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Physical Review A

Radial dependence of the angular momentum density of a paraxial optical vortex

Victor Victorovich Kotlyar, A. A. Kovalev, Alexey P. Porfirev

Published May 23, 2018• 15 citations
Publisher

Abstract

We obtain general analytic expressions for the linear and angular momentum density of an elliptically polarized paraxial laser beam. As a partial case, we obtain expressions for the linear and angular momentum density of a paraxial elliptically polarized optical vortex. It is shown that for an arbitrary vortex field with rotational symmetry and with circular polarization the effect of the ``angular tractor'' takes place, which means that the flux of light energy rotates around the optical axis clockwise or counterclockwise at different radii in the beam cross section. It is also shown that the axial component of the angular momentum density of the vortex light field changes its sign at the same radii in the beam cross section. Microparticles trapped in the vortex Bessel beam at different radii are experimentally shown to rotate simultaneously clockwise and counterclockwise.

DOI
10.1103/physreva.97.053833

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We obtain general analytic expressions for the linear and angular momentum density of an elliptically polarized paraxial laser beam. As a partial case, we obtain expressions for the linear and angular momentum density of a paraxial elliptically polarized optical vortex. It is shown that for an arbitrary vortex field with rotational symmetry and with circular polarization the effect of the ``angular tractor'' takes place, which means that the flux of light energy rotates around the optical axis clockwise or counterclockwise at different radii in the beam cross section. It is also shown that the axial component of the angular momentum density of the vortex light field changes its sign at the same radii in the beam cross section. Microparticles trapped in the vortex Bessel beam at different radii are experimentally shown to rotate simultaneously clockwise and counterclockwise.
The orbital angular momentum density of Bessel beams is calculated explicitly within a rigorous vectorial treatment. This allows us to investigate some aspects that have not been analysed previously, such as the angular momentum content of azimuthally and radially polarized beams. Furthermore, we demonstrate experimentally the mechanical transfer of orbital angular momentum to trapped particles in optical tweezers using a high-order Bessel beam. We set transparent particles of known dimensions into rotation, where the sense of rotation can be reversed by changing the sign of the singularity. Quantitative results are obtained for rotation rates. This paper's animations are available from the Multimedia Enhancements page.
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 demonstrate the transfer of orbital angular momentum to optically levitated microparticles in vacuum [1]. We prepare two-dimensional and three-dimensional optical potentials. In the former case the microparticle is placed within a Laguerre-Gaussian beam and orbits the annular beam profile with increasing angular velocity as the air drag coefficient is reduced. We explore the particle dynamics as a function of the topological charge of the levitating beam. Our results reveal that there is a fundamental limit to the orbital angular momentum that may be transferred to a trapped particle, dependent upon the beam parameters and inertial forces present. This effect was predicted theoretically [2] and can be understood considering the underlying dynamics arising from the link between the magnitude of the azimuthal index and the beam radius [3]. Whilst a Laguerre-Gaussian beam scales in size with azimuthal index `, recently we have created a “perfect” vortex beam whose radial intensity profile and radius are both independent of topological charge [4, 5]. As the Fourier transform of a perfect vortex yields a Bessel beam. Imaging a perfect vortex, with its subsequent propagation thus realises a complex three dimensional optical field. In this scenario we load individual silica microparticles into this field and observe their trajectories. The optical gradient and scattering forces interplay with the inertial and gravitational forces acting on the trapped particle, including the rotational degrees of freedom. As a result the trapped microparticle exhibits a complex three dimensional motion that includes a periodic orbital motion between the Bessel and the perfect vortex beam. We are able to determine the three dimensional optical potential in situ by tracking the particle. This first demonstration of trapping microparticles within a complex three dimensional optical potential in vacuum opens up new possibilities for fundamental studies of many-body dynamics, mesoscopic entanglement [6, 7], and optical binding [8, 9].
It is shown how the quantum mechanical mass flux and the electromagnetic fields of an electron Bessel vortex mode generate its intrinsic linear momentum and angular momentum properties. Although the corresponding volume density vectors due to the mass flux contain transverse vector components, their volume integrals are shown by explicit analysis to yield null results. The total linear and angular momenta are thus purely axial vectors. There are additional contributions associated with the vortex electric and magnetic fields and these too are shown to be purely axial vectors. Order of magnitude estimates are made in the context of a suggested experiment on the rotation of an optically levitated nanoparticle subject to an electron vortex.
We consider the optical rotation of the polarization of a linearly polarized probe field passing through an M-type atomic system by using the interaction between two vortex control fields and optical transitions. We investigate theoretically to generate the spatially dependent structured light with the atoms acting as a spatially varying circular birefringent medium. We show that the polarization and intensity distributions of the vector beam spatially vary by changing the orbital angular momentum (OAM) of the vortex control field.
Neglecting the electric field component along the light's propagation direction is a common practice, known as paraxial approximation. However, experimental evidence and theory on the head-on excitation of atoms by Laguerre-Gaussian beams reveal that the full vector character of the light field has to be taken into account. Optical vortices are a large family of fields, being Laguerre-Gaussian only one particular case. Now we extend the study of the applicability of the paraxial approximation to a broader set of optical vortices by considering the interaction of atoms with generalized Laguerre-Gaussian and generalized Bessel beams. We demonstrate that all vortex beams here considered look the same close to the beam axis, where the atom is placed. From this, we conclude that the paraxial approximation fails for the large set of vortex beams, and that a full understanding of the interaction of optical vortices with atoms requires the inclusion of all vector components of the electric field.