1. Introduction

Maxwell showed, based on the theory of electromagnetism, that when an electromagnetic wave interacts with the induced polarization of an object, it exerts force through its scattering and absorption. This force is sometimes called the ‘optical force’ and is classified into two types: One is the dissipative force, which arises from the transfer of light momentum to a substance by absorption and scattering, and the other is the gradient force arising from the electromagnetic interaction between induced polarization and incident light. The dissipative force is useful for pushing and transporting particles, and the gradient force is generally used for attracting and trapping particles. One of the most effective usages of optical force is particle trapping within a single focused laser beam demonstrated by Ashkin et al. in 1986 [1]. This technique has been developed as optical tweezers, which is now widely applied in a variety of research fields, to manipulate micrometer-scale objects such as biocells and polymer beads [2–5]. In recent years, the target of optical manipulation has been shifting to matter at the nanometer-scale that may lead to the development of high-precision sensing, innovative nanofabrication technologies, and efficient nanometrology. However, within the Rayleigh scattering regime, the optical force is approximately proportional to the volume of the object. Thus, overcoming the disturbance from the usual environment of nano-objects is difficult. To enhance the force, use of the evanescent field with a steep gradient of the electric field has been proposed [6,7]. Recently, the trapping of nano-objects associated with plasmonic resonance is being extensively studied [8–14]. As another approach, the use of resonance with transitions between electronic levels in nano-objects has been theoretically proposed [15–20]. In general, nanostructures have quantized electronic levels. Thus, the optical force can be enhanced resonantly if the incident light frequency coincides with their transition energies. Actually, successful trapping and transport of nano-objects using resonant laser light have been reported [21–25].

In the resonance optical manipulation of nano-objects, strong light intensity and/or a large absorption cross-section by the electronic resonance effect can easily cause optical nonlinear effects, including absorption saturation and inversion of the excitation probability. Such effects lead not only to minor corrections, but also to the enhancement of the trapping efficiency [20,26], which have been verified experimentally [27]. Furthermore, the resonant nonlinear optical response can considerably enhance the degrees of freedom to manipulate nano-objects. For example, strong excitation causes gain of objects, resulting in switching behavior of the direction of optical force [20] where two photons are emitted by stimulated emission when a photon of the same mode is absorbed, which leads to a pulling force according to the momentum conservation law. A similar mechanism working in the anteroposterior motion control near the metallic optical antenna has been theoretically proposed, and would lead to super-resolution trapping [28] by using a scheme analogous to a stimulated emission depletion (STED) microscope [29].

The purpose of this study is to demonstrate a scheme to realize rotation and its direction switching of nano-objects in macroscopic and nanoscopic areas. Transferring the angular momentum of light to a small object has potential applications for nano-electromechanical systems and chiral sensing [30,31]. In particular, the orbital angular momentum of light such as a Laguerre–Gaussian (LG) beam [32] causes the orbital rotation of targets. In a well-known demonstration of rotational manipulation, microparticles made an orbital motion along a ring-shaped area in which the field intensity of the LG beam was strong [33]. Recently, some demonstrations have been reported for rotational optical manipulation of nano-objects by plasmonic trapping [10,12]. However, at present, the full potential of optical manipulation for the rotational control in nanoscale regimes remains unknown. If we can realize the highly flexible rotational optical manipulation of nano-objects, such as nanoscale rotation control and rotational direction switching, all the basic elements of nano-object motion control (i.e., pushing, pulling, and rotating) will have been realized.

For the demonstration of the macroscopic rotation in this study, the orbital angular momentum of the LG beam is used, whereas, for the nanoscopic rotation, the conversion between the spin angular momentum and orbital angular momentum of light at metallic nano-complexes plays an essential role. In both the demonstrations, controlling the balance between the dissipative and the gradient forces by using the optical nonlinearity of nano-objects is key in rotation manipulation. The dissipative force is saturated with the increase in light intensity in the nonlinear regime, whereas the gradient force is increased, being not saturated with the intensity. We propose to use this mechanism to control the rotational motion in this study for the first time.

