## Abstract

The generation, propagation, and applications of different types of integer vector beams have been extensively investigated. However, little attention focuses on the photophysical and photomechanical properties of the fractional vector beam (FVB). Herein, we theoretically and experimentally investigate the spin angular momentum (SAM) separation and propagation characteristics of weakly focused FVBs. It is demonstrated that such a beam carrying no SAM leads to both the transverse separation of SAM and the special intensity patterns in the focal region. Furthermore, we study the intensity, SAM, and orbital angular momentum (OAM) distributions of the tightly focused FVBs. It is shown that both three-dimensional SAM and OAM are spatially separated in the focal region of tightly focused FVBs. We investigate the optical forces, spin torques, and orbital torques on a dielectric Rayleigh particle produced by the focused FVBs. The results reveal that asymmetrical spinning and orbiting motions of optically trapped particles can be realized by manipulating FVBs.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the past decades, singular optics has attracted broadly academic interest and technological applications [1]. Typical singular optics includes vortex beams having phase singularities [2] and vector beams possessing polarization singularities [3]. The former carries orbital angular momentum (OAM) when its topological charge is an integer [4], whereas the latter is characterized by an integer topological index [5]. Recently, vortex beams with non-integer topological charges, named as fractional vortex beams, have been reported many interesting and unique properties, such as alternating charge vortices [6], Hilbert’s hotel paradox [7], and vortex strength and beam propagation factor [8]. Analogously, vector beams with non-integer topological indexes, called as fractional vector beams (FVBs), result in the uncertainty of state of polarization (SoP) and then the formation of polarization singular lines [5,9,10], different from the polarization singularities of integer vector beams [5]. Moreover, the evolution of these polarization singular lines during propagation in free space leads to the special intensity pattern, SoPs distribution, and chains of polarization singularities [11].

It is well known that light carries both optical linear momentum and angular momentum [2,12,13], which have been exploited in various applications, including optical communications [14], quantum entanglement [15], and particle manipulation [16]. Generally, the optical angular momentum can be decomposed into two parts: the spin angular momentum (SAM) associated with the polarization of light and the OAM originated from the phase structure of light. Typically, a circularly polarized light possesses a SAM value of ${\pm} \hbar $ per photon (here $\hbar $ is the reduced Planck constant), which correspond to left-handed (LH) and right-handed (RH) circular polarizations, respectively. However, a linearly polarized light carries no SAM component because two circular polarization components are coupled within the beam itself and counteract with each other in free space. Interestingly, two SAM components in a linearly polarized light may split in inhomogeneous media when the beam is refracted, reflected, or scattered with the aid of spin-orbit interaction [17–19]. On the other hand, the focusing of the linearly polarized vector beam can cause the spatial separation of SAM. For examples, Chen *et al*. [20] demonstrated the radial splitting of SAM mediated by the phase vortex in a tightly focused space-variant linearly polarized vector vortex beam. Pan *et al*. [21] reported that the focusing of the asymmetric-sector-shaped vector beam with the localized linear polarization leads to the redistribution of the SAM at the focal plane. Liu *et al*. [22] realized the spin separation along the propagation direction by modulating the Pancharatnam-Berry phase. In the nonlinear optics regime, the redistribution of the SAM can be achieved by the interaction of a radially polarized beam with the anisotropic Kerr medium [23].

In principle, the optical linear momentum and angular momentum can transfer from the electromagnetic radiation to the tiny particle. The former leads to the optical force exerted on the particle. The SAM and OAM, in contrast, give rise to the spin and orbital torques on the particle, resulting in the rotation of the particle around its own axis and the optical axis, respectively [2,24]. These photomechanical effects can be fully exploited to stably trap and manipulate the particle. Up to now, the optical trapping and manipulation of tiny particles have been extensively investigated by the use of Gaussian beam [25], vortex beam [26], cylindrical vector beam [27,28], cylindrical vector vortex beam [16], fractional vortex beam [29], etc. However, little attention focuses on the optical manipulation of particles using the FVBs.

