Abstract

We study the optical properties of subwavelength metallic waveguides made of nanoscale apertures in a metal. We develop analytical expressions for the fundamental optical modes in apertures. The results are in excellent agreement with finite element calculations. This model provides a physical understanding of the role of non-perfect metallic walls, and of the shape and size of the apertures. They reveal the effect of the skin depth and of the surface plasmon polariton coupling on the waveguide modes. The nanoscopic origin of the increase of the cut-off wavelength due to the electromagnetic penetration depth in the metal is described. Simple expressions and universal curves for the effective index and the cut-off wavelength of the fundamental guided mode of any rectangular metallic waveguide are presented. The results provide an efficient tool for the design of nanoscale waveguides with real metal.

© 2007 Optical Society of America

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  3. H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
    [CrossRef] [PubMed]
  4. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
    [CrossRef] [PubMed]
  5. D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen "Beyond the Bethe limit: tunable enhanced light transmission through a single sub-wavelength aperture," Adv. Mater. 11, 860 (1999).
    [CrossRef]
  6. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972 (2001).
    [CrossRef]
  7. A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61 (2004).
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  8. F. Garcia de Abajo, "Light transmission through a single cylindrical hole in a metallic film," Opt. Express 10, 1475 (2002).
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. N. M. Arslanov, "The optimal form of the scanning near-field optical microscopy probe with subwavelength aperture," J. Opt. Soc. Am. A 8, 338 (2006).
    [CrossRef]
  13. M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguides for single-molecule analysis at high concentrations," Science 299, 682 (2003).
    [CrossRef] [PubMed]
  14. L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
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  16. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
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  17. F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials,"J. Opt. A: Pure and Appl. Opt. 7, S97 (2005).
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  24. L. Martín-Moreno, and F. J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619 (2004).
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    [CrossRef] [PubMed]
  37. 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 (2006).
    [CrossRef] [PubMed]
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2006

F. J. García-Vidal, L. Martín-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B 74,153411 (2006).
[CrossRef]

N. M. Arslanov, "The optimal form of the scanning near-field optical microscopy probe with subwavelength aperture," J. Opt. Soc. Am. A 8, 338 (2006).
[CrossRef]

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

P. Lalanne, and J.-P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nat. Mater. 2, 509 (2006).

J. Luo and C. Jiao "Effect of the lossy layer thickness of metal cylindrical waveguide wall on the propagation constant of electromagnetic modes," Appl. Phys. Lett. 88, 061115 (2006).
[CrossRef]

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
[CrossRef]

K. J. Webb and J. Li, "Analysis of transmission through small apertures in conducting films," Phys. Rev. B 73, 033401 (2006).
[CrossRef]

K. J. Webb and J. Li "Analysis of transmission through small apertures in conducting films," Phys. Rev. B 73, 033401 (2006).
[CrossRef]

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 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467 (2006).
[CrossRef] [PubMed]

2005

R. Gordon, and A. G. Brolo, "Increased cut-off wavelength for a subwavelength hole in a real metal," Opt. Express 13, 1933 (2005).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).
[CrossRef]

A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure and Appl. Opt. 7, S90 (2005).
[CrossRef]

P. Lalanne, J.-C. Rodier, and J.-P. Hugonin, "Surface plasmons of metallic surfaces perforated by nanohole arrays," J. Opt. A: Pure and Appl. Opt. 7, 422 (2005).
[CrossRef]

F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials,"J. Opt. A: Pure and Appl. Opt. 7, S97 (2005).
[CrossRef]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

E. Popov, M. Nevière, P. Boyer, and N. Bonod, "Light transmission through a subwavelength hole," Opt. Commun. 255, 338 (2005).
[CrossRef]

H. Shin, P. B. Catrysse, and S. Fan "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

2004

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61 (2004).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

K. L. van der Molen, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Influence of hole size on the extraordinary transmission through subwavelength hole arrays," Appl. Phys. Lett. 85, 4316 (2004).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, "Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands," Appl. Phys. B 79, 1 (2004).
[CrossRef]

L. Martín-Moreno, and F. J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619 (2004).
[CrossRef]

2003

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguides for single-molecule analysis at high concentrations," Science 299, 682 (2003).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 824, 424 (2003).

2002

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

F. Garcia de Abajo, "Light transmission through a single cylindrical hole in a metallic film," Opt. Express 10, 1475 (2002).

