Abstract

The electrostatic polarizability for both vertical and horizontal polarization of two conjoined half-cylinders partly buried in a substrate is derived in an analytical closed-form expression. Using the derived analytical polarizabilities we analyze the localized surface plasmon resonances of three important metal nanowire configurations: (1) a half-cylinder, (2) a half-cylinder on a substrate, and (3) a cylinder partly buried in a substrate. Among other results we show that the substrate plays an important role for spectral location of the plasmon resonances. Our analytical results enable an easy, fast, and exact analysis of many complicated plasmonic nanowire configurations including nanowires on substrates. This is important both for comparison with experimental data, for applications, and as benchmarks for numerical methods.

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  1. M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. Lond. 147, 145–181 (1857).
    [CrossRef]
  2. J. W. Strutt (Lord Rayleigh), “On the scattering of light by small particles,” Phil. Mag. 41, 447–454 (1871).
  3. L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr. 6, 1–62 (1890).
  4. G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 330, 337–445 (1908).
    [CrossRef]
  5. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  6. L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag. 36, 365–376 (1918).
  7. J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
    [CrossRef]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 2000).
  9. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
  10. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
    [CrossRef]
  11. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
    [CrossRef]
  12. W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007).
    [CrossRef]
  13. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007).
    [CrossRef]
  14. P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
    [CrossRef] [PubMed]
  15. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
    [CrossRef] [PubMed]
  16. F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
    [CrossRef]
  17. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
    [CrossRef] [PubMed]
  18. P. C. Waterman, “Surface fields and the T matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999).
    [CrossRef]
  19. A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994).
    [CrossRef]
  20. A. Salandrino, A. Alu, and N. Engheta, “Parallel, series, and intermediate interconnects of optical nanocircuit elements. 1. Analytical solution,” J. Opt. Soc. Am. B 24, 3007–3013 (2007).
    [CrossRef]
  21. M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. 24, 1267–1277 (2010).
    [CrossRef]
  22. H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. 102, 044105 (2007).
    [CrossRef]
  23. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).
  24. H. E. Lockwood, A Book of Curves (Cambridge University Press, 1963).
  25. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  26. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]

2010

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef] [PubMed]

M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. 24, 1267–1277 (2010).
[CrossRef]

2008

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

2007

A. Salandrino, A. Alu, and N. Engheta, “Parallel, series, and intermediate interconnects of optical nanocircuit elements. 1. Analytical solution,” J. Opt. Soc. Am. B 24, 3007–3013 (2007).
[CrossRef]

H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. 102, 044105 (2007).
[CrossRef]

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007).
[CrossRef]

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007).
[CrossRef]

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

2005

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

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef] [PubMed]

2003

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[CrossRef]

2000

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 2000).

1999

1994

A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994).
[CrossRef]

1972

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

1955

J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

1918

L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag. 36, 365–376 (1918).

1908

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 330, 337–445 (1908).
[CrossRef]

1890

L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr. 6, 1–62 (1890).

1871

J. W. Strutt (Lord Rayleigh), “On the scattering of light by small particles,” Phil. Mag. 41, 447–454 (1871).

1857

M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. Lond. 147, 145–181 (1857).
[CrossRef]

Alu, A.

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef] [PubMed]

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

Barnes, W. L.

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Christy, R. W.

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

Eisler, H.-J.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef] [PubMed]

Engheta, N.

Fahr, S.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Faraday, M.

M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. Lond. 147, 145–181 (1857).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).

Graener, H.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Halas, N. J.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007).
[CrossRef]

Hallermann, F.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Hecht, B.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef] [PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Johnson, P. B.

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

Kettunen, H.

H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. 102, 044105 (2007).
[CrossRef]

Lal, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007).
[CrossRef]

Lederer, F.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Link, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007).
[CrossRef]

Lockwood, H. E.

H. E. Lockwood, A Book of Curves (Cambridge University Press, 1963).

Lorenz, L.

L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr. 6, 1–62 (1890).

Maier, S. A.

