Abstract

We proposed a vertically-oriented dimer structure based on focused plasmonic trapping of metallic nanoparticle. Quantitative FDTD calculations and qualitative analysis by simplified dipole approximation revealed that localized surface plasmon coupling dominates in the plasmon hybridization, and the vertically-oriented dimer can effectively make use of the dominant longitudinal component of the surface plasmon virtual probe thus providing much stronger electric field in the gap. Furthermore, for practical application the top nanoparticle of the dimer can be replaced with an atomic force microscope tip which enables the precise control of the gap distance of the dimer. Therefore the proposed vertically-oriented dimer structure provides both the scanning capability and the extremely-high electrical field necessary for the high sensitivity Raman imaging.

© 2016 Optical Society of America

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References

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2016 (1)

2015 (1)

Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
[Crossref] [PubMed]

2014 (2)

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
[Crossref] [PubMed]

2013 (4)

A. Farhang, N. Bigler, and O. J. F. Martin, “Coupling of multiple LSP and SPP resonances: interactions between an elongated nanoparticle and a thin metallic film,” Opt. Lett. 38(22), 4758–4761 (2013).
[Crossref] [PubMed]

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

J. F. Shen, J. Wang, C. J. Zhang, C. J. Min, H. Fang, L. P. Du, S. W. Zhu, and X. C. Yuan, “Dynamic plasmonic tweezers enabled single-particle-film-system gap-mode Surface-enhanced Raman scattering,” Appl. Phys. Lett. 103(19), 191119 (2013).
[Crossref]

L. P. Du, D. Y. Tang, G. H. Yuan, S. B. Wei, and X. C. Yuan, “Emission pattern of surface-enhanced Raman scattering from single nanoparticle-film junction,” Appl. Phys. Lett. 102(8), 081117 (2013).
[Crossref]

2012 (3)

K. Uetsuki, P. Verma, P. Nordlander, and S. Kawata, “Tunable plasmon resonances in a metallic nanotip-film system,” Nanoscale 4(19), 5931–5935 (2012).
[Crossref] [PubMed]

A. Polemi and K. L. Shuford, “Distance dependent quenching effect in nanoparticle dimers,” J. Chem. Phys. 136(18), 184703 (2012).
[Crossref] [PubMed]

A. A. E. Saleh and J. A. Dionne, “Toward Efficient Optical Trapping of Sub-10-nm Particles with Coaxial Plasmonic Apertures,” Nano Lett. 12(11), 5581–5586 (2012).
[Crossref] [PubMed]

2011 (1)

M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
[Crossref]

2010 (5)

K. D. Alexander, K. Skinner, S. Zhang, H. Wei, and R. Lopez, “Tunable SERS in Gold Nanorod Dimers through Strain Control on an Elastomeric Substrate,” Nano Lett. 10(11), 4488–4493 (2010).
[Crossref] [PubMed]

N. A. Hatab, C. H. Hsueh, A. L. Gaddis, S. T. Retterer, J. H. Li, G. Eres, Z. Zhang, and B. Gu, “Free-standing optical gold bowtie nanoantenna with variable gap size for enhanced Raman spectroscopy,” Nano Lett. 10(12), 4952–4955 (2010).
[Crossref] [PubMed]

W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, “Trapping and Sensing 10 nm Metal Nanoparticles Using Plasmonic Dipole Antennas,” Nano Lett. 10(3), 1006–1011 (2010).
[Crossref] [PubMed]

Y. P. Wu and P. Nordlander, “Finite-Difference Time-Domain Modeling of the Optical Properties of Nanoparticles near Dielectric Substrates,” J. Phys. Chem. C 114(16), 7302–7307 (2010).
[Crossref]

S. Albaladejo, R. Gómez-Medina, L. S. Froufe-Pérez, H. Marinchio, R. Carminati, J. F. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18(4), 3556–3567 (2010).
[Crossref] [PubMed]

2009 (8)

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and Transport Velocities for a Particle in a Slot Waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
[Crossref] [PubMed]

