Abstract

Chain plasmonic waveguides are formed by linear arrays of metallic grains embedded in a dielectric matrix. Plasmonic structures of this kind have potential applications to subwavelength guiding, subwavelength imaging and SERS technology. We present qualitative analysis and numerical results for bound plasmonic modes propagating along the chain of closely spaced silver cylinders of subwavelength diameter. The dispersion relation and electromagnetic field structure of the modes are calculated by the cylindrical harmonic expansion method. We demonstrate that it is possible to match simultaneously both the frequency and wave number of the fundamental transverse mode and the first longitudinal mode in optical range. The application of dense chain of cylinders for optical switching between guided modes is discussed.

© 2014 Optical Society of America

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References

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  1. D.K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot. 4, 83–91 (2010).
    [Crossref]
  2. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998).
    [Crossref]
  3. Stefan A. Maier, Pieter G. Kik, and Harry A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
    [Crossref]
  4. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
    [Crossref] [PubMed]
  5. Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55(11), 3070–3077 (2007).
    [Crossref]
  6. M. Conforti and M. Guasoni, “Dispersive properties of linear chains of lossy metal nanoparticles,” J. Opt. Soc. Am. B 27(8), 1576–1582 (2010).
    [Crossref]
  7. S. M. Raeis Zadeh Bajestani, M. Shahabadi, and N. Talebi, “Analysis of plasmon propagation along a chain of metal nanospheres using the generalized multipole technique,” J. Opt. Soc. Am. B 28(4), 937–943 (2011).
    [Crossref]
  8. N. A. Giannakis, J. E. Inglesfield, A. K. Jastrzebski, and P. R. Young, “Photonic modes of a chain of nanocylinders by the embedding method,” J. Opt. Soc. Am. B 30(6), 1755–1764 (2013).
    [Crossref]
  9. I.L. Rasskazov, S.V. Karpov, and V.A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. 38(22), 4743–4746 (2013).
    [Crossref] [PubMed]
  10. E. Smith and G. Dent, Modern Raman Spectroscopy: A Practical Approach (John Wiley and Sons, 2005).
  11. K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Spectroscopy: a Brief Perspective. In Surface-Enhanced Raman Scattering: Physics and Applications (Springer, 2006).
    [Crossref]
  12. S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Phot. 2, 438–442 (2008).
    [Crossref]
  13. V. Markel and A. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Physical Review B 75, 111 (2007)
    [Crossref]
  14. C. R. Simovski and E. A. Yankovskaya, “Propagation of light along the waveguide of silver nano-cylinders,” Proc. SPIE 5927, Plasmonics: Metallic Nanostructures and Their Optical Properties III, 59271K (2005)
    [Crossref]
  15. B. Rolly, N. Bonod, and B. Stout, “Dispersion relations in metal nanoparticle chains: necessity of the multipole approach,” J. Opt. Soc. Am. B 29, 1012 (2012)
    [Crossref]
  16. W. Zakowicz, “Two coupled dielectric cylindrical waveguides,” J. Opt. Soc. Am. A 14(3), 580–587 (1997).
    [Crossref]
  17. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13(3), 483–493 (1996).
    [Crossref]
  18. R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A 14(7), 1500–1504 (1997).
    [Crossref]
  19. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52,(10) 2603–2611 (2004).
    [Crossref]
  20. S. Belan, S. Vergeles, and P. Vorobev, “Adjustable subwavelength localization in a hybrid plasmonic waveguide,” Opt. Express 21(6), 7427–7438 (2013).
    [Crossref] [PubMed]
  21. V.E. Babicheva, S.S. Vergeles, P.E. Vorobev, and S. Burger, “Localized surface plasmon modes in a system of two interacting metallic cylinders,” J. Opt. Soc. Am. B 29, 1263–1269 (2012).
    [Crossref]
  22. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (NIST National Institute of Standards and Technology & Cambridge University Press, 2010).
  23. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972).
    [Crossref]
  24. E.N. Economou, “Surface Plasmons in Thin Films,” Phys.Rev. 182(2), 539 (1969).
  25. D.J. Griffins, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2d ed., 2005)
  26. V. Twersky, “Elementary function representation of the Schlomilch series,” Arch. Rational Mech. Anal. 8(1), 323–332 (1961).
    [Crossref]
  27. C.M. Linton, “The Greens function for the two-dimensional Helmholtz equation in periodic domains,” Journal of Engineering Mathematics 33(4), 377–402 (1998).
    [Crossref]
  28. D. V. Evans and R. Porter, “Trapping and near-trapping by arrays of cylinders in waves,” Journal of Engineering Mathematics 35(1–2), 149–179 (1999).
    [Crossref]

