{"paper_id":"3d2aa9aa-2d95-4227-93ef-e346b4d070e9","body_text":"Optomechanical Motions of Gold Dimer’s Spin, Rotation and Revolution Manipulated by Bessel Beam | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Optomechanical Motions of Gold Dimer’s Spin, Rotation and Revolution Manipulated by Bessel Beam Chao-Kang Liu, Yun-Cheng Ku, Mao-Kuen Kuo, Jiunn-Woei Liaw This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4386749/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The optomechanical motions of a pair of optically bound gold nanoparticles (GNPs), in fluid manipulated by a Bessel beam are studied. Since a Bessel beam possesses orbital angular momentum (OAM) and spin angular momentum (SAM) simultaneously, complicated rigid-body motions of the dimer can be induced. The mechanism involves the equilibrium between the optical force with the reactive drag force exerted by the fluid. Our results demonstrate that the 2D planar motion includes the rotation of the dimer around its center of mass (COM) and the orbital revolution of the COM around the optical axis. Additionally, each individual GNP undergoes spinning. The directions of the GNPs’ spin and the orbital revolution of COM depend on the handedness and the order (topological charge) of Bessel beam, respectively. Nevertheless, the rotation direction of the dimer depends on the size of GNP. In the case of a smaller dimer, the direction of dimer’s rotation with respect to the COM is consistent with the handedness of the light. Conversely, a larger dimer performs a reverse rotation, accompanied by a precession during the orbital revolution. There are multiple turning points in the radius of the GNP for the alternating rotation of the dimer caused by positive or negative optical torque. Our finding may provide an insight to the optomechanical manipulation of optical vortexes on the motions of GNP clusters. Bessel beam gold nanoparticles dimer orbital angular momentum spin angular momentum optical force optical torque drag force Stokes’ law Maxwell’s stress tensor optical binding MMP rotation orbital revolution Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction In recent decades, the motions of microparticels or nanoparticles manipulated by light through the light-matter interaction has gained significant attention [ 1 – 9 ]. In particular, the study of using plasmon-enhanced optical force to manipulate gold nanoparticles (GNPs) or silver nanoparticles (SNPs) for optical tweezers attracted a lot of attention [ 1 , 5 – 16 ]. For example, a laser beam, a linearly polarized light (LPL), can induce an optical binding for a pair of GNPs with a stable distance of integer multiples of the effective wavelength of light in the medium, where the central line of the dimer is perpendicular to the polarization of LPL [ 11 , 12 ]. In addition, a circularly polarized light (CPL) can induce the spin of a single GNP or gold nanorod (GNR) significantly due to the absorption of photon’s spin angular momentum (SAM) [ 1 , 3 , 13 – 15 ]. Although a CPL has only SAM but without orbital angular momentum (OAM) intrinsically, CPL may induce the optical binding between two individual GNPs to form a dimer with a stable distance, and drives them to rotate around their center of mass (COM) accompanied with individual spinning [ 16 ]. Of interesting, the rotation’s direction of dimer not only depends on the handedness of CPL but also the size of GNPs [ 17 – 19 ]. For a smaller GNP/SNP dimer, the rotational direction is always consistent with the handedness of CPL. In contrast, for a larger GNP/SNP dimer a reverse rigid-body rotation around the COM may be induced due to the negative optical torque caused by higher-order modes [ 17 – 19 ]. Moreover, using CPL to manipulate multiple SNPs, GNPs or GNRs have also been studied [ 4 , 7 , 15 , 18 , 20 ]. For example, the planetary gear motion of optical bound hexagonal lattice composed of multiply SNPs driven by CPL were investigated by Refs. [ 7 , 15 ]. Recently, the study of using an optical vortex, e.g. Laguerre-Gaussian beam or Bessel beam, to manipulate dielectric microparticles, GNPs or SNPs has also drawn considerable attention [ 21 – 39 ]. Since an optical vortex, a structured light, possesses both SAM and OAM simultaneously, the optomechanical motions of these particles become more versatile. For example, utilizing Bessel beam to capture a single GNP or GNR onto an orbit to perform an orbital revolution with spinning [ 25 , 26 ]. The observation of spin-orbit interaction (SOI) of the electromagnetic (EM) field through the light interacting with these plasmonic nanostructures is noted; a part of SAM of optical vortex may be converted into OAM via the light-matter interaction [ 23 , 26 ]. Moreover, a plasmonic dimer, two optically bound SNPs, may perform an orbital motion in an optical field with transverse phase gradient [ 5 , 28 ]. In this paper, we focus on the study of the two-dimensional (2D) planar motions of two identical GNPs manipulated by different-order Bessel beams. We use the multiple multipole (MMP) method to simulate the coupled EM field of two discrete GNPs interacting with an incident Bessel beam, and then utilize the results to obtain the optical force and torque upon individual GNP [ 14 , 16 , 23 , 26 ]. Subsequently, we calculate the trajectories of these GNPs based on their equations of motion in terms of the optical force and the drag fore of fluid. Furthermore, the trajectory of the two GNPs’ COM will be analyzed. In addition, the terminal spin of each GNP due to the balance of the optical torque with the drag torque will be investigated. Figure 1 illustrates the configuration of the motions (spin, rotation and orbital revolution) of a GNP dimer manipulated by a Bessel beam. Herein, we assume that the sizes of both spherical GNPs are identical and the medium is water with a refractive index of n = 1.33. The trajectories of the two GNPs and their COM are depicted in Fig. 1 for demonstration. Due to the optical binding between the two GNPs, they will be bound together eventually to form a dimer with a stable distance about λ/n , regardless of their initial positions [ 5 , 16 , 28 ]. Additionally, the dimer may undergo rotation around its COM, which in turn experiences an orbital revolution around the z -axis, aligning with the optical axis of a normally incident Bessel beam. The size-dependent rigid-body rotation of the dimer will also be studied. Moreover, a precession in COM’s orbital revolution may be induced, if the directions of rotation and revolution are opposite. Method We use the MMP method to simulate the coupled EM field of two discrete GNPs interacting with an incident Bessel beam, and utilize the results to obtain the optical force upon individual GNP. In this paper, we assume that a pair of GNPs are constrained within a specific plane by a glass substrate, enabling 2D planar motion. The optical forces F i ( i = 1 or 2) exerted on the two discrete GNPs can be obtained by calculating the surface integral of Maxwell’s stress tensor T in terms of EM field, $${{\\mathbf{F}}_i}=\\int_{{{S_i}}}^{{}} {{\\mathbf{n}} \\cdot {\\mathbf{T}}dS}$$ 1 and $${\\mathbf{T}}=\\frac{1}{2}\\operatorname{Re} \\left( {\\varepsilon {\\mathbf{E}} \\otimes \\overline {{\\mathbf{E}}} +\\mu {\\mathbf{H}} \\otimes \\overline {{\\mathbf{H}}} - \\frac{1}{2}{\\mathbf{I}}\\left( {\\varepsilon {\\mathbf{E}} \\cdot \\overline {{\\mathbf{E}}} +\\mu {\\mathbf{H}} \\cdot \\overline {{\\mathbf{H}}} } \\right)} \\right)$$ 2 where the upper bar denotes the conjugate and Re is the real part. In Eq. ( 2 ), ε and µ are the permittivity and permeability of the medium, respectively. Subsequently, we calculate the equation of motion of individual GNP in terms of the optical force and the reactive drag fore of fluid. GNPs’ motions obey the equation of motion based on Newton’s second law as follows, $${m_i}\\frac{{{d^2}{{\\mathbf{x}}_i}}}{{d{t^2}}}= - {\\eta _i}\\frac{{d{{\\mathbf{x}}_i}}}{{dt}}+{{\\mathbf{F}}_i}$$ 3 , In the above equation, x i ( i = 1 or 2) represents the position vector of the i -th GNP’s center, and F i is the optical force upon it, where - η i d x i / dt is the drag force of medium according to Stokes’ first law. Here, the coefficient η i is 6π µ v a i with the viscosity µ v and the radius a i of the i -th GNP. Furthermore, the trajectory of the two GNPs’ COM will be analyzed. The position vector and velocity of COM are defined as $${{\\mathbf{x}}_c}=\\frac{1}{M}\\sum\\nolimits_{i}^{{}} {{m_i}{{\\mathbf{x}}_i}}$$ 4 , $${{\\mathbf{v}}_c}=\\frac{1}{M}\\sum\\nolimits_{i}^{{}} {{m_i}\\frac{{d{{\\mathbf{x}}_i}}}{{dt}}}$$ 5 , where M is the total mass of these GNPs; M = Σ m i . The z component of the angular speed of orbital revolution Ω c is defined as $${\\Omega _c}=\\frac{{{{\\mathbf{x}}_c} \\times {{\\mathbf{v}}_c}}}{{{{\\left| {{{\\mathbf{x}}_c}} \\right|}^2}}} \\cdot {{\\mathbf{e}}_z}$$ 6 , where \\({\\left|{\\mathbf{x}}_{c}\\right|}^{2}\\) is the radius of the orbit. The z component of the angular speed of rotation Ω p of the i -th GNP w. r. t. COM is $${\\Omega _p}=\\frac{{\\left( {{{\\mathbf{x}}_i} - {{\\mathbf{x}}_c}} \\right) \\times \\left( {{{\\mathbf{v}}_i} - {{\\mathbf{v}}_c}} \\right)}}{{{{\\left| {{{\\mathbf{x}}_i} - {{\\mathbf{x}}_c}} \\right|}^2}}} \\cdot {{\\mathbf{e}}_z}$$ 7 The optical spin torque M S,i on the i -th GNP is defined $${{\\mathbf{M}}_{S,i}}=\\int_{{{S_i}}}^{{}} {{\\mathbf{r}} \\times \\left( {{\\mathbf{T}} \\cdot {\\mathbf{n}}} \\right)dS}$$ 8 , where r is the position vector of any point on the surface of S i with respect to the center of each GNP. According to Stokes’ second law, the z component of the terminal spin angular speed of the i -th GNP is calculated by using optical spin torque M S,i $${\\omega _{S,i}}=\\frac{{{{\\mathbf{e}}_z} \\cdot {{\\mathbf{M}}_{S,i}}}}{{8\\pi {\\mu _v}a_{{}}^{3}}}$$ 9 , where a = a i . In addition, the optical rotation torque M O,i with respect to COM on the i -th GNP is defined $${{\\mathbf{M}}_{O,i}}=\\left( {{{\\mathbf{x}}_i} - {{\\mathbf{x}}_c}} \\right) \\times \\int_{{{S_i}}}^{{}} {{\\mathbf{T}} \\cdot {\\mathbf{n}}dS}$$ 10 . Based on Eq. ( 10 ), the terminal rotation angular speed can be determined by assuming equilibrium between the total optical rotation torque and the total resistant torque caused by the drag forces from the medium upon the two GNPs. The electric field of an incident right-handed (RH) Bessel beam of the l -th order in the cylindrical coordinates, $${{\\mathbf{E}}^i}\\left( {r,\\theta ,z} \\right)={E_0}\\left( {j\\frac{{{k_z}}}{{{k_r}}}{J_l}\\left( {{k_r}r} \\right){e^{jl\\theta }}\\left( {{{\\mathbf{e}}_x}+j{{\\mathbf{e}}_y}} \\right)+{J_{l+1}}\\left( {{k_r}r} \\right){e^{j\\left( {l+1} \\right)\\theta }}{{\\mathbf{e}}_z}} \\right){e^{j{k_z}z}}$$ 11 , where the superscript i represents incident light, E 0 is the amplitude, and J l is the Bessel function of the l -th order. Here, the time-harmonic factor of Maxwell’s equations is exp (- jωt ); ω is the angular frequency. The cone angle α of Bessel beam is \\(\\alpha ={\\tan ^{ - 1}}\\left( {{{{k_r}} \\mathord{\\left/ {\\vphantom {{{k_r}} {{k_z}}}} \\right. \\kern-0pt} {{k_z}}}} \\right)\\) , where the wavenumber k is \\({k^2}=k_{r}^{2}+k_{z}^{2}\\) . Similarly, the electric field of an incident left-handed (LH) Bessel beam is $${{\\mathbf{E}}^i}\\left( {r,\\theta ,z} \\right)={E_0}\\left( {j\\frac{{{k_z}}}{{{k_r}}}{J_l}\\left( {{k_r}r} \\right){e^{jl\\theta }}\\left( {{{\\mathbf{e}}_x} - j{{\\mathbf{e}}_y}} \\right) - {J_{l - 1}}\\left( {{k_r}r} \\right){e^{j\\left( {l - 1} \\right)\\theta }}{{\\mathbf{e}}_z}} \\right){e^{j{k_z}z}}$$ 12 . Results and Discussion The motions of GNP dimers of different sizes manipulated by LH or RH Bessel beams of different orders are studied. The motions of two individual and identical GNPs in water under the irradiation of 800-nm Bessel beams of different orders, l = 0, 1 or 2, propagating in z direction are simulated by using particles’ dynamic equations of motion in terms of the optical forces, Eq. ( 3 ). The optical forces exerted on the two GNPs are determined by calculating the surface integrals of Maxwell’s stress tensors of each GNP, where the EM field is simulated by MMP method [ 23 , 26 ]. The wavelength of light is 800 nm; the effective wavelength in water is about 601 nm. The dielectric constant of gold at a wavelength of 800 nm, cited from Ref. [ 40 ], is utilized for the following simulation. The intensity distributions of Bessel beams of l = 0, 1 or 2 are plotted in Fig. S1 (Supplementary Material), where the radii of the first ring with the peak intensity of these Bessel beams are 2115 nm, 1020 nm and 1690 nm, respectively. Throughout this paper, the amplitude of Bessel beam is \\({E_0}={\\text{11}}{\\text{.89 (}}\\frac{{{\\text{MV}}}}{{\\text{m}}}{\\text{)}}\\) . Figure 2 a shows the trajectory of the two identical GNPs of a = 100 nm irradiated by RH Bessel beams of l = 0 with a cone angle of α = 10°. The result of a = 150 nm is shown in Fig. 2 b. Our data shows that regardless of the initial positions of the two GNPs they will inevitably become optically bound to form a dimer rotating around their COM. Additionally, the COM simultaneously performs an orbital revolution around the optical axis. The distance between the two GNPs is approximately 600 nm, which is close to the effective wavelength of light in water. The lateral scattering field from two initially separated GNPs provides optical binding to confine them for the formation of a dimer with a fixed distance. In addition, each GNP spins individually. The corresponding orbital radius of COM, angular speeds of spin, rotation and revolution of GNPs are shown in Figs. 2 c and 2 d. Although the Bessel beam of l = 0 only has SAM without OAM, the orbital revolution of dimer’s COM is still induced. It implies that a part of SAM of light is converted into OAM exerted on the two GNPs through the light-matter interaction. For smaller GNPs ( a = 100 nm), the rotational direction of the dimer is consistent with the handedness of light. However, the rotational direction of a larger GNP ( a = 150 nm) dimer is against the direction of SAM, because the negative optical torque is induced. This phenomenon has been discussed in the previous studies [ 17 – 19 ]. The average terminal angular speeds of spin, rotation and revolution of GNP dimer induced by Bessel beams of l = 0, 1 and 2 are listed in Table I, when the steady-state motions of the two light-driven GNPs are reached. The dynamic motions of the two GNPs induced by Bessel beam of l = 0 are referred to Visualization 1. In contrast, the orbital revolution of a smaller GNP dimer of a = 50 nm is nearly absent due to the small scattering cross section, as shown in Fig. S2 (Supplementary Material); the SOI effect can almost be neglected. The terminal angular speeds of a = 50 nm are listed in Table SI (Supplementary Material). Figures 3 a to 3 d show the trajectories of two GNPs of a = 100 nm or 150 nm irradiated by a LH or RH Bessel beams of l = 1 with α = 10°. Since the order l (topological charge) is positive, the direction of the orbital revolution of the dimer’s COM is CCW. The corresponding orbital radius of COM and the angular speeds of spin, rotation and revolution of GNP dimer versus time versus time are depicted in Figs. 3 e to 3 h. For a LH Bessel beam of l = 1 irradiating two smaller initially separated GNPs ( a = 100 nm), the rotational direction of the dimer is CW because of the LH Bessel beam, as shown in Figs. 3 a and 3 e. All the spins, dimer’s rotation and COM’s orbital revolution display quasi-periodic motions. The average rotation and revolution speeds of this dimer of a = 100 nm irradiated by LH Bessel beam are Ω p = -1.21 kHz and Ω c = 19.92 kHz, respectively, as shown in Fig. 3 e and Table I. The variations, shown in Figs. 3 a and 3 e, indicate that there is a precession in the orbital motion. This precession in COM’s orbital revolution could be due to the opposite directions of the rotation and the revolution. The periods of the variation in Fig. 3 e are 0.0237 ms for Ω c , Ω p and the orbital radius, and 0.0474 ms for ω s , respectively; the corresponding frequencies of fluctuation are 42.2 kHz and 21.1 kHz, respectively. For the case of RH Bessel beam of l = 1, and a = 100 nm, the rotation direction of this smaller dimer is CCW because of the RH Bessel beam. Since the directions of the rotation and revolution are consistent (CCW), this is no precession in the COM’s orbital revolution, resulting in a circle orbit for this dimer’s COM with equal angular speeds for both rotation and revolution, as shown in Figs. 3 b and 3 f. The dynamic motions of the two GNPs induced by Bessel beam of l = 1 are referred to Visualization 2. For a bigger GNP dimer ( a = 150 nm) irradiated by a RH Bessel beam of l = 1, the reverse rotation of dimer occurs due to the negative optical torque, leading to the inconsistent directions of the rotation (CW) and revolution (CCW), as shown in Figs. 3 d and 3 h. As a result, a precession in the orbital revolution is observed for this case. Again, the variations of orbits, shown in Figs. 3 d and 3 h, indicate the precession in the orbital motion. The periods of the variation in Fig. 3 h are 0.0173 ms for Ω c , Ω p and orbital radius, and 0.0346 ms for ω s , respectively; the corresponding frequencies of fluctuation are 57.8 kHz and 28.9 kHz, respectively. The average rotation and revolution angular speeds of this dimer of a = 150 nm irradiated by RH Bessel beam are − 13.98 kHz and 15.49 kHz, respectively (Fig. 3 (h) and Table I). This inconsistent behavior of the bigger dimer’s rotation and revolution (Figs. 3 d and 3 h) is different from that of a smaller one (Figs. 3 b and 3 f), as both are irradiated by the same RH Bessel beam. On the other hand, Figs. 3 c and 3 g show that the trajectories of two GNPs are circles (inner and outer circles), as irradiated by LH Bessel beam, optically bound together to form a dimer performing a rigid-body rotation and circling motion. Hence, the directions and the angular speeds of the rotation of dimer’s COM and the orbital revolution are the same; CCW and 28.06 kHz. In contrast, for the case of a LH Bessel beam of l = 1, the directions of the rotation and revolution of 150-nm GNP dimer are consistent (CCW); the precession is absent as shown in Figs. 3 c and 3 g. The rotation direction of this bigger dimer opposes the handedness of Bessel beam, whereas the direction of the orbital revolution is guided by the OAM of a Bessel beam of a positive order. These complicated motions involve the SOI facilitated by light-scattering of the multimode of the two GNPs. Note that all these terminal angular speeds of GNP dimer, as listed in Table I, are linearly proportional to the intensity of Bessel beam. The results of a GNP dimer of a = 50 nm irradiated by a LH or RH Bessel beam of l = 1 are shown in Fig. S3 (Supplementary Material). Figure 4 a shows the optical force and torque exerted on two GNPs of a = 150 nm with a distance D irradiated by an 800-nm right-handed circularly polarized (RCP) plane wave propagating in the z direction. The first and second stable distances between them are 592 nm and 1179 nm, respectively, which are the points of zero cross with a negative slope of F r ; these values correspond to the integer multiples of the effective wavelength in the medium. Figure 4 b shows the size effect of GNP on the optical torque e z ⋅ M O driving the rotation of an optical bound dimer caused by a RCP plane wave. There are multiple turning points in the radius of the GNP for the alternating rotation of the dimer caused by the negative optical torque. There are two turning points of the GNP’s radius for the forward/reverse rotation, a = 122 nm and 179 nm. The negative optical torque upon a bigger plasmonic dimer provided by a RCP plane wave has been studied [ 19 ]. This phenomenon of the reverse rotation of dimer w. r. t. the handedness of light could be related to SOI. In contrast, the optical spin torque e z ⋅ M S is always positive for various size, induced by a RCP plane wave. The trajectories of a GNP dimer’s COM, manipulated by a LH or RH Bessel beam of l = 2 are shown in Fig. 5 . The corresponding orbital radius of COM and the angular speeds of spin, rotation and revolution of GNP dimer versus time are shown in Fig. S3 (Supplementary Material), and the terminal values are listed in Table I. No matter the initial positions of the two separate GNPs are, they will eventually come together to form a dimer by the optical binding and perform rigid-body motions. In addition, the COM moves in an orbital trajectory. For the cases of a smaller dimer ( a = 100 nm) irradiated by a LH Bessel beam of l = 2 or a larger dimer ( a = 150 nm) by a RH one, the directions of rotation and revolution are opposite, and a precession is induced, as shown in Figs. 5 a and 5 d. In contrast, for the cases of a smaller dimer ( a = 100 nm) irradiated by a RH Bessel beam or a larger dimer ( a = 150 nm) by a LH one, the directions of the rotation and revolution are the same. Therefore, there is no precession, as shown in