Theory of paraxial optical skyrmions

Vector light beams, characterized by a spatially varying polarization, can exhibit localized structures reminiscent of the skyrmions familiar from the study of magnetic media. We present a theory of such skyrmions within paraxial optics, exploiting mathematical analogies with the study of superfluids, especially the A phase of superfluid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mtext/> <mml:mi>He</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . The key feature is the skyrmion field, which, together with the underlying skyrmion vector potential, determines the properties of the skyrmions and, more generally, the polarization structure of every paraxial vector beam. In addition to structures with integer skyrmion numbers, we find polarization patterns with non-integer skyrmion numbers; these seem to have no analogue in other fields of physics.