K. Wang, E. Schonbrun, and K. B. Crozier, “Propulsion of Gold Nanoparticles with Surface Plasmon Polaritons: Evidence of Enhanced Optical Force from Near-Field Coupling between Gold Particle and Gold Film,” Nano Lett. 9(7), 2623–2629 (2009).
[Crossref] [PubMed]

K. Fujita, S. Ishitobi, K. Hamada, N. I. Smith, A. Taguchi, Y. Inouye, and S. Kawata, “Time-resolved observation of surface-enhanced Raman scattering from gold nanoparticles during transport through a living cell,” J. Biomed. Opt. 14(2), 024038 (2009).
[Crossref] [PubMed]

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
[Crossref]

W. Li, P. H. Camargo, X. Lu, and Y. Xia, “Dimers of silver nanospheres: facile synthesis and their use as hot spots for surface-enhanced Raman scattering,” Nano Lett. 9(1), 485–490 (2009).
[Crossref] [PubMed]

E. G. Bortchagovsky, S. Klein, and U. C. Fischer, “Surface plasmon mediated tip enhanced Raman scattering,” Appl. Phys. Lett. 94(6), 063118 (2009).
[Crossref]

W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam,” Opt. Lett. 34(6), 722–724 (2009).
[Crossref] [PubMed]

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and Transport Velocities for a Particle in a Slot Waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
[Crossref] [PubMed]

2008 (2)

E. Bailo and V. Deckert, “Tip-enhanced Raman spectroscopy of single RNA strands: towards a novel direct-sequencing method,” Angew. Chem. Int. Ed. Engl. 47(9), 1658–1661 (2008).
[Crossref] [PubMed]

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[Crossref] [PubMed]

2007 (3)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

F. Le, N. Z. Lwin, N. J. Halas, and P. Nordlander, “Plasmonic interactions between a metallic nanoshell and a thin metallic film,” Phys. Rev. B 76(16), 165410 (2007).
[Crossref]

N. Papanikolaou, “Optical properties of metallic nanoparticle arrays on a thin metallic film,” Phys. Rev. B 75(23), 235426 (2007).
[Crossref]

2006 (4)

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31(11), 1726–1728 (2006).
[Crossref] [PubMed]

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
[Crossref] [PubMed]

G. Lévêque and O. J. F. Martin, “Optical interactions in a plasmonic particle coupled to a metallic film,” Opt. Express 14(21), 9971–9981 (2006).
[Crossref] [PubMed]

E. C. Le Ru and P. G. Etchegoin, “Rigorous justification of the [E](4) enhancement factor in Surface Enhanced Raman Spectroscopy,” Chem. Phys. Lett. 423(1-3), 63–66 (2006).
[Crossref]

2005 (3)

A. Pinchuk and G. Schatz, “Anisotropic polarizability tensor of a dimer of nanospheres in the vicinity of a plane substrate,” Nanotechnology 16(10), 2209–2217 (2005).
[Crossref] [PubMed]

C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-enhanced Raman scattering from individual au nanoparticles and nanoparticle dimer substrates,” Nano Lett. 5(8), 1569–1574 (2005).
[Crossref] [PubMed]

C. Oubre and P. Nordlander, “Finite-difference time-domain studies of the optical properties of nanoshell dimers,” J. Phys. Chem. B 109(20), 10042–10051 (2005).
[Crossref] [PubMed]

2004 (3)

P. Nordlander and E. Prodan, “Plasmon Hybridization in Nanoparticles near Metallic Surfaces,” Nano Lett. 4(11), 2209–2213 (2004).
[Crossref]

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120(1), 357–366 (2004).
[Crossref] [PubMed]

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

2003 (1)

V. V. Gozhenko, L. G. Grechko, and K. W. Whites, “Electrodynamics of spatial clusters of spheres: Substrate effects,” Phys. Rev. B 68(12), 125422 (2003).
[Crossref]

1999 (1)

L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402(6763), 785–787 (1999).
[Crossref]

1998 (1)

E. L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys., A Mater. Sci. Process. 66(7), S75–S78 (1998).
[Crossref]

1997 (1)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Albaladejo, S.

Alexander, K. D.

