Theoretical investigation on asymmetrical spinning and orbiting motions of particles in a tightly focused power-exponent azimuthal-variant vector field

Yingdi Zhang; Yuxiong Xue; Zhuqing Zhu; Guanghao Rui; Yiping Cui; Bing Gu

doi:10.1364/OE.26.004318

1. Introduction

It is well known that optical linear momentum (LM) and angular momentum (AM) are the main dynamical properties of electromagnetic waves, which play a crucial role in light-matter interactions [1–4]. The LM of light can be separated into two contributions [5]: $\overset{⇀}{P} = {\overset{⇀}{P}}_{spin} + {\overset{⇀}{P}}_{orb}$ . Here the spin linear momentum (SLM) ${\overset{⇀}{P}}_{spin}$ is proportional to the curl of the spin density of the light field, whereas the orbital linear momentum (OLM) ${\overset{⇀}{P}}_{orb}$ is proportional to the radiation force on a Rayleigh particle [6,7]. Analogously, the optical AM of an electromagnetic wave can be decomposed into two parts: $\overset{⇀}{J} = {\overset{⇀}{J}}_{spin} + {\overset{⇀}{J}}_{orb}$ . The spin angular momentum (SAM) ${\overset{⇀}{J}}_{spin}$ arises from the polarization of light, resulting in the rotation of a particle around its own axis [8], whereas the orbital angular momentum (OAM) ${\overset{⇀}{J}}_{orb}$ is associated with the phase structure of light, causing the rotation of the particles around the optical axis [1,9].

Theoretically, the optical LM and AM can transfer from the electromagnetic radiation to the small particle. The former results in the optical force exerted on the particle, whereas the latter gives rise to the optical torque on the particle, causing the rotation of the particle. Experimentally, the optical LM and AM densities can be quantitatively measured by placing a small particle in the light field and observing its motion trajectory [6,9–13]. This optical trapping is a useful technique for noncontact and noninvasive manipulation of small particles using a focused laser beam [14]. Over the past decades, researchers have reported various motion trajectories of trapped particles. For examples, Allen et al. [15] observed that optical vortex beams carrying OAM can drive particles to orbit around the optical axis. O’Neil et al. [9] experimentally examined the motion of particles trapped off axis in an optical tweezers and confirmed that the OAM and SAM are associated with the particles’ orbital motion and spinning motion, respectively. Zhao et al. [16] demonstrated that focusing a beam carrying SAM can induce an OAM to drive the orbital motion of a micron-sized metal particle off the beam axis. Cao et al. [17] presented the spin-controlled orbital motion in tightly focused high-order Laguerre-Gaussian beams. Li et al. [18] realized the non-axial spinning and orbiting motions of absorptive particles in vortex beams with circular or radial polarizations. Note that the above-mentioned particles’ motions centered on the optical axis have the axial-symmetry. However, exploiting particles’ motion with broken axial symmetry may have both academic interest and technological applications, although this issue is seldom in the literature so far [19].

Polarization, as an intrinsic and fundamental vectorial nature of light, plays an important role in focusing properties, propagation behaviors, and light-matter interactions. In recent years, manipulating polarization of light field has received extensively attentions due to the fact that cylindrical vector fields with spatially variant states of polarization (SoPs) have fascinating properties and novel applications in optical trapping [20], optical micro- machining [21], optical microscopy [22], etc. Researchers have experimentally generated a variety of types of cylindrical vector fields with the desired polarization distribution, such as radially and azimuthally polarized beams [23,24], hybrid polarized vector fields [25], Poincaré sphere beams [26], and vector vortex beams [27]. Except for the axial-symmetry light fields, the axial-symmetry-broken light fields, for examples, asymmetric-sector-shaped vector beams [28] and asymmetric Laguerre-Gaussian beams [29], have been reported. With the aid of diffractive optical elements, all kinds of novel focused fields have been reported, including optical bottle [30], optical cage [31], optical chain [32], optical needle [33], and dark channel [34]. Moreover, the optical AM of vector fields has been investigated [35–37]. Besides, the optical trapping characteristics of different vector fields, including cylindrical vector fields [38], hybridly polarized vector fields [24,35], full Poincaré beams [39], and elliptically polarized vector fields [40], have been exploited.

