1. Introduction

Noble metal nanoparticles exhibit a strong interaction with light due to the excitation of collective oscillations of their free electrons, which is known as localized surface plasmon resonance (LSPR) [1–3]. On the resonance, the plasmon oscillation gives rise to an intense near field on the nanoparticle surface and a strong scattering light in the far field. The plasmonic characteristics strongly depend on not only the nanoparticle size and shape but also the configurations of the particles, e.g., their separation, orientation and so on [4–7]. In particular, when two nanoparticles are close to each other, their plasmon coupling occurs owing to the strong interactions [5–7], resulting in an appearance of two plasmon modes different from individual constituents. The plasmon resonances and the near-field enhancement effects of these hybridized modes are significantly changed by the nanoparticle configurations [8–12]. In other words, we can efficiently control the plasmon characteristics through the configurations with the plasmon coupling.

The plasmon coupling between nanoparticles drastically enhances the electromagnetic (EM) field in the narrow gap separating them. Over many decades, extensive investigations on the basis of plasmon coupling have been carried out, promoting the applications of plasmonics, such as surface-enhanced fluorescence [13,14], surface-enhanced Raman scattering [15,16], chemical sensors and biosensors [17,18], and high harmonic generation [19,20]. Furthermore, the plasmon coupling strongly enhances optical forces between two nanoparticles [21,22]. This interaction force provides a possible approach for assembly control of the nanoparticles [23–25].

The optical forces are induced by the linear momentum transfer between light and matter, and they have been widely applied to optical manipulations [26–32]. Light has not only the linear momentum but also angular momentum [32–38]. The angular momentum carried by light can produce an optical torque via scattering or absorption, which has been firstly demonstrated by Beth more than 80 years ago [38]. The optical torque is able to rotate objects in a controlled manner. It has been attracting widespread attention since it improves the mechanical degree of freedom to manipulate the objects, with a variety of applications in atomic and molecular physics, nanotechnology and biology [39–44].

Here, we study, for the first time, the direct relation between the optical torque and plasmon hybridization between twisted metal nanorods. Our numerical simulations indicate that the behaviors of the interaction optical torque at hybridized modes are different from that of an isolated nanorod, which depend on not only the gap size but also the twisted angle between the nanorods. This interaction optical torque implements the rotations to mutually perpendicular and parallel alignments of nanorods with the light excitations of different hybridized modes. Thus, the torques due to the plasmon hybridization would play an important role to control the plasmonic characteristics and functions.

2. Methods

Optical torque was studied using the finite element method simulations with the Maxwell stress tensor (MST). Another approach based on the dipole approximation is also presented in Appendix Section A.1. In order to simply excite and observe the plasmon hybridized modes, we chose a dimer of twisted nanorods as shown in Fig. 1. The two nanorods are separated by gap size d, and they are twisted by angle θ. The diameter (D) and length (L) of both nanorods were 40 nm and 120 nm, respectively. The long axis of Rod 1 in Fig. 1 was fixed along the y-axis. Conversely, Rod 2 was rotatable for changing the twisted angle. The dimer was irradiated by a y-polarized plane wave to propagate along the z-axis from the positive to the negative direction. We chose the linearly polarized light without any intrinsic angular momentum, which simplifies the analysis of physical mechanism on the optical torque. The surrounding medium was air. The optical torque on the nanorods was calculated from the conservation law of angular momentum, which can be expressed as [45]

 

Fig. 1. Schematic illustration of twisted gold nanorods with gap size d and twisted angle θ. The diameter (D) and length (L) of both nanorods are 40 nm and 120 nm, respectively.

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3. Results and discussion

In the twisted nanorod dimer, the gap size d and twisted angle θ are crucial parameters to decide its configuration. Figure 2(a) shows the plasmon resonance spectra of an isolated nanorod and its dimer at a twisted angle of π/12 with different gap sizes d. As the gap size decreases, the resonance of individual nanorods, i.e., the isolated nanorod, starts to split and two resonance peaks shift to shorter and longer wavelengths due to the plasmon coupling between the nanorods. We also observe similar spectral behavior in the twisted angle dependence for the dimer with a gap size of 1 nm, as shown in Fig. 2(b). As the twisted angle increases and decreases, the separation between two resonance peaks decreases and increases, respectively. At the twisted angle of π/2, the dimer exhibits only one resonance peak even with the gap size of 1 nm, in other words, no plasmon coupling occurs. These results show that the plasmon coupling between twisted nanorods can be controlled by not only the gap size but also the twisted angle.

