Abstract

We investigate the coupling of gap plasmons in various configurations of neighboring metallic nanowires. Starting with the basic element defining a gap plasmon, consisting of two neighboring silver wires, we study the energy splitting and symmetry properties of hybridized plasmons resulting from the interaction of two wire pairs. The system is shown to display non-avoided crossings of hybridized modes, and it evolves at short distances towards a degenerate system consisting of four wires arranged in a square, where two new gap plasmons emerge from redshifted higher-energy modes. The gap modes of three neighboring wires are also described in a continuous transition from a coplanar configuration to an equilateral triangle arrangement. The interaction between wire pairs is shown to be weak enough to prevent efficient transfer of plasmon signal from a pair to the other one, which is beneficial to avoid crosstalking, but not to produce waveguide couplers. The coupling is significantly increased by placing a wire of rectangular cross section in between the wire pairs, thus allowing us to achieve large plasmon-signal transfers within propagation distances below the attenuation length. Our results can find application in the design of signal-processing devices based upon gap plasmons.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2009 (1)

A. Manjavacas and F. J. García de Abajo, "Robust plasmon waveguides in strongly interacting nanowire arrays," Nano Lett. 9, 1285-1289 (2009).
[CrossRef]

2008 (4)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, "A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation," Nat. Photon. 2, 496-500 (2008).
[CrossRef]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Mart’?n-Moreno, and F. J. García-Vidal, "Guiding and focusing of electromagnetic fields with wedge plasmon polaritons," Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

F. J. Garc’?a de Abajo and M. Kociak, "Probing the photonic local density of states with electron energy loss spectroscopy," Phys. Rev. Lett. 100, 106804 (2008).
[CrossRef] [PubMed]

G. Veronis and S. Fan, "Crosstalk between three-dimensional plasmonic slot waveguides," Opt. Express 16, 2129-2140 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

2005 (1)

2003 (2)

F. J. Garc’?a de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

2002 (1)

F. J. García de Abajo and A. Howie, "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Phys. Rev. B 65, 115418 (2002).
[CrossRef]

2001 (1)

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures," Phys. Rev. B 63, 125417 (2001).
[CrossRef]

2000 (1)

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

1999 (1)

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

1975 (2)

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - II: Theory," IEEE Trans. Microwave Theory Tech. 23, 429-433 (1975).
[CrossRef]

1972 (1)

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

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Aussenegg, F. R.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Berini, P.

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures," Phys. Rev. B 63, 125417 (2001).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

Bourillot, E.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Bozhevolnyi, S. I.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Mart’?n-Moreno, and F. J. García-Vidal, "Guiding and focusing of electromagnetic fields with wedge plasmon polaritons," Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Christy, R. W.

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

Conway, J. A.

Dereux, A.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Echenique, P. M.

F. J. Garc’?a de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

Fan, S.

Garc’ia de Abajo, F. J.

F. J. Garc’?a de Abajo and M. Kociak, "Probing the photonic local density of states with electron energy loss spectroscopy," Phys. Rev. Lett. 100, 106804 (2008).
[CrossRef] [PubMed]

F. J. Garc’?a de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

García de Abajo, F. J.

A. Manjavacas and F. J. García de Abajo, "Robust plasmon waveguides in strongly interacting nanowire arrays," Nano Lett. 9, 1285-1289 (2009).
[CrossRef]

F. J. García de Abajo and A. Howie, "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Phys. Rev. B 65, 115418 (2002).
[CrossRef]

García-Vidal, F. J.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Mart’?n-Moreno, and F. J. García-Vidal, "Guiding and focusing of electromagnetic fields with wedge plasmon polaritons," Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, "A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation," Nat. Photon. 2, 496-500 (2008).
[CrossRef]

Girard, C.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Gotschy, W.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Goudonnet, J. P.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Harel, E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Howie, A.

F. J. García de Abajo and A. Howie, "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Johnson, P. B.

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

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Kociak, M.

F. J. Garc’?a de Abajo and M. Kociak, "Probing the photonic local density of states with electron energy loss spectroscopy," Phys. Rev. Lett. 100, 106804 (2008).
[CrossRef] [PubMed]

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Krenn, J. R.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Lacroute, Y.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Laluet, J. Y.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Leitner, A.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Manjavacas, A.

A. Manjavacas and F. J. García de Abajo, "Robust plasmon waveguides in strongly interacting nanowire arrays," Nano Lett. 9, 1285-1289 (2009).
[CrossRef]

Mart’in-Moreno, L.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Mart’?n-Moreno, and F. J. García-Vidal, "Guiding and focusing of electromagnetic fields with wedge plasmon polaritons," Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

McIsaac, P. R.

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - II: Theory," IEEE Trans. Microwave Theory Tech. 23, 429-433 (1975).
[CrossRef]

Meltzer, S.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Moreno, E.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Mart’?n-Moreno, and F. J. García-Vidal, "Guiding and focusing of electromagnetic fields with wedge plasmon polaritons," Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, "A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation," Nat. Photon. 2, 496-500 (2008).
[CrossRef]

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, "A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation," Nat. Photon. 2, 496-500 (2008).
[CrossRef]

Requicha, A. A. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Rivacoba, A.

F. J. Garc’?a de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

Rodrigo, S. G.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Mart’?n-Moreno, and F. J. García-Vidal, "Guiding and focusing of electromagnetic fields with wedge plasmon polaritons," Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef] [PubMed]

Sahni, S.

