Abstract

We study the optical force arising when isolated gold nanowire pairs and metamaterials with a gold nanowire pair in the unit cell are illuminated with laser radiation. Firstly, we show that isolated nanowire pairs are subject to much stronger optical forces than nanospheres due to their stronger electric and magnetic dipole resonances. We also investigate the properties of the optical force as a function of the length of the nanowires and of the distance between the nanowires. Secondly, we study the optical force in a metamaterial that consists of a periodic array of nanowire pairs. We show that the ratio of the size of the unit cell to the length of the nanowires determines whether the electric dipole resonance leads to an attractive or a repulsive force, and we present the underlying physical mechanism for this effect.

© 2010 OSA

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  6. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
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  7. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
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  8. 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, 3042–3044 (2005).
    [CrossRef] [PubMed]
  9. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature (London) 462, 633–636 (2009).
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  10. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
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  11. A. J. Hallock, P. L. Redmond, and L. E. Brus, “Optical forces between metallic particles,” Proc. Natl. Acad. Sci. U.S.A. 102, 1280–1284 (2005).
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  12. P. Chu and D. L. Mills, “Laser-induced forces in metallic nanosystems: The role of plasmon resonances,” Phys. Rev. Lett. 99, 127401 (2007).
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  13. C. Rockstuhl and H. P. Herzig, “Wavelength-dependent optical force on elliptical silver cylinders at plasmon resonance,” Opt. Lett. 29, 2181–2183 (2004).
    [CrossRef] [PubMed]
  14. K. Halterman, J. M. Elson, and S. Singh, “Plasmonic resonances and electromagnetic forces between coupled silver nanowires,” Phys. Rev. B 72, 075429 (2005).
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  15. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  16. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  17. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter 10, 4785–4809 (1998).
    [CrossRef]
  18. D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [CrossRef] [PubMed]
  19. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the electric resonance of split-ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
    [CrossRef]
  20. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
    [CrossRef] [PubMed]
  21. J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
    [CrossRef]
  22. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
    [CrossRef] [PubMed]
  23. G. Dolling, C. Enkrich, M. Wegener, and C. M. Soukoulis, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
    [CrossRef] [PubMed]
  24. V. M. Shalaev, W. S. 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, 3356–3358 (2005).
    [CrossRef]
  25. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  26. M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
    [CrossRef]
  27. H.-S. Park, A. Agarwal, N. A. Kotov, and O. D. Lavrentovich, “Controllable side-by-side and end-to-end assembly of Au nanorods by lyotropic chromonic materials,” Langmuir 24, 13833–13837 (2008).
    [CrossRef] [PubMed]
  28. J. Zhou, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31, 3620–3622 (2006).
    [CrossRef] [PubMed]
  29. P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
    [CrossRef]
  30. R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16, 18131–18144 (2008).
    [CrossRef] [PubMed]
  31. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
    [CrossRef] [PubMed]
  32. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Planar designs for electromagnetically induced transparency in metamaterials,” Opt. Express 17, 5595–5605 (2009).
    [CrossRef] [PubMed]
  33. N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
    [CrossRef] [PubMed]
  34. N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
    [CrossRef] [PubMed]

2010 (1)

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

2009 (4)

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

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Planar designs for electromagnetically induced transparency in metamaterials,” Opt. Express 17, 5595–5605 (2009).
[CrossRef] [PubMed]

2008 (6)

R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16, 18131–18144 (2008).
[CrossRef] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

H.-S. Park, A. Agarwal, N. A. Kotov, and O. D. Lavrentovich, “Controllable side-by-side and end-to-end assembly of Au nanorods by lyotropic chromonic materials,” Langmuir 24, 13833–13837 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

2007 (1)

P. Chu and D. L. Mills, “Laser-induced forces in metallic nanosystems: The role of plasmon resonances,” Phys. Rev. Lett. 99, 127401 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (7)

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, 3042–3044 (2005).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, and C. M. Soukoulis, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. S. 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, 3356–3358 (2005).
[CrossRef]

K. Halterman, J. M. Elson, and S. Singh, “Plasmonic resonances and electromagnetic forces between coupled silver nanowires,” Phys. Rev. B 72, 075429 (2005).
[CrossRef]

A. J. Hallock, P. L. Redmond, and L. E. Brus, “Optical forces between metallic particles,” Proc. Natl. Acad. Sci. U.S.A. 102, 1280–1284 (2005).
[CrossRef] [PubMed]

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

2004 (3)

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the electric resonance of split-ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

C. Rockstuhl and H. P. Herzig, “Wavelength-dependent optical force on elliptical silver cylinders at plasmon resonance,” Opt. Lett. 29, 2181–2183 (2004).
[CrossRef] [PubMed]

2002 (1)

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

1998 (4)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter 10, 4785–4809 (1998).
[CrossRef]

S. Chu, “Nobel lecture: The manipulation of neutral particles,” Rev. Mod. Phys. 70, 685–706 (1998).
[CrossRef]

C. Cohen-Tannoudji, “Nobel lecture: Manipulating atoms with photons,” Rev. Mod. Phys. 70, 707–719 (1998).
[CrossRef]

W. D. Phillips, “Nobel lecture: Laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70, 721–741 (1998).
[CrossRef]

1972 (1)

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

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Agarwal, A.

