The occurence of "vortex structures" in laser beams can be related to Faraday rotation and the Falaco Soliton structures observed in fluid discontinuities.
A topological view of electromagnetism indicates that there are two distinct
concepts of Topological Torsion and Topological Spin, and both have handedness properties that can be described by
However, even the classic study of characteristic (signal) solutions to
Maxwell's equations indicates that the "polarization" due to Optical Activity
and the "polarization" due to Faraday rotation are not the same. The first is
reciprocal and the second is not reciprocal. For each effect taken separately,
a complex number representation is possible, and the phase velocities of the
characteristic polynomial for the singular (signal) solutions to Maxwell's
equations are doubly degenerate.
The usual attempts to analyze the polarization problem assume that the wave
functions can be written as complex functions with amplitude and phase.
However, when both Faraday polarization and the polarization of Optical Activity
occur simultaneously it has been shown that the singular (signal) solutions to
Maxwell's equations do not admit representation as complex numbers, but instead
require a quaternion
representation . No longer are the phase velocities doubly degenerate, but
the different states of polarization have phase velocites that are distinct for
each direction of propagation and for each handedness.
In addition to the singular characteristic (phase) solutions, there are extremal
(phase) solutions which exhibit "defects" and topological quantum numbers.
Topological Torsion reduces to the classic Faraday polarization when the
extremal torsion quantum number is zero. Topological Spin reduces to the
classic polarization of Optical Activity when the extremal spin quantum number
literature states "Phase singularities in laser beams carry angular
momentum due to the associated helical wavefront structure. This angular
momentum can be transferred to absorbing particles trapped in the beam, setting
them into rotation." From a topological point of view it would appear
permissable to associate the polarization of Faraday rotation with "orbital
angular momentum" and the polarization of Optical Activity to "spin angular
An entangled state can be constructed from the defect modes.