1. Introduction

Understanding of the interaction between light and matter has been developed enormously over the past three decades. Since the photon has linear momentum, light will push on any object by means of scattering optical force. In the field of optical trapping or optical tweezers, a focused light beam can be used to trap particles with gradient force arising from the inhomogeneity of the optical field [1,2]. As optical tweezers are not always successful in trapping metallic particles, plasmonic tweezers, based on surface plasmon polaritons, were proposed for trapping and manipulating metallic particles [3,4]

On the other hand, a gradientless light beam as the tractor beam was proposed to pull particles backward all the way towards the light source due to maximal forward scattering via interference of the radiation multipoles [5,6]. Also, optical pulling force can be realized when two oppsitely directed beams are applied [7,8]. Most previous researches covered passive dielectric materials, but gain medium has not been studied in depth. Gain medium is modeled by a complex permittivity with an active imaginary part, which may reverse the mechanical effect [9,10]. For instance, charged particles travelling through the gain medium will be accelerated rather than decelerated, and the gain medium itself is also subjected to the force generated by the stimulated radiation [11]. Negative radiative pressure on the frozen atoms caused by optical gain has been observed experimentally [12]. Gain particles tend to be pulled towards the incident plane wave due to light amplification [9,13]. If we incorporate a gain medium into a nanoscale photodynamic system, an additional degree of freedom will be added to optical micromanipulation.

In this paper, we study the optical pulling force on the coated subwavelength nanoparticles consisting of active core and metallic shell. In the subwavelength region, nonlocal effect for the metal component should be taken into account. Actually, Nonlocality arises from electron interactions in the metal medium and leads to spatial dispersion [14–16]. Moreover, nonlocality reduces the effect of geometric imperfection [17], makes the resonant position blueshift [18–23], and enhances the optical bistability [24–26]. Since the nanoparticles are in the long-wavelength range, one can take one step forward to establish the nonlocal effective medium theory for the investigation of equivalent permittivity of core-shell nanoparticles [27]. Then, the equivalent permittivity for the gain core-plasmonic shell nanoparticles shall be adopted to analyze the behavior of scattering optical pulling/pushing force. Incidentally, optical “tractor beam” for the coated nanoparticle with plasmonic core and passive shell was optimized by first-order Bessel beams with approximate polarization [28]. Compared with such tractor beams, we take into account the effect of nonlocality or spatial dispersion in the core-shell subwavelength nanoparticles, and the negative scattering optical force in our case results from the gain materials with the incident plane wave instead of the Bessel beams. In addition, for coated particles with metallic shells, there are two resonant modes, arising from hybridization of the inner and outer surface plasmons [29]. And, one of the dipole resonant scattering and the cloaking scattering based on plasmonic shell can be used as the bright and dark modes to induce the Fano-like scattering [30]. Moreover, when the nonlocality is considered, the longitudinal modes, which may be prohibited in the local system, can be excited above/below the plasma frequency/wavelength. Mutual interferences among dipole longitudinal modes can also result in Fano-like scattering response. We shall show that in these two Fano-resonance regions, negative optical scattering force arises, indicating Fano-resonance-induced negative optical force [31–34]. We believe that the realization of negative optical force with nanosizes can open up the possibility of pulling ultrasmall particles.

2. Model and theories

Let us consider the external electric field $E=E_0(\omega )e^{ikz}{e}^{-i\omega t}{\widehat{e}}_x$ lighting upon the spherical coated nanoparticle with inner radius $a$ and outer radius $b$embedded in the host medium with relative permittivity ${\epsilon }_h$, as shown in Fig. 1. In addition, the relative permittivity of gain core is described by ${\epsilon }_c={\epsilon }_{g1}+i{\epsilon }_{g2}$, whereas ${\epsilon }_{g2}<0$ corresponds to material gain, and the nonlocal shell has spatial dispersive permittivity ${\epsilon }_s(\omega ,k)$. For simplicity, the size of the nanosphere is assumed to be much smaller than the incident wavelength, and hence the retardation effect is neglected and long-wavelength approximation can be adopted [27,35,36].