2. Model and theoretical method

In the following numerical demonstrations, we calculate the time-averaged optical force exerted on a three-level nanoparticle (NP) that is modeled by the typical three-level molecule including vibronic levels, as depicted in Fig. 1(a). The induced polarization includes nonlinear terms to describe up to the inversion of excitation probability. The expression of the time-averaged optical force was derived from the Lorentz force formula as [34],

 

Fig. 1. Schematic of the model of incident beam and metallic nano-complex. (a) Images of rotational optical manipulation with an LG beam. The beam waist is set to 600 nm. (b) Image of nanoscale rotational optical manipulation inside tetramer metallic structures with a circularly polarized plane wave. The metallic structures are assumed to be four gold panels forming nanogaps on the glass substrate. The size of each panel is 75$\times$60$\times$20 nm$^3$. (The whole system is divided into cells of size 5$\times$5$\times$5 nm$^3$ for DDA calculation.) The gap distance between the facing edges of gold panels is equal to $2 \times$ [the length of diagonal line of a face of a cell]. The size of the NP is set to 5$\times$5$\times$5 nm$^3$, which is the size of one cell. Although the explicit shape of the NP cannot be given in this model, the shape dependence of the force is almost negligible in this size regime. In both cases (a) and (b), incident lights propagate to the negative direction of the z-axis.

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The NP polarization should be determined by the total electric field. Then, we assume the following Hamiltonian of the NP with isotropic dipole moments:

By employing the mean-field approximation, we can solve Eq. (8) and Eq. (4) in a self-consistent way; then, we obtain the NP polarization and the total electric field. Here, we assume that the directions of the NP’s dipole moments coincide with those of the background electric field resonant to each transition. In this solution, if we consider up to the higher-order correlations in the master equation, it is possible to take into account the effects of the nonlinear optical response in the NP beyond the perturbation regime so that the inversion of the excitation probability can be treated. Substituting the obtained $\boldsymbol {E}$ and $\boldsymbol {P}$ into Eq. (1), we obtain the optical force, including the nonlinear effect of the NP.

With respect to the parameters of the NPs, we assume a realistic fluorescent dye and that the resonance energies for the 0–1 and 0–2 transitions are 1.80 and 1.85 eV, respectively. The dipole moments of the NP for the 1–0 and 2–0 transitions are set to 10 Debye assuming the molecular aggregate with a size of $5^3$ nm$^3$, and the nonradiative population decay constants for the 1–0 and 2–1 transitions are set to 1$\mu$eV and 20 meV, respectively. The pure dephasing constant for the excited levels is assumed to be 2 meV. Even if this value is much larger, it does not significantly affect the population creation in excited levels when the corresponding wide line widths of incident lasers are considered. In the present theoretical method, the above value is chosen for a reasonable demonstration when monochromatic incident light is used. Note that the effect of the radiative population decay is automatically incorporated in our theory using the self-consistent calculation between the NP and the light field based on the renormalized Green’s function that includes the geometric information of the metallic structures.

3. Results and discussion

3.1 Rotation switching by optical nonlinearity with LG beam

Before rotation control within the nanoscale area, we demonstrate macroscopic rotation by using a normal LG beam ($s$ = 0, $l$ = 1) with paraxial approximation as the manipulation light [32], where $s$ and $l$ are the handedness of the circular polarization and azimuthal mode index, respectively. Here, we confine the radial mode $p$ to 0, and assume the LG beam resonance to be the 0–1 transition. The LG beam induces the following two types of forces on the NP. One is a dissipative force along the Poynting vector circulating around a singular point at the LG beam center and along the z-axis direction. Another force is the gradient force attracting the NP to the bright ring of the LG beam around z = 0 (see Fig. 2(a)). In contrast with conventional rotational manipulation using an LG beam (under non-resonant conditions), under resonant conditions, the LG beam induces a strong dissipative force. Although this force makes the particle rotate, it pushes the particle out of the focal spot that cannot be overcome by the gradient force to attract the particle toward the bright ring in a linear optical response regime (Fig. 2(c)). However, in a nonlinear regime, the increase in the dissipative force is saturated, whereas the strength of the gradient force increases linearly with the light intensity. As a result, for a sufficient intensity of light, the particle is trapped in the bright ring at a certain position with positive z. Figure 2(d) shows the force map for a light intensity of 100 kW/cm$^2$ where the dissipative force is much stronger than the gradient force (see the weak-intensity region in Fig. 2(c)) and the particle cannot be attracted. By contrast, Fig. 2(e) shows the case of strong light intensity (10 MW/cm$^2$) where the gradient force attracts the particle toward the bright ring, overcoming the dissipative force at z = 50 nm (where the gradient and dissipative forces along the z-direction are balanced) for the present parameters (see Appendix A). Furthermore, the gradient force along the y-axis is much stronger than the dissipative force in Fig. 2(e) for this intensity, and hence, the particle rotates, being attracted toward the bright ring. Detailed examinations of the three-dimensional behavior of the NP for both linear and nonlinear cases are provided in Appendix A. In the present demonstration, the absolute value of exerted force on the NP is weak for the stable trapping, overcoming the thermal disturbance from the environment because we assume small NP for the simplicity of calculation. However, in the experiments, one can use larger dye aggregates [38] or dye-doped nano-polymers in which many dyes are doped [27]. The optical force is roughly proportional to the NP volume in the Rayleigh scattering regime, and hence, the possibility of stable trapping is subject to the choice of sample. (Although the laser intensity of the level of 10 MW/cm$^2$ is often used in the experiments, it can be weaker depending on the NP size.)