 

Fig. 2. Plasmon resonance spectra of an isolated nanorod and its dimer: (a) gap size dependence (θ = π/12) and (b) twisted angle dependence (d = 1 nm). The solid lines, dashed lines, and dotted lines show the extinction, absorption and scattering cross sections, respectively. The vertical dashed lines show the resonance peak wavelength of the isolated nanorod.

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The coupling between two nanorods has been described as an elegant physical picture which is plasmon hybridization in analogy with molecular orbital theory [8–12]. Each nanorod can be approximated as a dipole with energy U. The plasmon coupling between two dipole modes generates two new plasmon modes: anti-bonding mode and bonding mode (see Fig. 3). These two hybridized modes are corresponding to a higher energy U₊ and a lower energy U₋ respectively, which are produced by splitting from the degenerate energy levels. For the anti-bonding mode, two dipoles are arranged in a parallel manner, i.e., in phase, leading to a constructive superposition of the dipole moment. It is therefore referred to as a bright plasmon mode that shows a large cross section (see Fig. 2). Also, the charges with the same sign at the edges of nanorods result in a higher energy mode due to the charge repulsion. However, for the binding mode, the dipoles are in an anti-parallel alignment that shows opposite charges at the edges of the nanorods, reducing the charge repulsion and leading to a lower energy. In this case, the destructive superposition of the dipole moment produces a dark plasmon mode that exhibits a small cross section (see Fig. 2). In the dimer of twisted nanorods (see Fig. 3), the decrease of the twisted angle, as well as the gap size, would relieve the charge repulsion for bonding mode but inverse for anti-bonding mode, resulting in the LSPR redshift and blueshift, respectively. Additionally, at the twisted angle of π/2, the charge repulsion between two nanorods vanishes away due to the orthogonal dipoles, so that plasmon hybridization does not occur.

 

Fig. 3. Plasmon hybridization diagram in nanorod dimer with twisted angle θ. The polarization direction of the incident light is along the longitudinal axis of the bottom nanorod.

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Figure 4 shows the optical torque applied to such twisted nanorods. The behaviors of the optical torque strongly depend on the gap size and the twisted angle between the nanorods. The magnitude of the torque on the whole dimer at the resonance wavelength is similar to that on the isolated nanorod, even in the case of the strong plasmon coupling between the nanorods, i.e., small gap size and twisted angle. However, the optical torque on each nanorod in the dimer can be remarkably enhanced by the plasmon coupling, leading to more than 10 times as large as that on the isolated nanorod. It should be noted that the extinction cross section at the redshifted plasmon resonance (bonding mode) is much smaller than that at the blueshifted plasmon resonance (anti-bonding mode) in Fig. 2, whereas the magnitude of the optical torque at the redshifted resonance is much larger than that at the blueshifted resonance in Figs. 4(b) and 4(d).

 

Fig. 4. Optical torque acting on (a, c) the whole dimer and (b, d) on each nanorod: (a, b) gap size dependence (θ = π/12) and (c, d) twisted angle dependence (d = 1 nm). The dashed and dotdashed lines show the peak wavelengths at the redshifted resonance. The left- and right- side insets in Fig. 4(d) show the rotation directions of Rod 2 at the blueshifted and redshifted resonances, respectively.

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Let us discuss the spectral shape and the mechanism of the optical torques on the twisted nanorods. When a linearly polarized field E illuminates an isolated nanorod, the time-averaged optical torque on the induced dipole moment p of the nanorod can be simply expressed as [46,47]

