Spin Angular Momentum Density Research Articles

Analysis of the Polarization Distribution and Spin Angular Momentum of the Interference Field Obtained by Co-Planar Beams with Linear and Circular Polarization

Interference of two and four light beams with linear or circular polarization is studied analytically and numerically based on the Richards–Wolf formalism. We consider such characteristics of the interference fields as the distribution of intensity, polarization, and spin angular momentum density. The generation of light fields with 1D and 2D periodic structure of both intensity and polarization is demonstrated. We can control the periodic structure both by changing the polarization state of the interfering beams and by changing the numerical aperture of focusing. We consider examples with a basic configuration, as well as those with a certain symmetry in the polarization state of the interfering beams. In some cases, increasing the numerical aperture of the focusing system significantly affects the generated distributions of both intensity and polarization. Experimental results, obtained using a polarization video camera, are in good agreement with the simulation results. The considered light fields can be used in laser processing of thin films of photosensitive (as well as polarization-sensitive) materials in order to create arrays of various ordered nano- and microstructures.

Optical manipulation in Rubidium-87 atomic medium using Kerr field: Dynamic insights into two-interfering electromagnetic waves

In this manuscript, we investigate the intricate dynamical properties arising from the interaction of two-interfering electromagnetic waves within a Rubidium-87 four-level N-type atomic medium. The study employs a probe field to interact with the atomic medium while incorporating two control fields one of which would be the Kerr field to modulate the system dynamics. The key parameters under scrutiny include the probe field detuning (ΔP/γ), control field parameters such as Rabi frequencies and detunings (Ω1,2/γ, Δ1,2/γ), and the angular orientation (θ) between the two-interfering waves. The research delves into the Energy density W, Momentum density P, Spin Angular momentum density S, Absorption rate Arate, Gradient force Fgrad, and Torque T associated with the two-interfering waves within the atomic medium. Through systematic variations in the aforementioned parameters, significant insights are gained into the nuanced behaviour of the system. The results showcase a rich tapestry of variations in the dynamical properties, shedding light on the underlying physics governing the interaction of electromagnetic waves in this unique atomic medium. This investigation not only contributes to the fundamental understanding of the dynamics within complex atomic systems but also holds potential applications in fields such as quantum information processing and quantum optics. The findings of this manuscript pave the way for further exploration and utilization of these intricate phenomena in the realm of advanced technologies and quantum communication.

Directional difference of the spin angular-momentum density and spin vectors in tightly focused bichromatic optical Lissajous beams

Directional difference of the spin angular-momentum density and spin vectors in tightly focused bichromatic optical Lissajous beams

Optimization method to construct multiple acoustic vortices for holograms

In this study, the novel optimization method of holograms based on spin density is developed by introducing a new degree of freedom—spin angular momentum density into the direct search method. This approach leverages the programmability of numerous pixels in holograms and eliminates the need for meticulous management of amplitude and phase at discrete points, facilitating the construction of multiple first-order acoustic vortices. We efficiently construct a pattern of 40 vortices within a plane and vortices with different intensities and structures using this approach. We also illustrate the practical application of our design methodology by showcasing its ability to manipulate particles through the transfer of angular momentum from vortices to particles. This work not only advances our understanding of acoustic vortices but also explore a new degree of freedom in designing holograms, potentially enhancing the capacity of holograms.

Enhancing the Spin Hall Effect of Cylindrically Polarized Beams.

Two linked gear wheels in a micromachine can be simultaneously rotated in opposite directions by using a laser beam that has in its section areas the spin angular momentum (SAM) of the opposite sign. However, for instance, a cylindrical vector beam has zero SAM in the focus. We alter a cylindrical vector beam so as to generate areas in its focus where the SAM is of opposite signs. The first alteration is adding to the cylindrical vector beam a linearly polarized beam. Thus, we study superposition of two rotationally symmetric beams: those with cylindrical and linear polarization. We obtain an expression for the SAM and prove two of its properties. The first property is that changing superposition coefficients does not change the shape of the SAM density distribution, whereas the intensity changes. The second property is that maximal SAM density is achieved when both beams in the superposition have the same energy. The second perturbation is adding a spatial carrier frequency. We study the SAM density of a cylindrical vector beam with a spatial carrier frequency. Due to periodic modulation, upon propagation in space, such a beam is split into two beams, having left and right elliptic polarization. Thus, in the beam transverse section, areas with the spin of different signs are separated in space, which is a manifestation of the spin Hall effect. We demonstrate that such light beams can be generated by metasurfaces, with the transmittance depending periodically on one coordinate.