In this work, we report the experimental generation of a power-exponent azimuthal- variant (PE-AV) vector field and numerically study its focusing properties. Moreover, we numerically investigate the three-dimensional distributions of linear and angular momenta, and optical force and torque on a dielectric Rayleigh particle produced by the tightly focused vector field. We demonstrate the asymmetrical spinning and orbiting motions of trapped Rayleigh particles by the use of a tightly vector field with PE-AV SoPs.

2. Generation of the PE-AV vector field

The electric field distribution of a PE-AV vector field can be expressed as

\vec{E} (ρ, ϕ) = E_{0} \exp (- \frac{ρ^{2}}{ω^{2}}) [\cos (\frac{ϕ^{2}}{2 π}) {\vec{e}}_{x} + \sin (\frac{ϕ^{2}}{2 π}) {\vec{e}}_{y}] .

Here E₀ is an amplitude constant, ω is the radius of the incident beam, ρ is the polar radius, and ϕ is the azimuthal angle.

{\overset{⇀}{e}}_{x}

and

{\overset{⇀}{e}}_{y}

are the unit vectors in the Cartesian coordinate system (x, y).

To generate the PE-AV vector fields, we adopt a universal approach in a common path interferometer implemented with a spatial light modulator (SLM), based on the wavefront reconstruction and the Poincaré sphere [23,25,41]. It should be noted that the additional phase in a computer-controlled SLM is set as δ(ρ, ϕ) = ϕ²/(2π). The experimental arrangement and the detail description can be found elsewhere [23,25,41]. Using a CW laser beam at a wavelength of 532 nm, we experimentally generate the PE-AV vector field with its radius of ω = 5.30 ± 0.05 mm. Moreover, we use the quarter-wave retarder polarizer method [42] to characterize the Stokes parameters of the generated vector field.

Figure 1(a) shows the normalized Stokes parameters of the generated PE-AV vector field. The imperfection of the measured patterns mainly arises from both the instability of the system for generating vector fields [43] and the diffraction stripes after synthesis of the interference path [23,25]. Using the measured Stokes parameters, we computed the distribution of SoPs for equally spaced points displayed in Fig. 1(b). For the sake of comparison, the corresponding theoretical results are shown in Fig. 1(c). Obviously, the measured SoPs are consistent with the numerical simulations, although some points of the experimental SoPs do not match the theoretical results very well. To quantitatively evaluate the measured SoPs with the desired results, we computed both the ellipticity and the orientation angle of every point in the transverse plane of the generated vector field, which have the maximum errors of 15% and 13%, respectively. This difference is anticipated for the following two reasons: (i) laser jitter results in beam instability; and (ii) there is the bias in measuring strength components utilizing the polarizer and wave plate. As shown in Fig. 1, the intensity pattern of the generated vector field has a Gaussian profile, in contrary to the radially polarized beam that has a central dark spot arising from the polarization singularity at the center. More interestingly, the cylindrical PE-AV vector field exhibits a cylindrical- symmetry-broken distribution of SoPs shown in Fig. 1(b), quite different from the radially polarized beam with linear polarization aligned in the radial direction and the axial-symmetry-broken light fields [28,29].

Fig. 1 (a) Experimentally measured Stokes parameters of the generated PE-AV vector field. (b) Experimentally measured and (c) theoretically predicted SoP distributions.

