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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11783–11793
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Optical forces in twisted split-ring-resonator dimer stereometamaterials

Chaojun Tang, Qiugu Wang, Fanxin Liu, Zhuo Chen, and Zhenlin Wang  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11783-11793 (2013)
http://dx.doi.org/10.1364/OE.21.011783


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Abstract

We numerically investigate the optical forces in stereometamaterials composed of two-dimensional arrays of two spatially stacked split ring resonators with a twisted angle. At the hybridized magnetic resonances, we obtain both attractive and repulsive relative optical forces, which can be further exploited to control the separation between the two split ring resonators. Due to the strongest inductive coupling achieved for a twist angle of 180°, an attractive relative force as high as ~1200 piconewtons is realized at illumination intensities of 50 mW/µm2. We show that a quasi-static dipole-dipole interaction model could predict well the characteristic and magnitude of the relative optical forces. We also demonstrate that although the optical force exerted on each of the split ring resonators could be oriented in a direction opposite to the propagation wave vector, the mass center of the two resonators is always pushed away from the light source.

© 2013 OSA

1. Introduction

Since Ashkin and associates discovered that a single, tightly focused laser beam could be used to capture small dielectric particles [1

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

], optical force generated from the gradient of the light field has been widely used in optical tweezers for non-invasive three dimensional trapping and manipulation of atoms, microscale dielectric particles and biomolecules [2

2. S. Chu, “Laser manipulation of atoms and particles,” Science 253(5022), 861–866 (1991). [CrossRef] [PubMed]

9

9. T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-Binding Photonic Molecule Modes of Resonant Bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999). [CrossRef]

]. Attributing to the advancement in nanofabrication techniques, it has been recently demonstrated in experiments that the strongly enhanced gradient of the optical field close to nanophotonic waveguides could lead to greatly increased optical forces [10

10. M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express 13(20), 8286–8295 (2005). [CrossRef] [PubMed]

12

12. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high- double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett. 97(2), 023903 (2006). [CrossRef] [PubMed]

]. For example, when a freestanding dielectric strip waveguide was placed in the evanescent field of an eigenmode propagating in a second waveguide, it was subjected to an optical force of a few piconewtons (pN) using milliwatt input power levels [13

13. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009). [CrossRef]

,14

14. J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009). [CrossRef] [PubMed]

]. The optical force can be further enhanced by several orders of magnitude, up to a few nanonewtons per milliwatt, using various types of photonic resonators such as double ring structures [15

15. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009). [CrossRef]

17

17. G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express 19(3), 2782–2790 (2011). [CrossRef] [PubMed]

] and micro disk resonators [18

18. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1(7), 416–422 (2007). [CrossRef]

,19

19. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103(22), 223901 (2009). [CrossRef] [PubMed]

]. Nowadays, such gradient optical forces have been intensively investigated in dielectric structures for many applications, such as optomechanical wavelength and energy conversion [12

12. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high- double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett. 97(2), 023903 (2006). [CrossRef] [PubMed]

], all-optical wavelength routing [20

20. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009). [CrossRef]

], tunable optical filter [21

21. T. Stomeo, M. Grande, G. Rainò, A. Passaseo, A. D’Orazio, R. Cingolani, A. Locatelli, D. Modotto, C. De Angelis, and M. De Vittorio, “Optical filter based on two coupled PhC GaAs-membranes,” Opt. Lett. 35(3), 411–413 (2010). [CrossRef] [PubMed]

], and actuating optomechanical devices [22

22. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010). [CrossRef]

].

On the other hand, optical energy could be confined to deep subwavelength scale at a plasmonic resonance [23

23. A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102(4), 043904 (2009). [CrossRef] [PubMed]

], resulting in significantly enhanced optical field strength and gradient of light field, which therefore will greatly enhance the optical force. Such enhanced optical forces have been investigated in various resonant plasmonic nanostructures and metamaterials, for example, gold nanoparticle dimers [24

24. R. A. Nome, M. J. Guffey, N. F. Scherer, and S. K. Gray, “Plasmonic interactions and Optical Forces between Au Bipyramidal Nanoparticle Dimers,” J. Phys. Chem. A 113(16), 4408–4415 (2009). [CrossRef] [PubMed]

], silver nanoaggregates [25

25. H. Xu and M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89(24), 246802 (2002). [CrossRef] [PubMed]

], coupled metal planar waveguides [26

26. D. Woolf, M. Loncar, and F. Capasso, “The forces from coupled surface plasmon polaritons in planar waveguides,” Opt. Express 17(22), 19996–20011 (2009). [CrossRef] [PubMed]

], hybrid plasmonic waveguides [27

27. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011). [CrossRef] [PubMed]

], slot waveguides of hyperbolic metamaterials [28

28. Y. He, S. He, J. Gao, and X. Yang, “Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Express 20(20), 22372–22382 (2012). [CrossRef] [PubMed]

