# Rotation of optically bound particle assembly due to scattering induced
spin-orbit coupling of light

Yukihiro Tao  Department of Material Engineering Science, Graduate School of
Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka
560-8531, Japan  Tomohiro Yokoyama<EMAIL_ADDRESS>Department of Material Engineering Science, Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Hajime Ishihara  Department of Material Engineering Science, Graduate School
of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka
560-8531, Japan  Department of Physics and Electronics, Graduate school of
Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai,
Osaka 599-8531, Japan  Center for Quantum Information and Quantum Biology,
Institute for Open and Transdisciplinary Research Initiatives, Osaka
University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

###### Abstract

The optical binding of many particles has great potential to achieve the wide-
area formation of a “crystal” of small materials. Unlike conventional optical
binding, where the whole assembly of targeted particles is irradiated with
light, if one can indirectly manipulate remote particles using a single
trapped particle through optical binding, the degrees of freedom to create
ordered structures will be greatly enhanced. In this Letter, we theoretically
investigate the dynamics of the assembly of gold nanoparticles that is
manipulated using a single particle trapped by a focused laser. As a result,
we demonstrate that the spin–orbit coupling and angular momentum generation of
light via scattering induce the assembly and rotational motion of particles
through indirect optical force. This result opens the possibility of creating
ordered structures with a wide area and manipulating them, controlling local
properties using scanning laser beams.

##  I Introduction

Optical manipulation is a non-contact method to capture various micro-scale
objects Ashkin et al. (1986), such as metals, semiconductors, dielectrics,
organic materials, and living cells Zhang and Liu (2008). Due to this variety
of trappable objects, laser trapping has been developed for a wide range of
research fields Li et al. (2010); Fazal and Block (2011). One significant
development in optical manipulation is the trapping of multiple particles. For
instance, a holographical technique can form various chains of microparticles
Curtis et al. (2002); Grier and Roichman (2006). On a glass substrate, total
reflection can provide a trapping force over a wide area Mellor and Bain
(2006); Mellor et al. (2006); Taylor et al. (2008). In addition to such design
of incident light, micro-scale fabrication has also achieved the trapping of
many particles, e.g., plasmonic structures Righini et al. (2007); Pang and
Gordon (2012) and photonic crystals Rahmani and Chaumet (2006); Yang et al.
(2009); Jaquay et al. (2013). The formation of an ordered monolayer of
particles at a liquid–liquid interface has also been reported Aveyard et al.
(2002); Park and Furst (2008).

Optical binding is a key concept for the optical manipulation of many
particles Depasse and Vigoureux (1994); Forbes et al. (2020). Optically
induced polarizations cause attractive or repulsive forces between the
particles, which results in an ordering of the particles Demergis and Florin
(2012). Yan et al. investigated the formation of a nanoparticle array under
wide-area laser irradiation with circular Han et al. (2018) and linear
polarizations Yan et al. (2013, 2014), where the ordering of particles
depended on the type of polarization. The spin angular momentum (SAM) of
circular polarization gives a torque to the array, although the mechanism of
this torque transfer is still unclear.

Recent reports by Kudo et al. Kudo et al. (2016); Wang et al. (2016); Kudo et
al. (2018) demonstrated new possibilities of optical manipulation and optical
binding. They examined the trapping of many nanoparticles by a tightly focused
single laser, where trapped particles showed tetragonal or hexagonal ordered
arrays depending on the polarization. These arrays were beyond the irradiation
area, like a crystal growing Sugiyama et al. (2007). Moreover, outside of the
focal area, polystyrene particles form additional horns Kudo et al. (2016) and
gold particles show revolution and swarming dynamics Kudo et al. (2018). This
implies that focal irradiation and scattering fields from the particles cause
self-organized and indirect optical binding, which is in significant contrast
with conventional optical binding with a wide area of irradiation. The
mechanism of such indirect binding of multiple particles is unknown at
present. Its elucidation will lead to an unconventional scheme for creating
and manipulating wide-area ordered structures, implementing finely controlled
local properties with scanning beams and rich extension of optical binding by
a combination of multiple focused lasers.

In this Letter, we theoretically investigates the mechanism of the optical
binding and dynamics of nanoparticles due to indirect optical force under the
irradiation of a single, tightly focused, and circularly polarized laser.
Considering gold nanoparticles, we numerically demonstrate the revolution of
surrounding particles. The binding position is determined by the field
intensity, whereas the revolution is explained by a “spin–orbit (SO) coupling”
of light. An analysis based on the SAM and orbital angular momentum (OAM) of
light elucidates three factors: large scattering and suppressed absorption
cross-sections, conversion of SAM to OAM by a single particle, and imbalance
of positive and negative OAM generation. The first and second factors are
determined by the individual particle’s properties, which accelerates the
revolution. The revolution should relate to a vortical flow of the momentum of
light. The optical current discussed by Berry Berry (2009) describes such
momentum flow of light. The interference between incident and scattered field
clearly exhibits a vortical structure of the optical current, as shown in
Appendix B. The last factor is affected by the particle configurations as well
as properties. Our results imply that the imbalance of OAM will reveal the
revolution direction.

##  II Model

Considering the experiment reported in Ref. Kudo et al. (2018), we assume the
following model and conditions. We consider nanoscale spherical gold particles
(refractive index: $n_{\rm Au}\simeq 0.258+6.97i$) in a water solvent ($n_{\rm
w}\simeq 1.33$) on a glass substrate. 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 Gaussian beam Richards and Wolf (1959); Zhao et al. (2007);
Novotny and Hecht (2006) with an approximate spot size of $2\omega_{0}\simeq
800\,\mathrm{nm}$. 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 $NA=n_{\rm w}\sin(2/(k\omega_{0}))\simeq 0.996$. The focal point is set on
the substrate surface. The incident light propagates from water to glass (see
Figs. 1(a) and (b)). The particles move and are optically manipulated on the
substrate. An effect of charge on the particles is not essential and is not
considered. The details of the methodology are summarized at the end of this
Letter, after the concluding remarks.