In this work, we report the spatial separation of angular momenta in focused FVBs for optical manipulation. First, we derive the analytical expressions for the electric field of the weakly focused FVB, which is given by the modal superposition of integer vector beams. Second, we theoretically analyze and experimentally verify both the transverse separation of SAM and the special intensity patterns of FVBs in the focal region. Third, we investigate the three-dimensional (3D) distributions of the intensity, SAM, and OAM of the tightly focused FVBs at the focal region. Last, we numerically study the optical forces and torques on a dielectric Rayleigh particle produced by the tightly focused FVBs. Owing to the transverse separation of SAM and OAM of tightly focused FVBs, we realize the asymmetrical spinning and orbiting motions of trapped particles.

## 2. Generation of FVBs

In a cylindrical coordinate system, the FVB at the initial plane can be expressed as [5]

Here, *ρ* is the polar radius, *ϕ* is the azimuthal angle, the non-integer *α* is the topological index. *A*(*ρ*) stands for the axially symmetric amplitude distribution of the beam. For the uniform-intensity illumination, we have *A*(*r*)=*E*_{0} within the region of 0≤*ρ*≤*R*, where *R* is the radius of the vector beam.

For the non-integer *α*, following the method presented by Berry [6], the non-integer phase distribution at the initial plane can be expanded into the Fourier series

Defining the parameters of ${A_n} = \frac{{\sin (2\pi \alpha )}}{{\pi ({\alpha ^2} - {n^2})}}$ and ${B_n} = \frac{{2{{\sin }^2}(\pi \alpha )}}{{\pi ({\alpha ^2} - {n^2})}}$, we rewrite the FVB as

Clearly, the FVB is expressible as a modal superposition of all integer vector beams.

To experimentally generate FVBs, we adopt a flexible method in a common path interferometer with the aid of a spatial light modulator (SLM) [5]. Note that the additional phase in a computer-controlled SLM is set as *δ*(*ρ*,*ϕ*)=*αϕ*. Using a CW laser beam at the wavelength of *λ*=532 nm, we generate the FVBs with the radius of *R*=2.67 ± 0.01 mm. Adopting the quarter-wave retarder polarizer method [30], we measure the Stokes parameters of the generated FVBs. As a result, the distributions of both SoP and SAM are visualized.

Figures 1(b) and 1(d) show the measured intensity patterns and SoP distributions of FVBs with different values of *α*, respectively. For comparison, the corresponding theoretical results are displayed in Figs. 1(a) and 1(c). Obviously, the experimentally measured results are in good agreement with the theoretically predictions. Different from a central dark spot for the integer vector beam (e.g., *α*=1), the FVB has a polarization singular line from the center along the + *x* direction. Moreover, this polarization singular line in the half-integer vector beam (e.g., *α*=0.5) is clearer than that in arbitrary FVB (e.g., *α*=0.7).

## 3. SAM separation in weakly focused FVBs

Based on the vectorial Rayleigh-Sommerfeld formulas under the paraxial approximation [31], one obtains the electric field for the paraxial focusing of the FVB in free space along the + *z* direction as

Here *η*=-*kz*/[2*f*(*z *+ *f*)], *γ*=*kr*/(*z *+ *f*), *k*=2*π*/*λ*, and *f* is the focal length of the thin lens. Note that the coordinate origin *z*=0 is at the lens’ geometrical focus.

Substituting Eqs. (3) and (4) into Eq. (5), and using the Bessel integral identity [32], we derive the electric field vector as

*J*(·) is the

_{n}*n*-order Bessel function of the first kind.

Specially, on the postfocal plane of the lens (*z*=0), we get the focal field as

Here, $\gamma_{0}=k r / f \, \textrm{and} \,\, \omega_{0}=\lambda f /(2 \textrm{R})$ is the beam waist at the focus. Correspondingly, $z_{0}=k \omega_{0}^{2} / 2$ is the Rayleigh length of the FVB.

When one expands the Bessel function into a series and recalls the incomplete gamma function $\Gamma (\xi ,h) = \int_h^\infty {{e^{ - x}}{x^{\xi - 1}}dx}$, the integral calculations of Eqs. (6) and (7) obtain

As described by Eqs. (10) and (11), the focused FVB can be expressed as the modal superposition of all integer vector beams with the complex weight coefficient *c _{n}*, i.e., ${\vec{E}_\alpha } = \sum\limits_n {{c_n}} {\vec{E}_n}$. For instance, the focused FVB

*α*=0.7 depicted in Fig. 2 is modal decomposition of integer vector beams with

*n*=0,1,2,···. It is shown that both the intensity pattern and SoP distribution of the focused FVB are different from those of focused integer vector beams. By altering the

*α*value of a FVB (alternatively, tuning the weight coefficients of integer vector beams in the modal superposition), one could manipulate both the intensity pattern and SoP distribution of the focused FVB.

