Abstract

Plasmonic metamolecules have many peculiar properties and have been applied in biological and materials science such as in reconfigurable 3D building blocks for complex nano-architectures and imaging probes for high-resolution sensing. In these applications, fast detection of the bond angles of the sub-wavelength metamolecules is highly desired. However, angle detection is not the same as orientation detection. The two orientations must be determined simultaneously, and common orientation sensors can only measure one. In this work, we propose and demonstrate a method to resolve the bond angle of a plasmonic metamolecule composed of three spherical nanoparticles. The detection of the bond angle is achieved via modulation depth analysis of polarization-resolved dark-field images. The underlying mechanism is found to be the opposing responses of the longitudinal and transversal bonding modes to the polarization variation of the incident light. In addition, the spectrally degenerate structures are further distinguished by the spot center localization method. This method may pave the way for practical application of plasmonic metamolecules.

© 2017 Optical Society of America

1. INTRODUCTION

Plasmonic resonators made of metallic nanoparticle (NP) assembly, are also known as plasmonic metamolecules due to similarities to molecules [13]. Both fundamental properties and possible applications of the plasmonic metamolecule have been studied and demonstrated, such as the suppression of radiation losses [1], data storage [2], Fano resonance [36], plasmon induced transparency [7], quantum tunneling effect [8], circular dichroism [9] and reconfigurable device [10]. On the other hand, metamolecules have also been exploited as imaging probes because of their nonbleaching and nonblinking properties and their excellent biocompatibility [1115]. Typically, to obtain the orientation information of the target, asymmetric metallic gold nanorods, instead of NPs, are used [1113,16]. In these applications, the detection of the bond angle of the metamolecules is sometimes needed for non-destructive, real time monitoring of the structural dynamics. However, it is difficult to detect the angle compared to the detection of an orientation sensor, because an angle is determined by two orientations and they need to be measured simultaneously. Therefore, non-contact far-field resolution of the bond angle of the plasmonic molecule is highly desired.

To determine the angle information, another pathway is to detect the overall response of the structure instead of a specific mode (e.g., Fano mode or longitude mode of a rod) and extract the angle information via further data analysis. Modulation depth (M) analysis method is a good candidate to realize this target. In this method, the M value is usually obtained based on the analysis of the intensity evolution of the polarization-resolved images. This method was originally applied to extract the orientation and conformation of a single conjugated polymer chain based on absorption [17], and it was later used to analyze the nanorod orientation by photothermal imaging, which is also based on absorption [12].

In this work, we use modulation depth analysis to resolve the bond angle of a plasmonic metamolecule. Gold NPs are employed to build the metamolecules [8,1820]. The minimum NPs to form an angle is three. Here, we focus our study on the most basic plasmonic metamolecule: a trimer composed of three linked particles. To mimic the case when the metamolecule is dynamically deformed, we use atomic force microscopy (AFM) manipulation techniques to assemble gold NP metamolecules with different bond angles [Fig. 1(a)] [2123]. The benefit of using AFM assembly is that the structure is controllable and the bond angle of the metamolecule can be tuned and measured precisely (2°). In addition, we can choose nanoparticles with similar sizes based on the particle morphology and scattering spectra before the assembly process.

 figure: Fig. 1.

Fig. 1. Illustration of sample assembly and M value measurement. (a) Schematic diagram of assembly method for the Au NP cluster via AFM manipulation. (b) Dark-field image of the fabricated sample. Scale bar: 20 μm. Inset shows the AFM image of an assembled metamolecule composed of three 60 nm Au NPs. Bond angle θ=110°. Scale bar: 150 nm. (c) Scattering images of the metamolecule when it is the brightest and the darkest as the polarization angle φ of the incident light being rotated from 0 to 180° anticlockwise, where 0° is along the horizontal axis. Wavelength range of the signal collected by the camera is 450–800 nm. Measured M value is 0.395. (d) Simulated scattering intensity versus incident light polarization angle φ of the same metamolecule. Simulated M value is 0.409.

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The scattering images of the metamolecule projected onto the varying incident polarization directions are taken to extract the M value. The bond angles of these metamolecules θ are then obtained with the modulation depth analysis technique to quantitatively determine the θM relation. In addition to the measurement of bond angle, we also discussed a method to determine the orientation of the metamolecules. Cases exist in which structures with different orientations have the same bond angle (called spectrally degenerate structures, similar to the degenerate states in quantum mechanics). We further distinguished these degenerate structures by the image spot center localization method. These results with plasmonic metamolecules may find applications in dynamic structure assembly and high-resolution sensing.

2. RESULT AND DISCUSSION

One of the samples fabricated is shown in Fig. 1(b). The scattering images are taken using a commercial dark-field microscope (Olympus IX73P2F). Therefore, the AFM image and the dark-field scattering image are correlated to ensure that the bond angle measured by an optical method can be directly compared with the AFM result. The modulation depth analysis method can be described by the equation

M=ImaxIminImax+Imin,
where Imax and Imin represent the maximum and minimum intensities, respectively, of the images of the structure under study as the incident polarization is rotated. Here, we use the M method in the analysis of dark-field scattering imaging, and the original equation is modified slightly to be more intuitive. The dark-field images when the 110°-angle metamolecule is most bright and most dark are taken, and the value M is calculated accordingly, as shown in Fig. 1(c). The simulation results of the scattering intensity versus incident polarization angle φ are illustrated in Fig. 1(d). We can see that the value M obtained from the simulation result is almost the same as that extracted from experimental result.

