Abstract

The mechanical forces associated with surface currents are widely overlooked and point to a new family of plasmonically-driven processes. Here, we investigate the Lorentz forces acting on a free electron gas that is bound to the surface of a nanowire. We demonstrate that appreciable mechanical forces are produced by longer illumination wavelengths between longitudinal and transverse absorption resonances via the excitation of chiral hybrid plasmon modes. We are the first to associate plasmonic activity as the underlying mechanism for nanowire rotation, which explains prior experimental results. The presence of chiral hybrid plasmon modes yields the greatest net translation and torque forces. The asymmetric plasmon behavior subsequently affects the complex nonlinear dynamics of plasmonic nonspherical nanoparticles in fluids.

© 2014 Optical Society of America

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References

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2014 (4)

W.-Y. Tsai, J.-S. Huang, and C.-B. Huang, “Selective trapping or rotation of isotropic dielectric microparticles by optical near field in a plasmonic archimedes spiral,” Nano letters14, 547–552 (2014).
[Crossref] [PubMed]

A. Lehmuskero, Y. Li, P. Johansson, and M. Käll, “Plasmonic particles set into fast orbital motion by an optical vortex beam,” Opt. Express22, 4349–4356 (2014).
[Crossref] [PubMed]

J.-W. Liaw, W.-J. Lo, and M.-K. Kuo, “Wavelength-dependent longitudinal polarizability of gold nanorod on optical torques,” Opt. Express22, 10858–10867 (2014).
[Crossref] [PubMed]

M. Moocarme, J. L. Dominguez-Juarez, and L. T. Vuong, “Ultralow-intensity magneto-optical and mechanical effects in metal nanocolloids,” Nano Letters14, 1178–1183 (2014).
[Crossref] [PubMed]

2013 (6)

Z. Fang, Y.-R. Zhen, O. Neumann, A. Polman, F. J. Garca de Abajo, P. Nordlander, and N. J. Halas, “Evolution of light-induced vapor generation at a liquid-immersed metallic nanoparticle,” Nano Letters13, 1736–1742 (2013). PMID: .
[PubMed]

J. J. Arcenegui, P. Garcia-Sanchez, H. Morgan, and A. Ramos, “Electro-orientation and electrorotation of metal nanowires,” Phys. Rev. E88, 063018 (2013).
[Crossref]

Z. Yan, M. Pelton, L. Vigderman, E. R. Zubarev, and N. F. Scherer, “Why single-beam optical tweezers trap gold nanowires in three dimensions,” ACS nano7, 8794–8800 (2013).
[Crossref] [PubMed]

Z. Yan and N. F. Scherer, “Optical vortex induced rotation of silver nanowires,” J. Phys. Chem. Lett.4, 2937–2942 (2013).
[Crossref]

Z. Yan, R. a. Shah, G. Chado, S. K. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS nano7, 1790–1802 (2013).
[Crossref] [PubMed]

A. Lehmuskero, R. Ogier, T. Gschneidtner, P. Johansson, and M. Kll, “Ultrafast spinning of gold nanoparticles in water using circularly polarized light,” Nano Letters13, 3129–3134 (2013).
[Crossref] [PubMed]

2012 (4)

N. D. Singh, M. Moocarme, B. Edelstein, N. Punnoose, and L. T. Vuong, “Anomalously-large photo-induced magnetic response of metallic nanocolloids in aqueous solution using a solar simulator,” Opt. Express20, 19214–19225 (2012).
[Crossref] [PubMed]

Z. Yan, J. E. Jureller, J. Sweet, M. J. Guffey, M. Pelton, and N. F. Scherer, “Three-dimensional optical trapping and manipulation of single silver nanowires,” Nano letters12, 5155–5161 (2012).
[Crossref] [PubMed]

Z. Yan, J. Sweet, J. E. Jureller, M. J. Guffey, M. Pelton, and N. F. Scherer, “Controlling the position and orientation of single silver nanowires on a surface using structured optical fields,” ACS nano6, 8144–8155 (2012).
[Crossref] [PubMed]