For each nanorod in the dimer, the optical torque is determined by the scattered field from the other nanorod together with the incident light. In Figs. 4(b) and 4(d), the directions of optical torque on the two nanorods are opposite while their magnitudes are close to each other. In other words, each nanorod dominantly experiences an optical torque due to the angular momentum of the EM field arising from the interaction between the two nanorod-plasmons, i.e., interaction optical torque. Therefore, the sign of the optical torque on the nanorods should be decided by the phase difference between the dipole moments of the two nanorods. For the anti-bonding mode (blueshifted resonance), the two dipoles are in phase, producing a negative torque on Rod 1 and a positive torque on Rod 2. Conversely, for the bonding mode (redshifted resonance), the dipoles are out of phase. The optical torques on Rod 1 and Rod 2 are positive and negative, respectively. Moreover, because the bonding plasmon mode has much smaller scattering loss than the anti-bonding mode in Fig. 2, its near field between two nanorods can be more strongly enhanced, as shown in Fig. 5. This results in a much larger magnitude of the interaction optical torque on the redshifted resonance even with its smaller extinction cross section. Additionally, in the case of the two nanorods parallel and perpendicular to each other, i.e., θ = 0, π/2, the EM field does not possess the angular momentum, resulting in no optical torque on the nanorods. Such spectral behavior of the optical torque on each nanorod can be confirmed by another approach based on the dipole approximation (see Appendix Section A.1 for more detail).

 

Fig. 5. (a) Twisted angle dependence of electric field intensity enhancement at the gap center (d = 1 nm). The dot-dashed lines show the peak wavelengths at the redshifted resonance. (b, c) Distributions of the intensity enhancement at the resonance wavelength of 610 nm and 885 nm, respectively, with the twisted angle of π/12 at the gap center plane.

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As shown in the inset of Fig. 4(d), the optical torque on Rod 2 for the blueshifted resonance produces anti-clockwise rotations, whereas the torques for the redshifted resonance yield the opposite rotations. As the Rod 1 is fixed along the y-axis, the former and the latter lead to mutually perpendicular and parallel alignments of the nanorods, i.e., θ = π/2 and 0, respectively. For this optical alignment dependent on the incident light wavelength, the torque on the Rod 2 should be large enough to overcome its rotational Brownian motion. The average energy of the Brownian motion associated with each degree of freedom can be expressed as k_BT/2 in a system at thermodynamic equilibrium, where k_B is the Boltzmann constant and T is the absolute temperature [45]. In this system, the temperature T will increase due to the light absorption of the nanorods which depends on their configurations. To simplify the discussion, we keep T at 300 K. We define the rotational potential energy as

Figure 6 shows the rotational potentials on Rod 2 for its twisted angle θ at different incident light wavelengths on the redshifted and blueshifted resonances. The Rod 1 is fixed along the y-axis. Supercontinuum white lights with a uniform power over the spectrum from 500 nm to 645 nm (blueshifted resonance) and 645 nm to 1000 nm (redshifted resonance) are assumed to be the incident lights. Their illumination with the same intensity generates the potential wells with similar depths, which show the potential minimums at different twisted angles. To identify the equilibrium positions, we should compare the rotational potential energy with the Brownian motion energy. For an incident intensity of 10 mW/μm², the potential depths are more than 30 times as large as the Brownian motion energy k_BT/2. These potential wells enable the perpendicular and parallel alignments of the nanorods with angle fluctuations of ~±9° and ~±4°, respectively.

 

Fig. 6. Rotational potentials in units of k_BT (T = 300K) for Rod 2 in the dimer (d = 1 nm) as a function of the twisted angle at different incident light wavelengths. Supercontinuum white lights with a uniform power over the spectrum from 500 nm to 645 nm (blueshifted resonance) and 645 nm to 1000 nm (redshifted resonance) are assumed to be the incident lights.

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To achieve such optical alignments in an experiment, the Rod 1 should be embedded on the surface of a transparent substrate, e.g., silica (see Appendix Section A.3). Furthermore, the sign of optical force, consisting of interaction optical force and radiation pressure, on the Rod 2 should be considered to discuss the possibility of the optical alignments. In the configuration in Fig. 1, the Rod 2 experiences optical force toward the Rod 1 over the wavelength of the incident light (see Appendix Section A.4 for more detail). Thus, the Rod 2 in a solution dropped on the substrate surface can approach the Rod 1. Additionally, we can choose other metals that can further enhance the interaction optical torque to facilitate the optical alignment. For example, silver is another common noble metal to provide LSPRs in the visible spectral range, which is well separated from its interband transitions in comparison with the gold LSPRs [1]. The plasmon resonance of the silver dimer is sharper than that of the gold dimer because of its low ohmic losses and electronic interband transitions. This further enhances the peak magnitude of the interaction optical torque.

