1. Introduction

As A. Ashkin firstly reported that the laser induced optical force could trap particles [1], the gradient optical force has been utilized in optical trapping [2] and manipulation of nanoparticles [3]. Recently, optical gradient forces in plasmonic metamaterials [4–10] and waveguiding structures [11] were researched based on their characteristics of strong local field. The significant enhancement of optical gradient force, due to plasmonic confinement of the local optical field and tailored field region below the diffraction limit, can realize many interesting applications in principle, such as manipulating nanoparticles [5,12] and structure reconfiguration [13,14]. Moreover, it was reported that optical force can be enhanced by various plasmon resonances. From example, strong negative optical pressure can be induced in plasmonic cavities by optical magnetic resonance and it was reversed with the material replaced by perfect electrical conductors [7,9]. Repulsive and attractive forces in gold nanowire pairs were found as results of electric dipolar and magnetic dipolar resonances [15]. Hybridized magnetic dipolar resonances in a set of stereo-metamaterials could generate large optical force up to ~1200 piconewtons at an illumination intensity of $50\mu m^{2}m{W}^{-1}$ [16]. Fano-type resonance would enhance the optical force several times greater than the radiation pressure in planar dielectric metamaterials [17] and plasmonic nanorod heterodimers [18]. Besides the resonances above, a unique dipolar resonance, toroidal dipole, has been demonstrated in metamaterials, which may contribute to the application efforts of enhanced plasmonic force.

Toroidal dipole was put forward by Zel’dovich in 1957 [19]. Naturally, it is weaker than conventional electric and magnetic dipolar responses. Nevertheless, Zheludev reported that the toroidal dipolar response could be dominant in a specific metamaterial [20]. Subsequently, various toroidal metamaterials were proposed to study the toroidal dipolar response [21–25]. Recently, XL Zhang et al. came up with optical force to identify the existence of toroidal dipole [26]. The purpose of this paper is to explore the optical force induced in a toroidal dipolar resonance system and to study the force property enhanced by toroidal dipolar resonance in comparison with common dipolar resonances, e.g., magnetic dipole. A double-disk metastructure with toroidal dipolar response is considered to investigate the optical force based on its unique local field characteristic for both electric and magnetic field confinements [22]. Numerical results show the optical force enhanced by the toroidal dipolar resonance can exceed the forces enhanced by conventional magnetic dipolar and quadrupolar responses in this metastructure. Moreover, the force acted on a 5-nm spherical nanoparticle is studied to reveal the annular optical trapping capability.

2. Schematic and numerical method

Figure 1(a) demonstrates the schematic of the double-disk metastructure with disk radius r = 500 nm, thickness t = 40 nm, and disk-to-disk gap g = 10 nm. Open boundaries are applied in all x-, y-, and z- directions. The incident light along the x direction is z-polarized with electric-field amplitude of 1 V/m. Silver is chosen for the disks, described by the Drude dispersion model $\epsilon \left(\omega \right)={\epsilon }_{\infty }-{\omega }_p{}^2/\left({\omega }^2+i\omega \gamma \right)$with${\epsilon }_{\infty }=6.0$,${\omega }_p=1.5\times {10}^{16}rad/s$, and $\gamma =7.73\times {10}^{13}rad/s$ After the electromagnetic fields are numerically calculated by a 3D Finite-Difference-Time-Domain method, the optical force is evaluated using surface integral [27]

 

Fig. 1 (a) Schematic of double-disk system, note that the incident configuration is plotted in coordinates and origin is at the structure center. (b) The side view shows that an attractive optical force will be exerted on the upper disk at toroidal resonance, as the red arrow indicated. (c) Optical force exerted on a dielectric nanoparticle in water solution.

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where the time-averaged Maxwell Stress Tensor (MST) is

In Eqs. (1) and (2), the footnotes i, j, $\alpha$and$\beta$represent x, y, or z component of physical quantities. S is a closed surface around object and n is the outward unit normal vector on S. In addition, E and H represent electric and magnetic fields respectively,${\epsilon }_r$and${\mu }_r$are the relative permittivity and permeability of the background respectively.