], discrete spherical invisibility cloak [29

29. P. C. Chaumet, A. Rahmani, F. Zolla, and A. Nicolet, “Electromagnetic forces on a discrete spherical invisibility cloak under time-harmonic illumination,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(5), 056602 (2012). [CrossRef] [PubMed]

], and dielectric slab waveguides with a metamaterial cladding [30

30. V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Enhancing optical gradient forces with metamaterials,” Phys. Rev. Lett. 110(5), 057401 (2013). [CrossRef] [PubMed]

]. It has also been theoretically demonstrated that a strong attractive near-field optical force may be generated when a plasmonic metamaterial film consisting of periodic arrays of asymmetric split-ring slits is illuminated in close proximity to a dielectric or metal surface, providing an optically controlled adhesion mechanism mimicking the gecko toe [31

31. J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Optical gecko toe: Optically controlled attractive near-field forces between plasmonic metamaterials and dielectric or metal surfaces,” Phys. Rev. B 85(20), 205123 (2012). [CrossRef]

]. In addition, some specially shaped metallic nanostructures (artificial magnetic atoms), such as split ring resonators (SRRs) [32

32. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005). [CrossRef] [PubMed]

], rod pairs or cut-wire pairs [33

33. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef] [PubMed]

,34

34. H.-K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15(3), 1076–1083 (2007). [CrossRef] [PubMed]

] and overlapped double triangles [35

35. J. Pan, Z. Chen, Z. D. Yan, Z. S. Cao, P. Zhan, N. B. Ming, and Z. L. Wang, “Symmetric and anti-symmetric magnetic resonances in double-triangle nanoparticle arrays fabricated via angle-resolved nanosphere lithography,” AIP Adv. 1(4), 042114 (2011). [CrossRef]

], have been demonstrated to mimic optical magnetism by supporting circular current plasmonic modes. These magnetic atoms are of special interest since the induced currents and accumulated charges can play significant roles in manipulating the optical forces. For example, at the electric and magnetic dipole resonances of a metamaterial with a gold nanowire pairs, Coulomb interactions between the charges accumulated at the ends of the nanowires are predicted to generate repulsive and attractive optical forces, respectively [36

36. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18(25), 25665–25676 (2010). [CrossRef] [PubMed]

]. More recently, it has been demonstrated that the attractive Ampere’s force between the in-phase currents induced in the neighboring SRRs [37

37. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2011). [CrossRef] [PubMed]

] and the radiation torque between two twisted SRRs [38

38. M. Liu, D. A. Powell, and I. V. Shadrivov, “Chiral meta-atoms rotated by light,” Appl. Phys. Lett. 101(3), 031105 (2012). [CrossRef]

,39

39. M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Twists and turns for metamaterials,” arXiv:1301.5960 [physics.optics].

] could be utilized to change the metamaterial structure with elastic feedback and thus dynamically tune its effective properties.

In this paper, we investigate the optical forces in a set of stereometamaterials, each having unit cells composed of two stacked identical SRRs with different twist angles [40

40. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

]. Due to the strong coupling between the stacked SRRs, two spectrally split magnetic resonances arise from the hybridization of the original state of an individual SRR. The optical force on each of the SRRs is numerically calculated using the Maxwell stress tensor formalism [41

41. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

]. We show that the relative force between the two SRRs could be either attractive or repulsive, depending on the current and magnetic field distributions at the respective resonances. Furthermore, using a quasi-static dipole-dipole interaction model, we demonstrate that at optical frequencies the electric dipole-dipole interaction exceeds the magnetic dipole-dipole interaction by several orders of magnitude and thus plays a dominant role in the relative force. It is found that a strong attractive force as high as ~1,200 pN could be generated in a 180°-twisted SRR dimer metamaterial at the 190 THz resonant frequency with an intensity of 50 mW/μm2. We anticipate that our results could be helpful to understand the optical forces in artificial metamaterials and may be useful to optimize the design of magnetoelastic metamaterials [37

37. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2011). [CrossRef] [PubMed]

].

2. Results and discussions

The geometry of the stereometamaterials, together with their design parameters, are schematically illustrated in Fig. 1
Fig. 1 Schematic of a stereometamaterial composed of two stacked identical SRRs with the defined geometrical parameters: l = 200 nm, t = 30 nm, g = 40 nm, s = 120 nm, h = 40 nm. The two SRRs are twisted at an angle φ with respect to one another. The periods in both x and y directions are p = 500 nm. The light is normally incident on the stereometamaterials, with its wave vector k, electric field E, and magnetic field H parallel to the z, x, and y axes, respectively. The optical forces exerted on the upper and lower SRRs are denoted by F1 and F2, respectively.
. Each unit cell consists of two stacked identical SRRs, which are twisted at an angle φ with respect to each other. In the present study, numerical simulations are preformed based on a commercial finite element method (Comsol Multiphysics). The SRRs are assumed to be surrounded by a homogeneous dielectric with ε = 1 (that is, air). Silver is described as a dispersive medium with the complex dielectric parameters taken from experimental data by Johnson and Christy [42

42. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

]. A normally incident light with its wave vector k along the z-direction and its electric field E along the x-direction is used. A cuboid simulation domain containing a translation unit cell with periodic boundary conditions on the faces parallel to the propagation direction is utilized to mimic a stereometamaterial array of infinite extent. Two faces perpendicular to the propagation direction are terminated with Perfectly Matched Layers to absorb reflected and transmitted light. Such a numerical model can simultaneously provide data on the optical spectra of the structure and fully three-dimensional electromagnetic field distributions. Within the framework of classical electrodynamics, the components of the total time-averaged force F acting on the object can be calculated using a surface integral [41

41. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

]:
Fi=STijnjdS,
(1)
where <Tij> is the time-averaged Maxwell stress tensor defined by
Tij=12Re[εrε0(EiEj*12mEmEm*)+μrμ0(HiHj*12mHmHm*)],
(2)
footnotes i, j and m stand for the x, y, or z component of physical quantity, S is a bounding surface around the object, n is the unit normal vector to this surface, E and H are electric and magnetic fields, εr and µr are the relative permittivity and permeability of the surrounding medium. By extending the integration surface S to a rectangular parallelepiped that solely encloses the upper or lower SRRs, with walls along each of the four periodic boundaries and outside each of the two free surfaces of the SRRs, the optical forces exerted on the upper and lower SRRs (F1 and F2 as indicated in Fig. 1) can be straightforwardly calculated via the stress tensor integral Eq. (1). For reconfigurable metamaterials exploiting the optical force, for example, magnetoelastic metamaterials [37

37. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2011). [CrossRef] [PubMed]

], it is more appropriate to consider the relative force Frel = (F1F2)/2 from which we can determine whether the optical forces will increase the separation between the SRRs (repulsive force) or decrease the separation (attractive force).

We first consider the optical forces in the stereo-SRR dimer metamaterials with a twist angle φ = 0°. Figure 2(a)
Fig. 2 (a) Calculated absorption spectra of a stereometamaterials with a twist angle of φ = 0°. (b) and (c) Surface current density distributions and the alignments of the magnetic dipoles (m1 and m2) and electric dipoles (p1 and p2) at respective resonances of ω- and ω+. Arrows and color maps (blue smaller and yellow larger) indicate the direction and the magnitude of the surface current density, respectively. (d) The simulated relative force Frel = (F1 - F2)/2 as a function of frequency. The incident field intensity is assumed to be 50 mW/µm2.
shows the calculated absorption spectrum. For the 0°-twisted SRR dimer metamaterial, the electric component of the incident light can excite the magnetic resonance in each of the two identical SRRs. It has already been demonstrated experimentally and theoretically that the strong inductive coupling between two stacked SRRs could lead to the formation of two new plasmonic modes [40

40. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

]. In our case, two conspicuous resonances observed at the frequencies of ω- = 200 THz and ω+ = 242 THz in Fig. 2(a) are such antisymmetric and symmetric modes that arise from the hybridization of the original state of an individual SRR. To intuitively demonstrate these spectral characteristics, surface current density distributions at these two resonances are calculated and shown in Fig. 2(b). It is seen that the induced circular currents along the two SRRs are oppositely wound at the lower resonance frequency ω-. In contrast, at the higher resonance frequency ω+ the currents in the two SRRs are in phase [Fig. 2(c)]. With the electric E and magnetic field H distributions obtained from the model, the optical forces F1 and F2 exerted on the upper and lower SRRs are respectively evaluated via the Maxwell stress tensor integral Eq. (1), in which the incident field intensity is assumed to be a laboratory available value of 50 mW/µm2. Then, the relative force Frel is calculated and plotted as a function of frequency [Fig. 2(d)]. It is apparent that the dispersion of the relative force is linked to variations in the stereometamaterial’s absorption spectra and has two local extrema, corresponding to the absorption peaks at the frequencies of ω- and ω+. The relative force resonant at ω- = 200 THz is attractive (Frel < 0) across the spectral range from 190 THz to 210 THz, reaching a peak magnitude of approximately 650 pN. However, at the resonance frequency of ω+ = 242 THz the relative force, having a relatively wide bandwidth and a maximum value of ~100 pN, is repulsive (Frel > 0) and would push one SRR away from the other.

For the interpretation of the optical forces, we implement a similar model for gold nanowire pairs reported in [36

36. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18(25), 25665–25676 (2010). [CrossRef] [PubMed]

], which takes into account simultaneously electric and magnetic dipole-dipole interactions. In this model, the upper (lower) SRR is substituted by two dipoles: an electric dipole p1 (p2) oriented parallel to the SRR base and a magnetic dipole m1 (m2) oriented perpendicular to the SRR plane. We limit ourselves to the dipole-dipole interactions within the same unit cell, ignoring possible interactions between different unit cells. However, we will show that this model can excellently capture the main physics embodied in our observations, and enables us to clarify the role of the Coulomb force and the Ampere force played in the relative optical force Frel.