To find the correspondence between the metamolecule bond angle and the value M, more metamolecules are fabricated, and three with bond angles of 90°, 110°, and 180° are shown in Fig. 2(a). The corresponding normalized polarization traces are illustrated in Fig. 2(b). The normalization is carried out by dividing each curve by the sum of the maximum and minimum of the curve. The variation of the traces as a function of bond angles is obvious. The magnified dark-field images for different polarization angles are also shown in the insets. The simulated results with bond angles varying from 90–180° are presented in Fig. 2(c), where all the traces have a sine function shape. These results agree well with the experimental results. Figure 2(d) is a plot of all experimentally measured M values and, simultaneously, the results extracted from the simulations for comparison. Ideally, because of the structural symmetry, M=1 and 0 should represent bond angles of 180° and 60°, respectively. However, since the single NP mode and background signal always exist, in our setup, the maximum M obtained is around 0.7 for a 180°-angle metamolecule. Very high consistency can be found between these experimental and simulated results, which indicates that this method is highly accuracy, especially when the bond angle is away from 180°. We attribute this high accuracy mainly to the high signal/noise ratio of the polarization-resolved dark-field images, as can be seen from the insets in Fig. 2(b).

 figure: Fig. 2.

Fig. 2. Determining the metamolecule bond angle and orientation based on modulation depth analysis. (a) Trimers with varying angles assembled. (b) Experimental results of trimers with angles 180°, 110°, and 87°. Insets show the dark-field image evolution of the individual trimers as a function of illumination polarization angle. (c) Simulated polarization dependence of total scattering intensity for trimers with varying angles. The minimum of each curve indicates the orientation of the trimer angle bisector, i.e., the orientation of the metamolecule. (d) Modulation depth M of trimer with different bond angles. Dots represent the experimental data, and dashed curve is extracted from simulation results in (c).

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The underlying mechanism of this technique can be understood through modal analysis based on the polarization-resolved spectra. More metamolecules were fabricated for spectra analysis, and one of them with bond angle of 120° is shown in Fig. 3. Guiding the scattering signal to a spectrometer, we measured the polarization-resolved scattering spectra of these trimers, as illustrated in Fig. 3(b). From these spectra, we can see that each spectrum has three modes: an anti-bonding mode at 550nm, a transversal bonding mode from 610–650 nm, and a longitudinal bonding mode around 680–715 nm, where the transversal and longitudinal directions are perpendicular and parallel to the base of the isosceles triangular trimers [24,25], respectively. It is also obvious that both the transversal and longitudinal bonding modes are sensitive to the incident polarization angle, although the longitudinal bonding mode is the dominant component in the spectra. Simulation results are also shown in Figs. 3(d)3(e) for comparison, and good consistency with the experimental results can be found. To show the mode polarization dependence in an explicit way, we extracted the peak intensities of each spectrum and demonstrated the mode intensity versus polarization relation in Figs. 3(c) (experiment) and 3(f) (simulation). Apparently, these two mode traces have a cosine function relation, where however, the polarization traces of the longitudinal and transversal bonding modes have a π-phase shift (π/2-polarization angle difference), that is, an opposite response to the polarization rotation. The reason for this phase shift is that the metamolecule studied here is a symmetric structure. Specifically, the longitudinal bonding mode is always perpendicular to the angle bisector, while the transversal bonding mode has two components along the two arms of the angle, and the vector sum of these two components is parallel to the angle bisector, shown in Fig. 3(a) as an example. Alternatively, this result can also be understood via the relationship of the longitudinal and transversal bonding modes, which are typically proportional to cos2(φ) and sin2(φ), respectively. More data with metamolecules with other bond angles can be found in Supplement 1.

 figure: Fig. 3.

Fig. 3. Modal analysis of the scattering spectra of the metamolecules. (a) AFM images of the trimer with bond angle θ=120°. Arrows illustrate the typical transversal bonding mode (red) and the longitudinal bonding mode (blue). (b) Polarization-dependent scattering spectra of the metamolecule obtained from experiment. (c) Spectral peak intensities versus incident polarization angle φ. (d)–(f) Corresponding simulated results for comparison. Results for trimers with other bond angles can be found in Supplement 1.

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We can summarize the above results and discussion as follows:

I=Il+It=Cl+Alcos2(φ+φ0)+Ilb+Ct+Atcos2(φ+π2+φ0)+Itb,
where I, Il, and It are the intensities of the total, longitudinal bonding mode, and transversal bonding mode, respectively. Cl and Ct are the offset constants, which are approximately equal to the amplitude Al and At, respectively. Usually, Al is larger than At. φ0 denotes the initial polarization angle of the excitation light. Ilb and Itb are the background signals of the longitudinal and transversal bonding modes. This background signal may have several originations: anti-bonding mode, residual bonding modes excited by the depolarized focusing light, and substrate scattering. It is relatively uniform, and can therefore be expressed as Ib=Ilb+Itb. The maximum and minimum intensities can be obtained in a straightforward manner:
Imax=Cl+Ct+(AlAt)+IbImin=Cl+Ct(AlAt)+Ib.

