We investigated the optical binding force in a plasmonic heterodimer structure consisting of two nano-disks. It is found that when illuminated by a tightly focused radially polarized beam (RPB), the plasmon modes of the two nano-disks are strongly hybridized, forming bonding/antibonding modes. An interesting observation of this setup is that the direction of the optical binding force can be controlled by changing the wavelength of illumination, the location of the dimer, the diameter of the nano-disks, and the dimer gap size. Further analysis yields that the inhomogeneous polarization state of RPB can be utilized to readily control the bonding type of plasmon modes and distribute the underlying local field confined in the gap (the periphery) of the dimer, leading to a positive (negative) optical binding force. Our findings provide a clear strategy to engineer optical binding forces via changes in device geometry and its illumination profile. Thus, we envision a significant role for our device in emerging nanophotonics structures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Since the pioneer work of Arthur Ashkin, optical tweezer has become a powerful tool in a broad range of disciplines and given birth to intriguing applications in atom cooling , photonic force microscopy , and DNA dynamics analysis . Trapping and manipulation of single particles with optical tweezer are usually attributed to combined actions of the scattering and gradient forces . When two or more particles are present in an intense light field, an optical binding force appears between the particles due to the electric field gradient which is enhanced by multiple scatterings [5,6]. The magnitude of this optical binding force becomes significantly large especially for metallic nanoparticles when their localized surface plasmons are excited [7–10]. Until now, the optical binding force has been intensively studied in various of nanoparticle clusters such as dimers [8,11–13], tetramers , disk-ring pairs , and particle-on-film structures , which has promised applications in surface-enhanced Raman spectroscopy , particles sorting , and controllable aggregation of nanoparticles .
Physically, the optical binding force is dominated by mode coupling between nearby plasmonic particles . Thus, having the ability to deterministically control this mode coupling would allow one to engineer the optical binding force including its direction. For example, if we consider a dimer, the binding force direction can be reversed by changing the coupling from bonding to antibonding configurations , controlling the coupling of Mie resonators via the surrounding media and the polarization of excitation [12,13], and selecting the coupling configuration between head-to-tail and side-by-side [7,19]. For more complex plasmonic clusters, the Fano resonance resulted from the mode hybridization has been employed to achieve binding force reversal with the aid of in-phase and out-of-phase mode coupling around the Fano dip [14,20–22]. In addition, higher order mode interactions have been successfully utilized to mediate the optical binding force [15,23]. Notably, these aforementioned strategies primarily relied on molding the geometry or increasing the complexity of the structure. It is highly desirable to develop efficient, yet simpler ways to generate and control optical binding force. In this work, we achieve the reversal of the optical binding force in a heterodimer structure by applying a radially polarized beam (RPB) to manipulate the mode coupling. Due to the quite nonuniform polarization state of RPB, optical binding force reversals can be realized simply by changing the excitation wavelength or the position of the dimer. Our results provide a flexible way to manipulate nanoparticles with reversible optical binding forces and open up the possibility for applications in the creation of unconventional optical matters.
2. Results and discussions
The plasmonic nanoparticle under consideration is a heterodimer, which is defined by radii of two Ag nano-disks r1, r2and the gap of dimer g [see the lower right inset of Fig. 1(a)]. The heights of two nano-disks is kept at 30 nm. A radially polarized beam (RPB) is focused onto the dimer through an objective lens with NA = 0.9. The averaged intensity on the top surface plane of the dimer is 1 mW/µm2 which is obtain through dividing the beam power by the beam area. The upper right inset of Fig. 1(a) shows the intensity distribution of this tightly focused beam on the top surface of the dimer, which is calculated by MATLAB codes based on the vector diffraction theory of Richards–Wolf . To model the plasmonic response of the dimer, we perform a full wave simulation based on the finite-difference time-domain method (FDTD Solutions, Lumerical Inc., Canada). The permittivity of Ag are taken from the experimental data of Palik  and the dimer is assumed freestanding in the air (n = 1). The perfectly matched layers as absorbing boundaries are used to mimic nanoparticles placed in an infinitely large free space. In the middle panel of Fig. 1(b), we show the scattering spectrum of the Ag dimer (r1=250 nm, r2=150 nm, and g = 30 nm) when the RPB is coaxially focused on the center of the left-disk (this location is denoted x = y = 0 nm). As seen, three pronounced peaks appear in the scattering spectrum locating at wavelengths of 550 nm, 790 nm, and 970 nm. To unravel the origin of these peaks, scattering spectra of the individual right- and left-disks are shown in the top and bottom panels of Fig. 1(b), respectively. As shown in the insets of Fig. 1(b), we calculate the charge distributions at the resonant wavelengths by Gauss’s law and adopt the strongest real part with the phase shift method . Due to the symmetry of the polarization state, RPB can excite plasmonic dark modes such as the quadrupole mode and radially breathing mode (RBM) which are normally unattainable when illuminated by a plane wave [27,28]. It can be seen from the insets of Fig. 1(b), the RBM and quadrupole modes (Q) are excited by the RPB when the individual left-disk locates at the beam center and 600 nm away from beam center, respectively. Also, the RPB can produce quadrupole and dipolar modes (D) in the right-disk. The plasmon hybridization theory is invoked to interpret the origin of plasmon modes in the dimer : (1) The presence of right-disk introduces a symmetry breaking to the dimer structure, therefore, a quadrupole mode is excited in the left-disk, which though locates at the beam center. This quadrupole mode couples with the dipole mode of the right-disk, resulting in hybrid modes I and II of the dimer. (2) The RBM binds with the quadrupole mode of the right-disk, forming modes III and IV of the dimer.