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

H.-S. Park, A. Agarwal, N. A. Kotov, and O. D. Lavrentovich, “Controllable side-by-side and end-to-end assembly of Au nanorods by lyotropic chromonic materials,” Langmuir 24, 13833–13837 (2008).
[CrossRef] [PubMed]

Aydin, K.

Baehr-Jones, T.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

Baena, J. D.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Brus, L. E.

A. J. Hallock, P. L. Redmond, and L. E. Brus, “Optical forces between metallic particles,” Proc. Natl. Acad. Sci. U.S.A. 102, 1280–1284 (2005).
[CrossRef] [PubMed]

Cai, W. S.

Capasso, F.

Chen, L.

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

Chettiar, U. K.

Christy, R. W.

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

Chu, P.

P. Chu and D. L. Mills, “Laser-induced forces in metallic nanosystems: The role of plasmon resonances,” Phys. Rev. Lett. 99, 127401 (2007).
[CrossRef] [PubMed]

Chu, S.

S. Chu, “Nobel lecture: The manipulation of neutral particles,” Rev. Mod. Phys. 70, 685–706 (1998).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, “Nobel lecture: Manipulating atoms with photons,” Rev. Mod. Phys. 70, 707–719 (1998).
[CrossRef]

Dolling, G.

Drachev, V. P.

Economou, E. N.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Planar designs for electromagnetically induced transparency in metamaterials,” Opt. Express 17, 5595–5605 (2009).
[CrossRef] [PubMed]

R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16, 18131–18144 (2008).
[CrossRef] [PubMed]

J. Zhou, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31, 3620–3622 (2006).
[CrossRef] [PubMed]

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the electric resonance of split-ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

Elson, J. M.

K. Halterman, J. M. Elson, and S. Singh, “Plasmonic resonances and electromagnetic forces between coupled silver nanowires,” Phys. Rev. B 72, 075429 (2005).
[CrossRef]

Enkrich, C.

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Fedotov, V. A.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Garcia-Garcia, J.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Gay-Balmaz, P.

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

Giessen, H.

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Gondarenko, A.

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

Hallock, A. J.

A. J. Hallock, P. L. Redmond, and L. E. Brus, “Optical forces between metallic particles,” Proc. Natl. Acad. Sci. U.S.A. 102, 1280–1284 (2005).
[CrossRef] [PubMed]

Halterman, K.

K. Halterman, J. M. Elson, and S. Singh, “Plasmonic resonances and electromagnetic forces between coupled silver nanowires,” Phys. Rev. B 72, 075429 (2005).
[CrossRef]

Herzig, H. P.

Hochberg, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter 10, 4785–4809 (1998).
[CrossRef]

Ibanescu, M.

Jackson, J. D.

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

Jelink, L.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Joannopoulos, J. D.

Johnson, P. B.

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

Johnson, S. G.

Kafesaki, M.

R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16, 18131–18144 (2008).
[CrossRef] [PubMed]

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the electric resonance of split-ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

Kastel, J.

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Katsarakis, N.

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the electric resonance of split-ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

Kildishev, A. V.

Kippenberg, T. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

Koschny, T.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Planar designs for electromagnetically induced transparency in metamaterials,” Opt. Express 17, 5595–5605 (2009).
[CrossRef] [PubMed]

R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16, 18131–18144 (2008).
[CrossRef] [PubMed]

J. Zhou, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Unifying approach to left-handed material design,” Opt. Lett. 31, 3620–3622 (2006).
[CrossRef] [PubMed]

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the electric resonance of split-ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

Kotov, N.

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

Kotov, N. A.

H.-S. Park, A. Agarwal, N. A. Kotov, and O. D. Lavrentovich, “Controllable side-by-side and end-to-end assembly of Au nanorods by lyotropic chromonic materials,” Langmuir 24, 13833–13837 (2008).
[CrossRef] [PubMed]

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Lavrentovich, O. D.

H.-S. Park, A. Agarwal, N. A. Kotov, and O. D. Lavrentovich, “Controllable side-by-side and end-to-end assembly of Au nanorods by lyotropic chromonic materials,” Langmuir 24, 13833–13837 (2008).
[CrossRef] [PubMed]

Li, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

Lipson, M.

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

Liu, N.

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Loncar, M.

Malloy, K. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Mansfield, J. F.

