The field of values of Jones matrices: classification and special cases | Zendy

Julio C. Gutiérrez-Vega | Zendy

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The field of values of Jones matrices: classification and special cases

Author(s) -

Julio C. Gutiérrez-Vega

Publication year - 2020

Publication title -

proceedings of the royal society a mathematical physical and engineering sciences

Language(s) - English

Resource type - Journals

eISSN - 1471-2946

pISSN - 1364-5021

DOI - 10.1098/rspa.2020.0361

Subject(s) - eigenvalues and eigenvectors , polarization (electrochemistry) , geometric phase , homogeneous space , physics , algebraic number , optics , mathematics , mathematical analysis , geometry , quantum mechanics , chemistry

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.

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