Abstract

Using the effective-index approach and an explicit expression for the propagation constant of gap surface plasmon polaritons (G-SPPs) obtained for moderate gap widths, we introduce a normalized waveguide parameter characterizing the mode field confinement and obtain the corresponding expressions for various (gap, trench and V-groove) G-SPP based waveguides. Usage of the obtained relations is investigated with a finite-element method, demonstrating that waveguides with different dimensions and operating at different wavelengths, but having the same normalized parameter, exhibit very similar field confinement. These relations allow one to design G-SPP waveguides for single-mode operation supporting a well-confined fundamental mode.

© 2008 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
  2. E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
    [CrossRef]
  3. S. I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467-9476 (2006).
    [CrossRef] [PubMed]
  4. K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
    [CrossRef]
  5. K. Tanaka, M. Tanaka, and T. Sugiyama, "Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguides," Opt. Express 13, 256-266 (2005).
    [CrossRef] [PubMed]
  6. S. H. Chang, T. C. Chiu, and C.-Y. Tai, "Propagation characteristics of the supermode based on two coupled semi-infinite rib plasmonic waveguides," Opt. Express 15, 1755-1761 (2007).
    [CrossRef] [PubMed]
  7. I. V. Novikov and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002).
    [CrossRef]
  8. D. K. Gramotnev and D. F. P. Pile, "Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface," Appl. Phys. Lett. 85, 6323-6325 (2004).
    [CrossRef]
  9. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
    [CrossRef] [PubMed]
  10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonantors," Nature 440, 508-511 (2006).
    [CrossRef] [PubMed]
  11. V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
    [CrossRef] [PubMed]
  12. E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, "Channel plasmon-polaritons: modal shape, dispersion, and losses," Opt. Lett. 31, 3447-3449 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
  15. R. Zia, A. Chandran, and M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005).
    [CrossRef] [PubMed]
  16. H. Kogelnik and V. Ramaswamy, "Scaling rules for thin-film optical waveguides," Appl. Opt. 13, 1857- (1974).
    [CrossRef] [PubMed]
  17. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
    [CrossRef]
  18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  19. M. Yan and M. Qiu, "Guided plasmon polariton at 2D metal corners," J. Opt. Soc. Am. B 24, 2333-2342 (2007).
    [CrossRef]

2007 (4)

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
[CrossRef]

S. H. Chang, T. C. Chiu, and C.-Y. Tai, "Propagation characteristics of the supermode based on two coupled semi-infinite rib plasmonic waveguides," Opt. Express 15, 1755-1761 (2007).
[CrossRef] [PubMed]

M. Yan and M. Qiu, "Guided plasmon polariton at 2D metal corners," J. Opt. Soc. Am. B 24, 2333-2342 (2007).
[CrossRef]

2006 (3)

2005 (4)

2004 (1)

D. K. Gramotnev and D. F. P. Pile, "Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface," Appl. Phys. Lett. 85, 6323-6325 (2004).
[CrossRef]

2003 (1)

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

2002 (1)

I. V. Novikov and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002).
[CrossRef]

1977 (1)

1969 (1)

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Boltasseva, A.

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
[CrossRef]

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, "Effective-index modeling of channel plasmon polaritons," Opt. Express 14, 9467-9476 (2006).
[CrossRef] [PubMed]

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

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, "Channel plasmon-polaritons: modal shape, dispersion, and losses," Opt. Lett. 31, 3447-3449 (2006).
[CrossRef] [PubMed]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, "Integrated optical components utilizing long-range surface plasmon polaritons," J. Lightwave Technol. 23, 413-422 (2005).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Brongersma, M. L.

Burns, W. K.

Chandran, A.

Chang, S. H.

Chiu, T. C.

Devaux, E.

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
[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 resonantors," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Ebbesen, T. W.

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
[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 resonantors," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Economou, E. N.

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Garcia-Vidal, F. J.

Gramotnev, D. K.

