1. Introduction

Optical manipulation is a non-contact method of capturing various micro-scale objects [1], such as metals, semiconductors, dielectrics, organic materials, and living cells [2,3]. Due to the variety of trappable objects, laser trapping has been developed for a wide range of research fields [4,5]. One significant development in the optical manipulation is the trapping of multiple particles. For instance, a holographical technique can form chains of microparticles [6,7]. The total reflection on a glass substrate can provide a trapping force over a wide area [8–10]. In addition to the design of incident light, micro-scale fabrication has also achieved the trapping of many particles, e.g., plasmonic structures [11,12], and photonic crystals [13–15]. The formation of an ordered monolayer of particles at a liquid–liquid interface has also been reported [16,17]. Particle rotation is also an important development in the optical manipulation. A nonlinearity under a single focal laser can cause a rotation of single or a few particles [18,19]. The evanescent field also can generate an unconventional rotational optical force owing to the imaginary Poynting momentum or reactive power [20–24], where the particles can show a rotational orbital motion even though the angular momentum of light is absent [20,24]. The demonstration of particle rotation has also been reported for biological living cells [25].

Optical binding is a key concept for the optical manipulation of many particles [26,27]. Optically induced polarizations cause attractive or repulsive forces between the particles, which results in an ordering of the particles with a finite distance [28]. Yan and coworkers investigated the formation of a nanoparticle array under wide-area laser irradiation with circular [29] and linear polarizations [30,31], where the ordering of particles depended on the type of polarization. The spin angular momentum (SAM) of circular polarized light gives torque to an array and causes its rotation although the mechanism of this torque transfer is still unclear. Parker et al. recently analyzed the rotational dynamics of the optical array in terms of SAM and the orbital angular momentum (OAM) of the scattered light [32]. They pointed out that generation of the OAM in the scattered light causes positive and negative torques.

The control of the dynamics of optically bound particles is a significant subject. The modulation of the irradiation area by the focus of a laser beam has many possibilities. For instance, experiments by Kudo et al. [33–36] demonstrated some new possibilities of the optical manipulation and optical binding of nanoparticles. They examined the trapping of many nanoparticles by a tightly focused single laser, where trapped particles showed tetragonal or hexagonal ordered arrays depending on their polarization. These arrays were extended over the irradiation area, like a growing crystal [37]. Moreover, outside of the focal area, polystyrene particles formed additional horns [33] and gold particles showed revolution and swarming dynamics [35]. Very recent experiment by Huang et al. [36] demonstrated one-by-one manipulation of particles at outside of the focal area when a linear polarized laser is irradiated. These experiments suggest the many possibilities of indirect optical binding within the restricted irradiation area, where a contribution of the scattering field from the particles is enhanced. This is in significant contrast with conventional optical binding through a wide area of irradiation.

The mechanism of the indirect optical binding of multiple particles is unknown at present. Understanding this mechanism will lead to an unconventional scheme for creating and manipulating wide-area ordered structures, implementing finely controlled local properties with scanning beams, and the rich extension of optical binding by combining multiple focused lasers.

In this study, we theoretically investigate the optical binding and dynamics of nanoparticles due to the indirect optical force under the irradiation of a single tightly focused laser. Further, we consider the circularly polarized laser by a reason we will explain later (see Figs. 1(a) and (b)). Considering gold nanoparticles, we demonstrate numerically the revolution of surrounding particles. The focal laser irradiation directly and strongly traps a central particle. It provides a well-defined rotational axis for the SAM and OAM of light. Under the tightly focused laser, interference between the scattered light from the central particle and the incident light, causes the indirect optical force, leading to the surrounding particles becoming bound and revolved. This indirect mechanism for manipulating the surrounding particles is different to direct optical binding.

 

Fig. 1. (a)Schematic view of optical trapping and binding by a single focused laser and (b)Model of the numerical simulation for the optical binding of multiple particles restricted on a glass substrate. (c)Schematic explanation for the SAM and OAM of the incident and scattered fields.

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We consider two factors concerning the angular momenta: The first one is the “spin–orbit (SO) coupling” of light by respective particles accompanied with a conversion of the incident SAM $\sigma = +1$ to the OAM $l=+2$ with $\sigma = -1$ (spin-flip). The second is the generation of the OAM due to the array structure without a spin flip. The indirect optical force under the tightly focused laser is related to the “SO coupling”, whereas the direct optical force under the wide-area irradiation is in agreement with the generation of OAM [32].

Our motivation is based on the wide-area formation and control of a two-dimensional ordered particle array. Under a strongly focused laser, single or several particles are stably trapped to form a core structure. The array grows beyond the irradiation area by indirect optical binding. An understanding of the indirect optical binding mechanism is the purpose of this study. For wide-area formation, a highly symmetric core structure is favorable. The shape of the focal spot is isotropic (elliptic) for the circular (linear) polarization. Consequently, a circularly polarized focused laser was needed for the purpose.

At a point of the optical torque, its dependence on the particle’s order, number, and distance has been investigated [29,38]. The analysis based on the SAM and OAM shows a correspondence between the direction of optical torque and the sign of OAM generation [32] for conventional optical binding, whereas the SO coupling of light in the scattering is marginal. With indirect optical binding, the incident focal laser includes the spin-flip component due to lens “scattering” (see Fig. 1(c)). It induces an interference with the spin-flip component of scattered fields caused by the SO coupling. Then, we find a correlation between the SO coupling and the optical force on the bound particles out of the focal area.