 

Fig. 1 Schematic of the coated nanosphere embedded in the host medium with relative permittivity ${\epsilon }_h$. The coated particle is composed of a gain core with inner radius $a$ and the relative permittivity ${\epsilon }_{\text{c}}$, and a nonlocal plasmonic shell with outer radius $b$ and${\epsilon }_s(k,\omega )$.

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The electrostatic potentials in the core and the host medium can be well solved with Laplace’s equation. However, to achieve the electrostatic potential inside the shell region due to nonlocality, we can introduce the semiclassical infinite barrier model (SCIB) [27, 35]. In general, the field intensity$E$and the displacement $D$ are related through the relation for nonlocal shell

The spirit of the SCIB is to extend the nonlocal shell to all system and solve the Possion-type equation for the displacement potential $D(r)=-\nabla {\phi }_D(r).$ By assuming charge sources located on the inner and outer interfaces, one yields,

Taking the Fourier transform of Eq. (2), we have

On the other hand, by taking the Fourier transform of Eq. (1), we have an alternative expression as ${\phi }_D(k)={\epsilon }_s(k,\omega )\phi (k)$ in $k$space. Here, the nonlocal shell is described by the hydrodynamic model [16,23]

As a consequence, one obtain the electric potential $\phi (k)$ in$k$space,

By using the inverse Fourier transform, we express the displacement potential and electric potential in $r$space inside the nonlocal shell as,

Then, the general expressions for the electric potentials and displacement potential can be written as,

To determine the coefficients, the continuity of the electrical potential and the displacement potential should be matched on the two interfaces [35]. By considering the boundary conditions at $r=a$ and $r=b$, one yields

with the volume fraction $f=a^3/b^3$.

For such a core-shell system, the electric polarizability including the radiation correction can be written as [6,9,39]

On the other hand, in the long-wavelength approximation, we may derive the equivalent permittivity ${\epsilon }_{eq}$ of the core-shell nanospheres containing the gain core and nonlocal shell. The equivalent permittivity ${\epsilon }_{eq}$ can be obtained self-consistently with the dipolar factor $D$. If we identify ${\epsilon }_h$ with ${\epsilon }_{eq}$, then the dipolar factor $D$in Eq. (10) becomes a self-consistency equation, which readily implies that $D=0$ [40]. As a result, one yields the equivalent permittivity of the coated nanospheres containing nonlocal shell,

Note that Eq. (13) is an original effective medium theory for inhomogeneous nonlocal system. This new formula will be adopted to analyze the behavior of the optical force.

If we do not take into account the nonlocal property of metal shell, we can get$G_a=G_b=G_{ab}={\epsilon }_s(\omega )$, and Eq. (13) is naturally reduced to local case,

3. Results and discussion

We are now in a position to present some numerical results. The relevant parameters are ${\epsilon }_{g1}=2.1025$for the gain core, and${\epsilon }_b=5$, ${\omega }_p=1.367\times {10}^{16}{s}^{-1}$ (or${\lambda }_p=137.8nm$),$\gamma =2.733\times {10}^{13}{s}^{-1}$for the metal shell [38,41,42]. Without loss of generality, we set the outer radius $b=10nm$and we assume that the host medium is vacuum with ${\epsilon }_h=1$. In addition, we normalized the force by $F_0=\pi b^2{S}_{inc}/c$ ($S_{inc}$is the power flow density of the incident wave) [9].