The formula of the Coulomb force Fp1p2 could be derived from the interaction between the electric dipolar moments of p1 and p2, oscillating at frequency ω [43

43. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

]
Fp1p2=[p2eiωtE(p1,r,ω)],
(3)
where E(p1,r,ω) is the electric field generated from the electric dipole p1 [43

43. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

]
E(p1,r,ω)=14πε0εr[(1ikr)3(rp1)r^p1r3(kr)2(rp1)r^p1r3]eiωt,
(4)
r^is the unit vector pointing from the dipole to the field point at distance r, εr is the relative permittivity of the homogeneous medium in which the dipole is embedded, k=εrω/c is the magnitude of the wave vector, and is gradient operator. For the 0°-twisted SRR dimer metamaterial, the excited two electric dipoles align anti-parallel at the resonance frequency of ω- and parallel at the resonance frequency of ω+ [40

40. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

]. As shown in Figs. 2(b) and 2(c), the two electric dipoles point along the x-direction (i.e.,p1=p1x^,p2=p2x^) and are spatially separated with a distance r along the z-direction (i.e., r=rz^). In this special case, thanks to rp1=0, the Eq. (3) could be expressed as
Fp1p2=3p1p2e2iωt4πε0εrr4[1ikr23(kr)2+i3(kr)3]eikrz^.
(5)
In quasi-static approximation (i.e., kr ≈0), the later three terms in Eq. (5) could be ignored, and thus the Coulomb force Fp1p2 averaged on a time cycle is simplified as

Fp1p2=3Re(p1*p2)8πε0εrr4z^.
(6)

In a similar way, we can also derive the formula of the Ampere force Fm1m2 from the interaction between the magnetic dipolar moments of m1 and m2,
Fm1m2=[m2eiωtB(m1,r,ω)],
(7)
where the magnetic flux density B(m1,r,ω) can be obtained by replacing p1 and 1/ε0εr in Eq. (4) with m1 and µ0µr, respectively [44

44. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74(3), 033402 (2006). [CrossRef]

]
B(m1,r,ω)=μ0μr4π[(1ikr)3(rm1)r^m1r3(kr)2(rm1)r^m1r3]eiωt,
(8)
For the 0°-twisted SRR dimer metamaterial, the excited two magnetic dipoles point along the z-direction (i.e., m1=m1z^,m2=m2z^) and are spatially separated with a distance r along the z-direction (i.e., r=rz^). By substitutingrm1=rm1 and Eq. (8) into Eq. (7), the time-averaged Ampere force is readily expressed as
Fm1m2=6Re(m1*m2)μ0μr8πr4z^.
(9)
In the calculations, the electric and magnetic dipolar moments p1, p2, m1, and m2 could be obtained from the local electromagnetic fields available in the numerical model [36

36. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18(25), 25665–25676 (2010). [CrossRef] [PubMed]

]
p1=V1(Dxε0Ex)dV,p2=V2(Dxε0Ex)dV,m1=iωV1[x(Dyε0Ey)y(Dxε0Ex)]dV,m2=iωV2[x(Dyε0Ey)y(Dxε0Ex)]dV,
(10)
where V1 (V2) indicates the volume integration should be carried out in the upper (lower) SRR, Dx, Dy, and Dz are the x, y and z component of the electric displacement vector. Here, the effective separated distance r between the two dipoles is taken to be 154 nm, yielding the best fitting to the numerical results. Together with the above derived Eqs. (6) and (9), the Coulomb force Fp1p2 arising from the electric dipole-dipole interaction and the Ampere force Fm1m2 arising from the magnetic dipole-dipole interaction are calculated for the 0°-twisted SRR dimer metamaterial and plotted as a function of frequency in Fig. 3(a)
Fig. 3 (a) The Coulomb force Fp1p2 and Ampere force Fm1m2 as a function of frequency for the 0°-twisted SRR dimer metamaterial calculated from Eq. (6) and Eq. (9), respectively, in which only the electric or magnetic dipole-dipole interaction is considered. (b) The relative optical force obtained from the dipole-dipole interaction theoretical model (solid line), which is the sum of the Coulomb force and the Ampere force, Frel = Fp1p2 + Fm1m2. For direct comparison, the simulated relative force is again plotted (line with open squares).
.

