Abstract

The interaction between individual plasmonic nanoparticles plays a crucial role in tuning and shaping the surface plasmon resonances of a composite structure. Here, we demonstrate that the detailed character of the coupling between plasmonic structures can be captured by a modified “circuit” model. This approach is generally applicable and, as an example here, is applied to a dolmen-like nanostructure consisting of a vertically placed gold monomer slab and two horizontally placed dimer slabs. By utilizing the full-wave eigenmode expansion method (EEM), we extract the eigenmodes and eigenvalues for these constituting elements and reduce their electromagnetic interaction to the structures’ mode interactions. Using the reaction concept, we further summarize the mode interactions within a “coupling” matrix. When the driving voltage source imposed by the incident light is identified, an equivalent circuit model can be constructed. Within this model, hybridization of the plasmonic modes in the constituting nanostructure elements is discussed. The proposed circuit model allows the reuse of powerful circuit analysis techniques in the context of plasmonic structures. As an example, we derive an equivalent of Thévenin’s theorem in circuit theory for nanostructures. Applying the equivalent Thévenin’s theorem, the well-known Fano resonance is easily explained.

© 2013 Optical Society of America

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2013 (2)

2012 (4)

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

Y. Francescato, V. Giannini, and S. A. Maier, “Plasmonic systems unveiled by Fano resonances,” ACS Nano6(2), 1830–1838 (2012).
[CrossRef] [PubMed]

B. Hourahine and F. Papoff, “The geometrical nature of optical resonances: from a sphere to fused dimer nanoparticles,” Meas. Sci. Technol.23(8), 084002 (2012).
[CrossRef]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

2011 (7)

N. Verellen, P. Van Dorpe, C. Huang, K. Lodewijks, G. A. E. Vandenbosch, L. Lagae, and V. V. Moshchalkov, “Plasmon line shaping using nanocrosses for high sensitivity localized surface plasmon resonance sensing,” Nano Lett.11(2), 391–397 (2011).
[CrossRef] [PubMed]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5), RS0E02 (2011).
[CrossRef]

B. Gallinet and O. J. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B83(23), 235427 (2011).
[CrossRef]

B. Gallinet and O. J. Martin, “Influence of electromagnetic interactions on the line shape of plasmonic Fano resonances,” ACS Nano5(11), 8999–9008 (2011).
[CrossRef] [PubMed]

V. Giannini, Y. Francescato, H. Amrania, C. C. Phillips, and S. A. Maier, “Fano resonances in nanoscale plasmonic systems: a parameter-free modeling approach,” Nano Lett.11(7), 2835–2840 (2011).
[CrossRef] [PubMed]

F. Papoff and B. Hourahine, “Geometrical Mie theory for resonances in nanoparticles of any shape,” Opt. Express19(22), 21432–21444 (2011).
[CrossRef] [PubMed]

2010 (1)

T. J. Davis, D. E. Gómez, and K. C. Vernon, “Simple model for the hybridization of surface plasmon resonances in metallic nanoparticles,” Nano Lett.10(7), 2618–2625 (2010).
[CrossRef] [PubMed]

2009 (2)

T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B79(15), 155423 (2009).
[CrossRef]

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
[CrossRef] [PubMed]

2008 (3)

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett.101(4), 047401 (2008).
[CrossRef] [PubMed]

A. Alù, M. E. Young, and N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B77(14), 144107 (2008).
[CrossRef]

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett.101(4), 043901 (2008).
[CrossRef] [PubMed]

2007 (4)

Y. Schols and G. A. E. Vandenbosch, “Separation of horizontal and vertical dependencies in a surface/volume integral equation approach to model quasi 3-D structures in multilayered media,” IEEE Trans. Antennas Propag.55(4), 1086–1094 (2007).
[CrossRef]

F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett.7(3), 729–732 (2007).
[CrossRef] [PubMed]

