Non-paraxial 3d Polarization in 4d Light Fields

Abstract

Recently, research has mainly been focused on structured singular light in the paraxial regime, as also presented in previous chapters. Here, investigations were concentrated on fields shaped in 2d or 3d space, in which the typically elliptical polarization of light is purely transverse, i.e., two-dimensional (2d polarization). Hence, the light field oscillates solely within the plane orthogonal to the beam’s propagation direction with \(\vec {E}(x,y,z) = [E_x(x,y,z), E_y(x,y,z)]^T\). Respective studies have lead to pathbreaking findings enabling advanced applications as well as new insights into the general and not yet fully explored, nor understood, nature of light (see Chaps. 2, 3). However, a major current challenge is the moving from these 3d fields of 2d polarization to 4d structured light of 3d polarization with \(\vec {\mathcal {{E}}}(x,y,z) = [\mathcal {E}_x(x,y,z), \mathcal {E}_y(x,y,z), \mathcal {E}_z(x,y,z)]^T\), exploiting all three electric field components, i.e., transverse as well as longitudinal ones. These fields do not only give new insights into the fundamentals of light, but also open new horizons for applied optics: 4d fields allow for structures of nano-scale complexity whereby their 3d polarization nature is an enriching feature for handling, e.g., polarization sensitive materials. The crucial longitudinal components are non-negligible for non-paraxial light fields, fulfilling the vectorial Helmholtz equation (Eq. (2.7)), as in tightly focused beams. These fields are formed by the application of focusing optical components imparting a numerical aperture (\(\mathrm {NA}\)) larger or equal to 0.7. In this case, radial components of the input light field are transformed into significant longitudinal focal field contributions [1,2,3]. Considering this effect, some studies have been performed representing first steps towards the investigation and future application of 4d structured fields. For example, tightly focusing 2d polarization structures enabled shaping focal intensity configurations [4,5,6,7,8] as optical needles. Furthermore, longitudinal field components facilitate the formation of complex 3d polarization topologies surrounding optical singularities as optical cones, twisted ribbons and Möbius strips. These topologies were as firstly predicted by Freund in 2005 [9] and, ten years later, experimentally proven by Bauer et al. [10].