1. Introduction

It is of great interest to examine opto-mechanical effects when optical resonances happen in photonic nanostructures and metamaterials. Effects of various optical and photonic resonances on the optical forces and optical micromanipulations have been reported [1–7]. Near the resonances, despite of the increase of force magnitude by the strong field enhancement, the force direction can also be tuned via the relative phase delay between the constituting optical fields. In some cases, the interacting force between nanostructures changes from positive to negative, indicating mutual attraction or repulsion of nearby nano-objects. For instance, two adjacent silicon waveguides may repel or attract each other by controlling the phase relationship of the guided modes [1]. A pair of photonic crystal slabs can be optically tuned to attract or repel each other [2]. A metallic parallel plate cavity exhibits enhanced and reversed optical pressure by an anomalous magnetic resonance [3, 4]. Moreover, a pair of side-by-side nanorods could generate attractive or repulsive forces by electric or magnetic resonances [5].

Recently, many works have revealed a variety of plasmonic nanostructures that sustain Fano resonances, including single asymmetric particles, core-shell structures, nanoholes, metal-dielectric-metal waveguides, and particle clusters and chains [8–21]. The Fano resonances in such plasmonic nanostructures are simply regarded as the photonic analogies to the original one in quantum systems [22]. Classical oscillator model [13], phenomenological mode [14], coupled-mode formalism [15], as well as ab initio theory [16] have been developed to analysis them. However, to the best of our knowledge, there is by far no report of the Fano resonance effect on the optical forces among such plasmonic systems.

In this work, we present such a study and show that a dipole-quadrupole (DQ) Fano resonance [17, 18] can dramatically affect the optical binding force, in both magnitude and direction. We numerically examine the optical forces on a gold nanorod heterodimer which is designated to support a DQ Fano resonance (see Fig. 1) by simultaneously overlapping a dipole mode and a quadrupole mode spectrally and spatially. The dipole mode of the short nanorod can act as a continuum while the relatively high-Q quadrupole mode in the long nanorod as the sub-radiative discrete level (e.g, dark mode) [8, 17, 18]. Interestingly, near the Fano dip, the binding force between the two nanorods becomes negative, indicating an attraction to repulsion transition and possible equilibrium configuration. We stress that this is not possible for a homodimer under the parallel configuration. It is because that the anti-parallel mode is totally “dark” in such case and not excitable by a normal incidence plane wave polarized along the dimer axes [7,23].

 

Fig. 1 The geometry of the system and the incoming light configuration. The plasmonic dimer consists of two nanorods with $L_1=100$nm, $L_2=280$ nm, and gap $g=10$nm. The diameter of the nanorods is $d=40$nm.

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2. Numerical approaches and benchmark

The numerical calculations are performed by employing an electromagnetic computational tool based on the finite integral technique (FIT) [24]. We use the optical constant of gold from Ref [25]. The interested volume is meshed with a resolution of $2$nm along all the three coordinates and the edge and gap region is fine-meshed to $0.5$nm. A Gaussian time pulse with appropriate central frequency and bandwidth coming from the $\widehat{y}$direction (i.e., transverse to the dimer axis as shown in Fig. 1) illuminates the dimer with $E$ field $\widehat{z}$-polarized. The spatial distribution of the light fields in the frequency domain is obtained by the Fourier transformation of the FIT time-domain results. The optical force $F^{(i)}$on each individual object is then evaluated via a surface integral

In Eq. (1), $S^{(i)}$ must be a closed surface exclusively containing the $i$-th object and $\widehat{n}$is the outward normal vector on $S^{(i)}$. For convenience, we have chosen a rectangular parallelepiped box to enclose the target nanorod $i$($i=1,2$), with walls along the principle coordinate axis. In all calculations, the optical forces were monitored by changing the size of the box and regarded as convergence when the fluctuation is below 1%.

Additionally, to justify the calculations by this FIT plus MST (FIT-MST) method, we have compared the results to the ones (dots in Fig. 1) by the method of discrete dipole approximation (DDA), in which the optical forces are calculated by the Lorentz formula [23, 26]. The total dipole number is$N=60040$ (~$8$nm³/dipole) for the DDA calculations. Good agreements of the results by the FIT-MST (curves) and DDA (symbols) are seen for both the optical cross sections and the optical forces in Figs. 2(a) and 2(b), respectively. With respect to the force magnitude in this work, our calculations are effectively in accord to an illuminating plane wave with intensity $I_0=1$mW/µm².