Figs. 5 b and 5 c; the trajectories of the two GNPs are circles. The terminal angular speeds of the rotation and revolution of GNP dimer are the same constants, as shown in Fig. S4 (Supplementary Material) and Table I. Again, the reverse rigid-body rotation is observed for larger GNPs ( a = 150 nm), caused by the negative optical torque. This size-dependent behavior of dimer’s rotation induced by a Bessel beam of l = 2 is similar to that by a Bessel beam of l = 1, shown in Fig. 3 . We found that these rigid-body motions of a larger plasmonic dimer with a precession induced by a Bessel beam are more complicated than those observed by Refs. [ 5 , 28 ]. Table I. The average terminal angular speeds of spin, rotation and revolution of GNP dimer of a = 100 nm or 150 nm induced by LH/RH 800-nm Bessel beams of l = 0, 1, and 2 with a cone angle of α = 10°. (kHz) LH RH Order Radius a Ω C Ω P ω S Ω C Ω P ω S l = 0 100 nm -0.05 -0.05 -16.87, -16.86 0.05 0.05 16.87, 16.86 150 nm 0.19 0.19 -8.46, -8.40 -0.19 -0.19 8.46, 8.40 l = 1 100 nm 19.92 -1.21 -32.90 19.91 19.91 35.73, 32.66 150 nm 28.06 28.06 -18.64, -15.50 15.49 -13.98 17.15 l = 2 100 nm 10.18 -2.75 -23.06 11.47 11.47 26.00, 22.00 150 nm 12.75 12.75 -13.03, -10.25 7.89 -7.55 11.69 Conclusion The optomechanical 2D motions of an optically bound GNP dimer manipulated by Bessel beams of different orders were studied. We calculated the equation of motion of two individual GNPs in terms of the optical force and drag fore of fluid to analyze their trajectories. Our results demonstrate that they will come together eventually to form a dimer with a stable distance due to the optical binding between the two GNPs, regardless of their initial positions. Since a Bessel beam has OAM and SAM simultaneously, a complicated rigid-body motions of the GNP dimer are observed, including GNPs’ spin, dimer’s rotation and COM’s orbital revolution. The directions of the spin and the orbital revolution depend on the handedness and the sign of the order (topological charge) of Bessel beam, respectively. Nevertheless, the rotation direction of the dimer w. r. t. COM varies with the size of GNPs. In the case of a smaller dimer (e.g. a = 100 nm), the direction of dimer’s rotation is consistent with the handedness of the incident light. In contrast, the rotational direction of a larger dimer ( a = 150 nm) is opposite to that of the incident light’s handedness. This is because the negative optical torque upon a bigger GNP dimer is induced by CPL [ 19 ]. The phenomenon could be due to the SOI induced by the multimode of bigger GNPs. There are multiple turning points in the radius of the GNP for the alternating rotation. Furthermore, we found that a precession may be induced in COM’s orbital revolution as the directions of the rotation and the revolution are opposite. Our finding may provide an insight to the gear motions of GNP clusters manipulated by an optical vortex. Declarations Research funding : This work was supported by the National Science and Technology Council, Taiwan (MOST 110-2221-E-182-039-MY3, 111-2221-E-002-138). Author contributions : All authors have accepted responsibility for the entire content of this manuscript and approved its submission. Conflict of interest : Authors state no conflicts of interest. Data availability : The datasets generated and/or analyzed during the current study can be obtained on request from the corresponding author. References Lehmuskero A, Ogier R, Gschneidtner T, Johansson P, Käll M (2013) Ultra-fast spinning of gold nanoparticles in water using circularly polarized light. Nano Lett. 13(7):3129–3134. 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Yan Z, Scherer NF (2013) Optical vortex induced rotation of silver nanowires. J. Phys. Chem. Lett. 4:2937−2942. Simpson SH, Hanna S (2009) Optical angular momentum transfer by Laguerre–Gaussian beams. J. Opt. Soc. Am. A 26(3):625−638. Simpson SH, Hanna S (2010) Orbital motion of optically trapped particles in Laguerre–Gaussian beams. J. Opt. Soc. Am. A 27(9):2061−2071. Lehmuskero A, Li Y, Johansson P, Käll M (2014) Plasmonic particles set into fast orbital motion by an optical vortex beam. Opt Express 22(4):4349−4356. Mishra SR (1991) A vector wave analysis of a Bessel beam. Opt. Commun. 85:159−161. Garces-Chavez V, McGloin D, Melville H, et al. (2002) Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam. Nature 419:145−147. Mitri FG (2017) Reverse orbiting and spinning of a Rayleigh dielectric spheroid in a J0 Bessel optical beam. J. Opt. Soc. Am. B 34:2169−2178. Hakobyan D, Brasselet E (2015) Optical torque reversal and spin-orbit rotational Doppler shift experiments. Opt. Express 23:31230−31239. Johnson PB, Christy RW (1972) Optical constants of the noble metals. Phys. Rev. B 6:4370−4379. Supplementary Video Supplementary Video is not available with this version Additional Declarations No competing interests reported. Supplementary Files Supplementarymaterials8.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-4386749\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":true,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":302298637,\"identity\":\"c9840c4d-f74c-46dd-8b92-ef337f1fe389\",\"order_by\":0,\"name\":\"Chao-Kang Liu\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"National Taiwan University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Chao-Kang\",\"middleName\":\"\",\"lastName\":\"Liu\",\"suffix\":\"\"},{\"id\":302298638,\"identity\":\"1614eb10-dc29-4c81-81c7-6e1956b5ed4b\",\"order_by\":1,\"name\":\"Yun-Cheng Ku\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"National Taiwan University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Yun-Cheng\",\"middleName\":\"\",\"lastName\":\"Ku\",\"suffix\":\"\"},{\"id\":302298639,\"identity\":\"8c2d9bab-795f-4790-ba52-272b507afe4e\",\"order_by\":2,\"name\":\"Mao-Kuen Kuo\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"National Taiwan University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Mao-Kuen\",\"middleName\":\"\",\"lastName\":\"Kuo\",\"suffix\":\"\"},{\"id\":302298641,\"identity\":\"3a9cf1b5-dd36-43cb-9a03-ebe6d9899bdd\",\"order_by\":3,\"name\":\"Jiunn-Woei Liaw\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+0lEQVRIiWNgGAWjYHACZoYHBgwM/EDWgQdQIQmCWhKAWiQbgFoSEFoMCGgBkgYHgARRWnTbzz42SCiwS9x87fBDoC12dhsOMB+8zcPwJ7EBhxazM+nGCQkGyYnbbqcZALUkJ284wJZszcNggFvLgTTmAwkGzLnbbieAtBxINjjAYyYN1JKLU8v5ZyAt9bmbZ6d/gGrh/4Zfy400ZqDDDudukM4B22IHtIWNgJZnzAYJBsfrZ9zOKQBal5wgeZjN2HKOgXE9boelMUt8+FNtzD87ffOHDxV29nzHmx/eeFMhZ4xDBzowYEhsYIYwiAf2JKgdBaNgFIyCEQIAVM9aD7ePCYYAAAAASUVORK5CYII=\",\"orcid\":\"\",\"institution\":\"Chang Gung University\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Jiunn-Woei\",\"middleName\":\"\",\"lastName\":\"Liaw\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2024-05-08 05:46:50\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-4386749/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-4386749/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":56621937,\"identity\":\"1210f6d6-4a63-4245-ac53-7ec2dfe4e603\",\"added_by\":\"auto\",\"created_at\":\"2024-05-16 18:12:36\",\"extension\":\"png\",\"order_by\":1,\"title\":\"Figure 1\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":1193984,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eConfiguration of optically bound GNP dimer’s motions of spin, rotation and orbital revolution induced by a Bessel beam. If the directions of rotation and revolution are opposite, a precession is induced in COM’s orbital revolution.\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"floatimage1.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4386749/v1/864bdb6225453861ac5ab110.png\"},{\"id\":56621942,\"identity\":\"e1be1478-244f-425c-ba8d-0fece723cc55\",\"added_by\":\"auto\",\"created_at\":\"2024-05-16 18:12:37\",\"extension\":\"png\",\"order_by\":2,\"title\":\"Figure 2\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":485123,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eThe trajectories of GNP dimer of (a) \\u003cem\\u003ea\\u003c/em\\u003e= 100 nm and (b) 150 nm, respectively, irradiated by RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e= 0. (c) and (b) The corresponding orbital radius of COM, and the angular speeds of spin, rotation and revolution of GNPs of \\u003cem\\u003ea\\u003c/em\\u003e= 100 nm and 150 nm, respectively, versus time. The orbital radius of COM is about 2.09 mm for both cases.\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"floatimage2.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4386749/v1/786017956694524b64e961f1.png\"},{\"id\":56622231,\"identity\":\"e93530bf-d968-4a3d-8cac-f3b7322ada69\",\"added_by\":\"auto\",\"created_at\":\"2024-05-16 18:20:36\",\"extension\":\"png\",\"order_by\":3,\"title\":\"Figure 3\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":1525403,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eThe trajectories of a smaller GNP dimer of \\u003cem\\u003ea\\u003c/em\\u003e=100 nm, irradiated by a (a) LH and (b) RH 800-nm Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e= 1 with a cone angle \\u003cem\\u003eα = \\u003c/em\\u003e10°. The trajectories of a larger GNP dimer of \\u003cem\\u003ea\\u003c/em\\u003e= 150 nm, irradiated by a (c) LH and (d) RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e= 1. Blue and red lines: trajectories of two GNPs, and black line: trajectory of their COM. (e) to (h) The orbital radius of COM and the angular speeds of spin, rotation and revolution of GNP dimer of \\u003cem\\u003ea\\u003c/em\\u003e= 100 nm or 150 nm versus time, corresponding to (a) to (d), respectively. The periods of the variation in (e) are 0.0237 ms (orbital radius, rotation and revolution), and 0.0474 ms (spin), respectively. The periods of the variation in (h) are 0.0173 ms (orbital radius, rotation and revolution), and 0.0346 ms (spin), respectively.\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"floatimage3.