Hence the M factor is determined by

M=ImaxIminImax+IminAlAtAl+At+Ib.

Using the data shown in Fig. 3 and Supplement 1, we can estimate the M value. For example, if Ib is neglected, the M value for the 180° trimer should be close to 1. But when Ib is included, this value is estimated to be 0.7, which is the same as the measured result in Fig. 2. We can see that the total image intensity maximum decreases with the decrease of bond angle θ while the total intensity minimum increases. Therefore, the modulation depth M will drop with the decrease of θ.

From Fig. 3, we can also understand more intuitively why we cannot directly obtain the metamolecule angle from the polarization-resolved spectra, as people usually did in the study of nano-protractors [13]. In these spectra, it seems plausible that all metamolecule angles can be determined by the maximum and minimum of the bonding modes, but this rule works only when it is already known that the 0° axis is along the horizontal direction, which is not satisfied in most of the situations.

Because we have already obtained the θM relation, it is straightforward to determine the bond angle of an unknown metamolecule via measurement of M. For daily use, we only need to measure the two dark-field images of the metamolecule when the intensity is most bright and most weak, and θ is then determined with no need to repeat all the spectra measurements as done in Fig. 3. Indeed, these spectra are only useful when one needs to calibrate the microscopic system. Therefore, the working efficiency of this method should be much higher than those based on spectral analysis.

In addition to the bond angle, the orientation of the metamolecule, that is, the opening direction of the trimer, can be determined in a straightforward manner. Actually, it can be seen from Fig. 2 that the minimum of the total scattering intensity is simply the orientation of the metamolecule. This fact can be understood from Fig. 3, where we have pointed out that the longitudinal bonding mode dominates the other modes and the longitudinal and transversal bonding modes have opposite responses to the polarization variation. Therefore, the minimum in each curve of Fig. 2(b) corresponds to the minimum of the longitudinal bonding mode and the maximum of the transversal bonding mode, which can be used to determine the orientation of the metamolecule.

Typically, however, two metamolecules with opposite orientations have the same bond angles and, hence, the same modes. We call these metamolecules spectrally degenerate structures because they have the same M value and the same spectra response. One such example is shown in Fig. S2 in Supplement 1. Notice that this degeneracy has also widely existed in the applications using a rod as a nano-probe [11,12,16]. Herein, we use the spot center localization method to lift this degeneracy, which has been proven to be effective in other works [26,27]. It is well know that the detail of a sub-wavelength-scale trimer cannot be seen directly. However, a commonly neglected fact is that, when the polarization angle of the incident light rotates from one arm of the bond angle to the other arm, the spot center of the far-field image will shift, where the direction of the shift reveals the orientation of the metamolecule being observed.

In Fig. 4(a), we use two 90°-angle metamolecules (L-shape and 7-shape) as examples to illustrate how to distinguish the spectrally degenerate trimers. We rotate the polarization angle of the incident light from the direction in which the scattering image of the trimer is the brightest, that is, the longitudinal bonding mode is the strongest. If the polarization angle is rotated anticlockwise, for the L-shape trimer, the center of the equivalent dipole (red spot) will shift to the lower-right direction first (0° polarization) and then turn back to the opposite direction (90° polarization), while for the 7-shape trimer, it will shift to the upper-left direction first (0° polarization) and then turn back (90° polarization). The mechanism of this shifting is that when the incident polarization is along any arm of the trimer, both the longitudinal and transversal bonding modes can be excited [25], and they have different dependence on the incident polarization. Therefore, the near-field interference between the two modes changes with the variation of incident polarization. This illustration is confirmed by the simulation results in Fig. 4(b) (far-field) and in Figs. S4 and S5 in Supplement 1 (near-field). From these simulation results we can see that the mode center does have an obvious shift under different incident polarization.

 figure: Fig. 4.

Fig. 4. Removal of structure degeneracy of symmetric metamolecules. (a) Illustration of the spot center localization method. Polarization of the incident light is rotated from horizontal to vertical direction. Blue spot denotes the equivalent dipole location of the longitudinal bonding mode when the polarization angle is 135°. The red and magenta spots represent the equivalent dipole location after mode interference under 0° and 90° polarized incident light, respectively. (b) Simulated electric field distribution of the longitudinal and transversal bonding modes recorded from the monitor plane 500 nm above the L-shape metamolecule, which is highlighted in the black dashed box in (a). Red arrows represent the polarization angle of the incident light. White dashed lines denotes the reference frame, with the horizontal (x) axis and vertical (y) axis along the two arms of the trimer. X and Y are the coordinates of the spot center. (c) AFM image of assembled structure: an L-shape metamolecule whose bond angle is 90°. The inset is the enlarged view of the metamolecule. The two dimers on the left and the bottom work as the reference frame to correct image drift. (d) Measured results of the spot center shift under the polarization angle 0° (red) and 90° (magenta). Simulated results are also shown in green circles, by integrating all the modes. The dashed arrows denote the moving path of the spot center with polarization angles rotated anticlockwise. The results of 7-shape metamolecule can be obtained by rotating images (b), (c), and (d) 180°.

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The observed spot is basically the superposition of these modes. Therefore, the center of the spot will also shift in the same way with the varying polarization angle. Since the spot-center shift is very small, for real applications, it is better to compare the two images when the polarizations of the incident light are along the two arms, and the spots are most separated.