 

Fig. 2. Rotational manipulation with an LG beam. (a) Intensity map of incident LG beam. The black dot indicates the sample position (x,y)=(0 nm, 200 nm) for calculating (c). (b) Schematic of two-color excitation. Manipulation light is the LG beam ($s$ = 0, $l$ = 1) and is resonant to the 0–1 transition, and phase-averaged counter-plane waves (pump light) are resonant to the 0–2 transition. The polarizations of pump and manipulation lights are along the y- and x-axis, respectively. The pump light does not cause the optical force acting on the NP. (c) Dissipative and gradient forces on the NP as functions of LG beam intensity. The x- and y-direction components of optical force are the dissipative and gradient forces, respectively. Dissipative force (blue line) and gradient force (green dotted line) without pump light, and dissipative force (red line) and gradient force (yellow line: covered by the dotted green line) with pump light (20 kW/cm$^2$). (d,e) Optical force map with only the LG beam. Incident LG beam ($s$ = 0, $l$ = 1) is resonant to the 0–1 transition. Each intensity is (d) 100 kW/cm$^2$ and (e) 10 MW/cm$^2$, and the maps are calculated at (d) z = –500 nm and (e) z = –50 nm. White arrows show the force vectors in the x–y plane. Color bars show the force in the z-direction. (f) Case with pump light. The manipulation light and pump light intensity are fixed at 10 MW/cm$^2$ and 20 kW/cm$^2$, respectively. The map is calculated at z = +50nm.

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Here, a particularly notable effect is the behavior of nano-objects under the pump beam to create a gain of the objects. When the NP is irradiated by the pump beam, as illustrated in Fig. 2(b), the direction of rotation and that along the z-axis are inverted. In this case, the direction of dissipative forces is inverted both along the z-axis and along the orbital direction because the phase of the induced polarization is inverted with the inversion of excitation probability of the state $\left | \right . 1 \rangle$, but the sign of the gradient force is not changed and is much stronger than the dissipative force. Thus, the particle is still trapped at the position with a certain negative z (z = –50 nm for the present parameters) and rotated. In this process, stimulated emission occurs, where the absorption of one photon induces the emission of two photons. Thus, the observed effect here seems consistent with the conservation laws of orbital angular momentum, though the present calculation of optical force is based on the classical treatment of the light field. Considering the success of the optical trapping by nonlinear response in the experiment [27], we can expect that the above-predicted phenomenon will occur in reality.