The SAM and OAM of light are the main sources of particle dynamics in optical manipulation. Hence, the SO coupling of light is also an attractive and worthy subject for optical manipulation. Especially, the dynamics out of the irradiation area expand the subject to “generalized optical binding” under designed and multiple laser irradiations.

The paper is organized by four sections: Introduction, Model and Simulation parameters, Results and Discussions, and Conclusions. In Sec. 2, we describe the model for dynamics of nanoparticles with the optical force, Brownian random force, and hydrodynamic effect. The optical force is evaluated from the generalized Mie theory. In Sec. 3 for the simulated results, we discuss dynamics and several analysis. The last section is devoted to conclusions.

2. Model and simulation parameters

Considering the experiment reported in Ref. [35], we assume the following model and conditions. We consider spherical gold nanoparticles in a water solvent. The refractive indices of particles and solvent are $n_{\rm Au} \simeq 0.258 + 6.97 i$ and $n_{\rm w} \simeq 1.33$, respectively. The diameter of particles is $d = 200\, {\mathrm{nm}}$. The wavelength of the laser is $\lambda = 2\pi /k = 1064\, {\mathrm{nm}}$ in a vacuum. A single focal incident laser is modeled by a quasi-Gaussian beam [39,40]. In our simulation, the irradiation size for the incident laser with $2\omega _0 = 660\, {\mathrm{nm}}$ has effectively an $800\, {\mathrm{nm}}$ diameter on the focal plane. This is related to the maximal half-angle $\theta _{\rm max}$ by $\omega _0 \sim 2/(k \tan \theta _{\rm max})$. The numerical aperture is estimated as ${\rm NA} = n_{\rm w} \sin ( 2/(k\omega _0) )\simeq 0.996$. For the dynamics, we set the laser power as $P = 100\, \mathrm {mW}$ in the simulation area $S$. This assumption corresponds to the experimental setup [35]. We consider the particle dynamics only on a glass substrate at $z = 0$ when their motion is restricted on the substrate by the incident laser (Fig. 1(b)). The focal point is set at $z = - d/2$. Effects of the zeta potential $\zeta _{\rm p} = -100\, \mathrm {mV}$ on the particles and the electrolytic concentration $\rho _{\rm ch} = 10^{-5}\, \mathrm {mol/l}$ with the valence $Z=1$ in the solvent. As an assumption, the dynamics are sufficiently faster than an increase in the number of bound particles.

The following results qualitatively explain several aspects of the experiment Ref. [35]. A stable optical binding and rotation with hexagonal order even outside of the focal area is found. The particle distance is determined by the wavelength in the solvent and in agreement with the experiment. The rotation speed is also reasonable for the experimental observation. We point out the significance of the focus degree of freedom in the incident laser, namely NA. The NA can switch the rotational direction of the array without changing the particles and solvent conditions. The focal height is also another control parameter [41].

The effect of heat generation by light irradiation is another central interest. Generally, the heat generation disturbs the optical trapping and manipulation [42–46]. A temperature elevation of water solvent near the irradiated nanoparticles is investigated in detail experimentally [42,46] and theoretically [47,48]. The temperature elevation should happen in the vicinity of particles in the irradiation area. However, it is not significant for the hexagonal array and binding distance when the laser power is strong enough for indirect optical binding. The temperature elevation fastens the revolution due to the viscosity modulation. In recent studies, a concept to utilize such heat generation on the nanoparticles has been developed as opto-thermoelectric tweezers [49,50]. By separating the thermoelectric effect in the irradiation area and the indirect optical manipulation at the outside of the area, our study will open new possibilities for optical manipulation.

The optical force on particle $i$ is evaluated based on the electromagnetic field on the particle surface. The field is calculated self-consistently by the generalized Mie theory and T-matrix method [51–53]:

The integral is over the surface of particle $i$ [54]. The simulation of particle dynamics follows the Langevin equation,

To analyze the simulated results, we consider the SO coupling of light, the OAM generation, and the optical current [58]. The correlation between SAM and OAM has previously been discussed for rotational optical manipulation by the optical vortex with OAM [59]. It should be noted that even when light with finite SAM and zero OSM is injected, a finite OAM in the scattered light is generated by the finite size and the geometrically arrayed structure because its rotation generates the OAM inescapably. Hence, the SO coupling is inevitable.

We simulated the dynamics of nanoscale spherical gold particles in a water solvent on a glass substrate. For the optical force, a single focal incident laser with counterclockwise circular polarization was applied. The electric field $\boldsymbol {E}_{\rm inc} (\boldsymbol {r})$ in Eq. (1) was modeled as a quasi-Gaussian beam. The incident field is expanded by the vector spherical harmonics (VSH) functions, $\boldsymbol {M}_{nmp,i}^{(1)}$ and $\boldsymbol {N}_{nmp,i}^{(1)}$, as

Here, the superscript $(1)$ in $\boldsymbol {M}_{nmp,i}^{(1)}$ and $\boldsymbol {N}_{nmp,i}^{(1)}$ represents the spherical Bessel function in the radial part to describe the incident field. The subscript $i$ denotes that these VSH functions are located at $\boldsymbol {r}_i$, i.e., the position of particle $i$. The index $p = {\rm e,m}$ corresponds to the TE and TM modes. $u_{nmp,i}$ and $v_{nmp,i}$ are the expansion coefficients, as determined by localized approximation [51]. The expansion of $\boldsymbol {E}_{\rm inc}$ by $\boldsymbol {M}_{nmp,i}^{(1)}$ and $\boldsymbol {N}_{nmp,i}^{(1)}$ is applied to its scattering by particle $i$.

