The concept of field of values (FoV), also known as the numerical range, is applied to the 2 × 2 Jones matrices used in polarization optics. We discover the relevant interplay between the geometric properties of the FoV, the algebraic properties of the Jones matrices and the representation of polarization states on the Poincaré sphere. The properties of the FoV reveal hidden symmetries in the relationships between the eigenvectors and eigenvalues of the Jones matrices. We determine the main mathematical properties of the FoV, discuss the special cases that are relevant to polarization optics, and describe its application to calculate the Pancharatnam-Berry phase introduced by an optical system to the input state.

中文翻译：

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琼斯矩阵的值域：分类和特殊情况

值场 (FoV) 的概念，也称为数值范围，适用于偏振光学中使用的 2 × 2 琼斯矩阵。我们发现了 FoV 的几何特性、琼斯矩阵的代数特性和 Poincaré 球上极化状态的表示之间的相关相互作用。FoV 的属性揭示了琼斯矩阵的特征向量和特征值之间的关系中隐藏的对称性。我们确定了 FoV 的主要数学特性，讨论了与偏振光学相关的特殊情况，并描述了其在计算由光学系统引入到输入状态的 Pancharatnam-Berry 相位的应用。