J. Eur. Opt. Society-Rapid Publ. 2025, 21, 29 Ó The Author( s), published by EDP Sciences, 2025 https:// doi. org / 10.1051 / jeos / 2025024 Available online at: https:// jeos. edpsciences. org
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE
Interpretation of three-dimensional polarization states through the smart decomposition of the polarization matrix
José J. Gil 1,*
, Andreas Norrman 2, Ari T. Friberg 2, and Tero Setälä 2
1 Group of Photonic Technologies, University of Zaragoza, Pedro Cerbuna 12, 50009, Zaragoza, Spain 2 Center for Photonics Sciences, University of Eastern Finland, PO Box 111, FI-80101 Joensuu, Finland
Received 24 April 2025 / Accepted 14 May 2025
Abstract. A complete description of a three-dimensional( 3D) polarization state is provided by the two most significant eigenstates of the polarization matrix, together with the two indices of polarimetric purity. By means of the so-called smart decomposition, such information can be arranged to represent the state as a combination of two components, one partially polarized( active component) and one unpolarized. Contrary to what happens for two-dimensional( 2D) polarization states( with the electric field fluctuating within a fixed plane), whose active component is constituted by a single totally polarized state, in the general case of 3D polarization states the active component is given by a weighted incoherent composition of the two above-mentioned eigenstates. We show that a detailed description of the intensity and spin anisotropies is encompassed by the active component of the state, which admits a simple interpretation and geometric representation. In addition, it is found that the degree of nonregularity can be viewed as a distance of the state to a regular state.
Keywords: Polarization, Nonparaxial light, Smart decomposition, Nonregular polarization states.
1 Introduction
The polarization properties of an electromagnetic wave at a given point in space are represented by the associated 3 3 polarization matrix( also called coherency matrix). Leaving aside the particular well-known case of partially polarized states whose electric field fluctuates in a fixed plane( hereafter called two-dimensional( 2D) polarization states), recent advances in the study and control of light related to near-field phenomena, tightly focused beams, and nanotechnology in general, have stimulated much interest in true three-dimensional( 3D) polarization states [ 1 – 13 ]. Paralleling this progress, interpretations of the polarization matrix from algebraic, statistical, and geometrical points of view have been dealt with in a number of contributions from both theoretical and applied perspectives [ 14 – 28 ]. While the polarization matrix describes local polarization effects, the 3D polarization features of spatially distributed( random and deterministic) fields have also been actively studied [ 29 – 37 ].
In this work, the frame for the physical interpretation of genuine 3D polarization states is set by the so-called smart decomposition, which expresses the 3 3 polarization matrix as an incoherent composition of two totally polarized states( called pure states) and a totally unpolarized
* Corresponding author: ppgil @ unizar. es
3D state [ 21, 38 ]. This approach provides a new view on certain properties, like the degree of nonregularity [ 39 ], obtained previously through other procedures. In general, the analysis relies on the fact that the unpolarized component does not carry any information on the shape of the polarization object, composed of the intensity ellipsoid and the spin vector of the state [ 17, 27, 40 ]. The component which is not unpolarized, hereafter called the active component, depends on the specific configuration of the pair of eigenstates associated with the largest eigenvalues of the polarization matrix, and it determines the intensity and spin anisotropies of the polarization state. This means that, up to a scale factor fixed by the relative weight of the unpolarized component, all polarization descriptors, including the nine 3D Stokes parameters, the spin vector, the degree of nonregularity, etc., are provided by the active component.
The structure and main points of the work are as follows. Section 2 is devoted to introducing the theoretical framework necessary to describe the original results presented in Sections 3 – 5, which include the definition of the active component of a polarization state as a representative of the intensity and spin anisotropies in a simple and condensed manner. The smart decomposition( without using this name) was introduced for the first time in [ 21 ] and then also described in [ 38 ]; nevertheless, some aspects like the indices of polarimetric purity( IPP) of the active
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