When the particles are isolated from each other, the scattered field is approximately given by

The multiple scatterings between the particles combine the T-matrices $\hat {t}_i$ and result in generalized T-matrix elements $\hat {T}_{ij}$ [51–53]. An extension of the vector, $\vec {C}_{\rm inc} = (\vec {c}_{{\rm inc},1}, \vec {c}_{{\rm inc},2}, \ldots )^{\rm t}$, shows a simple formulation of the multiple scattering:

The evaluated coefficients $\vec {c}_{{\rm sca},i} = (\{ A_{nmp,i}, B_{nmp,i} \})^{\rm t}$ give a full scattered field

The Maxwell’s stress tensor $\bar {T}_{\rm E/H}$ is defined as

For the simulation of particle dynamics, we restrict it only to the $z=0$ plane due to the glass substrate. The substrate might cause the reflection of the incident laser and the friction of particle motion. They would modulate the simulated results but are not essential in the present study.

3. Results and discussions

3.1 Dynamics of nanoparticles

Figures 2(a1)–(h1) show a snapshot of the intensity of the total electric field with $N_{\rm p} = 4$–$10$ particles and their trajectories under the tightly focused circularly polarized laser. At the origin, one particle is trapped directly and strongly by the incident laser. The trapped particle scatters the incident light, which causes a ring-shape oscillation of the light intensity due to the interference. The ring-shaped intensity binds the particles at the distance $\Delta \simeq \lambda /n_{\rm w}$ from the center particle. This is a mechanism of indirect binding.

 

Fig. 2. (a1)–(h1)Intensity profiles of the total electric field with $N_{\rm p} = 4$ (a), $6$ (b), $7$ (c), $8$ (d,e), $9$ (f,g), and $10$ (h) on the $z=0$ plane. The incident laser is tightly focused with circular polarization. The focal point is at the origin. Black (red) lines indicate trajectories of particles in $0.5\, \mathrm {msec}$ ($10\, \mathrm {msec}$). The surrounding bound particles show revolution dynamics (see Visualization 1, Visualization 2, Visualization 3, Visualization 4, Visualization 5, and Visualization 6 for $N_{\rm p} = 4$ in (a), $6$ in (b), $7$ in (c), $8$ in (d,e), and $10$ in (h), respectively) The field intensity is normalized by the incident field $E_0$ at the origin. (a2)–(h2)Rotational angle of the surrounding particles and (a3)–(h3)Particle distance from the origin in time. The respective line colors correspond to those of panel (a2)–(h2). Respective insets indicate particle configurations at $5\, \mathrm {msec}$. The simulation parameters are as follows: The particles are Au with $d=200\, {\mathrm{nm}}$ diameter in a water solvent. The zeta potential due to the charge on the particles is $\zeta _{\rm p} = -100\, \mathrm {mV}$. The electrolytic concentration in the water is $\rho _{\rm ch} = 10^{-5}\, \mathrm {mol/l}$ with the valence $Z=1$. The incident focal laser is with $\omega _0 = 330\, {\mathrm{nm}}$ and $\lambda = 1064\, {\mathrm{nm}}$. The total laser power is about $P = 100\, \mathrm {mW}$.

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Let us consider an increase in the number of particles one by one. The bound particles revolve by the indirect optical force and find a stable configuration. The stable configuration depends on the particle charge and the solvent condition. The direction of revolution accords with the optical current.

When $N_{\rm p} \le 6$, the revolution speed of surrounding particles at $\Delta \approx 870\, {\mathrm{nm}}$ is almost irrelevant to the number of particles (Figs. 2(a2,b2)). The particle distance from the center particle also does not depend on the particle number (Figs. 2(a3,b3)). When $N_{\rm p} = 7$, the speed is slightly increased as shown in Fig. 2(c2). The seven particles stably hold a hexagonal configuration, as shown in an inset of Fig. 2(c), which indicates that our simulation reproduces the experimental observation in Ref. [35] well.

For $N_{\rm p} = 8$ and $9$, we consider two initial distributions: $6 + 1$ or $2$ in Figs. 2(d,f), and $7 + 0$ or $8 + 0$ in Figs. 2(e,g). Both cases show the stable revolution (see also the insets). The outer particles in Figs. 2(d,f) are pushed out to the third local maximum of the field intensity at $\Delta \approx 1720\, {\mathrm{nm}}$. The outer particle does not reach the second local maximum because the particles possess the electric charge, and the repulsive force due to the Coulomb interaction pushes out the outer particle. The optical force due to the scattered light from the outer particles is weak when the outer particle is at the third local maximum. Hence the hexagonal configuration of the particles at the center and first local maximum is stable. We find that the revolution of particles is only in a positive direction under a tightly focal laser in contrast to wide-area irradiation. If the effect of particle charge was weak or absent and the laser power was weak, outer particle could stochastically go into the second local maximum. This particle interacted with the particles at the first local maximum by the multiple scattering of light and an enlarged optical force by the multiple scattering disturbed the hexagonal configuration for the trapped central particle and the bound particles at the first local maximum. Then, strong fluctuations of the particles are caused. In one case, after the fluctuation, the particle at the second local maximum is stably bound in the first local maximum and results in $7 + 0$ or $8 + 0$ configurations. In other case, the particle is stochastically kicked out to the third local maximum. Therefore, for both situations of the presence and absence of the charge on particles, the optical binding at the second local maximum is unstable. For $N_{\rm p} = 10$ with the hexagonal distribution in Fig. 2(h), the dynamics are almost the same as those in Figs. 2(d,f).