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3. Tightly focusing property of the PE-AV vector field

Due to the asymmetrical SoPs distribution of the PE-AV vector field shown in Fig. 1(a), it is expected that this vector field may have peculiar focusing properties compared with that of the radially polarized beam [44]. Considering that the vector field described by Eq. (1) is tightly focused by a high numerical aperture (NA) objective that obeys the sine condition of sinθ = ρ/f (where f is the focal length of the high NA objective), its pupil apodization functions along the x and y axes at the objective aperture can be written as

A_{x} (θ, ϕ) = E_{0} \exp (- β^{2} \frac{\sin^{2} θ}{\sin^{2} α}) \cos (\frac{ϕ^{2}}{2 π}),

A_{y} (θ, ϕ) = E_{0} \exp (- β^{2} \frac{\sin^{2} θ}{\sin^{2} α}) \sin (\frac{ϕ^{2}}{2 π}),

where β is the pupil filling factor defined as the ratio of the pupil radius to the beam waist ω.

According to the Richards-Wolf vectorial diffraction method [45], the three-dimensional electric field distribution in the focal region can be expressed as [46]

\begin{array}{l} E_{x} (r, φ, z) = - \frac{i k f}{2 π} \int_{0}^{Θ} \int_{0}^{2 π} \sin θ \sqrt{\cos θ} e^{i k (z \cos θ + r \sin θ \cos (ϕ - φ))} \\ \times [A_{x} (θ, ϕ) (\cos^{2} ϕ \cos θ + \sin^{2} ϕ) \\ + A_{y} (θ, ϕ) \cos ϕ \sin ϕ (\cos θ - 1)] d θ d ϕ, \end{array}

\begin{array}{l} E_{y} (r, φ, z) = - \frac{i k f}{2 π} \int_{0}^{Θ} \int_{0}^{2 π} \sin θ \sqrt{\cos θ} e^{i k (z \cos θ + r \sin θ \cos (ϕ - φ))} \\ \times [A_{x} (θ, ϕ) \cos ϕ \sin ϕ (\cos θ - 1) \\ + A_{y} (θ, ϕ) (\cos^{2} ϕ + \sin^{2} ϕ \cos θ)] d θ d ϕ, \end{array}

\begin{array}{l} E_{z} (r, φ, z) = - \frac{i k f}{2 π} \int_{0}^{Θ} \int_{0}^{2 π} \sin θ \sqrt{\cos θ} e^{i k (z \cos θ + r \sin θ \cos (ϕ - φ))} \\ \times [A_{x} (θ, ϕ) \cos ϕ \sin θ + A_{y} (θ, ϕ) \sin ϕ \sin θ] d θ d ϕ . \end{array}

Here k = 2πn₁/λ is the wave number, λ is the wavelength, n₁ is the refractive index in the image space, and Θ = arcsin(NA/n₁) is the maximal converging angle determined by the NA of the objective lens.

To explore the tightly focusing properties of the PE-AV vector field, we perform the numerical simulation using Eqs. (2)-(6). As an example, Fig. 2 illustrates the intensity patterns of the focused PE-AV vector field in the transverse plane for z = 0 (top row) and in the longitudinal plane for x = 0 (bottom row), by taking the parameters as λ = 532 nm, NA = 1.26, n₁ = 1.33, and β = 1. Figures 2(a)-2(c) show the transverse, longitudinal, and global intensity patterns of the vector field at the focus in the x-y plane (z = 0). The intensity patterns are normalized by the maximum of the global intensity. It is shown that the longitudinal intensity of this vector field is small compared with the transverse intensity. Furthermore, numerical simulation indicates that the transverse intensity mainly originates from its x-component owing to the localized linear polarization of the input vector field mainly aligned in the x-direction [see Fig. 1(b)]. Figures 2(d)-2(f) display the intensity patterns through the focus in the z-y plane, which is normalized by the maximum of the global intensity. By calculating the Stokes parameters of the electric field, we map the polarization projections on the x-y and z-y planes superimposed with the intensity patterns shown in Figs. 2(a) and 2(d), respectively. Completely different from the intensity patterns of the radially polarized beam having a sharp focal spot, of the azimuthally polarized beam with a hollow dark spot [44], or of the azimuthal-variant vector fields exhibiting a flower-like pattern [47], the focused PE-AV vector field forms a double half-moon shaped pattern with a two-fold rotational symmetry shown in Fig. 2(c), which consists of low intensity at the center surrounded by regions of high intensity. On the other hand, the focused PE-AV vector field has abundant SoPs [including right-handed (RH) and left-handed (LH) elliptical polarizations] distributions in the cross section of the field shown in Figs. 1(a) and 1(d), dramatically different from the localized linearly polarization distribution of the focused radially polarized beam [44]. Moreover, the polarization projection on the x-y plane exhibits a Taiji-like pattern. It is noteworthy that the SoPs distributions displayed in Figs. 2(a) and 2(d) have a pseudo-two-fold rotational symmetry because the chirality changes from LH polarization to RH polarization.