K. D. Alexander, K. Skinner, S. Zhang, H. Wei, and R. Lopez, “Tunable SERS in Gold Nanorod Dimers through Strain Control on an Elastomeric Substrate,” Nano Lett. 10(11), 4488–4493 (2010).
[Crossref] [PubMed]

Armelles, G.

Avlasevich, Y.

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
[Crossref]

Badenes, G.

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
[Crossref] [PubMed]

Bailo, E.

E. Bailo and V. Deckert, “Tip-enhanced Raman spectroscopy of single RNA strands: towards a novel direct-sequencing method,” Angew. Chem. Int. Ed. Engl. 47(9), 1658–1661 (2008).
[Crossref] [PubMed]

Bigler, N.

Block, S. M.

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

Bortchagovsky, E. G.

E. G. Bortchagovsky, S. Klein, and U. C. Fischer, “Surface plasmon mediated tip enhanced Raman scattering,” Appl. Phys. Lett. 94(6), 063118 (2009).
[Crossref]

Camargo, P. H.

W. Li, P. H. Camargo, X. Lu, and Y. Xia, “Dimers of silver nanospheres: facile synthesis and their use as hot spots for surface-enhanced Raman scattering,” Nano Lett. 9(1), 485–490 (2009).
[Crossref] [PubMed]

Carminati, R.

Chen, W.

Crozier, K. B.

K. Wang, E. Schonbrun, and K. B. Crozier, “Propulsion of Gold Nanoparticles with Surface Plasmon Polaritons: Evidence of Enhanced Optical Force from Near-Field Coupling between Gold Particle and Gold Film,” Nano Lett. 9(7), 2623–2629 (2009).
[Crossref] [PubMed]

Dasari, R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Deckert, V.

E. Bailo and V. Deckert, “Tip-enhanced Raman spectroscopy of single RNA strands: towards a novel direct-sequencing method,” Angew. Chem. Int. Ed. Engl. 47(9), 1658–1661 (2008).
[Crossref] [PubMed]

Dionne, J. A.

A. A. E. Saleh and J. A. Dionne, “Toward Efficient Optical Trapping of Sub-10-nm Particles with Coaxial Plasmonic Apertures,” Nano Lett. 12(11), 5581–5586 (2012).
[Crossref] [PubMed]

Du, L.

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Du, L. P.

L. P. Du, D. Y. Tang, G. H. Yuan, S. B. Wei, and X. C. Yuan, “Emission pattern of surface-enhanced Raman scattering from single nanoparticle-film junction,” Appl. Phys. Lett. 102(8), 081117 (2013).
[Crossref]

