## Abstract

We investigate the spin properties of a family of cylindrical vector vortex beams under a focusing condition. The spin-orbit interaction is demonstrated by comparing the energy flow and spin flow density of the focused field to those of the incident field. This spin-orbit interaction is analyzed to construct the desired distribution of spin angular momentum for optical manipulation. The structured spin angular momentum of the focused field can transfer to the optical torque for the non-magnetic absorptive particle. The influences of polarization topological charge, vortex topological charge and wavelength on optical torque in the hot-spot of focused field are summarized for three typical particles. Such results may be exploited in practical optical manipulation, particularly for optically induced rotations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Over the past few decades, considerable interest has grown in the optical manipulation of microscopic objects because it provides the capability for noncontact and noninvasive manipulation [1–6]. Light has been well-known to carry both momentum and angular momentum (AM) since the seminal works of Poynting [7]. The AM of light can induce an optical torque and thus cause the rotation of a trapped particle [8–12]. Generally, the AM of light can be decomposed into two parts: the spin angular momentum (SAM) and the orbital angular momentum (OAM) [13–18]. The SAM is an intrinsic property of light, which is associated with the helicity of the light polarization. The SAM would cause the rotation of a particle with respect to its own axis. The OAM, on the other hand, is associated with the vortex phase of structured light. It thus causes the rotation of particles with respect to the optical axis [19]. Recently, it was demonstrated that the AM densities in structured optical fields present unusual features, such as the transverse SAM [20–27]. This extraordinary phenomenon, which is a fundamental property of a laterally-confined optical field, has attracted considerable attention in recent years. The appearance of the transverse SAM, which can be transferred to small particles along a nonaxial direction, produces a torque on the particles and would allow for additional rotation degrees of freedom for optical manipulation.

Polarization is an important property of light, and plays a crucial role on a variety of light-matter interactions [28,29]. In addition to the ordinary scalar optical beams, vector beams with spatially variant polarization states have attracted great interest because of their fascinating focusing properties and novel applications. In particular, the local SAM of a focused field is strongly dependent on the incident polarization [30–34]. In this paper, we perform theoretically investigation on the SAM of focused fields of cylindrical vector vortex beams (CVVBs) from the perspective of spin-basis geometric rotational transformation [35–37]. The spin-orbit interaction is validated by comparing the energy flow and the spin flow density of the focused field with that of the incident field, and is then employed to construct the distribution of SAM. It was demonstrated that particles can be trapped at the hot spots of the focused field, where both the longitudinal and transverse components of the SAM can be modulated flexibly. The effects of different parameters, including the polarization topological charge (PTC), vortex topological charge (VTC), and wavelength on optical torque are investigated for silicon, silver and gold particles. The results provide a general guidance for optical manipulation and show that optically-induced rotation may be modulated flexibly in future applications.

## 2. Transversal and longitudinal SAM of focused field for CVVBs

The transverse electric field components of CVVBs propagating in the z-direction can be expressed in cylindrical coordinates (*r*, φ, z) with corresponding directional unit vectors $\left({\widehat{e}}_{r},{\widehat{e}}_{\phi},{\widehat{e}}_{z}\right)$ as [36]

_{0}is set at 0 throughout this paper. Here

*A*is the envelope of amplitude and generally acts as a Bessel-Gaussian function [30]. The superscript T indicates the transpose of a matrix.

_{r}*n*and

*m*denote the PTC and the VTC, respectively. The superscript and subscript of the electric field vector represent the cylindrical coordinate and the transverse field, respectively. By employing the Maxwell’s theorem $\nabla \cdot {\tilde{E}}^{c}\left(\overrightarrow{r}\right)=0$, the longitudinal electric field component can be calculated to be

*k*is the wavevector in the free space. Note that 1/

*kr*<<1 for a macroscopic Bessel-Gaussian beams. Moreover, by considering the singularity (polarization and phase) properties and the physical meaning because the physical quantities containing 1/

*kr*is divergence at the center, the longitudinal electric field, including 1/

*kr*, can be ignored in Eq. (2). Because k≃10

^{7}for the visible light, the magnitude of the longitudinal electric field component is much less than that of the transverse electric field component, which means that the longitudinal electric field component can be ignored and the transverse property of electromagnetic wave is also fulfilled. The above field can be translated into

Firstly, we investigate the spin properties of the focused field in free space, which can be easily can be understood theoretically. The high-numerical aperture (NA) focusing theorem implies that the light is parallel-transported along its geometrical-optics ray and refracted by the objective with the state of polarization invariance in its local coordinates [35–37]. The far-field spectrum can be expressed with spherical angles (*θ*, *φ*) with respect to the focal point as

The first and second items in Eq. (5) are derived from the contributions of the ${\widehat{\sigma}}^{+}$ and ${\widehat{\sigma}}^{-}$ components of the incident light, respectively. The focused field is calculated using a two-dimensional Fourier integral over the spectrum. Although the incident light is local linear polarization, the partial waves of incident plane with different positions can give rise to phase difference, which will affect and produce the transverse and longitudinal electric field components in the laboratory frame. The phase difference can lead to the local SAM, and the longitudinal component can present orbital properties, which are known as the spin-to-orbit AM conversion.

