van de Hulst Essay: Geometric-phase portrayal of electromagnetic scattering by a three-dimensional object in free space

TL;DR

This essay introduces a geometric-phase framework for electromagnetic scattering by a 3D object in free space, representing plane-wave illumination on the Poincaré sphere and defining symmetric and asymmetric Poincaré spinors for the far-zone scattered field. It develops a formalism based on vector spherical wavefunctions to express the scattered field, its far-field amplitude, and direction-dependent Stokes parameters, then defines two direction-dependent geometric phases to compare incident and scattered waves. The approach is illustrated with six homogeneous isotropic spheres under various boundary conditions, showing that geometric-phase densities reveal richer, more polarization- and boundary-condition-sensitive information than standard differential scattering metrics. The results suggest strong potential for inverse-scattering applications and for enriching forward-scattering analyses through polarization- and boundary- dependent geometric phases.

Abstract

The concept of geometric phase was applied to initiate the geometric-phase portrayal of electromagnetic scattering by a three-dimensional object in free space. Whereas the incident electromagnetic field is that of an arbitrarily polarized plane wave, the direction-dependent far-zone scattering amplitude is used to define direction-dependent Stokes parameters for the scattered field. Both symmetric and asymmetric Poincaré spinors are formulated to characterize the polarization states of incident plane wave and the far-zone scattering amplitude, and two different geometric phases are defined therefrom. Density plots of both geometric phases were calculated for six different homogeneous isotropic spheres with different linear constitutive properties and boundary conditions: dielectric-magnetic spheres (non-dissipative and dissipative), impedance spheres, perfect electrically conducting spheres, charged dielectric-magnetic spheres, dielectric-magnetic spheres with topologically insulating surface states, and isotropic chiral spheres. The incident plane waves were taken to be linearly and circularly polarized, for the sake of illustration. Numerical results revealed that geometric-phase density plots possess significantly richer features than their counterparts for the differential scattering efficiency. The geometric-phase portrayals exhibit enhanced sensitivity to changes in the size and composition of the scatterer, the boundary conditions, and the incident polarization state, suggesting promise for inverse-scattering problems.

Figures (11)

• Figure 1: Schematic illustrating the boundary-value problem.
• Figure 2: $Q_{\rm D}$, $\Psi^{\rm s}_{\rm sca}$, and $\Psi^{\rm a}_{\rm sca}$ as functions of $\theta$ and $\phi$ for a non-dissipative dielectric-magnetic sphere with charge-free surface. The sphere of size parameter $k_{ 0} a=5$ is made of a material with $\varepsilon_{\rm r}=3$ and $\mu_{\rm r}=1.3$.
• Figure 3: Same as Fig. \ref{['Fig:DMlossless']} except that $\varepsilon_{\rm r}=3(1+0.1i)$ and $\mu_{\rm r}=1.3(1+0.1i)$.
• Figure 4: Same as Fig. \ref{['Fig:DMlossless']} except that $\varepsilon_{\rm r}=3(-1+0.1i)$ and $\mu_{\rm r}=1.3(-1+0.1i)$.
• Figure 5: $Q_{\rm D}$, $\Psi^{\rm s}_{\rm sca}$, and $\Psi^{\rm a}_{\rm sca}$ as functions of $\theta$ and $\phi$ for an impedance sphere of size parameter $k_{ 0} a=5$ and relative surface impedance $\eta_{\rm s}=4$.