2013 (3)

2012 (2)

2011 (1)

2010 (2)

M. Conforti and M. Guasoni, “Dispersive properties of linear chains of lossy metal nanoparticles,” J. Opt. Soc. Am. B 27(8), 1576–1582 (2010).
[Crossref]

D.K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot. 4, 83–91 (2010).
[Crossref]

2008 (1)

S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Phot. 2, 438–442 (2008).
[Crossref]

2007 (2)

V. Markel and A. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Physical Review B 75, 111 (2007)
[Crossref]

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55(11), 3070–3077 (2007).
[Crossref]

2004 (1)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52,(10) 2603–2611 (2004).
[Crossref]

2003 (2)

Stefan A. Maier, Pieter G. Kik, and Harry A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

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

1999 (1)

D. V. Evans and R. Porter, “Trapping and near-trapping by arrays of cylinders in waves,” Journal of Engineering Mathematics 35(1–2), 149–179 (1999).
[Crossref]

1998 (2)

C.M. Linton, “The Greens function for the two-dimensional Helmholtz equation in periodic domains,” Journal of Engineering Mathematics 33(4), 377–402 (1998).
[Crossref]

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998).
[Crossref]

1997 (2)

1996 (1)

1972 (1)

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

1969 (1)

E.N. Economou, “Surface Plasmons in Thin Films,” Phys.Rev. 182(2), 539 (1969).

1961 (1)

V. Twersky, “Elementary function representation of the Schlomilch series,” Arch. Rational Mech. Anal. 8(1), 323–332 (1961).
[Crossref]

Atwater, H. A.

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

Atwater, Harry A.

Stefan A. Maier, Pieter G. Kik, and Harry A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

Aussenegg, F. R.

Babicheva, V.E.

Belan, S.

Boisvert, R. F.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (NIST National Institute of Standards and Technology & Cambridge University Press, 2010).

Bonod, N.

Borghi, R.

Bozhevolnyi, S.I.

D.K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot. 4, 83–91 (2010).
[Crossref]

Burger, S.

Christy, R.

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

Clark, C. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (NIST National Institute of Standards and Technology & Cambridge University Press, 2010).

Conforti, M.

Dent, G.

E. Smith and G. Dent, Modern Raman Spectroscopy: A Practical Approach (John Wiley and Sons, 2005).

Economou, E.N.

E.N. Economou, “Surface Plasmons in Thin Films,” Phys.Rev. 182(2), 539 (1969).

Evans, D. V.

D. V. Evans and R. Porter, “Trapping and near-trapping by arrays of cylinders in waves,” Journal of Engineering Mathematics 35(1–2), 149–179 (1999).
[Crossref]

Frezza, F.

Giannakis, N. A.

Gori, F.

Gramotnev, D.K.

D.K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot. 4, 83–91 (2010).
[Crossref]

Griffins, D.J.

D.J. Griffins, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2d ed., 2005)

Guasoni, M.

Hao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55(11), 3070–3077 (2007).
[Crossref]

Harel, E.

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

Inglesfield, J. E.

Jastrzebski, A. K.

Johnson, P.

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

Karpov, S.V.

Kawata, S.

S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Phot. 2, 438–442 (2008).
[Crossref]

Kik, P. G.

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

Kik, Pieter G.

Stefan A. Maier, Pieter G. Kik, and Harry A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

Kneipp, H.

K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Spectroscopy: a Brief Perspective. In Surface-Enhanced Raman Scattering: Physics and Applications (Springer, 2006).
[Crossref]

Kneipp, K.