Fig. 2 Normalized intensity patterns of the focused PE-AV vector field in the transverse plane for z = 0 (top row) and in the longitudinal plane for x = 0 (bottom row), by taking λ = 532 nm, NA = 1.26, n₁ = 1.33, and β = 1. (a) and (d) also show the polarization projections (white: LH polarization; black: RH polarization) on the x-y and z-y planes, respectively.

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4. LM of the tightly focused PE-AV vector field

For a monochromatic light field, the cycle-averaged LM density is given by [7,48]

\vec{P} = \frac{ε_{0}}{2 \bar{ω}} Im [{\vec{E}}^{*} \times (\nabla \times \vec{E})],

where ε₀ is the vacuum permittivity,

\bar{ω} = k c

is the circular frequency of light, and c is the velocity of light in vacuum. Note that the LM

\overset{⇀}{P}

is related to the Poynting vector

\overset{⇀}{S}

by

\overset{⇀}{P} = \overset{⇀}{S} / c^{2}

. Interestingly, the density of the LM can be separated into spin and orbital parts as [47]

{\vec{P}}_{spin} = \frac{ε_{0}}{4 \bar{ω}} \nabla \times Im [{\vec{E}}^{*} \times \vec{E}],

{\vec{P}}_{orb} = \frac{ε_{0}}{2 \bar{ω}} Im [{\vec{E}}^{*} \cdot (\nabla) \vec{E}] .

Here the SLM

{\overset{⇀}{P}}_{spin}

, also named as the solenoidal spin part, is proportional to the curl of the spin density of the light field. Whereas the orbital part

{\overset{⇀}{P}}_{orb}

equals the canonical momentum density, which is proportional to the radiation force on a Rayleigh particle [6,7].

The LM properties of the tightly focused PE-AV vector field are investigated by numerical simulations using Eqs. (7)-(9) with λ = 532 nm, NA = 1.26, n₁ = 1.33, β = 1, and z = 0. The top and bottom rows of Fig. 3 respectively show the distributions of the transverse and longitudinal LM densities in the x-y plane, which are normalized by the maximum value of the LM density displayed in Fig. 3. In Figs. 3(a)-3(c), the magnitudes and directions of the transverse LMs are illustrated by the colorbar and arrows, respectively. On the contrary, the positive (or negative) value of the longitudinal LMs means that its direction is along the + z (or –z) direction. It is noteworthy that there is no transverse LM at the focal plane for a localized linearly polarized vector field with axial symmetry [35,49,50]. However, the focused EP-AV vector field carries transverse LM flow at the focal plane (see the top row of Fig. 3), owing to this localized linearly polarized vector field with broken axial symmetry. Moreover, compared with the longitudinal component of the LM, the transverse LM at the focal plane is high enough to play a non-negligible role in the total LM. Besides, the distribution of the LM density is not uniform. For instance, the distributions of the transverse SLM and OLM, as shown in Figs. 3(a) and 3(b), exhibit S-like and Taiji-like patterns, respectively. The distribution of the longitudinal LM has a double half-moon shaped pattern in the transverse plane shown in Fig. 3(f). In particular, these transverse and longitudinal LMs (i.e., energy flux) mainly flow to the particular directions and transfer from the electromagnetic radiation to the particle, resulting in the radiation force exerted on the particle. This property can be used for optical trapping, as we will demonstrate in Sec. 6.