3.2 Optical force for rotation in nanoscale area

In the above demonstration, the orbital size is macroscopic and it is impossible to rotate particles within a nanoscale area around the beam axis because there is no light intensity, owing to the donut shape of the LG beam. In order to realize nanoscale rotational manipulation, a different mechanism is necessary. Thus, here, we demonstrate a scheme utilizing the localized optical vortex induced by the metallic nano-complex, as illustrated in Fig. 1(b). For the calculation of optical force exerted on the NP in the vicinity of the metallic nano-complex, we consider the explicit geometrical information of the system such as the metallic shape and the NP position because the coupling strength between the plasmonic field and the NP strongly depends on their geometries. We explicitly consider the nonlinear optical response of NPs interacting with localized plasmons, while the nonlinear effect and temperature dependence of the gold dielectric function are not considered, because the possible saturation or broadening effects do not change the essence of the results in the considered intensity region, although a small quantitative modification is expected [39,40]. It has been theoretically revealed that the excited plasmon modes inside the gap of the metal complex are determined by the symmetry of the metallic structures and optical angular momentum of incident light [41]. The trimer and tetramer of the metallic nanostructure have symmetries that effectively create an optical vortex by circularly polarized waves; thus, we employ the tetramer structure because of its better matching with the discrete dipole approximation (DDA) modeling (see Section 2). Here, the circularly polarized plane waves are injected from the above. We assume that two incident plane waves (manipulation light and pump light) have spin angular momentum ($s$=+1 or –1) and energies resonant to the 0–1 and 0–2 transitions, respectively. The rotation direction of the NP induced by incident light depends on its spin angular momentum. It should be noted that the spin angular momentum of light can cause the orbital motion of the NP by utilizing the metallic nano-complex as shown below. Figure 3(a) shows the force map inside the tetramer structure in the x–y plane, where incident light is the only manipulation light and its spin angular momentum and intensity are $s=+1$ and 100 kW/cm$^2$, respectively. In this case, the dissipative force saturates and the gradient force is dominant to move the NP; therefore, the rotational force is suppressed and the NP is attracted toward the gaps where the field intensity is strong. On the other hand, under two-beam excitation, the pump intensity enables us to adjust the ratio of the dissipative force to the gradient force, as shown in Fig. 2(c). Figure 3(b) shows a force map where we assume the pump light to have the same spin angular momentum ($s$=+1) as the manipulation light and both intensities to be 100 kW/cm$^2$. In this case, the inversion of the excitation probability occurs, resulting in reversion of the force induced by the manipulation light. As a result, although both incident lights have the same spin angular momentum, the rotation direction by the manipulation light becomes opposite to that of the pump light, and hence, their rotational forces (dissipative forces) cancel each other out. However, in the case of different spin angular momenta, their rotational forces reinforce each other because the rotation direction by the manipulation light becomes parallel to that of the pump light in this case. Thus, the orbital motion of the NP is realized, as shown in Fig. 3(c). Here, we should note that we can avoid pointing the optical force outward by slightly red-detuning the manipulation light energy to the resonance of the 0–1 transition. We discuss such an adjustment in detail in Appendix B. Note that this rotational force is several tens of times stronger than that in the case of using the LG beam without metallic structures. Thus, one can realize rotational manipulation with stronger optical force in nanoscale area and can control its direction selectively according to the resonance lines and the presence/absence of pump light.

 

Fig. 3. Optical force map inside the metallic nano-complex. Black arrows indicate force vectors in the x–y plane. Color bars show the magnitude of the optical force. Manipulation light and pump light energies are resonant to the 0–1 and 0–2 transitions, respectively. (a) Case with only the manipulation light. The intensity is 100 kW/cm$^2$ and it has spin angular momentum of $s$ = +1. (b) Case with both pump light and manipulation light. Both lights have spin angular momentum of $s$ = +1 and their intensities are 100 kW/cm$^2$. (c) Case with both pump light and manipulation light. Pump light and manipulation light have spin angular momentum of $s$ = –1 and $s$ =+1, respectively, and their intensities are 100 kW/cm$^2$. In this case, the manipulation light energy is slightly red-detuned (1.798 eV). Arrows drawing circles below (b,c) show the rotation direction of optical force caused by pump light and manipulation light, respectively.

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3.3 Optical current in nanoscale area

Finally, we show the Poynting vector (optical current) inducing the dissipative force exerted on the NP [42,43] in order to better understand why the optical force rotating the NP inside the metallic nano-complex works. The optical current is classified into orbital and spin parts [43]. Figures 4(a) and (b) shows the orbital part of time-averaged optical current inside the metallic nano-complex when circularly polarized light with $s$ = +1 is irradiated. Under these conditions, the optical current flows clockwise inside the metallic complex. Here, the excited localized field modes sustained in the metal complex determine the generation of the localized optical vortex depending on the angular momentum and the energy of incident light. In the case of the tetramer structure irradiated with circularly polarized light, only the fundamental mode makes the vortex. If there is no inversion of the excitation probability, the rotation direction of the NP merely obeys the optical current, whereas the rotation direction becomes opposite if the excitation probability of the NP is inverted. This clearly explains the effect shown in Fig. 3(c).

 

Fig. 4. Optical current map inside the metallic nano-complex. White arrows denote the current vectors in the x–y plane. Color bars show the field intensity. (a) Incident energy is 1.80 eV and its spin angular momentum is $s$ = +1. (b) Incident energy is 1.85 eV and its spin angular momentum is $s$ = +1.