As indicated by the solid line in Fig. 3(a), the sign of Fp1p2 is negative at the resonance frequency of ω- and positive at the resonance frequency of ω+. This is exactly in consistence with the fact that the two transversely coupled and anti-parallel aligned electric dipoles [ω-, Fig. 2(b)] should generate an attractive force, while a repulsive force is expected in the two transversely coupled and parallel aligned electric dipoles [ω+, Fig. 2(c)]. However, a quite different situation is found for the Ampere force Fm1m2, in which the two magnetic dipoles are longitudinally coupled [see Figs. 2(b) and 2(c)]. The two anti-parallel aligned magnetic dipoles should generate a repulsive force, while the two anti-parallel magnetic dipoles generate an attractive force [dashed line in Fig. 3(a)]. Therefore, the sign of Fm1m2 is opposite to that of Fp1p2 at the frequencies of ω- and ω+. As already mentioned above, in the dipole-dipole interaction model the relative optical force is actually the sum of the Coulomb force and the Ampere force, Frel = Fp1p2 + Fm1m2. Figure 3(b) shows the relative optical force obtained from the dipole-dipole interaction theoretical model [solid line]. For direct comparison, the numerical result is again plotted in Fig. 3(b) [line with open symbols]. It is seen that the result obtained from the dipole-dipole interaction theoretical model is in a good agreement with the numerical result, except for slight differences around ω+ resonance. Because the Coulomb force Fp1p2 exceeds the Ampere force Fm1m2 by one order of magnitude, the dispersion of the sum of these two forces or the relative optical force [Fig. 3(b)] is found to mainly follow variations in the Fp1p2 spectra [Fig. 3(a)], suggesting that the Coulomb force Fp1p2 plays a key role in the relative optical force.

The absorption spectra and the relative optical force in the stereo-SRR dimer metamaterials with a twist angle φ = 180° are further studied. Figure 4(a)
Fig. 4 (a) Calculated absorption spectra of a stereometamaterials with a twist angle of φ = 180°. Inset: surface current distributions at the respective resonance frequency ω- and ω+. (b) The Coulomb force Fp1p2 and Ampere force Fm1m2 as a function of frequency calculated from Eq. (6) and Eq. (9), respectively. (c) The relative optical force as a function of frequency calculated from the numerical simulations (line with open squares) and obtained from the theoretical dipole-dipole interaction model (solid line). The incident field intensity is assumed to be 50 mW/µm2.
shows that the inductive coupling between two SRR elements again leads to the splitting of the magnetic resonance of the individual SRR. A feature of note is that the lower (ω- = 190 THz) and higher resonance frequency (ω+ = 252 THz) for the 180°-twisted SRR dimer metamaterial has a respective spectral blueshift and redshift with respect to the two resonances observed for the 0°-twisted SRR dimer metamaterial. A larger spectra splitting observed here indicates that the inductive coupling for φ = 180° is much stronger than that for φ = 0°. Shown in the insets of Fig. 4(a) are the surface current density distributions at these two resonances. It is seen that for φ = 180° the induced currents circulating along the two SRRs are in-phase and anti-phase at the respective resonance frequency ω- and ω+, a feature quite different from that for the 0°-twisted SRR dimer metamaterial [Fig. 2(b) and 2(c)]. From the surface current distributions at the resonance frequency ω-, we can deduce that the induced electric dipole moments (p1 and p2) in the upper and lower SRRs align anti-parallel along the x-direction, while the magnetic dipole moments (m1 and m2) in the two SRRs align parallel along the z-direction [left inset of Fig. 4(a)]. Similarly, at the resonance frequency ω+ the excited electric dipole moments and magnetic dipole moments align parallel and anti-parallel, respectively [right inset of Fig. 4(a)].

Figure 4(b) shows the Coulomb force Fp1p2 and the Ampere force Fm1m2 for the 180°-twisted SRR dimer metamaterial as a function of frequency, obtained from Eqs. (6) and (9). It is seen that the dispersion of the Coulomb force Fp1p2 for φ = 180° [solid line in Fig. 4(b)] is similar to that for φ = 0° [solid line in Fig. 3(a)]. An attractive force (Fp1p2 < 0) and a repulsive force (Fp1p2 > 0) is generated from the transverse interaction between two anti-parallel and parallel aligned electric dipoles at the resonance frequency ω- and ω+, respectively. On the other hand, the two parallel and anti-parallel aligned magnetic dipoles are transversely coupled, leading to the attractive Ampere force (Fm1m2 < 0) and repulsive Ampere force (Fm1m2 > 0) at the resonance frequency ω- and ω+, respectively [dashed line in Fig. 4(b)]. But unlike the case for φ = 0° that the longitudinal magnetic interaction counteracts the transverse electric interaction [Fig. 3(a)], these two interactions contribute positively in the 180°-twisted SRR dimer metamaterial, leading to the largest spectral splitting (the strongest inductive coupling) in the 180°-twisted SRR dimer system [40

40. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

].

