电磁学中的双索引几何相位
The double-indexed geometric phase for electromagnetics
浏览论文内容
中文总结 AI 辅助
本文提出双索引几何相位（DIGP）以扩展潘查拉坦姆相位，解析其对称性与几何解释，验证其可用于非共传播平面波比较，能解析薄膜及三维物体的偏振相关特征，为偏振表征与逆散射提供互补框架。
中文摘要 AI 辅助
本文基于一族庞加莱旋量构建了双索引几何相位（DIGP），以扩展通常用于描述光学系统中偏振态相对于参考偏振态演化的潘查拉坦姆相位。分析结果确立了双索引几何相位的对称性：当庞加莱球上纬度的索引 $p\in\mathbb{R}$ 反转时、同一球上经度的索引 $q\in\mathbb{R}$ 反转时，双索引几何相位的变化，以及其对 $q$ 的周期性依赖关系。对于庞加莱球上的闭合回路，双索引几何相位可通过与 $p$ 相关的地理坐标映射及球心所对的立体角来进行几何解释。尽管几何相位通常用于比较自由空间中沿相同方向传播的两个平面波，但双索引几何相位有望在比较两个非共传播平面波时包含相关信息，这一点已通过平面薄膜的镜面反射和三维物体的远场散射得到验证。薄膜镜面反射与透射的双索引几何相位图可解析布拉格现象、法布里-珀罗共振、结构手性、各向异性及缺陷，这些细节在反射率和透射率图中并不明显。三维物体平面波散射的方向依赖双索引几何相位图可能揭示出微分散射效率图中缺失的强极向和方位角结构。主成分分析表明，由同一组偏振测量计算得到的一组双索引几何相位图构成了一个协调的数据族，而非一系列独立的可观测量。双索引几何相位概念为偏振态分辨表征和逆散射提供了一个互补框架。
英文摘要
The double-indexed geometric phase (DIGP) based on a family of Poincaré 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 Poincaré 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 Poincaré 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--Pérot 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.