Chiral nanoparticle separation and discrimination using radially polarized circular Airy vortex beams with orbital-induced spin angular momentum.

We report orbit-induced localized spin angular momentum associated with optical spin-orbit interactions in tightly focused radially polarized circular Airy vortex beams and demonstrate their potential for separation and discrimination of chiral nanoparticles. We find that variations in spin angular momentum density endow these beams with positive and negative annular optical chirality density. Utilizing these extraordinary distributions, particles having different chirality parameters can be separated and discriminated by using two degrees of freedom, i.e., radial trapping position and azimuthal rotation. We also discuss the impacts of longitudinal optical force and topological charge on manipulating chiral particles. Additionally, we conduct a comparative analysis of the optical trapping of a non-chiral particle. Our work is expected to deepen the understanding of spin-orbit interactions and provide valuable insight into vortex beam interactions with chiral particles.

Spin angular momentum of Gaussian beams with several polarization singularities

We study a paraxial vector Gaussian beam with several polarization singularities located on a circle. Such a beam is superposition of a cylindrically polarized Laguerre-Gaussian beam and a linearly polarized Gaussian beam. It is found that although polarization in the initial plane is linear, alternating regions with the different-sign spin angular momentum density are generated upon free-space propagation, showing that a spin Hall effect arises. For an arbitrary transverse plane, it is shown that the spin angular momentum magnitude is maximal on a certain-radius circle. We obtain an approximate expression for the distance to the transverse plane where the spin angular momentum density is maximal. Besides, we derive an optimal radius of the singularity-containing circle in the initial plane for which the maximal spin angular momentum density can be achieved upon propagation. It is revealed that in this case, the energies of the Laguerre-Gaussian beam and the Gaussian beam are equal to each other. We also obtain an expression for the orbital angular momentum density and find it to be defined by the spin angular momentum density, multiplied by –m/2, with m being the upper index of the Laguerre-Gaussian beam, equal to the number of the polarization singularities. An analogy with plane waves reveals that the spin Hall effect arises due to different divergence rates of the linearly polarized Gaussian beam and the cylindrically polarized Laguerre-Gaussian beam.

Dynamical and topological properties of the spin angular momenta in general electromagnetic fields

Spin angular momenta play important roles in light–matter interactions, leading to the emergence of the spin Hall effect and topological quasiparticles in modern optics. The typical approach is to decompose the spins of plane electromagnetic waves into longitudinal and transverse components, yet this description is not easily transferable to more structured electromagnetic environments. Here, we developed a field theory to reveal the physical origin and topological properties of longitudinal and transverse spins for arbitrary electromagnetic waves (including water waves and acoustic waves) in both near-field and free space. For electromagnetic waves carrying intrinsic helicity, we observed the emergence of helicity-dependent transverse spin possessing helicity-dependent spin-momentum locking. To verify that the number of spin-momentum locking states coincides with the spin Chern number, we experimentally measured the three-dimensional spin angular momentum densities of Bloch-type optical skyrmions. Our findings yield valuable insight for constructing spin-based field theory and exploiting optical topological quasiparticle-based applications.

Generation of Light Fields with Controlled Non-Uniform Elliptical Polarization When Focusing on Structured Laser Beams