At first, we show the evolution of the equivalent permittivity with our nonlocal effective medium theory [see Eq. (13)] for different gain in the core in Fig. 2. It is evident that the real part of equivalent permittivity $Re({\epsilon }_{eq})$ always exhibits a ripple-like line shape independent of the gain value. However, the resonant magnitude of the real part gradually increases as ${\epsilon }_{g2}$ is increased [see Fig. 2(a)-2(c)], and achieves the maximum at the singular plasmon resonant wavelength $\lambda =330nm$ for ${\epsilon }_{g2}\approx -0.0706$ [see Fig. 2(d)]. As the gain is increased after the singular point, the magnitude of the real part degrades due to excess gain [36]. On the other hand, the imaginary part $Im({\epsilon }_{eq})$ behaves as a bell with positive value for small gain [see Fig. 2(a) and 2(b)], and has a sharp resonance at the singular point [see Fig. 2(d)]. After that, it is worth noting that $Im({\epsilon }_{eq})$ is now negative in the resonant region [Figs. 2(e) and 2(f)]. We shall see that it is possible to achieve large negative optical force on the coated nanosphere with gain core and plasmonic shell especially near the surface plasmon resonance of the coated nanoparticles.

 

Fig. 2 Evolution of the equivalent permittivity as gain is increased with Eq. (13), from (a) to (f). Parameters: ${\epsilon }_{g1}=2.1025$ and $f=0.03$. The real and imaginary parts are denoted by the red solid lines and blue dash-dotted lines. The black dash line represents the value of $-2{\epsilon }_h$.

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Then, we study the optical force and the equivalent permittivity for the coated nanospheres as a function of the incident wavelength for fixed core gain ${\epsilon }_{g2}\approx -0.1$ in Fig. 3. Such a choice of the gain may be much realistic, which requires the gain threshold to be $G_{th}\text{=}\text{-}4\pi Im(n_G)/{\lambda }_0\text{=}1.322\times {10}^{4}c{m}^{-1}$ [43], where $n_G=\sqrt{{\epsilon }_c}$, with plasmonic resonant wavelength ${\lambda }_0\text{=}327.7nm$.

 

Fig. 3 Normalized optical force and equivalent permittivity as a function of the incident wavelength with$f=0.03$ (a-c) and $f=0.3$ (e-f) in nonlocal (red solid lines) and local (blue dash-dotted lines) cases.

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There are two plasmonic resonant modes, i.e., the sphere-like symmetric dipole mode at resonant long-wavelength ${\lambda }_{+}$; and the cavity-like antisymmetric dipole mode at resonant short-wavelength ${\lambda }_{-}$ [38,41,42]. At these two surface plasmon resonant wavelengths, the optical forces are found to be greatly enhanced positively or negatively, as expected. In detail, for small volume fractions such as $f=0.03$ [see Fig. 3(a)], one observes large negative (positive) force at ${\lambda }_{-}$ (${\lambda }_{+}$); whereas one has large positive (negative) force at ${\lambda }_{-}$ (${\lambda }_{+}$) for large volume fractions such as $f=0.3$ [see Fig. 3(d)]. Note that in the short-wavelength region, the loss of nonlocal metallic shell is small, and singular plasmonic reasonance takes place. As a consequence, one observes the ripple-behavior of $Re({\epsilon }_{eq})$, which is lack due to large loss at long-wavelength resonant wavelength [see Fig. 3(b) and 3(e)].

In addition, for small volume fractions $f$, the metallic properties will prevail in the coated nanospheres, resulting in large negative $Im({\epsilon }_{eq})$ [see Fig. 3(c)] and negative optical force [see Fig. 3(a)] in the short plasmonic resonant wavelength. To one’s interest, the normalized negative optical force for our coated nanospheres can be as large as 40~100, which is about two orders of magnitude for the dielectric Mie sphere [9]. On the contrary, for large volume fractions, $Im({\epsilon }_{eq})$ takes large positive values near the short plasmonic resonant wavelength [see Fig. 3(f)]. Hence, large positive optical force is found [see Fig. 3(d)].

For comparison, we also consider the local case in Fig. 3. Acutally, the introduction of nonlocality results in much large negative force in comparison with the local one.

