Circular lines of circular polarization in three dimensions, and their transverse-field counterparts

A superficially paradoxical phenomenon is associated with curved c lines, on which the polarization of a three-dimensional optical field in space is purely circular. If the c line has fixed index (signed half-integer winding number), its two intersections with an observation plane, each with the same index, coalesce and annihilate when the c line is tangent to the plane, seeming to contradict the conservation of index. But there is no paradox: for the associated C line (distinct from the c line) on which the field transverse to the observation plane is circularly polarized, the two intersections have opposite indices, so their total index is zero, which is conserved during the annihilation. The different geometries of the c and C lines are studied in detail for model fields where the c line is a circle.