The relative optical force calculated directly from the numerical simulations and obtained from the theoretical dipole-dipole interaction model (that is the sum of the Coulomb force and the Ampere force) are plotted as functions of frequency in Fig. 4(c). In essence, the optical force obtained from the theoretical dipole-dipole interaction model [solid line in Fig. 4(c)] matches the numerical results [line with open symbols in Fig. 4(c)]. Like the relative optical force for the 0°-twisted SRR dimer metamaterial, the relative optical force present for the 180°-twisted SRR dimer metamaterial is attractive (Frel < 0) at the lower resonance frequency ω- and repulsive (Frel > 0) at the higher resonance frequency ω+. Notably, as seen from Fig. 4(b) the magnitude of the Coulomb force is much larger than the Ampere force, which implies that the relative optical force in the 180°-twisted SRR dimer metamaterial is dominated by the transverse electric dipole-dipole interaction. In particular, due to the strongest inductive coupling occurred in the 180°-twisted SRR dimer metamaterial, an attractive force as high as ~1,200 pN could be achieved at the resonance frequency ω- = 190 THz.

It has been reported that the two resonance branches (ω- and ω+) in the optical spectrum of the stereometamaterials first tend to converge and subsequently shift away from one another with increasing the twist angle φ [40

40. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

, 46

46. D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83(23), 235420 (2011). [CrossRef]

]. Based on the above results and discussions, it is evident that the optical forces in twisted SRR dimer metamaterial depend on the spatial arrangement of the building constituents. Therefore, we plot the simulated dispersion of the relative optical forces in Fig. 5(a)
Fig. 5 (a) The simulated relative optical force Frel = (F1 - F2)/2 (a) and common force Fcomm = F1 + F2 (b) as a function of frequency and twist angle φ. The black dashed lines in (a) indicate the boundaries at which the optical forces equal zero.
. For clarity, the black dashed lines are used in Fig. 5(a) to outline the boundary between the attractive force (Frel < 0) and repulsive force (Frel > 0). The regions bounded with the dashed lines and the respective coordinate axes represent repulsive characteristics of the relative optical force, while the left region represents an attractive relative optical force. It is further seen that the relative optical forces at resonances show an X-shaped dispersion as the twist angle φ is varied from 0 to 180° and the dispersion branches intersect at a twist angle φ = 60° at a frequency of ω- = ω+ = 210 THz. For the twist angle varying in the range from φ = 0° to φ = 20°, the relative optical forces keep attractive at ω- and repulsive at ω+. However, the optical forces at both resonance branches only exhibit attractive characteristic for φ within the range from 30° to 130°. When the twist angle φ is further increased from 140° to 180°, the repulsive relative optical force reappears at the higher resonance frequency ω+. Figure 5(a) also shows that in the whole range of the twist angles, the attractive optical force far exceeds the repulsive optical force and that the magnitude of the relative optical force could be controlled by simply varying the twist angles. For example, for the 180°-twisted SRR dimer metamaterial the attractive optical force could reach a maximum of ~1200 pNat ω- = 190 THz, whereas the maximum value of the repulsive optical force at ω+ = 252 THz is only ~100 pN.

It is found that in the twisted SRR dimer metamaterials the optical force exerted on the upper or lower SRRs (F1 or F2) could be oriented in a direction opposite to the propagation wave vector (data not shown here), i.e., could drive the SRRs towards the light source. However, as a whole the two SRRs in a unit cell are always pushed away from the light source. To more clearly demonstrate this, the common optical force Fcomm is defined as Fcomm = F1 + F2 to describe the optical force acting on the center of mass of the two SRRs. As shown in Fig. 5(b), the common optical force, with its maximum magnitude of ~80 pN, is also resonant at the frequency ω- and ω+, and exhibits an X-shaped dispersion. Specifically, the sign of Fcomm is observed to be positive for any twist angles. This directly indicates that the incident plane wave always pushes the mass center of the two SRRs forward, i.e., along the same direction of the propagation wave vector.

3. Conclusion

In summary, we have numerically investigated optical forces in twisted SRR dimer metamaterials. The optical forces are found to be resonant at the symmetric and antisymmetric magnetic resonances that arise from the hybridization of the origin state of the individual SRR. Both attractive and repulsive relative optical forces could be achieved at the resonance frequency for relatively small twist angles [0°, 20°] or large twist angles [140°,180°], while the resonant relative optical force only exhibits the attractive characteristic within the twist angle ranging from 30° to 130°. In particular, the attractive relative optical force as high as ~1,200 pN could be obtained in a 180°-twisted SRR dimer metamaterial at 190 THz with an intensity of 50 mW/μm2. A dipole-dipole interaction model in the quasistatic approximation is further proposed to interpret the relative optical force. It is found that the Coulomb force arising from the electric dipolar moments interaction is much stronger than the Ampere force arising from the magnetic dipolar moments interactions and thus contributes most to the relative optical force. We have also demonstrated that although the optical force exerted on each of the SRRs could be oriented in a direction opposite to the propagation wave vector, the common force always pushes the mass center of the two SRRs away from the light source. We suggest that our results could be helpful to understand the optical forces in artificial metamaterials and may be useful to optimize the design of magnetoelastic metamaterials [37

37. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2011). [CrossRef] [PubMed]

].

