1. Introduction

The optical forces induced by the light-matter interaction as an important tool for noninvasive manipulation of mesoscopic objects has been widely used in biology, biochemistry, and physics since the radiation force was firstly applied to accelerate and trap microscopic particles by Ashkin in 1970s [1]. In recent years, the nonconservative pulling force induced by the gradientless laser beam, so-called the tractor beam has attracted tremendous attention since it can be used to drag small particles towards the laser source [2–9]. Chen et al. theoretically investigated the optical pulling force acting on the polystyrene beads by using a Bessel beam [2]. The underlying physical mechanism is the maximization of the forward scattering via the interference among different radiation multipole resonant modes, which in turn introduces a backward recoil force on the nanoparticles [2]. The tractor beam can be realized when the recoil force overcomes the incident force caused by the momentums transformed from light to the particles. Therefore, effectively improving the forward scattering and suppressing the backward scattering of the particles is a good choice to enhance the optical pulling force [10–14]. In all these researches, noble metal (plasmonic) nanoparticles such as silver (Ag),gold (Au) nanoparticles, some high refractive index dielectric nanoparticles such as silicon (Si) nanoparticles and some relative core-shell nanoparticles composed of both kinds of materials, become the most popular candidates because these nanoparticles can exhibit distinct multipole resonant modes interference in the spectrum ranging from the visible to the near infrared compared with some low refractive index dielectric particles, such as the polystyrene and silica beads.

Noble metal nanoparticles can sustain the localized surface plasmon resonances (LSPR) which correspond to the collective oscillations of the electrons in the surface of the plasmonic nanoparticles. Thanks to their capability of strong field localization and enhancement associated with LSPRs, enhanced optical forces can be realized in the optical trapping of plasmonic nanoparticles [15–19]. In addition to the improvement of the optical trapping force at the optical resonance, Li et al. theoretically predict that a transverse optical scattering force exerted on an Ag, Au or silica/Au core shell can be also strengthened by the Fano resonance resulting from the interference between the electric dipole (ED) and quadrupole (EQ) modes [20]. The lateral optical scattering force, which is totally different from the gradient force induced by the tightly focused laser beam or LSPR enhanced electric field, can be transferred from optical pulling force to optical pushing force by tuning the geometric sizes of the plasmonic nanoparticles and the incident wavelength. Recently Gao et al. also demonstrated the Fano resonance-induced optical pulling force acting on active plasmonic nanoparticles can be improved by two orders of magnitude compared with a single active nanoparticle with the radiation of a uniform plane wave [11]. However, for plasmonic nanoparticles, the optical resonance enhanced optical pulling forces are mainly caused by the electric component of the light because the plasmonic nanoparticles hardly have responses to the magnetic component to the incident light in the optical spectrum due to the absence of the field inside the nanoparticles.

Unlike the plasmonic nanoparticles, some dielectric nanoparticles with high refractive indices such as Si, germanium (Ge) and gallium arsenide (GaAs) nanoparticles can support not only electric resonances (ED, EQ, etc.) but also magnetic resonances (magnetic dipole mode (MD), magnetic quadrupole mode (MQ), etc.) [21–26]. The effects originate from the optical induced electric and magnetic multipole moments which correspond to the collective oscillation of the polarized charges and the circular displacement currents inside the nanoparticles respectively [21]. Therefore, rich types of resonances and the interferences between different resonant modes in dielectric nanoparticles provide a new platform for studying the effects of the electric, magnetic, and the interferences among different resonant modes on the optical forces [13,27–29]. In this theoretical paper, the tunable optical forces including the optical incident forces and the recoil forces on the Au/Si core-shell nanoparticles at the optical resonances are investigated with the multipole expansion technique based on the Mie scattering theory. This type of hybrid nanoparticles have been proved that the electric resonant mode and the anapole mode can be flexibly tuned by changing the relative geometric sizes of the core and the shell to realize the scattering transparency and the ideal magnetic dipole scattering [30–32], which implies that the interference interactions among different resonant modes can also be efficiently modulated. Our results demonstrate explicitly that the total scattering force, including optical incident and optical recoil forces can be tuned and enhanced remarkably by tuning the interference between neighboring resonant modes. Importantly, the maximum of the total optical force acting on the core-shell nanoparticles mainly depends on the purity of the resonance mode instead of the resonant scattering modes of the hybrid system due to the enhancement of the negative recoil force induced by the interference between adjacent optical resonant modes.