3. Result and discussion

In our previous work [22], it was demonstrated that double-disk system could exhibit an intriguing toroidal dipolar response in addition to the conventional magnetic dipolar resonance in the nearby spectrum. Based on this design, here we investigate the optical force enhanced by the toroidal plasmon resonance, as schematically shown in Fig. 1(b). Figure 2 shows the resonant responses and optical forces exerted on the upper disk. Black and red solid lines in Fig. 2(a) represent the frequency dependence of local electric-field magnitude at coordinates (500nm, 0nm, 0nm) and (0nm, 0nm, 0nm), where $f_M=75THz$,$f_Q=124THz$, and $f_T=155THz$ refer to frequencies of magnetic dipolar resonance, magnetic quadrupolar resonance, and toroidal resonance [22]. Magnetic field maps at these resonance frequencies are inserted in Fig. 2(a). Figure 2(c) presents the scattering power of multipolar moments, indicating that the toroidal moment is dominant over the scattering power at $f_T=155THz$. Figure 2(b) shows the calculated optical forces per unit power density acting on the upper disk as a function of frequency. The optical forces are decomposed into three axial components. The blue solid line Fx is a result of conventional optical pressure exerted on the upper disk, which is multiplied by 100 for eye guide. The positive value indicates that Fx is along the + x propagation direction of incident light. The black solid line Fy equals zero, attributed to the symmetric configuration. The main force component is Fz, shown as the red solid line in Fig. 2(b). For all three resonance frequencies, strong negative optical forces (Fz) emerge with significant enhancements due to different local field confinements. Interestingly, the optical force at the toroidal resonance frequency reaches a competitive value of $-182.5pN\mu m^{2}m{W}^{-1}$ as compared with the ones of$-170.0pN\mu m^{2}m{W}^{-1}$at magnetic dipolar resonance and$-83.1pN\mu m^{2}m{W}^{-1}$at magnetic quadrupolar resonance.

 

Fig. 2 (a) Simulated local electric field amplitude at (0, 0, 0) and (500, 0, 0) coordinate positions, with origin of the coordinate located at the structural center. Inset pictures show H field maps on z = 0 plane at frequencies of$f_M$,$f_Q$and$f_T$respectively. (b) Calculated optical forces. (c) Scattering power of decomposed multipoles.

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For an explicit description of the optical forces, we implement the total optical force [Eq. (1)] into two parts, a Coulomb electric force${〈F^e〉}_t={{\displaystyle {∯}_s〈T_{\alpha \beta }^e〉}}_{t}ds$and an Ampere magnetic ${〈F^m〉}_t={{\displaystyle {∯}_s〈T_{\alpha \beta }^m〉}}_{t}ds$for$T_{{}_{\alpha \beta }}^e={\epsilon }_0\left(E_{\alpha }{E}_{\beta }-EE{\delta }_{\alpha \beta }/2\right)$$T_{{}_{\alpha \beta }}^m={\epsilon }_0\left(H_{\alpha }{H}_{\beta }-HH{\delta }_{\alpha \beta }/2\right)$ [7]. Table 1 shows the optical force components of Coulomb electric and Ampere magnetic contributions in the subwavelength system. The role of the kinetic energy of electrons makes strong negative Coulomb force (i.e.,$-226.2pN\mu m^{2}m{W}^{-1}$for toroidal dipolar resonance, $-208.9pN\mu m^{2}m{W}^{-1}$for magnetic dipolar resonance), as compared with positive Ampere force ($43.8pN\mu m^{2}m{W}^{-1}$for toroidal one,$38.9pN\mu m^{2}m{W}^{-1}$for magnetic one), the total forces for both magnetic and toroidal dipolar resonances are$-170.0pN\mu m^{2}m{W}^{-1}$and$-182.5pN\mu m^{2}m{W}^{-1}$, respectively.

Table 1. Numerically Calculated Optical Forces for Ampere and Coulomb Contributions at Magnetic (M-) and Toroidal (T-) Dipolar Resonances.

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To understand the mechanism of how optical force can be modulated by the toroidal resonance, we plot the force in dependence of gap distance g in Fig. 3 while disk radius equals 500 nm and thickness is kept at 40 nm. Figure 3(a) shows the variation of E field strength in a color map as a function of frequency and gap distance g. The gap distance influences the interaction of the double disks and then changes the toroidal resonant frequency and intensity obviously. As the gap increases, the frequency of toroidal resonance has a blue shift, while the local E field becomes weaker and thus reduces the optical gradient force [Fig. 3(b)]. Therefore, it is clear that a narrow gap is much beneficial for the toroidal resonance to enhance the optical force.

 

Fig. 3 (a) Local E-field amplitude probed at structural center as a function of frequency and gap distance. (b) The optical force on upper disk for gap distance g = 5, 10, 20, and 30 nm.

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On the other hand, influences of other geometric parameters on the optical force are studied. The correlation between disk radius r and toroidal-mode enhanced optical force shown in Fig. 4(a) indicates that the force increases steady with disk radius, because the redshift resonance with increasing disk radius strengthen the plasmon interaction between disks. Figure 4(b) shows the relationship of thickness t and toroidal-mode enhanced optical force, the thickness t would not change the toroidal resonance frequency but can strengthen the optical force up to$-392.2pN\mu m^{2}m{W}^{-1}$. The parametric results indicate that we could modulate the optical force enhanced by toroidal resonance so as to utilize it in nanoscale applications, such as mechanical nano-manipulation and force sensor.

 

Fig. 4 The optical force independence of geometric parameters. (a) Keeping gap g = 10 nm and thickness t = 40 nm but changing the radius of disks. (b) For various thickness t while maintaining gap g = 10 nm and radius r = 500 nm.