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4. Conclusion

Using the resonant optical response in the optical manipulation easily causes optical nonlinearity. Involving nonlinear effects leads to the enhancement of both trapping efficiency and the degrees of freedom of NP manipulation, which has been demonstrated both theoretically and experimentally [20,26,27]. Here, it is shown that using nonlinear effects enables one to flexibly control the ratio between dissipative and gradient forces by the light intensity according to the resonance lines of individual NPs. This leads to the selective switching of NP rotations according to their resonance line; furthermore, by introducing the effect of the localized surface plasmons, nanoscale rotation and switching of its direction become possible. In this paper, we demonstrated the effects for a specific model of the metallic nano-complex. However, similar effects can be expected for different structures, and hence, further study for a variety of types of metallic structures is desired. For realizing the proposed phenomena in experiments, further examinations are necessary to see the influences of the interaction between the employed model molecule and environment. For example, in the presence of the metallic structures, it is desirable to consider multiple effects including those arising from the temperature elevation such as the thermophoretic force. However, some studies have reported highly efficient trappings by using metallic structures overcoming such effects [44,45], which encourage us to further study the proposed phenomena for efficient manipulation in the presence of the thermal effects. Further, experimental studies using NPs with more solid electronic systems (e.g., quantum dots and nano-diamonds with NV-centers) are expected to show the reality of the proposed mechanism. If the manipulation scheme suggested here can be realized, the basic elements of manipulating NPs (i.e., pushing, pulling, and rotating) will be completed, and this will open up new technologies not only for nanofabrication but also for nano-opto-mechanics involving chiral materials and highly sensitive and selective chiral sensing.

Appendix A: Three-dimensional rotational optical manipulation

To discuss stereoscopic rotational optical manipulation, we need to note the balance between the dissipative force and gradient force in the $x-y$ plane and the $z$ direction, respectively. In this section, we show three-dimensional rotational optical manipulation with an LG beam for three cases: weak excitation case (manipulation light intensity: 100kW/cm$^2$, pump light intensity: 0W/cm$^2$), strong excitation case (manipulation light intensity: 10MW/cm$^2$, pump light intensity: 0W/cm$^2$), and pump excitation case (manipulation light intensity: 10MW/cm$^2$, pump light intensity: 20kW/cm$^2$). In the following results, we focus on the rotational force in the $x-y$ plane at the force balance point in the $z$ direction ($z_\textrm {stop}$).

In the weak excitation case, at any position, the dissipative force is larger than the gradient force both in the $x-y$ plane and the $z$ direction. In particular, the strong dissipative force works near the bright ring. As in Fig. 5(a), the NP is blown off by the dissipative force drawing on the helical trajectory. Therefore, $z_\textrm {stop}$ is absent and it is difficult to trap the NP in the first place.

 

Fig. 5. (a) Schematic of stereoscopic rotational manipulation during weak excitation. The trapping point in the $z$ direction is absent ($z_\textrm {stop}$ does not exist). Green lines indicate an image of trajectory for the NP. (b–d) Optical force map during weak excitation. Manipulation light and pump light intensities are 100 kW/cm$^2$ and 0 W/cm$^2$, respectively. White arrows show the optical force in the $x-y$ plane. Color bars represent the optical force in the $z$ direction. (b) Optical force map at $z$ = +500 nm. (c) Optical force map at $z$ = 0 nm. (d) Optical force map at $z$ = –500 nm.

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Under strong excitation, the dissipative force is saturated with the increase in the light intensity because of the nonlinearity, whereas the gradient force is not saturated. Thus, the gradient force becomes much larger than the dissipative force in the $x-y$ plane. First, the NP is trapped at the bright ring in the $x-y$ plane. Then, it is carried until z$_\textrm {stop}$ ($\sim$ –50nm) and it rotates on the bright ring at the $z_\textrm {stop}$. The $z_\textrm {stop}$ depends on the incident intensity and the optical property of the NP. The rotation direction at $z_\textrm {stop}$ is counter-clockwise, as illustrated in Fig. 6(e).