The revolution speed depends on the number of surrounding particles. In Figs. 2(c,d,f), the particles bound at the first maximum have the same speed, whereas the outer bound particles are much slower than those at the first maximum. This means there is no correlation between the particles at the first and third local maximum. The slow dynamics are due to a decrease in field intensity, hence the binding distance fluctuates more. For $7 + 0$ or $8 + 0$ configurations in Figs. 2(e,g), the surrounding particles are faster or slower than the hexagonal configuration, which is attributed to the multiple scattering and the Coulomb interaction between the bound particles. The bound distance is slightly longer than the hexagonal configuration due to the particle charge. The charge can switch the revolution direction (positive and negative optical torque) by the particle distance [29]. However, in the range of the present condition, the revolution direction is only positive.

We also examine modulations of the laser power and the temperature of water solvent in Supplementary Document. When the laser power is decreased to $P = 0.1\, \mathrm {mW}$, the Brownian random force by the solvent is dominant and the ordered optical binding is not found stably, whereas the strong laser with $P = 1\, \mathrm {W}$ does not change the dynamics except a revolution speed (no figure). If the particle had no electric charge, an aggregation of the particles occured under a stronger and wider laser irradiation. The temperature elevation is not important for the indirect optical binding although the revolution becomes fast owing to the viscosity modulation.

The simulated results show a quantitative agreement with the experiment. Considering a laser power of $P = 100\, \mathrm {mW}$ ($P/S = 0.34\, \mathrm {MW/cm^2}$ for wide-area irradiation) is reasonable to the experiment [35,46]. The particle distance is $\Delta \simeq 870\, {\mathrm{nm}}$ in the simulation, which is determined by the wavelength of optical binding laser in the solvent ($\lambda \approx 800\, {\mathrm{nm}}$). In the experiment by Kudo et al. [35], the particle distance under a focal circularly polarized laser with $P \approx 60\, \mathrm {mW}$ laser power was $\Delta \simeq 830\, {\mathrm{nm}}$, which is in reasonable agreement with our simulation. The revolution speed of particles, $2 \mathrm {cycle}/ 10\, \mathrm {msec}$ in Figs. 2(a2)–(h2), is too fast to observe in a real-time measuremant. Such fast dynamics are also observed in the vicinity of the laser irradiation (see a movie in Ref. [35]). A reduction of the laser power to $P = 30\, \mathrm {mW}$ decreases the speed to $0.25 \mathrm {cycle}/ 10\, \mathrm {msec}$. Consequently, the simulated dynamics become reasonable for the experimental observation [35].

A center trapped particle and six surrounding bound particles scatter the light, which reduce the mode purity of the optical vortices and their correlation are reduced [60–63]. Then, the rescattering light by the secondary trapped (bound) particles disturbs the hexagonal binding. If an outer particle was bound at the second local maximum of field intensity, the effect of rescattering light is enlarged owing to the short particle distance. It can be observed when the particle charge is negligible. The enhancement of the effect of rescattering light causes large fluctuations of the hexagonal configuration of seven inner particles. Then, a reconfiguration of all the particles happens. In some cases, an exchange between surrounding particles and the central particle occurs. In such processes, the binding at the second local maximum is not stable. When the outer particle is stored at the third local maximum, the rescattering light from the outer particles does not have sufficient strength. Hence the hexagonal configuration of seven inner particles and the outer particle binding are stable enough.

The bound particles might rotate (spin) respectively in addition to their revolution. The latter is observable, whereas it is difficult to observe the former. However, the spin of particles should also be induced by the SAM and OAM of light. The spin of respective particles is not scope of this study. Han et al. have discussed an influence of the particle spin and shown that the spin torque is much smaller than that of the orbital revolution. Hence we focus on the orbital revolution and dynamics of bound particles.

From the results, we find that the binding position is located at the maxima of field intensity oscillation. The optical force is determined by the integral of time-averaged Maxwell’s stress tensor as shown in Eq. (2), where an interference between the incident and scattered field contributes to the optical force. The time-averaged Maxwell’s stress tensor is divided into three components: $\bar {T}_{E/H} = \bar {T}_{{\rm inc},E/H} + \sum _i \bar {T}_{{\rm sca},i,E/H} + \bar {T}_{{\rm interference},E/H}$. Here, $\bar {T}_{{\rm sca},i,E/H}$ is given by the scattered field from the particle $i$. For the hexagonal configuration of seven particles, $\bar {T}_{{\rm inc},E/H}$ and $\bar {T}_{{\rm sca},i,E/H}$ cause the radial and out-of-plane forces, whereas the interference term contributes to the azimuthal force. Hence, the revolution of bound particles is induced by the multiple scattering and interaction between the bound particles. These indirect optical forces at the outside of the focal area depend on the configuration of bound particles. It is favorable to achieve more arrays of the particles. In the case of linear polarized laser, the particles are aligned perpendicular to the polarization. The result is a reasonable explanation for the experimental observations [35,36].