Acknowledgments

This work is financially supported by the State Key Program for Basic Research of China (SKPBRC) under Grant No. 2012CB921501 and National Nature Science Foundation of China (NSFC) under Grant Nos. 11104136, 11104135 and 11174137. Z. L. Wang also acknowledges partial support from NSFC under Grant Nos. 91221206 and 51271092 and from SKPBRC under Grant No. 2013CB632703.

References and links

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A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]

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F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5(6), 318–321 (2011). [CrossRef] [PubMed]

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K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [CrossRef] [PubMed]

7.

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett. 63(12), 1233–1236 (1989). [CrossRef] [PubMed]

8.

R. Omori, T. Kobayashi, and A. Suzuki, “Observation of a single-beam gradient-force optical trap for dielectric particles in air,” Opt. Lett. 22(11), 816–818 (1997). [CrossRef] [PubMed]

9.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-Binding Photonic Molecule Modes of Resonant Bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999). [CrossRef]

10.

M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express 13(20), 8286–8295 (2005). [CrossRef] [PubMed]

11.

M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30(22), 3042–3044 (2005). [CrossRef] [PubMed]

12.

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high- double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett. 97(2), 023903 (2006). [CrossRef] [PubMed]

13.

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009). [CrossRef]

14.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009). [CrossRef] [PubMed]

15.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009). [CrossRef]

16.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009). [CrossRef] [PubMed]

17.

G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express 19(3), 2782–2790 (2011). [CrossRef] [PubMed]

18.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1(7), 416–422 (2007). [CrossRef]

19.

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103(22), 223901 (2009). [CrossRef] [PubMed]

20.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009). [CrossRef]

21.

T. Stomeo, M. Grande, G. Rainò, A. Passaseo, A. D’Orazio, R. Cingolani, A. Locatelli, D. Modotto, C. De Angelis, and M. De Vittorio, “Optical filter based on two coupled PhC GaAs-membranes,” Opt. Lett. 35(3), 411–413 (2010). [CrossRef] [PubMed]

22.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010). [CrossRef]

23.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102(4), 043904 (2009). [CrossRef] [PubMed]

24.

R. A. Nome, M. J. Guffey, N. F. Scherer, and S. K. Gray, “Plasmonic interactions and Optical Forces between Au Bipyramidal Nanoparticle Dimers,” J. Phys. Chem. A 113(16), 4408–4415 (2009). [CrossRef] [PubMed]

25.

H. Xu and M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89(24), 246802 (2002). [CrossRef] [PubMed]

26.

D. Woolf, M. Loncar, and F. Capasso, “The forces from coupled surface plasmon polaritons in planar waveguides,” Opt. Express 17(22), 19996–20011 (2009). [CrossRef] [PubMed]

27.

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011). [CrossRef] [PubMed]

28.

Y. He, S. He, J. Gao, and X. Yang, “Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Express 20(20), 22372–22382 (2012). [CrossRef] [PubMed]

29.

P. C. Chaumet, A. Rahmani, F. Zolla, and A. Nicolet, “Electromagnetic forces on a discrete spherical invisibility cloak under time-harmonic illumination,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(5), 056602 (2012). [CrossRef] [PubMed]

30.

V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Enhancing optical gradient forces with metamaterials,” Phys. Rev. Lett. 110(5), 057401 (2013). [CrossRef] [PubMed]

31.

J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Optical gecko toe: Optically controlled attractive near-field forces between plasmonic metamaterials and dielectric or metal surfaces,” Phys. Rev. B 85(20), 205123 (2012). [CrossRef]

32.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005). [CrossRef] [PubMed]

33.

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef] [PubMed]

34.

H.-K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15(3), 1076–1083 (2007). [CrossRef] [PubMed]

35.

J. Pan, Z. Chen, Z. D. Yan, Z. S. Cao, P. Zhan, N. B. Ming, and Z. L. Wang, “Symmetric and anti-symmetric magnetic resonances in double-triangle nanoparticle arrays fabricated via angle-resolved nanosphere lithography,” AIP Adv. 1(4), 042114 (2011). [CrossRef]

36.

R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18(25), 25665–25676 (2010). [CrossRef] [PubMed]

37.

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2011). [CrossRef] [PubMed]

38.

M. Liu, D. A. Powell, and I. V. Shadrivov, “Chiral meta-atoms rotated by light,” Appl. Phys. Lett. 101(3), 031105 (2012). [CrossRef]

39.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Twists and turns for metamaterials,” arXiv:1301.5960 [physics.optics].

40.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

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46.