Note here that there is also an orbit-to-spin AM conversion in the focusing process. To demonstrate this property of spin-orbit interaction, we employ the relationship between the momentum density ($\tilde{p}$) and the energy flow ($\tilde{P}$) [14]. Due to the decomposition stemming from quantum mechanics and the requirements of special relativity, the momentum density, which is proportional to the energy flow, can be decomposed into the spin flow density (SFD) and the orbital flow density (OFD) [14–17]. The relationship can be expressed mathematically as $\tilde{p}={\tilde{p}}_{s}+{\tilde{p}}_{o}\propto \tilde{P}/c$, where the SFD ${\tilde{p}}_{s}$ and the OFD ${\tilde{p}}_{o}$ represent the contributions of the OAM and SAM component to the momentum density, respectively. The symbol *c* is the velocity of light in vacuum. To calculate the energy flow of the incident light, we firstly give the expression of the magnetic field in the cylindrical coordinates as

Then, the energy flow then can be calculated to be

*ɛ*and

*μ*denote the permittivity and permeability of material around the field, respectively. The superscript * indicates the complex conjugate of a field. Note that the quantities containing 1/

*kr*are ignored. The SAM can be expressed approximately as $\tilde{S}\propto \partial {\left|{A}_{r}\right|}^{2}/\partial r{\left[\mathrm{sin}\left[2\left(n-1\right)\phi \right],-\mathrm{cos}\left[2\left(n-1\right)\phi \right],0\right]}^{T}$ and the SFD of the incident light can then be calculated in a form

The longitudinal energy flow and SFD component of the focused field of the proposed beams are illustrated in Fig. 1 when the PTC *n* = 5 and the VTC has values of by 0 and 1. Note that the SFD of the focused field is affected significantly by the VTC. From a global perspective, the integral of SFD in the incident plane vanishes whether the vortex phase exists or not, which means there is not net spin momentum in the incident plane. In the focused plane, there only exist the longitudinal components of energy flow and SFD when the vortex phase is absent. The pattern of SFD in the focal plane is modulated by sin^{2}[(*n*-1)*φ*] at the azimuthal direction [Fig. 1(b)]. Meanwhile, the patterns of energy flow and SFD of focused field are modulated by the radial Bessel function owing to the cylindrical symmetrical focusing. The net spin momentum vanishes by integral the SFD over the focal plane, which is in accordance with that of the incident light. When the vortex phase is added, the transverse components appear. The longitudinal SFD is now modulated by cos^{2}[(*n*-1)*φ*] and emerges petal-shaped pattern with 2|*n*-1| hot spots [Fig. 1(d)]. Note that there is a net negative spin momentum by integrating the longitudinal SFD over the focal plane, which is different from that of incident light. Therefore, we can conclude that part of the OFD in the incident light is transformed into the SFD of the focused field during the focusing, i.e. the orbit-to-spin AM conversion.

The spin-orbit interaction can affect the SAM distribution strongly in the focused field. For example, in the absence of the vortex phase (i.e., *m* = 0), the intensities and phases of the ${\widehat{\sigma}}^{+}$ and ${\widehat{\sigma}}^{-}$ components of the focused field are shown in Figs. 2(a)-2(d). By reexamining Eq. (5) and applying the properties of the Bessel function B* _{p}*(-

*z*) = (−1)

*B*

^{p}*(*

_{p}*z*) for an integer

*p*, it can be determined that the intensity distributions of ${\widehat{\sigma}}^{+}$ and ${\widehat{\sigma}}^{-}$ components are identical, with the exception of a reversed vortex phase for each PTC. In addition, the absolute value of VTC of the helical components is equal to

*n*. This phenomenon arises because of the symmetry of ${\widehat{\sigma}}^{+}$ and ${\widehat{\sigma}}^{-}$ components as illustrated by Eq. (5). However, if VTC≠0, an additional vortex phase difference of 2

*mφ*would then be introduced into the field, which would lead to the separation of the two helical components. The sum of the VTCs of these two helical components is 2

*n*, which is in accordance with the result for

*m*= 0. The angular separation of the two helical components and vortex phase difference can obviously be seen in Figs. 2(e)-2(h), where the PTC and the VTC are set to 5 and 1, respectively. All these intensities exhibit ring-like patterns with 2|

*n*-1| lobes (or petals) that are arranged symmetrically on the ring. However, increasing the VTC would cause the ${\widehat{\sigma}}^{-}$ component to be dominant and the focused field would then converge to a zero-order Bessel wave packet. When the VTC is scaled up even further, it will then return to a ring-shaped distribution.