3.2 SAM/OAM analysis

We analyze the SAM and OAM components of scattered light on the upper and lower celestial hemispheres (inset in Fig. 3(a)-(b)) at enough far position from the bound particles, $r = R_{\rm c} \gg d$:

Here, $\boldsymbol {e}_{\sigma,l} (\theta, \phi )$ is the mode of the electric field with spin $\sigma$ and vortex $l$ with respect to the $z$-axis. $C^{(0)}$ is a normalization factor to satisfy $\sum _{\sigma,l} {C_{\sigma,l}}^2 =1$. On the upper hemisphere, the electromagnetic field is only out-going waves. On the other hand, on the lower, the incident focal laser has in-coming angular momentum. The subtraction in Eq. (13) is to reduce the in-coming component and to consider only the out-going angular momentum. In this definition, because the analysis is examined at far position, the longitudinal fields by the bound particles are suppressed. Although the longitudinal fields contribute to the revolving dynamics, the SAM and OAM analysis is only for the transverse fields.

 

Fig. 3. Analysis and comparison of the optical force between a tightly focused laser ($\omega _0 = 330\, {\mathrm{nm}}$ for (a,c,e)) and an intermediated focused laser ($\omega _0 = 2000\, {\mathrm{nm}}$ for (b,d,f)) when $N_{\rm p} = 7$ with the hexagonal configuration. The other parameters are the same as those in Fig. 2. (a,b) Coefficients of SAM and OAM components of out-going electromagnetic field with $N_{\rm p} = 0,1,7$ when $\sigma = +1$ circularly polarized light is applied. For $N_{\rm p} = 7$, the particle distance is $\Delta = 870\, {\mathrm{nm}}$ in (a) and $800\, {\mathrm{nm}}$ in (b). Panel (a2) and (b2) are log scale plots of (a1) and (b1), respectively. Inset indicates a schematic image of upper and lower hemispheres to analyze the SAM and OAM. (c,d,e,f) Radial $F_{\rm radial}$ and azimuthal optical forces $F_{\rm azimuth}$ on the surrounding particles, SO coupling of light $C_{\rm SO} = C_-/C_+$, and imbalance of OAM generation $C_{\rm OG} = C_> - C_<$ when the particle distance $\Delta$ is tuned. Thick purple and green lines are for $\omega _0 = 330\, {\mathrm{nm}}$ and $2000\, {\mathrm{nm}}$, respectively. The optical forces in panel (c) and (d) are divided by $F_0 = (\lambda /(2\pi ))^2 (P/Sv)$ with the laser power $P$. $v = c/n_{\rm w}$ is the velocity of light in the solvent. $S$ indicates a simulation area for particle dynamics with $-16.1 \lambda /(2\pi ) \le x,y \le 16.1 \lambda /(2\pi )$. In panel (d) and (f), a wide-area irradiation with $\omega _0 = 6250\, {\mathrm{nm}}$ is also examined (light blue thick lines). Thin red and blue lines are 10 times the values of purple and light blue thick lines for clear representation. (g,h) Gray-scale plot for the optical force, SO coupling, and OAM generation in ($\omega _0$-$\Delta$) plane. Particle distance dependence of the coefficients when $N_{\rm p} =7$. The optical forces in (g) are normalized by $F_0$ with a fixed laser power $P$ in the area $S$ for various $\omega _0$. The SO coupling is modulated by subtracting the values at $\Delta = 30\, \mu \mathrm {m}$, $\delta C_{\rm SO} = C_{\rm SO}(\Delta ) - C_{\rm SO}^{\rm (far)}$. Circles in each panel indicate the binding position when $\omega _0 = 330\, {\mathrm{nm}}$ and $2000\, {\mathrm{nm}}$ for neutral particle. Solid line is a guide to indicate $\omega _0 = \Delta$. (i)Gray-scale plot for the azimuthal component of the time-averaged Poynting vector $\langle S \rangle _{\rm azimuth} = {\rm Re}(\boldsymbol {E} \times \boldsymbol {H}^*)_{\rm azimuth}/2$ when $N_{\rm p} = 1$. The plot is normalized by $E_0 H_0$. Inset is a plot of $|S_{\rm azimuth}|$ at position $\Delta$ when $\omega _0 = 330\, {\mathrm{nm}}$.

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The coefficients $C_{\sigma, l}$ can define quantities for the SO coupling and OAM generation as follows. The SO coupling yields a flip of the SAM from $\sigma = +1$ to $-1$ with an increase of the OAM by $+2$. For the up and down spin (right and left circular polarization) coefficients,

We consider an increase of the total angular momentum (TAM), $\sigma + l$. For an incidence of $(\sigma,l) = (+1,0)$ and $(-1,+2)$, the positive OAM generation is given as

The imbalance of OAM generation is evaluated as

These definitions are summarized in Fig. 3(e). Note that for the OAM generation without the spin-flip, $C_{\pm }$ is not affected. The spin-flip by the SO coupling does not change the TAM. Hence, $C_{\rm SO}$ and $C_{\rm OG}$ are independent from each other.

In the case of a plane wave, the light has no OAM ($l = 0$) [39]. On the contrary, a focal laser with right circular polarization has a slight $(\sigma,l) = (-1,+2)$ component in addition to the majority $(+1,0)$ component. This is due to “scattering” by the lens. Figure 1(c) depicts the spin-flip schematically. The incident focal laser with $\omega _0 = 330\, {\mathrm{nm}}$ consists of $C_{+1,0} \approx 0.9884$ and $C_{-1,+2} \approx 0.1518$ (see Fig. 3(a)). Note that the sum of their squares is unity. With an increase of $\omega _0$, $C_{-1,+2}$ decreases to zero.