4. Conclusions

We have presented a study of the relation between the interaction optical torque and plasmon hybridization in a dimer of twisted metal nanorods. Firstly, we have explained the physical description of plasmon hybridization in the dimer, which depends on not only the gap size but also the twisted angle between the nanorods. Then, we have demonstrated the interaction optical torque is highly associated with the plasmon hybridization by comparing the optical torques on the nanorods with the plasmon resonance spectra. This clearly shows the physical mechanism of the interaction optical torque based on the plasmon-hybridization-induced dipole momentum in the nanorods. The torque behaviors are decided by the relative phase between the coupled dipole moments of the two nanorods. In order to show how one can apply our findings to practical experiments and applications, we have obtained the rotational potential by using the calculated optical torques on Rod 2 in comparison with the Brownian motion energy. When the two-coupled dipoles are in phase and out of phase, i.e., hybridized anti-bonding and bonding modes, their rotations lead to mutually perpendicular and parallel alignments, respectively. In addition, we have discussed the experimental realization for the configuration of the simulation model. The optical alignments depending on the hybridized modes with the different resonance wavelength regions would dynamically control the plasmonic characteristics and functions, e.g., the EM field enhancement and chirality, through the nanoparticle configurations with the plasmon coupling. Thus, our findings will open a new route to all-optical active plasmonic and metamaterial devices.

Appendix

A.1 Calculation of optical torque by dipole approximation (DA) method

DA method is a widely utilized method to calculate the optical torque. A nanorod can be approximated as a dipole with polarizability of α. The induced dipole moment p of the nanorod illuminated by an arbitrary monochromatic incident electric field E can be expressed as

(6)$${\textbf p}\textrm{ = }{\boldsymbol \alpha }{\textbf E ,}$$

Since the transverse polarizability of a nanorod is negligible [see Fig. 8(a)], we only consider the longitudinal polarizability α_l. Figure 8(b) shows the real part and imaginary part of α_l. Hence, with the incident field of y-polarized plane light E to propagate along –z-axis as shown in Fig. 7, the time-averaged optical torque on an isolated nanorod rotated with an angle of θ can be obtained by [46,47]

(7)$${{\boldsymbol {N}}_{\textrm{isolated}}}\textrm{ = }\frac{\textrm{1}}{\textrm{2}}\textrm{Re[}{\textbf p}{ \times }{{\textbf E}^{\ast }}\textrm{] = } - \frac{\textrm{1}}{\textrm{4}}\textrm{Re[}{\alpha _l}\textrm{] sin2}\theta {|{\textbf E} |^\textrm{2}}{{\boldsymbol e}_\textrm{z}},$$

where e_z is the unit vector along the z-axis.

 

Fig. 7. Schematic illustration of an isolated nanorod and twisted gold nanorods with gap size d and twisted angle θ illuminated by a y-polarized light. L = 120 nm, D = 40 nm.

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Fig. 8. (a) Plasmon resonance spectra of an isolated Au nanorod at longitudinal mode and transverse mode. (b) Longitudinal polarizability of the Au nanorod calculated by the finite element method. The length and diameter of the Au nanorods were 120 nm and 40 nm, respectively.

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Equation (7) shows a good agreement with the results calculated by the MST method (see Fig. 11). According to Eq. (7), the direction of optical torque is decided by the sign of the real part of α_l. Under the situation of 0<θ<π/2, Re[α_l] is positive in the longer wavelength region than the resonance peak wavelength. The phase difference between p and the E is in the range of 0 to π/2. The optical torque is negative. In the shorter wavelength region, Re[α_l] is negative, that is, the phase difference between p and the incident electric field E is in the range of π/2 to π. The optical torque is positive. Additionally, when the phase difference between p and the incident electric field E is π/2, i.e., Re[α_l] = 0, the optical torque is 0.

