  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.