For the arbitrary light field in nonmagnetic media, the SAM and OAM densities can be written as [33]

*ε*

_{0}is the permittivity of free space,

*μ*

_{0}is the vacuum permeability,

*ϖ*=

*kc*is the circular frequency of light,

*c*is the speed of light in vacuum, and ${\vec{E}^ \ast }$ and ${\vec{H}^ \ast }$ denote the complex conjugate of the electric field $\vec{E}$ and the magnetic field $\vec{H}$, respectively. Clearly, the SAM and OAM densities depend on both the electric and magnetic fields. However, compared with the dominant interaction between the electric field with nonmagnetic particles, the magnetic field acts insignificant on them. Therefore, we only consider the SAM and OAM originated from the electric field $\vec{E}$ and its corresponding photomechanical effect.

For a 3D electric field, the normalized three orthogonal SAM and OAM components in Cartesian coordinates can be expressed as

It is noted that the formula of ${\vec{E}^\ast } \cdot (\nabla )\vec{E} = ({\vec{E}^\ast } \cdot \nabla )\vec{E}$ is used to derive Eq. (16) from Eq. (13). A few literatures mistakenly regard ${[{\vec{E}^\ast } \cdot (\nabla )\vec{E}]_j} = \sum\nolimits_j {E_j^\ast } {\nabla _j}{E_j}$ [33]. In fact, the correct notation is ${\vec{E}^\ast } \cdot (\nabla )\vec{E} + {\vec{E}^\ast } \times (\nabla \times \vec{E}) = {\vec{E}^\ast } \cdot (\nabla \vec{E})$.

As described by Eqs. (14) and (15), the normalized SAM and OAM densities of the 3D electric field contain the transverse and longitudinal components. In particular, for the transverse electric field described by Eqs. (3) and (4) or (10) and (11), the FVB does not carry OAM, whereas the SAM density along the light propagating direction is proportional to the third Stokes parameter. Clearly, this longitudinal SAM is associated with the polarization of the light field. Specially, a LH (or RH) circularly polarized light has a normalized SAM density of $\tilde{J}_z^{\textrm{SAM}} ={+} 1$ (or $- 1$). For a linearly polarized light, one obtains $\tilde{J}_z^{\textrm{SAM}} = 0$. Intuitively, the FVB with localized linear polarization carries no SAM and OAM components. Surprisingly, as we will demonstrate below, the focused FVB leads to the transverse splitting of the angular momentum because of the modal superposition of all integer vector beams.

Experimentally, to investigate the propagation behaviors and SAM characteristics of the weakly focused FVBs, the generated FVB is focused by a thin lens with *f*=150 mm, producing the beam waist at the focus *ω*_{0}≈15.5 μm (correspondingly, the Rayleigh length *z*_{0}=1.42 mm). A detector (CinCam CMOS-1202) is scanned across the lens’ geometrical focus along the optical beam axis using a computer-controlled translation stage. Hence, the intensity patterns of the focused FVB at different positions are recorded.

Figures 3(a) and 3(c) show the simulated intensity patterns and SAM distributions of the FVBs with different values of *α* at the focal plane (*z*=0), respectively. The corresponding experimental results are illustrated in Figs. 3(b) and 3(d). Clearly, the theoretical simulations are consistent with the experimental observations, indicating that our theoretical analysis is correct. Because the focused FVB can be regarded as the modal superposition of all integer vector beams [see Eqs. (10) and (11) or Fig. 2], the FVB exhibits a rich focal field distribution. As shown in Fig. 3(a), the intensity patterns form circular and elliptical bright spots for *α*=0.5 and 0.7, respectively. On the contrary, the focal fields of the FVBs with *α*=1.3 and 1.5 are split into two symmetric spots, which are different from the doughnut-shaped focal field with circularly symmetry for the radially polarized beam (i.e., *α*=1) [3,32]. It is concluded that the focal field distribution of the FVB can be flexibly engineered by tuning the non-integer topological index *α*.