D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83(23), 235420 (2011). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(160.3918) Materials : Metamaterials
(120.4880) Instrumentation, measurement, and metrology : Optomechanics

ToC Category:
Metamaterials

History
Original Manuscript: March 6, 2013
Revised Manuscript: April 9, 2013
Manuscript Accepted: May 1, 2013
Published: May 7, 2013

Citation
Chaojun Tang, Qiugu Wang, Fanxin Liu, Zhuo Chen, and Zhenlin Wang, "Optical forces in twisted split-ring-resonator dimer stereometamaterials," Opt. Express 21, 11783-11793 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11783


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References

  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett.24(4), 156–159 (1970). [CrossRef]
  2. S. Chu, “Laser manipulation of atoms and particles,” Science253(5022), 861–866 (1991). [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11(5), 288–290 (1986). [CrossRef] [PubMed]
  4. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987). [CrossRef] [PubMed]
  5. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics5(6), 318–321 (2011). [CrossRef] [PubMed]
  6. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum.75(9), 2787–2809 (2004). [CrossRef] [PubMed]
  7. M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical binding,” Phys. Rev. Lett.63(12), 1233–1236 (1989). [CrossRef] [PubMed]
  8. R. Omori, T. Kobayashi, and A. Suzuki, “Observation of a single-beam gradient-force optical trap for dielectric particles in air,” Opt. Lett.22(11), 816–818 (1997). [CrossRef] [PubMed]
  9. T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, “Tight-Binding Photonic Molecule Modes of Resonant Bispheres,” Phys. Rev. Lett.82(23), 4623–4626 (1999). [CrossRef]
  10. M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,” Opt. Express13(20), 8286–8295 (2005). [CrossRef] [PubMed]
  11. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett.30(22), 3042–3044 (2005). [CrossRef] [PubMed]
  12. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high- double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett.97(2), 023903 (2006). [CrossRef] [PubMed]
  13. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3(8), 464–468 (2009). [CrossRef]
  14. J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4(8), 510–513 (2009). [CrossRef] [PubMed]
  15. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009). [CrossRef]
  16. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009). [CrossRef] [PubMed]
  17. G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19(3), 2782–2790 (2011). [CrossRef] [PubMed]
  18. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1(7), 416–422 (2007). [CrossRef]
  19. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett.103(22), 223901 (2009). [CrossRef] [PubMed]
  20. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009). [CrossRef]
  21. T. Stomeo, M. Grande, G. Rainò, A. Passaseo, A. D’Orazio, R. Cingolani, A. Locatelli, D. Modotto, C. De Angelis, and M. De Vittorio, “Optical filter based on two coupled PhC GaAs-membranes,” Opt. Lett.35(3), 411–413 (2010). [CrossRef] [PubMed]
  22. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics4(4), 211–217 (2010). [CrossRef]
  23. A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett.102(4), 043904 (2009). [CrossRef] [PubMed]
  24. R. A. Nome, M. J. Guffey, N. F. Scherer, and S. K. Gray, “Plasmonic interactions and Optical Forces between Au Bipyramidal Nanoparticle Dimers,” J. Phys. Chem. A113(16), 4408–4415 (2009). [CrossRef] [PubMed]
  25. H. Xu and M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett.89(24), 246802 (2002). [CrossRef] [PubMed]
  26. D. Woolf, M. Loncar, and F. Capasso, “The forces from coupled surface plasmon polaritons in planar waveguides,” Opt. Express17(22), 19996–20011 (2009). [CrossRef] [PubMed]
  27. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett.11(2), 321–328 (2011). [CrossRef] [PubMed]
  28. Y. He, S. He, J. Gao, and X. Yang, “Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Express20(20), 22372–22382 (2012). [CrossRef] [PubMed]
  29. P. C. Chaumet, A. Rahmani, F. Zolla, and A. Nicolet, “Electromagnetic forces on a discrete spherical invisibility cloak under time-harmonic illumination,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.85(5), 056602 (2012). [CrossRef] [PubMed]
  30. V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Enhancing optical gradient forces with metamaterials,” Phys. Rev. Lett.110(5), 057401 (2013). [CrossRef] [PubMed]
  31. J. Zhang, K. F. MacDonald, and N. I. Zheludev, “Optical gecko toe: Optically controlled attractive near-field forces between plasmonic metamaterials and dielectric or metal surfaces,” Phys. Rev. B85(20), 205123 (2012). [CrossRef]
  32. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95(20), 203901 (2005). [CrossRef] [PubMed]
  33. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett.30(24), 3356–3358 (2005). [CrossRef] [PubMed]
  34. H.-K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express15(3), 1076–1083 (2007). [CrossRef] [PubMed]
  35. J. Pan, Z. Chen, Z. D. Yan, Z. S. Cao, P. Zhan, N. B. Ming, and Z. L. Wang, “Symmetric and anti-symmetric magnetic resonances in double-triangle nanoparticle arrays fabricated via angle-resolved nanosphere lithography,” AIP Adv.1(4), 042114 (2011). [CrossRef]
  36. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express18(25), 25665–25676 (2010). [CrossRef] [PubMed]
  37. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater.11(1), 30–33 (2011). [CrossRef] [PubMed]
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