T. S. Troutman, J. K. Barton, and M. Romanowski, “Optical coherence tomography with plasmon resonant nanorods of gold,” Opt. Lett.32(11), 1438–1440 (2007).
[CrossRef] [PubMed]

A. Alú, A. Salandrino, and N. Engheta, “Coupling of optical lumped nanocircuit elements and effects of substrates,” Opt. Express15(21), 13865–13876 (2007).
[CrossRef] [PubMed]

2006 (1)

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B110(14), 7238–7248 (2006).
[CrossRef] [PubMed]

2005 (3)

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B72(15), 155412 (2005).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
[CrossRef] [PubMed]

K. Aslan, J. R. Lakowicz, and C. D. Geddes, “Plasmon light scattering in biology and medicine: new sensing approaches, visions and perspectives,” Curr. Opin. Chem. Biol.9(5), 538–544 (2005).
[CrossRef] [PubMed]

2004 (1)

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

2003 (3)

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett.90(5), 057401 (2003).
[CrossRef] [PubMed]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science302(5644), 419–422 (2003).
[CrossRef] [PubMed]

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A.100(23), 13549–13554 (2003).
[CrossRef] [PubMed]

1998 (1)

F. J. Demuynck, G. A. E. Vandenbosch, and A. R. Van de Capelle, “The expansion wave concept–Part I: Efficient calculation of spatial Green's functions in a stratified dielectric medium,” IEEE Trans. Antennas Propag.46, 397–406 (1998).
[CrossRef]

1997 (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science275(5303), 1102–1106 (1997).
[CrossRef] [PubMed]

1992 (1)

G. A. E. Vandenbosch and A. R. Van de Capelle, “Mixed-potential integral expression formulation of the electric field in a stratified dielectric medium-application to the case of a probe current source,” IEEE Trans. Antennas Propag.40(7), 806–817 (1992).
[CrossRef]

1982 (1)

A. G. Ramm, “Mathematical foundations of the singularity and eigenmode expansion methods (SEM and EEM),” J. Math. Anal. Appl.86(2), 562–591 (1982).
[CrossRef]

1976 (1)

C. E. Baum, “Emerging technology for transient and broadband analysis and synthesis of antennas and scatterers,” Proc. IEEE64(11), 1598–1616 (1976).
[CrossRef]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124(6), 1866–1878 (1961).
[CrossRef]

1954 (1)

V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev.94(6), 1483–1491 (1954).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys.330(3), 377–445 (1908).
[CrossRef]

Aizpurua, J.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett.90(5), 057401 (2003).
[CrossRef] [PubMed]

Aktsipetrov, O. A.

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

Alú, A.

Alù, A.

A. Alù, M. E. Young, and N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B77(14), 144107 (2008).
[CrossRef]

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett.101(4), 043901 (2008).
[CrossRef] [PubMed]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
[CrossRef] [PubMed]

Ameloot, M.

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

Amrania, H.

V. Giannini, Y. Francescato, H. Amrania, C. C. Phillips, and S. A. Maier, “Fano resonances in nanoscale plasmonic systems: a parameter-free modeling approach,” Nano Lett.11(7), 2835–2840 (2011).
[CrossRef] [PubMed]

Aslan, K.

K. Aslan, J. R. Lakowicz, and C. D. Geddes, “Plasmon light scattering in biology and medicine: new sensing approaches, visions and perspectives,” Curr. Opin. Chem. Biol.9(5), 538–544 (2005).
[CrossRef] [PubMed]

Bankson, J. A.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A.100(23), 13549–13554 (2003).
[CrossRef] [PubMed]

Barton, J. K.

Baum, C. E.

C. E. Baum, “Emerging technology for transient and broadband analysis and synthesis of antennas and scatterers,” Proc. IEEE64(11), 1598–1616 (1976).
[CrossRef]

Baumberg, J. J.

Biris, C. G.

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

Bryant, G. W.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett.90(5), 057401 (2003).
[CrossRef] [PubMed]

Centini, M.