To investigate such behavior qualitatively, we consider the optical force for the nanosphere with the corresponding equivalent permittivity of the nonlocal coated nanoparticles. In the long-wavelength limit, we can approximate the optical force to be,

Thus, the optical force will be enhanced at two surface plasmon resonances, $Re({\epsilon }_{eq})=-2{\epsilon }_h$, as shown in Fig. 3(b) and 3(e). Equation (15) includes two terms: the second term due to the scattering is always positive, and the direction of force is fully determined by the first term, which originates from the generation of power in the gain core. In general, to obtain a negative optical force, $Im({\epsilon }_{eq})$must be negative and large enough. However, by substituting $Re({\epsilon }_{eq})=-2{\epsilon }_h$ into Eq. (15) and ignoring the second term for sufficiently small$kb$, we get$〈F〉=6\pi {\epsilon }_0{\epsilon }_h^{2}k{b}^3{E}_0^2/Im({\epsilon }_{eq})$ . It indicates that the resonant optical force is inversely proportional to $Im({\epsilon }_{eq})$. For small volume fractions at short-wavelength plasmon resonance wavelength, the imaginary part of equivalent permittivity $Im({\epsilon }_{eq})$ with our nonlocal effective medium theory is $-0.0675$ in Fig. 3(c), which is smaller than $-0.08365$ predicted by the local formula Eq. (14). Therefore, one obtains stronger negative optical force due to nonlocality. Similar analysis can be performed for large volume fractions at long-wavelength resonant wavelength. Incidentally, the positive optical force may become weak when one takes into account the nonlocality or spatial dispersion. This is just due to the fact that positive $Im({\epsilon }_{eq})$ with nonlocal effective medium theory is larger than that predicted with the local one.

The dependence of two resonant wavelengths and the corresponding resonant optical forces on the volume fraction $f$ are shown in Fig. 4. It is also evident that with increasing $f$, the resonant long wavelength ${\lambda }_{+}$ exhibits red-shift with both nonlocal effective medium theory and local theory, and the introduction of nonlocality makes ${\lambda }_{+}$ blueshift [see Fig. 4(a)]. However, as $f$ increases, the resonant short wavelength ${\lambda }_{-}$ is decreased monotonically with local theory. Again, nonlocality brings the resonant short wavelength ${\lambda }_{-}$ blue-shift, but it exhibits non-monotonic behavior with $f$ [see Fig. 4(b)]. Actually, since the dipole mode at the resonant short wavelength ${\lambda }_{-}$is more cavity-like, and the corresponding field-enhancement is expected near the inner core. For small volume fractions $f$ or small inner cores, the nonlocal effect prevails, resulting in dramatic blue-shift. This shift is not evident in Fig. 4(a) because the nonlocal effect does not dominate the surface plasmon for the relatively large outer surface.

 

Fig. 4 Two plasmonic resonant wavelengths (blue lines), and corresponding normalized resonant optical forces (red lines) with increasing volume fraction$f$, in nonlocal (solid lines) and local (dash-dotted lines) cases, respectively.

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As for the optical force at the corresponding surface resonant long wavelength [see Fig. 4(c)], with increasing $f$, the pushing optical force is increased, achieves positive maximum and then suddenly changes direction. After that, one has the pulling optical force and even maximal negative force at $f\approx 0.18$. Further increasing $f$ results in decreased resonant pulling force. Again, with nonlocal effective medium theory, one predicts larger pulling optical force than that based on local effective medium theory. Similar behavior of the optical force can be found at the resonant short-wavelength if one decreases $f$ [see Fig. 4(d)]. Thus, the introduction of nonlocal effect here makes the resonant position shift and the amplitude change. Besides that, the level of gain required to get singular resonance in local case is much smaller than the nonlocal case (not shown here). From this Figure, we conclude that the coated nanosphere with gain core and metallic shell provides more degrees of freedom to manipulate the nanoparticles.

