Abstract

We study the propagation characteristics of optical signals in waveguides composed of linear periodic arrangements of metallic nanoparticles embedded in a dielectric host. We find the complex Bloch band diagram for the guided modes including material losses by employing Mie scattering theory as well as coupled dipole approximations. The results of the model are validated through finite element solution of the Maxwell’s equations.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
    [CrossRef]
  3. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
    [CrossRef]
  4. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
    [CrossRef]
  5. D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
    [CrossRef]
  6. R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
    [CrossRef]
  7. C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver nanoparticles and its possible application,” Phys. Rev. E 72, 066606 (2005).
    [CrossRef]
  8. D. S. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. 31, 98–100 (2006).
    [CrossRef] [PubMed]
  9. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
    [CrossRef]
  10. A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic nanoparticles as a subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
    [CrossRef]
  11. V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
    [CrossRef]
  12. A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78, 035403 (2008).
    [CrossRef]
  13. J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
    [CrossRef]
  14. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
    [CrossRef] [PubMed]
  15. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
    [CrossRef] [PubMed]
  16. J. M. Gerardy and M. Ausloos, “Absorption spectrum of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
    [CrossRef]
  17. J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  18. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  19. Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
    [CrossRef]
  20. L. Lewin, Structural Properties of Polylogarithms (American Mathematical Society, 1991).
  21. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [CrossRef]
  22. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  23. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), p. 29.
  24. M. Davanco, Y. Uhzhumov, and G. Shvetz, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681–9691 (2007).
    [CrossRef] [PubMed]
  25. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).
  26. J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
    [CrossRef]
  27. D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
    [CrossRef]
  28. http://www.comsol.com.

2008 (2)

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78, 035403 (2008).
[CrossRef]

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

2007 (2)

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

M. Davanco, Y. Uhzhumov, and G. Shvetz, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681–9691 (2007).
[CrossRef] [PubMed]

2006 (5)

D. S. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. 31, 98–100 (2006).
[CrossRef] [PubMed]

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic nanoparticles as a subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

2005 (3)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
[CrossRef]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver nanoparticles and its possible application,” Phys. Rev. E 72, 066606 (2005).
[CrossRef]

2004 (2)

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
[CrossRef]

2003 (1)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

2000 (1)

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

1998 (1)

1996 (1)

Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

1989 (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

1982 (1)

J. M. Gerardy and M. Ausloos, “Absorption spectrum of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

1980 (1)

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

1972 (1)

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

Alú, A.

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic nanoparticles as a subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Ausloos, M.

J. M. Gerardy and M. Ausloos, “Absorption spectrum of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Aussenegg, F. R.

Boffi, D.

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Bohren, C. F.

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

Brongersma, M. L.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Christy, R. W.

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

Citrin, D. S.

D. S. Citrin, “Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium,” Opt. Lett. 31, 98–100 (2006).
[CrossRef] [PubMed]

D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
[CrossRef]

Conforti, M.

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Davanco, M.

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Engheta, N.

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic nanoparticles as a subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Gastaldi, L.

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Gerardy, J. M.

J. M. Gerardy and M. Ausloos, “Absorption spectrum of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Govyadinov, A. A.

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78, 035403 (2008).
[CrossRef]

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Huffman, D. R.

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

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

Johnson, P. B.

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

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Koenderink, A. F.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

Krenn, J. R.

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

Leitner, A.

Lewin, L.

L. Lewin, Structural Properties of Polylogarithms (American Mathematical Society, 1991).

Lyanders, O.

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), p. 29.

Markel, V. A.

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78, 035403 (2008).
[CrossRef]

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

Nédélec, J. -C.

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Polman, A.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

Quinten, M.

Sarychev, A. K.

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

Shan, N. C.

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Shore, R. A.

R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
[CrossRef]

Shvetz, G.

Simovski, C. R.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver nanoparticles and its possible application,” Phys. Rev. E 72, 066606 (2005).
[CrossRef]

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

Tretyakov, S. A.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver nanoparticles and its possible application,” Phys. Rev. E 72, 066606 (2005).
[CrossRef]

Uhzhumov, Y.

Van Duyne, R. P.