Figure 1:  (a) Schematic view of optical trapping and binding by a single
focused laser accompanying by the generation of OAM, and (b) model of the
numerical simulation for $d=200\,\mathrm{nm}$ diameter Au particles in water.
(c)-(g) Intensity profiles of total electric field with $N_{\rm p}=2$ (c), $5$
(d), $7$ (e), and $8$ (f,g) on the $z=0$ plane. The incident laser is tightly
focused with circular polarization. The focal point is at the origin. Black
lines in the panels indicate trajectories of particles in a finite period.

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 Mackowski (1994); Mackowski and
Mishchenko (1996, 2011):

$\bm{E}_{\rm tot}(\bm{r})=\bm{E}_{\rm inc}(\bm{r})+\sum_{i=1}^{N_{\rm
p}}\bm{E}_{{\rm sca},i}(\bm{r}),$ (1)

where $\bm{E}_{\rm inc}$ and $\bm{E}_{{\rm sca},i}$ represent the incident
light and scattered light from particle $i$, respectively. $N_{\rm p}$ is the
number of particles. The optical force $\bm{F}_{i}$ is calculated from the
total electromagnetic field $\bm{E}_{\rm tot}$ via Maxwell’s stress tensor as

$\bm{F}_{i}=\oint_{S_{i}}d\Omega\cdot\left(\bar{T}_{\rm E}+\bar{T}_{\rm
B}\right).$ (2)

The integral is over the surface of particle $i$ Datsyuk and Pavlyniuk (2015).
The simulation of particle dynamics follows the Langevin equation,

$m\frac{d^{2}\bm{r}_{i}}{dt^{2}}=-\zeta\frac{d\bm{r}_{i}}{dt}+\bm{F}_{i}+\bm{\xi}_{i},$
(3)

with $\zeta=3\pi\eta_{\rm w}d$ being the friction coefficient. $\bm{\xi}_{i}$
represents the Gaussian random force due to the Brownian motion of water
molecules. All particles are identical and their mass is $m$. The viscosity of
water is assumed to be $\eta_{\rm w}=0.890\,\mathrm{mPa\cdot s}$ at room
temperature. The optical force on one particle accelerates the other particles
due to the hydrodynamic interaction. This is accounted for by the
Ront–Prager–Yamakawa mobility tensor Rotne and Prager (1969); Yamakawa (1970);
Happel and Brenner (1983) (see method), which gives additional velocities to
particle $i$ from the external force acting on the other $j\neq i$:

$\Delta\bm{v}_{i}=\sum_{j\neq i}\tilde{\bm{\mu}}_{ij}\bm{F}_{j}.$ (4)

##  III Results and discussion

###  III.1 Dynamics of gold particles

We use the simulated results to reveal the indirect mechanism. The multiple
scattering and interference with the incident field are essential. The binding
distance approximately corresponds to the wavelength of light. The dynamics is
faster with an increase in the number of bound particles. The observed motion
is elucidated based on the “SO coupling” of light and the optical current
Berry (2009), as explained below. The correlation between SAM and OAM has
previously been discussed for the rotational optical manipulation by the
optical vortex with OAM Tamura et al. (2019). However, it should be noted that
SO coupling plays an essential role even when light with only SAM is injected
if the targeted matter system has a geometrical structure.

Figures 1(c)–(g) show a snapshot of the intensity of the total electric field
with $N_{\rm p}=2\sim 8$ particles and their trajectories when the circular
polarization of the incident laser is applied. At the center of the focal
area, one particle is optically trapped directly by the incident laser. The
trapped particle causes a scattering of incident laser. Then, due to
interference, the light intensity shows a ring-shaped oscillation (Fig. 1(c)
and Fig. 4 in Appendix A). This oscillation results in the binding of
surrounding particles by an indirect mechanism in the vicinity of the local
maximum of the intensity at $r\simeq\lambda/n_{\rm w}$ from the center
particle. The map of the optical force on an additional small particle shown
in Fig. 5(b) in Appendix B clearly indicates the position of binding due to
the indirect mechanism. How the added particles are bounded one-by-one is
shown in Figs. 1(d)–(f). For six surrounding particles, the bound position
from the center is $\Delta\approx 853.4\,\mathrm{nm}$. The intervals between
the particles are approximately equivalent with $\lambda/n_{\rm w}$. The
particles show a hexagonal distribution, which indicates that our simulation
well reproduces the experimental observation in Ref. Kudo et al. (2018). When
$N_{\rm p}\geq 8$, we find two semi-stable distributions, as shown in Figs.
1(f) and (g). Figure 1(g) shows the optical binding at the second neighboring
position (see also the binding position in Fig. 5(d) in Appendix B). This is
not at the local maximum of the intensity and suggests the indirect mechanism
of binding. As a side note, the case of a linear polarized laser is discussed
in Appendix C, where the particles are aligned in a direction perpendicular to
the polarization. This result is a good demonstration that our simulation
reasonably explains experimental observations Kudo et al. (2018).

The center particle at the focal point is strongly trapped and hardly moves,
whereas the other surrounding particles revolve by the optical force. The
lines in Figs. 1(c)–(g) indicate a slight trajectory of the revolution. The
direction of revolution accords with the optical current shown in Figs. 5(a)
and (c), which is in contrast with the map of force on an additional small
particle in Figs. 5(b) and (d). The speed of revolving particles increases
with the number of surrounding particles. This acceleration is attributed to
an enhancement of the multiple scattering of light. When $N_{\rm p}=7$ in Fig.
1(e), the hexagonal distribution is approximately maintained during the
revolution, although the particles are affected by the random force. The
motions are shown in Movie of Appendix. In Fig. 1(g), the second neighboring
particle is much slower than the first neighboring particles because of the
decreased light intensity.

