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

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2013 (1)

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]

2012 (4)

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]

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

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]

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]

2011 (6)

G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19(3), 2782–2790 (2011).
[CrossRef] [PubMed]

D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B83(23), 235420 (2011).
[CrossRef]

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]

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]

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

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics5(6), 318–321 (2011).
[CrossRef] [PubMed]

2010 (3)

2009 (10)

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]

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]

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]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3(8), 464–468 (2009).
[CrossRef]

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]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

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]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

2007 (2)

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]

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]

2006 (2)

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]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B74(3), 033402 (2006).
[CrossRef]

2005 (4)

2004 (1)

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

2002 (1)

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

1999 (1)

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]

1997 (1)

1991 (1)

S. Chu, “Laser manipulation of atoms and particles,” Science253(5022), 861–866 (1991).
[CrossRef] [PubMed]

1989 (1)

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

1987 (1)

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

1986 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1970 (1)

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

Ashkin, A.

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

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]

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

Baets, R.

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]

Bartal, G.

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]

Bjorkholm, J. E.

Block, S. M.

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics5(6), 318–321 (2011).
[CrossRef] [PubMed]

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

Boltasseva, A.

Burger, S.

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]

Burns, M. M.

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

Cai, W.

Cao, Z. S.

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]

Capasso, F.

Chaumet, P. C.

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]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Chen, Z.

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]

Chettiar, U. K.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Chu, S.

Cingolani, R.

D’Orazio, A.

De Angelis, C.

De Vittorio, M.

De Vlaminck, I.

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]

Drachev, V. P.

Dziedzic, J. M.

Eichenfield, M.

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]

Enkrich, C.

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]

Fazal, F. M.

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics5(6), 318–321 (2011).
[CrossRef] [PubMed]

Fournier, J. M.

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

Gao, J.

Genov, D. A.

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]

Giessen, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Ginis, V.

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]

Golovchenko, J. A.

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

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Grande, M.

Gray, S. K.

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]

Guffey, M. J.

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]

Hannam, K.

D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B83(23), 235420 (2011).
[CrossRef]

He, S.

He, Y.

Ibanescu, M.

Ishikawa, A.

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]

Jimba, Y.

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]

Joannopoulos, J. D.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Johnson, S. G.

Käll, M.

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

Kildishev, A. V.

Kivshar, Y. S.

D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B83(23), 235420 (2011).
[CrossRef]

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

Kobayashi, T.

Koenderink, A. F.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B74(3), 033402 (2006).
[CrossRef]

Koschny, T.

Koschny, Th.

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]

Kuramochi, E.

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]

Kuwata-Gonokami, M.

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]

Lagae, L.

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]

Lapine, M.

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

Lee, S.

Li, M.

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3(8), 464–468 (2009).
[CrossRef]

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]

Lin, Q.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

Linden, S.

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]

Lipson, M.

G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19(3), 2782–2790 (2011).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Liu, H.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Liu, M.

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

Liu, N.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Liu, Y.

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]

Locatelli, A.

Loncar, M.

MacDonald, K. F.

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]

Maes, B.

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]

Manipatruni, S.

Michael, C. P.

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]

Ming, N. B.

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]

Mitsugi, S.

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]

Miyazaki, H.

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]

Modotto, D.

Mukaiyama, T.

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]

Neuman, K. C.

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

Nicolet, A.

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]

Nome, R. A.

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]

Notomi, M.

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]

Omori, R.

Oulton, R. F.

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]

Painter, O.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

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]

Pan, J.

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]

Passaseo, A.

Perahia, R.

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]

Pernice, W. H. P.

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]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3(8), 464–468 (2009).
[CrossRef]

Polman, A.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B74(3), 033402 (2006).
[CrossRef]

Povinelli, M. L.

Powell, D. A.

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

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

D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B83(23), 235420 (2011).
[CrossRef]

Rahmani, A.

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]

Rainò, G.

Roels, J.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics4(4), 211–217 (2010).
[CrossRef]

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]

Rosenberg, J.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

Sarychev, A. K.

Scherer, N. F.

