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arXiv:2610.40523

On the gauge-invariant orbital angular momentum of vector optical vortices in the paraxial approximation and beyond

E. V. Maliutina, E. A. Vashukevich, V. A. Leonov, D. D. Reshetnikov, T. Yu. Golubeva

physics.opticsquant-phSubmitted Oct 29, 2026• Announced Oct 30, 2026
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Abstract

In this paper we present a gauge-invariant formalism for describing the spin and orbital angular momenta of scalar and vector optical vortices. The correction term, arising from consistent incorporation of the transverse vector potential's nonlocality, restores gauge invariance and makes it possible to separate the total angular momentum into spin and orbital components. We propose a method for calculating the orbital and spin angular momenta of multimode vector vortex fields and show that the orbital angular momentum of such beams contains two qualitatively different components. The polarization-independent contribution to the orbital angular momentum is related to the topological charge that determines the order of the phase singularity, while the polarization-dependent one is proportional to the helical topological charge that determines the spatial inhomogeneity of the polarization. We also discuss the quantization procedure for fields with spatially inhomogeneous polarization in terms of modes with a helical topological charge, and examine the limits of applicability of the proposed formalism.

AI Summary

The paper addresses gauge-invariant separation of electromagnetic angular momentum into spin and orbital contributions for optical vortices with spatially varying polarization, including corrections beyond the paraxial limit. It constructs the transverse vector potential using the Helmholtz projector, expands the inverse Laplacian in transverse derivatives, and evaluates angular-momentum integrals for Laguerre–Gaussian profiles. The resulting expressions are factorially divergent asymptotic series in the nonparaxiality parameter $\xi=\sqrt{2}/(k_0^2\rho_0^2)$. Retaining a surface term, described as a Belinfante correction, gives $J_z=L_z+S_z-G_z$. After redistributing this correction between integrated spin and orbital quantities according to their helicity and topological-charge dependence, Borel summation yields analytical expressions recovering $L_z=l$ and $S_z=\sigma$ in the paraxial limit, with overall normalization suppressed. For a two-component vector vortex with circularly polarized components of charges $l_1,l_2$, the reported paraxial result is $L_z=l+w\sigma$, where $l=(l_1+l_2)/2$ and $w=(l_1-l_2)/2$. This distinguishes a common-phase contribution from one associated with rotation of the polarization basis. A multimode extension uses polarization-resolved mean charges. Quantization is formulated through a change from circular-polarization/OAM modes to radial and azimuthal vector-vortex modes. The construction assumes slowly varying longitudinal amplitudes and primarily treats fields near the beam waist. The paper states that separate local spin and orbital densities remain unavailable, that the expansions require regularization, and that its mode-operator commutation relations apply for $z\ll z_R$. No experimental demonstration is reported.

Little sign of AI-generated textOverclaimsUnclear noveltyStrong assumptionsPossible error

Central claim may not hold: The correction is added to spin and orbital quantities already defined gauge-invariantly, and the subsequent conventional allocation of terms does not establish that gauge invariance determines the claimed physical separation. The fixed-frequency, fixed-kz ansatz violates the free-space dispersion relation for nonzero transverse wavevectors, omitting longitudinal evolution at the same order as the retained nonparaxial corrections. Equation (n_laplacian) has an extra azimuthal factor exp(i phi) already at n=0 compared with direct differentiation of the stated scalar-vortex potential. Appendix A's proposed Borel integral diverges at its zero lower endpoint for every nonnegative integer w and therefore cannot have the asserted limit of one. The stated polarization basis gives Lz=-w for a pure right-circular field with phase exp(-iw phi), contradicting Lz=w sigma and exposing a reversed circular-amplitude conversion.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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11 pages, 1 figure