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A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and Transport Velocities for a Particle in a Slot Waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
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F. Le, N. Z. Lwin, N. J. Halas, and P. Nordlander, “Plasmonic interactions between a metallic nanoshell and a thin metallic film,” Phys. Rev. B 76(16), 165410 (2007).
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E. L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys., A Mater. Sci. Process. 66(7), S75–S78 (1998).
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N. A. Hatab, C. H. Hsueh, A. L. Gaddis, S. T. Retterer, J. H. Li, G. Eres, Z. Zhang, and B. Gu, “Free-standing optical gold bowtie nanoantenna with variable gap size for enhanced Raman spectroscopy,” Nano Lett. 10(12), 4952–4955 (2010).
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W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, “Trapping and Sensing 10 nm Metal Nanoparticles Using Plasmonic Dipole Antennas,” Nano Lett. 10(3), 1006–1011 (2010).
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H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
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C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-enhanced Raman scattering from individual au nanoparticles and nanoparticle dimer substrates,” Nano Lett. 5(8), 1569–1574 (2005).
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K. Fujita, S. Ishitobi, K. Hamada, N. I. Smith, A. Taguchi, Y. Inouye, and S. Kawata, “Time-resolved observation of surface-enhanced Raman scattering from gold nanoparticles during transport through a living cell,” J. Biomed. Opt. 14(2), 024038 (2009).
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K. Fujita, S. Ishitobi, K. Hamada, N. I. Smith, A. Taguchi, Y. Inouye, and S. Kawata, “Time-resolved observation of surface-enhanced Raman scattering from gold nanoparticles during transport through a living cell,” J. Biomed. Opt. 14(2), 024038 (2009).
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K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
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C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-enhanced Raman scattering from individual au nanoparticles and nanoparticle dimer substrates,” Nano Lett. 5(8), 1569–1574 (2005).
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M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
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K. Uetsuki, P. Verma, P. Nordlander, and S. Kawata, “Tunable plasmon resonances in a metallic nanotip-film system,” Nanoscale 4(19), 5931–5935 (2012).
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K. Fujita, S. Ishitobi, K. Hamada, N. I. Smith, A. Taguchi, Y. Inouye, and S. Kawata, “Time-resolved observation of surface-enhanced Raman scattering from gold nanoparticles during transport through a living cell,” J. Biomed. Opt. 14(2), 024038 (2009).
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A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
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E. G. Bortchagovsky, S. Klein, and U. C. Fischer, “Surface plasmon mediated tip enhanced Raman scattering,” Appl. Phys. Lett. 94(6), 063118 (2009).
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K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
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C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-enhanced Raman scattering from individual au nanoparticles and nanoparticle dimer substrates,” Nano Lett. 5(8), 1569–1574 (2005).
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F. Le, N. Z. Lwin, N. J. Halas, and P. Nordlander, “Plasmonic interactions between a metallic nanoshell and a thin metallic film,” Phys. Rev. B 76(16), 165410 (2007).
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E. C. Le Ru and P. G. Etchegoin, “Rigorous justification of the [E](4) enhancement factor in Surface Enhanced Raman Spectroscopy,” Chem. Phys. Lett. 423(1-3), 63–66 (2006).
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C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
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A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and Transport Velocities for a Particle in a Slot Waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
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A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and Transport Velocities for a Particle in a Slot Waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
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Li, J. H.

N. A. Hatab, C. H. Hsueh, A. L. Gaddis, S. T. Retterer, J. H. Li, G. Eres, Z. Zhang, and B. Gu, “Free-standing optical gold bowtie nanoantenna with variable gap size for enhanced Raman spectroscopy,” Nano Lett. 10(12), 4952–4955 (2010).
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H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
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H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
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F. Le, N. Z. Lwin, N. J. Halas, and P. Nordlander, “Plasmonic interactions between a metallic nanoshell and a thin metallic film,” Phys. Rev. B 76(16), 165410 (2007).
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Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
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Martin, O. J. F.

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L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402(6763), 785–787 (1999).
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Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
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C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
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J. F. Shen, J. Wang, C. J. Zhang, C. J. Min, H. Fang, L. P. Du, S. W. Zhu, and X. C. Yuan, “Dynamic plasmonic tweezers enabled single-particle-film-system gap-mode Surface-enhanced Raman scattering,” Appl. Phys. Lett. 103(19), 191119 (2013).
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A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
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A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
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K. Uetsuki, P. Verma, P. Nordlander, and S. Kawata, “Tunable plasmon resonances in a metallic nanotip-film system,” Nanoscale 4(19), 5931–5935 (2012).
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P. Nordlander and E. Prodan, “Plasmon Hybridization in Nanoparticles near Metallic Surfaces,” Nano Lett. 4(11), 2209–2213 (2004).
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C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-enhanced Raman scattering from individual au nanoparticles and nanoparticle dimer substrates,” Nano Lett. 5(8), 1569–1574 (2005).
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M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
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P. Nordlander and E. Prodan, “Plasmon Hybridization in Nanoparticles near Metallic Surfaces,” Nano Lett. 4(11), 2209–2213 (2004).
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M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
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M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
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N. A. Hatab, C. H. Hsueh, A. L. Gaddis, S. T. Retterer, J. H. Li, G. Eres, Z. Zhang, and B. Gu, “Free-standing optical gold bowtie nanoantenna with variable gap size for enhanced Raman spectroscopy,” Nano Lett. 10(12), 4952–4955 (2010).
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M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
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M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
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Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
[Crossref] [PubMed]

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Shen, J. F.

