The double-indexed geometric phase for electromagnetics
The double-indexed geometric phase (DIGP) based on a family of Poincar\'e spinors was formulated to extend the Pancharatnam phase, commonly used to describe the evolution of the polarization state in an optical system relative to a reference polarization state. Analytical results establish the symmetries of the DIGP under reversal of the index $p\in\mathbb{R}$ for the latitude on the Poincar\'e sphere, the phase change induced by reversal of the index $q\in\mathbb{R}$ for the longitude on the same sphere, and the periodic dependence on $q$. For closed loops on the Poincar\'e sphere, the DIGP can be interpreted geometrically through a $p$-dependent mapping of both geographic coordinates and the associated solid angle subtended at the center of the sphere. Although the geometric phase is usually applied to compare two plane waves propagating in the same direction in free space, the DIGP is expected to contain information when comparing two non-co-propagating plane waves, as demonstrated by specular reflection by a planar thin film and far-zone scattering by a three-dimensional object. DIGP maps for specular reflection and transmission by thin films resolve Bragg phenomenons, Fabry--P\'erot resonances, structural chirality, anisotropy, and defects with detail not evident in reflectance and transmittance maps. A map of direction-dependent DIGP for plane-wave scattering by a three-dimensional object may reveal strong polar and azimuthal structure absent from the map of differential scattering efficiency. Principal component analysis indicates that a collection of DIGP maps, all calculated from the same set of polarimetric measurements, form a coordinated data family rather than a collection of independent observables. The DIGP concept offers a complementary framework for polarization-state-resolved characterization and inverse scattering.