Topological interference: singular optics, optical vortices and
polarization singularities
Mark Dennis, University of Bristol, UK
Understanding of complicated spatial patterns emerging from physical phenomena is
frequently aided by insight from topology: the isolated places where some fundamental
physical quantity -- such as optical phase in a complicated interference or diffraction
pattern -- is undefined (or singular) organize the rest of the field. Such defects play an
important role in condensed matter physics, such as dislocations in crystals, disclinations
in liquid crystals, and quantized vortices in superfluids.
Similar defects occur in optical fields. In scalar wave patterns, the optical phase is
undefined at points in 2D, and lines in 3D, in general whenever 3 or more waves
interfere. This happens when the intensity is zero, and such nodes are called optical
vortices as they are vortices of optical energy flow. These optical vortices are a generic
feature of optical speckle patterns, and their spatial configuration can be controlled
holographically in 3D to create closed loops, lines and knots.
In optical fields whose polarization varies with position, these vortices are replaced by
polarization singularities: loci where the polarization azimuth undefined (e.g. the
polarization ellipse is linear). Several branches of classical optics can be reformulated in
terms of these singularities, in particular crystal optics, and the polarization pattern of
skylight (possibly known to the Vikings).
Numerically-calculated tangle of vortex lines in a three-dimensional speckle pattern. The
tangle is projected into the xz-plane, with the white curves depicting closed loops, the red
curves infinite lines. (Figure courtesy of Kevin O'Holleran, University of Glasgow.)
Fisheye photograph of the full sky hemisphere, with observed polarization directions (red
lines) and theoretical fit (blue curves). The polarization fingerprint pattern has a
singularity (point where polarization direction is undefined) above the sun.