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Viitanen, A. J.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver nanoparticles and its possible application,” Phys. Rev. E 72, 066606 (2005).
[CrossRef]

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Xu, Y. -l.

Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Yaghjian, A. D.

R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
[CrossRef]

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

Zhao, J.

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Electron. Lett. (1)

R. A. Shore and A. D. Yaghjian, “Travelling electromagnetic waves on linear periodic arrays of lossless spheres,” Electron. Lett. 41, 578–580 (2005).
[CrossRef]

Math. Comput. (1)

Y.-l. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Nano Lett. (1)

D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
[CrossRef]

Nature Mater. (1)

J. N. Anker, W. P. Hall, O. Lyanders, N. C. Shan, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Numer. Math. (2)

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

D. Boffi, M. Conforti, and L. Gastaldi, “Modified edge finite elements for photonic crystals,” Numer. Math. 105, 249–266 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (10)

J. M. Gerardy and M. Ausloos, “Absorption spectrum of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

A. Alú and N. Engheta, “Theory of linear chains of metamaterial/plasmonic nanoparticles as a subdiffraction optical nanotrasmission lines,” Phys. Rev. B 74, 205436 (2006).
[CrossRef]

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of nanoparticles,” Phys. Rev. B 75, 085426 (2007).
[CrossRef]

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78, 035403 (2008).
[CrossRef]

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

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

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003).
[CrossRef]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Phys. Rev. E (1)

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Resonator mode in chains of silver nanoparticles and its possible application,” Phys. Rev. E 72, 066606 (2005).
[CrossRef]

Science (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef] [PubMed]

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Other (6)

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), p. 29.

L. Lewin, Structural Properties of Polylogarithms (American Mathematical Society, 1991).

http://www.comsol.com.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Real part of the roots β ( ω ) of the dispersion relations (18, 19). Dotted curves, lossless metal; circles and crosses, lossy metal.

Fig. 2
Fig. 2

Imaginary part of the roots β ( ω ) of the dispersion relations (18, 19). Dotted curves, lossless metal; circles and crosses, lossy metal. The horizontal line ω d / ( 2 π c ) 0.169 indicates the zero group velocity of the transverse mode T 1 in the lossless case.

Fig. 3
Fig. 3

Transverse mode T 1 . Real and imaginary parts of the roots β ( ω ) of the dispersion relations (18) (solid line) and results of finite element simulation (circles). Insets show three slices ( x = 0 , z = 0 , y = 0 planes) of the norm of the magnetic field for the transverse mode at ω d / ( 2 π c ) = 0.16 .

Fig. 4
Fig. 4

Longitudinal mode L. Real and imaginary parts of the roots β ( ω ) of the dispersion relations (19) (solid line) and results of finite element simulation (circles). Insets show three slices ( x = 0 , z = 0 , y = 0 planes) of the norm of the magnetic field for the transverse mode at ω d / ( 2 π c ) = 0.183 .

Equations (31)

