Séries divergentes, aproximantes de Padé e espalhamento da luz

Abstract

Light scattering is a very important physical phenomenon when it comes to the study of radiation-matter interaction. This phenomenon can be approached to discover scattering patterns of certain materials, such as determining crystal structures using X-rays, or to produce materials with specific scattering characteristics that result in useful applications, such as metamaterials. From a more formal theoretical perspective, the various equations do not always offer analytical solutions and are, in most cases, quite intractable. Therefore, it is common to resort to approximation methods when seeking to solve scattering problems. Among them, the perturbative methods stands out, which consists of assuming that the scattered field can be written as an infinite power series. The problem is then replaced by several smaller ones, presumably more manageable, which allow finding approximate solutions for a series of physically interesting cases. The Born series is the most commonly used to represent the scattered field in these problems. For scatterings in which light interacts weakly with the material (weak scattering), truncating this infinite series at the first non-zero power term (first Born approximation) already represents a good approximation of the problem solution. However, for scatterings where light interacts strongly with the material (strong scattering), we observe that the first Born approximation fails to describe the result. This fragility of the Born approximation occurs in both Hermitian and non-Hermitian materials, sometimes being more pronounced in the latter. Hence, the need to seek approximate methods that provide good results for both weak and strong scatterings, as well as for any type of material. Padé approximants emerge as a promising tool for this purpose since they usually generate larger convergence regions when compared to the respective Born series. In this work, we apply Padé approximants to a series of physically interesting problems. For certain parameter choices, replacing the analytical solution with the Born series tends to diverge. From the analysis performed, we verify that Padé approximants are an extremely useful tool for describing light scattering in both weak and strong regimes, as well as in non-Hermitian materials.