Cones, spirals, and Möbius strips, in elliptically polarized light

The orientation of the ellipses in elliptically polarized light generally varies throughout space. In three dimensions, the orientation of an ellipse may be described by a 3-frame in which one frame axis is along the major axis of the ellipse, a second frame axis is along the minor axis of the ellipse, and the third frame axis is along the normal to the ellipse. These three axes are shown to generate cones, spirals, and Möbius strips, characterized by a total of 27 different topological indices. For ordinary ellipses (the vast majority) that are not on singular lines of circular or linear polarization 21 indices are non-zero. These indices, if independent, could collectively divide the field into 2²¹ = 2,097,152 structurally different volumes separated by singular surfaces on which an index becomes undefined. We have, however, found selection rules that reduce the number of independent configurations to 140,608, and have demonstrated that large numbers of configurations appear in practice by harvesting more than 10,000 different configurations in a simulated random field. This structural proliferation is intrinsic to spatially varying elliptically polarized light, and is also found in non-random ellipse fields such as optical lattices. Other systems described locally by spatially varying 3-frames, such as liquid crystals or the dielectric constants of random media, may show a similar degree of structural proliferation.