 

Fig. 2 Optical features associated with the DQ Fano resonance. (a) Optical extinction cross sections and optical scattering cross sections. (b) Optical forces between the two nanorods and along the k-direction.

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3. Results and discussion

Figure 2(a) shows the scattering cross section (dashed line) and the extinction cross section (solid line) of the plasmonic heterodimer shown in Fig. 1. The geometry parameters are listed in the figure caption. The peaks around the wavelength of$\lambda =715$ nm ($f_{{}_{\text{DQ}}}^{-}=420$THz) and $\lambda =635$nm ($f_{\text{DQ}}^{+}=473$THz) come from the hybridized DQ plasmon modes. The plasmonic DQ Fano resonance featuring an asymmetric line shape in the optical cross section is spectrally between these two modes and can also be regarded as a result from the interference of the fundamental plasmonic modes of the two nanorods [17, 18]. The identification of the Fano resonance is confirmed by the optical spectrum of individual nanorods (figures not shown here). The dipole resonance of the short nanorod is at $f_D^{(s)}=463$THz and that of the long nanorod is at much lower frequency $f_D^{(l)}=262$THz. However, the quadrupole (or second-order) resonance of the long nanorod is around $f_Q^{(l)}=443$THz, very close to $f_D^{(s)}$. The situation can be further corroborated by the current distributions as shown in Fig. 3 which shows the current density (arrows) in the $yz$plane through the nanorod center. By the polarization of the incoming light, the current is dominated by its $\widehat{z}$-component and we show it as the contour in the figures. It is clearly seen in Fig. 3 that inside the investigated frequency range, there is no (one) node in the short (long) nanorod. This is actually true for all frequencies between $f_{\text{DQ}}^{-}$ and $f_{\text{DQ}}^{+}$. It is noteworthy that in some cases, the mode excited in the long nanorod is not strictly a quadrupole but rather a general high-order mode. However, for convenience we retain the notation of quadrupole throughout this paper.

 

Fig. 3 Snapshot of steady-state current density in the $yz$plane through the nanorod center. (a)–(d) are for the frequencies of 473 THz, 448 THz, 420 THz, and 385 THz, respectively, corresponding to the positions marked with (A), (B), (C), and (D) in Fig. 2(b). The contour is for the dominating$z$component of the current.

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The key part of our results is contained in Fig. 2(b) where the k-direction scattering force $F_{scat}=F_y^{(1)}+F_y^{(2)}$on the whole dimer and the inter-particle force $F_{bind}=F_z^{(1)}-F_z^{(2)}$between the two nanorods are plotted against the light frequency. Notice that we have labeled the short nanorod in the left as 1 and the other one as 2 (see Fig. 1). It is straightforward to see that with respect to the defined coordinates, positive (negative) $F_{bind}$represents attraction (repulsion) of the two nanorods in the $\widehat{z}$-direction.

Figure 2(b) demonstrates that the k-direction scattering force spectrum (black solid curve) has a line shape similar to the extinction cross section (black solid curve) shown in Fig. 2(a). This is a consequence of usual radiation pressure which is basically determined by the total extinction cross section and the differential cross section [27]. The binding force spectrum (red dashed curve), however, looks much more similar to the scattering cross section, particularly around the Fano dip near $f=448$THz. These similarities are reasonable since the defined$F_{bind}$is dominated by the scattering process whereas $F_{scat}$is more directly relevant to both light scattering and absorption. As a matter fact, optical force can be regard as a quantity explicitly determined by the near-field properties which are correlated to the far-field features at Fano resonances [10].