K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Spectroscopy: a Brief Perspective. In Surface-Enhanced Raman Scattering: Physics and Applications (Springer, 2006).
[Crossref]

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. 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.

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52,(10) 2603–2611 (2004).
[Crossref]

Leitner, A.

Linton, C.M.

C.M. Linton, “The Greens function for the two-dimensional Helmholtz equation in periodic domains,” Journal of Engineering Mathematics 33(4), 377–402 (1998).
[Crossref]

Lozier, D. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (NIST National Institute of Standards and Technology & Cambridge University Press, 2010).

Maier, S. A.

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

Maier, Stefan A.

Stefan A. Maier, Pieter G. Kik, and Harry A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

Markel, V.

V. Markel and A. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Physical Review B 75, 111 (2007)
[Crossref]

Markel, V.A.

Meltzer, S.

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

Moskovits, M.

K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Spectroscopy: a Brief Perspective. In Surface-Enhanced Raman Scattering: Physics and Applications (Springer, 2006).
[Crossref]

Olver, F. W. J.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (NIST National Institute of Standards and Technology & Cambridge University Press, 2010).

Ono, A.

S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Phot. 2, 438–442 (2008).
[Crossref]

Porter, R.

D. V. Evans and R. Porter, “Trapping and near-trapping by arrays of cylinders in waves,” Journal of Engineering Mathematics 35(1–2), 149–179 (1999).
[Crossref]

Quinten, M.

Raeis Zadeh Bajestani, S. M.

Rasskazov, I.L.

Requicha, A. A.

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

Rolly, B.

Santarsiero, M.

Sarychev, A.

V. Markel and A. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Physical Review B 75, 111 (2007)
[Crossref]

Schettini, G.

Shahabadi, M.

Simovski, C. R.

C. R. Simovski and E. A. Yankovskaya, “Propagation of light along the waveguide of silver nano-cylinders,” Proc. SPIE 5927, Plasmonics: Metallic Nanostructures and Their Optical Properties III, 59271K (2005)
[Crossref]

Smith, E.

E. Smith and G. Dent, Modern Raman Spectroscopy: A Practical Approach (John Wiley and Sons, 2005).

Stout, B.

Talebi, N.

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52,(10) 2603–2611 (2004).
[Crossref]

Twersky, V.

V. Twersky, “Elementary function representation of the Schlomilch series,” Arch. Rational Mech. Anal. 8(1), 323–332 (1961).
[Crossref]

Vergeles, S.

Vergeles, S.S.

Verma, P.

S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Phot. 2, 438–442 (2008).
[Crossref]

Vorobev, P.

Vorobev, P.E.

Yankovskaya, E. A.

C. R. Simovski and E. A. Yankovskaya, “Propagation of light along the waveguide of silver nano-cylinders,” Proc. SPIE 5927, Plasmonics: Metallic Nanostructures and Their Optical Properties III, 59271K (2005)
[Crossref]

Yasumoto, K.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52,(10) 2603–2611 (2004).
[Crossref]

Young, P. R.

Zakowicz, W.

Zhao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55(11), 3070–3077 (2007).
[Crossref]

Arch. Rational Mech. Anal. (1)

V. Twersky, “Elementary function representation of the Schlomilch series,” Arch. Rational Mech. Anal. 8(1), 323–332 (1961).
[Crossref]

IEEE Trans. Antennas Propag. (2)

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55(11), 3070–3077 (2007).
[Crossref]

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52,(10) 2603–2611 (2004).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (5)

Journal of Engineering Mathematics (2)

C.M. Linton, “The Greens function for the two-dimensional Helmholtz equation in periodic domains,” Journal of Engineering Mathematics 33(4), 377–402 (1998).
[Crossref]

D. V. Evans and R. Porter, “Trapping and near-trapping by arrays of cylinders in waves,” Journal of Engineering Mathematics 35(1–2), 149–179 (1999).
[Crossref]

Nat. Mater. (1)

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. 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. Phot. (2)

D.K. Gramotnev and S.I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot. 4, 83–91 (2010).
[Crossref]

S. Kawata, A. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nat. Phot. 2, 438–442 (2008).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (2)

Stefan A. Maier, Pieter G. Kik, and Harry A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

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

Phys.Rev. (1)

E.N. Economou, “Surface Plasmons in Thin Films,” Phys.Rev. 182(2), 539 (1969).

