Optical Möbius Strips in Three Dimensional Ellipse Fields: Lines of Circular Polarization

The major and minor axes of the polarization ellipses that surround singular lines of circular polarization in three dimensional optical ellipse fields are shown to be organized into Möbius strips. These strips can have either one or three half-twists, and can be either right-or left-handed. The normals to the surrounding ellipses generate cone-like structures. Two special projections, one new geometrical, and seven new topological indices are developed to characterize the rather complex structures of the Möbius strips and cones. These eight indices , together with the two well-known indices used until now to characterize singular lines of circular polarization, could, if independent, generate 16, 384 geometrically and topologically distinct lines. Geometric constraints and 13 selection rules are discussed that reduce the number of lines to 2, 104, some 1, 150 of which have been observed in practice; this number of different C lines is ∼ 350 times greater than the three types of lines recognized previously. Statistical probabilities are presented for the most important index combinations in random fields. It is argued that it is presently feasible to perform experimental measurements of the Möbius strips and cones described here theoretically.