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

E i = l m { a i , l m ( n ) m l m 1 ( n ) + b i , l m ( n ) n l m 1 ( n ) } ,
H i = k M i μ 0 ω l m { b i , l m ( n ) m l m 1 ( n ) + a i , l m ( n ) n l m 1 ( n ) } ,
E ( n ) = l m { a l m ( n ) m l m 1 ( n ) + b l m ( n ) n l m 1 ( n ) + c l m ( n ) m l m 3 ( n ) + d l m ( n ) n l m 3 ( n ) } .
c l m ( n ) = Γ l ( n ) a l m ( n ) ,
d l m ( n ) = Δ l ( n ) b l m ( n ) ,
a l m ( n ) = c l m ( n ) / Γ l = a i , l m ( n ) + v n p q { T p q l m ( v , n ) c p q ( v ) + C p q l m ( v , n ) d p q ( v ) } .
b l m ( n ) = d l m ( n ) / Δ l = b i , l m ( n ) + v n p q { C p q l m ( v , n ) c p q ( v ) + T p q l m ( v , n ) d p q ( v ) } .
b i , 1 m ( n ) = d 1 m ( n ) / Δ 1 v n q = 1 1 T 1 q 1 m ( v , n ) d 1 q ( v ) .
b i , 10 ( n ) = Δ 1 1 d 10 ( n ) v n T 1010 ( v n ) d 10 ( v ) = U 1010 ( n ) d 10 ( n ) ,
b i , 1 1 ( n ) = U 1 11 1 ( n ) d 1 1 ( n ) + U 111 1 ( n ) d 11 ( n ) ,
b i , 11 ( n ) = U 1 111 ( n ) d 1 1 ( n ) + U 1111 ( n ) d 11 ( n ) ,
T 1010 ( n ) = i 3 2 e i k M | d n | k M | d n | + 3 2 e i k M | d n | ( k M | d n | ) 2 + i 3 2 e i k M | d n | ( k M | d n | ) 3 ,
T 1 111 ( n ) = i 3 4 e i k M | d n | k M | d n | 9 4 e i k M | d n | ( k M | d n | ) 2 i 9 4 e i k M | d n | ( k M | d n | ) 3 ,
T 1111 ( n ) = i 3 4 e i k M | d n | k M | d n | 3 4 e i k M | d n | ( k M | d n | ) 2 i 3 4 e i k M | d n | ( k M | d n | ) 3 .
b ̂ i , 10 ( k ) = U ̂ 1010 ( k ) d ̂ 10 ( k ) ,
b ̂ i , 1 1 ( k ) = U ̂ 1111 ( k ) d ̂ 1 1 ( k ) + U ̂ 1 111 ( k ) d ̂ 11 ( k ) ,
b ̂ i , 11 ( k ) = U ̂ 1 111 ( k ) d ̂ 1 1 ( k ) + U ̂ 1111 ( n ) d ̂ 11 ( k ) ,
U ̂ 1010 ( k ) = Δ 1 1 n = 1 T 1010 ( n ) e i k n n = 1 T 1010 ( n ) e + i k n = 0 ,
0 = Δ 1 1 + i 3 2 Li 1 ( e i ( k M d k ) ) + Li 1 ( e i ( k M d + k ) ) k M d 3 2 Li 2 ( e i ( k M d k ) ) + Li 2 ( e i ( k M d + k ) ) ( k M d ) 2 i 3 2 Li 3 ( e i ( k M d k ) ) + Li 3 ( e i ( k M d + k ) ) ( k M d ) 3 ,
E = n = n 10 3 ( n ) e i k n ,
H = k M i μ 0 ω n = m 10 3 ( n ) e i k n .
U ̂ 1111 ( k ) + U ̂ 1 111 ( k ) = 0 ,
0 = Δ 1 1 + 3 Li 2 ( e i ( k M d k ) ) + Li 2 ( e i ( k M d + k ) ) ( k M d ) 2 + i 3 Li 3 ( e i ( k M d k ) ) + Li 3 ( e i ( k M d + k ) ) ( k M d ) 3 .
E = n = ( n 1 1 3 ( n ) + n 11 3 ( n ) ) e i k n ,
H = k M i μ 0 ω n = ( m 1 1 3 ( n ) + m 11 3 ( n ) ) e i k n .
1 + α ( ω ) 4 π d 3 ϵ m ϵ 0 Σ T ( ω , β ) = 0 ,
1 2 α ( ω ) 4 π d 3 ϵ m ϵ 0 Σ L ( ω , β ) = 0 ,
× ε 1 × H ̃ = ( ω c ) 2 H ̃     in   Ω .
find   ω R   such   that   0 ̸ H V : Ω ε 1 ( + i k ) × H ( + i k ) × ϕ ¯ = ( ω c ) 2 Ω H ϕ ¯ ,     ϕ V ,
find   β C   such   that   0 ̸ H V : Ω ε 1 ( × H ) ( × ϕ ) ( ω c ) 2 Ω H ϕ = i β { Ω ε 1 k ̂ ( H × × ϕ ) Ω ε 1 k ̂ ( ϕ × × H ) } β 2 Ω ε 1 { H ϕ ( k ̂ H ) ( k ̂ ϕ ) } ,     ϕ V ,
( A ω 2 c 2 B ) x = i β ( C D ) x + β 2 F x .

Metrics