Interestingly, as shown in Figs. 3(c) and 3(d), the weakly focused FVB causes the LH and RH circularly polarization components to be spatially split into two parts, resulting in the transverse separation of the SAM distribution at the focal plane. The SAM distributions of two separated parts with a pseudo two-fold rotational symmetry have opposite signs (i.e., the positive and negative SAM densities are transversely split). For comparison, the focused radially polarized beam (i.e., *α*=1) with localized linear polarization carries no SAM component because its field distribution preserves the initial polarization (alternatively, SAM) at any propagation position [32]. The SAM separation of the FVB in the focal region arises from the modal superposition of all integer vector beams (see Fig. 2). It is noteworthy that the focusing of the FVB only causes the LH and RH circular polarization components to be spatially split and redistributes the SAM distribution, whereas the total SAM is zero and remains the conservation.

Figure 4 shows the simulated and measured intensity patterns and SAM distributions of a FVB with *α*=0.7 at five typical positions (i.e., *z*=-2*z*_{0}, -*z*_{0}, 0, *z*_{0}, and 2*z*_{0}). Obviously, the experimental results agree with the numerical simulations. The focused FVB with *α*=0.7 forms a half-moon shaped pattern with the complex SAM distribution outside the focal plane and has an elliptical-shaped bright spot at the focal plane. Furthermore, the intensity patterns before and after the focal plane are mirror-image symmetry. Although the transverse separation of positive and negative SAMs in the focal region, the FVB carrying no SAM preserves the SAM conservation at any propagation position.

## 4. SAM and OAM separations in tightly focused FVBs

It is assumed that the FVB described by Eqs. (3) and (4) is tightly focused by a high-numerical-aperture (*NA*) objective lens. Adopting the Richards-Wolf vectorial diffraction method [34,35] and the Bessel integral identity [32], we obtain the 3D electric field in the focal region of an aplanatic lens as

*NA*and the refractive index $n_0^m$ in the image space. The coefficient

*E*

_{0}is given by ${{|E_0|}^2}=P/{({\pi\varepsilon_0}{{cn}^m_o}{R^2})}$, where

*P*is the incident laser power of the FVB.

Figures 5(a)–5(c) show the transverse, longitudinal, and total intensity patterns of tightly focused FVBs with different values of α at the focal plane, by taking *λ*=532 nm, *NA*=0.85, ${n_0^m=1}$, and *z*=0. The intensity patterns are normalized by the maximum of the corresponding total intensity. Similar to the case of weak focusing (see Fig. 2), the tightly focused FVB can also be regarded as the modal superposition of all focused integer vector beams. Accordingly, the tightly focused FVB shown in Fig. 5 exhibits a rich focal field distribution. Quietly different from the transvers field of the radially polarized beam exhibiting a doughnut-shaped distribution [35], as shown in Fig. 5(a), the transverse intensity pattern of the tightly focused FVB is split into two parts with a two-fold rotational symmetry. In addition, different from the transverse field in the case of weak focusing, the tightly focused FVB is a 3D electric field with a weaker longitudinal field component than that of radially polarized beam. Briefly speaking, by adjusting the controllable parameter *α* (i.e., changing the weight of integer vector beams in the modal superposition), the asymmetric focal field distribution of the tightly focused FVB can be flexibly engineered.