Figure 2(b) further shows that near the low-frequency resonance $f_{{}_{\text{DQ}}}^{-}=420$THz (marked as C), the two nanorods attract each other due to bonding hybridization where the electric currents at the nanorod ends across the gap are in phase [see Fig. 3(c)]. On the other hand, around the Fano dip (near the point marked as B), the $F_{bind}$ spectrum develops an asymmetric line shape which, more importantly, crosses the zero point. In other words, the binding force is reversed. Particularly, the two nanorods repel each other in a frequency window as marked by the vertical dashed lines which fall in between $f_{{}_{\text{DQ}}}^{-}$ and $f_{\text{DQ}}^{+}$ . We attribute this to the fact that inside this regime, the excitations in the two nanorods are nearly out-of-phase, resulting from the Fano interference. Indeed, Fig. 3(a) shows that the currents across the gap is basically out-of-phase at $f=473$THz. Figure 3(b) shows a rather similar but dipole-weakened version of Fig. 3(a) when the frequency goes down to$f=448$THz. These two cases correspond respectively to the points A and B as marked in Fig. 2(b), for which the binding force vanishes ($F_{bind}\approx 0$). For frequencies between these two points, the binding forces are all negative. We notice that at a frequency much lower than $f_{{}_{\text{DQ}}}^{-}$[marked as D in Fig. 2(b)], the binding force also reverses slightly. This is due to the destructive interference effect of a dipole-dipole configuration [17]. Figure 3(d) shows such a case for $f=385$THz where the current in the long nanorod represents the first-order mode.

To examine how the forces are mediated by the DQ Fano resonance, we plot in Fig. 4 the force spectra for different gap distance $g$between the two nanorods. Actually, introducing a variation of the gap $g$is to spatially tune the separation of dipole and quadrupole resonances. We note that the peaks on the force spectra shift in consistent with the plasmon hybridization prediction [13]. As the gap $g$increases, the multiple scattering interactions between the nanorods are expected to weaken. One of the consequences is that the two DQ-hybridized modes come closer in their spectral positions, converging to the nearly-overlapped dipole and quadrupole resonances of the individual nanorods, which is around $f=463$THz. Accompanying such shifting, the k-direction force $F_{scat}$increases because the spectrum sharpens [see Fig. 4(b)] and becomes the simple summation of those on individual (isolated)nanorod. Meanwhile, the binding force [see Fig. 4(a)] gradually diminishes and the force reversal phenomenon finally disappears for $g\approx 50$nm as the Fano resonance is killed in absence of strong interaction.

 

Fig. 4 Spectra of optical forces for nanorod heterodimers in the parallel configuration. (a) The total scattering force and (b) the relative interparticle force. The lengths are kept as $L_1=100$nm and $L_2=250$nm.

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Figure 5 shows the force spectrum variation when one of the nanorod’s length changes. Changing$L_1$is to spectrally tune the dipole resonance while changing $L_2$for the quadrupole one. We have kept the gap $g=10$nm in such processes during the calculation. Note that the dipole resonance shifts faster than the quadrupole one does when the nanorod length is varied [19]. Not surprisingly, as the dipole and quadrupole resonances are spectrally separated, their interaction (hybridization) is modulated. As a consequence, the optical force spectra vary accordingly. For instance, when the quadrupole resonance is red-shifted by increasing $L_2$ from $260$nm to $300$ nm, i.e., tuning $f_Q^{(l)}$from $455$THz to $431$THz, the force spectra redshift as a whole and the force reversal survives robustly (see the left column of Fig. 5). This is partially because of the rather broad dipole continuum which persistently overlaps with the quadrupole resonance. However, when the dipole resonance is red-shifted by increasing $L_1$(e.g., tuning $f_D^{(s)}$from $450$THz to $400$THz with fixed $f_Q^{(l)}=462$THz for$L_2=250$nm), the binding force reversal weakens and becomes almost invisible for $L_1=150$nm [see Fig. 5(b)]. This is because that the spectral separation for the case of $f_D^{(s)}=400$THz and $f_Q^{(l)}=462$THz is quite large and the DQ interference becomes negligible.

 

Fig. 5 Same as Fig. 2 while keeping the gap $g=10$nm but varying the nanorod length. (a) and (c) for $L_1=100$ nm and various$L_2$; (b) and (d) for $L_2=250$nm and various$L_1$.

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4. Conclusion

We have shown that dipole-quadrupole Fano resonance can dramatically affect the optical binding force and even lead to a reversal. The binding force spectrum features asymmetric line shape and therefore represents an important consequence of plasmonic Fano resonance. The results we obtained are relevant to near-field optical micromanipulation and to the study of “optical matters” in plasmonics. It is possible to experimentally fabricate the long nanorod as the tip of a fixed near-field optical tweezer over a substrate and use it to control other particles which are like the short nanorod in our model [7]. The attraction-to-repulsion transition may be easily observed by measuring the particles relative motion to the tip. Flexible control of magnitude and direction of the optical binding force opens the door for the observation of collective phenomena of nanoparticles and the design of new materials and devices.