Geometric descriptions for the polarization for nonparaxial optical fields: a tutorial

Abstract

This article provides an overview of the local description of polarization for nonparaxial fields, for which all Cartesian components of the vector field are significant. The polarization of light at each point is characterized by a 3 × 3 polarization matrix, as opposed to the 2 × 2 matrix used in the study of polarization for paraxial light. For nonparaxial light, concepts like the degree of polarization, the Stokes parameters and the Poincaré sphere have generalizations that are either not unique or not trivial. This work aims to clarify some of these discrepancies and provide a framework that highlights the similarities and differences with the description for the paraxial regimes. Particular emphasis is placed on geometric interpretations.

Optical helicity of unpolarized light

“…which gives the identical optical helicity contribution as the method of averaging the optical helicity densities of two orthogonal linear polarizations (10). This alternative method of calculation equally proves that there is an optical helicity density which is 2D polarization independent in 3D vortex beams.…”

“…This alternative method of calculation equally proves that there is an optical helicity density which is 2D polarization independent in 3D vortex beams. It is also important to highlight that the middle term in square brackets in (12) is the 2D polarization independent optical helicity density term [i.e., ( 6), ( 9), (10), and ( 13)] and so we may also conclude that even when the input source field is 2D circularly polarized this 2D polarization independent 023524-3 contribution survives and is therefore robust again spin-orbit interactions of light [5].…”

“…For example, taking a paraxial 2D structured vortex and tightly focusing it to create a 3D structure, the ensuing optical helicity density contribution (10) generated is completely independent of the 2D polarization state of the input paraxial structured mode. The 2D polarization independent optical helicity density (10) is plotted in Fig. 1 at w 0 .…”

“…1 because of the fact this is the radial index predominantly implemented in experiments involving LG beams. However, (10) is general and applies to any mode ( , p). Increasing p has two effects: firstly, (2p + 2) rings of helicity density are produced in general and secondly, we increase the relative magnitude of the longitudinal fields responsible for the 2D polarization independent helicity density because increasing p leads to larger transverse field gradients and thus larger longitudinal field components.…”

“…The nonparaxial nature and 3D structure of spatially confined optical fields has led to some remarkable light-matter interactions [4][5][6][7], which are especially striking when compared to the prevailing textbook description of light-matter interactions in terms of propagating plane waves. Because all three spatial components of the field vector generally play a role in nonparaxial light, the theory of polarization has been extended to the 3D case [8][9][10]. While generic 2D polarization is described by four Stokes parameters, 3D polarization is characterized by nine polarization parameters.…”