The time averaged optical force exerted on each single disk is evaluated by Maxwell’s stress tensor (MST) method 
It has been demonstrated that RPB can serve as a powerful tool to tailor the relative weight of multipolar resonances in nanoparticles , or even select the plasmon mode type simply by changing the location of a nanoparticle in the focal field of RPB . Therefore, it is expected that the Fbind reversal can be achieved by controlling the binding type of plasmon modes via moving the dimer in a tightly focused RPB. We first keep the y position of the dimer (r1=250 nm, r2=150 nm, and g = 30 nm) to be 0 nm and examine the scattering spectra of the dimer at different x displacements as shown in Fig. 3(a). A dramatic change in the scattering spectrum is observed when the dimer is moved away from the beam center, implying the inhomogeneous polarization state of RPB can work as a function to tune the plasmon modes of two disks and the coupling between them. As shown in Fig. 3(b), it is found the obvious change of optical binding force spectra companying with x position change. Figure 3(c) displays Fbind versus Δx at the excitation wavelength of 975 nm. Reversals of Fbind are found around x=-500 nm, -240 nm, 300 nm, and 700 nm. In the left panels of Figs. 3(d)–3(h), we show the charge distributions at Δx=-800 nm, -400 nm, 0 nm, 500 nm, and 800 nm, respectively. It is apparent that the mode of the right-disk remains dipolar, while that of the left-disk changed between horizontally and vertically aligned quadrupole modes. For positive Fbind, attractive Coulomb interactions are seen from the charge maps in the left panels of Figs. 3(d), 3(f), and 3(h). These positive Fbind are also confirmed from the field enhancement maps in the right panels of Figs. 3(d), 3(f), and 3(h), where the local field is highly confined in the gap, rendering attraction between the two disks. On the contrary, when Fbind is negative, more intense local field resides at the periphery of the dimer than that in the gap [see right panels of Figs. 3(e) and 3(g)]. As a result, the optical gradient force pushes two disks away. However, a counter-intuitive scenario is also found in the left panel of Fig. 3(g), two disks have a weak bonding mode even though Fbind is negative. The possible reason is that when off the resonance of mode I, more homopolar charges accumulate at both sides of the gap, distributing the local field away from the gap. Note that, such opposite scenario resulted from bonding modes is also found in other heterodimer structures when the dipole and multipole interaction takes place [11,15].
Subsequently, we move the dimer along y direction while keeping its x position at 0 nm. Figure 4(a) displays the scattering spectra of the dimer as a dependence of y position. It is clearly seen that the dimer has an obvious change in the scattering when moved away from the beam center. For example, at the wavelength of 830 nm, it experiences the scattering dip, peak and dip in succession. The underlying variation of mode coupling leads to a dramatical change of binding force spectra shown in Fig. 4(b). Especially, multi-reversals of Fbind are achieved by changing y position at the wavelength of 830 nm as shown in Fig. 4(c). The charge and field enhancement maps are presented in Figs. 4(d)–4(f), which corresponds to the y positions of 0 nm, 350 nm, and 800 nm, respectively. As seen from the left panels of Figs. 4(d)–4(f), with increasing y, the dimer keeps a quadrupole and dipole interaction, nevertheless, has the switching between anti-bonding and bonding mode interactions. As a consequence, the anti-bonding mode interaction produces a repulsive Coulomb force and makes considerable local field distribution at the periphery of the dimer which tends to push the two disks away, leading to a negative Fbind [see Figs. 4(d) and 4(f)]. The bonding mode interaction has an attractive Coulomb force and squeezes the local field in the gap which pulls two disks towards each other, resulting in a positive Fbind [see Fig. 4(e)].