J. F. Shen, J. Wang, C. J. Zhang, C. J. Min, H. Fang, L. P. Du, S. W. Zhu, and X. C. Yuan, “Dynamic plasmonic tweezers enabled single-particle-film-system gap-mode Surface-enhanced Raman scattering,” Appl. Phys. Lett. 103(19), 191119 (2013).
[Crossref]

Shen, Z.

Z. Shen and L. Su, “Plasmonic trapping and tuning of a gold nanoparticle dimer,” Opt. Express 24(5), 4801–4811 (2016).
[Crossref]

Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
[Crossref] [PubMed]

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Shi, W.

Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
[Crossref] [PubMed]

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

Shuford, K. L.

A. Polemi and K. L. Shuford, “Distance dependent quenching effect in nanoparticle dimers,” J. Chem. Phys. 136(18), 184703 (2012).
[Crossref] [PubMed]

Skinner, K.

K. D. Alexander, K. Skinner, S. Zhang, H. Wei, and R. Lopez, “Tunable SERS in Gold Nanorod Dimers through Strain Control on an Elastomeric Substrate,” Nano Lett. 10(11), 4488–4493 (2010).
[Crossref] [PubMed]

Smith, N. I.

K. Fujita, S. Ishitobi, K. Hamada, N. I. Smith, A. Taguchi, Y. Inouye, and S. Kawata, “Time-resolved observation of surface-enhanced Raman scattering from gold nanoparticles during transport through a living cell,” J. Biomed. Opt. 14(2), 024038 (2009).
[Crossref] [PubMed]

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E. L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys., A Mater. Sci. Process. 66(7), S75–S78 (1998).
[Crossref]

Su, L.

Sun, M.

H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
[Crossref] [PubMed]

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K. Fujita, S. Ishitobi, K. Hamada, N. I. Smith, A. Taguchi, Y. Inouye, and S. Kawata, “Time-resolved observation of surface-enhanced Raman scattering from gold nanoparticles during transport through a living cell,” J. Biomed. Opt. 14(2), 024038 (2009).
[Crossref] [PubMed]

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C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-enhanced Raman scattering from individual au nanoparticles and nanoparticle dimer substrates,” Nano Lett. 5(8), 1569–1574 (2005).
[Crossref] [PubMed]

Tang, D. Y.

L. P. Du, D. Y. Tang, G. H. Yuan, S. B. Wei, and X. C. Yuan, “Emission pattern of surface-enhanced Raman scattering from single nanoparticle-film junction,” Appl. Phys. Lett. 102(8), 081117 (2013).
[Crossref]

Torrado, J. F.

Uetsuki, K.

K. Uetsuki, P. Verma, P. Nordlander, and S. Kawata, “Tunable plasmon resonances in a metallic nanotip-film system,” Nanoscale 4(19), 5931–5935 (2012).
[Crossref] [PubMed]

Urbach, H. P.

Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
[Crossref] [PubMed]

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

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K. Uetsuki, P. Verma, P. Nordlander, and S. Kawata, “Tunable plasmon resonances in a metallic nanotip-film system,” Nanoscale 4(19), 5931–5935 (2012).
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M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[Crossref] [PubMed]

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
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H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
[Crossref] [PubMed]

Wang, J.

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

J. F. Shen, J. Wang, C. J. Zhang, C. J. Min, H. Fang, L. P. Du, S. W. Zhu, and X. C. Yuan, “Dynamic plasmonic tweezers enabled single-particle-film-system gap-mode Surface-enhanced Raman scattering,” Appl. Phys. Lett. 103(19), 191119 (2013).
[Crossref]

Wang, K.

K. Wang, E. Schonbrun, and K. B. Crozier, “Propulsion of Gold Nanoparticles with Surface Plasmon Polaritons: Evidence of Enhanced Optical Force from Near-Field Coupling between Gold Particle and Gold Film,” Nano Lett. 9(7), 2623–2629 (2009).
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Wang, S.

H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
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Wang, Y.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
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K. D. Alexander, K. Skinner, S. Zhang, H. Wei, and R. Lopez, “Tunable SERS in Gold Nanorod Dimers through Strain Control on an Elastomeric Substrate,” Nano Lett. 10(11), 4488–4493 (2010).
[Crossref] [PubMed]

Wei, S. B.