Chichkov, B. N.

Clercq, B. D.

Davis, T. J.

T. J. Davis, D. E. Gómez, and K. C. Vernon, “Simple model for the hybridization of surface plasmon resonances in metallic nanoparticles,” Nano Lett.10(7), 2618–2625 (2010).
[CrossRef] [PubMed]

T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B79(15), 155423 (2009).
[CrossRef]

De Clercq, B.

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

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J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett.90(5), 057401 (2003).
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S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
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N. Verellen, P. Van Dorpe, C. Huang, K. Lodewijks, G. A. E. Vandenbosch, L. Lagae, and V. V. Moshchalkov, “Plasmon line shaping using nanocrosses for high sensitivity localized surface plasmon resonance sensing,” Nano Lett.11(2), 391–397 (2011).
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K. Aslan, J. R. Lakowicz, and C. D. Geddes, “Plasmon light scattering in biology and medicine: new sensing approaches, visions and perspectives,” Curr. Opin. Chem. Biol.9(5), 538–544 (2005).
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P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B110(14), 7238–7248 (2006).
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A. Lovera, B. Gallinet, P. Nordlander, and O. J. Martin, “Mechanisms of Fano resonances in coupled plasmonic systems,” ACS Nano7(5), 4527–4536 (2013).
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Y. Francescato, V. Giannini, and S. A. Maier, “Plasmonic systems unveiled by Fano resonances,” ACS Nano6(2), 1830–1838 (2012).
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V. Giannini, Y. Francescato, H. Amrania, C. C. Phillips, and S. A. Maier, “Fano resonances in nanoscale plasmonic systems: a parameter-free modeling approach,” Nano Lett.11(7), 2835–2840 (2011).
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N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
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A. Lovera, B. Gallinet, P. Nordlander, and O. J. Martin, “Mechanisms of Fano resonances in coupled plasmonic systems,” ACS Nano7(5), 4527–4536 (2013).
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B. Gallinet and O. J. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B83(23), 235427 (2011).
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B. Gallinet and O. J. Martin, “Influence of electromagnetic interactions on the line shape of plasmonic Fano resonances,” ACS Nano5(11), 8999–9008 (2011).
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V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
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[CrossRef] [PubMed]

N. Verellen, P. Van Dorpe, C. Huang, K. Lodewijks, G. A. E. Vandenbosch, L. Lagae, and V. V. Moshchalkov, “Plasmon line shaping using nanocrosses for high sensitivity localized surface plasmon resonance sensing,” Nano Lett.11(2), 391–397 (2011).
[CrossRef] [PubMed]

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
[CrossRef] [PubMed]

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F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett.7(3), 729–732 (2007).
[CrossRef] [PubMed]

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S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science275(5303), 1102–1106 (1997).
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A. Lovera, B. Gallinet, P. Nordlander, and O. J. Martin, “Mechanisms of Fano resonances in coupled plasmonic systems,” ACS Nano7(5), 4527–4536 (2013).
[CrossRef] [PubMed]

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
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F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett.7(3), 729–732 (2007).
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E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science302(5644), 419–422 (2003).
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Panoiu, N. C.

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
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B. Hourahine and F. Papoff, “The geometrical nature of optical resonances: from a sphere to fused dimer nanoparticles,” Meas. Sci. Technol.23(8), 084002 (2012).
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F. Papoff and B. Hourahine, “Geometrical Mie theory for resonances in nanoparticles of any shape,” Opt. Express19(22), 21432–21444 (2011).
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Phillips, C. C.

V. Giannini, Y. Francescato, H. Amrania, C. C. Phillips, and S. A. Maier, “Fano resonances in nanoscale plasmonic systems: a parameter-free modeling approach,” Nano Lett.11(7), 2835–2840 (2011).
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L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A.100(23), 13549–13554 (2003).
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A. Alú, A. Salandrino, and N. Engheta, “Coupling of optical lumped nanocircuit elements and effects of substrates,” Opt. Express15(21), 13865–13876 (2007).
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N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
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Silhanek, A. V.