We study the sharp focusing of the input structured light field that has a non-uniform elliptical polarization: the parameters of the ellipse depend on the position in the input plane (we limited ourselves to the dependence only on the angular variable). Two types of non-uniformity were considered. The first type corresponds to the situation when the semi-axes of the polarization ellipse are fixed while the slope of the major semi-axis changes. The second type is determined by the situation when the slope of the major semi-axis of the polarization ellipse is constant, and the ratio between the semi-axis changes (we limited ourselves to the trigonometric dependence of this ratio on the polar angle). Theoretical and numerical calculations show that in the case of the first type of non-uniformity, if the tilt angle is a multiple of the polar angle with an integer coefficient, then the intensity distribution has rotational symmetry, and the energy flow is radially symmetric and has the negative direction near the optical axis. In this second case, the intensity symmetry is not very pronounced, but with an odd dependence of the ratio of the semi-axes of the polarization ellipse, the focused field at each point has a local linear polarization, despite the rather complex form of the input field. In addition, we investigate the distribution of the longitudinal component of the Poynting vector. The obtained results may be used for the formation of focused light fields with the desired distributions of polarization, Poynting vector density, or spin angular momentum density in the field of laser manipulation and laser matter interaction.

Transfers of the OAM and SAM densities of Bessel beam into the orbital and spin torques upon gold nanorod

The optomechanical interplay of a Bessel beam (an optical vortex) with a gold nanorod (GNR) in water is theoretically studied. The optical force and torque exerted on the GNR are analyzed using Maxwell's stress tensor to investigate the transfer rates of the orbital angular momentum (OAM) and the spin angular momentum (SAM) densities of Bessel beam into the orbital and spin torques upon GNR, respectively. From the numerical analysis, we find that the ratio of the orbital torque to the spin torque on the GNR matches the ratio of OAM to SAM of Bessel beam, corresponding to the order (topological charge) of the Bessel beam. This is to say that the transfer rate of the OAM density of Bessel beam into the orbital torque exerted on GNR nearly equals that of the SAM density into the spin torque. It implies that the spin-orbit interaction (SOI) of the optical vortex beam via a GNR is very weak. In addition, the spin torque is nearly equal to the corresponding zilch torque, the surface integral of Lipkin's zilch stress tensor. This could be because a GNR can be regarded as a perfect electric dipole particularly when the wavelength of the incident light is close to and little longer than the longitudinal surface plasmon resonance of GNR; the dipole's orientation is along the long axis of GNR. In contrast, the SOI via a spherical gold nanoparticle (GNP) is strong due to no preferred orientation for the equivalent dipole; a part of SAM of the optical vortex is transferred to the orbital torque upon GNP. Our results demonstrate that GNR, functioning as an ideal electric dipole, is a highly effective probe for detecting the OAM and SAM densities of an optical vortex individually through its distinct orbital and spin motions, without any crosstalk.

Spin angular momentum and angular harmonics spectrum of two-order polarization vortices at the tight focus

We investigate the spin angular momentum (SAM) of two-order cylindrical vector beams at the tight focus. Such beams are a generalization of the conventional cylindrical vector beams since the transverse field components on the Cartesian axes have different polarization orders. Using the Richards-Wolf approximation, we derive an expression for the longitudinal component of the SAM density distribution. We show that if the polarization indices are of different parity then, an optical spin Hall effect arises at the tight focus, meaning that alternating areas with positive and negative SAM are originated although the initial light field is linearly polarized. We study the angular harmonics spectrum of all the components of the focused light field and determine the predominant angular harmonics. Then, neglecting the insignificant harmonics, we define the shape of the longitudinal component of the SAM density distribution and demonstrate the possibility of generating a focal SAM distribution where the areas with positive and negative SAM reside on a circle as alternating pairs or separated into different half-circles.

Spin Hall Effect in Paraxial Vectorial Light Beams with an Infinite Number of Polarization Singularities.

Elements of micromachines can be driven by light, including structured light with phase and/or polarization singularities. We investigate here a paraxial vector Gaussian beam with an infinite number of polarization singularities residing evenly on a straight line. The intensity distribution is derived analytically and the polarization singularities are shown to exist only in the initial plane and in the far field. The azimuthal angle of the polarization singularities is shown to increase in the far field by π/2. We obtain the longitudinal component of the spin angular momentum (SAM) density and show that it is independent of the azimuthal angle of the polarization singularities. Upon propagation in free space, an infinite number of C-points is generated, where polarization is circular. We show that the SAM density distribution has a shape of four spots, two with left and two with right elliptic polarization. The distance to the transverse plane with the maximal SAM density decreases with decreasing distance between the polarization singularities in the initial plane. Generating such alternating areas with positive and negative SAM density, despite linear polarization in the initial plane, manifests the optical spin Hall effect. Application areas of the obtained results include designing micromachines with optically driven elements.