L. P. Du, D. Y. Tang, G. H. Yuan, S. B. Wei, and X. C. Yuan, “Emission pattern of surface-enhanced Raman scattering from single nanoparticle-film junction,” Appl. Phys. Lett. 102(8), 081117 (2013).
[Crossref]

Wen, W.

H. Wang, T. Liu, Y. Huang, Y. Fang, R. Liu, S. Wang, W. Wen, and M. Sun, “Plasmon-driven surface catalysis in hybridized plasmonic gap modes,” Sci. Rep. 4, 7087 (2014).
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V. V. Gozhenko, L. G. Grechko, and K. W. Whites, “Electrodynamics of spatial clusters of spheres: Substrate effects,” Phys. Rev. B 68(12), 125422 (2003).
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B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
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Y. P. Wu and P. Nordlander, “Finite-Difference Time-Domain Modeling of the Optical Properties of Nanoparticles near Dielectric Substrates,” J. Phys. Chem. C 114(16), 7302–7307 (2010).
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W. Li, P. H. Camargo, X. Lu, and Y. Xia, “Dimers of silver nanospheres: facile synthesis and their use as hot spots for surface-enhanced Raman scattering,” Nano Lett. 9(1), 485–490 (2009).
[Crossref] [PubMed]

Yang, A. H. J.

A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and Transport Velocities for a Particle in a Slot Waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
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A. H. J. Yang, T. Lerdsuchatawanich, and D. Erickson, “Forces and Transport Velocities for a Particle in a Slot Waveguide,” Nano Lett. 9(3), 1182–1188 (2009).
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A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
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Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Yuan, G. H.

L. P. Du, D. Y. Tang, G. H. Yuan, S. B. Wei, and X. C. Yuan, “Emission pattern of surface-enhanced Raman scattering from single nanoparticle-film junction,” Appl. Phys. Lett. 102(8), 081117 (2013).
[Crossref]

Yuan, X.

Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
[Crossref] [PubMed]

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
[Crossref] [PubMed]

Yuan, X. C.

L. P. Du, D. Y. Tang, G. H. Yuan, S. B. Wei, and X. C. Yuan, “Emission pattern of surface-enhanced Raman scattering from single nanoparticle-film junction,” Appl. Phys. Lett. 102(8), 081117 (2013).
[Crossref]

J. F. Shen, J. Wang, C. J. Zhang, C. J. Min, H. Fang, L. P. Du, S. W. Zhu, and X. C. Yuan, “Dynamic plasmonic tweezers enabled single-particle-film-system gap-mode Surface-enhanced Raman scattering,” Appl. Phys. Lett. 103(19), 191119 (2013).
[Crossref]

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M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
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Zhan, Q.

Zhang, C. J.

J. F. Shen, J. Wang, C. J. Zhang, C. J. Min, H. Fang, L. P. Du, S. W. Zhu, and X. C. Yuan, “Dynamic plasmonic tweezers enabled single-particle-film-system gap-mode Surface-enhanced Raman scattering,” Appl. Phys. Lett. 103(19), 191119 (2013).
[Crossref]

Zhang, S.

K. D. Alexander, K. Skinner, S. Zhang, H. Wei, and R. Lopez, “Tunable SERS in Gold Nanorod Dimers through Strain Control on an Elastomeric Substrate,” Nano Lett. 10(11), 4488–4493 (2010).
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Zhang, W.

W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, “Trapping and Sensing 10 nm Metal Nanoparticles Using Plasmonic Dipole Antennas,” Nano Lett. 10(3), 1006–1011 (2010).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
[Crossref] [PubMed]

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
[Crossref] [PubMed]

C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
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N. A. Hatab, C. H. Hsueh, A. L. Gaddis, S. T. Retterer, J. H. Li, G. Eres, Z. Zhang, and B. Gu, “Free-standing optical gold bowtie nanoantenna with variable gap size for enhanced Raman spectroscopy,” Nano Lett. 10(12), 4952–4955 (2010).
[Crossref] [PubMed]