The angular momentum properties of the tightly focused FVBs are investigated by numerical simulations using Eqs. (14) and (15) with *λ*=532 nm, *NA*=0.85, *n*m 0 = 1, and *z*=0. Figures 6(a) and 6(b) [or 6(c) and 6(d)] show the distributions of the normalized transverse and longitudinal SAM (or OAM) densities at the focal plane, respectively. Note that the transverse OAM is the vector superposition of the *x*-component and *y*-component OAMs. The magnitude of the transverse SAM (or OAM) densities is defined as ${\tilde{J}_T} = {(\tilde{J}_x^2 + \tilde{J}_y^2)^{1/2}}$. In Figs. 6(a) and 6(c), the magnitudes and directions of the transverse SAM and OAM are respectively illustrated by the colorbar and arrows. On the contrary, the positive (or negative) values of the longitudinal SAM and OAM indicate that their directions are along the + *z* (or -*z*) direction. As described by Eqs. (14) and (15), the longitudinal SAM and OAM densities are directly related to the transverse electric field, while the appearance of the transverse SAM and OAM is attributed to the existence of longitudinal field component. As shown in Figs. 6(a) and 6(b) [or 6(c) and 6(d)], the tightly focused FVB makes the spatial separation of 3D focal fields based on the modal superposition of different integer-order vector vortex beams [see Eqs. (17)–(19)], leading to the transverse and longitudinal separations of SAM (or OAM) distribution in the focal region, respectively. Moreover, it is found that both SAM and OAM are conserved in *x*-, *y*, *z*-components, and all components are equal to zero. Unlike the radially polarized beam only having a circular symmetric transverse angular momentum distribution, the 3D SAM and OAM distributions of the tightly focused FVB are spatially split into two parts with a pseudo two-fold rotational symmetry. Different from the transverse separation of longitudinal SAM in the case of weak focusing [see Fig. 3(c)], the spatial separation of 3D SAM and OAM occurs in the tightly focused FVB, due to the partial conversion of SAM to OAM. It should be noted that the tightly focusing of a FVB only results in the redistribution of 3D SAM and OAM in the focal region, but the total angular momentum is conserved. Different from the beam having purely transverse OAM [36], the tightly focused FVBs do not possess a purely transverse OAM, but only cause the transverse separation of OAM.

## 5. Optical forces on Rayleigh particles produced by tightly focused FVBs

The above-mentioned results demonstrate that the tightly focused FVB gives rise to the spatial separation of both 3D SAM and OAM densities. Therefore, such a beam can be exploited for optical manipulation of tiny particles. We consider that a spherical dielectric particle (*a*<<*λ*, where *a* is the radius of the particle) is located at the focal region of the focused FVB. The transfer of the linear momentum from light to the Rayleigh particle induces an optical force on the particle as [37,38]

Here, *ε*_{2} and *ε*_{1} are the permittivities of the particle and the surrounding medium, respectively.

When a FVB is tightly focused onto a Rayleigh particle, the optical forces exerted on the Rayleigh particle can be calculated numerically using Eq. (20). Without loss of generality, the parameters in the force and torque analyses are taken to be *λ*=532 nm, *NA*=0.85, *a*=40 nm, *ε*_{2}=2.53 (for polypropylene), *ε*_{1}=1.77 (for water), and *P*=100 mW.

Figures 7(a)–7(c) illustrate the optical force distributions on the particle produced by tightly focused FVBs with different values of *α* on the *x*-*y* plane (*z*=0), *z*-*x* plane (*y*=0), and *z*-*y* plane (*x*=0), respectively. Note that the transverse force shown in Fig. 7(a) is the vector superposition of the forces in *x*- and *y*-directions, and its magnitude is given as ${{({F_x^2}+{F_y^2})}^{1/2}}$ And that the positive (or negative) longitudinal force means that its direction is in the + *z* (or -*z*) direction. The numerical simulations show that the radiation force is negligible compared with the gradient force, and hence the total force mainly arises from the gradient force. Note that the symmetrically transverse force produced by a radially polarized beam always directs to the position of the focal point to produce a force balance [27]. Interestingly, as shown in Fig. 7(a), the asymmetrically transverse force produced by tightly focused FVBs at the focal plane traps the particle in two symmetric centrifugal regions. That is, the Rayleigh particle can be trapped in two spots of intensity maximum shown in Fig. 5(c). Moreover, by tuning the *α* value of FVBs, the spatial distribution of the trapped particles can be precisely controlled and adjusted in principle. Figures 7(b) and 7(c) illustrate the distributions of the longitudinal forces in the *z*-*x* plane (*y*=0) and *z*-*y* plane (*x*=0), respectively. Different from the cylindrically symmetric longitudinal force produced by the radially polarized beam [27], the longitudinal force distributions of FVBs have axial asymmetry. As shown in Figs. 7(b) and 7(c), the particle can be easily trapped by the tightly focused FVB at the focal plane due to the existence of the equilibrium position. As a result, the particle forms a stable 3D trap at two points on the focal plane.