We found strong SO coupling by a particle or particle assembly. Figures 3(a) and (b) exhibit the coefficient $C_{\sigma,l}$ of light scattered with $N_{\rm p} =1$ and $7$ for the strongly focused and intermediate focused focal lasers, respectively. For a single particle, the scattered light under the $\omega _0 = 330\, {\mathrm{nm}}$ tightly focused laser has $C_{1,0} \approx 0.9509$ and $C_{-1,+2} \approx 0.3094$, and the others are zero (Fig. 3(a)). Here, the TAM is conserved by the rotational symmetry with respect to the $z$-axis. The presence of OAM in the scattered field is attributed to the size of the particle (and lens) although it is difficult to find its orbital motion and rotation.

When $N_{\rm p} \ge 2$, the rotational symmetry is broken, and the TAM is not conserved. In Fig. 2(c), the system has 6-fold rotational symmetry, and the scattered light consists of $l= \pm 6 m$ with $\sigma = +1$ as shown in Figs. 3(a). Here, $m$ is an integer. A spin-flip also occurs, and $C_{-1,2 \pm 6m}$ is generated, where the TAM is distributed from $j=1$ to only $1 \pm 6 m$. We examine an intermediate focal laser with $\omega _0 = 2000\, {\mathrm{nm}}$ and a wide-area irradiation laser with $\omega _0 = 6250\, {\mathrm{nm}}$ for a comparison between the indirect and direct optical binding mechanisms. We find a large OAM generation for $\omega _0 = 2000\, {\mathrm{nm}}$ in Fig. 3(b). The distribution of TAM (mainly OAM) is imbalanced. When $\omega _0 = 330\, {\mathrm{nm}}$, the coefficient of $j=1-6$ is larger than that of $j=1+6$, whereas $j=1+6$ is slightly larger than $j=1-6$ for $\omega _0 = 2000\, {\mathrm{nm}}$. It is consistent with the rotation of assembly if the particles and photons follow Newton’s third law of motion. When $N_{\rm p} = 5$ and $9$ with square and octagonal configurations, the OAM generation shows $j = 1 \pm 4m$ and $1 \pm 8m$, respectively.

When a right circular polarized laser is focused, the fields have three components on the focal plane: right circular polarization with no optical vortex ($s=+1,l=0$), left circular one with the second order vortex ($s=-1,l=+2$), and perpendicular polarization with the first order vortex ($s=0,l=+1$) [21,39,65]. These three components contribute to the revolving dynamics of bound particles. In the vicinity of bound particles, the longitudinal field is induced. The longitudinal component also contributes to the dynamics. The fields are fully considered in the present simulation based on the generalized Mie theory and T-matrix method. We analyze the SAM and OAM on the far hemispheres defined by Eqs. (12) and (13). The longitudinal field is suppressed on the hemispheres. Hence we obtain only $(s,l)=(+1,\pm 6m)$ and $(-1,2 \pm 6m)$ components for the hexagonal configuration. The analysis of perpendicular polarization component in the focal laser and the longitudinal component by the scattering is a significant subject for the indirect and generalized optical binding. It is one of the next scopes of this project.

Figure 3(c) represents the radial and azimuthal forces, $F_{\rm radial}$ and $F_{\rm azimuth}$, on the surrounding particle as an increase in the particle distance $\Delta$ under the tightly focused laser. The radial force on the surrounding particle oscillates clearly with exponential decay. If the particles are electric neutral, the (first) bound distance is about $0.85\, \mu \mathrm {m}$. The azimuthal force shows non-periodic oscillation and is only positive. Note that the forces are relatively large peaks even at $\Delta > \omega _0$ (or outside of the effective irradiation area) and are suppressed strongly around $\Delta = 1\, \mu \mathrm {m}$, which is several times larger than the irradiation size.

For large $\omega _0$, $F_{\rm radial}$ indicates complicated dependence on $\Delta$ in Fig. 3(d). Owing to a normalization of the laser power, the optical force tends to be large when the laser is tightly focused. When $\omega _0 = 2000\, {\mathrm{nm}}$, we find two binding distances at $\Delta \approx 410\, {\mathrm{nm}}$ and $800\, {\mathrm{nm}}$ although our simulation demonstrates only $\Delta \approx 800\, {\mathrm{nm}}$ for charged particles. At the binding distance, $F_{\rm azimuth}$ provides the negative torque. The optical force is observed to be suppressed at the boundary of the irradiation area ($\Delta \approx \omega _0$) in contrast to the tightly focused case. For both $\omega _0 = 2000\, {\mathrm{nm}}$ and $6250\, {\mathrm{nm}}$, $F_{\rm radial}$ and $F_{\rm azimuth}$ exhibit similar (but opposite sign) oscillation behaviors with each other, which suggests their correlation in the direct optical binding. $F_{\rm azimuth}$ also shows the sign change. These features are the differences between indirect and direct optical binding.