V. K. Valev, W. Libaers, U. Zywietz, X. Zheng, M. Centini, N. Pfullmann, L. O. Herrmann, C. Reinhardt, V. Volskiy, A. V. Silhanek, B. N. Chichkov, C. Sibilia, G. A. Vandenbosch, V. V. Moshchalkov, J. J. Baumberg, and T. Verbiest, “Nanostripe length dependence of plasmon-induced material deformations,” Opt. Lett.38(13), 2256–2258 (2013).
[CrossRef] [PubMed]

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

Sobhani, H.

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
[CrossRef] [PubMed]

Sonnefraud, Y.

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
[CrossRef] [PubMed]

Soukoulis, C. M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Stafford, R. J.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A.100(23), 13549–13554 (2003).
[CrossRef] [PubMed]

Sutherland, D. S.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett.90(5), 057401 (2003).
[CrossRef] [PubMed]

Troutman, T. S.

Valev, V. K.

V. K. Valev, W. Libaers, U. Zywietz, X. Zheng, M. Centini, N. Pfullmann, L. O. Herrmann, C. Reinhardt, V. Volskiy, A. V. Silhanek, B. N. Chichkov, C. Sibilia, G. A. Vandenbosch, V. V. Moshchalkov, J. J. Baumberg, and T. Verbiest, “Nanostripe length dependence of plasmon-induced material deformations,” Opt. Lett.38(13), 2256–2258 (2013).
[CrossRef] [PubMed]

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

Van de Capelle, A. R.

F. J. Demuynck, G. A. E. Vandenbosch, and A. R. Van de Capelle, “The expansion wave concept–Part I: Efficient calculation of spatial Green's functions in a stratified dielectric medium,” IEEE Trans. Antennas Propag.46, 397–406 (1998).
[CrossRef]

G. A. E. Vandenbosch and A. R. Van de Capelle, “Mixed-potential integral expression formulation of the electric field in a stratified dielectric medium-application to the case of a probe current source,” IEEE Trans. Antennas Propag.40(7), 806–817 (1992).
[CrossRef]

Van Dorpe, P.

N. Verellen, P. Van Dorpe, C. Huang, K. Lodewijks, G. A. E. Vandenbosch, L. Lagae, and V. V. Moshchalkov, “Plasmon line shaping using nanocrosses for high sensitivity localized surface plasmon resonance sensing,” Nano Lett.11(2), 391–397 (2011).
[CrossRef] [PubMed]

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
[CrossRef] [PubMed]

Vandenbosch, G. A.

Vandenbosch, G. A. E.

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5), RS0E02 (2011).
[CrossRef]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

N. Verellen, P. Van Dorpe, C. Huang, K. Lodewijks, G. A. E. Vandenbosch, L. Lagae, and V. V. Moshchalkov, “Plasmon line shaping using nanocrosses for high sensitivity localized surface plasmon resonance sensing,” Nano Lett.11(2), 391–397 (2011).
[CrossRef] [PubMed]

Y. Schols and G. A. E. Vandenbosch, “Separation of horizontal and vertical dependencies in a surface/volume integral equation approach to model quasi 3-D structures in multilayered media,” IEEE Trans. Antennas Propag.55(4), 1086–1094 (2007).
[CrossRef]

F. J. Demuynck, G. A. E. Vandenbosch, and A. R. Van de Capelle, “The expansion wave concept–Part I: Efficient calculation of spatial Green's functions in a stratified dielectric medium,” IEEE Trans. Antennas Propag.46, 397–406 (1998).
[CrossRef]

G. A. E. Vandenbosch and A. R. Van de Capelle, “Mixed-potential integral expression formulation of the electric field in a stratified dielectric medium-application to the case of a probe current source,” IEEE Trans. Antennas Propag.40(7), 806–817 (1992).
[CrossRef]

Vandendriessche, S.