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4386749/v1/9b71b4266fb6ed1554409652.png\"},{\"id\":56621941,\"identity\":\"1afe3af1-af6c-4be2-87b4-6be812b04ffb\",\"added_by\":\"auto\",\"created_at\":\"2024-05-16 18:12:37\",\"extension\":\"png\",\"order_by\":4,\"title\":\"Figure 4\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":277186,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003e(a) Optical force and torque on GNP dimer of\\u003cem\\u003e a\\u003c/em\\u003e= 150 nm irradiated by an 800-nm RCP plane wave versus distance \\u003cem\\u003eD \\u003c/em\\u003ebetween two GNPs. The first and second stable distances are 592 nm and 1179 nm, respectively; the points with zero cross and a negative slope of \\u003cem\\u003eF\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003er\\u003c/em\\u003e\\u003c/sub\\u003e. (b) The first stable distance \\u003cem\\u003eD\\u003c/em\\u003e and the corresponding optical torque (green line) on a GNP dimer of various the radius \\u003cem\\u003ea\\u003c/em\\u003e of GNP. The grey zone indicates the negative optical torque \\u003cstrong\\u003ee\\u003c/strong\\u003e\\u003csub\\u003ez\\u003c/sub\\u003e×\\u003cstrong\\u003eM\\u003c/strong\\u003e\\u003csub\\u003e\\u003cem\\u003eO\\u003c/em\\u003e\\u003c/sub\\u003e, which can induce a reverse rigid-body rotation of dimer \\u003cem\\u003ew. r. t. \\u003c/em\\u003ethe handedness of the incident light; the negative optical-torque zone is between \\u003cem\\u003ea\\u003c/em\\u003e= 123 nm and 178 nm.\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"floatimage4.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4386749/v1/a3c5c87b621dfd6569e92f87.png\"},{\"id\":56621939,\"identity\":\"ef46d9ad-c241-4884-b22c-ea7ae3caf07f\",\"added_by\":\"auto\",\"created_at\":\"2024-05-16 18:12:37\",\"extension\":\"png\",\"order_by\":5,\"title\":\"Figure 5\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":900760,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eThe trajectories of two GNPs of \\u003cem\\u003ea\\u003c/em\\u003e= 100 nm and the COM, irradiated by a \\u0026nbsp;(a) LH and (b) RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e= 2, where \\u003cem\\u003el\\u003c/em\\u003e= 800 nm and \\u003cem\\u003eα = \\u003c/em\\u003e10°. The motions of two GNPs of \\u003cem\\u003ea\\u003c/em\\u003e= 150 nm and the COM, irradiated by a (c) LH and (d) RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e= 2. Blue and red lines: trajectories of two GNPs, and black line: trajectory of their COM.\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"floatimage5.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4386749/v1/3c4357785e5f7e0353e4ccf5.png\"},{\"id\":58044218,\"identity\":\"91cc6af4-d270-4fef-a92f-a2c5cd1ef966\",\"added_by\":\"auto\",\"created_at\":\"2024-06-10 11:10:18\",\"extension\":\"pdf\",\"order_by\":0,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":5928479,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"manuscript.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4386749/v1/69862693-69bd-4907-9320-9d4684be16f2.pdf\"},{\"id\":56622232,\"identity\":\"6c4aa28b-d641-437e-8f5a-09cd40e1ec3e\",\"added_by\":\"auto\",\"created_at\":\"2024-05-16 18:20:37\",\"extension\":\"docx\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"supplement\",\"size\":6376744,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"Supplementarymaterials8.docx\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4386749/v1/be5cb8b650abf5c9a6660f53.docx\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"Optomechanical Motions of Gold Dimer’s Spin, Rotation and Revolution Manipulated by Bessel Beam\",\"fulltext\":[{\"header\":\"Introduction\",\"content\":\"\\u003cp\\u003eIn recent decades, the motions of microparticels or nanoparticles manipulated by light through the light-matter interaction has gained significant attention [\\u003cspan additionalcitationids=\\\"CR2 CR3 CR4 CR5 CR6 CR7 CR8\\\" citationid=\\\"CR1\\\" class=\\\"CitationRef\\\"\\u003e1\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR9\\\" class=\\\"CitationRef\\\"\\u003e9\\u003c/span\\u003e]. In particular, the study of using plasmon-enhanced optical force to manipulate gold nanoparticles (GNPs) or silver nanoparticles (SNPs) for optical tweezers attracted a lot of attention [\\u003cspan citationid=\\\"CR1\\\" class=\\\"CitationRef\\\"\\u003e1\\u003c/span\\u003e, \\u003cspan additionalcitationids=\\\"CR6 CR7 CR8 CR9 CR10 CR11 CR12 CR13 CR14 CR15\\\" citationid=\\\"CR5\\\" class=\\\"CitationRef\\\"\\u003e5\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR16\\\" class=\\\"CitationRef\\\"\\u003e16\\u003c/span\\u003e]. For example, a laser beam, a linearly polarized light (LPL), can induce an optical binding for a pair of GNPs with a stable distance of integer multiples of the effective wavelength of light in the medium, where the central line of the dimer is perpendicular to the polarization of LPL [\\u003cspan citationid=\\\"CR11\\\" class=\\\"CitationRef\\\"\\u003e11\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR12\\\" class=\\\"CitationRef\\\"\\u003e12\\u003c/span\\u003e]. In addition, a circularly polarized light (CPL) can induce the spin of a single GNP or gold nanorod (GNR) significantly due to the absorption of photon\\u0026rsquo;s spin angular momentum (SAM) [\\u003cspan citationid=\\\"CR1\\\" class=\\\"CitationRef\\\"\\u003e1\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR3\\\" class=\\\"CitationRef\\\"\\u003e3\\u003c/span\\u003e, \\u003cspan additionalcitationids=\\\"CR14\\\" citationid=\\\"CR13\\\" class=\\\"CitationRef\\\"\\u003e13\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR15\\\" class=\\\"CitationRef\\\"\\u003e15\\u003c/span\\u003e]. Although a CPL has only SAM but without orbital angular momentum (OAM) intrinsically, CPL may induce the optical binding between two individual GNPs to form a dimer with a stable distance, and drives them to rotate around their center of mass (COM) accompanied with individual spinning [\\u003cspan citationid=\\\"CR16\\\" class=\\\"CitationRef\\\"\\u003e16\\u003c/span\\u003e]. Of interesting, the rotation\\u0026rsquo;s direction of dimer not only depends on the handedness of CPL but also the size of GNPs [\\u003cspan additionalcitationids=\\\"CR18\\\" citationid=\\\"CR17\\\" class=\\\"CitationRef\\\"\\u003e17\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR19\\\" class=\\\"CitationRef\\\"\\u003e19\\u003c/span\\u003e]. For a smaller GNP/SNP dimer, the rotational direction is always consistent with the handedness of CPL. In contrast, for a larger GNP/SNP dimer a reverse rigid-body rotation around the COM may be induced due to the negative optical torque caused by higher-order modes [\\u003cspan additionalcitationids=\\\"CR18\\\" citationid=\\\"CR17\\\" class=\\\"CitationRef\\\"\\u003e17\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR19\\\" class=\\\"CitationRef\\\"\\u003e19\\u003c/span\\u003e]. Moreover, using CPL to manipulate multiple SNPs, GNPs or GNRs have also been studied [\\u003cspan citationid=\\\"CR4\\\" class=\\\"CitationRef\\\"\\u003e4\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR7\\\" class=\\\"CitationRef\\\"\\u003e7\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR15\\\" class=\\\"CitationRef\\\"\\u003e15\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR18\\\" class=\\\"CitationRef\\\"\\u003e18\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR20\\\" class=\\\"CitationRef\\\"\\u003e20\\u003c/span\\u003e]. For example, the planetary gear motion of optical bound hexagonal lattice composed of multiply SNPs driven by CPL were investigated by Refs. [\\u003cspan citationid=\\\"CR7\\\" class=\\\"CitationRef\\\"\\u003e7\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR15\\\" class=\\\"CitationRef\\\"\\u003e15\\u003c/span\\u003e].\\u003c/p\\u003e \\u003cp\\u003eRecently, the study of using an optical vortex, e.g. Laguerre-Gaussian beam or Bessel beam, to manipulate dielectric microparticles, GNPs or SNPs has also drawn considerable attention [\\u003cspan additionalcitationids=\\\"CR22 CR23 CR24 CR25 CR26 CR27 CR28 CR29 CR30 CR31 CR32 CR33 CR34 CR35 CR36 CR37 CR38\\\" citationid=\\\"CR21\\\" class=\\\"CitationRef\\\"\\u003e21\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR39\\\" class=\\\"CitationRef\\\"\\u003e39\\u003c/span\\u003e]. Since an optical vortex, a structured light, possesses both SAM and OAM simultaneously, the optomechanical motions of these particles become more versatile. For example, utilizing Bessel beam to capture a single GNP or GNR onto an orbit to perform an orbital revolution with spinning [\\u003cspan citationid=\\\"CR25\\\" class=\\\"CitationRef\\\"\\u003e25\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR26\\\" class=\\\"CitationRef\\\"\\u003e26\\u003c/span\\u003e]. The observation of spin-orbit interaction (SOI) of the electromagnetic (EM) field through the light interacting with these plasmonic nanostructures is noted; a part of SAM of optical vortex may be converted into OAM via the light-matter interaction [\\u003cspan citationid=\\\"CR23\\\" class=\\\"CitationRef\\\"\\u003e23\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR26\\\" class=\\\"CitationRef\\\"\\u003e26\\u003c/span\\u003e]. Moreover, a plasmonic dimer, two optically bound SNPs, may perform an orbital motion in an optical field with transverse phase gradient [\\u003cspan citationid=\\\"CR5\\\" class=\\\"CitationRef\\\"\\u003e5\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR28\\\" class=\\\"CitationRef\\\"\\u003e28\\u003c/span\\u003e].\\u003c/p\\u003e \\u003cp\\u003eIn this paper, we focus on the study of the two-dimensional (2D) planar motions of two identical GNPs manipulated by different-order Bessel beams. We use the multiple multipole (MMP) method to simulate the coupled EM field of two discrete GNPs interacting with an incident Bessel beam, and then utilize the results to obtain the optical force and torque upon individual GNP [\\u003cspan citationid=\\\"CR14\\\" class=\\\"CitationRef\\\"\\u003e14\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR16\\\" class=\\\"CitationRef\\\"\\u003e16\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR23\\\" class=\\\"CitationRef\\\"\\u003e23\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR26\\\" class=\\\"CitationRef\\\"\\u003e26\\u003c/span\\u003e]. Subsequently, we calculate the trajectories of these GNPs based on their equations of motion in terms of the optical force and the drag fore of fluid. Furthermore, the trajectory of the two GNPs\\u0026rsquo; COM will be analyzed. In addition, the terminal spin of each GNP due to the balance of the optical torque with the drag torque will be investigated. Figure\\u0026nbsp;\\u003cspan refid=\\\"Fig1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e illustrates the configuration of the motions (spin, rotation and orbital revolution) of a GNP dimer manipulated by a Bessel beam. Herein, we assume that the sizes of both spherical GNPs are identical and the medium is water with a refractive index of \\u003cem\\u003en\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1.33. The trajectories of the two GNPs and their COM are depicted in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e for demonstration. Due to the optical binding between the two GNPs, they will be bound together eventually to form a dimer with a stable distance about \\u003cem\\u003eλ/n\\u003c/em\\u003e, regardless of their initial positions [\\u003cspan citationid=\\\"CR5\\\" class=\\\"CitationRef\\\"\\u003e5\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR16\\\" class=\\\"CitationRef\\\"\\u003e16\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR28\\\" class=\\\"CitationRef\\\"\\u003e28\\u003c/span\\u003e]. Additionally, the dimer may undergo rotation around its COM, which in turn experiences an orbital revolution around the \\u003cem\\u003ez\\u003c/em\\u003e-axis, aligning with the optical axis of a normally incident Bessel beam. The size-dependent rigid-body rotation of the dimer will also be studied. Moreover, a precession in COM\\u0026rsquo;s orbital revolution may be induced, if the directions of rotation and revolution are opposite.