In order to calculate the optical torque acting on each nanorod in the dimer of twisted nanorods, we need to calculate the electric field on each nanorod by the dipole discrete approximation. Base on the dipole radiation, each component of the electric field on each nanorod can be expressed as

(8)$${{\textbf E}_\textrm{1}}\textrm{ = }\left( \begin{array}{c} - \frac{{A\textrm{ + }{A^\textrm{2}}\textrm{exp[}ik\textrm{(}d\textrm{ + }D)\textrm{]}}}{{\textrm{1} - {A^\textrm{2}}\textrm{co}{\textrm{s}^\textrm{2}}\theta }}\textrm{sin}\theta \textrm{cos}\theta {\textrm{E}_\textrm{y}}\\ \textrm{ }\frac{{\textrm{exp[}ik\textrm{(}d\textrm{ + }D)\textrm{] + }A\textrm{co}{\textrm{s}^\textrm{2}}\theta }}{{\textrm{1} - {A^\textrm{2}}\textrm{co}{\textrm{s}^\textrm{2}}\theta }}{\textrm{E}_\textrm{y}}\textrm{ }\\ \textrm{ }\\ \textrm{ 0} \end{array} \right)\textrm{ ,}$$

(9)$${{\textbf E}_\textrm{2}}\textrm{ = }\left( \begin{array}{c} \textrm{ 0}\\ \frac{{\textrm{1} + A\textrm{exp[}ik\textrm{(}d\textrm{ + }D)\textrm{]}}}{{\textrm{1} - {A^\textrm{2}}\textrm{co}{\textrm{s}^\textrm{2}}\theta }}{\textrm{E}_\textrm{y}}\\ \textrm{ 0} \end{array} \right),$$

(10)$$A\textrm{ = }\frac{{{\alpha _l}\textrm{exp[}ik\textrm{(}d\textrm{ + }D)\textrm{]}}}{{\textrm{4}\pi \varepsilon \textrm{(}d + D\textrm{)}}}\textrm{ [}{k^\textrm{2}}\textrm{ + }\frac{{ik\textrm{(}d + D\textrm{)} - \textrm{1}}}{{{{\textrm{(}d + D\textrm{)}}^\textrm{2}}}}\textrm{],}$$

where θ is the twisted angle in the dimer, k is the wavenumber, ɛ is the permittivity of the surrounding medium, d is the gap size, and D is the diameter of nanorods. ∗ denotes the operation of conjugation. E_y is the y component of the incident light. Thus, the optical torques on Rod 1 and Rod 2 can be expressed as

(11)$${{\boldsymbol N}_\textrm{1}}\textrm{ = }\frac{\textrm{1}}{\textrm{4}}\textrm{Re[}{\alpha _l}\frac{{{{\textrm{(}A\textrm{ + }{A^\textrm{2}}\textrm{exp[}ik\textrm{(}d + D\textrm{)])}}^{\ast }}\textrm{(exp[}ik\textrm{(}d + D\textrm{)] + }A\textrm{co}{\textrm{s}^\textrm{2}}\theta \textrm{)}}}{{{{|{\textrm{1} - {A^\textrm{2}}\textrm{co}{\textrm{s}^\textrm{2}}\theta } |}^\textrm{2}}}}\textrm{] sin2}\theta {|{\textbf E} |^\textrm{2}}\textrm{ }{{\boldsymbol e}_\textrm{z}},$$

(12)$${{\boldsymbol N}_\textrm{2}}\textrm{ = } - \frac{\textrm{1}}{\textrm{4}}\textrm{Re[}{\alpha _l}\textrm{] (}\frac{{{{|{\textrm{1 + }A\textrm{exp[}ik\textrm{(}d\textrm{ + }D)\textrm{]}} |}^\textrm{2}}}}{{{{|{\textrm{1} - {A^\textrm{2}}\textrm{co}{\textrm{s}^\textrm{2}}\theta } |}^\textrm{2}}}}\textrm{)sin2}\theta {|{\textbf E} |^\textrm{2}}{{\boldsymbol e}_\textrm{z}}.$$