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In the previous section, we have confirmed there are enhanced optical forces under the special toroidal resonance in double-disk system. Interestingly, plasmonic force has its advantages in trapping both dielectric and metallic particles due to the local field enhancement under surface plasmon resonance, which has been investigated experimentally as well as theoretically [28–31]. In the following section, we will study the optical force acted on a dielectric nanoparticle while the toroidal resonance is excited [Fig. 1(c)]. The disk radius, thickness, and gap distance between double disks are 500 nm, 40 nm, and 10 nm, respectively. A 5-nm-diameter spherical particle with refractive index n = 1.6 is used for investigating the optical trapping characteristic at$z=0nm$plane, and the surrounding medium is water solution with refractive index n_w = 1.33 [11]. It is noticed that the 5-nm-diameter particle has a negligible perturbation to plasmon resonances, due to its small volume as well as low refractive index near the background medium (water).

Figure 5(a) shows the calculated optical force acted on the spherical nanoparticle at different locations on the x axis. The black solid line represents the force enhanced by magnetic dipolar resonance, and the nanoparticle surfers a trapping force pointing to the structural center. The potential is also plotted in Fig. 5(b) by black solid line, which exhibits a minimum value of$-8.29K_{B}T\mu m^2{W}^{-1}$ in the geometric center (a value much larger than the potential of Brownian motion). However, the electromagnetic concentration capability of the toroidal resonance in the double-disk system shows that the E field concentrated in the structural center while H field in a vortex distribution [22], and thus implying a different trapping behavior for the toroidal resonance (i.e., an annulus trapping). Figure 5(a) shows that an enhanced repulsive force, under the toroidal resonance, will be acted on the nanoparticle setting at the position of structural center, and it is zero for positions of $x=\pm 300nm$. With the calculated potential shown by the red line in Fig. 5(b), we can justify that its trapping positions are on the potential valleys ($-6.34K_{B}T\mu m^2{W}^{-1}$for$x=+300nm$, and$-5.76K_{B}T\mu m^2{W}^{-1}$ for$x=-300nm$). Note that the rapid changes near$x=\pm 500nm$for both toroidal and magnetic dipolar resonances are due to the edge effect as electromagnetic field changed rapidly on the charge-accumulated disk edges. Figures 5(c) and 5(d) show the calculated force along the y axis. Since the incident light is along the x axis while its polarization is parallel to the z axis (Fig. 1), the local E field of magnetic dipolar resonance will be concentrated in two sides of disks perpendicular to the x axis and the local H field concentrated in the gap will be parallel to the y axis [Fig. 2(c)]. As a result of this local field characteristic, negligible optical force under magnetic dipolar resonance will be exerted on nanoparticle setting on the y axis. On the other hand, since field distribution of toroidal resonance reveals a vortex property, the force acted on nanoparticle positioned along the y axis is similar to the case for nanoparticle positioned along the x axis [red solid line shown in Fig. 5(c)]. Moreover, the trapping positions along the y axis are$-2.03K_{B}T\mu m^2{W}^{-1}$for$y=300nm$and$-2.62K_{B}T\mu m^2{W}^{-1}$ for$y=-300nm$. Consequently, the xy-plane trapping region for a nanoparticle under the toroidal resonance is an annular pattern, and the unique trapping property could be used in certain nanoscale occasions.

 

Fig. 5 Optical force and corresponding potential on a 5-nm-diameter spherical nanoparticle setting in different axial positions. (a) and (b) Optical force along the x axis (F_x dominates the optical force) and its potential (calculated by the formula$U={\displaystyle \int F\cdot dr}$), the red solid line represents the toroidal-mode enhanced force and black solid line indicates the magnetic-dipole-mode enhanced one. (c) and (d) Optical force along the y axis and its potential.

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4. Conclusion

In conclusion, we have numerically investigated the optical force based on the toroidal resonance in the double-disk system by the MST. The results show the optical force enhanced by toroidal dipolar resonance can reach a value of$-182.5pN\mu m^{2}m{W}^{-1}$, larger than the forces enhanced by both magnetic dipolar and quadrupolar resonances. We analyzed both the Coulomb and Ampere contributions to the total optical force, and then geometric dependences of the toroidal-resonance-based optical force were discussed. Finally, the optical forces acted on a 5-nm-diameter spherical particle were calculated when it was suspending in the double-disk gap under a water background. With a trapping potential comparison between the magnetic and toroidal dipolar resonances co-existed in the same metastructure, it is found that the unique nanoparticle trapping position for toroidal mode is in an annular shape and can be utilized in specific nanoscale manipulations of tiny particles. We believe that this result will be useful in understanding the significant optical force enhanced by the intriguing toroidal resonance, and thus finding applications for the nanoscale trapping.