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e\"},{\"header\":\"Method\",\"content\":\"\\u003cp\\u003eWe use the MMP method to simulate the coupled EM field of two discrete GNPs interacting with an incident Bessel beam, and utilize the results to obtain the optical force upon individual GNP. In this paper, we assume that a pair of GNPs are constrained within a specific plane by a glass substrate, enabling 2D planar motion. The optical forces \\u003cb\\u003eF\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e (\\u003cem\\u003ei\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1 or 2) exerted on the two discrete GNPs can be obtained by calculating the surface integral of Maxwell\\u0026rsquo;s stress tensor \\u003cb\\u003eT\\u003c/b\\u003e in terms of EM field,\\u003cdiv id=\\\"Equ1\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ1\\\" name=\\\"EquationSource\\\"\\u003e\\n$${{\\\\mathbf{F}}_i}=\\\\int_{{{S_i}}}^{{}} {{\\\\mathbf{n}} \\\\cdot {\\\\mathbf{T}}dS}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e1\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e \\u003cp\\u003eand\\u003cdiv id=\\\"Equ2\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ2\\\" name=\\\"EquationSource\\\"\\u003e\\n$${\\\\mathbf{T}}=\\\\frac{1}{2}\\\\operatorname{Re} \\\\left( {\\\\varepsilon {\\\\mathbf{E}} \\\\otimes \\\\overline {{\\\\mathbf{E}}} +\\\\mu {\\\\mathbf{H}} \\\\otimes \\\\overline {{\\\\mathbf{H}}} - \\\\frac{1}{2}{\\\\mathbf{I}}\\\\left( {\\\\varepsilon {\\\\mathbf{E}} \\\\cdot \\\\overline {{\\\\mathbf{E}}} +\\\\mu {\\\\mathbf{H}} \\\\cdot \\\\overline {{\\\\mathbf{H}}} } \\\\right)} \\\\right)$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e2\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e \\u003cp\\u003ewhere the upper bar denotes the conjugate and Re is the real part. In Eq.\\u0026nbsp;(\\u003cspan refid=\\\"Equ2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e), \\u003cem\\u003eε\\u003c/em\\u003e and \\u003cem\\u003e\\u0026micro;\\u003c/em\\u003e are the permittivity and permeability of the medium, respectively. Subsequently, we calculate the equation of motion of individual GNP in terms of the optical force and the reactive drag fore of fluid. GNPs\\u0026rsquo; motions obey the equation of motion based on Newton\\u0026rsquo;s second law as follows,\\u003cdiv id=\\\"Equ3\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ3\\\" name=\\\"EquationSource\\\"\\u003e\\n$${m_i}\\\\frac{{{d^2}{{\\\\mathbf{x}}_i}}}{{d{t^2}}}= - {\\\\eta _i}\\\\frac{{d{{\\\\mathbf{x}}_i}}}{{dt}}+{{\\\\mathbf{F}}_i}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e3\\u003c/div\\u003e\\u003c/div\\u003e,\\u003c/p\\u003e \\u003cp\\u003eIn the above equation, \\u003cb\\u003ex\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e (\\u003cem\\u003ei\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1 or 2) represents the position vector of the \\u003cem\\u003ei\\u003c/em\\u003e-th GNP\\u0026rsquo;s center, and \\u003cb\\u003eF\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e is the optical force upon it, where -\\u003cem\\u003eη\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e \\u003cem\\u003ed\\u003c/em\\u003e\\u003cb\\u003ex\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e /\\u003cem\\u003edt\\u003c/em\\u003e is the drag force of medium according to Stokes\\u0026rsquo; first law. Here, the coefficient \\u003cem\\u003eη\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e is 6π\\u003cem\\u003e\\u0026micro;\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ev\\u003c/em\\u003e\\u003c/sub\\u003e\\u003cem\\u003ea\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e with the viscosity \\u003cem\\u003e\\u0026micro;\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ev\\u003c/em\\u003e\\u003c/sub\\u003e and the radius \\u003cem\\u003ea\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e of the \\u003cem\\u003ei\\u003c/em\\u003e-th GNP. Furthermore, the trajectory of the two GNPs\\u0026rsquo; COM will be analyzed. The position vector and velocity of COM are defined as\\u003cdiv id=\\\"Equ4\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ4\\\" name=\\\"EquationSource\\\"\\u003e\\n$${{\\\\mathbf{x}}_c}=\\\\frac{1}{M}\\\\sum\\\\nolimits_{i}^{{}} {{m_i}{{\\\\mathbf{x}}_i}}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e4\\u003c/div\\u003e\\u003c/div\\u003e,\\u003cdiv id=\\\"Equ5\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ5\\\" name=\\\"EquationSource\\\"\\u003e\\n$${{\\\\mathbf{v}}_c}=\\\\frac{1}{M}\\\\sum\\\\nolimits_{i}^{{}} {{m_i}\\\\frac{{d{{\\\\mathbf{x}}_i}}}{{dt}}}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e5\\u003c/div\\u003e\\u003c/div\\u003e,\\u003c/p\\u003e \\u003cp\\u003ewhere \\u003cem\\u003eM\\u003c/em\\u003e is the total mass of these GNPs; \\u003cem\\u003eM\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;Σ \\u003cem\\u003em\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e. The \\u003cem\\u003ez\\u003c/em\\u003e component of the angular speed of orbital revolution Ω\\u003csub\\u003e\\u003cem\\u003ec\\u003c/em\\u003e\\u003c/sub\\u003e is defined as\\u003cdiv id=\\\"Equ6\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ6\\\" name=\\\"EquationSource\\\"\\u003e\\n$${\\\\Omega _c}=\\\\frac{{{{\\\\mathbf{x}}_c} \\\\times {{\\\\mathbf{v}}_c}}}{{{{\\\\left| {{{\\\\mathbf{x}}_c}} \\\\right|}^2}}} \\\\cdot {{\\\\mathbf{e}}_z}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e6\\u003c/div\\u003e\\u003c/div\\u003e,\\u003c/p\\u003e \\u003cp\\u003ewhere \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\({\\\\left|{\\\\mathbf{x}}_{c}\\\\right|}^{2}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the radius of the orbit. The \\u003cem\\u003ez\\u003c/em\\u003e component of the angular speed of rotation Ω\\u003csub\\u003e\\u003cem\\u003ep\\u003c/em\\u003e\\u003c/sub\\u003e of the \\u003cem\\u003ei\\u003c/em\\u003e-th GNP \\u003cem\\u003ew. r. t.\\u003c/em\\u003e COM is\\u003cdiv id=\\\"Equ7\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ7\\\" name=\\\"EquationSource\\\"\\u003e\\n$${\\\\Omega _p}=\\\\frac{{\\\\left( {{{\\\\mathbf{x}}_i} - {{\\\\mathbf{x}}_c}} \\\\right) \\\\times \\\\left( {{{\\\\mathbf{v}}_i} - {{\\\\mathbf{v}}_c}} \\\\right)}}{{{{\\\\left| {{{\\\\mathbf{x}}_i} - {{\\\\mathbf{x}}_c}} \\\\right|}^2}}} \\\\cdot {{\\\\mathbf{e}}_z}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e7\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e \\u003cp\\u003eThe optical spin torque \\u003cb\\u003eM\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003eS,i\\u003c/em\\u003e\\u003c/sub\\u003e on the \\u003cem\\u003ei\\u003c/em\\u003e-th GNP is defined\\u003cdiv id=\\\"Equ8\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ8\\\" name=\\\"EquationSource\\\"\\u003e\\n$${{\\\\mathbf{M}}_{S,i}}=\\\\int_{{{S_i}}}^{{}} {{\\\\mathbf{r}} \\\\times \\\\left( {{\\\\mathbf{T}} \\\\cdot {\\\\mathbf{n}}} \\\\right)dS}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e8\\u003c/div\\u003e\\u003c/div\\u003e,\\u003c/p\\u003e \\u003cp\\u003ewhere \\u003cb\\u003er\\u003c/b\\u003e is the position vector of any point on the surface of \\u003cem\\u003eS\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e with respect to the center of each GNP. According to Stokes\\u0026rsquo; second law, the \\u003cem\\u003ez\\u003c/em\\u003e component of the terminal spin angular speed of the \\u003cem\\u003ei\\u003c/em\\u003e-th GNP is calculated by using optical spin torque \\u003cb\\u003eM\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003eS,i\\u003c/em\\u003e\\u003c/sub\\u003e\\u003cdiv id=\\\"Equ9\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ9\\\" name=\\\"EquationSource\\\"\\u003e\\n$${\\\\omega _{S,i}}=\\\\frac{{{{\\\\mathbf{e}}_z} \\\\cdot {{\\\\mathbf{M}}_{S,i}}}}{{8\\\\pi {\\\\mu _v}a_{{}}^{3}}}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e9\\u003c/div\\u003e\\u003c/div\\u003e,\\u003c/p\\u003e \\u003cp\\u003ewhere \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;\\u003cem\\u003ea\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003ei\\u003c/em\\u003e\\u003c/sub\\u003e. In addition, the optical rotation torque \\u003cb\\u003eM\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003eO,i\\u003c/em\\u003e\\u003c/sub\\u003e with respect to COM on the \\u003cem\\u003ei\\u003c/em\\u003e-th GNP is defined\\u003cdiv id=\\\"Equ10\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ10\\\" name=\\\"EquationSource\\\"\\u003e\\n$${{\\\\mathbf{M}}_{O,i}}=\\\\left( {{{\\\\mathbf{x}}_i} - {{\\\\mathbf{x}}_c}} \\\\right) \\\\times \\\\int_{{{S_i}}}^{{}} {{\\\\mathbf{T}} \\\\cdot {\\\\mathbf{n}}dS}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e10\\u003c/div\\u003e\\u003c/div\\u003e.\\u003c/p\\u003e \\u003cp\\u003eBased on Eq.\\u0026nbsp;(\\u003cspan refid=\\\"Equ10\\\" class=\\\"InternalRef\\\"\\u003e10\\u003c/span\\u003e), the terminal rotation angular speed can be determined by assuming equilibrium between the total optical rotation torque and the total resistant torque caused by the drag forces from the medium upon the two GNPs.