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]

Schmidt, F.

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]

Shadrivov, I. V.

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

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

D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B83(23), 235420 (2011).
[CrossRef]

Shalaev, V. M.

Smythe, E. J.

Soukoulis, C. M.

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]

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]

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]

Stomeo, T.

Suzuki, A.

Takeda, K.

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]

Tang, H. X.

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3(8), 464–468 (2009).
[CrossRef]

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]

Taniyama, H.

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]

Tassin, P.

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]

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]

Van Thourhout, D.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics4(4), 211–217 (2010).
[CrossRef]

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]

Veretennicoff, I.

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]

Wang, Z. L.

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]

Wegener, M.

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]

Wiederhecker, G. S.

G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19(3), 2782–2790 (2011).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Woolf, D.

Xu, H.

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

Yan, Z. D.

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).
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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).
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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]

Yin, X.

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]

Yuan, H.-K.

Zhan, P.

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]

Zhang, J.

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]

Zhang, S.

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).
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Zhang, X.

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).
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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]

Zhao, R.

Zheludev, N. I.

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]

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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).
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Zhu, S.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
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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]

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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]

AIP Adv. (1)

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).
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Appl. Phys. Lett. (1)

M. Liu, D. A. Powell, and I. V. Shadrivov, “Chiral meta-atoms rotated by light,” Appl. Phys. Lett.101(3), 031105 (2012).
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J. Phys. Chem. A (1)

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]

Nano Lett. (1)

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]

Nat. Mater. (1)

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

Nat. Nanotechnol. (1)

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]

Nat. Photonics (7)

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
[CrossRef]

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]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3(8), 478–483 (2009).
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D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics4(4), 211–217 (2010).
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N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics3(3), 157–162 (2009).
[CrossRef]

Nature (1)

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
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Opt. Express (6)

Opt. Lett. (5)

Phys. Rev. B (4)

D. A. Powell, K. Hannam, I. V. Shadrivov, and Y. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B83(23), 235420 (2011).
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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]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

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).
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Phys. Rev. Lett. (9)

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

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]

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]

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

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).
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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).
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[CrossRef] [PubMed]

Other (4)

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

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].

H. F. Harmuth, Sequency Theory-Foundations and Applications (Academic Press, 1977).

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

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Figures (5)

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.

Fig. 2
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.

Fig. 3
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).

Fig. 4
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.

Fig. 5
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.

Equations (10)

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F i = S T ij n j dS,
T ij = 1 2 Re[ ε r ε 0 ( E i E j * 1 2 m E m E m * )+ μ r μ 0 ( H i H j * 1 2 m H m H m * ) ],
F p1p2 =[ p 2 e iωt E( p 1 ,r,ω ) ],
E( p 1 ,r,ω )= 1 4π ε 0 ε r [ ( 1ikr ) 3(r p 1 ) r ^ p 1 r 3 ( kr ) 2 (r p 1 ) r ^ p 1 r 3 ] e iωt ,
F p1p2 = 3 p 1 p 2 e 2iωt 4π ε 0 ε r r 4 [ 1ikr 2 3 ( kr ) 2 + i 3 ( kr ) 3 ] e ikr z ^ .
F p1p2 = 3Re( p 1 * p 2 ) 8π ε 0 ε r r 4 z ^ .
F m1m2 =[ m 2 e iωt B( m 1 ,r,ω ) ],
B( m 1 ,r,ω )= μ 0 μ r 4π [ ( 1ikr ) 3(r m 1 ) r ^ m 1 r 3 ( kr ) 2 (r m 1 ) r ^ m 1 r 3 ] e iωt ,
F m1m2 = 6Re( m 1 * m 2 ) μ 0 μ r 8π r 4 z ^ .
p 1 = V 1 ( D x ε 0 E x ) dV, p 2 = V 2 ( D x ε 0 E x ) dV, m 1 =iω V 1 [ x( D y ε 0 E y )y( D x ε 0 E x ) ] dV, m 2 =iω V 2 [ x( D y ε 0 E y )y( D x ε 0 E x ) ] dV,

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