To confirm the principle of the image center localization method, we fabricated an L-shape trimer as shown in Fig. 4(c). Two dimers are also fabricated at a distance to the left and the bottom of the trimer, acting as the reference frame to correct the image drift (see the line profiles of the AFM image in Supplement 1). The measured spot centers, when they are most separated under different incident polarization, are plotted as red and magenta stars in Fig. 4(d). The simulation results of the spot center with varying polarization angles are also shown in Fig. 4(d) as green circles for comparison, while the dashed arrows denote the moving path of the spot center with the incident polarization angle rotated anticlockwise. The result of the 7-shape trimer can be obtained by rotating Figs. 4(b)4(d) 180°. Obviously, the spot center of the two trimers moves reversely with the polarization rotated anticlockwise, which demonstrates that the spot center localization method is capable of distinguishing the spectrally degenerated structures.

To this point, the methods proposed in this paper have succeeded in determining both the bond angle θ and the orientation of a metamolecule. If the plasmonics metamolecule is reconfigurable, then this method can monitor the real time angle variation. For future applications, this metamolecule can be attached to, for example, an organic texture to subsequently detect the angle of the target, or even dynamically monitor the bending, rotating, or deforming of the target.

The detection uncertainty of the bond angle primarily originates from the fluctuation of the constituent NP sizes. An asymmetric metamolecule composed of NPs with different sizes may lead to a spurious result, so NPs with perfectly spherical shapes and identical sizes should be used to improve the precision of the angle detection [28]. From Eq. (4) we can see that careful management of the background signal is also important to improve the detecting accuracy. The method presented here also works for a three-dimensional object. In addition, normal-incidence dark-field microscopy can help to improve both the performance and the efficiency of the angle detection [28].

There are also limitations of this method. As can be seen from the above analysis, if the target angle is from 0°–60°, it cannot be detected using this technique. An angle of 60° can still be detected, but the orientation cannot be determined because of the spectral symmetry. When the M value is 0, no minimum can be found, and thus we cannot determine the orientation.

3. CONCLUSION

In conclusion, we have demonstrated a method to resolve the bond angle of a plasmonic metamolecule comprising three spherical NPs. Detection of the bond angle is realized based on the modulation depth analysis method. The polarization-resolved dark-field images of the metamolecule are recorded to obtain the M value. The bond angle is then extracted through the θM relation. In addition, we can also determine the orientation of the metamolecule by the minimum of the modulation curve. The spectrally degenerate structures are further distinguished by the spot center localization method. This technique will further facilitate the applications of plasmonic metamolecules, especially in nanophotonics such as high-resolution imaging, artificial nano-material construction, and nano sensing.

Funding

National Natural Science Foundation of China (NSFC) (11374037, 11674032, 11774035); Global Networking Talent Program (NT3.0), Ministry of Education (MOE) in Taiwan.

Acknowledgment

We thank Prof. Xiaoqin Li from University of Texas at Austin for helpful discussions.

 

See Supplement 1 for supporting content.

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References

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  1. V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
    [Crossref]
  2. T. S. Kao, S. D. Jenkins, J. Ruostekoski, and N. I. Zheludev, “Coherent control of nanoscale light localization in metamaterial: creating and positioning isolated subwavelength energy hot spots,” Phys. Rev. Lett. 106, 085501 (2011).
    [Crossref]
  3. P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
    [Crossref]
  4. F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
    [Crossref]
  5. N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
    [Crossref]
  6. K. Q. Le, A. Alu, and J. Bai, “Multiple Fano interferences in a plasmonic metamolecule consisting of asymmetric metallic nanodimers,” J. Appl. Phys. 117, 023118 (2015).
    [Crossref]
  7. P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1, 131–134 (2012).
    [Crossref]
  8. J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
    [Crossref]
  9. S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
    [Crossref]
  10. A. Kuzyk, R. Schreiber, H. Zhang, A. O. Govorov, T. Liedl, and N. Liu, “Reconfigurable 3D plasmonic metamolecules,” Nat. Mater. 13, 862–866 (2014).
    [Crossref]
  11. C. Sonnichsen and A. P. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5, 301–304 (2005).
    [Crossref]
  12. W.-S. Chang, J. W. Ha, L. S. Slaughter, and S. Link, “Plasmonic nanorod absorbers as orientation sensors,” Proc. Natl. Acad. Sci. USA 107, 2781–2786 (2010).
    [Crossref]
  13. F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
    [Crossref]
  14. Editorial, “Artifacts of light,” Nat. Methods10, 1135 (2013).
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2016 (2)

J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
[Crossref]

L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
[Crossref]

2015 (1)

K. Q. Le, A. Alu, and J. Bai, “Multiple Fano interferences in a plasmonic metamolecule consisting of asymmetric metallic nanodimers,” J. Appl. Phys. 117, 023118 (2015).
[Crossref]

2014 (2)

A. Kuzyk, R. Schreiber, H. Zhang, A. O. Govorov, T. Liedl, and N. Liu, “Reconfigurable 3D plasmonic metamolecules,” Nat. Mater. 13, 862–866 (2014).
[Crossref]

N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
[Crossref]

2013 (2)

F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
[Crossref]

F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
[Crossref]

2012 (2)

P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1, 131–134 (2012).
[Crossref]

