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 demonstrate that the differences between Faraday rotation where the 3-form of Topological Torsion, A^F, is not zero, and Optical Activity where the 3-form of Topological Spin, A^G, is not zero.
The 1-form, k, whose line integral generates the phase function, Theta, 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
This result is counter to the popular view that ''spin'' is related to circular
When the 3-forms A^F and A^G are closed in a exterior differential sense, then
their integrals over closed domains form deformable (topological) invariants
with values whose ratios are rational (quantized). There are two types of
optical phase defects. The first type of defect is related to Faraday rotation,
circular polarization, and topological torsion, with A^F not equal to zero. The
second type of defect is related to Optical Activity, linear polarization, and
topological spin, with A^G not equal to zero.
It would appear appropriate to describe the first type of defect as Optical
Vortices with defects of rotational shears. The second type of phase defect is
related to dislocations which involve translational shears.