## 6. Torque on Rayleigh particles produced by tightly focused FVBs

It is well known that the transfer of SAM and OAM from light to the particle results in the optical spin and orbital torques, which causes the particle to rotate around its own axis and the optical beam axis, respectively [2,24]. When the external electric field $\vec{E}$ varies harmonically in time, the time-averaged spin torque is given by [39]

At the focal plane, the transverse force drives the trapped particle to move around the optical beam axis. Therefore, the orbital torque can be expressed as

To investigate the trajectory of the trapped particles, we calculate the longitudinal spin torque *T _{z}^{spin}* and the longitudinal orbital torque

*T*exerted on the particle, by taking the same parameters in Fig. 7. Figures 8(a) and 8(b) illustrate the distributions of

_{z}^{orb}*T*and

_{z}^{spin}*T*on the particle produced by tightly focused FVBs with different values of

_{z}^{orb}*α*at the focal plane, respectively. Here, the positive (or negative) values of

*T*and

_{z}^{spin}*T*denote that their directions are along the +

_{z}^{orb}*z*(or -

*z*) direction. As shown in Fig. 8(a), the distribution of

*T*is similar to that of the longitudinal SAM density displayed in Fig. 6(b), because the transfer of SAM from light to the particle causes the spin torque. As shown in Fig. 8(b), the

_{z}^{spin}*T*distributions on the particle produced by tightly focused FVBs are spatially separated with a two-fold rotational symmetry. In principle, the orbital torque is the result of OAM in light field transferred to the particle. However, comparing Figs. 8(b) with 6(d), it is found that the distribution of

_{z}^{orb}*T*is completely different from that of longitudinal OAM density. The reasons for this difference are as follows. At the focal plane of the tightly focused FVB, the asymmetric intensity distribution [see Fig. 5(c)] results in that the transverse force exerted on the particle is not completely along the radial direction [see Fig. 7(a)]. The azimuthal component of the transverse force can be regarded as the swirling force, producing the longitudinal orbital torque

_{z}^{orb}*T*[see Fig. 8(b)]. The motion trajectories of the trapped particle using tightly focused FVBs at the focal plane are shown in Fig. 8(c). The positive and negative

_{z}^{orb}*T*are separated transversely, resulting in the trapped particles rotating anticlockwise and clockwise around their own axis, respectively. Meanwhile, the particles located at different places rotate clockwise or anticlockwise around the optical beam axis, depending on the sign of

_{z}^{spin}*T*. Finally, the particles are driven by the longitudinal orbital torque

_{z}^{orb}*T*to the two equilibrium positions, where the

_{z}^{orb}*T*value is zero. Note that the trapped particles always have spin motion when the Brownian motion is ignored. For comparison, the result for a radially polarized beam (i.e.,

_{z}^{orb}*α*=1) shown in Fig. 8(c) indicates that the particle without any spin and orbital motions is directly captured on the optical beam axis.

## 7. Conclusions

In summary, we have investigated the spatial separation of SAM and OAM in focused FVBs for optical manipulation. Based on the vectorial Rayleigh-Sommerfeld formulas under the paraxial approximation, we presented the analytical expressions for the weakly focused field of the FVB, which is given by the modal superposition of all integer vector beams. We theoretically simulated and experimentally measured the evolution of the intensity patterns and the longitudinal SAM distributions in the focal region. It is demonstrated that such a beam carrying no SAM leads to both the transverse separation of SAM and the special intensity patterns in the focal region. Furthermore, we investigated the 3D distributions of the intensity, SAM, and OAM of the tightly focused FVBs at the focal region. It is shown the spatial separation of 3D SAM and OAM in the tightly focused FVB, due to the partial conversion of SAM to OAM. We numerically studied the optical force, spin torque, and orbital torque on a dielectric Rayleigh particle produced by the tightly focused FVBs. It is found that the trapped particles at the focal plane will asymmetrically spin and orbital motions in a tightly focused FVB. The presented work may find useful applications in optical trapping and manipulation, especially in optically induced rotation and motion.

## Funding

National Natural Science Foundation of China (11774055, 12074066); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_0084); Natural Science Foundation of Jiangsu Province (BK20181384).

## Disclosures

The authors declare no conflicts of interest.

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