Scattering from Spheres: A New Look into an Old Problem

Abstract

In this work, we introduce a theoretical framework to describe the scattering from spheres. In our proposed framework, the total field in the outer medium is decomposed in terms of inward and outward electromagnetic fields, rather than in terms of incident and scattered fields, as in the classical Lorenz–Mie formulation. The fields are expressed as series of spherical harmonics, whose combination weights can be interpreted as reflection and transmission coefficients, which provides an intuitive understanding of the propagation and scattering phenomena. Our formulation extends the previously proposed theory of non-uniform transmission lines by introducing an expression for impedance transfer, which yields a closed-form solution for the fields inside and outside the sphere. The power transmitted in and scattered by the sphere can be also evaluated with a simple closed-form expression and related with the modulus of the reflection coefficient. We showed that our method is fully consistent with the classical Mie scattering theory. We also showed that our method can provide an intuitive physical interpretation of electromagnetic scattering in terms of impedance matching and resonances, and that it is especially useful for the case of inward traveling spherical waves generated by sources surrounding the scatterer.

A Physical Framework to Interpret the Effects of High Permittivity Materials on Radiofrequency Coil Performance in Magnetic Resonance Imaging

“…where i is the imaginary unit, (r, ϑ, ϕ) are the three coordinates of a spherical coordinate system with the origin in the center of the sphere, B n is a spherical Bessel function of the n th order, P m n is the associated Legendre polynomial, and π nm and τ nm are sectorial functions [18]:…”

“…In this section, we recall the basic principles of a recently introduced formulation of the scattering from spheres [ 18 ]. In the following, we will consider only the Transverse Electric (TE) polarization.…”

“…To address this, we propose an analytical framework for the study of electromagnetic (EM) field propagation in HPM based on a recently published formulation of the scattering from spheres [ 18 ]. In such framework, the field is described as a superposition of elementary waves (modes), which are expressed in terms of spherical travelling waves in every layer of the sphere.…”

“…Following this approach, one can model field propagation using non-uniform transmission lines. The formulation introduced by [ 18 ] has enriched the established work on non-uniform transmission lines [ 23 ]–[ 27 ] by providing a simple formula for the impedance transport that allows one to express reflection and transmission coefficients in terms of the layered sphere impedances, enabling a straightforward engineering analysis. Here, we employ this formulation to investigate a system composed of a uniform sphere surrounded by a layer of HPM.…”

Electromagnetic Scattering and Its Applications: From Low Frequencies to Photonics

“…Electromagnetic scattering from the so-called canonical structures is one of the subjects covered in this Special Issue. In particular, a theoretical framework is presented in [1] to provide a description of scattering phenomena involving dielectric and conducting spheres by decomposing the total field in the outer space in terms of inward and outward electromagnetic fields, rather than in terms of incident and scattered fields, as in the classical Lorenz-Mie formulation. The method can provide an intuitive physical interpretation of electromagnetic scattering in terms of impedance matching and resonances.…”