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

Marques, R.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Martin, F.

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

Martin, O. J. F.

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Clarendon Press, Oxford, 1873).

Mills, D. L.

P. Chu and D. L. Mills, “Laser-induced forces in metallic nanosystems: The role of plasmon resonances,” Phys. Rev. Lett. 99, 127401 (2007).
[CrossRef] [PubMed]

N’Gom, M.

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

Nemat-Nasser, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Norris, T. B.

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

Osgood, R. M.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Ozbay, E.

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Panoiu, N. C.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Papasimakis, N.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Park, H.-S.

H.-S. Park, A. Agarwal, N. A. Kotov, and O. D. Lavrentovich, “Controllable side-by-side and end-to-end assembly of Au nanorods by lyotropic chromonic materials,” Langmuir 24, 13833–13837 (2008).
[CrossRef] [PubMed]

Penciu, R. S.

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter 10, 4785–4809 (1998).
[CrossRef]

Pernice, W. H. P.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Phillips, W. D.

W. D. Phillips, “Nobel lecture: Laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70, 721–741 (1998).
[CrossRef]

Povinelli, M. L.

Prosvirnin, S. L.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Redmond, P. L.

A. J. Hallock, P. L. Redmond, and L. E. Brus, “Optical forces between metallic particles,” Proc. Natl. Acad. Sci. U.S.A. 102, 1280–1284 (2005).
[CrossRef] [PubMed]

Ringnalda, J.

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter 10, 4785–4809 (1998).
[CrossRef]

Rockstuhl, C.

Roels, J.

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

Sarychev, A. K.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Singh, S.

K. Halterman, J. M. Elson, and S. Singh, “Plasmonic resonances and electromagnetic forces between coupled silver nanowires,” Phys. Rev. B 72, 075429 (2005).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Smythe, E. J.

Soukoulis, C. M.

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter 10, 4785–4809 (1998).
[CrossRef]

Tang, H. X.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

Tassin, P.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Planar designs for electromagnetically induced transparency in metamaterials,” Opt. Express 17, 5595–5605 (2009).
[CrossRef] [PubMed]

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

Van Thourhout, D.

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

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Wegener, M.

Weiss, T.

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Wiederhecker, G. S.

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

Xiong, C.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

Yuan, H. K.

Zaluzec, N. J.

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

Zhang, L.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Planar designs for electromagnetically induced transparency in metamaterials,” Opt. Express 17, 5595–5605 (2009).
[CrossRef] [PubMed]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

Zhang, S.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Zheludev, N. I.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Zhou, J.

Appl. Phys. Lett. (1)

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the electric resonance of split-ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004).
[CrossRef]

J. Appl. Phys. (2)

J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. 98, 033103 (2005).
[CrossRef]

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002).
[CrossRef]

J. Phys. Condens. Matter (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin wire structures,” J. Phys. Condens. Matter 10, 4785–4809 (1998).
[CrossRef]

Langmuir (1)

H.-S. Park, A. Agarwal, N. A. Kotov, and O. D. Lavrentovich, “Controllable side-by-side and end-to-end assembly of Au nanorods by lyotropic chromonic materials,” Langmuir 24, 13833–13837 (2008).
[CrossRef] [PubMed]

Nano Lett. (1)

M. N’Gom, J. Ringnalda, J. F. Mansfield, A. Agarwal, N. Kotov, N. J. Zaluzec, and T. B. Norris, “Single particle plasmon spectroscopy of silver nanowires and gold nanorods,” Nano Lett. 8, 3200–3204 (2008).
[CrossRef]

Nat. Mater. (1)

N. Liu, L. Langguth, T. Weiss, J. Kastel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Nat. Photonics (1)

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

Nature (London) (2)

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

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature (London) 456, 480–484 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. B (2)

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

K. Halterman, J. M. Elson, and S. Singh, “Plasmonic resonances and electromagnetic forces between coupled silver nanowires,” Phys. Rev. B 72, 075429 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

P. Chu and D. L. Mills, “Laser-induced forces in metallic nanosystems: The role of plasmon resonances,” Phys. Rev. Lett. 99, 127401 (2007).
[CrossRef] [PubMed]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

A. J. Hallock, P. L. Redmond, and L. E. Brus, “Optical forces between metallic particles,” Proc. Natl. Acad. Sci. U.S.A. 102, 1280–1284 (2005).
[CrossRef] [PubMed]

Rev. Mod. Phys. (3)

S. Chu, “Nobel lecture: The manipulation of neutral particles,” Rev. Mod. Phys. 70, 685–706 (1998).
[CrossRef]

C. Cohen-Tannoudji, “Nobel lecture: Manipulating atoms with photons,” Rev. Mod. Phys. 70, 707–719 (1998).
[CrossRef]

W. D. Phillips, “Nobel lecture: Laser cooling and trapping of neutral atoms,” Rev. Mod. Phys. 70, 721–741 (1998).
[CrossRef]

Science (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (2)

J. C. Maxwell, A Treatise on Electricity and Magnetism (Clarendon Press, Oxford, 1873).

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

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Figures (11)

Fig. 1
Fig. 1

The nanowire pair that is considered in this article for the purpose of studying the optical force in metamaterials. The propagation direction and polarization of the incident electromagnetic wave is indicated.