 

Fig. 6. (a) Schematic of stereoscopic rotational manipulation during strong excitation. The dissipative force and the gradient force in the $z$ direction balance in the negative $z$ area. Green lines indicate the trajectory for the NP. (b–e) Optical force map during strong excitation. Manipulation light and pump light intensities are 10 MW/cm$^2$ and 0 W/cm$^2$, respectively. White arrows show the optical force in the $x-y$ plane. Color bars represent the optical force in the $z$ direction. (b) Optical force map at $z$ = +500 nm. (c) Optical force map at $z$ = 0 nm. (d) Optical force map at $z$ = –50 nm. (e) Azimuthal component (dissipative force) of the optical force of (d) in the $x-y$ plane. Optical force in the $z$ direction is the same as (d).

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Under pump excitation, inversion of the excitation probability occurs and inverts the direction of the dissipative force. As shown in Fig. 7, the inversion of force direction occurs near the bright ring. The inversion of the dissipative force causes a shift in the point at which the dissipative force and the gradient force balance in the $z$ direction, and the sign of the $z_\textrm {stop}$ inverts (in this condition, $z_\textrm {stop} \sim$ +50 nm). With respect to the optical force in the $x-y$ plane, the gradient force works toward the bright ring, and the NP rotates clockwise (back reverse for Fig. 6(e)) at the $z_\textrm {stop}$ (see Fig. 7(e)).

 

Fig. 7. (a) Schematic of stereoscopic rotational manipulation during pump excitation. The dissipative force and the gradient force in the $z$ direction balance in the positive $z$ area. Green lines indicate the trajectory for the NP. (b–e) Optical force map during strong excitation. Manipulation light and pump light intensities are 10 MW/cm$^2$ and 20 kW/cm$^2$, respectively. White arrows show the optical force in the $x-y$ plane. Color bars represent the optical force in the $z$ direction. (b) Optical force map at $z$ = +50 nm. (c) Optical force map at $z$ = 0 nm. (d) Optical force map at $z$ = –100 nm. (e) Azimuthal component (dissipative force) of the optical force of (d) in the $x-y$ plane. Optical force in the $z$ direction is the same as (b). Rotation direction of dissipative force is the opposite as that shown in Fig. 6(d).

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Appendix B: Balance between dissipative force and gradient force under strong near field

To understand the balance between the dissipative force and gradient force exerted on an NP under the strong near field, we employ the formula of time-averaged optical force,

(9)$$\langle \boldsymbol{F}(\omega) \rangle =\frac{1}{2}\textrm{Re}{\bigl [} \int_{V} d\boldsymbol{r} [\nabla \boldsymbol{E}(\boldsymbol{r},\omega )^* ] \cdot \boldsymbol{P}_\textrm{NP}(\boldsymbol{r},\omega ){\bigr]}.$$

Here, we consider an evanescent field with a simple profile and an induced polarization represented by complex susceptibility as follows:

(10)$$\boldsymbol{E}(\boldsymbol{r},\omega) = \boldsymbol{E}(x,y) e^{i (k_{1}+ i k_{2}) z},$$

(11)$$\boldsymbol{P}_\textrm{NP}(\boldsymbol{r},\omega) = (\chi_{1} + i \chi_{2}) \boldsymbol{E}(\boldsymbol{r} ) = (\chi_{1} + i \chi_{2}) \boldsymbol{E}(x,y) e^{i (k_{1}+ i k_{2}) z},$$

where $k_1$ and $k_2$ denote wavenumber and extinction coefficients in the z direction, respectively; $\chi _{1}$ and $\chi _{2}$ represent real and imaginary parts of the NP’s susceptibility, respectively; and $E(x,y)$ has a real value. Substituting Eq. (10) and Eq. (11) into Eq. (9) and regarding the induced polarization as the point dipole, the optical force can be written as

(12)$$\langle \boldsymbol{F}(\omega) \rangle =\frac{1}{2} V_\textrm{NP} \left [ \frac{1}{2} \chi_{1} \boldsymbol{\nabla}_{x,y} |\boldsymbol{E}| ^{2} - \chi_{1} k_{2} |\boldsymbol{E}|^{2} \boldsymbol{n}_{z} + \chi_{2} k_{1} |\boldsymbol{E}|^{2} \boldsymbol{n}_{z} \right ] e^{- 2k_{2} z},$$

where $V_\textrm {NP}$ is the volume of the NP, $\boldsymbol {\nabla }_{x,y}$ is the two-dimensional gradient, and $\boldsymbol {n}_{z}$ is the unit vector along the $z$-axis. Then, we classify the terms of Eq. (12) into dissipative force and gradient force. If the wavenumber is real, namely the case of $k_{2}$=0, the optical force can be written as