2. Physical model and numerical methods

The physical model proposed in our work is schematically depicted in Fig. 1. An Au core-Si shell nanoparticle immersed in water (refractive index n = 1.33) is illuminated by a linearly polarized plane EM wave with the power intensity of 10 mW. In order to efficiently excite the electric and magnetic modes, the radius of the Si shell R_shell is fixed at 175 nm. The dielectric permittivity for gold and Si and their dispersion relations used in calculations are taken from the experimental data in [33] and [34], respectively.

 

Fig. 1 Schematic showing the Au core-Si shell nanoparticles excited with a linearly polarized plane EM wave. The linearly polarized plane wave illuminates on the Au core-Si shell nanoparticles along with x direction and the polarization direction is along with z direction, as shown in the inset.

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For spherical core-shell nanoparticles excited by a linearly polarized plane EM wave, the Mie coefficients $a_l$ and $b_l$ are given by [35–38]

Once the Mie coefficients are attained, the scattering, extinction and absorption efficiencies Q_sca, Q_ext and Q_abs are expressed by [35,36]

By combining expressions (6) and (7), the total optical scattering force F_total acting on the core-shell nanoparticles is given by [35,39]

Analytically, the total optical radiation force F_total can be decomposed into two components, namely, the incident forces (F_e^l and F_m^l) and the recoil forces (F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1) as follows [13]:

The former components F_e^l and F_m^l correspond to the contributions of the pure resonant modes such as electric modes and magnetic modes, and the latter three terms originate from the interference among different resonant modes (electric modes and magnetic modes), such as F_e^l_m^l, and adjacent orders (dipole, quadrupole, octupole modes, etc.) of the same kind of mode, such as F_e^l_e^l+1, and F_m^l_m^l+1. It can be clearly seen from expressions (12) - (15) that the optical forces, especially the recoil forces, exerted on the core-shell nanoparticles can be tuned by tailoring the interference between resonant modes with neighboring orders (F_e^l_e^l+1, and F_m^l_m^l+1) and both types of resonant modes with the same order (F_e^l_m^l).

3. Results and discussion

3.1 Tunable interference interaction among different resonant modes

Let us first examine the evolution of the scattering spectrum of the gold core-Si shell nanoparticles with the increase of the size of the Au core. Figure 2(a) presents the evolution of the total scattering efficiency spectrum of the Au core-silicon shell nanoparticles when R_core increases from 0 nm (a pure Si nanosphere) to 175 nm (a pure Au nanosphere). In order to distinguish the modulations of the size of the Au core on each resonant mode, we also present the contribution of each mode to the scattering efficiency of the core-shell nanoparticles as a function of the Au-core radius R_core and the wavelength, as shown in Figs. 2(b)-(e), which corresponds to ED, MD, EQ, and MQ modes respectively. Some higher order modes are negligible due to their weak responses in the considered optical spectrum range (0.4-1.8 μm). For comparison, the scattering efficiency spectra and the contributions of each mode of the pure Si and Au nanoparticles as the particle radii R_Si and R_Au increase from 40 to 175 nm are also shown in Fig. 2.

 

Fig. 2 Evolution of the total scattering efficiency spectra (the first column) and the contributions of ED mode (the second column), MD mode (the third column), EQ mode (the fourth column), and MQ mode (the fifth column) to the total scattering efficiency with the increase of the radii of the core for the Au core-Si shell nanoparticle (the upper panel), the Si nanoparticle (the middle panel) and the Au nanoparticle (the lower panel).

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For Au nanoparticles, one can see that only electric resonant modes (ED and EQ) can be excited while the magnetic modes are not supported in such plasmonic nanoparticles due to the localization of the polarized electrons in the surface of the nanoparticles, as shown in Figs. 2(k)-(o). Hence, only electric resonant modes inducing incident and recoil forces are considered for plasmonic nanoparticles. Additionally, it can be seen clearly that the interference interaction between ED and EQ modes is very weak (partly interference) and only exists in large sized nanoparticles due to the boarding of resonant modes with the increasing of radius which means that only tiny recoil forces can be realized with pure plasmonic nanoparticles [11,12]. For the case of Si nanoparticles, we can see from Fig. 2(f) that the degree of the interference between neighboring resonant modes is hardly modulated and keeps partly interference due to the red-shifts of all the resonant modes though ED, MD, EQ, and MQ can be excited efficiently as R_Si increases, as shown in Fig. 2(g)-(j), which indicates that it is hard to modulate the optical radiation forces, especially the recoil force, by tuning the interferences among different modes in particles with high refractive index [13].