\\u003c/p\\u003e \\u003cp\\u003eThe electric field of an incident right-handed (RH) Bessel beam of the \\u003cem\\u003el\\u003c/em\\u003e-th order in the cylindrical coordinates,\\u003cdiv id=\\\"Equ11\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ11\\\" name=\\\"EquationSource\\\"\\u003e\\n$${{\\\\mathbf{E}}^i}\\\\left( {r,\\\\theta ,z} \\\\right)={E_0}\\\\left( {j\\\\frac{{{k_z}}}{{{k_r}}}{J_l}\\\\left( {{k_r}r} \\\\right){e^{jl\\\\theta }}\\\\left( {{{\\\\mathbf{e}}_x}+j{{\\\\mathbf{e}}_y}} \\\\right)+{J_{l+1}}\\\\left( {{k_r}r} \\\\right){e^{j\\\\left( {l+1} \\\\right)\\\\theta }}{{\\\\mathbf{e}}_z}} \\\\right){e^{j{k_z}z}}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e11\\u003c/div\\u003e\\u003c/div\\u003e,\\u003c/p\\u003e \\u003cp\\u003ewhere the superscript \\u003cem\\u003ei\\u003c/em\\u003e represents incident light, \\u003cem\\u003eE\\u003c/em\\u003e\\u003csub\\u003e0\\u003c/sub\\u003e is the amplitude, and \\u003cem\\u003eJ\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003el\\u003c/em\\u003e\\u003c/sub\\u003e is the Bessel function of the \\u003cem\\u003el\\u003c/em\\u003e-th order. Here, the time-harmonic factor of Maxwell\\u0026rsquo;s equations is \\u003cem\\u003eexp\\u003c/em\\u003e(-\\u003cem\\u003ejωt\\u003c/em\\u003e); \\u003cem\\u003eω\\u003c/em\\u003e is the angular frequency. The cone angle \\u003cem\\u003eα\\u003c/em\\u003e of Bessel beam is \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\alpha ={\\\\tan ^{ - 1}}\\\\left( {{{{k_r}} \\\\mathord{\\\\left/ {\\\\vphantom {{{k_r}} {{k_z}}}} \\\\right. \\\\kern-0pt} {{k_z}}}} \\\\right)\\\\)\\u003c/span\\u003e\\u003c/span\\u003e, where the wavenumber \\u003cem\\u003ek\\u003c/em\\u003e is \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\({k^2}=k_{r}^{2}+k_{z}^{2}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e. Similarly, the electric field of an incident left-handed (LH) Bessel beam is\\u003cdiv id=\\\"Equ12\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ12\\\" name=\\\"EquationSource\\\"\\u003e\\n$${{\\\\mathbf{E}}^i}\\\\left( {r,\\\\theta ,z} \\\\right)={E_0}\\\\left( {j\\\\frac{{{k_z}}}{{{k_r}}}{J_l}\\\\left( {{k_r}r} \\\\right){e^{jl\\\\theta }}\\\\left( {{{\\\\mathbf{e}}_x} - j{{\\\\mathbf{e}}_y}} \\\\right) - {J_{l - 1}}\\\\left( {{k_r}r} \\\\right){e^{j\\\\left( {l - 1} \\\\right)\\\\theta }}{{\\\\mathbf{e}}_z}} \\\\right){e^{j{k_z}z}}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e12\\u003c/div\\u003e\\u003c/div\\u003e.\\u003c/p\\u003e\"},{\"header\":\"Results and Discussion\",\"content\":\"\\u003cp\\u003eThe motions of GNP dimers of different sizes manipulated by LH or RH Bessel beams of different orders are studied. The motions of two individual and identical GNPs in water under the irradiation of 800-nm Bessel beams of different orders, \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0, 1 or 2, propagating in \\u003cem\\u003ez\\u003c/em\\u003e direction are simulated by using particles\\u0026rsquo; dynamic equations of motion in terms of the optical forces, Eq.\\u0026nbsp;(\\u003cspan refid=\\\"Equ3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003e). The optical forces exerted on the two GNPs are determined by calculating the surface integrals of Maxwell\\u0026rsquo;s stress tensors of each GNP, where the EM field is simulated by MMP method [\\u003cspan citationid=\\\"CR23\\\" class=\\\"CitationRef\\\"\\u003e23\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR26\\\" class=\\\"CitationRef\\\"\\u003e26\\u003c/span\\u003e]. The wavelength of light is 800 nm; the effective wavelength in water is about 601 nm. The dielectric constant of gold at a wavelength of 800 nm, cited from Ref. [\\u003cspan citationid=\\\"CR40\\\" class=\\\"CitationRef\\\"\\u003e40\\u003c/span\\u003e], is utilized for the following simulation. The intensity distributions of Bessel beams of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0, 1 or 2 are plotted in Fig. \\u003cspan refid=\\\"MOESM1\\\" class=\\\"InternalRef\\\"\\u003eS1\\u003c/span\\u003e (Supplementary Material), where the radii of the first ring with the peak intensity of these Bessel beams are 2115 nm, 1020 nm and 1690 nm, respectively. Throughout this paper, the amplitude of Bessel beam is \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\({E_0}={\\\\text{11}}{\\\\text{.89 (}}\\\\frac{{{\\\\text{MV}}}}{{\\\\text{m}}}{\\\\text{)}}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e. Figure\\u0026nbsp;\\u003cspan refid=\\\"Fig2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003ea shows the trajectory of the two identical GNPs of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm irradiated by RH Bessel beams of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0 with a cone angle of \\u003cem\\u003eα\\u0026thinsp;=\\u003c/em\\u003e\\u0026thinsp;10\\u0026deg;. The result of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm is shown in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003eb. Our data shows that regardless of the initial positions of the two GNPs they will inevitably become optically bound to form a dimer rotating around their COM. Additionally, the COM simultaneously performs an orbital revolution around the optical axis. The distance between the two GNPs is approximately 600 nm, which is close to the effective wavelength of light in water. The lateral scattering field from two initially separated GNPs provides optical binding to confine them for the formation of a dimer with a fixed distance. In addition, each GNP spins individually. The corresponding orbital radius of COM, angular speeds of spin, rotation and revolution of GNPs are shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003ec and \\u003cspan refid=\\\"Fig2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003ed. Although the Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0 only has SAM without OAM, the orbital revolution of dimer\\u0026rsquo;s COM is still induced. It implies that a part of SAM of light is converted into OAM exerted on the two GNPs through the light-matter interaction. For smaller GNPs (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm), the rotational direction of the dimer is consistent with the handedness of light. However, the rotational direction of a larger GNP (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm) dimer is against the direction of SAM, because the negative optical torque is induced. This phenomenon has been discussed in the previous studies [\\u003cspan additionalcitationids=\\\"CR18\\\" citationid=\\\"CR17\\\" class=\\\"CitationRef\\\"\\u003e17\\u003c/span\\u003e\\u0026ndash;\\u003cspan citationid=\\\"CR19\\\" class=\\\"CitationRef\\\"\\u003e19\\u003c/span\\u003e]. The average terminal angular speeds of spin, rotation and revolution of GNP dimer induced by Bessel beams of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0, 1 and 2 are listed in Table I, when the steady-state motions of the two light-driven GNPs are reached. The dynamic motions of the two GNPs induced by Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0 are referred to Visualization 1. In contrast, the orbital revolution of a smaller GNP dimer of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;50 nm is nearly absent due to the small scattering cross section, as shown in Fig. S2 (Supplementary Material); the SOI effect can almost be neglected. The terminal angular speeds of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;50 nm are listed in Table SI (Supplementary Material).\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003eFigures \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ea to \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ed show the trajectories of two GNPs of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm or 150 nm irradiated by a LH or RH Bessel beams of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1 with \\u003cem\\u003eα\\u0026thinsp;=\\u003c/em\\u003e\\u0026thinsp;10\\u0026deg;. Since the order \\u003cem\\u003el\\u003c/em\\u003e (topological charge) is positive, the direction of the orbital revolution of the dimer\\u0026rsquo;s COM is CCW. The corresponding orbital radius of COM and the angular speeds of spin, rotation and revolution of GNP dimer versus time versus time are depicted in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ee to \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eh. For a LH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1 irradiating two smaller initially separated GNPs (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm), the rotational direction of the dimer is CW because of the LH Bessel beam, as shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ea and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ee. All the spins, dimer\\u0026rsquo;s rotation and COM\\u0026rsquo;s orbital revolution display quasi-periodic motions. The average rotation and revolution speeds of this dimer of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm irradiated by LH Bessel beam are Ω\\u003csub\\u003e\\u003cem\\u003ep\\u003c/em\\u003e\\u003c/sub\\u003e= -1.21 kHz and Ω\\u003csub\\u003e\\u003cem\\u003ec\\u003c/em\\u003e\\u003c/sub\\u003e\\u0026thinsp;=\\u0026thinsp;19.92 kHz, respectively, as shown in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ee and Table I. The variations, shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ea and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ee, indicate that there is a precession in the orbital motion. This precession in COM\\u0026rsquo;s orbital revolution could be due to the opposite directions of the rotation and the revolution. The periods of the variation in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ee are 0.0237 ms for Ω\\u003csub\\u003e\\u003cem\\u003ec\\u003c/em\\u003e\\u003c/sub\\u003e, Ω\\u003csub\\u003e\\u003cem\\u003ep\\u003c/em\\u003e\\u003c/sub\\u003e and the orbital radius, and 0.0474 ms for \\u003cem\\u003eω\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003es\\u003c/em\\u003e\\u003c/sub\\u003e, respectively; the corresponding frequencies of fluctuation are 42.2 kHz and 21.1 kHz, respectively. For the case of RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1, and \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm, the rotation direction of this smaller dimer is CCW because of the RH Bessel beam. Since the directions of the rotation and revolution are consistent (CCW), this is no precession in the COM\\u0026rsquo;s orbital revolution, resulting in a circle orbit for this dimer\\u0026rsquo;s COM with equal angular speeds for both rotation and revolution, as shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eb and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ef. The dynamic motions of the two GNPs induced by Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1 are referred to Visualization 2.