The optical force is evaluated by the square of the field, $|\bm{E}_{\rm
inc}+\sum_{i}\bm{E}_{{\rm sca},i}|^{2}$. The interference term is significant
for the binding and revolution, whereas $|\bm{E}_{\rm inc}|^{2}$ and
$\sum_{i}|\bm{E}_{{\rm sca},i}|^{2}$ might result in force in the radial and
out-of-plane directions. Because the scattered field is larger than or
comparable to the incident field at the binding position, the self-assembly
process is critical for indirect optical manipulation. It has the possibility
to achieve more various configurations of particles by utilizing the internal
degrees of freedom of light.

###  III.2 Angular momentum conversion

For the dynamics, we analyze the SAM and OAM components of scattered light on
the upper and lower celestial hemispheres at $r=R_{\rm c}\gg d$:

$\displaystyle C_{\sigma,l}$ $\displaystyle=$ $\displaystyle
C_{\sigma,l}^{\rm(u)}+C_{\sigma,l}^{\rm(l)}$ (5) $\displaystyle
C_{\sigma,l}^{\rm(u)}$ $\displaystyle=$
$\displaystyle\frac{1}{C^{(0)}}\int_{0}^{\pi/2}\sin\theta d\theta$ (6)
$\displaystyle\times\left|\int_{0}^{2\pi}d\phi\left\\{\bm{e}_{\sigma,l}(\theta,\phi)e^{iR_{\rm
c}}\right\\}^{\ast}\cdot\bm{E}_{\rm tot}(R_{\rm c},\theta,\phi)\right|,$
$\displaystyle C_{\sigma,l}^{\rm(l)}$ $\displaystyle=$
$\displaystyle\frac{1}{C^{(0)}}\int_{\pi/2}^{\pi}\sin\theta d\theta$ (7)
$\displaystyle\times\left|\int_{0}^{2\pi}d\phi\left\\{\bm{e}_{\sigma,l}(\theta,\phi)e^{iR_{\rm
c}}\right\\}^{\ast}\right.$ $\displaystyle\cdot\left\\{\bm{E}_{\rm tot}(R_{\rm
c},\theta,\phi)-\bm{E}_{\rm inc}(R_{\rm c},\theta,\phi)\right\\}\Big{|}.$

Here, $\bm{e}_{\sigma,l}(\theta,\phi)$ corresponds to the mode of the electric
field with spin $\sigma$ and vortex $l$ with respect to the $z$-axis (see
definition in Appendix D). $C^{(0)}$ is a normalization factor to satisfy
$\sum_{\sigma,l}{C_{\sigma,l}}^{2}=1$. Note that the subtraction in Eq. (7) is
to consider only the emission.

In the case of a plane wave, the light has no OAM ($l=0$) Zhao et al. (2007).
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. The incident laser consists of
$C_{+1,0}\approx 0.9750$ and $C_{-1,+2}\approx 0.2222$ when
$2\omega_{0}=800\,\mathrm{nm}$. Note that the sum of their squares is unity.

Figure 2:  (a) Coefficients of SAM and OAM components of scattered field with
$N_{\rm p}=1,7$ when $\sigma=+1$ circular polarized light is tightly focused
and applied. $N_{\rm p}=7$ and the other parameters correspond to Fig. 1(e).
(b) Log scale plot of (a). (c,d) Particle distance dependence of the
coefficients when $N_{\rm p}=7$. (e) Summary of the coefficients for the SAM
and OAM defined in Eqs. (8) and (9) and the sentence that follows them.

Scattering by a particle or particle assembly produces a further $C_{-1,+2}$
component. Figures 2(a) and (b) exhibit the coefficient $C_{\sigma,l}$ of
light scattered when $N_{\rm p}=1$ and $7$. For a single particle, the
scattered light shows $C_{-1,+2}>0.3$, and the others of $\sigma=-1$ are zero.
This represents a conversion of the SAM to the OAM under the conservation of
total angular momentum (TAM), $j=l+\sigma$. This is regarded as SO coupling
for the light. However, it is difficult to determine the orbital motion due to
a strong trapping.

In the situation of Figs. 1(c)–(g), the rotational symmetry is broken and the
TAM of light is not conserved. In Fig. 1(e), the system has a six-hold
rotational symmetry and the scattered light consists of $l=\pm 6m$ with
$\sigma=+1$ (where $m$ 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
$1\pm 6m$. Here, note that the distribution of TAM (mainly OAM) is imbalanced
and, the coefficient of $j=1-6$ is larger than that of $j=1+6$. This is
consistent with the rotation of assembly if the particles and photons follow
Newton’s third law of motion.

Here, we claim that the key elements of the particle dynamics are the spin-
flip by the “SO coupling” and the imbalance of OAM generation. Let us discuss
the SAM and OAM components in the parameter space of the particle distance and
the complex refractive index. We introduce

$\displaystyle C_{\pm}$ $\displaystyle=$
$\displaystyle\left[{\textstyle\sum_{l}}{C_{\sigma=\pm
1,l}}^{2}\right]^{\frac{1}{2}},$ (8) $\displaystyle C_{>(<)}$ $\displaystyle=$
$\displaystyle\left[{C_{+1,>(<)}}^{2}+{C_{-1,>(<)}}^{2}\right]^{\frac{1}{2}}$
(9)

with ${C_{+1,>(<)}}=[\sum_{l>0(<0)}{C_{+1,l}}^{2}]^{1/2}$ and
${C_{-1,>(<)}}=[\sum_{l>2(<2)}{C_{-1,l}}^{2}]^{1/2}$. Note that $C_{\pm}$ and
$C_{>(<)}$ are indicators of the extent of the spin-flip and imbalance of OAM
generation, respectively, which are summarized in Fig. 2(e).