We analyze the factors for the SO coupling of light, $C_{\rm SO} = C_-/C_+$, and OAM generation imbalance, $C_{\rm OG} = C_> - C_<$. In Figs. 3(e), the SO coupling strength $C_{\rm SO}$ oscillates with $\Delta$ for the tightly focused case. The oscillation behavior shows a correlation with $F_{\rm radial}$, which suggests that the $\sigma = -1$ component in the incident and scattered fields gives the indirect optical binding. If the surrounding particles are far from the irradiation area, then the spin-flip of light is attributed to only the center particle. Hence $\delta C_{\rm SO} = C_{\rm SO} - C_{\rm SO}^{\rm (far)}$ effectively provides the spin-flip caused by “particle array.” $\delta C_{\rm SO}$ can be positive and negative, the behavior of which qualitatively agrees with $F_{\rm radial}$. On the other hand, the behavior of $C_{\rm OG}$ does not have a clear correlation with the others. If the surrounding particles are far, $C_{\rm OG}^{\rm (far)} = 0$, thus the OAM generation by “array” is almost positive. It corresponds to $F_{\rm azimuth}$.

For the direct binding, the behavior of $C_{\rm SO}$ in Fig. 3(f) differs from that of $\omega _0 = 330\, {\mathrm{nm}}$. The peak positions and the sign of $\delta C_{\rm SO}$ cannot explain the optical forces in Figs. 3(d). The OAM generation $C_{\rm OG}$ also shows complicated behavior. When $\omega _0 = 2000\, {\mathrm{nm}}$, the relation between $C_{\rm OG}$ and the optical forces is not clear. However, we find a definite agreement between the sign changes of $F_{\rm azimuth}$ and $C_{\rm OG}$ for the wide-area irradiation ($\omega _0 = 6250\, {\mathrm{nm}}$). This result agrees with an intuitive understanding and has been discussed in Ref. [32].

To see the relationship between the optical force and the SAM/OAM analysis, we examine continuous modulation of the focal size $\omega _0$ for $F_{\rm radial}$, $F_{\rm azimuth}$, $C_{\rm SO}$, and $C_{\rm OG}$ in Figs. 3(g) and (h). Moreover, as a comparison, we examine the optical current which is proportional to the time-averaged Poynting vector $\langle \boldsymbol {S} \rangle = {\rm Re}(\boldsymbol {E} \times \boldsymbol {H}^*)/2$ in Fig. 3(i). Note that the Poynting vector is a time-dependent quantity, which is crucial for the optical manipulation with the reactive power [22,23]. However, the particle dynamics is much slower than the time-dependence of Poynting vector. Moreover, the reactive power and quantities for the optical manipulation is beyond our present scope. Therefore, we focus on the time-averaged optical force and Poynting vector. Figures visualize the difference of physics in $\omega _0 \gtrsim \Delta$ region for the direct optical binding, $\Delta \gtrsim \omega _0 > \lambda /2$ region with no binding, and $\omega _0 \approx \lambda /2$ region for indirect optical binding. In the direct binding region, the physics is $\lambda _{\rm w}$-periodic as a function of $\Delta$. Here, $\lambda _{\rm w} = \lambda /n_{\rm w} \approx 800\, {\mathrm{nm}}$ is the wavelength in water solvent. This might be attributed to an interaction between particles with equally induced polarization. On the other hand, in $\omega _0 \approx \lambda /2$, the physics is smooth since the scattered field from a center particle determines the dynamics of surrounding particles. However, at a tight focus region $\omega _0 \approx \lambda /2$, we find $(\lambda _{\rm w}/2)$-periodic behaviors. It might be attributed to the interference of spin-flipped components between the incident and scattered fields. For an intermediated $\omega _0$, we find no sign change from $F_{\rm radial} > 0$ to $F_{\rm radial} < 0$ as a function of $\Delta$ in Fig. 3(g1), thus no optical binding is obtained.

3.3 Scattering/absorption cross-section

The strength of SO coupling can be modulated by the particle parameters, e.g., the refractive index and diameter. To determine the correlation between SO coupling and the optical force under a tightly focused laser, we discuss the azimuthal optical force $F_{\rm azimuth}$, the scattering and absorption cross-sections $Q_{\rm sca/abs}$, and the SAM/OAM analysis for $C_{\rm SO/OG}$ in a plane of the complex refractive index, $\tilde {n} = n + i\kappa$, in Fig. 4. The cross-sections depend on the particle size strongly.

 

Fig. 4. (a,b) Profile of in-plane optical force $(F_x,F_y)$ on one of the surrounding particles in the plane of complex refractive index $\tilde {n} = n + i\kappa$ when $N_{\rm p} =7$ and tightly focused (a) and not focused (b) lasers are applied. The beam waists are $2\omega _0 = 0.8\, \mu \mathrm {m}$ and $12.5\, \mu \mathrm {m}$, respectively. The latter situation is a conventional setup for optical binding. The diameter of particles is $d=200\, {\mathrm{nm}}$. One particle is trapped at $r=0$, and the others are at $r=\Delta = 853.4\, {\mathrm{nm}}$ (see Fig. 2(c)). (c)Single particle cross-sections of the scattering $Q_{\rm sca}$ and absorption $Q_{\rm abs}$. The plot is normalized by $\pi R^2$, where $R$ is the radius. (d)“SO coupling” $C_{\rm SO} = C_-/C_+$ and imbalance of OAM generation $C_{\rm OG} = C_> - C_<$ evaluated from the coefficients of SAM and OAM.

