Optical Vortices

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 "polarization".

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 is zero.

Current 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 momentum".

An entangled state can be constructed from the defect modes.


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Last update 08/27/2001
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