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In this paper we present a gauge-invariant formalism for describing the spin and orbital angular momenta of scalar and vector optical vortices. The correction term, arising from consistent incorporation of the transverse vector potential's nonlocality, restores gauge invariance and makes it possible to separate the total angular momentum into spin and orbital components. We propose a method for calculating the orbital and spin angular momenta of multimode vector vortex fields and show that the orbital angular momentum of such beams contains two qualitatively different components. The polarization-independent contribution to the orbital angular momentum is related to the topological charge that determines the order of the phase singularity, while the polarization-dependent one is proportional to the helical topological charge that determines the spatial inhomogeneity of the polarization. We also discuss the quantization procedure for fields with spatially inhomogeneous polarization in terms of modes with a helical topological charge, and examine the limits of applicability of the proposed formalism.
The paper addresses gauge-invariant separation of electromagnetic angular momentum into spin and orbital contributions for optical vortices with spatially varying polarization, including corrections beyond the paraxial limit. It constructs the transverse vector potential using the Helmholtz projector, expands the inverse Laplacian in transverse derivatives, and evaluates angular-momentum integrals for Laguerre–Gaussian profiles.
The resulting expressions are factorially divergent asymptotic series in the nonparaxiality parameter ξ=2/(k02ρ02)\xi=\sqrt{2}/(k_0^2\rho_0^2)ξ=2​/(k02​ρ02​). Retaining a surface term, described as a Belinfante correction, gives Jz=Lz+Sz−GzJ_z=L_z+S_z-G_zJz​=Lz​+Sz​−Gz​. After redistributing this correction between integrated spin and orbital quantities according to their helicity and topological-charge dependence, Borel summation yields analytical expressions recovering Lz=lL_z=lLz​=l and Sz=σS_z=\sigmaSz​=σ in the paraxial limit, with overall normalization suppressed.
For a two-component vector vortex with circularly polarized components of charges l1,l2l_1,l_2l1​,l2​, the reported paraxial result is Lz=l+wσL_z=l+w\sigmaLz​=l+wσ, where l=(l1+l2)/2l=(l_1+l_2)/2l=(l1​+l2​)/2 and w=(l1−l2)/2w=(l_1-l_2)/2w=(l1​−l2​)/2. This distinguishes a common-phase contribution from one associated with rotation of the polarization basis. A multimode extension uses polarization-resolved mean charges. Quantization is formulated through a change from circular-polarization/OAM modes to radial and azimuthal vector-vortex modes.
The construction assumes slowly varying longitudinal amplitudes and primarily treats fields near the beam waist. The paper states that separate local spin and orbital densities remain unavailable, that the expansions require regularization, and that its mode-operator commutation relations apply for z≪zRz\ll z_Rz≪zR​. No experimental demonstration is reported.
We developed a theoretical formalism for the generation of optical vortices by phased arrays of atoms. Using the Jacobi–Anger expansion, we demonstrated the resulting field topology and determined the least number of individual atoms necessary for the generation of vortices with a given topological charge. Vector vortices were considered, taking into account both the spin and orbital angular momenta of electromagnetic fields. It was found for the vortex field that, in the far field limit, the spatial variation in spin-density matrix parameters—orientation and alignment—is independent of the distance to the radiation source.
The angular momentum of light is a useful resource for many applications. In specific physical architectures it can be considered as the sum of two independent terms, the spin and the orbital components, in analogy to particle systems. The spin angular momentum is related to the polarization of the optical beam, that is the direction of the oscillating electric field, whereas the orbital angular momentum is associated with the spatial distribution of the field. Being independent, spin and orbital angular momenta have been discovered and explored in separate contexts for many years, while only recently it has been considered the possibility to address both quantities on the same beam (or individual photons). The interaction between these two quantities gives rise to complex structures of the electromagnetic field, or to the so called classical entanglement in the domain of single photons. The research presented in this work aimed to show that combining spin and orbital angular momenta in light beams or single photons may be a useful tool for a variety of applications, with particular interest to the case of architectures characterized by spin-orbit interaction. This concept was made concrete through the design and the realization of several experiments, in the framework of singular optics, foundations of quantum mechanics, quantum information theory and quantum simulation.
Optical vortices became a hot topic since almost two decades ago, when it was recognized that Laguerre-Gaussian laser modes carry orbital angular momentum [Allen et al. Phys Rev A 45, 8185 (1992)] related with a screw phase dislocation, and different from the spin angular momentum associated to circular polarization. In 1995, this dynamical quantity was transferred to matter in an optical micromanipulation system for the first time [H. He, et al., Phys. Rev. Lett. 75, 826 (1995)], and since then, a number of studies on angular momentum of light have unveiled different interesting aspects on the subject. However, there are still open questions, which have arisen together with the generation of novel light beams, such as vector vortices, for instance. In contrast with scalar vortices, with usual polarization states (linear, circular, elliptical), the orientation and magnitude of the electric field of vector vortices (solutions of the vector wave equation) is a function of space and time. In this work, we present an experimental study of the local angular momentum density of a Bessel vector vortex of first order by means of an optical trap. For this purpose, we used different probe particles in order to sense the local contribution to the optical angular momentum in each region of the beam. But optical fields are not the only wave fields that may exhibit phase dislocations or singularities. There are close analogies between light and sound fields that can be exploited in order to get a better understanding of common phenomena and study new aspects in both branches of physics. Here we also present the first experimental demonstration and theoretical analysis of acoustical vortices in free field, with similar properties to those of the optical vortices, including the angular momentum that can be transferred to matter. The corresponding analogies and differences with the optical case turn out to be very enlightening for the understanding of the phenomenon of angular momentum in wave fields.
We investigate the propagation of an optical vector vortex weakly interacting with a coherently prepared atomic medium (phaseonium) in a three-level ΛΛΛ configuration. The vector beam consists of vortex pulse pairs with right- and left-circular polarizations, corresponding to opposite spin angular momentum (SAM), and carrying opposite orbital angular momentum (OAM) charges ±l\pm l±l. We show that during the propagation of the vortex pairs, analytically obtained in the linear regime, the medium inherits the topology of the vortex pair, mapping the OAM onto a spatially structured atomic coherence. This mapping produces 2∣l∣−2|l|-2∣l∣−fold azimuthal transparency structures that reshape the beam intensity from a ring into a petal-like pattern. The OAM-structured atomic coherence induces a corresponding optical anisotropy within the medium, which feeds back into the propagating vector beam, resulting in optical spin-orbit coupling manifested as SAM exchange, rotation, and evolution of polarization textures. Depending on the initial ground-state population of the phaseonium, the polarization state evolves between left-circular, linear, and right-circular polarizations.
In analogy with the separation of the total optical angular momentum into a spin and an orbital part in electrodynamics, we introduce a new concept of spin and orbital angular coherence momenta into the general coherence theory of vector electromagnetic fields. The properties of the newly introduced spin and orbital angular coherence momenta are investigated through the decomposition of the total coherence angular momentum into the sum of these two components, and their separate conservations have been derived for what is believed to be the first time.
We investigate the spatial and quantum intensity correlations between the probe and Stokes optical fields produced via four-wave mixing in a double-Λ configuration, when both incoming probe and control fields carry non-zero optical orbital angular momentum (OAM). We observed that the topological charge of the generated Stokes field obeyed the OAM conservation law. However, the maximum values and optimal conditions for the intensity squeezing between the probe and Stokes fields were largely independent of the angular momenta of the beams, even when these two fields had significantly different OAM charges. We also investigated the case of a composite-vortex pump field, containing two closely positioned optical vortices, and showed that the generated Stokes field carried the OAM corresponding to the total topological charge of the pump field, further expanding the range of possible OAM manipulation techniques.