Fig. 2
Fig. 2

(a) The common and (b) the relative optical force exerted on a pair of nanowires by an incident field of 50 mW / μm2. The length of the nanowires is L = 100 nm, their diameter is D = 25 nm, and the distance between the nanowires is d = 25 nm. The relative force exhibits two resonances: the magnetic dipole resonance with an attractive force and the electric dipole resonance with a repulsive force.

Fig. 3
Fig. 3

Electric (a) and magnetic (b) dipole moment of the charge distribution in an isolated nanowire pair with length L = 100 nm, diameter D = 25 nm, and interwire distance D = 25 nm. The insets show the charge and current distributions of the eigenmodes at 445 THz (electric dipole resonance) and at 410 THz (magnetic dipole resonance).

Fig. 4
Fig. 4

The relative optical force in an isolated nanowire pair as a function of length. All nanowires have diameter D = 25 nm and interwire distance d = 25 nm. Note that the force in the longer rods is exponentially larger than the force in the shortest sample (which has L = D = 25 nm and is therefore actually a pair of nanospheres).

Fig. 5
Fig. 5

(a) The relative optical force in an isolated nanowire pair as a function of frequency for different interwire distances. All nanowires have diameter D = 25 nm and length L = 100 nm. (b) The relative optical force at the resonance frequency as a function of the interwire distance.

Fig. 6
Fig. 6

Schematic of the array of nanowires studied in Sec. 4.

Fig. 7
Fig. 7

(a) The common and (b) the relative optical force exerted on a pair of nanowires in a 2D periodic array with unit cell 100 nm × 200 nm when illuminated by a laser field of 50 mW/μm2. The length of the nanowires is L = 100 nm, their diameter is D = 25 nm, and the distance between the nanowires is d = 25 nm. The relative force exhibits two resonances: the magnetic dipole resonance with an attractive force and the electric dipole resonance with a repulsive force.

Fig. 8
Fig. 8

Electric (a) and magnetic (b) dipole moment of the charge distribution in a 2D periodic nanowire pair metamaterial with unit cell 100 nm × 200 nm. The length of the nanowires is L = 100 nm, their diameter D = 25 nm, and the interwire distance equals D = 25 nm.

Fig. 9
Fig. 9

The relative optical force in a periodic nanowire pair metamaterial as a function of the unit cell size height h (for smaller unit cell height, the ends of the nanowires in neighbouring cells come closer). All nanowires have length L = 100 nm, diameter D = 25 nm, and interwire distance d = 25 nm. The unit cell size is 100 nm × h. Note that the optical force decreases significantly for metamaterials with smaller unit cell height.

Fig. 10
Fig. 10

The relative optical force in a periodic nanowire pair metamaterial as a function of the unit cell size height h (for smaller unit cell height, the ends of the nanowires in neighbouring cells come closer). All nanowires have length L = 200 nm, diameter D = 50 nm, and interwire distance d = 50 nm. The unit cell size is 200 nm × h. Note that the optical force reverses from attractive to repulsive for metamaterials with smaller unit cell height. The arrows mark the position of the magnetic dipole resonance.

Fig. 11
Fig. 11

Charge and current distribution in the nanowire pair metamaterial. (a) If the height of the unit cell is substantially larger than the length of the nanowires, the force at the magnetic dipole resonance frequency has the nature of a Coulomb force between the charges at the ends of the wires and is attractive for the magnetic dipole resonance. (b) If the height of the unit cell is only slightly larger than the length of the nanowires, the force at the magnetic dipole resonance frequency has the nature of an electric dipole-dipole interaction; the repulsive Lorentz force between the currents flowing in opposite directions can now be of comparable size or even dominate the optical force.

Equations (6)

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

F i = V ( ρ E i + ɛ ijk J j B k ) d V ,
F i = ɛ 0 μ 0 V S i t d V + V T ij x j d V ,
T ij = ɛ 0 ( E i E j 1 2 δ ij E 2 ) + 1 μ 0 ( B i B i 1 2 δ ij B 2 ) ,
F i = S T i j n i d S .
p z = V P z d V = V ( D z ɛ 0 E z ) d V .
m y = V ( z P x t x P z t ) d V = i ω V [ z ( D x ɛ 0 E x ) x ( D z ɛ 0 E z ) ] d V .

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