To verify this analysis, we present the SAM of the focused field accordingly. The SAM of the focused field can be calculated using

*n*= 5, the azimuthal and longitudinal components of the SAM for the focused field of the CVVB without and with the vortex phase (i.e. when

*m*= 0 and

*m*= 1, respectively) are summarized in Fig. 3. Note that the radial component of the SAM vanishes in these two cases.

From the preceding analysis, the field distributions of the ${\widehat{\sigma}}^{+}$ and ${\widehat{\sigma}}^{-}$ components are identical with the exception of their opposing vortex phases in the absence of the VTC. This causes the longitudinal SAM component to vanish [Fig. 3(b)], with only the azimuthal SAM appearing because of the generation of phase difference between the longitudinal and transverse field components [Fig. 3(a)]. When the VTC present, the two helical components separate, which leads to the appearance of the longitudinal SAM component [Fig. 3(d)]. In conclusion, the longitudinal SAM vanishes when the vortex phase is absence, whereas the longitudinal SAM appears if the vortex topological charge is nonzero. This is also a proof of orbit-to-spin AM conversion in the focusing from a local perspective. Meanwhile, it clearly demonstrates that the transverse SAM component, which is associated with the generation of longitudinal field, and the longitudinal SAM component, which stems from the separation of the two helical components, are both affected strongly by the spin-orbit interaction in a focusing configuration.

Subsequently, we consider the focusing of the CVVB in a typical optical trapping system. The light is focused through a two-layer interface between a glass substrate and water from the substrate side, which has a refractive index of 1.514. The NA is set at 1.4 based on the assumption of oil immersion. Comparing with the previously mentioned system, the Fresnel coefficients need be employed in the equation of far-field spectrum of focused electric field [38]. The spectrum can be rewritten as

*t*and

_{p}*t*, respectively. The calculated components of SAM when the PTC is equal to 5 and the VTC has values of 0 and 1, are illustrated in Fig. 4.

_{s}It is obvious that the components in the water are more concentrated than those corresponding components in the air because of the increase of refractive index. In addition, the primary influences of the interfacial effect in the focusing system would result in the presence of complex transmission coefficients for the radial and azimuthal components, and would lead to the difference of intensities and phases between these two components. The slight difference between the intensities of the radial and azimuthal polarizations would induce the longitudinal SAM while the phase shift between the complex transmission coefficients would mainly contribute to the node of the longitudinal SAM in the radial direction as shown in Figs. 4(c) and 4(f). The azimuthal SAMs are approximately the same as those in the previous cases with the exception of a cos[2(*n*-1)φ] modulation of their intensities. Because of the presence of the Fresnel coefficients, the radial SAM component also appears. Note here that, in both cases, the net radial component of SAM nearly vanishes by integrating over the whole focusing plane and is much smaller than the other two components. Thus, the mechanical effects of the radial SAMs on the trapped particles can be considered to be the perturbation and can be ignored. When the PTC = 5 and VTC = 0, the pattern of the longitudinal SAM component are modulated approximately by sin[2(*n*-1)φ], while the longitudinal SAM components for PTC = 5 and VTC = 1 are modulated by sin^{2}[(*n*-1)φ]. Therefore, by modulating the PTC and VTC, tuning of the structured SAM is possible. Because the mechanical torque is related to the local SAM in the hot-spots in the focused field, the SAM can be adjusted flexibly to achieve the desired optical rotation through focusing of the CVVBs.

## 3. Optical torque of nanoparticle in focusing field

The above analysis demonstrates that the orbital-to-spin AM conversion would give rise to a transverse SAM component along with the conventional longitudinal SAM component. Therefore, we can construct a structured SAM and then examine the mechanical consequences of its interactions with trapped particles.

Consider the nonmagnetic Rayleigh spherical particle of permittivity *ɛ*_{2} that is located in the focal plane, which has permittivity *ɛ*_{1}. The particle has a radius *a*, which is much smaller than the trapping wavelength. Therefore the particle can be considered to be an electric dipole, and the induced dipole moment is **p** = α**E**, where *α* is the polarizability [33, 39, 40].

