# Momentum-space geometric structure of helical evanescent waves and its implications on near-field directionality

## TL;DR

This paper uncovers a geometric structure in helical evanescent waves that links near-field and far-field properties, enabling the design of directional near-field sources through structured illumination.

## Contribution

It reveals a momentum-space geometric framework for helical evanescent waves and introduces a method to control near-field directionality using structured helical illumination.

## Key findings

- Helical evanescent waves have a momentum-space structure with shared vector orientations.
- Zeroes in the angular spectrum are helicity singularities where tangent lines intersect.
- Structured illumination allows full control of near-field directionality.

## Abstract

In this work, a momentum-space geometrical structure in helical evanescent electromagnetic waves is revealed. It is shown that for every helical evanescent wave on a helicity-dependent half tangent line in momentum space, the orientation of each of its field, spin, and Poynting vectors is the same. This geometric structure reveals itself as a remarkable relation between the far-field and near-field components of the angular spectrum. Any general evanescent wavevector is linked to two points on the $k_{\rho}=k_0$ circle of propagating wavevectors via two helicity-dependent tangent lines. Knowing the field on the $k_{\rho}=k_0$ circle of a general dipolar source is sufficient to determine its entire evanescent angular spectrum. Applying this concept, we gain insights into near-field directionality by showing that every zero in the angular spectrum is a helicity singularity where two half-tangent lines of opposite helicity intersect. A powerful method for synthetic design of near-field directional sources is also devised, using structured helical illumination to gain full control of the near-field directionality. The results provide a fundamental insight of helical evanescent waves and have implications in areas where chiral light-matter interaction plays a central role.