First, we examine the case that the particle distance $\Delta$ changes
virtually, as shown in Figs. 2(c) and (d). The oscillations of the
coefficients are seen in these figures. Here, we can see why the rotation is
strongly driven in the considered system. Figure 2(c) shows that the spin-flip
coefficients generating negative $l$, namely $C_{\pm 1,<}$, are enhanced
around $\Delta$ corresponding to the light wavelength in water and its half,
whereas the maximum positions of the coefficients generating positive $l$,
namely $C_{\pm 1,>}$ are shifted, i.e., oscillate in a different phase, and
$C_{\pm 1,>}$ becomes minimum around $\Delta\approx$ wavelength. This is why
the imbalance of OAM becomes maximum (namely, $C_{>}/C_{<}$ becomes minimum)
there. At the binding position of our calculation, the OAM is generated with
sufficient imbalance to drive the revolution of surrounding particles in a
particular direction. The oscillation period suggests that the interference of
multiple scattering of light by the particle’s configuration determines the
generation of OAM.

Figure 2(d) shows that $C_{-}/C_{+}$ is insensitive to $\Delta$, except in the
region of small $\Delta$, and that $C_{>}<C_{<}$ is maintained. The latter
observation agrees with the absence of negative torque for a tightly focused
laser, which is in contrast with the case of wide-area irradiation Han et al.
(2018). In the case of the wide-area irradiation, the interference between the
incident and first-order scattering fields from the respective particles is
significant rather than the multiple scattering in the present case; the maps
of optical current and force are shown in Fig. 6. The rotation behavior
arising from the interference of scattered light can also be discussed in
terms of the optical current. The relevant figures and discussion are provided
in Appendix B.

Figure 3:  (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
$2\omega_{0}=0.8\,\mathrm{\mu m}$ and $12.5\,\mathrm{\mu m}$, 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. 1(e)). (c,d) Spin-
flip ratio due to the “SO coupling” (c) and imbalance of OAM generation (d)
evaluated from the coefficients of SAM and OAM. (e,f) Single particle cross-
section of the scattering $C_{\rm sca}$ (e) and absorption $C_{\rm abs}$ (f).
The plot is normalized by $\pi R^{2}$, where $R$ is the radius.

As shown in Fig. 2 (d), $C_{-}/C_{+}$ is insensitive to the particle distance.
However, it remarkably depends on the particle material, specifically its
complex refractive index, as discussed below. Here, we discuss the spin-flip
and OAM imbalance ratios by examining the indirect optical force, scattering,
and absorption cross-sections in a plane of the complex refractive index.
Figure 3(a) exhibits the indirect optical force driving the surrounding
particles in the situation of Fig. 1(e). Positive force corresponds to the
counterclockwise direction in Fig. 1(e). The optical force is enlarged when
the refractive index is almost purely imaginary (perfect conductor), and the
imaginary part is $\kappa\approx 2.67$. The force is also enlarged around
$n\approx 6.5,\kappa\ll 1$.

For a tightly focused laser, the indirect optical force is only positive in
the $n$-$\kappa$ plane, whereas both positive and negative force is found when
the light is applied widely [Fig. 3(b)]. 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 negative torque in direct optical
binding with wide-area irradiation is consistent with Ref. Han et al. (2018).

The “SO coupling” is also examined in the $n$-$\kappa$ plane for the seven
particles in Fig. 3(c). The profile of spin-flip ratio $C_{-}/C_{+}$ shows
similar behavior to the indirect optical force, where a region of spin-flip
enhancement correlates with a large indirect force region. Meanwhile, the
difference of generated OAM, $C_{>}-C_{<}$, in Fig. 3(d) shows that the sign
changes and there are positive and negative peaks at $n\ll 1,\kappa\approx
6.5$ and $n\approx 8.6,\kappa\ll 1$, respectively. This result indicates that
negative torque can appear if the material is different, even in the case of
focused irradiation.

Figures 3(e) and (f) exhibit the scattering and absorption cross-sections of a
single particle in water, respectively, evaluated by the Mie coefficients
$a_{n}$ and $b_{n}$ Bohren and Huffman (1998) as follows:

$\displaystyle C_{\rm sca}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{k^{2}}\sum_{n=1}^{\infty}(2n+1)(|a_{n}|^{2}+|b_{n}|^{2}),$
(10) $\displaystyle C_{\rm ext}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{k^{2}}\sum_{n=1}^{\infty}(2n+1){\rm
Re}(a_{n}+b_{n}),$ (11) $\displaystyle C_{\rm abs}$ $\displaystyle=$
$\displaystyle C_{\rm ext}-C_{\rm sca}.$ (12)

The profile of $C_{\rm sca}$, especially in the enhancement parameter regions,
shows good agreement with the indirect optical force in Fig. 3(a). Meanwhile,
$C_{\rm abs}$ is not enlarged at $n\ll 1$. Therefore, by comparing Fig. 3(a)
with (c) and (e), we can see that the enhancements of scattering and “SO
coupling” are dominant factors in the indirect optical manipulation.

The above discussion gives a guideline for indirect and wide-area optical
manipulation by multiple scattering, such that one should prepare particles
and conditions with 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 multiple
scattering is suppressed. For the case of $d=200\,\mathrm{nm}$, $C_{\rm
sca}\gg C_{\rm abs}\approx 0$ is a better condition for indirect optical
manipulation. The particle size is also an important parameter. In Appendix E,
we investigate the indirect force and cross-sections for various diameters
(see Fig. 8). We find that the criterion to obtain large indirect optical
force is adaptable for $d\gtrsim 150\,\mathrm{nm}$.

##  IV Conclusions

In conclusion, we conducted numerical simulations of nanoparticles optically
trapped and bound by a single focused laser. The simulation revealed that the
scattered light from the strongly trapped center particle causes binding and
indirect optical force on the surrounding particles. Due to the multiple
scattering between all particles, the ordering and optical manipulation of
particles can be achieved beyond the focal irradiation area. Under circularly
polarized laser irradiation, a hexagonal ordering of particles with wavelength
distance is formed, and the surrounding particles revolve. The simulated
results qualitatively agree well with recent experimental observations Kudo et
al. (2018).