To observe the negative force region clearly, we provide a phase diagram in Fig. 5. The phase diagram clearly shows the transition from pushing to pulling force. For instance, for a given$f$, one may realize the reversal of optical action from repulsion/attraction to attraction/repulsion near the critical wavelength. As $f$ increases, the negative force region becomes broad because the gain core, which is responsible for producing the pulling optical force, dominates the whole nanoparticle. In addition, the negative optical force region extends to long wavelength. The black region shows extremely large negative optical force caused by these plasmonic resonant modes. For small volume fractions [see the insert in Fig. 5], as $f$decreases, a blue shift arises from nonlocality, similar to the case in Fig. 4(b).

 

Fig. 5 Normalized optical force $F/F_0$with respect to incident wavelength and volume fraction$f$ in nonlocal theory. Gray region indicates the parameter space for the pushing force, colored region indicates the pulling force. The circled region is magnified in the inset. The black region shows extremely large negative optical force much stronger than −15.

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It is known that Fano resonance may induce a pulling optical scattering force acting on plasmonic nanoparticles of large sizes when an approximate manipulating laser beam is adopted [30,31]. In general, the plasmonic Fano resonance is coupled to high-order modes such as the interference of simultaneously excited multipoles. Here, we take one step forward to investigate how Fano resonance affects the optical scattering force in the present coated subwavelength nanospheres with gain core and nonlocal plasmonic shell. Actually, in such a system, there are two kinds of Fano resonances due to different dipole modes, which are quite different from the Fano mechanisms due to high-order resonances [31]: one is Fano-like scattering due to the interaction of the cloaking and resonant scattering modes [30], and the other is multiple Fano-like profiles arising from the interference of excited longitudinal modes above/below the plasma frequency/wavelength [23].

To observe the Fano-like profile, we introduce scattering efficiency of the coated nanospheres, as defined as $Q_{sca}=8{\epsilon }_h^2{\left(2\pi b/{\lambda }_0\right)}^4{\left|D/b^3\right|}^2/3$. For scattering efficiency, there are not only two plasmonic symmetric and antisymmetric modes due to the hybridization between the dipole plasmon solid Ag sphere and that of Ag cavity, but also one cloaking mode with much smaller scattering efficiency, as shown in Fig. 6(a), which is similar as that for the local coated plasmonic nanoparticles [30]. With decreasing the aspect ratio, the cloaking and antisymmetric dipole states merge, and they can be served as the dark and bright modes, leading to a narrowband Fano resonance [30] [see Fig. 6(b)].

 

Fig. 6 (a) Dependence of scattering efficiency on incident wavelength and aspect ratio for core-shell spheres under nonlocal frameworks. The yellow and blue lines show resonant and cloaking modes, respectively. (b)-(d) show corresponding scattering efficiency, normalized optical force and the equivalent permittivity with$a=1.2nm(\eta =0.12)$, respectively.

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This Fano resonance, resulting from the dipole resonant mode and the dipole cloaking mode, is really beneficial for inducing the negative optical scattering force, as shown in Fig. 6(c). Here, we further estimate the magnitude of the optical scattering force. $F_0$ is about 0.105 pN, and hence $F$ is −0.2 pN, whose magnitude is slightly smaller than that reported for the large plasmonic nanoparticles [32]. From the viewpoint of nonlocal effective medium theory, it is evident that the scattering efficiency achieves the maximum when $Re({\epsilon }_{eq})$ is closer to the −2, and it is minimal as the the real part of the equivalent permittivity approaches to 1 [see Fig. 6(d)]. Since the surface resonant wavelength and the cloaking wavelength are almost the same for much small aspect ratio, we obtain a narrowband Fano-like response [see Fig. 6(b)]. In addition, near the resonant and cloaking wavelengths, $Im({\epsilon }_{eq})$is negative with magnified magnitude, resulting in the negative optical force [see Fig. 6(c)].