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

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Martin, O. J. F.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef] [PubMed]

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 330, 337–445 (1908).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).

Muhlschlegel, P.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef] [PubMed]

Murray, W. A.

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007).
[CrossRef]

Novotny, L.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

Paley, A. V.

A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994).
[CrossRef]

Pitkonen, M

M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. 24, 1267–1277 (2010).
[CrossRef]

Plessen, G. V.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Pohl, D. W.

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef] [PubMed]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef] [PubMed]

Radchik, A. V.

A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag. 36, 365–376 (1918).

Rockstuhl, C.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Salandrino, A.

Seifert, G.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Sihvola, A.

H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. 102, 044105 (2007).
[CrossRef]

Smith, G. B.

A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994).
[CrossRef]

Smolyaninov, I. I.

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[CrossRef]

Strutt, J. W.

J. W. Strutt (Lord Rayleigh), “On the scattering of light by small particles,” Phil. Mag. 41, 447–454 (1871).

Vagov, A. V.

A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 2000).

Wackerow, S.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Wallen, H.

H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. 102, 044105 (2007).
[CrossRef]

Waterman, P. C.

Zayats, A. V.

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[CrossRef]

Adv. Mater.

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007).
[CrossRef]

Ann. Phys.

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 330, 337–445 (1908).
[CrossRef]

Can. J. Phys.

J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

J. Appl. Phys.

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

H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. 102, 044105 (2007).
[CrossRef]

A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994).
[CrossRef]

J. Electromagn. Waves Appl.

M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. 24, 1267–1277 (2010).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

K. Dan. Vidensk. Selsk. Skr.

L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr. 6, 1–62 (1890).

Light Scattering by Small Particles

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 2000).

Nat. Mater.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef] [PubMed]

Nat. Photon.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007).
[CrossRef]

Phil. Mag.

L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag. 36, 365–376 (1918).

J. W. Strutt (Lord Rayleigh), “On the scattering of light by small particles,” Phil. Mag. 41, 447–454 (1871).

Phil. Trans. R. Soc. Lond.

M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. Lond. 147, 145–181 (1857).
[CrossRef]

Phys. Rev. B

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

Phys. Rev. Lett.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
[CrossRef] [PubMed]

Phys. Stat. Sol. (a)

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plamon polaritons in solar cells,” Phys. Stat. Sol. (a) 12, 2844–2861 (2008).
[CrossRef]

Science

P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef] [PubMed]

Other

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).

H. E. Lockwood, A Book of Curves (Cambridge University Press, 1963).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the partly buried double half-cylinder.

Fig. 2
Fig. 2

Cross section of the partly buried double half-cylinder geometry. In bipolar coordinates (u and v), the four different regions in the xy plane are given as ɛ1 : {−∞ < u < ∞ and π/2 < v < π}, ɛ2 : {−∞ < u < ∞ and −π < v < −π/2}, ɛ3 : {−∞ < u < ∞ and 0 < v < π/2}, and ɛ4 : {−∞ < u < ∞ and −π/2 < v < 0}.

Fig. 3
Fig. 3

Polarizability as a function of ɛr of a half-cylinder with ɛ1 = ɛr + 0.01i in a surrounding medium described by ɛ2 = ɛ3 = ɛ4 = 1.

Fig. 4
Fig. 4

Polarizability as a function of ɛr of a half-cylinder laying on a substrate. The di-electric constants of the half-cylinder, the substrate, and the superstrate are ɛ1 = ɛr + 0.01i, ɛ2 = ɛ4 = 1.52, and ɛ3 = 1, respectively.

Fig. 5
Fig. 5

Polarizability as a function of ɛr of a partly buried cylinder with ɛ1 = ɛ2 = ɛr +0.01i. The dielectric constants of the sub- and superstrate are ɛ4 = 1.52 and ɛ3 = 1, respectively.

Fig. 6
Fig. 6

Polarizability as a function of wavelength for (a) a silver half-cylinder, (b) a silver half-cylinder on a quartz substrate, and (c) a silver cylinder partly buried in a quartz substrate.