Fig. 3 Normalized transverse (top row) and longitudinal LMs (bottom row) of the tightly focused PE-AV vector field in the x-y plane. The magnitudes and directions of the transverse LMs are illustrated by the colorbar and arrows in (a)-(c), respectively. The parameters for calculations are λ = 532 nm, NA = 1.26, n₁ = 1.33, β = 1, and z = 0.

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5. AM of the tightly focused PE-AV vector field

Similar to the LM, the cycle-averaged AM density of a monochromatic light field is expressed as

\vec{J} = \vec{r} \times \vec{P},

where

\overset{⇀}{r}

is the particle’s position from the origin [1]. Also the AM of light can be separated into orbital and spin parts [51]:

{\vec{J}}_{spin} = \frac{ε_{0}}{2 \bar{ω}} Im [({\vec{E}}^{*} \times \vec{E}],

{\vec{J}}_{orb} = \frac{ε_{0}}{2 \bar{ω}} \vec{r} \times Im [{\vec{E}}^{*} \cdot (\nabla) \vec{E}] .

In details, the SAM arises from the polarization of light whereas the OAM is associated with the phase structure of light. The optical SAM and OAM result in the spin and orbital torques on the particles, causing the rotation of the particles around its own axis and the optical axis, respectively [1,8,9].

Figure 4 illustrates the distributions of the SAM, OAM, and AM densities in the focal plane of the focused EP-AV vector field, by taking the same parameters in Fig. 3. Here the OAM density is obtained by ${\overset{⇀}{J}}_{orb} = \overset{⇀}{J} - {\overset{⇀}{J}}_{spin}$ [9,51,52]. Note that the AM patterns are normalized by the maximum value of the AM density shown in Fig. 4. The transverse SAM exhibits a hollow-shaped pattern with radial direction, as shown in Fig. 4(a). Note that the direction of transverse SAM depends upon the propagation direction of light inherently regulating the phase correlation of transverse and longitudinal electric-field components [37]. As shown in Fig. 4(b), the transverse OAM has a non-uniform hollow circular distribution with clockwise direction. It should be noted that the contribution of transverse SAM is very small compared with that of transverse OAM. Consequently, the transverse AM exhibiting the non-uniform hollow circular pattern mainly originates from the contribution of OAM. Besides, the AM density is the vector superposition of the SAM and OAM densities. That is why the center hole of the transverse total AM shown in Fig. 4(c) is larger than those of SAM and OAM. The bottom row of Fig. 4 shows the longitudinal AM densities in the transverse plane. Here the positive (or negative) value of the longitudinal AMs means that its direction is along the + z (or –z) direction. Figures 4(d)-4(f) all have a Taiji-like pattern with a two-fold rotational symmetry in the x-y plane. Similar to the SoP distribution at the focal plane [see Fig. 2(a)], the direction of the longitudinal SAM is determined by the polarization handedness [37]. Figure 4(e) illustrates the longitudinal OAM in the transverse plane with a pseudo-two-fold rotational symmetry because two parts have opposite directions. The longitudinal AM having a Taiji-like pattern arises from the contributions of both longitudinal SAM and OAM, as shown in Fig. 4(f). Interestingly, as we will verify in Sec. 7, the AM (i. e., SAM and OAM) of the focused PE-VA vector field will transfer to the trapped particles. It gives rise to the optical torque on the particles, resulting in the rotation of the particles.

Fig. 4 Normalized transverse (top row) and longitudinal AMs (bottom row) of the tightly focused PE-AV vector field in the x-y plane. The magnitudes and directions of the transverse AMs are illustrated by the colorbar and arrows in (a)-(c), respectively. The parameters for calculations are the same as those in Fig. 3.