In the end, we aim at the other Fano-like profile in the scattering efficiency from the nonlocal coated nanospheres above/below the plasma frequency/wavelength, as shown in Fig. 7. Within the nonlocal theory, a series of high-frequency peaks appear above the plasma frequency. These peaks result from the excitation of confined longitudinal modes determined by the vanishing of the longitudinal permittivity [44]. These effects of spatial dispersion may change the scattering process dramatically and bring about the Fano-like resonances. In Fig. 7, we plot the scattering efficiency and scattering optical force as a function of the incident wavelength below the plasma wavelength$308nm$. Generally, the magnitudes of these subsidiary peaks due to the excitation of the longitudinal modes strongly depend on damping coefficients [14,23,44]. To observe the Fano-like resonance in this wavelength region, one should suppress the metallic damping [23]. It is evident that Fano-like scattering response takes place between $306nm$ and $307nm$ [see Fig. 7(a)]. In the proximity of the dip of the Fano-like resonance, the negative optical force is demonstrated [see Fig. 7(b)]. This kind of Fano profile can still be understood from the concept of effective permittivity [see Fig. 7(c)]. For instance, the real part of equivalent permittivity reveals a sharp Fano profile, with negative permittivity at about $306nm$, resulting in large scattering, and unit permittivity near $305nm$, leading to cloaking state. This Fano profile provides quick switching from a strong pushing force to a pulling force, which could be beneficial in the fine tuning of optical nanoparticles.

 

Fig. 7 (a) Scattering efficiency, (b) normalized optical force and (c) The real (solid line) and imaginary (dash-dotted line) parts of equivalent permittivity with a weaker damping coefficient above/below the plasma frequency/wavelength as a function of incident wavelength for$f=0.6$ in nonlocal theory.

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4. Conclusions

In conclusion, we have investigated the optical scattering force on coated plasmonic nanoparticles containing gain core and nonlocal metallic shell in the long-wavelength approximation. We find that negative optical scattering force or pulling force can be realized near the symmetric or antisymmetric dipole resonant wavelengths. When the nonlocality is taken into account, the magnitude can be larger than the one predicted with the local theory. To understand such behavior, we establish the nonlocal effective medium theory to derive the equivalent permittivity of the coated system. Especially, near the resonant wavelength, the imaginary part of equivalent permittivity can be very large, which indicates the gain may be magnified, resulting in negative scattering optical force. In addition, two examples of Fano-like scattering responses in coated nanospheres with nonlocal plasmonic shells are investigated, and the corresponding negative optical scattering forces are predicted. Our study opens up the possibility of exerting negative optical force on nanoparticles, and offers an alternative method to realize optical selection and sorting of plasmonic nanoparticles. It is worth noting that, for small particles, the Brownian motion and the gradient force are neglectful compared to the giant negative scattering force. Meanwhile, the incident power that we used is low, so the electric field is not strong. Thus the nonlinear effect due to the gain core can be neglected.

Here some comments are in order. More recently, light-induced pulling and pushing force by taking into account the synergy of optical force and photophoretic force [45]. The coupling between optics, thermotics, and mechanics provides us rich fields to understand the light-matter interactions. The interfacial thermal resistance in the metallic surface plays an important role in the photophoretic motion, which may break the synergy of optical force and photophoretic force [46]. It is of interest to study the effect of interfacial thermal resistance on the photophoretic force, and to check the synergic effect of optical and photophoretic force. Meanwhile, the gain medium can be described in terms of the nonlinear permittivity with negative losses. By taking into account the nonlinear effect in the gain medium, an exactly solvable electrodynamical model for surface plasmon amplification by stimulated emission of radiation (spaser) is proposed [47]. It is interesting to consider the nonlinear effect on the negative optical force. Our study offers novel ways for optical manipulation of nanoparticles, and even for applications in optical tractor beams, and so on.