Transfer of the momentum from light to the particle would induce an optical force on the particle given by **F** = (**p**·∇)**E** + (1/*c*)(∂**p**/∂t) × **B**, whereas transfer of the angular momentum produces an optical torque is given by **Γ** = **p** × **E**. If the external electric-field **E** varies harmonically in time, the time-averaged spin torque is [39,40]

For simplicity, we consider three type of nonmagnetic spherical Rayleigh particle (silicon, silver and gold) with radius *a* = 30 nm that are suspended in water. It can be expected that the particles can be trapped at the hot-spots of the focused field and the radial SAM components are regarded as perturbation, which means that only the azimuthal and longitudinal torques at the hot-spots need to be discussed here. The azimuthal torque versus the VTC while the PTC changes from 0 to 5 are shown in Figs. 5(a), 5(b) and 5(c) at the wavelength of 0.532μm. Note here that when the PTC n = 1 and φ_{0} = 0, the polarization state degrades to the radial polarization, which has been widely used in optical trapping [3]. The transverse and longitudinal SAM components are almost zero when the VTC = 0 and the azimuthal torques attain their local maxima when VTC = PTC at the hot-spots. When the PTC has a nonzero value, the azimuthal torques are negative at VTC = 0, and subsequently reach zero when the VTC = PTC. The azimuthal torques reach their local maxima when the VTC = PTC + 1, and finally converge to small values with further increases in the VTC. The longitudinal torque values of these particles are summarized in Fig. 5(b), 5(d) and 5(f). The original longitudinal torques are almost zero, which agrees well with the results of the previous analysis. Because the longitudinal SAM appears when the VTC increases, the corresponding transverse torque subsequently decreases. Furthermore, these transverse torques increase and reach their local maxima. The value of VTC that is required to reach the maximum is equal to PTC for each value of PTC, while the azimuthal torques consequently vanish. With further increases in the VTC, the longitudinal torques will decrease and reach approximately the same values as the corresponding azimuthal torques.

Subsequently, the optical torques can be optimized by modulating the wavelength. When the PTC value is fixed at 5 and the VTC values are 0 and 5, at which the local maximal azimuthal and longitudinal torques occur, respectively, the variations in these two torque components versus wavelength are as shown in Fig. 6. These torques, which are related to the absorption and dispersion properties of materials, are strongly affected by variations in the wavelength. For the Ag and Au particles, the torques reach maxima around the wavelength of 0.5 μm, while the wavelength at which maximal torque is achieved is lower than 0.4 μm for the dielectric silicon particle. Note here that the torque of the Au particle is two orders of magnitude greater than that of the other two particles, and the torque of the dielectric particle is the smallest. In this way, we can conclude that the metallic particles offer higher absorption than the dielectric particles, which can present higher torques. Therefore, the wavelength can also be tuned to achieve optical rotation.

## 4. Conclusions

In summary, we have performed a theoretical investigation of the spin properties of the focused field of a family of CVVBs. The spin-orbit interaction was demonstrated by comparing the energy flow and the spin flow density of the focusing field to the corresponding properties of the incident field. Using the theory of orbit-to-spin conversion, we constructed the structured spin angular momentum with a purely transverse SAM, a purely longitudinal SAM or a hybrid SAM by modulating the PTC and the VTC. The structured SAM would modulate the transverse or longitudinal torques flexibly via focusing of the proposed beams. The effects of the PTC, the VTC and the wavelength on the optical torques were also summarized for absorptive dielectric and metallic particles. When the PTC n = 1 and φ_{0} = 0, the incident light is degraded to the radial polarized light. There were no transverse or longitudinal SAM components when the VTC = 0 and the azimuthal torques reached their local maxima when the VTC = PTC at the hot-spots. When the PTC has a nonzero value, the transverse torque, which is mainly caused by the azimuthal SAM component, is dominant in the absence of the vortex phase. When the VTC increases, the longitudinal torque then appears. The condition required to reach the local minimum of the transverse torque is VTC = PTC, at which the longitudinal torque reaches a local maximum. With further increases in the VTC, the longitudinal and transverse torques both decrease and are approximately equal. Finally, we can optimize the optical torques of the absorptive particles by simply tuning the wavelength. These results will be helpful in the investigation of optical manipulation processes, particularly for optically-induced rotation and transfer of the optical SAM.

## Funding

National Natural Science Foundation of China (NSFC) (grants Nos. 61490712, 61427819, 61705135, 61622504, 11504244); The leading talents of Guangdong province program (grant 00201505); Natural Science Foundation of Guangdong Province (grant 2016A030312010); Shenzhen Science and Technology Innovation Commission (grants Nos. KQTD2015071016560101, KQTD2017033011044403, ZDSYS201703031605029); China Postdoctoral Science Foundation (grant 2017M622765).

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