Controlling the Spin Hall Effect in the Sharp Focus of an Axial Superposition of Two Optical Vortices with Left- and Right-Handed Circular Polarization

We consider sharp focusing of an axial superposition of two optical vortices with identical topological charges, but different amplitudes and circular polarizations of different signs. The ratio of the amplitudes of the two beams is a parameter. When this parameter changes, the polarization state of the superposition changes from linear polarization to right-hand circular polarization. Based on the Richards–Wolf theory, exact expressions are obtained for the longitudinal components of the spin angular momentum (SAM) density and orbital angular momentum (OAM) density at the focus of the considered superposition. It follows from these expressions that the sum of the total longitudinal components of the SAM and OAM is conserved upon focusing, and also that, due to the spin-orbit conversion, the total longitudinal component of the SAM decreases during focusing, while the total longitudinal component of the OAM increases by the same amount. By changing the ratio of the amplitudes of the constituent beams from 1 to 0, one can change the value of the spin-orbit conversion from zero (for linear polarization) to a maximum (for circular polarization). Also, by changing this parameter, one can control the spin Hall effect at the focus, which takes place at the focus of the considered beam. This study can be applied for controlling the rotation velocity of microparticles trapped in the focus.

Spin Hall Effect in the Paraxial Light Beams with Multiple Polarization Singularities.

Elements of micromachines can be driven by light, including structured light with phase and/or polarization singularities. We investigate a paraxial vectorial Gaussian beam with multiple polarization singularities residing on a circle. Such a beam is a superposition of a cylindrically polarized Laguerre-Gaussian beam with a linearly polarized Gaussian beam. We demonstrate that, despite linear polarization in the initial plane, on propagation in space, alternating areas are generated with a spin angular momentum (SAM) density of opposite sign, that manifest about the spin Hall effect. We derive that in each transverse plane, maximal SAM magnitude is on a certain-radius circle. We obtain an approximate expression for the distance to the transverse plane with the maximal SAM density. Besides, we define the singularities circle radius, for which the achievable SAM density is maximal. It turns out that in this case the energies of the Laguerre-Gaussian and of the Gaussian beams are equal. We obtain an expression for the orbital angular momentum density and find that it is equal to the SAM density, multiplied by -m/2 with m being the order of the Laguerre-Gaussian beam, equal to the number of the polarization singularities. We consider an analogy with plane waves and find that the spin Hall affect arises due to the different divergence between the linearly polarized Gaussian beam and cylindrically polarized Laguerre-Gaussian beam. Application areas of the obtained results are designing micromachines with optically driven elements.

Angular and Orbital Angular Momenta in the Tight Focus of a Circularly Polarized Optical Vortex

Based on the Richards-Wolf (RW) formalism, we obtain two different exact expressions for the angular momentum (AM) density of light in the focus of an optical vortex with a topological charge n and right circular polarization. One expression for the AM density is derived as the cross product of the position vector and the Poynting vector and has a nonzero value in the focus for an arbitrary integer n. Another expression for the AM density is equal to a sum of the orbital angular momentum (OAM) and the spin angular momentum (SAM) and, in the focus of a considered light field, is equal to zero at n = −1. These expressions are not equal at each point in space, but their 3D integrals are equal. Thus, we derive exact expressions for the AM, SAM and OAM densities in the focus of an optical vortex with right circular polarization and demonstrate that the identity for the densities AM = SAM + OAM is not valid. In addition, we show that the expressions for the strength vectors of the electric and magnetic field near the tight focus, obtained on the basis of the RW formalism, are exact solutions of Maxwell’s equations. Thus, the RW theory exactly describes the behavior of light near the tight focus in free space.

