Optical Vortices, Faraday Rotation, Optical Activity

Characteristic solutions to Maxwell's equations, constrained to equivalence classes by constitutive equations that have certain crystal symmetries, are represented by phase 1-forms, k, such that both 3-forms vanish: k^F = 0 and k^G = 0: Simple examples are given which show the di®erences between Faraday rotation where the 3-form of Topological Torsion is not zero, A^F 6= 0, and Optical Activity where the 3-form of Topological Spin is not zero, A^G 6= 0. The 1-form, k, whose line integral generates the phase function £; is not necessarily equal to the system 1-form of Action per unit charge, A, whose components form the vector and scalar potentials. The characteristic solutions demonstrate that linearly polarized solutions are to be associated with topological spin, and that circularly polarized solutions are to be associated topological torsion. This result is counter to the popular view that "spin" is related to circular polarization. When the 3-forms of A^F and A^G; are closed in a exterior di®erential sense, then their integrals over closed domains form deformable invariants with values whose ratios are rational (quantized). There are two types of optical phase defects. The ̄rst type of defect is related to Faraday rotation, circular polarization, and topological torsion, A^F 6= 0: The second type of defect is related to Optical Activity, linear polarization, and topological spin, A^G 6= 0: It would appear appropriate to describe the ̄rst type of defect as Optical Vortices with defects of rotational shears. The second type of phase defect is related to dislocations which involve translational