We then keep the left disk at the coordinate origin with the radius and dimer gap size unchanged (r1=250 nm, g = 30 nm) and study the Fbind reversal as the dependence on the radius of the right disk r2. Figures 5(a) and 5(b) shows the scattering and optical binding force spectra of the dimer when r2 increased from 75 nm to 200 nm. It is clearly seen that even at a fixed wavelength, e.g. 785 nm, one can tune the scattering on and off resonance by merely changing the disk size. We calculate Fbind versus r2 at the wavelength of 785 nm as shown in Fig. 5(c). It yields that the Fbind reversals are achieved at r2=129 nm and 176 nm. To clarify the origin of these Fbind reversals, we calculate the charge and field enhancement maps at r2=100 nm, 150 nm and 200 nm, as respectively shown in Figs. 5(d)–5(f). It is clearly seen that when r2 increases, the mode on the right-disk transits from a dipole to a quadrupole because of the phase retardation effect introduced by the inhomogeneous polarization state of RPB [33,34]. These mode changings contribute to the bonding interactions at r2=100 nm and 200 nm and make the local field confined in the gap [see Figs. 5(d) and 5(f)]. Therefore, the underlying attractive Coulomb and optical gradient forces work together to draw two disks closer (positive Fbind). On the other hand, it forms an anti-bonding interaction at r2=150 nm and distributes the local field at periphery of the dimer [see Fig. 5(e)]. The repulsive Coulomb and optical gradient forces push two disks apart (negative Fbind).
As the coupling of plasmon modes is strongly distance-dependent, we finally investigate the binding force as the dependence of the dimer gap size. Figure 6(a) displays the scattering spectra of the dimer (r1=250 nm, r2=150 nm) when the left-disk is placed at the coordinate origin and the gap size g is increased from 10 nm to 50 nm. The blue- and red-shifts, respectively, occur for the modes I and II as g increasing due to the coupling strength decreasing . As can be seen from Fig. 6(b), this decreased coupling strength also results in a decrease in the maximum optical binding force within our interested spectral range. We plot the Fbind as a function of g at the wavelength of 810 nm as shown in Fig. 6(c), which suggests the Fbind is gradually changed from positive to negative. As seen from the charge plot in Fig. 6(d), a bonding mode appears at g = 10 nm, because the dimer is off-resonance from mode II while approaches to mode I at the wavelength of 810 nm. This binding mode makes considerable local field confined in the gap, leading to a positive binding force [see right panel of Fig. 6(d)]. With increasing of g, the anti-binding mode (mode II) is supported in the dimer, which distribute the local field to the periphery of the dimer, leading to a negative binding force [see Figs. 6(e) and 6(f)]. Note that, the Fbind for g = 50 nm is stronger than that for g = 30 nm, mainly due to that, at the wavelength of 810 nm, the dimer with g = 50 nm is more on the resonance of mode II and has a stronger anti-bonding interaction.
In summary, we have systematically studied the optical binding force in a plasmonic heterodimer by using the Maxwell’s stress tensor (MST) method. Remarkably, the optical binding force direction can be reversed from positive to negative by distributing the local field from the gap to the periphery of the dimer. Besides the conventional technique which relies on changing the geometry, the reversal of optical binding force can be realized at a fixed wavelength simply by moving the dimer in the RPB focal field. We envision that our results can be utilized to manipulate the distribution of an aggregation of nanoparticles, which is beneficial for studies of nanoparticle interactions and surface-enhanced Raman spectroscopy.
Key Technologies Research and Development Program (2017YFA0303800); National Natural Science Foundation of China (11634010, 11874050, 61675170, 61675171, 61701303); Open Research Fund of CAS Key Laboratory of Spectral Imaging Technology (LSIT201913W); Fundamental Research Funds for the Central Universities (310201911FZ049, 3102019JC008); Shaanxi Provincial Key Research and Development Project (2018KW-009).
The authors declare no conflicts of interest.
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