D. K. Gramotnev and D. F. P. Pile, "Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface," Appl. Phys. Lett. 85, 6323-6325 (2004).
[CrossRef]

Hocker, G. B.

Kjaer, K.

Laluet, J.-Y.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
[CrossRef]

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
[CrossRef] [PubMed]

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

Larsen, M. S.

Leosson, K.

Maradudin, A. A.

I. V. Novikov and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Martin-Moreno, L.

Moreno, E.

Nikolajsen, T.

Novikov, I. V.

I. V. Novikov and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Pile, D. F. P.

D. K. Gramotnev and D. F. P. Pile, "Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface," Appl. Phys. Lett. 85, 6323-6325 (2004).
[CrossRef]

Qiu, M.

Rodrigo, S. G.

Sugiyama, T.

Tai, C.-Y.

Tanaka, K.

K. Tanaka, M. Tanaka, and T. Sugiyama, "Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguides," Opt. Express 13, 256-266 (2005).
[CrossRef] [PubMed]

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

Tanaka, M.

K. Tanaka, M. Tanaka, and T. Sugiyama, "Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguides," Opt. Express 13, 256-266 (2005).
[CrossRef] [PubMed]

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

Volkov, V. S.

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
[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 resonantors," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Yan, M.

Zia, R.

Appl. Opt. (1)

Appl. Phys. A (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channelling surface plasmons," Appl. Phys. A 89, 225-231 (2007).
[CrossRef]

Appl. Phys. Lett. (2)

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

D. K. Gramotnev and D. F. P. Pile, "Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface," Appl. Phys. Lett. 85, 6323-6325 (2004).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Wavelength selective nanophotonic components utilizing channel plasmon polaritons," Nano Lett. 7, 880-884 (2007).
[CrossRef] [PubMed]

Nature (1)

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

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. (1)

E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Phys. Rev. B (1)

I. V. Novikov and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Other (3)

H. Kogelnik and V. Ramaswamy, "Scaling rules for thin-film optical waveguides," Appl. Opt. 13, 1857- (1974).
[CrossRef] [PubMed]

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the G-SPP based waveguides under consideration.

Fig. 2.
Fig. 2.

The G-SPP mode effective index and its propagation length as a function of the width t of the air gap in gold for several light wavelengths calculated exactly [Eq. (1)] and using the analytic (moderate gap) approximation [Eq. (3)].

Fig. 3.
Fig. 3.

(a, b). The G-SPP mode field magnitude distributions and (c) their lateral mid-plane cross sections shown by a dotted line in (a) for two waveguide configurations having the same waveguide parameter Vgsp =1.73.

Fig. 4.
Fig. 4.

(a, b). The TPP mode field magnitude distributions and (c) their lateral mid-plane cross sections shown by a dotted line in (a) for two waveguide configurations having the same waveguide parameter V tpp=π.

Fig. 5.
Fig. 5.

(a, b). The CPP mode field magnitude distributions and (c) their lateral mid-plane cross sections shown by a dotted line in (a) for two waveguide configurations having the same waveguide parameter V cpp=1.34π.

Equations (8)

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tanh ( k z ( d ) t 2 ) = ( ε d k z ( m ) ) ( ε m k z ( d ) ) , with   k z ( m , d ) = k g s p 2 ε m , d k 0 2 and k 0 = 2 π λ ,
k g s p k 0 ε d + 0.5 ( k g s p 0 k 0 ) 2 + ( k g s p 0 k 0 ) 2 [ ε d ε m + 0.25 ( k g s p 0 k 0 ) 2 ]
with     k g s p 0 = 2 ε d t ε m .
k g s p k 0 ε d + 2 ε d ε d ε m k 0 t ( ε m ) .
V = w k 0 n 1 2 n 2 2 .
V g s p 2 w π ε d ε d ε m λ ε m ( 1 t 1 D ) .
V t p p 2 d π ε d ε d ε m λ w ε m .
V c p p 2 k 0 d ε d ε d ε m ε m tan ( θ 2 ) .

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