S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
[Crossref]

2011 (4)

T. S. Kao, S. D. Jenkins, J. Ruostekoski, and N. I. Zheludev, “Coherent control of nanoscale light localization in metamaterial: creating and positioning isolated subwavelength energy hot spots,” Phys. Rev. Lett. 106, 085501 (2011).
[Crossref]

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

L. Xiao, Y. Qiao, Y. He, and E. S. Yeung, “Imaging translational and rotational diffusion of single anisotropic nanoparticles with planar illumination microscopy,” J. Am. Chem. Soc. 133, 10838–10645 (2011).
[Crossref]

L. Chuntonov and G. Haran, “Trimeric plasmonic molecules: the role of symmetry,” Nano Lett. 11, 2440–2445 (2011).
[Crossref]

2010 (2)

W.-S. Chang, J. W. Ha, L. S. Slaughter, and S. Link, “Plasmonic nanorod absorbers as orientation sensors,” Proc. Natl. Acad. Sci. USA 107, 2781–2786 (2010).
[Crossref]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[Crossref]

2009 (1)

S. Kim, D. C. Ratchford, and X. Li, “Atomic force microscope nanomanipulation with simultaneous visual guidance,” ACS Nano 3, 2989–2994 (2009).
[Crossref]

2008 (2)

J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
[Crossref]

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[Crossref]

2007 (1)

B. M. Reinhard, S. Sheikholeslami, A. Mastroianni, A. P. Alivisatos, and J. Liphardt, “Use of plasmon coupling to reveal the dynamics of DNA bending and cleavage by single EcoRV restriction enzymes,” Proc. Natl. Acad. Sci. USA 104, 2667–2672 (2007).
[Crossref]

2006 (4)

G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
[Crossref]

D. W. Brandl, N. A. Mirin, and P. Nordlander, “Plasmon modes of nanosphere trimers and quadrumers,” J. Phys. Chem. B 110, 12302–12310 (2006).
[Crossref]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[Crossref]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

2005 (2)

C. Sönnichsen, B. M. Reinhard, J. Liphardt, and A. P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref]

C. Sonnichsen and A. P. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5, 301–304 (2005).
[Crossref]

2000 (1)

D. Hu, J. Yu, K. Wong, B. Bagchi, P. J. Rossky, and P. F. Barbara, “Collapse of stiff conjugated polymers with chemical defects into ordered, cylindrical conformations,” Nature 405, 1030–1033 (2000).
[Crossref]

1995 (1)

T. Junno, K. Deppert, L. Montelius, and L. Samuelson, “Controlled manipulation of nanoparticles with an atomic force microscope,” Appl. Phys. Lett. 66, 3627–3629 (1995).
[Crossref]

Aguirregabiria, G.

J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
[Crossref]

Aizpurua, J.

J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
[Crossref]

Albella, P.

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

Alivisatos, A. P.

B. M. Reinhard, S. Sheikholeslami, A. Mastroianni, A. P. Alivisatos, and J. Liphardt, “Use of plasmon coupling to reveal the dynamics of DNA bending and cleavage by single EcoRV restriction enzymes,” Proc. Natl. Acad. Sci. USA 104, 2667–2672 (2007).
[Crossref]

G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
[Crossref]

C. Sönnichsen, B. M. Reinhard, J. Liphardt, and A. P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref]

C. Sonnichsen and A. P. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5, 301–304 (2005).
[Crossref]

Alkilany, A. M.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[Crossref]

Al-Naib, I.

N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
[Crossref]

Alonso-Gonzalez, P.

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

Alu, A.

K. Q. Le, A. Alu, and J. Bai, “Multiple Fano interferences in a plasmonic metamolecule consisting of asymmetric metallic nanodimers,” J. Appl. Phys. 117, 023118 (2015).
[Crossref]

F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
[Crossref]

Arju, N.

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

Arzubiaga, L.

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

Azad, A. K.

S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
[Crossref]

Bagchi, B.

D. Hu, J. Yu, K. Wong, B. Bagchi, P. J. Rossky, and P. F. Barbara, “Collapse of stiff conjugated polymers with chemical defects into ordered, cylindrical conformations,” Nature 405, 1030–1033 (2000).
[Crossref]

Bai, J.

K. Q. Le, A. Alu, and J. Bai, “Multiple Fano interferences in a plasmonic metamolecule consisting of asymmetric metallic nanodimers,” J. Appl. Phys. 117, 023118 (2015).
[Crossref]

Barbara, P. F.

D. Hu, J. Yu, K. Wong, B. Bagchi, P. J. Rossky, and P. F. Barbara, “Collapse of stiff conjugated polymers with chemical defects into ordered, cylindrical conformations,” Nature 405, 1030–1033 (2000).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[Crossref]

Baxter, S. C.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[Crossref]

Bear, D. G.

G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
[Crossref]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Bitzer, A.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[Crossref]

Boneberg, J.

J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
[Crossref]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Born, N.

N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
[Crossref]

Brandl, D. W.

D. W. Brandl, N. A. Mirin, and P. Nordlander, “Plasmon modes of nanosphere trimers and quadrumers,” J. Phys. Chem. B 110, 12302–12310 (2006).
[Crossref]

Bratschitsch, R.

J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
[Crossref]

Casanova, F.

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

Chang, W.-S.