Based on the analysis of SAM and OAM, we revealed the mechanisms to enlarge
the revolution in the indirect optical manipulation: $C_{\rm sca}\gg C_{\rm
abs}$ and large “SO coupling” of light, which must be related with each other.
How a strong SO coupling contributes to the revolution is also elucidated
schematically in Fig. 9 in Appendix F. This is determined by the particle
properties , i.e., complex refractive index, diameter, shape, etc. Their
engineering may be possible by using core-shell structures or dye-doped
polymers. The imbalance of OAM generation is another important factor for both
indirect and direct optical manipulation, which is largely affected by the
particle placement.

As we discuss in Appendix B, the optical binding and dynamics of particles are
described well by the optical force and current, respectively. The structure
of optical current is induced by interference between the scattering and
incident fields. Thus, strong scattering enlarges the dynamics.

The present results indicate the possibility that a scanning single beam can
create an ordered pattern with local structures over a wide area by
controlling the polarization. If we use several beams simultaneously with
different phases, the degrees of freedom for designing the pattered structures
will be greatly enhanced. We suggest that such large degrees of freedom would
open the possibility of new optical manipulation, such as a phase transition
of the order of optical binding, as will be elaborated on in our next
publication.

##  V Method

The remainder of this Letter details our methodology. We simulated the
dynamics of nanoscale spherical gold particles in a water solvent on a glass
substrate. A single focal incident laser with counterclockwise circular
polarization was considered. The electric field was modeled as a Gaussian beam
Richards and Wolf (1959); Zhao et al. (2007); Novotny and Hecht (2006):

$\bm{E}_{\rm
inc}(r,\varphi,z)=-\frac{ikf}{2}\sqrt{\frac{n_{1}}{n_{2}}}E_{0}e^{-ikf}\left[\begin{matrix}I_{00}+I_{02}e^{i2\varphi}\\\
i(I_{00}-I_{02}e^{i2\varphi})\\\ -2iI_{01}e^{i\varphi}\end{matrix}\right]$
(13)

with

$\displaystyle I_{00}$ $\displaystyle=$ $\displaystyle\int_{0}^{\theta_{\rm
max}}d\theta f_{w}(\theta)\sqrt{\cos\theta}\sin\theta(1+\cos\theta)$ (14)
$\displaystyle\times J_{0}(kr\sin\theta)e^{ikz\cos\theta},$ $\displaystyle
I_{01}$ $\displaystyle=$ $\displaystyle\int_{0}^{\theta_{\rm max}}d\theta
f_{w}(\theta)\sqrt{\cos\theta}\sin^{2}\theta$ (15) $\displaystyle\times
J_{1}(kr\sin\theta)e^{ikz\cos\theta},$ $\displaystyle I_{02}$ $\displaystyle=$
$\displaystyle\int_{0}^{\theta_{\rm max}}d\theta
f_{w}(\theta)\sqrt{\cos\theta}\sin\theta(1-\cos\theta)$ (16)
$\displaystyle\times J_{2}(kr\sin\theta)e^{ikz\cos\theta},$

where $f_{w}(\theta)=\exp(-(\sin\theta/\sin\theta_{\rm max})^{2})$. $J_{n}(x)$
is the $n$-the order Bessel function. The parameters $f$, $\theta_{\rm max}$,
and $k$ represent the focal distance, maximal half-angle of light cone, and
wavenumber in a vacuum, respectively.

The incident field is expanded by the vector spherical harmonics (VSH)
functions, $\bm{M}_{nmp,i}^{(1)}$ and $\bm{N}_{nmp,i}^{(1)}$, as

$\displaystyle\bm{E}_{\rm inc}(\bm{r})$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{n_{\rm max}}\sum_{m=-n}^{+n}\sum_{p={\rm
e,m}}\left[u_{nmp,i}\bm{M}_{nmp,i}^{(1)}(\bm{r})\right.$ (17) $\displaystyle+$
$\displaystyle\left.v_{nmp,i}\bm{N}_{nmp,i}^{(1)}(\bm{r})\right].$

Here, the superscript $(1)$ in $\bm{M}_{nmp,i}^{(1)}$ and
$\bm{N}_{nmp,i}^{(1)}$ 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 $\bm{r}_{i}$, 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 Mackowski (1994). The expansion of $\bm{E}_{\rm inc}$ by
$\bm{M}_{nmp,i}^{(1)}$ and $\bm{N}_{nmp,i}^{(1)}$ is applied to its scattering
by particle $i$.

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

$\displaystyle\bm{E}_{{\rm sca},i}^{(0)}(\bm{r})$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{n_{\rm
max}}\sum_{m=-n}^{+n}\left[a_{n}u_{nmp,i}\bm{M}_{nmp,i}^{(3)}(\bm{r})\right.$
(18) $\displaystyle+$
$\displaystyle\left.b_{n}v_{nmp,i}\bm{N}_{nmp,i}^{(3)}(\bm{r})\right]$

with the Mie coefficients $a_{n}$ and $b_{n}$ being independent of $i$. Note
that the radial part of $\bm{M}_{nmp,i}^{(3)}$ and $\bm{N}_{nmp,i}^{(3)}$ is
given by the spherical Hankel function of the first kind to describe the
outward spherical wave. By introducing vectors $\vec{c}_{{\rm
inc},i}=(\\{u_{nmp,i},v_{nmp,i}\\})^{\rm t}$ and $\vec{c}_{{\rm sca},i}^{\
(0)}=(\\{a_{n}u_{nmp,i},b_{n}v_{nmp,i}\\})^{\rm t}$ for the incident and
scattered fields, respectively, the Mie coefficients give the T-matrix for a
single particle, $\vec{c}_{{\rm sca},i}^{\ (0)}=\hat{t}_{i}\vec{c}_{{\rm
inc},i}$.