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

Verbiest, T.

V. K. Valev, W. Libaers, U. Zywietz, X. Zheng, M. Centini, N. Pfullmann, L. O. Herrmann, C. Reinhardt, V. Volskiy, A. V. Silhanek, B. N. Chichkov, C. Sibilia, G. A. Vandenbosch, V. V. Moshchalkov, J. J. Baumberg, and T. Verbiest, “Nanostripe length dependence of plasmon-induced material deformations,” Opt. Lett.38(13), 2256–2258 (2013).
[CrossRef] [PubMed]

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

Verellen, N.

N. Verellen, P. Van Dorpe, C. Huang, K. Lodewijks, G. A. E. Vandenbosch, L. Lagae, and V. V. Moshchalkov, “Plasmon line shaping using nanocrosses for high sensitivity localized surface plasmon resonance sensing,” Nano Lett.11(2), 391–397 (2011).
[CrossRef] [PubMed]

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5), RS0E02 (2011).
[CrossRef]

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett.9(4), 1663–1667 (2009).
[CrossRef] [PubMed]

Vernon, K. C.

T. J. Davis, D. E. Gómez, and K. C. Vernon, “Simple model for the hybridization of surface plasmon resonances in metallic nanoparticles,” Nano Lett.10(7), 2618–2625 (2010).
[CrossRef] [PubMed]

T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B79(15), 155423 (2009).
[CrossRef]

Volski, V.

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5), RS0E02 (2011).
[CrossRef]

Volskiy, V.

V. K. Valev, W. Libaers, U. Zywietz, X. Zheng, M. Centini, N. Pfullmann, L. O. Herrmann, C. Reinhardt, V. Volskiy, A. V. Silhanek, B. N. Chichkov, C. Sibilia, G. A. Vandenbosch, V. V. Moshchalkov, J. J. Baumberg, and T. Verbiest, “Nanostripe length dependence of plasmon-induced material deformations,” Opt. Lett.38(13), 2256–2258 (2013).
[CrossRef] [PubMed]

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

Wang, Y.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett.101(4), 047401 (2008).
[CrossRef] [PubMed]

Warburton, P. A.

Wegener, M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

West, J. L.

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U. S. A.100(23), 13549–13554 (2003).
[CrossRef] [PubMed]

Young, M. E.

A. Alù, M. E. Young, and N. Engheta, “Design of nanofilters for optical nanocircuits,” Phys. Rev. B77(14), 144107 (2008).
[CrossRef]

Zhang, S.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett.101(4), 047401 (2008).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett.101(4), 047401 (2008).
[CrossRef] [PubMed]

Zhang, Z.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B72(15), 155412 (2005).
[CrossRef]

Zheng, X.

V. K. Valev, W. Libaers, U. Zywietz, X. Zheng, M. Centini, N. Pfullmann, L. O. Herrmann, C. Reinhardt, V. Volskiy, A. V. Silhanek, B. N. Chichkov, C. Sibilia, G. A. Vandenbosch, V. V. Moshchalkov, J. J. Baumberg, and T. Verbiest, “Nanostripe length dependence of plasmon-induced material deformations,” Opt. Lett.38(13), 2256–2258 (2013).
[CrossRef] [PubMed]

V. K. Valev, B. D. Clercq, X. Zheng, D. Denkova, E. J. Osley, S. Vandendriessche, A. V. Silhanek, V. Volskiy, P. A. Warburton, G. A. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “The role of chiral local field enhancements below the resolution limit of Second Harmonic Generation microscopy,” Opt. Express20(1), 256–264 (2012).
[CrossRef] [PubMed]

V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
[PubMed]

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
[CrossRef] [PubMed]

