## Optical forces in twisted split-ring-resonator dimer stereometamaterials |

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/µm^{2}. 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

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^{2}. 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

*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]

*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

**acting on the object can be calculated using a surface integral [41]:where <**

*F**T*> is the time-averaged Maxwell stress tensor defined byfootnotes

_{ij}*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,

*ε*and

_{r}*µ*are the relative permittivity and permeability of the surrounding medium. By extending the integration surface

_{r}*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 (

**and**

*F*_{1}**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**

*F*_{2}37. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. **11**(1), 30–33 (2011). [CrossRef] [PubMed]

**= (**

*F*_{rel}**–**

*F*_{1}**)/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).**

*F*_{2}*φ*= 0°. Figure 2(a) 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]

*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

**and**

*F*_{1}**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/µm**

*F*_{2}^{2}. Then, the relative force

**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**

*F*_{rel}*ω*and

^{-}*ω*. The relative force resonant at

^{+}*ω*= 200 THz is attractive (

^{-}**< 0) across the spectral range from 190 THz to 210 THz, reaching a peak magnitude of approximately 650**

*F*_{rel}*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 (

**> 0) and would push one SRR away from the other.**

*F*_{rel}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]

*p*(

_{1}*p*) oriented parallel to the SRR base and a magnetic dipole

_{2}*m*(

_{1}*m*) 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

_{2}**.**

*F*_{rel}**could be derived from the interaction between the electric dipolar moments of**

*F*_{p1p2}**and**

*p*_{1}**, oscillating at frequency**

*p*_{2}*ω*[43]where

**(**

*E***,**

*p*_{1}**,**

*r**ω*) is the electric field generated from the electric dipole

**[43]**

*p*_{1}*r*,

*ε*is the relative permittivity of the homogeneous medium in which the dipole is embedded,

_{r}*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]

*x*-direction (i.e.,

*r*along the

*z*-direction (i.e.,

*kr*≈0), the later three terms in Eq. (5) could be ignored, and thus the Coulomb force

**averaged on a time cycle is simplified as**

*F*_{p1p2}**from the interaction between the magnetic dipolar moments of**

*F*_{m1m2}**and**

*m*_{1}**,where the magnetic flux density**

*m*_{2}**(**

*B***,**

*m*_{1}**,**

*r**ω*) can be obtained by replacing

**and 1/**

*p*_{1}*ε*

_{0}

*ε*in Eq. (4) with

_{r}**and**

*m*_{1}*µ*

_{0}

*µ*, respectively [44

_{r}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]

*z*-direction (i.e.,

*r*along the

*z*-direction (i.e.,

*p**,*

_{1}

*p**,*

_{2}

*m**, and*

_{1}

*m**could be obtained from the local electromagnetic fields available in the numerical model [36*

_{2}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]

*V*(

_{1}*V*) indicates the volume integration should be carried out in the upper (lower) SRR,

_{2}*D*,

_{x}*D*, and

_{y}*D*are the

_{z}*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

**arising from the electric dipole-dipole interaction and the Ampere force**

*F*_{p1p2}**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) .**

*F*_{m1m2}**is negative at the resonance frequency of**

*F*_{p1p2}*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

^{+}**, 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**

*F*_{m1m2}**is opposite to that of**

*F*_{m1m2}**at the frequencies of**

*F*_{p1p2}*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,

^{+}**=**

*F*_{rel}**+**

*F*_{p1p2}**. 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**

*F*_{m1m2}*ω*resonance. Because the Coulomb force

^{+}**exceeds the Ampere force**

*F*_{p1p2}**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**

*F*_{m1m2}**spectra [Fig. 3(a)], suggesting that the Coulomb force**

*F*_{p1p2}**plays a key role in the relative optical force.**

*F*_{p1p2}*φ*= 180° are further studied. Figure 4(a) 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 (

^{-}*p*and

_{1}*p*) in the upper and lower SRRs align anti-parallel along the

_{2}*x*-direction, while the magnetic dipole moments (

*m*and

_{1}*m*) in the two SRRs align parallel along the

_{2}*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)].

^{+}**and the Ampere force**

*F*_{p1p2}**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**

*F*_{m1m2}**for**

*F*_{p1p2}*φ*= 180° [solid line in Fig. 4(b)] is similar to that for

*φ*= 0° [solid line in Fig. 3(a)]. An attractive force (

**< 0) and a repulsive force (**

*F*_{p1p2}**> 0) is generated from the transverse interaction between two anti-parallel and parallel aligned electric dipoles at the resonance frequency**

*F*_{p1p2}*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 (

^{+}**< 0) and repulsive Ampere force (**

*F*_{m1m2}**> 0) at the resonance frequency**

*F*_{m1m2}*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

**3**(3), 157–162 (2009). [CrossRef]

**< 0) at the lower resonance frequency ω**

*F*_{rel}^{-}and repulsive (

**> 0) at the higher resonance frequency**

*F*_{rel}*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.

^{-}*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

**3**(3), 157–162 (2009). [CrossRef]

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]

**< 0) and repulsive force (**

*F*_{rel}**> 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**

*F*_{rel}*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.

^{+}**or**

*F*_{1}**) 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**

*F*_{2}**is defined as**

*F*_{comm}**=**

*F*_{comm}**+**

*F*_{1}**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**

*F*_{2}*ω*and

^{-}*ω*, and exhibits an X-shaped dispersion. Specifically, the sign of

^{+}**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.**

*F*_{comm}## 3. Conclusion

^{2}. 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

**11**(1), 30–33 (2011). [CrossRef] [PubMed]

## Acknowledgments

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17. | G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express |

18. | M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics |

19. | M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. |

20. | J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics |

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. |

22. | D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics |

23. | A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. |

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 |

25. | H. Xu and M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. |

26. | D. Woolf, M. Loncar, and F. Capasso, “The forces from coupled surface plasmon polaritons in planar waveguides,” Opt. Express |

27. | X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. |

28. | Y. He, S. He, J. Gao, and X. Yang, “Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Express |

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. |

30. | V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Enhancing optical gradient forces with metamaterials,” Phys. Rev. Lett. |

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 |

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. |

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. |

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 |

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. |

36. | R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express |

37. | M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. |

38. | M. Liu, D. A. Powell, and I. V. Shadrivov, “Chiral meta-atoms rotated by light,” Appl. Phys. Lett. |

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 |

41. | J. D. Jackson, |

42. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

43. | L. Novotny and B. Hecht, |

44. | A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B |

45. | H. F. Harmuth, |

46. | D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B |

**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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