The multiple scatterings between the particles combine the T-matrices
$\hat{t}_{i}$ and result in generalized T-matrix elements $\hat{T}_{ij}$
Mackowski (1994); Mackowski and Mishchenko (1996, 2011). An extension of the
vector, $\vec{C}_{\rm inc}=(\vec{c}_{{\rm inc},1},\vec{c}_{{\rm
inc},2},\cdots)^{\rm t}$, shows a simple formulation of the multiple
scattering:

$\vec{C}_{\rm sca}=\left(\begin{matrix}\vec{c}_{{\rm sca},1}\\\ \vec{c}_{{\rm
sca},2}\\\
\vdots\end{matrix}\right)=\left(\begin{matrix}\hat{T}_{11}&\hat{T}_{12}&\\\
\hat{T}_{21}&\hat{T}_{22}&\\\ &&\ddots\end{matrix}\right)\vec{C}_{\rm inc}.$
(19)

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

$\displaystyle\bm{E}_{{\rm sca},i}(\bm{r})$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{n_{\rm
max}}\sum_{m=-n}^{+n}\left[A_{nmp,i}\bm{M}_{nmp,i}^{(3)}(\bm{r})\right.$ (20)
$\displaystyle+$
$\displaystyle\left.B_{nmp,i}\bm{N}_{nmp,i}^{(3)}(\bm{r})\right].$

## ACKNOWLEDGMENT

The authors thank Prof. H. Masuhara, Dr. T. Kudo, and Z.-H. Huang for their
fruitful discussions on their experimental results. The authors acknowledge
for all member of Collective Optofluidic Dynamics of Nanoparticles meeting
organized by Prof. Masuhara. This work was supported by JSPS KAKENHI Grant
Number 16K21732 in Scientific Research on Innovative Areas “Nano-Material
Optical-Manipulation”. T.Y. was supported by JSPS KAKENHI Grant Number
18K13484 and H.I. was supported by JSPS KAKENHI Grant Number 16H06504 and
18H01151.

##  Appendix A Profile of incident and total fields

In our discussion on indirect optical manipulation under tightly focused laser
irradiation, the scattered field by a central trapped particle is larger than
or comparable with the incident field at the binding position of surrounding
particles. Therefore, the multiple scattering of light between particles is
significant. To show the field profile explicitly in our setup with $NA\simeq
0.996$, we plot the intensities of total electric fields in Fig. 4 when
$N_{\rm p}=0$ (incident field), $1$, and $7$. The incident field is a Gaussian
beam applying localized approximation with the Gaussian beam constant $C_{\rm
GB}$ being $0.51$. Although this value is beyond an appropriate range of
approximation with $C_{\rm GB}=1/(k\omega_{0})$ to estimate the beam waist
$\omega_{0}$, it is chosen to produce $\omega_{0}\approx 800\,\mathrm{nm}$;
see the plot of $N_{\rm p}=0$ in Fig. 4.

When the light is applied to a single particle, the intensity oscillates as a
function of the distance $r$ from the focal center due to interference between
the incident and scattered fields. At $r\simeq 850\,\mathrm{nm}$, the
intensity indicates the first maximum, where the gradient force is zero and
the particle could be bound optically. At the local maximum, the total field
intensity is larger than the incident one. In the case of $N_{\rm p}=7$, the
profile plot of field intensity follows a bisector between two of the
surrounding particles (for the case of Fig. 1(e), along the $y$ axis). The
oscillations of intensity along the radial direction for $N_{\rm p}=1$ and $7$
are almost equivalent.

We also plot the intensity of the magnetic field. The magnetic field intensity
is also larger than or comparable with that of incident light. Therefore, a
scattering-induced component of the Poynting vector describing the optical
current is comparable with the component due to the incident field.

Figure 4:  Profile of the intensities of total electric and magnetic fields,
$|\bm{E}_{\rm tot}|^{2}$ and $|\bm{H}_{\rm tot}|^{2}$, for a circular
polarization when $N_{\rm p}=0,1$, and $7$. The focal point of the incident
laser is located at the origin. The intensities are normalized by the incident
field at the focal point, $|\bm{E}_{\rm inc}(r=0)|^{2}={E_{0}}^{2}$ and
$|\bm{H}_{\rm inc}(r=0)|^{2}={H_{0}}^{2}$. The particles are gold with
$d=200\,\mathrm{nm}$ diameter.

##  Appendix B Optical current vs. optical force

In the main text and Fig. 2, we examine the analysis of scattered light in
terms of the SAM and OSM to discuss the mechanism of revolution by the
indirect optical force. In our simulation with gold particles and a tightly
focused laser, the imbalance of generated OAM is consistent with the
revolution direction. To consider a mechanism to determine the revolution
direction, as shown in Fig. 5, we examine maps of the optical current
described by the Poynting vector $\bm{S}=\bm{E}^{*}\times\bm{H}/2$ and the
optical force acting on an additional small particle. Note that the optical
force acts on all the present particles. To consider an “ optical force at an
arbitrary position”, we introduce a particle, which is $d=10\,\mathrm{nm}$ to
avoid additional light scattering. The optical current describes the
contribution of the scattering force, whereas the optical force consists of
both the scattering and gradient forces.

Figure 5:  Poynting vectors indicating the optical current (left) and optical
force on a small additional particle (right panels) in the $z=0$ plane when
$N_{\rm p}=1$ (a,b) and $7$ (c,d) under tightly focused laser irradiation. The
incident laser has $\sigma=+1$ circular polarization. The beam waist is
$0.8\,\mathrm{\mu m}$. The other parameters of the laser and particles are the
same as those in Fig. 1(e) in the main text. In the right panels, the
additional particle is $d=10\,\mathrm{nm}$ to avoid multiple scattering by the
additional particle. The color scale displays the absolute value of vectors,
$\sqrt{F_{x}^{2}+F_{y}^{2}}$. The black circle indicates Au particles with
$d=200\,\mathrm{nm}$.

