Abstract

We demonstrate that a three-dimensional (3D) index-guided multimode plasmonic waveguide can be approximated to a two-dimensional (2D) lossy slab waveguide by using an effective-index method. It is found that this 2D approximation is more accurate when the width of the multimode waveguide increases. Such a 2D approximation can be used for a quicker and more efficient design of complicated multimode plasmonic devices. 1×N ultrasmall multimode interference splitters based on multimode surface plasmon waveguides are designed by using this 2D model and the designs are validated with a 3D finite-difference time-domain method.

© 2007 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons (Springer-Verlag, 1988).
  2. L. Liu, Z. Han, and S. He, "Novel surface plasmon waveguide for high integration," Opt. Express 13, 6645-6650 (2005).
    [CrossRef] [PubMed]
  3. Z. Han, L. Liu, and E. Forsberg, "Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons," Opt. Commun. 259, 690-695 (2006).
    [CrossRef]
  4. Z. Han, E. Forsberg, and S. He, "Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides," IEEE Photon. Technol. Lett. 19, 91-93 (2007).
    [CrossRef]
  5. 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, 508-511 (2006).
    [CrossRef] [PubMed]
  6. A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2000).
  7. T. Fujisawa and M. Koshiba, "Theoretical investigation of ultrasmall polarization-insensitive I/spl times/2 multimode interference waveguides based on sandwiched structures," IEEE Photon. Technol. Lett. 18, 1246-1248 (2006).
    [CrossRef]
  8. J. Takahara and T. Kobayashi, "Nano-optical waveguides breaking through diffraction limit of light," Proc. SPIE 5604, 158-172 (2004).
    [CrossRef]
  9. F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, "Propagation properties of guided waves in index-guided two-dimensional optical waveguides," Appl. Phys. Lett. 86, 211101-211103 (2005).
    [CrossRef]
  10. P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguide," Opt. Commun. 137, 41-45 (1997).
    [CrossRef]
  11. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightwave Technol. 12, 487-493 (1994).
    [CrossRef]
  12. 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]

2007 (1)

Z. Han, E. Forsberg, and S. He, "Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides," IEEE Photon. Technol. Lett. 19, 91-93 (2007).
[CrossRef]

2006 (3)

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, 508-511 (2006).
[CrossRef] [PubMed]

T. Fujisawa and M. Koshiba, "Theoretical investigation of ultrasmall polarization-insensitive I/spl times/2 multimode interference waveguides based on sandwiched structures," IEEE Photon. Technol. Lett. 18, 1246-1248 (2006).
[CrossRef]

Z. Han, L. Liu, and E. Forsberg, "Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons," Opt. Commun. 259, 690-695 (2006).
[CrossRef]

2005 (3)

2004 (1)

J. Takahara and T. Kobayashi, "Nano-optical waveguides breaking through diffraction limit of light," Proc. SPIE 5604, 158-172 (2004).
[CrossRef]

2000 (1)

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

1997 (1)

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguide," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

1994 (1)

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightwave Technol. 12, 487-493 (1994).
[CrossRef]

1988 (1)

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

Boltasseva, A.

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, 508-511 (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]

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, 508-511 (2006).
[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, 508-511 (2006).
[CrossRef] [PubMed]

Forsberg, E.

Z. Han, E. Forsberg, and S. He, "Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides," IEEE Photon. Technol. Lett. 19, 91-93 (2007).
[CrossRef]

Z. Han, L. Liu, and E. Forsberg, "Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons," Opt. Commun. 259, 690-695 (2006).
[CrossRef]

Fujisawa, T.

T. Fujisawa and M. Koshiba, "Theoretical investigation of ultrasmall polarization-insensitive I/spl times/2 multimode interference waveguides based on sandwiched structures," IEEE Photon. Technol. Lett. 18, 1246-1248 (2006).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

Han, Z.

Z. Han, E. Forsberg, and S. He, "Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides," IEEE Photon. Technol. Lett. 19, 91-93 (2007).
[CrossRef]

Z. Han, L. Liu, and E. Forsberg, "Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons," Opt. Commun. 259, 690-695 (2006).
[CrossRef]

L. Liu, Z. Han, and S. He, "Novel surface plasmon waveguide for high integration," Opt. Express 13, 6645-6650 (2005).
[CrossRef] [PubMed]

He, S.

Z. Han, E. Forsberg, and S. He, "Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides," IEEE Photon. Technol. Lett. 19, 91-93 (2007).
[CrossRef]

L. Liu, Z. Han, and S. He, "Novel surface plasmon waveguide for high integration," Opt. Express 13, 6645-6650 (2005).
[CrossRef] [PubMed]

Kjaer, K.

Kobayashi, T.

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, "Propagation properties of guided waves in index-guided two-dimensional optical waveguides," Appl. Phys. Lett. 86, 211101-211103 (2005).
[CrossRef]

J. Takahara and T. Kobayashi, "Nano-optical waveguides breaking through diffraction limit of light," Proc. SPIE 5604, 158-172 (2004).
[CrossRef]

Koshiba, M.