Tables (2)

Tables Icon

Table 1 Plasmon resonance conditions for vertically induced dipole moments.

Tables Icon

Table 2 Plasmon resonance conditions for horizontally induced dipole moments.

Equations (28)

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

ϕ 0 ( x , y ) = y { 1 for y > 0 ɛ 3 ɛ 4 for y < 0 ,
ϕ 0 ( u , v ) = sin v cosh u cos v { 1 for v > 0 ɛ 3 ɛ 4 for v < 0 .
ϕ i ( u , v ) = 0 ϕ ¯ i ( λ , v ) cos ( λ u ) d λ ,
ϕ ¯ i ( λ , v ) = 2 π 0 ϕ i ( u , v ) cos ( λ u ) d u .
α v = 4 π 0 λ s 3 ( λ ) d λ .
α v = 2 π ɛ 1 ɛ 2 ( ɛ 3 + ɛ 4 ) + ( ɛ 1 + ɛ 2 ) ɛ 3 ɛ 4 × [ ɛ 1 ɛ 2 ( ɛ 3 + ɛ 4 ) ( ɛ 1 + ɛ 2 ) ɛ 3 ɛ 4 2 ( ɛ 1 + ɛ 2 ) ɛ 3 ɛ 4 π 2 ln 2 ( A + B A B + 2 i A B ) ] ,
ɛ 1 ɛ 2 ( ɛ 3 + ɛ 4 ) + ( ɛ 1 + ɛ 2 ) ɛ 3 ɛ 4 = 0 .
ϕ 0 ( u , v ) = sinh u cosh u cos v .
ϕ i ( u , v ) = 0 ϕ ¯ i ( λ , v ) sin ( λ u ) d λ .
α h = 4 π 0 λ c 3 ( λ ) d λ .
α h = 2 π ɛ 1 + ɛ 2 + ɛ 3 + ɛ 4 [ ɛ 1 + ɛ 2 ɛ 3 ɛ 4 + 2 ( ɛ 1 + ɛ 2 ) π 2 ln 2 ( A + B A B + 2 i A B ) ] .
ɛ 1 + ɛ 2 + ɛ 3 + ɛ 4 = 0 .
ϕ 3 ff = 2 y 0 λ s 3 ( λ ) d λ + 0 c 3 ( λ ) d λ ,
α v = 4 π 0 λ s 3 ( λ ) d λ .
c 1 ( λ ) γ + s 1 ( λ ) ξ = c 3 ( λ ) γ + s 3 ( λ ) ξ , c 2 ( λ ) γ s 2 ( λ ) ξ = c 4 ( λ ) γ s 4 ( λ ) ξ , c 1 ( λ ) ( γ 2 + ξ 2 ) + s 1 ( λ ) 2 γ ξ = c 2 ( λ ) ( γ 2 + ξ 2 ) s 2 ( λ ) 2 γ ξ , c 3 ( λ ) = c 4 ( λ ) ,
ϕ 0 ( u , v ) v = [ cos v cosh u cos v + sin 2 v ( cosh u cos v ) 2 ] { 1 for v > 0 ɛ 3 ɛ 4 for v < 0 .
v ϕ 0 ( u , π 2 ) = 1 cosh 2 u , v ϕ 0 ( u , π 2 ) = ɛ 3 ɛ 4 cosh 2 u , v ϕ 0 ( u , π ) = 1 1 + cosh u , and v ϕ 0 ( u , π ) = ɛ 3 ɛ 4 ( 1 + cosh u )