However, as can be seen obviously from Figs. 2(b)-(e), not only the electric and magnetic modes (ED, MD, EQ, and MQ) can be excited sufficiently but also the resonant peak of each mode can be modulated by changing the size of the Au core. More importantly, close inspections of Figs. 2(b) and (d) show that the electric resonant modes (ED and EQ) are shifted to the long wavelength while the magnetic resonant modes (MD and MQ) keep almost unchanged when R_core increases from 0 to 42 nm, which implies that the interference between adjacent modes can be tailored flexibly by changing the radius of the gold core, especially the interference between ED and MD resonant modes. In order to verify this point, we provide four typical cases about the interference between the ED and MD modes with the increase of the Au core size, as shown in Figs. 3(a)-(d) which correspond to the partly interference (Type I: R_core = 0 nm), the totally interference (constructive interference between ED an MD modes) (Type II: R_core = 42 nm), the pure MD resonant mode (constructive interference between anapole and MD modes) (Type III:R_core = 67 nm), and the pure ED resonant mode (Type IV:R_core = 115 nm). An analogous phenomenon can also be observed for the EQ and MQ modes, as shown in Fig. 2(a). More interestingly, the MQ and MD resonant modes are blueshifted with further increase of the size of the Au core, which even leads to an interference between the MD and the EQ resonant modes.

 

Fig. 3 Four typical interferences between MD resonant mode and ED mode with the increase of R_core: (a) partly interference (R_core = 0 nm), (b) totally interference (R_core = 42 nm), (c) pure MD mode (R_core = 67 nm), and (d) pure ED mode (R_core = 115 nm).

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3.2 Reconfigurable optical radiation, incident and recoil forces

We now inspect the optical force acting on the Au core-Si shell nanoparticles with the increase of the size of the Au core. Figures 4(a)-(f) present the total optical force F_total, the incident forces (F_e^l and F_m^l) and the recoil forces (F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1) as a function of the Au core radius and the wavelength. For comparison, the evolutions of total optical force F_total, the five components (F_e^l, F_m^l, F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1) acting on Si and Au nanoparticles with the increase of the radius of the nanoparticles are also shown in Fig. 4. For Au nanoparticles, the total radiation force only originates from the contribution of incident force F_e^l and recoil force F_e^l_e^l+1 due to the absence of magnetic resonant modes, as shown in Figs. 4(m)-(r). For Si nanoparticles, the total radiation force is dominated not only by the electric and magnetic resonant modes (F_e^l and F_m^l) but also by the interference between adjacent resonant mode (F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1), as shown in Figs. 4(g)-(l). However, by referring to Figs. 2(k) and (f), we can see that the maximum total radiation forces acting on the Au and Si nanoparticles always exist in the resonant wavelengths (see Figs. 4(m)-(o) and Figs. 4(g)-(i)) and cannot be tailored by increasing the radii of both type of nanoparticles though certain recoil forces are induced by the partly interference between neighboring modes. For the Au core-Si shell nanoparticles, however, it can be obviously seen that the total radiation force F_total and the five components (F_e^l, F_m^l, F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1) are modulated by increasing the radius of the Au core, as shown in Figs. 4(a)-(f). Very differently from the pure Au and Si nanoparticles, the maximum radiation force acting on the core-shell nanoparticles does not exist in wavelength of the maximum scattering (denoted with a circle in Figs. 2(a), 4(a)) due to the enhanced recoil force originating from the interference between neighboring resonant modes, especially the interference between ED and MD modes (denoted with a circle in Fig. 4(d)).

 

Fig. 4 Total radiation forces F_total (the first column), incident forces F_e^l (the second column) and F_m^l (the third column), and the recoil forces F_e^l_m^l (the fourth column), F_e^l_e^l+1 (the fifth column), and F_m^l_m^l+1 (the sixth column) as a function of the wavelength and the radii of the core for the Au core-Si shell nanoparticle (the upper panel), the Si nanoparticle (the middle panel) and the Au nanoparticle (the lower panel).