2001

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972 (2001).
[CrossRef]

1999

J. A. Porto and F. J. Garcia-Vidal and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845 (1999).
[CrossRef]

D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen "Beyond the Bethe limit: tunable enhanced light transmission through a single sub-wavelength aperture," Adv. Mater. 11, 860 (1999).
[CrossRef]

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 667, 391 (1998).

1994

L. Novotny and C. Hafner "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094 (1994).
[CrossRef]

Arslanov, N. M.

N. M. Arslanov, "The optimal form of the scanning near-field optical microscopy probe with subwavelength aperture," J. Opt. Soc. Am. A 8, 338 (2006).
[CrossRef]

Baida, F. I.

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
[CrossRef]

F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, "Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands," Appl. Phys. B 79, 1 (2004).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 824, 424 (2003).

Belkhir, A.

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
[CrossRef]

F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, "Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands," Appl. Phys. B 79, 1 (2004).
[CrossRef]

Bonod, N.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

E. Popov, M. Nevière, P. Boyer, and N. Bonod, "Light transmission through a subwavelength hole," Opt. Commun. 255, 338 (2005).
[CrossRef]

Boyer, P.

E. Popov, M. Nevière, P. Boyer, and N. Bonod, "Light transmission through a subwavelength hole," Opt. Commun. 255, 338 (2005).
[CrossRef]

Bozhevolnyi, S. I.

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 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Brolo, A. G.

Capoulade, J.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Catrysse, P. B.

H. Shin, P. B. Catrysse, and S. Fan "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

Craighead, H. G.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguides for single-molecule analysis at high concentrations," Science 299, 682 (2003).
[CrossRef] [PubMed]

Degiron, A.

A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure and Appl. Opt. 7, S90 (2005).
[CrossRef]

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61 (2004).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 824, 424 (2003).

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 (2006).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Dintinger, J.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[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 (2006).
[CrossRef] [PubMed]

A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure and Appl. Opt. 7, S90 (2005).
[CrossRef]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61 (2004).
[CrossRef]

W. L. Barnes, A. Dereux and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 824, 424 (2003).

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972 (2001).
[CrossRef]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen "Beyond the Bethe limit: tunable enhanced light transmission through a single sub-wavelength aperture," Adv. Mater. 11, 860 (1999).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 667, 391 (1998).

Enoch, S.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Erland, J.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Fan, S.

H. Shin, P. B. Catrysse, and S. Fan "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

Foquet, M.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguides for single-molecule analysis at high concentrations," Science 299, 682 (2003).
[CrossRef] [PubMed]

Garcia de Abajo, F.

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials,"J. Opt. A: Pure and Appl. Opt. 7, S97 (2005).
[CrossRef]

L. Martín-Moreno, and F. J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619 (2004).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

J. A. Porto and F. J. Garcia-Vidal and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845 (1999).
[CrossRef]

García-Vidal, F. J.

F. J. García-Vidal, L. Martín-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B 74,153411 (2006).
[CrossRef]

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 667, 391 (1998).

Gordon, R.

F. J. García-Vidal, L. Martín-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B 74,153411 (2006).
[CrossRef]

R. Gordon, and A. G. Brolo, "Increased cut-off wavelength for a subwavelength hole in a real metal," Opt. Express 13, 1933 (2005).
[CrossRef] [PubMed]

Granet, G.

F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, "Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands," Appl. Phys. B 79, 1 (2004).
[CrossRef]

Grupp, D. E.

D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen "Beyond the Bethe limit: tunable enhanced light transmission through a single sub-wavelength aperture," Adv. Mater. 11, 860 (1999).
[CrossRef]

Hafner, C.

L. Novotny and C. Hafner "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094 (1994).
[CrossRef]

Harmsen, R. H.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

Hugonin, J.-P.

P. Lalanne, and J.-P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nat. Mater. 2, 509 (2006).

P. Lalanne, J.-C. Rodier, and J.-P. Hugonin, "Surface plasmons of metallic surfaces perforated by nanohole arrays," J. Opt. A: Pure and Appl. Opt. 7, 422 (2005).
[CrossRef]

Hvam, J. M.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Jiao, C.

J. Luo and C. Jiao "Effect of the lossy layer thickness of metal cylindrical waveguide wall on the propagation constant of electromagnetic modes," Appl. Phys. Lett. 88, 061115 (2006).
[CrossRef]

Klein Koerkamp, K. J.