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Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
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Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
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C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4, 2891 (2013).
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J. F. Shen, J. Wang, C. J. Zhang, C. J. Min, H. Fang, L. P. Du, S. W. Zhu, and X. C. Yuan, “Dynamic plasmonic tweezers enabled single-particle-film-system gap-mode Surface-enhanced Raman scattering,” Appl. Phys. Lett. 103(19), 191119 (2013).
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Nano Lett. (11)

Y. Zhang, J. Wang, J. Shen, Z. Man, W. Shi, C. Min, G. Yuan, S. Zhu, H. P. Urbach, and X. Yuan, “Plasmonic Hybridization Induced Trapping and Manipulation of a Single Au Nanowire on a Metallic Surface,” Nano Lett. 14(11), 6430–6436 (2014).
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K. Wang, E. Schonbrun, and K. B. Crozier, “Propulsion of Gold Nanoparticles with Surface Plasmon Polaritons: Evidence of Enhanced Optical Force from Near-Field Coupling between Gold Particle and Gold Film,” Nano Lett. 9(7), 2623–2629 (2009).
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W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, “Trapping and Sensing 10 nm Metal Nanoparticles Using Plasmonic Dipole Antennas,” Nano Lett. 10(3), 1006–1011 (2010).
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K. D. Alexander, K. Skinner, S. Zhang, H. Wei, and R. Lopez, “Tunable SERS in Gold Nanorod Dimers through Strain Control on an Elastomeric Substrate,” Nano Lett. 10(11), 4488–4493 (2010).
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M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[Crossref] [PubMed]

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
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[Crossref] [PubMed]

Y. Zhang, W. Shi, Z. Shen, Z. Man, C. Min, J. Shen, S. Zhu, H. P. Urbach, and X. Yuan, “A Plasmonic Spanner for Metal Particle Manipulation,” Sci. Rep. 5, 15446 (2015).
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J. Jiao, X. Wang, F. Wackenhut, A. Horneber, L. Chen, A. V. Failla, A. J. Meixner, and D. Zhang, “Polarization-dependent SERS at differently oriented single gold nanorods,” Chemphyschem: a European journal of chemical physics and physical chemistry 13, 952–958 (2012).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 (a) The proposed plasmonic trapping system. The incident light is linearly-polarized and is focused to a 45nm-thick silver film through a 1.49- NA objective lens. (b) Top-view of the plasmonic field (showing the z direction electrical-field distribution 10nm above the silver layer. z0 = 1μm and f 0 =1 . The laser wavelength is 532 nm.
Fig. 2
Fig. 2 Geometry of the dipole model: (a) single sphere; (b) vertical spheres dimer.
Fig. 3
Fig. 3 The force analysis for: a single 50nm-diameter gold nanosphere: (a) placed on an Ag film; (b) a sphere is added to (a). The schematic diagrams show the locations of particles in the plasmonic field. d pp and d pf indicate the dimer gap distance and the particle-film gap dimer distance respectively. The green arrows show the forces at the points on the x-z section circle of the sphere. The white arrows indicate the total force as a result by the integral of the sphere surface. The background maps correspond to the electrical field.
Fig. 4
Fig. 4 The x- and z- direction total force distribution at the radial direction for (a) single particle above the Ag film and (b) vertically-oriented dimer above the Ag film, the bottom particle is studied. The parameters are same to Fig. 3. Radius is the particle offset length to the SP-VP center. The force was obtained every 100 nm. Both the particle-film and the particle-particle distances are 10 nm. The incident power is 1 W. (c) The calculated trapping potential well along x direction for single particle (blue curve) and the bottom particle of the dimer (red curve). (d) The z- direction force distribution with different height (particle bottom to surface) the single particle at the center and off the Ag film.
Fig. 5
Fig. 5 The calculated electrical field enhancement in x-z plane, when: (a) no gold particles, (b) a vertical dimer placed on the Ag film and d pf = d pp =10nm , (c) a horizontal dimer placed on the Ag film, the gap distance is same to 10nm and the height of the gap center is same to (b), which is 65nm from the surface.
Fig. 6
Fig. 6 The electrical field enhancement when changing d pf (a) and d pp (b) individually. The enhancement factor is obtained at the point next to the particle which will present the highest field value.
Fig. 7
Fig. 7 Three plasmon hybridization modes: (a) dimer is far away from the film ( d pf >>R> d pp ); (b) the dimer gap distance is large and the one sphere is close to the film ( d pp >>R> d pf ); (c) dimer is close to the film ( d pp ~ d pf <R).
Fig. 8
Fig. 8 The FDTD simulation for the electrical field line distribution of (a) a gold sphere placed above a silver film ( d pf =10nm ) and (b) gold nanosphere dimer ( d pp =10nm ) placed above a gold film ( d pf =10nm ). The color scale is the electrical intensity and it is same color bar for both.