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6. Optical forces on Rayleigh particles produced by a tightly focused PE-AV vector field

It is well known that the transfer of the LM from light to the Rayleigh particle induces an optical force on the particle. Optical forces on a Rayleigh particle can be divided into two parts: the gradient force ${\vec{F}}^{grad}$ , which is proportional to the gradient of intensity, drives the particle toward the equilibrium position; the radiation force ${\vec{F}}^{radi}$ , which is proportional to the orbital part of the Poynting vector of the field [6,7], destabilizes the trap by pushing the particle away from the focal plane. Hence, the optical forces exerted on the Rayleigh particle can be expressed as [6,53]

{\vec{F}}^{grad} = \frac{1}{4} Re (α) \nabla | \vec{E} |^{2},

{\vec{F}}^{radi} = \frac{\bar{ω}}{ε_{0}} Im (α) {\vec{P}}_{orb},

where α is the complex polarizability given by

α = \frac{α_{0}}{1 - i α_{0} k^{3} / (6 π ε_{0})}, α_{0} = 4 π ε_{0} a^{3} \frac{n_{2}^{2} / n_{1}^{2} - 1}{n_{2}^{2} / n_{1}^{2} + 2} .

Here a is the radius of the spherical particle. n₂ and n₁ are the refractive indices of the particle and the medium surrounding the particle, respectively.

When a PE-AV vector field is tightly focused onto a dielectric Rayleigh particle, the optical forces exerted on the particle can be calculated numerically using Eqs. (13) and (14). It is assumed that the particle is immersed in water and the parameters in the force analysis are taken to be λ = 532 nm, NA = 1.26, β = 1, n₁ = 1.33, n₂ = 1.60, a = 30 nm, and a laser power of P = 100 mW.

Figure 5 displays the distributions of the transverse (top row) and longitudinal (bottom row) forces in the x-y plane (z = 0) and z-x plane (y = 0), respectively. Here the transverse force is the vector superposition of the x-direction and y-direction forces. Its magnitude is obtained as (F_x² + F_y²)^1/2. The magnitudes and directions of the transverse forces are illustrated by the colorbar and arrows in Figs. 5(a)-5(c), respectively. On the contrary, the positive (or negative) longitudinal forces mean that their direction is along the + z (or –z) direction. As shown in the top row of Fig. 5, the distributions of the gradient force has a non-uniform double-ring pattern in the transverse plane, whereas the corresponding radiation force exhibits a Taiji-like pattern which is consistent with the distribution of transverse OLM density [see Fig. 3(b)]. As shown in Fig. 5(b), the transverse radiation force mainly points to the azimuthal direction. Accordingly, this transverse radiation force can be regarded as the swirling force and drives the particle to move around the optical axis at the focal plane when the Brownian motion is ignored. It is noteworthy that the total force mainly originates from the gradient force because the magnitude of the radiation force is negligible compared with that of the gradient force. More importantly, as shown in Fig. 5(c), the transverse force at the focal plane directs two positions nearly located at the y-axis to produce a force balance. That is, the particle can be trapped in two spots of intensity maximum, as shown in Fig. 2(c). The bottom row of Fig. 5 shows the distributions of longitudinal forces in the z-x plane. It is shown that the longitudinal gradient force is about one order of magnitude larger than that of the radiation force. Hence, the particle can be easily trapped by the focused PE-AV vector field at the focal plane due to the existence of the equilibrium point. As a result, the dielectric Rayleigh particle forms a stable three-dimensional trap at two points in the focal region.

Fig. 5 Transverse (top row) and longitudinal (bottom row) force distributions produced by the tightly focused EP-AV vector field in the x-y plane (z = 0) and x-z plane (y = 0), respectively. The parameters for calculations are P = 100 mW, λ = 532 nm, NA = 1.26, β = 1, n₁ = 1.33, n₂ = 1.60, and a = 30 nm.

