Canonical and Poynting currents in propagation and diffraction of structured light: tutorial

The local propagation and the energy flux in structured optical fields are often associated with the Poynting vector. However, the local phase gradient (i.e., local wavevector) in monochromatic fields in free space is described by another fundamental quantity: the canonical momentum density. Distributions of the Poynting and canonical momentum densities can differ significantly from each other in structured fields. We examine the role of these quantities in the propagation and diffraction of structured optical fields, exemplified by various circularly polarized vortex beams carrying orbital angular momentum. We describe the canonical and Poynting momentum distributions in such beams, experimentally measure the local transverse momentum density by a Shack-Hartmann wavefront sensor, and investigate fine features of the diffraction of various vortex beams on a knife-edge aperture. In all cases, the measured local momentum density and local beam evolution are consistent with the canonical momentum distribution rather than the Poynting vector. Furthermore, we introduce the local angular velocity in vortex beams and determine the universal integral 𝜋 angle of azimuthal rotation in an arbitrary (yet circularly symmetric) propagating and diffracting vortex beam. Finally, we discuss the “supermomentum” and “backflow” effects; both of these phenomena are examples of superoscillations and are related to the properties of the canonical momentum. Our results reveal the profound role of the canonical momentum in the evolution of light and demonstrate the importance of distinguishing between it and the Poynting vector in structured light.