Equations (23)

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E t,z (ρ,φ,z)= if e i k 1 f 2π 0 θ max 0 2π E inc (θ,ϕ)cosϕ e i k 2 ρsinθcos(ϕφ) e i( k z1 k z2 ) z 0 t p e iz k z2 k 2 2 k 1 sin 2 θ cosθ dθdφ
RP= HG 10 n x + HG 10 n y
E inc = E 0 (2 x / ω 0 ) e ( x 2 + y 2 )/ ω 2 =(2 E 0 f/ ω 0 )sinθcosϕ e f 2 sin 2 θ/ ω 0 2
f ω (θ)= e sin 2 θ f 0 2 sin 2 θ max
E t,z (ρ,φ,z)= if e i k 1 f 2π 0 θ max 0 2π (2 E 0 f/ ω 0 ) f ω (θ) cos 2 ϕ e i k 2 ρsinθcos(ϕφ) e i( k z1 k z2 ) z 0 t p e iz k z2 k 2 2 k 1 sin 3 θ cosθ dθdϕ
E t,z (ρ,φ,z)= if e i k 1 f 2π 0 θ max 0 2π (2 E 0 f/ ω 0 ) f ω (θ) sin 2 ϕ e i k 2 ρsinθcos(ϕφ) e i( k z1 k z2 ) z 0 t p e iz k z2 k 2 2 k 1 sin 3 θ cosθ dθdϕ
E t,z (ρ,φ,z)= if e i k 1 f 2π 0 θ max 0 2π (2 E 0 f/ ω 0 ) f ω (θ) e i k 2 ρsinθcos(ϕφ) e i( k z1 k z2 ) z 0 t p e iz k z2 k 2 2 k 1 sin 3 θ cosθ dθdϕ
E t,z (ρ,φ,z)=if e i k 1 f 0 2π (2 E 0 f/ ω 0 ) f ω (θ) J 0 ( k 2 ρsinθ) t p e i( k z1 k z2 ) z 0 e iz k z2 k 2 2 k 1 sin 3 θ cosθ dθ
E LHC =A( r ) e x +i e y 2 = A( r ) 2 e iϕ ( e r +i e ϕ )
E RHC =A( r ) e x i e y 2 = A( r ) 2 e iϕ ( e r i e ϕ )
E Radial = E LHC e iϕ + E RHC e iϕ = 2 A( r ) e r
F = s T · n ds
U( r 0 )= r 0 F (r)dr
α(ω)=4π R 3 ϵ(ω) ϵ m ϵ(ω)+2 ϵ m
p=α ϵ 0 ϵ m E
E= 3(p·r)rp r 2 4π ϵ 0 ϵ m r 5
C abs (ω)=kImα(ω)
C sca (ω)= k 4 6π | α(ω) | 2
E 1 loc = E 0 + p 2 / (2π ϵ 0 ϵ m l 3 )
E 2 loc = E 0 + p 1 / (2π ϵ 0 ϵ m l 3 )
α 1 eff = α 1 1+ α 2 / (2π l 3 ) 1 α 1 α 2 / (2π l 3 ) 2
α 2 eff = α 2 1+ α 1 / (2π l 3 ) 1 α 1 α 2 / (2π l 3 ) 2
α eff = α 1 + α 2 + 2 α 1 α 2 / (2π l 3 ) 1 α 1 α 2 / (2π l 3 ) 2

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