Schider, G.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, "A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation," Nat. Photon. 2, 496-500 (2008).
[CrossRef]

Szkopek, T.

Veronis, G.

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Weeber, J. C.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider,W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999).
[CrossRef]

Zabala, N.

F. J. Garc’?a de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

Zhang, X.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, "A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation," Nat. Photon. 2, 496-500 (2008).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - I: Summary of results," IEEE Trans. Microwave Theory Tech. 23, 421-429 (1975).
[CrossRef]

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - II: Theory," IEEE Trans. Microwave Theory Tech. 23, 429-433 (1975).
[CrossRef]

Nano Lett. (1)

A. Manjavacas and F. J. García de Abajo, "Robust plasmon waveguides in strongly interacting nanowire arrays," Nano Lett. 9, 1285-1289 (2009).
[CrossRef]

Nat. Mater. (1)

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003).
[CrossRef] [PubMed]

Nat. Photon. (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, "A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation," Nat. Photon. 2, 496-500 (2008).
[CrossRef]

Nature (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (5)

F. J. Garc’?a de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Gap plasmon modes for one and two wire-pairs. (a)-(b) Schematic view of the geometry. (c)-(d) Photonic density of states (DOS) as a function of energy and momentum parallel to the wires for one and two wire-pairs. The inter-wire gap distance is d=10nm in both cases and the distance b between the two wire-pairs in (d) is 10 nm. The insets show the spatial distribution of the local density of states (LDOS) for the lowest-energy gap mode at a free-space light wavelength of 1550 nm. Brighter regions correspond to higher DOS and LDOS values.

Fig. 2.
Fig. 2.

(a) Evolution of the modes in two wire-pairs as a function of the distance between them for fixed q=10µm-1. The inset shows a zoom of the low-energy mode-crossing region. (b) Orientation of the electric field in the gap regions for two limiting geometries: (1–3) square symmetry (b=d=10nm) and (4,5) rectangular symmetry (bd=10nm). The numerical labels correspond to the modes signalled by arrows in (a). The symmetry increase in the square configuration leads to mode degeneracy, as shown in (3).

Fig. 3.
Fig. 3.

Spatial dependence of field produced by a line dipole. (a) Line of dipoles e iqz ŷ distributed along the z axis. (b) y component of the electric field produced by the dipoles of (a) along the x axis for a free-space wavelength λ=1605nm and q=10µm-1, with the dipoles embedded in silica (ε=2.08).

Fig. 4.
Fig. 4.

Gap modes in a wire trimer. (a)-(c) Photonic density of states (DOS) as a function of energy and momentum parallel to the wires for three different trimer-angles. The insets show the spatial distribution of the local density of states (LDOS) for the lowest-energy gap mode at a free-space light wavelength of 1550 nm. (d) Evolution of the gap modes with the trimer angle for fixed q=10µm-1. The insets show the orientation of the electric field in the gap regions for equilateral and coplanar trimers. Two of the modes are degenerate in the 3-fold symmetric case (angle=60°). The inter-wire gap distance is d=10nm in all cases.

Fig. 5.
Fig. 5.

Waveguide coupler consisting of an intermediate wire of rectangular cross section placed between the two gap-plasmon waveguides. (a) Schematic view of the geometry. (b) Fraction of power transferred between waveguides as a function of coupler length along the wires (solid curve). For comparison, we show the power transfer in two wire-pairs without intermediate coupler (dashed curve) for a distance b=20nm. The transversal dimensions of the coupler are 140nm×20nm. The four circular wires are arranged as in Fig. 1(b) with d=10nm and b=20nm.

Fig. 6.
Fig. 6.

Electric field for the modes of the symmetric 4-wire structure of Fig. 1(b) with b=d=10nm. The light wavelength is λ=1000nm in all cases. Mode labels 1–3 are in correspondence with those of Fig. 2(b).

Fig. 7.
Fig. 7.

Electric field for the modes of various trimer structures. The gap distance is d= 10nm and the light wavelength is λ=1000nm in all cases.

Equations (14)

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E(Rx̂)=2[k2K0(ΓR)ΓεRK1(ΓR)]eiqzŷ,
es=12[e1+e2],ea=12[e1e2].
E(r)=12[eiqszes(x,y)+eiqazea(x,y)]=12(eiqsz+eiqaz)e1(x,y)+12(eiqszeiqaz)e2(x,y).
T(L)=14eiqsLeiqaL2.
T(L)=sin2[(qsqa)L/2],
××Ej(r)ωj2c2𝓔(r)Ej(r)=0 .
××G(r,r,ω)ω2c2𝓔(r)G(r,r,ω)=Ic2δ(rr),
E(r,ω)=4πiωdrG(r,r,ω)j(r,ω).
G(r,r,ω)=ΣjEj(r)Ej(r)ω2ωj2+,γ0+ .
LDOS(r,ω)=j|Ej(r)n^|2δ(ωωj).
LDOS(r,ω)=2ωπIm{n^G(r,r,ω)n^},
LDOS(r0,ω)=ω2𝓔1/23π2c3+12π2ωIm{n̂Eref(r0,ω)},
E(r,ω)A(ω)Ej(rb),
A(ω)=2πiε1ωωj+iγdrEj(r)j(r,ω)

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