Conservation law for angular momentum based on optical field derivatives: Analysis of optical spin-orbit conversion

We present a theoretical framework for analyzing the loss of optical angular momentum (AM), including spin (SAM) and orbital (OAM) components, in light-matter interactions. Conventional SAM and OAM conservation laws rely on transverse field components, neglecting longitudinal fields and limiting applicability to vacuum. Our approach defines optical AM using time derivatives of the electric and magnetic fields, yielding a gauge-invariant formulation that includes both transverse and longitudinal components and explicitly incorporates charge and current densities. This enables a more complete description of AM dissipation in materials. We apply this framework to analyze spin-orbit conversion (SOC) in two scenarios: scattering of circularly polarized (CP) beams by a gold nanoparticle and focusing of CP and linearly polarized optical vortex beams by a lens. The results show that SOC depends on particle size and polarization, with notable OAM loss in larger particles and CP beam focusing. This framework enables the evaluation of previously overlooked SAM and OAM losses, providing a powerful tool for studying systems in which the analysis of AM losses is intrinsically important, such as chiral materials, as well as for designing photonic devices and exploring light-matter interactions at the nanoscale.