W.-S. Chang, J. W. Ha, L. S. Slaughter, and S. Link, “Plasmonic nanorod absorbers as orientation sensors,” Proc. Natl. Acad. Sci. USA 107, 2781–2786 (2010).
[Crossref]

Chen, F. F.

G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
[Crossref]

Chen, H.-T.

S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
[Crossref]

Chen, W. T.

P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1, 131–134 (2012).
[Crossref]

Chuntonov, L.

L. Chuntonov and G. Haran, “Trimeric plasmonic molecules: the role of symmetry,” Nano Lett. 11, 2440–2445 (2011).
[Crossref]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Deppert, K.

T. Junno, K. Deppert, L. Montelius, and L. Samuelson, “Controlled manipulation of nanoparticles with an atomic force microscope,” Appl. Phys. Lett. 66, 3627–3629 (1995).
[Crossref]

Dionne, J. A.

J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
[Crossref]

Esteban, R.

J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
[Crossref]

Fedotov, V. A.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[Crossref]

Garcia-Etxarri, A.

J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
[Crossref]

Gerion, D.

G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
[Crossref]

Goldsmith, E. C.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[Crossref]

Gole, A. M.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[Crossref]

Golmar, F.

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

Govorov, A. O.

A. Kuzyk, R. Schreiber, H. Zhang, A. O. Govorov, T. Liedl, and N. Liu, “Reconfigurable 3D plasmonic metamolecules,” Nat. Mater. 13, 862–866 (2014).
[Crossref]

Gray, J. W.

G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
[Crossref]

Ha, J. W.

W.-S. Chang, J. W. Ha, L. S. Slaughter, and S. Link, “Plasmonic nanorod absorbers as orientation sensors,” Proc. Natl. Acad. Sci. USA 107, 2781–2786 (2010).
[Crossref]

Halm, A.

J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
[Crossref]

Haran, G.

L. Chuntonov and G. Haran, “Trimeric plasmonic molecules: the role of symmetry,” Nano Lett. 11, 2440–2445 (2011).
[Crossref]

Hartsfield, T.

F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
[Crossref]

He, Y.

L. Xiao, Y. Qiao, Y. He, and E. S. Yeung, “Imaging translational and rotational diffusion of single anisotropic nanoparticles with planar illumination microscopy,” J. Am. Chem. Soc. 133, 10838–10645 (2011).
[Crossref]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Hillenbrand, R.

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P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
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N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
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L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
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S. Kim, D. C. Ratchford, and X. Li, “Atomic force microscope nanomanipulation with simultaneous visual guidance,” ACS Nano 3, 2989–2994 (2009).
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N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
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G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
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J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
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S. Kim, D. C. Ratchford, and X. Li, “Atomic force microscope nanomanipulation with simultaneous visual guidance,” ACS Nano 3, 2989–2994 (2009).
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F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
[Crossref]

F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
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W.-S. Chang, J. W. Ha, L. S. Slaughter, and S. Link, “Plasmonic nanorod absorbers as orientation sensors,” Proc. Natl. Acad. Sci. USA 107, 2781–2786 (2010).
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A. Kuzyk, R. Schreiber, H. Zhang, A. O. Govorov, T. Liedl, and N. Liu, “Reconfigurable 3D plasmonic metamolecules,” Nat. Mater. 13, 862–866 (2014).
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L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
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B. M. Reinhard, S. Sheikholeslami, A. Mastroianni, A. P. Alivisatos, and J. Liphardt, “Use of plasmon coupling to reveal the dynamics of DNA bending and cleavage by single EcoRV restriction enzymes,” Proc. Natl. Acad. Sci. USA 104, 2667–2672 (2007).
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J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
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F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
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N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
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D. W. Brandl, N. A. Mirin, and P. Nordlander, “Plasmon modes of nanosphere trimers and quadrumers,” J. Phys. Chem. B 110, 12302–12310 (2006).
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N. Born, I. Al-Naib, C. Jansen, T. Ozaki, R. Morandotti, and M. Koch, “Excitation of multiple trapped-eigenmodes in terahertz metamolecule lattices,” Appl. Phys. Lett. 104, 101107 (2014).
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V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
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S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
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E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
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V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
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F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
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S. Kim, D. C. Ratchford, and X. Li, “Atomic force microscope nanomanipulation with simultaneous visual guidance,” ACS Nano 3, 2989–2994 (2009).
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B. M. Reinhard, S. Sheikholeslami, A. Mastroianni, A. P. Alivisatos, and J. Liphardt, “Use of plasmon coupling to reveal the dynamics of DNA bending and cleavage by single EcoRV restriction enzymes,” Proc. Natl. Acad. Sci. USA 104, 2667–2672 (2007).
[Crossref]

C. Sönnichsen, B. M. Reinhard, J. Liphardt, and A. P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref]

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S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
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D. Hu, J. Yu, K. Wong, B. Bagchi, P. J. Rossky, and P. F. Barbara, “Collapse of stiff conjugated polymers with chemical defects into ordered, cylindrical conformations,” Nature 405, 1030–1033 (2000).
[Crossref]