D. B. Ruffner and D. G. Grier, “Optical conveyors: A class of active tractor beams,” Phys. Rev. Lett.109, 163903 (2012).
[Crossref] [PubMed]

2011 (1)

S. Zhang, H. Wei, K. Bao, U. Håkanson, N. J. Halas, P. Nordlander, and H. Xu, “Chiral surface plasmon polaritons on metallic nanowires,” Phys. Rev. Lett.107, 096801 (2011).
[Crossref] [PubMed]

2010 (3)

L. Tong, V. D. Miljkovic, and M. Kall, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Letters10, 268–273 (2010). PMID: .
[Crossref] [PubMed]

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett.105, 118103 (2010).
[Crossref] [PubMed]

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

2009 (1)

2008 (1)

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics2, 021875 (2008).
[Crossref]

2007 (6)

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express15, 17172–17205 (2007).
[Crossref] [PubMed]

P. Janhunen and A. Sandroos, “Simulation study of solar wind push on a charged wire: basis of solar wind electric sail propulsion,” Annales Geophysicae25, 755–767 (2007).
[Crossref]

K. Keshoju, H. Xing, and L. Sun, “Magnetic field driven nanowire rotation in suspension,” Appl. Phys. Lett.91, 123114 (2007).
[Crossref]

M. W. Knight, N. K. Grady, R. Bardhan, F. Hao, P. Nordlander, and N. J. Halas, “Nanoparticle-mediated coupling of light into a nanowire,” Nano Letters7, 2346–2350 (2007). PMID: .
[Crossref] [PubMed]

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
[Crossref]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B76, 035420 (2007).
[Crossref]

2006 (3)

G. Y. Slepyan, M. V. Shuba, S. A. Maksimenko, and A. Lakhtakia, “Theory of optical scattering by achiral carbon nanotubes and their potential as optical nanoantennas,” Phys. Rev. B73, 195416 (2006).
[Crossref]

B. Edwards, T. S. Mayer, and R. B. Bhiladvala, “Synchronous electrorotation of nanowires in fluid,” Nano letters6, 626–632 (2006).
[Crossref] [PubMed]

O. D. Velev and K. H. Bhatt, “On-chip micromanipulation and assembly of colloidal particles by electric fields,” Soft Matter2, 738–750 (2006).
[Crossref]

2005 (3)

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005).
[Crossref]

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett.95, 257403 (2005).
[Crossref] [PubMed]

W. Shelton, K. Bonin, and T. Walker, “Nonlinear motion of optically torqued nanorods,” Phys. Rev. E71, 036204 (2005).
[Crossref]

2002 (2)

K. Bonin, B. Kourmanov, and T. Walker, “Light torque nanocontrol, nanomotors and nanorockers,” Opt. Express10, 984–989 (2002).
[Crossref] [PubMed]

T. Hugel, N. Holland, A. Cattani, L. Moroder, M. Seitz, and H. Gaub, “Single-molecule optomechanical cycle,” Science296, 1103–1106 (2002).
[Crossref] [PubMed]

2001 (1)

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292, 912–914 (2001).
[Crossref] [PubMed]

1999 (1)

J. Weiner, V. Bagnato, S. Zilio, and P. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys.71, 1–85 (1999).
[Crossref]

1997 (2)

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci.94, 4853–4860 (1997).
[Crossref] [PubMed]

L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett.79, 645–648 (1997).
[Crossref]

1993 (1)

W. Petrich, M. Grieser, R. Grimm, A. Gruber, D. Habs, H. Miesner, D. Schwalm, B. Wanner, H. Wernoe, A. Wolf, R. Grieser, G. Huner, R. Klein, T. Kuhl, R. Neumann, and S. Schroder, “Laser cooling of stored high-velocity ions by means of the spontaneous force,” Phys. Rev. A48, 2127–2144 (1993).
[Crossref] [PubMed]

1985 (1)

A. Chowdhury, B. Ackerson, and N. Clark, “Laser-induced freezing,” Phys. Rev. Lett.55, 833–836 (1985).
[Crossref] [PubMed]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev.50, 115–125 (1936).
[Crossref]

Ackerson, B.