T. Fujisawa and M. Koshiba, "Theoretical investigation of ultrasmall polarization-insensitive I/spl times/2 multimode interference waveguides based on sandwiched structures," IEEE Photon. Technol. Lett. 18, 1246-1248 (2006).
[CrossRef]

Kusunoki, F.

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, "Propagation properties of guided waves in index-guided two-dimensional optical waveguides," Appl. Phys. Lett. 86, 211101-211103 (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, 508-511 (2006).
[CrossRef] [PubMed]

Larsen, M. S.

Laude, V.

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguide," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

Leosson, K.

Liu, L.

Z. Han, L. Liu, and E. Forsberg, "Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons," Opt. Commun. 259, 690-695 (2006).
[CrossRef]

L. Liu, Z. Han, and S. He, "Novel surface plasmon waveguide for high integration," Opt. Express 13, 6645-6650 (2005).
[CrossRef] [PubMed]

Lüsse, P.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightwave Technol. 12, 487-493 (1994).
[CrossRef]

Nikolajsen, T.

Raether, H.

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

Schüle, J.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightwave Technol. 12, 487-493 (1994).
[CrossRef]

Stuwe, P.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightwave Technol. 12, 487-493 (1994).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

Takahara, J.

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, "Propagation properties of guided waves in index-guided two-dimensional optical waveguides," Appl. Phys. Lett. 86, 211101-211103 (2005).
[CrossRef]

J. Takahara and T. Kobayashi, "Nano-optical waveguides breaking through diffraction limit of light," Proc. SPIE 5604, 158-172 (2004).
[CrossRef]

Tournois, P.

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguide," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

Unger, H. G.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightwave Technol. 12, 487-493 (1994).
[CrossRef]

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, 508-511 (2006).
[CrossRef] [PubMed]

Yotsuya, T.

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, "Propagation properties of guided waves in index-guided two-dimensional optical waveguides," Appl. Phys. Lett. 86, 211101-211103 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, "Propagation properties of guided waves in index-guided two-dimensional optical waveguides," Appl. Phys. Lett. 86, 211101-211103 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

Z. Han, E. Forsberg, and S. He, "Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides," IEEE Photon. Technol. Lett. 19, 91-93 (2007).
[CrossRef]

T. Fujisawa and M. Koshiba, "Theoretical investigation of ultrasmall polarization-insensitive I/spl times/2 multimode interference waveguides based on sandwiched structures," IEEE Photon. Technol. Lett. 18, 1246-1248 (2006).
[CrossRef]

J. Lightwave Technol. (2)

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightwave Technol. 12, 487-493 (1994).
[CrossRef]

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]

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 resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Opt. Commun. (2)

P. Tournois and V. Laude, "Negative group velocities in metal-film optical waveguide," Opt. Commun. 137, 41-45 (1997).
[CrossRef]

Z. Han, L. Liu, and E. Forsberg, "Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons," Opt. Commun. 259, 690-695 (2006).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

J. Takahara and T. Kobayashi, "Nano-optical waveguides breaking through diffraction limit of light," Proc. SPIE 5604, 158-172 (2004).
[CrossRef]

Other (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2000).

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

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic diagram of the cross section of IG-MPWs. (b) Equivalent 2D waveguide along the guiding direction. The light propagation is along the z axis. The dominant component of the electric field is along the y direction [in (a)], and thus the field is TE polarized for the equivalent 2D waveguide in (b).

Fig. 2
Fig. 2

(Color online) (a) Real and (b) imaginary parts of the effective index of different modes for the multimode plasmonic waveguides as the waveguide width varies while the thickness of the dielectrics is 100   nm . The solid curves are the values obtained from the H-form FD mode solver (without 2D approximation), whereas the dots are from the 2D analog.

Fig. 3
Fig. 3

(Color online) (a) Real and (b) imaginary parts of the effective index of different modes for the multimode plasmonic waveguides as the waveguide width varies while the thickness of the dielectrics is 50   nm . The solid curves are the values obtained from the H-form FD mode solver (without 2D approximation), whereas the dots are from the 2D analog.

Fig. 4
Fig. 4

(Color online) (a) Total electric field intensity inside the whole structure of a combination of a single-mode plasmonic waveguide and an IG-MPW. (b) Electric field intensity calculated with a 2D model.

Fig. 5
Fig. 5

(Color online) Total electric field intensity distribution inside the 1 × 2 MMI power splitter obtained from (a) a 2D TE FDTD based on the 2D approximation and (b) 3D FDTD for IG-MPWs.

Fig. 6
Fig. 6

(Color online) (a) Intensity distribution of the electric field from a 2D TE FDTD simulation based on 2D approximation in a 1 × 3 MMI splitter. (b) Electric field intensity distribution at the output ports. (c) Electric field intensity on the output plane obtained from the 3D FDTD simulation for the 3D real structure.

Equations (2)

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tanh ( 1 2 α i d ) = ε i α m ε m α i ,
α i = n eff 2 k 0 2 ε i k 0 2 , α m = n eff 2 k 0 2 ε m k 0 2 ,

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