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T. S. Kao, S. D. Jenkins, J. Ruostekoski, and N. I. Zheludev, “Coherent control of nanoscale light localization in metamaterial: creating and positioning isolated subwavelength energy hot spots,” Phys. Rev. Lett. 106, 085501 (2011).
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T. Junno, K. Deppert, L. Montelius, and L. Samuelson, “Controlled manipulation of nanoparticles with an atomic force microscope,” Appl. Phys. Lett. 66, 3627–3629 (1995).
[Crossref]

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P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

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P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

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J. A. Scholl, A. Garcia-Etxarri, G. Aguirregabiria, R. Esteban, T. C. Narayan, A. Leen Koh, J. Aizpurua, and J. A. Dionne, “Evolution of plasmonic metamolecule modes in the quantum tunneling regime,” ACS Nano 10, 1346–1354 (2016).
[Crossref]

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A. Kuzyk, R. Schreiber, H. Zhang, A. O. Govorov, T. Liedl, and N. Liu, “Reconfigurable 3D plasmonic metamolecules,” Nat. Mater. 13, 862–866 (2014).
[Crossref]

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E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

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J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
[Crossref]

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F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
[Crossref]

F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
[Crossref]

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B. M. Reinhard, S. Sheikholeslami, A. Mastroianni, A. P. Alivisatos, and J. Liphardt, “Use of plasmon coupling to reveal the dynamics of DNA bending and cleavage by single EcoRV restriction enzymes,” Proc. Natl. Acad. Sci. USA 104, 2667–2672 (2007).
[Crossref]

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L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
[Crossref]

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L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
[Crossref]

F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
[Crossref]

P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
[Crossref]

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F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
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S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
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C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[Crossref]

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W.-S. Chang, J. W. Ha, L. S. Slaughter, and S. Link, “Plasmonic nanorod absorbers as orientation sensors,” Proc. Natl. Acad. Sci. USA 107, 2781–2786 (2010).
[Crossref]

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P. Alonso-Gonzalez, M. Schnell, P. Sarriugarte, H. Sobhani, C. Wu, N. Arju, A. Khanikaev, F. Golmar, P. Albella, L. Arzubiaga, F. Casanova, L. E. Hueso, P. Nordlander, G. Shvets, and R. Hillenbrand, “Real-space mapping of Fano interference in plasmonic metamolecules,” Nano Lett. 11, 3922–3926 (2011).
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C. Sönnichsen, B. M. Reinhard, J. Liphardt, and A. P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
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E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
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L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
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Walther, M.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
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F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
[Crossref]

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P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1, 131–134 (2012).
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P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1, 131–134 (2012).
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L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
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L. Xiao, Y. Qiao, Y. He, and E. S. Yeung, “Imaging translational and rotational diffusion of single anisotropic nanoparticles with planar illumination microscopy,” J. Am. Chem. Soc. 133, 10838–10645 (2011).
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L. Sun, T. Ma, S.-C. Yang, D.-K. Kim, G. Lee, J. Shi, I. Martinez, G.-R. Yi, G. Shvets, and X. Li, “The interplay between optical bianisotropy and magnetism in plasmonic metamolecules,” Nano Lett. 16, 4322–4328 (2016).
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S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
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[Crossref]

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S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
[Crossref]

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P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1, 131–134 (2012).
[Crossref]

T. S. Kao, S. D. Jenkins, J. Ruostekoski, and N. I. Zheludev, “Coherent control of nanoscale light localization in metamaterial: creating and positioning isolated subwavelength energy hot spots,” Phys. Rev. Lett. 106, 085501 (2011).
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V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[Crossref]

Zhou, J.

S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
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M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
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J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
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C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[Crossref]

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L. Xiao, Y. Qiao, Y. He, and E. S. Yeung, “Imaging translational and rotational diffusion of single anisotropic nanoparticles with planar illumination microscopy,” J. Am. Chem. Soc. 133, 10838–10645 (2011).
[Crossref]

J. Appl. Phys. (1)

K. Q. Le, A. Alu, and J. Bai, “Multiple Fano interferences in a plasmonic metamolecule consisting of asymmetric metallic nanodimers,” J. Appl. Phys. 117, 023118 (2015).
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Nanophotonics (1)

P. C. Wu, W. T. Chen, K.-Y. Yang, C. T. Hsiao, G. Sun, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, “Magnetic plasmon induced transparency in three-dimensional metamolecules,” Nanophotonics 1, 131–134 (2012).
[Crossref]

Nat. Biotechnol. (1)

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Nat. Commun. (1)

S. Zhang, J. Zhou, Y.-S. Park, J. Rho, R. Singh, S. Nam, A. K. Azad, H.-T. Chen, X. Yin, A. J. Taylor, and X. Zhang, “Photoinduced handedness switching in terahertz chiral metamolecules,” Nat. Commun. 3, 1908 (2012).
[Crossref]

Nat. Mater. (1)

A. Kuzyk, R. Schreiber, H. Zhang, A. O. Govorov, T. Liedl, and N. Liu, “Reconfigurable 3D plasmonic metamolecules,” Nat. Mater. 13, 862–866 (2014).
[Crossref]

Nat. Methods (1)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[Crossref]

Nat. Nanotechnol. (2)