\\u003c/p\\u003e \\u003cp\\u003eFor a bigger GNP dimer (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm) irradiated by a RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1, the reverse rotation of dimer occurs due to the negative optical torque, leading to the inconsistent directions of the rotation (CW) and revolution (CCW), as shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ed and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eh. As a result, a precession in the orbital revolution is observed for this case. Again, the variations of orbits, shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ed and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eh, indicate the precession in the orbital motion. The periods of the variation in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eh are 0.0173 ms for Ω\\u003csub\\u003e\\u003cem\\u003ec\\u003c/em\\u003e\\u003c/sub\\u003e, Ω\\u003csub\\u003e\\u003cem\\u003ep\\u003c/em\\u003e\\u003c/sub\\u003e and orbital radius, and 0.0346 ms for \\u003cem\\u003eω\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003es\\u003c/em\\u003e\\u003c/sub\\u003e, respectively; the corresponding frequencies of fluctuation are 57.8 kHz and 28.9 kHz, respectively. The average rotation and revolution angular speeds of this dimer of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm irradiated by RH Bessel beam are \\u0026minus;\\u0026thinsp;13.98 kHz and 15.49 kHz, respectively (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003e(h) and Table I). This inconsistent behavior of the bigger dimer\\u0026rsquo;s rotation and revolution (Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ed and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eh) is different from that of a smaller one (Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eb and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ef), as both are irradiated by the same RH Bessel beam. On the other hand, Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ec and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eg show that the trajectories of two GNPs are circles (inner and outer circles), as irradiated by LH Bessel beam, optically bound together to form a dimer performing a rigid-body rotation and circling motion. Hence, the directions and the angular speeds of the rotation of dimer\\u0026rsquo;s COM and the orbital revolution are the same; CCW and 28.06 kHz. In contrast, for the case of a LH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1, the directions of the rotation and revolution of 150-nm GNP dimer are consistent (CCW); the precession is absent as shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003ec and \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003eg. The rotation direction of this bigger dimer opposes the handedness of Bessel beam, whereas the direction of the orbital revolution is guided by the OAM of a Bessel beam of a positive order. These complicated motions involve the SOI facilitated by light-scattering of the multimode of the two GNPs. Note that all these terminal angular speeds of GNP dimer, as listed in Table I, are linearly proportional to the intensity of Bessel beam. The results of a GNP dimer of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;50 nm irradiated by a LH or RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1 are shown in Fig. S3 (Supplementary Material).\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003eFigure\\u0026nbsp;\\u003cspan refid=\\\"Fig4\\\" class=\\\"InternalRef\\\"\\u003e4\\u003c/span\\u003ea shows the optical force and torque exerted on two GNPs of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm with a distance \\u003cem\\u003eD\\u003c/em\\u003e irradiated by an 800-nm right-handed circularly polarized (RCP) plane wave propagating in the \\u003cem\\u003ez\\u003c/em\\u003e direction. The first and second stable distances between them are 592 nm and 1179 nm, respectively, which are the points of zero cross with a negative slope of \\u003cem\\u003eF\\u003c/em\\u003e\\u003csub\\u003e\\u003cem\\u003er\\u003c/em\\u003e\\u003c/sub\\u003e; these values correspond to the integer multiples of the effective wavelength in the medium. Figure\\u0026nbsp;\\u003cspan refid=\\\"Fig4\\\" class=\\\"InternalRef\\\"\\u003e4\\u003c/span\\u003eb shows the size effect of GNP on the optical torque \\u003cb\\u003ee\\u003c/b\\u003e\\u003csub\\u003ez\\u003c/sub\\u003e\\u0026sdot;\\u003cb\\u003eM\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003eO\\u003c/em\\u003e\\u003c/sub\\u003e driving the rotation of an optical bound dimer caused by a RCP plane wave. There are multiple turning points in the radius of the GNP for the alternating rotation of the dimer caused by the negative optical torque. There are two turning points of the GNP\\u0026rsquo;s radius for the forward/reverse rotation, \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;122 nm and 179 nm. The negative optical torque upon a bigger plasmonic dimer provided by a RCP plane wave has been studied [\\u003cspan citationid=\\\"CR19\\\" class=\\\"CitationRef\\\"\\u003e19\\u003c/span\\u003e]. This phenomenon of the reverse rotation of dimer \\u003cem\\u003ew. r. t.\\u003c/em\\u003e the handedness of light could be related to SOI. In contrast, the optical spin torque \\u003cb\\u003ee\\u003c/b\\u003e\\u003csub\\u003ez\\u003c/sub\\u003e\\u0026sdot;\\u003cb\\u003eM\\u003c/b\\u003e\\u003csub\\u003e\\u003cem\\u003eS\\u003c/em\\u003e\\u003c/sub\\u003e is always positive for various size, induced by a RCP plane wave.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003eThe trajectories of a GNP dimer\\u0026rsquo;s COM, manipulated by a LH or RH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;2 are shown in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003e. The corresponding orbital radius of COM and the angular speeds of spin, rotation and revolution of GNP dimer versus time are shown in Fig. S3 (Supplementary Material), and the terminal values are listed in Table I. No matter the initial positions of the two separate GNPs are, they will eventually come together to form a dimer by the optical binding and perform rigid-body motions. In addition, the COM moves in an orbital trajectory. For the cases of a smaller dimer (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm) irradiated by a LH Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;2 or a larger dimer (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm) by a RH one, the directions of rotation and revolution are opposite, and a precession is induced, as shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003ea and \\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003ed. In contrast, for the cases of a smaller dimer (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm) irradiated by a RH Bessel beam or a larger dimer (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm) by a LH one, the directions of the rotation and revolution are the same. Therefore, there is no precession, as shown in Figs.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003eb and \\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003ec; the trajectories of the two GNPs are circles. The terminal angular speeds of the rotation and revolution of GNP dimer are the same constants, as shown in Fig. S4 (Supplementary Material) and Table I. Again, the reverse rigid-body rotation is observed for larger GNPs (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm), caused by the negative optical torque. This size-dependent behavior of dimer\\u0026rsquo;s rotation induced by a Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;2 is similar to that by a Bessel beam of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1, shown in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003e. We found that these rigid-body motions of a larger plasmonic dimer with a precession induced by a Bessel beam are more complicated than those observed by Refs. [\\u003cspan citationid=\\\"CR5\\\" class=\\\"CitationRef\\\"\\u003e5\\u003c/span\\u003e, \\u003cspan citationid=\\\"CR28\\\" class=\\\"CitationRef\\\"\\u003e28\\u003c/span\\u003e].\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e \\u003cp\\u003e \\u003cb\\u003eTable I.\\u003c/b\\u003e The average terminal angular speeds of spin, rotation and revolution of GNP dimer of \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm or 150 nm induced by LH/RH 800-nm Bessel beams of \\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0, 1, and 2 with a cone angle of \\u003cem\\u003eα\\u0026thinsp;=\\u003c/em\\u003e\\u0026thinsp;10\\u0026deg;.\\u003c/p\\u003e \\u003cp\\u003e \\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"No\\\" id=\\\"Taba\\\" border=\\\"1\\\"\\u003e \\u003ccolgroup cols=\\\"8\\\"\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c3\\\" colnum=\\\"3\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c4\\\" colnum=\\\"4\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c5\\\" colnum=\\\"5\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c6\\\" colnum=\\\"6\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c7\\\" colnum=\\\"7\\\"\\u003e\\u003c/div\\u003e \\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c8\\\" colnum=\\\"8\\\"\\u003e\\u003c/div\\u003e \\u003cthead\\u003e \\u003ctr\\u003e \\u003cth align=\\\"left\\\" colspan=\\\"2\\\" nameend=\\\"c2\\\" namest=\\\"c1\\\"\\u003e \\u003cp\\u003e(kHz)\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colspan=\\\"3\\\" nameend=\\\"c5\\\" namest=\\\"c3\\\"\\u003e \\u003cp\\u003eLH\\u003c/p\\u003e \\u003c/th\\u003e \\u003cth align=\\\"left\\\" colspan=\\\"3\\\" nameend=\\\"c8\\\" namest=\\\"c6\\\"\\u003e \\u003cp\\u003eRH\\u003c/p\\u003e \\u003c/th\\u003e \\u003c/tr\\u003e \\u003c/thead\\u003e \\u003ctbody\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e \\u003cp\\u003eOrder\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003eRadius \\u003cem\\u003ea\\u003c/em\\u003e\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003eΩ\\u003csub\\u003eC\\u003c/sub\\u003e\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003eΩ\\u003csub\\u003eP\\u003c/sub\\u003e\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e\\u003cem\\u003eω\\u003c/em\\u003e\\u003csub\\u003eS\\u003c/sub\\u003e\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003eΩ\\u003csub\\u003eC\\u003c/sub\\u003e\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003eΩ\\u003csub\\u003eP\\u003c/sub\\u003e\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c8\\\"\\u003e \\u003cp\\u003e\\u003cem\\u003eω\\u003c/em\\u003e\\u003csub\\u003eS\\u003c/sub\\u003e\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\" morerows=\\\"1\\\" rowspan=\\\"2\\\"\\u003e \\u003cp\\u003e\\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;0\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e100 