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In order to verify this point, we inspect the modulation of the interference between neighboring resonant modes to the optical radiation forces F_total acting on the core-shell nanoparticles according to the four representative cases (Types I-IV) of interference between MD and ED mentioned above. The total optical scattering force and the five component F_e^l, F_m^l, F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1as a function of the wavelength for Types I-IV are shown in Fig. 5(a)-(d). As can be seen from these figures, F_total can be flexibly modulated by the recoil forces resulting from the interference between MD and ED modes by increasing the size of the Au core though F_total are dominated by the incident forces (F_e^l and F_m^l) because of the relative small recoil forces under the excitation of plane wave. It should be pointed out that the contributions of F_e^l_e^l+1 and F_m^l_m^l+1 to F_total at the MD resonant wavelength (indicated with dashed line in Fig. 3) can be neglected because of the weak interference between neighboring order resonant modes for the four cases considered. In general, the recoil force can be positive or negative, depending on the asymmetry scattering factor ˂cosθ ˃. However, from Figs. 5(a)-(d), one can see clearly that the recoil forces are mainly negative and the magnitude can be enhanced or suppressed by engineering the interference between MD and ED modes. By comparing Figs. 5(a) and (b), we can see clearly that the magnitude of F_e^l_m^l induced by the interference between ED and MD can be enhanced from 10 pN (partly interference between ED and MD) to 17 pN when the ED is fully interfered with the MD by increasing the size of the Au core. Therefore, as can be seen from Fig. 5 (b), the maxima of F_total appears at the MQ resonant wavelength (λ = 0.92 μm) rather than at the resonant scattering (λ = 1.28 μm, see Fig. 3(b)) due to the giant enhancement of F_e^l_m^l counteracting a large part of incident forces, which is totally different from the cases discussed in previous reports mentioned in the introduction [13].

 

Fig. 5 F_total and the components F_e^l, F_m^l, F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1 as a function of the wavelength for R_core = 0 nm (a), R_core = 42 nm (b), R_core = 67 nm (c), and R_core = 115 nm (d).

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From expressions (13)-(15), one can see that besides the enhancement of recoil force by strengthening the interference between ED and MD, we can also suppress the recoil force completely by weakening the interference interaction of ED and MD modes to realize a large optical radiation force which can be used to accelerate nanoparticles. In order to verify this point, we calculate the total optical force F_total and the five components F_e^l, F_m^l, F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1 exerted on the core-shell nanoparticle at pure MD and ED resonant modes, as shown in Figs. 5(c)-(d). For the case of the pure MD resonant mode, it can be seen from Fig. 5(c) that the total optical scattering force is dominated by the incident force F_m^l and keeps the same magnitude as that of Type I at MD resonant wavelength. It should be noted that the contribution of the MD resonant mode to the total scattering decreases gradually with the increase of the size of the Au core. Therefore, no obvious improvement can be achieved for the total scattering force though the recoil forces resulting from the interference are suppressed. However, a close inspection of Fig. 5(c) shows that the total optical scattering force at the ED resonant wavelength can be enhanced up to 20 pN due to the decrease of F_e^l_m^l. This point can be also confirmed by further increasing the size of the Au core up to 115 nm, where there is no overlap among the ED and other modes (see Fig. 3(d)). One can see from Fig. 5(d) that the total optical force can be further improved up to 25 pN at the pure ED resonant mode.

Besides the four typical cases mentioned above, we also investigate the evolutions of the total radiation force F_total and the components F_e^l, F_m^l, F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1at the MD and ED resonant wavelengths with the increase of the Au core. The results are presented in Figs. 6(a) and (b), respectively. As can be seen from both figures, the total scattering forces are mainly dominated by the incident forces F_e^l, F_m^l, and the recoil force F_e^l_m^l and the contributions of interference between adjacent-order modes, such as F_e^l_e^l+1and F_m^l_m^l+1, are negligible because of the weak interference interaction among these modes. Meanwhile, we can also see that the total optical scattering force acting on the core-shell nanoparticles at MD resonant mode decreases gradually with the increase of R_core. For the case of ED resonant mode, however, the total optical force on the core-shell can be improved by increasing R_core. This result is caused by the competition between ED and MD modes with the increase of the Au core, as shown in Fig. 2 (k). More interestingly, by comparing Figs. 6(a) and (b), we can see that the recoil forces are mainly dominated by the non-resonant mode rather than by the resonant mode, which is manifested as that the curves of F_e^l in Fig. 6(a) (F_m^l in Fig. 6(b)) and F_e^l_m^l follow the same tendency with the increase of R_core. It should be clarified here that a force peak existing in the curve of F_e^l at R_core = 94 nm in Fig. 6(a) is induced by the EQ resonant mode rather than by the ED mode.