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7. Torque on Rayleigh particles induced by a tightly focused PE-AV vector field

Now we turn our attention to the rotation movement of particles. It is noted that the transfer of the SAM and OAM from light to the particles leads to an optical torque, resulting in the rotation of the particles around its own axis and the optical beam axis, respectively [1, 8, 9]. As the external electric field varies harmonically in time, the time-averaged spin torque is [54]

{\vec{T}}^{spin} = \frac{1}{2} | α |^{2} Re [\frac{1}{α_{0}^{*}} \vec{E} \times {\vec{E}}^{*}] .

As discussed above, the transverse force drives the trapped particle to move around the optical beam axis at the focal plane. Accordingly, the orbital torque can be written as

{\vec{T}}^{orb} = \vec{r} \times \vec{F} .

Figure 6 illustrates the distributions of the transverse spin torque $T_{trans}^{spin}$ and longitudinal orbital torque $T_{z}^{orb}$ exerted on the particle in the focal plane, by taking the same parameters in Fig. 5. It is seen that the distribution of $T_{trans}^{spin}$ is in agreement with that of transverse SAM shown in Fig. 4(a). On the other hand, the longitudinal orbital torque $T_{z}^{orb}$ in the transverse plane exhibits a pinwheel-shaped pattern with two-fold rotational symmetry. Here the positive (or negative) value of $T_{z}^{orb}$ denotes that its direction is along the + z (or –z) direction. The motion trajectory of the trapped particles on the focal plane illustrated in Fig. 6(b) is described as follows. In the transverse plane of the tightly focused PE-AV vector field, the particle experiences a spin motion around its own axis nearly aligned in the radial direction at the focal plane, owing to the effect of the transverse spin torque shown in Fig. 6(a). At the same time, the particle located at different places rotate clockwise or anticlockwise, depending on the sign of the longitudinal orbital torque. Eventually, the particle is driven by the longitudinal orbital torque to the two equilibrium positions, where the longitudinal orbital torque is zero. Note that the trapped particle always has a spin motion when the Brownian motion is ignored.

Fig. 6 Transverse spin torque $T_{trans}^{spin}$ (a) and longitudinal orbital torque $T_{z}^{orb}$ (b) exerted on a Rayleigh particle induced by the tightly focused PE-AV vector field in the x-y plane (z = 0). The parameters for calculations are the same as those in Fig. 5.

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8. Conclusion

In summary, we have proposed and experimentally generated a new kind of localized linearly polarized vector fields, named a PE-AV vector field. We have numerically investigated the tight focusing properties of the PE-AV vector field based on the Richards–Wolf vectorial diffraction method. It is shown that the tightly focused field exhibits a double half-moon shaped pattern with the localized elliptical polarization in the cross section of field at the focal plane, owing to the asymmetrical SoPs distribution of this localized linearly polarized vector field. Furthermore, we have illustrated the three- dimensional distributions of SLM, OLM, SAM, and OAM densities in the focal region of the high NA objective. In particular, we have exploited the optical forces and torques of the tightly focused EP-AV vector field on a dielectric Rayleigh particle. It is found that the trapped particles on the focal plane will asymmetrically spin and orbit motions in a tightly focused PE-AV vector field. The findings reported in the work may find useful applications in optical manipulation, especially for optically induced rotation and motion.

Funding

National Natural Science Foundation of China (NSFC) (11474052, 11774055, 11504049); Natural Science Foundation of Jiangsu Province (BK20171364); National Key Laboratory of Science and Technology on Vacuum Technology and Physics (ZWK1608).

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Theoretical investigation on asymmetrical spinning and orbiting motions of particles in a tightly focused power-exponent azimuthal-variant vector field

Abstract

1. Introduction

2. Generation of the PE-AV vector field

3. Tightly focusing property of the PE-AV vector field

4. LM of the tightly focused PE-AV vector field

5. AM of the tightly focused PE-AV vector field

6. Optical forces on Rayleigh particles produced by a tightly focused PE-AV vector field

7. Torque on Rayleigh particles induced by a tightly focused PE-AV vector field

8. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (17)

Optics Express