Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect

We consider propagation of a paraxial beam carrying the spin angular momentum (polarization) and intrinsic orbital angular momentum (IOAM) in a smoothly inhomogeneous isotropic medium. It is shown that the presence of IOAM can dramatically enhance and rearrange the topological phenomena that previously were considered solely in connection to the polarization of transverse waves. In particular, the appearance of a new type of Berry phase that describes the parallel transport of the beam structure along a curved ray is predicted. We derive the ray equations demonstrating the splitting of beams with different values of IOAM. This is the orbital angular momentum Hall effect, which resembles the Magnus effect for optical vortices. Unlike the spin Hall effect of photons, it can be much larger in magnitude and is inherent to waves of any nature. Experimental means to detect the phenomena are discussed.     

The propagation of a polarized gaussian beam in a smoothly inhomogeneous isotropic medium

In the frame of the eikonal-based complex geometrical optics, which describes the phase front and cross section of a light beam using the quadratic expansion of a complex-valued eikonal, we investigate the transverse deflections of a polarized Gaussian beam (GB) in a smoothly inhomogeneous isotropic medium, which is called the spin Hall effect of the beam. The linear complex-valued eikonal terms are introduced firstly to describe the polarization-dependent transverse shifts of the beam in the inhomogeneous medium. We find that the polarization-dependent transverse shifts of the beams include two parts: one originates from the coupling between the spin angular momentum and the extrinsic orbital angular momentum due to the curve trajectory of the center of gravity of the polarized GB, and the other from the coupling between the spin angular momentum and the intrinsic orbital angular momentum due to the rotation of the beam with respect to the central ray.     

Proposals for the generation of angular momentum from non-uniformly polarized beams

Several optical arrangements using non-uniformly polarized fields are proposed for generating beams with spin and/or orbital angular momentum. By choosing adequately the input beam polarization and the characteristics of the different proposed set-ups we can control the overall angular momentum of the output beam at will. The orbital angular momentum is analyzed with the beam moments theory and the spin term is evaluated using the averaged s₃ Stokes parameter.     

Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle

We observe the spinning and orbital motion of a microscopic particle trapped within a multiringed light beam that arises from the transfer of the spin and orbital components of the light's angular momentum. The two rotation rates are measured as a function of the distance between the particle and the axis of the trapping beam. The radial dependence of these observations is found to be in close agreement with the accepted theory.     

Light with a twist in its tail

Polarized light is a phenomenon familiar to anyone with a pair of polaroid sunglasses. Optical components that change the nature of the polarization from linear to circular are common in any undergraduate laboratory. Probably only physicists know that circularly polarized light carries with it an angular momentum that results from the spin of individual photons. Few physicists realize, however, that a light beam can also carry orbital angular momentum associated not with photon spin but with helical wavefronts. Beams of this type have been studied only over the last decade. In many instances orbital angular momentum behaves in a similar way to spin. But this is not always so: orbital angular momentum has its own distinctive properties and its own distinctive optical components. This article outlines the general behaviour of such beams; how they can be used to rotate microscopic particles; how they interact with nonlinear materials; the role they play in atom-light interactions and how the rotation of such beams results in a measurable frequency shift.     

Spin-to-orbital angular momentum conversion and spin-polarization filtering in electron beams

We propose the design of a space-variant Wien filter for electron beams that induces a spin half-turn and converts the corresponding spin angular momentum variation into orbital angular momentum of the beam itself by exploiting a geometrical phase arising in the spin manipulation. When applied to a spatially coherent input spin-polarized electron beam, such a device can generate an electron vortex beam, carrying orbital angular momentum. When applied to an unpolarized input beam, the proposed device, in combination with a suitable diffraction element, can act as a very effective spin-polarization filter. The same approach can also be applied to neutron or atom beams.     

Geometrical phase shift of the extrinsic orbital angular momentum density of light propagating in a helically wound optical fiber

We calculate the intrinsic spin and extrinsic orbital angular momentum densities of an electromagnetic plane wave propagating in a helically wound optical fiber. The geometrical phase shift of the extrinsic angular momentum density of light traveling in such a fiber is derived analytically and discussed in comparison with that of the intrinsic angular momentum density.     

Gauge-Invariant Spin and Orbital Angular Momentum of Laguerre--Gaussian Laser

We adopt a gauge-invariant definition to calculate the spin and orbital angular momenta of a so-called Ith order Laguerre-Gaussian laser. The results reveal that photons on the axis of the beam may carry an orbital angular momentum of （l - 1）h besides lh per photon. For the spin, we obtain a more reasonable expression proportional to the beam intensity instead of the gradient of the intensity as previously derived. We also discuss how to experimentally discriminate the angular momentum expressions given here and those commonly accepted in the literature.     

A simple analytical model of the angular momentum transformation in strongly focused light beams

A ray-optics model is proposed to describe the vector beam transformation in a strongly focusing optical system. In contrast to usual approaches based on the focused field distribution near the focal plane, we use the beam pattern formed immediately after the exit aperture. In this cross section, details of the output field distribution are of minor physical interest but proper allowance is made for transformation of the beam polarization state. This enables the spin and orbital angular momentum representations to be obtained, which are valid for any cross section of the transformed beam. Simple analytical results are available for a transversely homogeneous, circularly polarized incident beam confined by a circular aperture. Variations of the spin and orbital angular momenta of the output beam with change of the focusing strength are analyzed. The analytical results are in good qualitative and reasonable quantitative agreement with the results of numerical calculations performed for the Gaussian and Laguerre-Gaussian beams. The model supplies an efficient and physically transparent means for qualitative analysis of the spin-to-orbital angular momentum conversion. It can be generalized to incident beams with complex spatial and polarization structure.     

