Hidden order in optical ellipse fields: I. Ordinary ellipses

The principal axes of ordinary (i.e., nonsingular) ellipses in elliptically polarized light are organized into locally ordered structures that take the form of cones and full-twist Möbius strips. These generic features appear in a wide variety of ellipse fields that range from ordered optical lattices to random speckle patterns. The cones and Möbius strips can take a number of different forms, and are characterized by a total of 21 different topological indices. These indices divide the field into a large number of small grains with distinguishably different structures separated by singular boundaries across which an index changes sign. Using a general lowest order expansion of the optical field that involves some 20 parameters, analytical expressions are obtained for all indices. These expressions are used to prove the four selection rules that reduce the number of possible configurations from 2²¹ = 2,097,152 to 52³ = 140,608. Within the linear approximation there exist degeneracies that further reduce the number of configurations to 17,360. Of these, 1728 are of first order, and should be readily accessible to experiment. Structures similar to those in elliptically polarized light can be expected to be present in other fields, such as liquid crystals, whose basic elements can be described by ellipses or ellipsoids and should be sought in such fields.