Semispherical Conservation of Topological Charge in Scalar Diffraction Theory

Topological charge is an important characteristic of a laser beam and is closely related to orbital angular momentum. Although in most cases the charge is conserved when a beam propagates in space, the opposite examples are known. We formulate and prove a general theorem for conservation of topological charge, which involves not only charges in the plane perpendicular to the propagation axis, but also in planes parallel to the propagation axis. Charges of the second type can be identified to some approximation with spatiotemporal charge, which have been widely studied in recent years.