(13)$$\langle \boldsymbol{F}(\omega) \rangle =\frac{1}{2} V_\textrm{NP} \left [ \frac{1}{2} \chi_{1} \boldsymbol{\nabla}_{x,y} |\boldsymbol{E}| ^{2} + \chi_{2} k_{1} |\boldsymbol{E}|^{2} \boldsymbol{n}_{z} \right ],$$

where the first and second terms on the right-hand side correspond to gradient force and dissipative force, respectively. However, in the presence of the finite $k_{2}$, we should regard the first and second term on the right-hand side of Eq. (12) as the gradient force and the third term as the dissipative force. It should be noted that the gradient force and dissipative force under the near field depend on $\chi _{1}$ and $\chi _{2}$, respectively. This means that one can control over-trapping/exclusion of the NP under a plasmonic field by adjusting the balance of $\chi _{1}$ and $\chi _{2}$ with the tuning of laser frequencies.

In the non-resonant case, $\chi _{1}$ and $\chi _{2}$ are constant for light energy, whereas the resonant polarization in the linear optical response obeys dispersion-type $\chi _{1}$ and Lorentz-type $\chi _{2}$ and the ratio of $\chi _{1}$ to $\chi _{2}$ drastically changes near the resonant energy of the NP. Moreover, strong nonlinear optical effects such as absorption saturation and population inversion can cause the sign inversion of $\chi _{1}$ and $\chi _{2}$. Here, we regard the ratio of $\chi _{1}$ to $\chi _{2}$ as that of gradient force to dissipative force, and demonstrate it for three cases: weak excitation, strong excitation, and the stimulated emission. Figures 8(b)–(d) show the real and imaginary parts of induced polarization of the NP at the sample position indicated in Fig. 8(a), where the polarization includes resonant and non-resonant elements for 0–1 and 0–2 transitions induced by manipulation light. The parameter of the NP is the same as that in the main text. Under the weak excitation, the nonlinear optical effects can be ignored, and the polarization obeys the linear optical response. Therefore, $\chi _{1}$ and $\chi _{2}$ have the dispersion-type and Lorentz-type spectra, respectively. Moreover, it should be noted that the resonant energy (e.g., the peak energy of the $\chi _{2}$) is slightly red-shifted by the metal–NP interaction. Under strong excitation, $\chi _{2}$ is almost zero because the absorption and emission balance with each other owing to the absorption saturation. With respect to $\chi _{1}$, it becomes large as the manipulation light energy increases. This is because the absorption saturation effect suppresses the polarization with the 0–1 transition, and the polarization with the 0–2 transition starts to appear. Additionally, in the presence of the pump light, the signs of $\chi _{1}$ and $\chi _{2}$ are inverted compared to the weakly excited case. Here, the negative value of the $\chi _{2}$ represents not reduction but gain. In other words, the negative value indicates the generation of stimulated emission. The stimulated emission also causes the sign inversion of $\chi _{1}$.

 

Fig. 8. (a) Sample position inside the tetramer structure for calculating susceptibility $\chi _{1}$ and $\chi _{2}$ of the NP’s polarization induced by the manipulation light. The position is ($x,y,z$)=(–20, –20, 25 nm). (b–d) Spectra of susceptibility $\chi _{1}$ (red line) and $\chi _{2}$ (blue line) as functions of the manipulation light energy. (b) Case with weak excitation. Manipulation light intensity is 1 W/cm$^2$. (c) Case with strong excitation. Manipulation light intensity is 100 kW/cm$^2$. (d) Case with stimulated emission. Pump light energy and intensity are 1.85 eV and 100 kW/cm$^2$, respectively. Manipulation light intensity is 100 kW/cm$^2$.

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Inside the tetramer structure, the dissipative force causes orbital motion, and the gradient force points toward the hotspots. In Fig. 3(b) of the main text, we employ the red-detuned manipulation light and utilize the repulsive force (negative value of $\chi _{1}$) from the hotspots, as shown in Fig. 8(d).

Funding

Japan Society for the Promotion of Science (JP16H06504).

Acknowledgments

The authors thank K. Sasaki for his fruitful discussion. This work was supported by JSPS KAKENHI Grant Number JP16H06504 in Scientific Research on Innovative Areas "Nano-Material Optical-Manipulation."

Disclosures

The authors declare no conflicts of interest.

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