A. Chowdhury, B. Ackerson, and N. Clark, “Laser-induced freezing,” Phys. Rev. Lett.55, 833–836 (1985).
[Crossref] [PubMed]

Arcenegui, J. J.

J. J. Arcenegui, P. Garcia-Sanchez, H. Morgan, and A. Ramos, “Electro-orientation and electrorotation of metal nanowires,” Phys. Rev. E88, 063018 (2013).
[Crossref]

Arlt, J.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292, 912–914 (2001).
[Crossref] [PubMed]

Ashkin, A.

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci.94, 4853–4860 (1997).
[Crossref] [PubMed]

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005).
[Crossref]

Aussenegg, F. R.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett.95, 257403 (2005).
[Crossref] [PubMed]

Bagnato, V.

J. Weiner, V. Bagnato, S. Zilio, and P. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys.71, 1–85 (1999).
[Crossref]

Bao, K.

S. Zhang, H. Wei, K. Bao, U. Håkanson, N. J. Halas, P. Nordlander, and H. Xu, “Chiral surface plasmon polaritons on metallic nanowires,” Phys. Rev. Lett.107, 096801 (2011).
[Crossref] [PubMed]

Bardhan, R.

M. W. Knight, N. K. Grady, R. Bardhan, F. Hao, P. Nordlander, and N. J. Halas, “Nanoparticle-mediated coupling of light into a nanowire,” Nano Letters7, 2346–2350 (2007). PMID: .
[Crossref] [PubMed]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev.50, 115–125 (1936).
[Crossref]

Bhatt, K. H.

O. D. Velev and K. H. Bhatt, “On-chip micromanipulation and assembly of colloidal particles by electric fields,” Soft Matter2, 738–750 (2006).
[Crossref]

Bhiladvala, R. B.

B. Edwards, T. S. Mayer, and R. B. Bhiladvala, “Synchronous electrorotation of nanowires in fluid,” Nano letters6, 626–632 (2006).
[Crossref] [PubMed]

Bian, R. X.

L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett.79, 645–648 (1997).
[Crossref]

Bonin, K.

W. Shelton, K. Bonin, and T. Walker, “Nonlinear motion of optically torqued nanorods,” Phys. Rev. E71, 036204 (2005).
[Crossref]

K. Bonin, B. Kourmanov, and T. Walker, “Light torque nanocontrol, nanomotors and nanorockers,” Opt. Express10, 984–989 (2002).
[Crossref] [PubMed]

Borghese, F.

Bryant, P.

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292, 912–914 (2001).
[Crossref] [PubMed]

Cattani, A.

T. Hugel, N. Holland, A. Cattani, L. Moroder, M. Seitz, and H. Gaub, “Single-molecule optomechanical cycle,” Science296, 1103–1106 (2002).
[Crossref] [PubMed]

Chado, G.

Z. Yan, R. a. Shah, G. Chado, S. K. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS nano7, 1790–1802 (2013).
[Crossref] [PubMed]

Chang, D. E.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B76, 035420 (2007).
[Crossref]

Chowdhury, A.

A. Chowdhury, B. Ackerson, and N. Clark, “Laser-induced freezing,” Phys. Rev. Lett.55, 833–836 (1985).
[Crossref] [PubMed]

Clark, N.

A. Chowdhury, B. Ackerson, and N. Clark, “Laser-induced freezing,” Phys. Rev. Lett.55, 833–836 (1985).
[Crossref] [PubMed]

Denti, P.

Desyatnikov, A. S.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett.105, 118103 (2010).
[Crossref] [PubMed]

Dholakia, K.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics2, 021875 (2008).
[Crossref]

L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292, 912–914 (2001).
[Crossref] [PubMed]

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics2, 021875 (2008).
[Crossref]

Ditlbacher, H.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett.95, 257403 (2005).
[Crossref] [PubMed]

Dominguez-Juarez, J. L.

M. Moocarme, J. L. Dominguez-Juarez, and L. T. Vuong, “Ultralow-intensity magneto-optical and mechanical effects in metal nanocolloids,” Nano Letters14, 1178–1183 (2014).
[Crossref] [PubMed]

Edelstein, B.