Though the DA method does not include the effect of the edges in the gap between the nanorods, it is a helpful method to qualitatively reveal the physics of the surface-plasmon-enhanced optical torque between the twisted nanorods. Figure 9 shows the optical torques on the Rod1 and Rod 2 calculated by Eqs. (11) and (12), respectively. The shapes of the curves show a similar tendency with the results calculated by the MST method in Fig. 4(d). For the optical torque on Rod 2, we can see the enhancement of optical torque is from the enhancement of the electric field illuminated on the nanorods by comparing the Eqs. (12) and (7). In the redshifted region, because the real part of polarizability changes slowly, as shown in Fig. 8(b), the enhancement of the electric field dominates the change of the optical torque. Thus, the peak of optical torque is corresponding to the plasmon coupling resonance. In the blueshifted region, the real part of polarizability varies drastically, determining the variation tendency of optical torque. At the peak wavelength of the plasmon resonance of an individual nanorod, we can find the optical torque is 0 in Fig. 9(b) because the real part of polarizability is 0. However, the DA method cannot completely describe the characteristics of the near field in the dimer, such as the polarization state distribution in the near field. Therefore, the optical torque shows different results from that in Fig. 4(d) (N₂). For the optical torque on Rod 1, we can see the electric field on Rod 1 is not a linear polarized light from Eq. (8). In this case, the optical torque calculated by Eq. (11) is decided by not only the real part of polarizability but also the imaginary part of polarizability. Hence, in the blueshifted resonance region, the spectral behavior of optical torque is different from the optical torque on Rod 2. On the other hand, in the redshifted resonance region, it is similar to the optical torque on Rod 2 because the enhancement of the EM field dominates the variation of optical torque. The peak of optical torque is in accordance with the plasmon coupling resonance as well.

 

Fig. 9. Optical torques on Rod 1(a) and Rod 2 (b) calculated by the DA method as a function of wavelength in the dimer with 10 nm gap size.

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Figure 10 demonstrates the different results of optical torque between the MST method and DA method. When the gap size d is larger than ≈70 nm, the results from the two methods are in good agreement with each other. However, in the case of gap size smaller than ≈70 nm, the DA method is not accurate to calculate the optical torque in the dimer due to the emergence of plasmon coupling at the edges in the gap.

 

Fig. 10. Comparison of the optical torques acting on the whole structure (a), Rod 1(b) and Rod2 (c) calculated by MST and DA method as a function of gap size in the dimer with 710 nm wavelength and π/12 twisted angle.

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A.2 Optical torque on an isolated nanorod

 

Fig. 11. Optical torque on an isolated nanorod with different rotation angles θ. The solid line and triangle with different colors represent the torques calculated by the DA method and MST method, respectively.

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A.3 Experimental realization

Firstly, a nanorod array is embedded on the surface of a transparent substrate, e.g., silica, by etching trenches in the substrate. By controlling the depth of the trenches and the thickness of metal evaporated into the trenches, we can get a nanorod array a few nanometers below the surface of the substrate. The roughness of the surface can also be controlled within a few nanometers. Such fabrication has been realized by electron beam lithography and reactive ion etching in some paper already [48,49]. Then, we put nanorods dispersed in a solvent on the nanorod array. The dispersed nanorods would be aligned by controlling the incident light wavelength.

A.4 The optical force between twisted Au nanorods

As the optical force along the x-axis and y-axis relative to that along the z-axis can be neglected, we only consider the optical force along the z-axis. From Figs. 12 and 13, we can see the optical force shows a similar spectral behavior with the optical torque. The interaction optical force can be strongly enhanced by the plasmon coupling at the redshifted plasmon resonance, highly depending on the gap size and the twisted angle. At the redshifted resonance, a very strong attractive force can be generated between the two nanorods. At the blueshifted resonance, it is difficult to determine whether it is an attractive or repulsive force. However, due to the scattering force from the incident light, the total optical force on Rod 2 almost always keeps a negative value which is toward Rod 1. Thus, in an experiment that the bottom nanorod is fixed, the top nanorod can approach the bottom nanorod.

 

Fig. 12. Optical force acting on (a) Rod 1 and (b) Rod 2 with different gap sizes. The black line shows the optical force on an isolated Au nanorod. The twisted angle was fixed as π/12.

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Fig. 13. Optical force acting on (a) Rod1 and (b) Rod 2 with different twisted angles. The gap size was fixed as 1 nm.

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Funding

Precursory Research for Embryonic Science and Technology (JPMJPR15PA); Japan Society for the Promotion of Science (JP18K18992, JP19H02533, JP19H04670).

Acknowledgments

The authors are grateful to R. Fukuhara for many useful discussions. The authors acknowledge H. Sugimoto for primary calculations.

Disclosures

The authors declare no conflicts of interest.

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