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Korlach, J.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguides for single-molecule analysis at high concentrations," Science 299, 682 (2003).
[CrossRef] [PubMed]

Kuipers, L.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).
[CrossRef]

K. L. van der Molen, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Influence of hole size on the extraordinary transmission through subwavelength hole arrays," Appl. Phys. Lett. 85, 4316 (2004).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Kumar, L. K. S.

F. J. García-Vidal, L. Martín-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B 74,153411 (2006).
[CrossRef]

Lalanne, P.

P. Lalanne, and J.-P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nat. Mater. 2, 509 (2006).

P. Lalanne, J.-C. Rodier, and J.-P. Hugonin, "Surface plasmons of metallic surfaces perforated by nanohole arrays," J. Opt. A: Pure and Appl. Opt. 7, 422 (2005).
[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 (2006).
[CrossRef] [PubMed]

Lamrous, O.

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
[CrossRef]

Lenne, P.-F.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Leosson, K.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Levene, M. J.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguides for single-molecule analysis at high concentrations," Science 299, 682 (2003).
[CrossRef] [PubMed]

Lezec, H. J.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61 (2004).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972 (2001).
[CrossRef]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen "Beyond the Bethe limit: tunable enhanced light transmission through a single sub-wavelength aperture," Adv. Mater. 11, 860 (1999).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 667, 391 (1998).

Li, J.

K. J. Webb and J. Li, "Analysis of transmission through small apertures in conducting films," Phys. Rev. B 73, 033401 (2006).
[CrossRef]

K. J. Webb and J. Li "Analysis of transmission through small apertures in conducting films," Phys. Rev. B 73, 033401 (2006).
[CrossRef]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972 (2001).
[CrossRef]

Luo, J.

J. Luo and C. Jiao "Effect of the lossy layer thickness of metal cylindrical waveguide wall on the propagation constant of electromagnetic modes," Appl. Phys. Lett. 88, 061115 (2006).
[CrossRef]

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Martín-Moreno, L.

F. J. García-Vidal, L. Martín-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B 74,153411 (2006).
[CrossRef]

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials,"J. Opt. A: Pure and Appl. Opt. 7, S97 (2005).
[CrossRef]

L. Martín-Moreno, and F. J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619 (2004).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

Moreau, A.

F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, "Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands," Appl. Phys. B 79, 1 (2004).
[CrossRef]

Moreno, E.

F. J. García-Vidal, L. Martín-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B 74,153411 (2006).
[CrossRef]

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Nevière, M.

E. Popov, M. Nevière, P. Boyer, and N. Bonod, "Light transmission through a subwavelength hole," Opt. Commun. 255, 338 (2005).
[CrossRef]

Novotny, L.

L. Novotny and C. Hafner "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094 (1994).
[CrossRef]

Pellerin, K. M.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972 (2001).
[CrossRef]

Pendry, J. B.

F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials,"J. Opt. A: Pure and Appl. Opt. 7, S97 (2005).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

J. A. Porto and F. J. Garcia-Vidal and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845 (1999).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, P. Boyer, and N. Bonod, "Light transmission through a subwavelength hole," Opt. Commun. 255, 338 (2005).
[CrossRef]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Porto, J. A.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

J. A. Porto and F. J. Garcia-Vidal and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845 (1999).
[CrossRef]

Prangsma, J. C.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

Rigneault, H.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Rodier, J.-C.

P. Lalanne, J.-C. Rodier, and J.-P. Hugonin, "Surface plasmons of metallic surfaces perforated by nanohole arrays," J. Opt. A: Pure and Appl. Opt. 7, 422 (2005).
[CrossRef]

Sandtke, M.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

Segerink, F. B.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

K. L. van der Molen, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Influence of hole size on the extraordinary transmission through subwavelength hole arrays," Appl. Phys. Lett. 85, 4316 (2004).
[CrossRef]

Shin, H.

H. Shin, P. B. Catrysse, and S. Fan "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

Skovgaard, P. M. W.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Thio, T.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, "Theory of extraordinary Optical Transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972 (2001).
[CrossRef]

D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen "Beyond the Bethe limit: tunable enhanced light transmission through a single sub-wavelength aperture," Adv. Mater. 11, 860 (1999).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 667, 391 (1998).

Turner, S. W.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguides for single-molecule analysis at high concentrations," Science 299, 682 (2003).
[CrossRef] [PubMed]

van der Molen, K. L.

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).
[CrossRef]

K. L. van der Molen, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Influence of hole size on the extraordinary transmission through subwavelength hole arrays," Appl. Phys. Lett. 85, 4316 (2004).
[CrossRef]

van Hulst, N. F.

