Spin angular momentum and optical chirality of Poincaré vector vortex beams
Kayn A. Forbes
Abstract
Abstract The optical chirality and spin angular momentum of structured scalar vortex beams has been intensively studied in recent years. The pseudoscalar topological charge <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:math> of these beams is responsible for their unique properties. Constructed from a superposition of scalar vortex beams with topological charges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mtext>A</mml:mtext> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mtext>B</mml:mtext> </mml:msub> </mml:mrow> </mml:math> , cylindrical vector vortex beams are higher-order Poincaré modes which possess a spatially inhomogeneous polarization distribution. Here we highlight the highly tailorable and exotic spatial distributions of the optical spin and chirality densities of these higher-order structured beams under both paraxial (weak focusing) and non-paraxial (tight focusing) conditions. Our analytical theory can yield the spin angular momentum and optical chirality of each point on any higher-order or hybrid-order Poincaré sphere. It is shown that the tunable Pancharatnam topological charge <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mtext>P</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mtext>A</mml:mtext> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mtext>B</mml:mtext> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and polarization index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mtext>B</mml:mtext> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mtext>A</mml:mtext> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> of the vector vortex beam plays a decisive role in customizing their spin and chirality spatial distributions. We also provide the correct analytical equations to describe a focused, non-paraxial scalar Bessel beam.