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The minor axes of, and the normals to, the polarization ellipses thatsurround singular lines of linear polarization in three dimensional opticalellipse fields are shown to be organized into Mobius strips and into structureswe call rippled rings (r-rings). The Mobius strips have two full twists, andcan be either right- or left-handed. The major axes of the surrounding ellipsesgenerate cone-like structures. Three orthogonal projections that give rise to15 indices are used to characterize the different structures. These indices, ifindependent, could generate 839,808 geometrically and topologically distinctlines; selection rules are presented that reduce the number of lines to 8,248,some 5,562 of which have been observed in a computer simulation. Statisticalprobabilities are presented for the most important index combinations in randomfields. It is argued that it is presently feasible to perform experimentalmeasurements of the Mobius strips, r-rings, and cones described heretheoretically.

Optical M0bius Strips in Three Dimensional Ellipse Fields: Lines of Linear Polarization