Physical Review B (1)

V. Markel and A. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Physical Review B 75, 111 (2007)
[Crossref]

Other (5)

C. R. Simovski and E. A. Yankovskaya, “Propagation of light along the waveguide of silver nano-cylinders,” Proc. SPIE 5927, Plasmonics: Metallic Nanostructures and Their Optical Properties III, 59271K (2005)
[Crossref]

E. Smith and G. Dent, Modern Raman Spectroscopy: A Practical Approach (John Wiley and Sons, 2005).

K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Spectroscopy: a Brief Perspective. In Surface-Enhanced Raman Scattering: Physics and Applications (Springer, 2006).
[Crossref]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (NIST National Institute of Standards and Technology & Cambridge University Press, 2010).

D.J. Griffins, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2d ed., 2005)

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Figures (6)

Fig. 1
Fig. 1 Charge distribution at wave number q = π/L for a) the fundamental mode and b) the first longitudinal mode.
Fig. 2
Fig. 2 a) The periodic grid metal cylinders. b) The continuous metal film of the same thickness.
Fig. 3
Fig. 3 The metal permittivity at resonance frequency for the first two guided modes of the chain waveguide in comparison with the resonant permittivity in system of two cylinders, see Eq. (1). We choose q = π/L to suppress as much as possible the retardation effects. Note, that the surface charge distribution has two nodes for the transverse mode and four nodes for the longitudinal one at q = π/L, see Fig 1. It follows from quasi-static solution [21] that the difference in the charge distribution leads to the longitudinal mode should have indeed higher frequency than the transverse one.
Fig. 4
Fig. 4 The dispersion diagram for a dense periodic chain of silver cylinders embedded in air. The frequency dependence of silver permittivity is taken from experimental data [23]. a) The set of parameters R = 25 nm, h = 5 nm corresponds to touching of two dispersion curves. b) The case of very dense array, R = 25 nm, h = 1 nm. The point of intersection of the dispersion branches corresponds to frequency ω = 2.96 eV (λ = 418 nm) and wave number q = 0.426π/L.
Fig. 5
Fig. 5 Spatial distribution of electromagnetic field for first two guided mode of dense chain of silver cylinders at wave number q = π/L. The radius of the cylinders is R = 25 nm and the gap width is h = 1 nm. (a) and (c) normalised distributions of total electric and magnetic field of transverse mode; (b) ans (d) normalised distributions of total electric and magnetic field of longitudinal mode.
Fig. 6
Fig. 6 a) The dispersion relation of fundamental mode calculated by numerical solving the dispersion equation with different values of truncated index: N = 1 (dipole approximation) and N = 15 (full multipole calculations) b) The frequency of the longitudinal mode at qL/π = 0.4: comparison between the 2D model of infinite cylinders (dashed line), 3D chain of rods (asterisks) and 3D chain of spheroids (dots). The longest semi-axis of spheroids is b. The height of cylinders is 2b. The transversal geometrical parameters was chosen to be R = 25nm and h = 5nm. Thus, whereas our 2D model is good approximation for chain of cylinders at moderate aspect ratios, it is weaker approximation for chain of prolate spheroids. Solution for 3D model was obtained using COMSOL.

Equations (27)