The optical current of the incident laser is almost along the incident
direction, and the in-plane components are negligible even for a tightly
focused laser (not shown). With the scattered field, the optical current
exhibits the spatial structures. When one particle is trapped at the focal
point, the optical current of the total field shows a vortical structure
according to the incident $\sigma=+1$ polarization in Fig. 5(a), whereas the
optical force is in the radial direction in Fig. 5(b). This is due to much
stronger gradient force than the scattering force. The optical force indicates
the binding position clearly, while the optical current does not. Therefore,
for the binding, one should discuss the optical force, whereas for the
dynamics or indirect manipulation, the optical current gives reasonable
information. The optical current with seven particles in Fig. 5(c) shows only
slight difference from that with one particle in (a) at a point of the
vortical flow to explain the revolution. The qualitative difference is an
enhancement of the optical currents by multiple scattering in the vicinity of
surrounding particles. This result qualitatively explains the direction of
revolution in Fig. 1(e) in the main text, whereas the force in Fig. 5(d) is
not useful to discuss the dynamics.

Figure 6:  Poynting vectors indicating the optical current (left) and optical
force on a small additional particle (right panels) in the $z=0$ plane when
$N_{\rm p}=1$ (a,b) and $7$ (c,d) under wide-area irradiation. The incident
laser has $\sigma=+1$ circular polarization. The beam waist is
$12.5\,\mathrm{\mu m}$. The other parameters of the laser and particles are
the same as those in Fig. 1(e) in the main text. The upper four panels show
the case of tightly focused laser irradiation. The lower four panels are the
non-focused case. In the right panels, the additional particle is
$d=10\,\mathrm{nm}$ to avoid multiple scattering by the additional particle.
The color scale displays the absolute value of vectors,
$\sqrt{F_{x}^{2}+F_{y}^{2}}$. The black circle indicates Au particles with
$d=200\,\mathrm{nm}$.

Figures 6(a)–(d) demonstrate widely focused irradiation. When $N_{\rm p}=1$ in
Fig. 6(a), the interference between the incident and scattered fields
indicates the changes of vortex direction of the optical current with the
radial distance $r$ from the focal point. This is also found in the optical
force in Fig. 6(b). Such behaviors qualitatively agree with the observation of
negative torque reported in Ref. Han et al. (2018). When $N_{\rm p}=7$, the
optical current implies a rotation of individual particles in Fig. 6(c),
whereas the dynamics of whole assembly is not readable. The optical force in
Fig. 6(d) shows the next binding positions, which are slightly different from
the second stable position in Fig. 6(b). Neither Figs. 6(c) nor (d) exhibit
binding or revolution, unlike the tightly focused case. For the direct optical
manipulation with a wide irradiation, one should first examine the optical
current and force by the scattering from a single particle rather than
multiple scattering.

The optical current suggests the absence and presence of negative torque by
tight and wide focusing, respectively. Thus, negative torque would be obtained
by properly tuning the particle distance by a given charge or other
techniques.

##  Appendix C Linear polarization

In contrast to circular polarization, a linear polarized laser does not have
SAM. Then, indirect optical manipulation by linear polarization must show a
qualitative difference from the circular polarized laser discussed in the main
text. Here, we consider the indirect optical binding by a focused linear
polarized laser and demonstrate a stable alignment of multiple particles,
which agrees with the experiment by Kudo et al. Kudo et al. (2018).

The parameters of the laser are the same as those for circular polarization.
Figure 7 shows the stable position of bound particles when $N_{\rm p}$
increases one by one. The polarization direction is the $x$-direction. In the
case of linear polarization, the bound particles are fixed in a finite period
and change their positions by the random force. Thus, Fig. 7 shows an example
of possible configurations. When $N_{\rm p}=2$, the “second” particle is
trapped at a slightly shifted position from the perpendicular direction (on
the $y$-axis), which may be attributed to the shape of incident laser shown in
the inset of Fig. 7(a). We can also find other stable trapping positions. The
“third” particle is trapped at another position. When the number of particles
increases, their shifts from the $y$-axis are reduced and they tend to be
aligned on a line. Such configurations were found in an experiment Kudo et al.
(2018).

For the circular polarization, we investigate the analysis of scattered fields
in terms of the SAM and OAM and reveal that the “SO coupling” of light and
imbalance of the generated OAM are significant for the revolution of the
surrounding particles. Meanwhile, for linear polarization, the indirect
binding is static. Although the scattering of linear polarized light by the
bound particles also causes OAM, it is balanced, which follows the stable
optical binding shown in Fig. 7.

Figure 7:  Intensity of total electric field with $N_{\rm p}=2$ (a), $3$ (b),
and $5$ (c) on the $z=0$ plane when an $x$-linearly polarized and focused
single laser is applied. The focal point of the incident laser is located at
the origin. The particles are stably trapped. The inset of panel (a) shows the
incident field.

##  Appendix D Definition of $\bm{e}_{\sigma,l}(\theta,\phi)$ for scattered
field

We analyze the scattered light from all particles in terms of the SAM and OAM.
The scattered light is a spherical wave at a sufficiently far position.
Therefore, to define the axis of angular momenta, the scattered field must be
converted to a plane wave. We consider a fictitious lens for this conversion.
Through the lens, the field at ($r=R_{\rm c},\theta,\phi$) is rewritten as

$\bm{E}_{\rm tot}\to\hat{R}_{y}^{-1}\hat{R}_{z}^{-1}\bm{E}_{\rm tot}$ (21)

with

$\displaystyle\hat{R}_{z}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\cos\phi&-\sin\phi&0\\\
\sin\phi&\cos\phi&0\\\ 0&0&1\end{matrix}\right),$ (22)
$\displaystyle\hat{R}_{y}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\cos\phi&0&\sin\phi\\\ 0&1&0\\\
-\sin\phi&0&\cos\phi\end{matrix}\right)$ (23)

being the rotation matrices with respect to the $z$ and $y$-axes. The
converted field is projected onto $\bm{e}_{\sigma=\pm 1}=(1,\pm i,0)^{\rm
t}/\sqrt{2}$ for $\sigma$ SAM with $l$ OAM, $\bm{e}_{\sigma}e^{il\phi}$. Thus,
the unit vector $\bm{e}_{\sigma,l}(\theta,\phi)$ in Eqs. (6) and (7) is given
as

$\bm{e}_{\sigma,l}(\theta,\phi)=\hat{R}_{z}\hat{R}_{y}\bm{e}_{\sigma}e^{il\phi}.$
(24)

##  Appendix E Particle size dependence of indirect optical force and cross-
sections

In the main text, we discussed the correlation of the indirect optical force
and cross-sections in the $n$-$\kappa$ plane when the particle diameter is
fixed at $d=200\,\mathrm{nm}$ in Fig. 3. Then, we noted that for the indirect
mechanism, the scattering rather than the absorption is essential. Here, we
examine this criterion when the particle size is changed.