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
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V. K. Valev, B. De Clercq, C. G. Biris, X. Zheng, S. Vandendriessche, M. Hojeij, D. Denkova, Y. Jeyaram, N. C. Panoiu, Y. Ekinci, A. V. Silhanek, V. Volskiy, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Distributing the optical near-field for efficient field-enhancements in nanostructures,” Adv. Mater.24(35), OP208–OP215 (2012).
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[CrossRef]

F. J. Demuynck, G. A. E. Vandenbosch, and A. R. Van de Capelle, “The expansion wave concept–Part I: Efficient calculation of spatial Green's functions in a stratified dielectric medium,” IEEE Trans. Antennas Propag.46, 397–406 (1998).
[CrossRef]

Y. Schols and G. A. E. Vandenbosch, “Separation of horizontal and vertical dependencies in a surface/volume integral equation approach to model quasi 3-D structures in multilayered media,” IEEE Trans. Antennas Propag.55(4), 1086–1094 (2007).
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N. Verellen, P. Van Dorpe, C. Huang, K. Lodewijks, G. A. E. Vandenbosch, L. Lagae, and V. V. Moshchalkov, “Plasmon line shaping using nanocrosses for high sensitivity localized surface plasmon resonance sensing,” Nano Lett.11(2), 391–397 (2011).
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G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5), RS0E02 (2011).
[CrossRef]

Science (3)

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004).
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E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science302(5644), 419–422 (2003).
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Small (1)

V. K. Valev, A. V. Silhanek, B. De Clercq, W. Gillijns, Y. Jeyaram, X. Zheng, V. Volskiy, O. A. Aktsipetrov, G. A. E. Vandenbosch, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “U-shaped switches for optical information processing at the nanoscale,” Small7(18), 2573–2576 (2011).
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X. Zheng, N. Verellen, V. K. Valev, V. Volskiy, D. Denkova, L. O. Herrmann, C. Blejean, J. J. Baumberg, A. V. Silhanek, G. A. E. Vandenbosch, and V. V. Moshchalkov, “ Nanoantenna modeled as N-port network: bridging surface plasmon modes and nano-circuits,” submitted (2013).

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Figures (4)

Fig. 1
Fig. 1

Dolmen structure. The topology, the depth profile and SEM image of a gold dolmen nanostructure (L1 = 160 nm, W1 = 110 nm, L2 = 135 nm, W2 = 100 nm, S = 55 nm, G = 20 nm, H = 50 nm). The white bar in the SEM image represents 100 nm.

Fig. 2
Fig. 2

Eigenmodes, eigenvalues and plasmonic response of the composite dolmen structure. The first five eigenmodes are extracted for the dolmen structure and their charge density distribution is shown in the insets of (a) for the L1, L3, and L5 modes and (e) for the L2 and L4 modes (blue and red indicate negative and positive charge, resp.). The real (solid) and the imaginary (dash) parts of their eigenvalues are presented in (a,e) and (b,f), resp. The coupling coefficients are shown in (c) and (g). The measured extinction cross sections of the dolmen structure are shown in (d) and (h) for Y and X polarized incident light, resp.

Fig. 3
Fig. 3

The equivalent circuit model of the dolmen structure excited by Y-polarized incident light. (a) shows the real (solid) and imaginary (dashed) parts of the self-coupling of the monomer’s dipolar mode (cyan, left column), the dolmen’s L1 and L3 modes (red and green, resp., middle column), and the dimer’s quadrupolar mode (pink, right column). The eigenvalues of the L1 and L3 modes directly extracted as in Eq. (17) (the red and green lines in Figs. 2(a) and 2(b)) are shown as grey lines in (a). The charge density distributions of the dimer’s quadrupolar mode, the monomer’s dipolar mode, and their hybridized modes are shown in the insets of (a). (b) shows the real (solid) and imaginary (dashed) parts of the mutual coupling. Knowing the self-coupling and mutual coupling, as well as the driving voltage source, an equivalent circuit for the dolmen structure can be constructed as shown in (c). The Thévenin’s equivalent of the equivalent circuit is also presented in (c). As a result, the current circulating in each loop can be found as in (d). The reference directions of the currents are shown in (c).