v ϕ ¯ 0 ( λ , π 2 ) = λ ξ , v ϕ ¯ 0 ( λ , π 2 ) = ɛ 3 λ ɛ 4 ξ , v ϕ ¯ 0 ( λ , π ) = λ γ ξ , and v ϕ ¯ 0 ( λ , π ) = ɛ 3 λ ɛ 4 γ ξ .
ϕ ¯ i ( λ , v ) v = λ [ c i ( λ ) sinh ( λ v ) + s i ( λ ) cosh ( λ v ) ] .
ɛ 1 [ c 1 ( λ ) ξ + s 1 ( λ ) γ + 1 ξ ] = ɛ 3 [ c 3 ( λ ) ξ + s 3 ( λ ) γ + 1 ξ ] , ɛ 2 [ c 2 ( λ ) ξ + s 2 ( λ ) γ + ɛ 3 ɛ 4 ξ ] = ɛ 4 [ c 4 ( λ ) ξ + s 4 ( λ ) γ + ɛ 3 ɛ 4 ξ ] , ɛ 1 [ c 1 ( λ ) 2 ξ γ + s 1 ( λ ) ( γ 2 + ξ 2 ) + 1 ξ γ ] = ɛ 2 [ c 2 ( λ ) 2 ξ γ + s 2 ( λ ) ( γ 2 + ξ 2 ) + ɛ 3 ɛ 4 ξ ] , ɛ 3 s 3 ( λ ) = ɛ 4 s 4 ( λ ) .
s 3 ( λ ) = 1 γ ξ ( ɛ 1 ɛ 4 ɛ 2 ɛ 3 ) 2 + ( ɛ 1 + ɛ 2 ) [ ɛ 1 ɛ 2 ( ɛ 3 + ɛ 4 ) ɛ 3 ɛ 4 ( ɛ 1 + ɛ 2 + ɛ 3 + ɛ 4 ) + ɛ 2 ɛ 3 2 + ɛ 1 ɛ 4 2 ] γ 2 ( ɛ 1 ɛ 4 ɛ 2 ɛ 3 ) 2 ξ 2 + ( ɛ 1 + ɛ 2 + ɛ 3 + ɛ 4 ) [ ɛ 1 ɛ 2 ( ɛ 3 + ɛ 4 ) + ( ɛ 1 + ɛ 2 ) ɛ 3 ɛ 4 ] γ 2 .
ϕ 3 ff = 2 π y 2 0 λ c 3 ( λ ) d λ ,
α h = 4 π 0 λ c 3 ( λ ) d λ .
v ϕ 0 ( u , v ) = sinh u sin v ( cosh u cos v ) 2 .
v ϕ 0 ( u , π ) = v ϕ 0 ( u , π ) = v ϕ 0 ( u , 0 ) = 0 ,
v ϕ 0 ( u , ± π 2 ) = ± sinh u cosh 2 u v ϕ ¯ 0 ( λ , ± π 2 ) = ± λ γ .
ɛ 1 [ c 1 ( λ ) ξ + s 1 ( λ ) γ + 1 γ ] = ɛ 3 [ c 3 ( λ ) ξ + s 3 ( λ ) γ + 1 γ ] , ɛ 2 [ c 2 ( λ ) ξ + s 2 ( λ ) γ 1 γ ] = ɛ 4 [ c 4 ( λ ) ξ + s 4 ( λ ) γ 1 γ ] , ɛ 1 [ c 1 ( λ ) 2 ξ γ + s 1 ( λ ) ( γ 2 + ξ 2 ) ] = ɛ 2 [ c 2 ( λ ) 2 ξ γ + s 2 ( λ ) ( γ 2 + ξ 2 ) ] , ɛ 2 s 3 ( λ ) = ɛ 4 s 4 ( λ ) .
c 3 ( λ ) = 1 γ ξ ( ɛ 1 ɛ 4 ɛ 2 ɛ 3 ) 2 ξ 2 + ( ɛ 1 + ɛ 2 ɛ 3 ɛ 4 ) [ ɛ 1 ɛ 2 ( ɛ 3 + ɛ 3 ɛ 4 ) + ( ɛ 1 + ɛ 2 ) ɛ 3 ɛ 4 ] γ 2 ( ɛ 1 ɛ 4 ɛ 2 ɛ 3 ) 2 ξ 2 + ( ɛ 1 + ɛ 2 + ɛ 3 + ɛ 4 ) [ ɛ 1 ɛ 2 ( ɛ 3 + ɛ 3 ɛ 4 ) + ( ɛ 1 + ɛ 2 ) ɛ 3 ɛ 4 ] γ 2 .

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