nm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e-0.05\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e-0.05\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e-16.87, -16.86\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e0.05\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003e0.05\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c8\\\"\\u003e \\u003cp\\u003e16.87, 16.86\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e150 nm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e0.19\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e0.19\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e-8.46, -8.40\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e-0.19\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003e-0.19\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c8\\\"\\u003e \\u003cp\\u003e8.46, 8.40\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\" morerows=\\\"1\\\" rowspan=\\\"2\\\"\\u003e \\u003cp\\u003e\\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;1\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e100 nm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e19.92\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e-1.21\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e-32.90\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e19.91\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003e19.91\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c8\\\"\\u003e \\u003cp\\u003e35.73, 32.66\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e150 nm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e28.06\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e28.06\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e-18.64, -15.50\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e15.49\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003e-13.98\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c8\\\"\\u003e \\u003cp\\u003e17.15\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c1\\\" morerows=\\\"1\\\" rowspan=\\\"2\\\"\\u003e \\u003cp\\u003e\\u003cem\\u003el\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;2\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e100 nm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e10.18\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e-2.75\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e-23.06\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e11.47\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003e11.47\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c8\\\"\\u003e \\u003cp\\u003e26.00, 22.00\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003ctr\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e \\u003cp\\u003e150 nm\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e \\u003cp\\u003e12.75\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e \\u003cp\\u003e12.75\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e \\u003cp\\u003e-13.03, -10.25\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e \\u003cp\\u003e7.89\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c7\\\"\\u003e \\u003cp\\u003e-7.55\\u003c/p\\u003e \\u003c/td\\u003e \\u003ctd align=\\\"left\\\" colname=\\\"c8\\\"\\u003e \\u003cp\\u003e11.69\\u003c/p\\u003e \\u003c/td\\u003e \\u003c/tr\\u003e \\u003c/tbody\\u003e \\u003c/colgroup\\u003e \\u003c/table\\u003e\\u003c/div\\u003e \\u003c/p\\u003e\"},{\"header\":\"Conclusion\",\"content\":\"\\u003cp\\u003eThe optomechanical 2D motions of an optically bound GNP dimer manipulated by Bessel beams of different orders were studied. We calculated the equation of motion of two individual GNPs in terms of the optical force and drag fore of fluid to analyze their trajectories. Our results demonstrate that they will come together eventually to form a dimer with a stable distance due to the optical binding between the two GNPs, regardless of their initial positions. Since a Bessel beam has OAM and SAM simultaneously, a complicated rigid-body motions of the GNP dimer are observed, including GNPs\\u0026rsquo; spin, dimer\\u0026rsquo;s rotation and COM\\u0026rsquo;s orbital revolution. The directions of the spin and the orbital revolution depend on the handedness and the sign of the order (topological charge) of Bessel beam, respectively. Nevertheless, the rotation direction of the dimer \\u003cem\\u003ew. r. t.\\u003c/em\\u003e COM varies with the size of GNPs. In the case of a smaller dimer (e.g. \\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;100 nm), the direction of dimer\\u0026rsquo;s rotation is consistent with the handedness of the incident light. In contrast, the rotational direction of a larger dimer (\\u003cem\\u003ea\\u003c/em\\u003e\\u0026thinsp;=\\u0026thinsp;150 nm) is opposite to that of the incident light\\u0026rsquo;s handedness. This is because the negative optical torque upon a bigger GNP dimer is induced by CPL [\\u003cspan citationid=\\\"CR19\\\" class=\\\"CitationRef\\\"\\u003e19\\u003c/span\\u003e]. The phenomenon could be due to the SOI induced by the multimode of bigger GNPs. There are multiple turning points in the radius of the GNP for the alternating rotation. Furthermore, we found that a precession may be induced in COM\\u0026rsquo;s orbital revolution as the directions of the rotation and the revolution are opposite. Our finding may provide an insight to the gear motions of GNP clusters manipulated by an optical vortex.\\u003c/p\\u003e \"},{\"header\":\"Declarations\",\"content\":\"\\u003cp\\u003e\\u003cstrong\\u003eResearch funding\\u003c/strong\\u003e: This work was supported by the National Science and Technology Council, Taiwan (MOST\\u0026nbsp;110-2221-E-182-039-MY3, 111-2221-E-002-138).\\u0026nbsp;\\u003c/p\\u003e\\n\\u003cp\\u003e\\u003cstrong\\u003eAuthor contributions\\u003c/strong\\u003e:\\u0026nbsp;All authors have accepted responsibility for the entire content of this manuscript and approved its submission.\\u0026nbsp;\\u003c/p\\u003e\\n\\u003cp\\u003e\\u003cstrong\\u003eConflict of interest\\u003c/strong\\u003e: Authors state no conflicts of interest.\\u0026nbsp;\\u003c/p\\u003e\\n\\u003cp\\u003e\\u003cstrong\\u003eData availability\\u003c/strong\\u003e: The datasets generated and/or analyzed during the current study can be obtained on request from the corresponding author.\\u003c/p\\u003e\"},{\"header\":\"References\",\"content\":\"\\u003col\\u003e\\n\\u003cli\\u003eLehmuskero A, Ogier R, Gschneidtner T, Johansson P, K\\u0026auml;ll M (2013) Ultra-fast spinning of gold nanoparticles in water using circularly polarized light. 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(2002) Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam. Nature 419:145\\u0026minus;147.\\u003c/li\\u003e\\n\\u003cli\\u003eMitri FG (2017) Reverse orbiting and spinning of a Rayleigh dielectric spheroid in a J0 Bessel optical beam. J. Opt. Soc. Am. B 34:2169\\u0026minus;2178.\\u003c/li\\u003e\\n\\u003cli\\u003eHakobyan D, Brasselet E (2015) Optical torque reversal and spin-orbit rotational Doppler shift experiments. Opt. Express 23:31230\\u0026minus;31239.\\u003c/li\\u003e\\n\\u003cli\\u003eJohnson PB, Christy RW (1972) Optical constants of the noble metals. Phys. Rev. B\\u003cstrong\\u003e \\u003c/strong\\u003e6:4370\\u0026minus;4379.\\u003c/li\\u003e\\n\\u003c/ol\\u003e\"},{\"header\":\"Supplementary Video\",\"content\":\"\\u003cp\\u003eSupplementary Video is not available with this version\\u003c/p\\u003e\"}],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":true,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":false,\"hideJournal\":true,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":false,\"isAuthorSuppliedPdf\":false,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":false,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true},\"keywords\":\"Bessel beam, gold nanoparticles, dimer, orbital angular momentum, spin angular momentum, optical force, optical torque, drag force, Stokes’ law, Maxwell’s stress tensor, optical binding, MMP, rotation, orbital revolution\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-4386749/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-4386749/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eThe optomechanical motions of a pair of optically bound gold nanoparticles (GNPs), in fluid manipulated by a Bessel beam are studied. Since a Bessel beam possesses orbital angular momentum (OAM) and spin angular momentum (SAM) simultaneously, complicated rigid-body motions of the dimer can be induced. The mechanism involves the equilibrium between the optical force with the reactive drag force exerted by the fluid. Our results demonstrate that the 2D planar motion includes the rotation of the dimer around its center of mass (COM) and the orbital revolution of the COM around the optical axis. Additionally, each individual GNP undergoes spinning. The directions of the GNPs\\u0026rsquo; spin and the orbital revolution of COM depend on the handedness and the order (topological charge) of Bessel beam, respectively. Nevertheless, the rotation direction of the dimer depends on the size of GNP. In the case of a smaller dimer, the direction of dimer\\u0026rsquo;s rotation with respect to the COM is consistent with the handedness of the light. Conversely, a larger dimer performs a reverse rotation, accompanied by a precession during the orbital revolution. There are multiple turning points in the radius of the GNP for the alternating rotation of the dimer caused by positive or negative optical torque. Our finding may provide an insight to the optomechanical manipulation of optical vortexes on the motions of GNP clusters.\\u003c/p\\u003e\",\"manuscriptTitle\":\"Optomechanical Motions of Gold Dimer’s Spin, Rotation and Revolution Manipulated by Bessel Beam\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2024-05-16 18:12:32\",\"doi\":\"10.21203/rs.3.rs-4386749/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true}}],\"origin\":\"\",\"ownerIdentity\":\"a83cd10d-1e45-404d-979e-6f473db25be7\",\"owner\":[],\"postedDate\":\"May 16th, 2024\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"posted\",\"subjectAreas\":[],\"tags\":[],\"updatedAt\":\"2024-07-01T04:29:18+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2024-05-16 18:12:32\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-4386749\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-4386749\",\"identity\":\"rs-4386749\",\"version\":[\"v1\"]},\"buildId\":\"qtupq5eGEP_6zYnWcrvyt\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}