Figure 8:  Indirect optical force and cross-sections in the plane of complex
refractive index, $\tilde{n}=n+i\kappa$, for $d=50\,\mathrm{nm}$(a) to
$350\,\mathrm{nm}$(f), excluding $d=200\,\mathrm{nm}$, when $N_{\rm p}=7$; the
case of $d=200\,\mathrm{nm}$ is shown in Fig. 3 in the main text. The optical
force is projected in the angular direction and evaluated on one surrounding
particle at $r=\Delta=853.4\,\mathrm{nm}$. Both cross-sections are plotted in
a unit of $\pi R^{2}$ with $R=100\,\mathrm{nm}$.

Figure 8 demonstrates the indirect optical force when $N_{\rm p}=7$ and the
scattering and absorption cross-sections of a single particle for
$d=50-350\,\mathrm{nm}$, excluding $d=200\,\mathrm{nm}$; the case of
$d=200\,\mathrm{nm}$ is shown in the main text. Note that the cross-sections
in Fig. 8 are normalized by $\pi(100\,\mathrm{nm})^{2}$ for any $d$. The
optical force is evaluated when the surrounding particles are at
$r=853.4\,\mathrm{nm}$. The force tends to increase with the particle size.
When $d<200\,\mathrm{nm}$, the optical force is much smaller than that of
$d=200\,\mathrm{nm}$, which corresponds to the depression of $C_{\rm sca}$. At
$d>200\,\mathrm{nm}$, the indirect optical force clearly increases with $d$,
whereas the scattering cross-section is increased only slightly. Then, the
scattered field intensity is not affected significantly by the diameter.
Meanwhile, the optical force increases according to the particle volume. The
stable binding distance $\Delta$ from the focal point is slightly affected by
the particle size, and one must evaluate the self-consistently. However, we
fixed the position of surrounding particles at $r=\Delta=853.4\,\mathrm{nm}$
because the structures of force in the $n$-$\kappa$ plane shown in Fig. 3 in
the main text and Figs. 8(a1)–(f1) exhibit no significant change.

The absorption cross-section also depends on the diameter. For large
particles, the absorption cross-section tends to be relatively smaller than
the scattering one. In the $d<200\,\mathrm{nm}$ region, however, we can see a
$C_{\rm sca}<C_{\rm abs}$ region in the plane. From the figures of $d=50$ and
$100\,\mathrm{nm}$, the behavior of indirect optical force in the $n$-$\kappa$
plane is similar to the absorption cross-section, whereas the structures of
optical force correspond to those of $C_{\rm sca}$ for $d\geq
150\,\mathrm{nm}$. Then, we can say that the scattering cross-section is an
essential factor for a large indirect optical force when the scatterers have
sufficient size. However, for tiny particles, $C_{\rm sca}$ is too small and
the absorption dominates the optical force.

The analysis of the SAM and OAM components of “emitted” light can also be
examined. However, discussion on the SAM and OAM with small or large particles
is essentially equivalent to that the main text.

##  Appendix F Scattering of light with spin–orbit coupling

The flip of SAM also explains the acceleration of surrounding particles.
Figure 9 shows the mechanism schematically. When $l=0$ OAM is scattered from
the center particle, the scattered fields at six surrounding particles have
the same phase [see Fig. 9(a)]. Then, the Huygens–Fresnel principle from the
surrounding particles results in light emission in the radial direction, as
shown in Fig. 9(c). Meanwhile, for $l=2$ shown in Fig. 9(b), the scattered
fields at the neighboring two surrounding particles have $4\pi/6$ phase
differences. This phase difference causes a tilt of wavefront of emitted
light, as in Fig. 9(d). This tilt of “light emission” causes a scattering
force and accelerates the surrounding particles.

If the “SO coupling” of light at a single particle is enlarged, the ratio of
$l=2$ scattering from the center particle against $l=0$ scattering increases.
Therefore, by this schematic understanding, stronger SO coupling results in
stronger indirect optical force, which is reasonable from the similarity
between the profiles of Figs. 3(a) and (c) in the main text.

Figure 9:  Phase profile of $l=0$ (a) and $l=2$ OAM (b) incident components.
Black circles show the particles with $d=200\,\mathrm{nm}$ when
$\Delta=853.4\,\mathrm{nm}$. The curve indicating a phase jump at $\pm\pi$ is
artificial redundancy in (b). (c,d) Schematic explanation of force by the
light scattering when $l=0$ (c) and $l=2$ (d) based on the phase of light and
the Huygens–Fresnel principle.

##  Appendix G Description of movie

The movie in this shows a numerical simulation of the dynamics of gold
particles in a water solvent. The parameters of the simulation are the same as
those of Fig. 1(e) in the main text. In this movie, we assume a slight charge
on the particles, which might exist experimentally. However, the presence of
charge on the particles only slightly changes the particle distance and is not
essential for the dynamics.

## ACKNOWLEDGMENT

The authors thank Prof. H. Masuhara, Dr. T. Kudo, and Z.-H. Huang for their
fruitful discussions on their experimental results. The authors acknowledge
for all member of Collective Optofluidic Dynamics of Nanoparticles meeting
organized by Prof. Masuhara. This work was supported by JSPS KAKENHI Grant
Number 16K21732 in Scientific Research on Innovative Areas “Nano-Material
Optical-Manipulation”. T.Y. was supported by JSPS KAKENHI Grant Number
18K13484 and H.I. was supported by JSPS KAKENHI Grant Number 16H06504 and
18H01151.

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