Orbital and Field Angular Momentum in the Nucleon

The nucleon spin problem raises experimental and theoretical questions regarding the contribution of the orbital angular momentum of the quarks to the total spin of the nucleon. In this article we examine the commutation relationships of various operators that contribute to the total angular momentum of the nucleon. We find that the sum of the orbital plus gluon field angular momenta should satisfy the angular momentum commutators, at least up to the one-loop level. This, requirement on the sum of these operators imposes a non-trivial restriction on the form of the color electric and magnetic fields. This is similar to the magnetic monopole/electric charge system where it is only the sum of the orbital plus field angular momentum that satisfies the correct commutation relationships.     

The angular momentum of light: optical spanners and the rotational frequency shift

An introduction is given to the concepts of the spin and orbital angular momentum of light beams. Both spin and orbital angular momentum can be transferred from a light beam to particles held within optical tweezers, so forming an optical spanner. Each also give rise to a frequency shift when the light beam is rotated. This arises because quarter or half-wave plates and /2 or mode converters play equivalent roles for spin and orbital angular momentum respectively.     

Beam moments and angular momentum in non-uniformly polarized beams

The angular momentum of non-uniformly totally polarized beams is investigated using methods from the beam characterization approach. The relationship between the elements of the beam matrix for the two components of the field and the angular momentum is given. The unconventional distribution of the polarization across the beam profile could result in contributions to both the spin and orbital terms of the angular momentum. To illustrate this, a particular example with a vortex beam is considered.     

Optical forces and torques in nonuniform beams of light

The spin angular momentum in an elliptically polarized beam of light plays several noteworthy roles in optical traps. It contributes to the linear momentum density in a nonuniform beam, and thus to the radiation pressure exerted on illuminated objects. It can be converted into orbital angular momentum, and thus can exert torques even on optically isotropic objects. Its curl, moreover, contributes to both forces and torques without spin-to-orbit conversion. We demonstrate these effects experimentally by tracking colloidal spheres diffusing in elliptically polarized optical tweezers. Clusters of spheres circulate deterministically about the beam's axis. A single sphere, by contrast, undergoes stochastic Brownian vortex circulation that maps out the optical force field.     

Management of the orbital angular momentum of vortex beams in a quadratic nonlinear interaction

Light intensity control of the orbital angular momentum of the fundamental beam in a quadratic nonlinear process is theoretically and numerically presented. In particular we analyzed a seeded second harmonic generation process in presence of orbital angular momentum of the interacting beams due both to on axis and off axis optical vortices. Examples are proposed and discussed.     

Focusing of elliptically polarized Gaussian beams through an annular high numerical aperture

Based on the vectorial Debye theory,the focusing properties of the Gaussian beam through an annular high numerical aperture are studied numerically,including the intensity,the phase and the orbital angular momentum properties.Then the influence of certain parameters on the focusing properties is also investigated.It is shown that sub-wavelength elliptical light spots can be obtained.And there exists a vortex in the longitudinal component of the focused field even though the incident beam is Gaussian beam,indicating that the spin angular momentum of the elliptically polarized Gaussian beam is converted into the orbital angular momentum by the focusing.     

Spin-orbit interaction and evolution of optical eddies in perturbed weakly directing optical fibers

The transformation of the angular momentum of an optical eddy in a weakly directing perturbed optical fiber is analyzed within the spin-orbit operator representation. The case of fibers with anisotropy of the core and cladding materials and the case of fibers with an elliptic cross section are considered. The spectrum of polarization corrections to the scalar propagation constant is determined for fibers of two types. For both the strongly anisotropic and elliptic fibers, the spin angular momentum of the linearly polarized LV eddy is suppressed and the orbital angular momentum is characterized by simple oscillations with a beating length dependent only on the spin-orbit parameter of an unperturbed fiber. The orbital and spin angular momenta of the circularly polarized CV eddy in the anisotropic fiber interchange in the elliptic fiber. The orbital angular momentum can be completely restored in the strongly anisotropic fiber, whereas only the spin angular momentum is completely restored in the elliptic fiber.     

Poincaré-sphere equivalent for light beams containing orbital angular momentum

The polarization state of a light beam is related to its spin angular momentum and can be represented on the Poincaré sphere. We propose a sphere for light beams in analogous orbital angular momentum states. Using the Poincaré-sphere equivalent, we interpret the rotational frequency shift for light beams with orbital angular momentum [Phys. Rev. Lett. 80, 3217 (1998)] as a dynamically evolving geometric phase.     

Orbital angular momentum and nonparaxial light beams

The simple relationship between total angular momentum and energy and the seemingly natural separation of the angular momentum into spin and orbital components in the paraxial approximation, are investigated for a general nonparaxial form of monochromatic beam with near cylindrical symmetry.     

Separation of spin angular momentum in space-variant linearly polarized beam

We show that the spin angular momentum (SAM) flux in a space-variant linearly polarized beam can be separated in the focal plane. Such a beam carries only orbital angular momentum (OAM) and develops a net SAM flux upon focusing. The radial splitting of the SAM flux density is mediated by the phase vortex (or OAM) and can be controlled by the topological charge of the phase vortex. Optical trapping experiments verify the separation of the SAM flux density. The proposed approach enriches the manipulation of the angular momentum of light fields and inspires more designs of focus engineering, which would benefit optical micromanipulation of microscopic particles.     

Resistance of nondiffracting vortex beam against amplitude and phase perturbations

The revival of the nondiffracting vortex beam after its interaction with the 2D on axis obstacle is examined. We show that the phase topology and the spatial distribution of the orbital angular momentum of the beam transmitted through the obstacle regenerate to the initial form during further free propagation. We verify that the healing effect appears even if the interaction is accompanied by the exchange of the orbital angular momentum.