Fig. 4
Fig. 4

Illustration of Thévenin’s theorem for coupled nanostructures. In (a1), only the monomer is present and excited by an incoming plane wave. The polarization and propagation direction of the incident plane wave are denoted by the blue and red arrows, resp. In (a2), the equivalent circuit model of (a1) is plotted with the open-circuit voltage explicitly defined. (a3) shows the corresponding the real (solid) and imaginary (dashed) parts of the currents circulating in the monomer’s loop j 1 M(1) ( ω ) and the dimer’s loop j 1 D(1) ( ω ) . In (b1), the monomer structure is excited by the quadrupolar mode in the dimer structure with an amplitude of j 1 D ( ω ) . The equivalent circuit of (b1) is illustrated in (b2). The real (solid) and imaginary (dashed) parts of the currents in the monomer’s loop j 1 M(2) ( ω ) and the dimer’s loop j 1 D(2) ( ω ) due to the excitation of the quadrupolar mode are plotted in (b3). As in Fig. 3, the monomer loop and its associated quantities are always denoted by the cyan color, while the dimer loop is specified by the pink color.

Equations (44)

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E tot ( r,ω )= E inc ( r,ω )+ E scat ( r,ω ), rentire space.
E tot ( r,ω )= J M ( r,ω ) iω( ε( r,ω ) ε 0 ) , r V M .
E scat ( r,ω )=iω μ 0 V M G ¯ MM ( r,r',ω ) J M ( r',ω )dv' iω μ 0 V D G ¯ MD ( r,r',ω ) J D ( r',ω )dv' ,r V M .
E tot ( r,ω )= J D ( r,ω ) iω( ε( r,ω ) ε 0 ) , r V D ,
E scat ( r,ω )=iω μ 0 V D G ¯ DD ( r,r',ω ) J D ( r',ω )dv' iω μ 0 V M G ¯ DM ( r,r',ω ) J M ( r',ω )dv' , r V D .
C MM ( J M ( r',ω ) )+ C MD ( J D ( r',ω ) )= E inc ( r,ω ), r V M ,
C DM ( J M ( r',ω ) )+ C DD ( J D ( r',ω ) )= E inc ( r,ω ), r V D .
C MM ( J M ( r',ω ) )= J M ( r,ω ) iω( ε( r,ω ) ε 0 ) +iω μ 0 V M G ¯ MM ( r,r',ω ) J M ( r',ω )dv' , r V M ,
C DD ( J D ( r',ω ) )= J D ( r,ω ) iω( ε( r,ω ) ε 0 ) +iω μ 0 V D G ¯ DD ( r,r',ω ) J D ( r',ω )dv' , r V D ,
C MD ( J D ( r',ω ) )=iω μ 0 V D G ¯ MD ( r,r',ω ) J D ( r',ω )dv' , r V M ,
C DM ( J M ( r',ω ) )=iω μ 0 V M G ¯ DM ( r,r',ω ) J M ( r',ω )dv' , r V D .
C MM | J n M ( r,ω ) = λ n M ( ω )| J n M ( r,ω ) , r V M ,
C DD | J n D ( r,ω ) = λ n D ( ω )| J n D ( r,ω ) , r V D .