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).
[CrossRef]

K. L. van der Molen, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Influence of hole size on the extraordinary transmission through subwavelength hole arrays," Appl. Phys. Lett. 85, 4316 (2004).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Van Labeke, D.

F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).
[CrossRef]

F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, "Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands," Appl. Phys. B 79, 1 (2004).
[CrossRef]

van Nieuwstadt, J. A. H.

J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, "Strong modification of the Nonlinear optical response of metallic subwavelength hole arrays," Phys. Rev. Lett. 97, 146102 (2006).
[CrossRef] [PubMed]

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 (2006).
[CrossRef] [PubMed]

Webb, K. J.

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H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P.-F. Lenne, "Enhancement of single-molecule fluorescence detection in subwavelength apertures," Phys. Rev. Lett. 95, 117401 (2005).
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A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure and Appl. Opt. 7, S90 (2005).
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P. Lalanne, and J.-P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nat. Mater. 2, 509 (2006).

Nature

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Nature (London)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 667, 391 (1998).

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H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002).
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Figures (8)

Fig. 1.
Fig. 1.

Schematic of 1D planar metallic waveguide (a) and 2D rectangular metallic waveguide (d). The fundamental TM mode of planar 1D waveguides is composed of two surface plasmons waves (red). They are weakly coupled in wide waveguides (wλ, (b)) and strongly coupled in narrow waveguides (wλ, (c))

Fig. 2.
Fig. 2.

Effective index of the fundamental TM mode of a planar metallic waveguide (εd = 1) as a function of the width w for εm = -50.

Fig. 3.
Fig. 3.

Skin depth of the metal δ = λ ( 2 π ε m ( λ ) ) (dark) as a function of the wavelength. The skin depth is nearly constant and equal to δ = c/ωp (red) between 0.5 and 20 μm.

Fig. 4.
Fig. 4.

Real (left) and imaginary (right) parts of the effective index of the fundamental TM mode of a planar metallic waveguide as a function of the wavelength λ. Dark solid line: effective index of a single metal/dielectric surface plasmon polariton. The metal permittivity is given by a Drude model: ε m = 1 ω p 2 ω 2 + iωγ , ω p = 1.2 × 10 16 s 1 , γ = 1.2 × 10 14 s 1 . The waveguide width is w = 100 nm.

Fig. 5.
Fig. 5.

Real (left) and imaginary (right) parts of the effective index of the fundamental TE10 mode of a rectangular metallic waveguide as a function of the wavelength λ. The widths of the waveguide is wx = 300 nm and wy = 200 nm. The results of the perfect metal (dark), finite element calculation (blue) and coupled SPP approximation (Eq. (7) and (8)) (red) are compared.

Fig. 6.
Fig. 6.

Cut-off wavelength of the fundamental guided mode TE10 as a function of wy , for a silver rectangular waveguide with wx = 270 nm. The approximated model from Eq. (11) (solid curve) is compared to reference [31] (red points), and to the perfect metal case (horizontal dashed line).

Fig. 7.
Fig. 7.

Universal curves for the real part of the effective index of the fundamental guided mode TE10 for three different wavelengths (λ/δ = 100, 200 and 500). Dimensions of the rectangular waveguide (wx ,wy ) are normalized to the metal skin depth δ.

Fig. 8.
Fig. 8.

Universal curves for the cut-off wavelength of the fundamental guided mode TE10 of rectangular metallic waveguides. Dimensions and wavelengths are normalized to the metal skin depth δ.

Equations (11)

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

k xd ε d [ 1 e ik xd w 1 + e ik xd w ] + k xm ε m = 0
k xm ε m = k xd ε d ( ik xd w 2 ) .
k xd 2 [ 1 + k xd 2 ( ε m ε d ) 2 w 2 4 ] = ( ε d ε m ) k 0 2
n 1 D = κ k 0 = ε d ( 1 + λ πw ε m 1 + ε d ε m ) 1 2
n w = ε d ( 1 + 2 δ c w )
tan ( k xd w x 2 ) = i k xm k xd
n 2 D = κ k 0 = ε d ( λ 2 w x ) 2
w x = w x ( 1 + λ πw x ( ε d ε m ) + ( λ 2 w x ) 2 )
w x = w x + 2 δ
n 2 D = ε d ( 1 + 2 δ w y ) ( λ 2 ( w x + 2 δ ) ) 2
λ c = 2 ( w x + 2 δ ) ε d ( 1 + 2 δ w y )

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