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Figure 4(a) shows the indirect optical force driving the revolution of surrounding particles when the particles are $N_{\rm p} = 7$ with hexagonal order and electrically neutral. $F_{\rm azimuth}$ is enlarged when the refractive index is purely imaginary (perfect conductor), and $\kappa \approx 2.7$. Another peak of the force is found around $n \approx 6.5, \kappa \ll 1$. For a tightly focused laser, the indirect optical force $F_{\rm azimuth}$ is only positive in the $n$-$\kappa$ plane. $F_{\rm azimuth}$ is also only positive for a tuning of the particle distance $\Delta$ as shown in Fig. 3(c2). In Fig. 4(b), on the other hand, both positive and negative forces are observed when the light is applied widely. This is a clear difference between indirect and direct optical manipulation. The negative force is enlarged at $n \approx 5.1, \kappa \ll 1$. The presence of a negative torque in direct optical binding with wide-area irradiation is consistent with Ref. [29].

The refractive index modulates the light scattering and absorption at each particle. Their cross-sections are evaluated from the Mie coefficients, [64] $a_n$ and $b_n$, as

Figure 4(c) represents $Q_{\rm sca}$ and $Q_{\rm abs}$ of single particle in water. The profile of scattering cross-section $Q_{\rm sca}$ shows a good agreement with the indirect optical force $F_{\rm azimuth}$. They are enlarged strongly at $n \ll 1, \kappa \approx 6.5$ region. There are another two peaks at $(n,\kappa ) \approx (5,0)$ and $(7,0)$. The absorption cross-section $Q_{\rm abs}$ also has enhancement regions in the $n$-$\kappa$ plane. However, the peak positions are not at $n \approx 0$ nor $\kappa \approx 0$, hence $Q_{\rm abs}$ does not explain the enhancement of $F_{\rm azimuth}$.

The discussion above gives a guideline for indirect and wide-area optical manipulation by multiple scattering. One should prepare particles and conditions with a large scattering cross-section rather than absorption. To transfer the momentum or angular momentum of photons to the particles directly, both strong scattering and absorption are suitable. However, if particles show a strong absorption, the light is extinct quickly and scattering is suppressed. When $d=200\, {\mathrm{nm}}$, $Q_{\rm sca} \gg Q_{\rm abs} \approx 0$ is a better condition for indirect optical manipulation. The particle size is also an important parameter. When the particle size is tuned, we can find that the criterion to obtain a large indirect optical force is adaptable for $d \gtrsim 150\, {\mathrm{nm}}$.

The SAM/OAM analysis in the $n$-$\kappa$ plane is also examined in Fig. 4(d). The profile of SO coupling $C_{\rm SO} = C_- /C_+$ shows similar behavior to $F_{\rm azimuth}$ in Fig. 4(a), whereas the OAM generation $C_{\rm OG} = C_> - C_<$ has different peak positions. Hence, for the tightly focused laser irradiation, the SO coupling makes a significant contribution to the indirect optical binding. $C_{\rm OG}$ indicates positive and negative regions. It implies that the negative torque would appear if the material was different, even when irradiation was focused.

4. Conclusions

In summary, we conducted numerical simulations of the dynamics of nanoparticles trapped and bound optically under a single tightly focused laser. We concluded that the scattered light from the strongly trapped center particle caused the indirect optical binding and revolution of surrounding particles out of the focal area. The increase in the number of bound particles enlarged the multiple scattering between the particles and caused an acceleration of the revolution. Our simulation demonstrated qualitative and partially quantitative agreement with the experimental observations [35], e.g., a hexagonal configuration with wavelength distance for circular polarization. Under the tightly focused laser, the SO coupling of light became essential and the interference between the spin-flipped components in both incident and scattered fields determined the binding position indirectly. It is different from the wide-area irradiation discussed in previous studies [29,32]. Hence the numerical aperture should be an important parameter, and we found three regions: wide-area irradiation $\omega _0 \gtrsim \Delta$ for direct binding, focused irradiation $\Delta \gtrsim \omega _0 > \lambda /2$ for no binding, and tightly focused irradiation $\omega _0 \approx \lambda /2$ for indirect binding.

Based on the analysis of SAM and OAM, we pointed out that the SO coupling of light, $C_{\rm SO} = C_-/C_+$, and the imbalance of OAM generation, $C_{\rm OG} = C_> - C_<$, are essential factors to understand the optical binding and particle dynamics. In the wide-area irradiation region, the sign of $C_{\rm OG}$ indicated agreement with the positive/negative torque. However, $C_{\rm SO}$ is indifferent. For indirect binding in the tightly focused irradiation region, $C_{\rm SO}$ became essential in addition to $C_{\rm OG}$. To enhance the multiple scattering, a large scattering cross section is favorable. The SO coupling due to the light scattering can be modulated by particle properties, i.e., the complex refractive index, diameter, and shape. It might be possible to apply core-shell structures or dye-doped polymers.

With the indirect optical binding, the in-plane scattered field from the directly trapped particle(s) caused the binding and manipulation. Hence the indirect binding array should be $f$-fold rotational symmetric with a well-defined center. Such $f$-fold symmetric structures were observed in the experiments [33,35]. By controlling the in-plane “scattered” field, indirect optical binding could open a rich range of possibilities. Simultaneous use of multiple focal lasers, which is the scope of our next paper, is a powerful candidate to develop indirect optical binding. For example, one focal laser with bound particles stops the revolution of indirectly bound particles at another focal laser. Then, scanning a single laser could create an arbitrarily shaped configuration of particles. Moreover, the combination of two circular polarizations would cause a transition between the hexagonal and tetragonal configurations. Such large degrees of freedom will open the doors to new optical manipulation.