold for any arbitrary localized source and can readily be applied beyond the dipole approximation. Finally, a general result allows the prediction of the quality of focusing of the time-reversed fields as a function of the electrical permittivity and the magnetic permeability tensors in homogeneous anisotropic media, which contributes to the understanding of time reversal in complex media such as super-resolution enabling metamaterials.