 

Fig. 6 Evolutions of F_total and the components F_e^l, F_m^l, F_e^l_m^l, F_e^l_e^l+1, and F_m^l_m^l+1 at MD resonant mode (a) and ED resonant mode (b) when R_core increases from 0 nm to 115 nm.

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In order to further explore the underlying physical insights about the modulation of the optical scattering force induced by the interference between ED and MD, we also investigate the far-field scattering patterns of the Au core-Si shell nanoparticles at resonant wavelengths of the four cases considered. Figures 7(a) and (b) present the radiation patterns of Type I and II at the MD resonant mode. It can be seen that the far-field scattering are dominated by the forward scattering and only a small part of backward scattering can be found for the case of Type I, as shown in Fig. 7(a). Therefore, as can be seen from Fig. 5(a), a certain backward recoil force can be induced by the partial interference between ED and MD modes. This fact can be explicitly manifested when the radius of Au core increases up to 42 nm, where the ED resonant mode is completely overlapped with MD resonant mode. It can be clearly seen that the far-field scattering is almost dominated by the forward scattering, as shown in Fig. 7(b). Importantly, the intensity of the forward scattering for Type II can be improved 50% compared with that of Type I. Hence, one can see that the recoil force can be prominently enhanced from 10 pN for the case of Type I to 17 pN for the case of Type II.

 

Fig. 7 Far-field scattering pattern for Type I (a), Type II (b), Type III (c), and Type IV (d) in E plane and H plane. All far-field scattering intensities are normalized with the maximum far-field scattering intensity of Type II. The incident direction of the beam is the same as the inset in Fig. 1.

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With the further increase of size of the Au core, however, we can see from Fig. 7(c) that the backward scattering can be improved and the forward scattering is suppressed obviously when the interference interaction between ED and MD is reduced, which results in a very small recoil force at the pure MD resonant mode, as shown in Fig. 5(c). An analogous phenomenon can also be found for the pure ED resonant mode, as shown in Fig. 7(d). One can see that the backward scattering can be further enhanced and the intensity of backward scattering is almost the same as that of the forward scattering, which leads to a 0 pN recoil force at the pure ED resonant mode. Meanwhile, the total scattering force acting on the core-shell nanoparticles can be improved for the case of Type IV due to the suppression of the backward recoil force.

4. Summary

In conclusion, we investigate theoretically the optical scattering forces acting on the Au core-Si shell nanoparticles with the illumination of the plane EM wave by using the multipolar expansion method based on the Mie scattering theory. It is found that the optical recoil force and the incident force can be tuned flexibly by tailoring the interference interaction between neighboring resonant modes, which in turn results in a configuration of the total optical scattering force acting on the core-shell nanoparticles. The recoil force induced by the fully interference between ED and MD can be enhanced up to 17 pN which is almost twice larger than that of partial interference (pure Si nanoparticles). The improved recoil force can counteract a large part of the incident forces, which causes that the maximum optical scattering force exerted on the core-shell nanoparticles is located at the MQ mode, but not at the wavelength where ED and MD modes overlap completely. On the other hand, the optical scattering force can be improved in some degree by suppressing the recoil force, which can be realized by reducing the interference between ED and MD modes by properly adjusting the size of the Au core. In addition, the far-field scattering patterns of the core-shell nanoparticles with different Au core sizes indicate clearly that the intensities of the forward and the backward scattering, which dominate magnitude of the recoil force, can also be modulated by the interference between ED and MD modes induced by increasing the size of the Au core. In the experimental point of view, the synthesis of Au core-Si shell nanoparticles is a big challenge currently. It may as well be anticipated for other noble metal-high refractive index materials core-shell nanoparticles, such as the gold core-titania shell nanoparticles, that can be synthesized easily [40]. Some noble metal core-shell nanoparticles [41] with the interference between adjacent resonant modes can also be used in experiments to investigate the modulation of interference between neighboring resonant modes on the optical forces. The results presented in this work can shed light on the dynamics interaction between hybrid core-shell nanoparticles and the light. Other applications, such as the enhancement of the optical pulling force or the pushing force in the optical tractor beam, can also be realized by using Au core-Si shell nanoparticles.