Edwards, B.

B. Edwards, T. S. Mayer, and R. B. Bhiladvala, “Synchronous electrorotation of nanowires in fluid,” Nano letters6, 626–632 (2006).
[Crossref] [PubMed]

Fang, Z.

Z. Fang, Y.-R. Zhen, O. Neumann, A. Polman, F. J. Garca de Abajo, P. Nordlander, and N. J. Halas, “Evolution of light-induced vapor generation at a liquid-immersed metallic nanoparticle,” Nano Letters13, 1736–1742 (2013). PMID: .
[PubMed]

Garca de Abajo, F. J.

Z. Fang, Y.-R. Zhen, O. Neumann, A. Polman, F. J. Garca de Abajo, P. Nordlander, and N. J. Halas, “Evolution of light-induced vapor generation at a liquid-immersed metallic nanoparticle,” Nano Letters13, 1736–1742 (2013). PMID: .
[PubMed]

Garcia-Sanchez, P.

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S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
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L. Tong, V. D. Miljkovic, and M. Kall, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Letters10, 268–273 (2010). PMID: .
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Morgan, H.

J. J. Arcenegui, P. Garcia-Sanchez, H. Morgan, and A. Ramos, “Electro-orientation and electrorotation of metal nanowires,” Phys. Rev. E88, 063018 (2013).
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Z. Fang, Y.-R. Zhen, O. Neumann, A. Polman, F. J. Garca de Abajo, P. Nordlander, and N. J. Halas, “Evolution of light-induced vapor generation at a liquid-immersed metallic nanoparticle,” Nano Letters13, 1736–1742 (2013). PMID: .
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W. Petrich, M. Grieser, R. Grimm, A. Gruber, D. Habs, H. Miesner, D. Schwalm, B. Wanner, H. Wernoe, A. Wolf, R. Grieser, G. Huner, R. Klein, T. Kuhl, R. Neumann, and S. Schroder, “Laser cooling of stored high-velocity ions by means of the spontaneous force,” Phys. Rev. A48, 2127–2144 (1993).
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Z. Fang, Y.-R. Zhen, O. Neumann, A. Polman, F. J. Garca de Abajo, P. Nordlander, and N. J. Halas, “Evolution of light-induced vapor generation at a liquid-immersed metallic nanoparticle,” Nano Letters13, 1736–1742 (2013). PMID: .
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J. J. Arcenegui, P. Garcia-Sanchez, H. Morgan, and A. Ramos, “Electro-orientation and electrorotation of metal nanowires,” Phys. Rev. E88, 063018 (2013).
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V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett.105, 118103 (2010).
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Z. Yan, R. a. Shah, G. Chado, S. K. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS nano7, 1790–1802 (2013).
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Z. Yan, R. a. Shah, G. Chado, S. K. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS nano7, 1790–1802 (2013).
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ACS nano (3)

Z. Yan, J. Sweet, J. E. Jureller, M. J. Guffey, M. Pelton, and N. F. Scherer, “Controlling the position and orientation of single silver nanowires on a surface using structured optical fields,” ACS nano6, 8144–8155 (2012).
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Z. Yan, R. a. Shah, G. Chado, S. K. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS nano7, 1790–1802 (2013).
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L. Tong, V. D. Miljkovic, and M. Kall, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Letters10, 268–273 (2010). PMID: .
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Z. Fang, Y.-R. Zhen, O. Neumann, A. Polman, F. J. Garca de Abajo, P. Nordlander, and N. J. Halas, “Evolution of light-induced vapor generation at a liquid-immersed metallic nanoparticle,” Nano Letters13, 1736–1742 (2013). PMID: .
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Nat. Photonics (2)

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

Phys. Rev. B (2)