Controlling the Abrupt Autofocusing of Circular Airy Vortex Beam via Uniaxial Crystal

The propagation of many kinds of structured light beams in uniaxial crystal has been investigated. However, the investigation of the evolution of these structured light beams after the uniaxial crystal is lacking. In this paper, an evolution formula of a light beam after passing through a uniaxial crystal is derived. Based on the formula, controlling the autofocusing of a circular Airy vortex beam (CAVB) via a uniaxial crystal is studied. It is found that a uniaxial crystal can prolong the focal length of the autofocusing. By changing the crystal length, the relative weight of the left- and right-hand circular polarization components and the relative value between the orbital and spin angular momentum densities of the beam’s focal plane can be adjusted flexibly. In addition, other optical elements can be inserted between the crystal and the focus to further adjust the focal plane field distribution. The influences of inserting x- and y-polarization polarizers on the intensity distribution are calculated as examples.

Effect of optical spatial coherence on localized spin angular momentum density in tightly focused light [Invited

Optical coherence is one of the most fundamental characteristics of light and has been viewed as a powerful tool for governing the spatial, spectral, and temporal statistical properties of optical fields during light-matter interactions. In this work, we use the optical coherence theory developed by Emil Wolf as well as the Richards-Wolf's vectorial diffraction method to numerically study the effect of optical coherence on the localized spin density of a tightly focused partially coherent vector beam. We find that both the transverse spin and longitudinal spin, with the former induced by the out-of-phase longitudinal field generated during strong light focusing and the latter induced by the vortex phase in the incident beam, are closely related to the optical coherence of the incident beam, i.e., with the decrease of the transverse spatial coherence width of the incident beam, the magnitude of the spin density components decreases as well. The numerical findings are interpreted well with the two-dimensional degrees of polarization between any two of the three orthogonal field components of the tightly focused field. We also explore the roles of the topological charge of the vortex phase on enhancing the spin density for the partially coherent tightly focused field. The effect of the incident beam's initial polarization state is also discussed.

Near-Field Behaviors of Internal Energy Flows of Free-Space Electromagnetic Waves Induced by Electric Point Dipoles

Both orbital and spin energy fluxes constitute the internal flows decomposed from a Poynting vector. For generic electromagnetic waves propagating through source-free media, these energy fluxes are quadratic in field variables so that their properties are not easily predictable. Notwithstanding, their near-field behaviors play important roles in nanoscale photonics. For time-oscillatory fields, we found two hitherto-overlooked distinctions between the two internal flows. The first is an unequal level between them because the vorticity of an orbital energy flux plays a role comparable to a spin energy flux itself. The second is regarding the electric-magnetic dual symmetry in handling the two internal flows, whence the reactive helicity plays a role as important as the electromagnetic helicity. By helicity conservation, an inter-electric-magnetic transport is possible for the spin angular momentum density, while the electric and magnetic constituents of orbital energy fluxes admit only respective intra-electric and intra-magnetic transports. We have tested the validities of all these claims by exemplarily taking the electromagnetic fields induced by an electric point dipole, either a linear or a circular one. We have thus made new contributions not only in deriving explicit forms of the internal energy flows but also in revealing the magnetic activities hidden under the electromagnetic waves induced by electric point dipoles.

Diffraction of an off-axis vector-beam by a tilted aperture

Manifestations of orbital angular momentum induced effects in the diffraction of a radially polarized vector beam by an off-axis tilted aperture are studied both experimentally and theoretically. Experiments were carried out to extract the degree of circular polarization, which was shown to be proportional to the on-axis component of the spin angular momentum density. We report a clear separation of the regions of dominance of the right and left circular polarizations associated with positive and negative topological charges respectively, which is reminiscent of the standard vortex-induced transverse shift, albeit in the diffraction scenario. The experimental results are supported by model simulations and the agreement is quite satisfactory. The results are useful to appreciate the orbit-orbit related effects due to unavoidable misalignment problems (especially for vortex beams).

Dependence of spin angular momentum density on helicity of ellipse field singularities

We numerically demonstrate the role of helicity of ellipse field singularities on spin angular momentum density. We consider both bright and dark ellipse field singularities to demonstrate this effect. It is found that as the helicity of C-point inverted, the spin angular momentum density undergoes 180° rotation. In case of bright C-point the z-component of the spin angular momentum density has always a central maxima irrespective of value of polarization singularity index, whereas for dark C-points there is always a central minima in the z-component of the spin angular momentum density. The number of minima around the central dark point in the longitudinal component for dark C-point singularity with C-point index (IC) is given by (|2IC−2|).