G. L. Liu, Y. Yin, S. Kunchakarra, B. Mukherjee, D. Gerion, S. D. Jett, D. G. Bear, J. W. Gray, A. P. Alivisatos, L. P. Lee, and F. F. Chen, “A nanoplasmonic molecular ruler for measuring nuclease activity and DNA footprinting,” Nat. Nanotechnol. 1, 47–52 (2006).
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F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alu, and X. Q. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nat. Nanotechnol. 8, 95–99 (2013).
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F. Shafiei, C. H. Wu, Y. W. Wu, A. B. Khanikaev, P. Putzke, A. Singh, X. Q. Li, and G. Shvets, “Plasmonic nano-protractor based on polarization spectro-tomography,” Nat. Photonics 7, 367–372 (2013).
[Crossref]

J. Merlein, M. Kahl, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics 2, 230–233 (2008).
[Crossref]

Nature (1)

D. Hu, J. Yu, K. Wong, B. Bagchi, P. J. Rossky, and P. F. Barbara, “Collapse of stiff conjugated polymers with chemical defects into ordered, cylindrical conformations,” Nature 405, 1030–1033 (2000).
[Crossref]

Phys. Rev. Lett. (2)

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[Crossref]

T. S. Kao, S. D. Jenkins, J. Ruostekoski, and N. I. Zheludev, “Coherent control of nanoscale light localization in metamaterial: creating and positioning isolated subwavelength energy hot spots,” Phys. Rev. Lett. 106, 085501 (2011).
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Proc. Natl. Acad. Sci. USA (2)

B. M. Reinhard, S. Sheikholeslami, A. Mastroianni, A. P. Alivisatos, and J. Liphardt, “Use of plasmon coupling to reveal the dynamics of DNA bending and cleavage by single EcoRV restriction enzymes,” Proc. Natl. Acad. Sci. USA 104, 2667–2672 (2007).
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Science (1)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
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Supplementary Material (1)

NameDescription
» Supplement 1       Extended data

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Figures (4)

Fig. 1.
Fig. 1. Illustration of sample assembly and M value measurement. (a) Schematic diagram of assembly method for the Au NP cluster via AFM manipulation. (b) Dark-field image of the fabricated sample. Scale bar: 20 μm. Inset shows the AFM image of an assembled metamolecule composed of three 60 nm Au NPs. Bond angle θ=110°. Scale bar: 150 nm. (c) Scattering images of the metamolecule when it is the brightest and the darkest as the polarization angle φ of the incident light being rotated from 0 to 180° anticlockwise, where 0° is along the horizontal axis. Wavelength range of the signal collected by the camera is 450–800 nm. Measured M value is 0.395. (d) Simulated scattering intensity versus incident light polarization angle φ of the same metamolecule. Simulated M value is 0.409.
Fig. 2.
Fig. 2. Determining the metamolecule bond angle and orientation based on modulation depth analysis. (a) Trimers with varying angles assembled. (b) Experimental results of trimers with angles 180°, 110°, and 87°. Insets show the dark-field image evolution of the individual trimers as a function of illumination polarization angle. (c) Simulated polarization dependence of total scattering intensity for trimers with varying angles. The minimum of each curve indicates the orientation of the trimer angle bisector, i.e., the orientation of the metamolecule. (d) Modulation depth M of trimer with different bond angles. Dots represent the experimental data, and dashed curve is extracted from simulation results in (c).
Fig. 3.
Fig. 3. Modal analysis of the scattering spectra of the metamolecules. (a) AFM images of the trimer with bond angle θ=120°. Arrows illustrate the typical transversal bonding mode (red) and the longitudinal bonding mode (blue). (b) Polarization-dependent scattering spectra of the metamolecule obtained from experiment. (c) Spectral peak intensities versus incident polarization angle φ. (d)–(f) Corresponding simulated results for comparison. Results for trimers with other bond angles can be found in Supplement 1.
Fig. 4.
Fig. 4. Removal of structure degeneracy of symmetric metamolecules. (a) Illustration of the spot center localization method. Polarization of the incident light is rotated from horizontal to vertical direction. Blue spot denotes the equivalent dipole location of the longitudinal bonding mode when the polarization angle is 135°. The red and magenta spots represent the equivalent dipole location after mode interference under 0° and 90° polarized incident light, respectively. (b) Simulated electric field distribution of the longitudinal and transversal bonding modes recorded from the monitor plane 500 nm above the L-shape metamolecule, which is highlighted in the black dashed box in (a). Red arrows represent the polarization angle of the incident light. White dashed lines denotes the reference frame, with the horizontal (x) axis and vertical (y) axis along the two arms of the trimer. X and Y are the coordinates of the spot center. (c) AFM image of assembled structure: an L-shape metamolecule whose bond angle is 90°. The inset is the enlarged view of the metamolecule. The two dimers on the left and the bottom work as the reference frame to correct image drift. (d) Measured results of the spot center shift under the polarization angle 0° (red) and 90° (magenta). Simulated results are also shown in green circles, by integrating all the modes. The dashed arrows denote the moving path of the spot center with polarization angles rotated anticlockwise. The results of 7-shape metamolecule can be obtained by rotating images (b), (c), and (d) 180°.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

M=ImaxIminImax+Imin,
I=Il+It=Cl+Alcos2(φ+φ0)+Ilb+Ct+Atcos2(φ+π2+φ0)+Itb,
Imax=Cl+Ct+(AlAt)+IbImin=Cl+Ct(AlAt)+Ib.
M=ImaxIminImax+IminAlAtAl+At+Ib.

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