J m M ( r,ω ), J n M ( r,ω ) = V' J m M ( r,ω ) J n M ( r,ω )dv' ={ 1 m=n 0 mn , J m D ( r,ω ), J n D ( r,ω ) = V' J m D ( r,ω ) J n D ( r,ω )dv' ={ 1 m=n 0 mn .
C( J( r,ω ) )= E inc ( r,ω ).
C( J( r,ω ) )= J( r,ω ) iω( ε( r,ω ) ε 0 ) +iω μ 0 V G ¯ ( r,r',ω )J( r,ω )dv' , rV.
C| J n ( r,ω ) = λ n ( ω )| J n ( r,ω ) , rV, J m ( r,ω ), J n ( r,ω ) = V' J m ( r,ω ) J n ( r,ω )dv' ={ 1 m=n 0 mn .
J( r,ω )= n c n ( ω ) J n ( r,ω ) .
c n ( ω )= J n ( r,ω ), E inc ( r,ω ) λ n ( ω ) .
J( r,ω )= j 1 M ( ω ) J 1 M ( r,ω )+ j 1 D ( ω ) J 1 D ( r,ω ).
C MM ( j 1 M ( ω ) J 1 M ( r',ω ) )+ C MD ( j 1 D ( ω ) J 1 D ( r',ω ) )= E inc ( r,ω ), r V M ,
C DM ( j 1 M ( ω ) J 1 M ( r',ω ) )+ C DD ( j 1 D ( ω ) J 1 D ( r',ω ) )= E inc ( r,ω ), r V D .
( c MM ( ω ) c MD ( ω ) c DM ( ω ) c DD ( ω ) )( j 1 M ( ω ) j 1 D ( ω ) )=( e 1 M ( ω ) e 1 D ( ω ) ).
c MM ( ω )= V M J 1 M ( r',ω ) C MM ( J 1 M ( r',ω ) )dv = λ 1 M ( ω ),
c DD ( ω )= V D J 1 D ( r',ω ) C DD ( J 1 D ( r',ω ) )dv = λ 1 D ( ω ).
c MD ( ω )= V D J 1 M ( r',ω ) C MD ( J 1 D ( r',ω ) )dv ,
c DM ( ω )= V D J 1 D ( r',ω ) C DM ( J 1 M ( r',ω ) )dv .
( C MD ) mn =iω μ 0 V g M m ( r )[ V' G ¯ MD ( r,r',ω ) f D n ( r' )dv' ] dv, r' V D ,r V M ,
( C DM ) mn =iω μ 0 V g D m ( r )[ V' G ¯ DM ( r,r',ω ) f M n ( r' )dv' ] dv, r' V M ,r V D .
e 1 M ( ω )= V M J 1 M ( r',ω ) E inc ( r,ω )dv ,
e 1 D ( ω )= V D J 1 D ( r',ω ) E inc ( r,ω )dv .
( c MM ( ω ) c MD ( ω ) c DM ( ω ) c DD ( ω ) )( j 1 M ( ω ) j 1 D ( ω ) )=λ( ω )( j 1 M ( ω ) j 1 D ( ω ) ).
λ( ω )= ( λ 1 M ( ω )+ λ 1 D ( ω ) )± ( λ 1 M ( ω ) λ 1 D ( ω ) ) 2 +4 c MD ( ω ) c DM ( ω ) 2 .
c MM ( ω ) j 1 M ( ω )= e 1 M ( ω ) c MD ( ω ) j 1 D ( ω ).
j 1 D(1) ( ω )=0,
j 1 M(1) ( ω )= e 1 M ( ω ) c MM ( ω ) .
e oc D ( ω )= c DM ( ω ) j 1 M(1) ( ω ).
j 1 D(2) ( ω )= j 1 D ( ω ).
j 1 M(2) ( ω )= c MD ( ω ) c MM ( ω ) j 1 D(2) ( ω ).
c D ( ω )= c DD ( ω )+ c DM ( ω )( c MD ( ω ) c MM ( ω ) ).
j 1 M ( ω )= j 1 M(1) ( ω )+ j 1 M(2) ( ω ).
e oc D ( ω )= c D ( ω ) j 1 D ( ω ).
j 1 D ( ω )= e oc D ( ω ) c D ( ω ) .
j 1 D ( ω )= e oc D ( ω ) c D ( ω ) = c DM ( ω ) e 1 M ( ω ) c DD ( ω ) c MM ( ω ) c MD ( ω ) c DM ( ω ) .

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