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

ε m ε d ~ 2 R / h + 1 4 R / h + 1 R / h .
H z out = m = + n = + A n m n ( 1 ) ( k r m ) e in φ m ,
H z in = n = + B n m n ( κ r m ) e in φ m ,
C ˜ n A n + ν = + A ν F ˜ n ν = 0 ,
n ( k r m ) e in φ m = ν = + e i π 2 ( n ν ) sign ( m ) n ν ( χ d | m | L ) 𝒥 ν ( χ d r ) e i ν φ ,
H z out = n = + e in φ ( A n 0 n ( k r ) + 𝒥 n ( k r ) m = , m 0 + ν = + e i π 2 ( ν n ) sign ( m ) A ν m ν n ( k | m | L ) ) .
H z out = n = + e in φ ( A n n ( k r ) + 𝒥 n ( k r ) ν = + F n ν A ν ) .
F n ν ( q ) = m = , m 0 + e i m q L e i π 2 ( ν n ) sign ( m ) ν n ( k | m | L ) .
E r = i c ε ω 1 r H z φ , E φ = i c ε ω H z r .
{ A n n ( k R ) + 𝒥 n ( k R ) ν = + F n ν A ν = n ( κ R ) B n , < n < + k 1 [ A n n ( k R ) + 𝒥 n ( k R ) ν = + F n ν A ν ] = κ 1 n ( κ R ) B n .
C n A n + ν = + A ν F n ν = 0 ,
C n = κ n ( k R ) n ( κ R ) + k n ( k R ) n ( κ R ) κ 𝒥 n ( k R ) n ( κ R ) + k 𝒥 n ( k R ) n ( κ R ) ,
S n = m = 1 + n ( k m L ) ( e i m q L + ( 1 ) n e i m q L ) .
C ˜ n A n + ν = + A ν F ˜ n ν = 0 ,
C ˜ n = κ 𝒴 n ( k R ) n ( κ R ) + k 𝒴 n ( k R ) n ( κ R ) κ 𝒥 n ( k R ) n ( κ R ) + k 𝒥 n ( k R ) n ( κ R ) ,
t n = A n + A n , 0 n < +
l n = A n A n , 1 n < +
C ˜ n t n + ν = 0 + ( 1 δ ν 0 2 ) ( F ˜ n ν + F ˜ n , ν ) t ν = 0 ,
C ˜ n l n + ν = 1 + ( F ˜ n ν F ˜ n , ν ) l ν = 0 ,
( C ˜ | C ˜ | 1 + | C ˜ | 1 / 2 F ˜ | C ˜ | 1 / 2 ) A ˜ = 0 ,
det [ C ˜ | C ˜ | 1 + | C ˜ | 1 / 2 F ˜ | C ˜ | 1 / 2 ] = 0 .
H z out ( r , φ ) = i n = + e in φ ( 𝒴 n ( k r ) C ˜ n 𝒥 n ( k r ) ) A n
H z out ( r , φ ) = i n = + e in φ ( 𝒴 n ( k R ) C ˜ n 𝒥 n ( k R ) ) n ( κ r ) n ( κ R ) A n
S 0 = 1 2 i π [ C + log k 2 p ] 2 i γ 0 L 2 i ( k 2 + 2 q 2 ) p 3 L ζ ( 3 ) 2 i L m = , m 0 + ( 1 γ m 1 p | m | k 2 + 2 q 2 2 p 3 | m | 3 ) ,
S 2 n = 2 i e 2 in α 0 γ 0 L 2 i m = 1 + ( e 2 in α m γ m L + e 2 in α m γ m L ( 1 ) n m π ( k 2 m p ) 2 n ) 2 i ( 1 ) n π ( k 2 p ) 2 n ζ ( 2 n + 1 ) + i n π + i π m = 1 n ( 1 ) m 2 2 m ( n + m 1 ) ! ( 2 m ) ! ( n m ) ! ( p k ) 2 m B 2 m ( q p ) ,
S 2 n 1 = 2 i e i ( 2 n 1 ) α 0 γ 0 L + 2 i m = 1 + ( e i ( 2 n 1 ) α m γ m L e i ( 2 n 1 ) α m γ m L + i ( 1 ) n q L n m 2 π 2 ( k 2 m p ) 2 n 1 ) + + 2 ( 1 ) n q L n π 2 ( k 2 p ) 2 n 1 ζ ( 2 n + 1 ) 2 π m = 0 n 1 ( 1 ) m 2 2 m ( n + m 1 ) ! ( 2 m + 1 ) ! ( n m 1 ) ! ( p k ) 2 m + 1 B 2 m + 1 ( q p ) ,
p = 2 π L , q m = q + m p , γ m = i k 2 q m 2 , α m = arcsin q m k .

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