G. Y. Slepyan, M. V. Shuba, S. A. Maksimenko, and A. Lakhtakia, “Theory of optical scattering by achiral carbon nanotubes and their potential as optical nanoantennas,” Phys. Rev. B73, 195416 (2006).
[Crossref]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B76, 035420 (2007).
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W. Shelton, K. Bonin, and T. Walker, “Nonlinear motion of optically torqued nanorods,” Phys. Rev. E71, 036204 (2005).
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S. Zhang, H. Wei, K. Bao, U. Håkanson, N. J. Halas, P. Nordlander, and H. Xu, “Chiral surface plasmon polaritons on metallic nanowires,” Phys. Rev. Lett.107, 096801 (2011).
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Soft Matter (1)

O. D. Velev and K. H. Bhatt, “On-chip micromanipulation and assembly of colloidal particles by electric fields,” Soft Matter2, 738–750 (2006).
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S. Strogatz, Nonlinear Dynamics and Chaos (Perseus Books Group, 1994).

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1

(a) System geometry: a gold nanowire (75-nm diameter, rounded-cap, 1025-nm length) is aligned about spherical coordinates θ and ϕ, and a y-polarized electromagnetic wave (1 W/cm2) travels in the −z direction. (b) The electric field profile surrounding a nanowire resulting from superposed angular momentum modes m = 0, ±1 with propagation lengths kz = 0, π/2, and π.

Fig. 2
Fig. 2

Characteristics of a nanowire in the xy plane for varying orientation angle ϕ with θ = 90°. (a) Absorption cross-section. (b) Scattering cross-section, ((a), (b) spectra offset for clarity). (c) Net longitudinal force in the direction of light propagation, −Fz, associated with the plasmonic Lorentz force. (d) In-plane torque Tz produced by plasmons fM (solid lines) and the induced electric dipole fE (dashed lines).

Fig. 3
Fig. 3

Forces on the nanowire surface produced by the surface currents (red arrows) and the norm of the surface magnetic field (surface colormap) for oblique geometries that excite the chiral hybrid plasmonic mode, where (a) θ = 15°, ϕ = 90° (b) θ = 30°, ϕ = 75° (c) θ = 60°, ϕ = 60° at λ = 1071nm.

Fig. 4
Fig. 4

Net (volume-integrated) Lorentz forces associated with nanowire surface currents fM in (a) x-direction, (b) y-direction, and (c) z-directions for illuminating wavelengths λ = (i) 441nm (ii) 517nm (iii) 625nm (iv) 790nm (v) 1071nm.

Fig. 5
Fig. 5

far-field radiation patterns for a nanowire aligned (a) in the xz plane with θ =75°, ϕ = 0°, (b) θ = 60°, ϕ = 60° for wavelengths λ = 441nm, 517nm, 625nm, 790nm, and 1071nm.

Fig. 6
Fig. 6

Torque around the (a) x-axis, (b) y-axis, and (c) z-axis at (i) 441nm, (ii) 535nm, (iii) 625nm, (iv) 790nm, and (v) 1071nm.

Fig. 7
Fig. 7

Phase portrait of the torque forces calculated at λ = 1071nm. Arrows in the x and y directions indicate torque that rotates the nanowire in the ϕ and θ directions respectively.

Equations (9)

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

E j ( r ) = 1 k j m { [ m s a j , m F j , m + k m ( ) k j , m ( ) k j b j , m F j , m ] i s ^ [ k j , m ( ) a j , m F j , m + m k m ( ) k j s b j , m F j , m ] ϕ ^ + [ k j , m ( ) ] 2 k j b j , m F j , m z ^ } e i k m ( ) z + i m ϕ
H j ( r ) = i ω μ 0 m { [ k m ( ) k j , m ( ) k j a j , m F j , m + m s b j , m F j , m ] i s ^ [ m k m ( ) k j s a j , m F j , m + k j , m ( ) F j , m ] ϕ ^ + [ k j , m ( ) ] 2 k j a j , m F j , m z ^ } e i k m ( ) z + i m ϕ
f Lorentz = ρ [ E * + ( v × B * ) ] + c . c .
= ε 0 ( E ) E * + J × B * + c . c .
= f E + f M
T M = V r × f M d V ,
T E = V r × f E d V = V P × E * d V + c . c . ,
A θ = sin ϕ T x + cos ϕ T y
A ϕ = T z .

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