Abstract

A numerical study of the complex propagation constants of a surface plasmon polariton gap waveguide (SPGW) that was nanometric in size is performed by the method of lines (MoL). The validity of the code based on the MoL is examined by comparing the present results with those calculated using a volume integral equation, which is a completely different numerical technique from the MoL. The dependences of the complex propagation constants on the sizes of the SPGWs are investigated in detail and the fundamental propagation characteristics of SPGWs are revealed. Three kinds of SPGW structures (slab-slab, slab-plate and staggered slab-slab) are examined with a view to reducing the attenuation constants and the spot size to nanometric size. It is found that the nanometric field confinement can be controlled by using the staggered slab-slab structure of SPGW without a large change in the propagation constants.

© 2009 Optical Society of America

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2008 (4)

2007 (5)

2006 (4)

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 013101 (2006).
[CrossRef]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef]

L. Chen, J. Shakya, and M. Lipson, "Subwavelength confinement in an integrated metal slot waveguide on silicon," Opt. Lett. 31, 2133-2135 (2006).
[CrossRef] [PubMed]

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

2005 (6)

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]

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

G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359-3361 (2005).
[CrossRef] [PubMed]

G.  Veronis and S.  Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett.  87, 131102 (2005).
[CrossRef] [PubMed]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, and M. Fukui, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005).
[CrossRef]

2004 (2)

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 (2004).
[CrossRef]

B. Wang and G. P. Wang, "Surface plasmon polariton propagation in nanoscale metal gap waveguides," Opt. Lett. 29, 1992-1994 (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]

2001 (1)

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

2000 (1)

1997 (1)

1993 (1)

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).

1986 (1)

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).

Atwater, H. A.

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Berini, P.

Bozhevolnyi, S. I.

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).

Chang, S. H.

Chen, L.

Chiu, T. C.

Devaux, E.

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

Dionne, J. A.

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef]

Ebbesen, T. W.

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

Fan, S.

Feigenbaum, E.

Fukui, M.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 013101 (2006).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, and M. Fukui, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Gramotnev, D. K.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 013101 (2006).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, and M. Fukui, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[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 (2004).
[CrossRef]

Han, Z.

Haraguchi, M.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 013101 (2006).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, and M. Fukui, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

He, S.

Hoffman, G. B.

Jung, J.

Katayama, K.

K. Tanaka, M. Tanaka, K. Katayama, and D. Miyahara, "Propagation constants of guided waves in surface plasmon polariton gap waveguides excited through an I-shaped aperture," C. R. Phys. 9, 16- 23 (2008).
[CrossRef] [PubMed]

Kik, P. G.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Kobayashi, T.

Laluet, J.-Y.

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

Lezec, H. J.

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef]

Lipson, M.

Liu, L.

Maier, S. A.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Matsuzaki, Y.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Meltzer, S.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Minh, T. T.

Miyahara, D.

K. Tanaka, M. Tanaka, K. Katayama, and D. Miyahara, "Propagation constants of guided waves in surface plasmon polariton gap waveguides excited through an I-shaped aperture," C. R. Phys. 9, 16- 23 (2008).
[CrossRef] [PubMed]

Morimoto, A.

Ogawa, T.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, and M. Fukui, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005).
[CrossRef]

Okamoto, T.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 013101 (2006).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, and M. Fukui, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Orenstein, M.

Pile, D. F. P.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, "Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap," J. Appl. Phys. 100, 013101 (2006).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, and M. Fukui, "Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding," Appl. Phys. Lett. 87, 061106 (2005).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[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 (2004).
[CrossRef]

Pregla, R.

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).

Reano, R. M.

Requicha, A. A. G.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Rogge, U.

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).

Satuby, Y.

Shakya, J.

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).

Sugiyama, T.

Tai, C.

Takahara, J.

Taki, H.

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).

Tanaka, K.

K. Tanaka, M. Tanaka, K. Katayama, and D. Miyahara, "Propagation constants of guided waves in surface plasmon polariton gap waveguides excited through an I-shaped aperture," C. R. Phys. 9, 16- 23 (2008).
[CrossRef] [PubMed]

T. T. Minh, K. Tanaka, and M. Tanaka, "Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines," Opt. Express 16, 9378-9390 (2008).
[CrossRef]

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]

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, K. Katayama, and D. Miyahara, "Propagation constants of guided waves in surface plasmon polariton gap waveguides excited through an I-shaped aperture," C. R. Phys. 9, 16- 23 (2008).
[CrossRef] [PubMed]

T. T. Minh, K. Tanaka, and M. Tanaka, "Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines," Opt. Express 16, 9378-9390 (2008).
[CrossRef]

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]

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]

Vernon, K. C.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Veronis, G.

Volkov, V. S.

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

Wang, B.

Wang, G. P.

Yamagishi, S.

Yamaguchi, K.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Adv. Mater. (1)

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Figures (12)

Fig. 1.
Fig. 1.

Geometry of the cross section of the I-shaped structure of the SPGW which is created in the metallic material of size A×B. A narrow-gap region of a x × a y is sandwiched between two wide-gap regions of b x × b y in the metallic materials in the x-y plane. The permittivities of the metal and the space inside the waveguide are given by ε 2 and ε 1, respectively.

Fig. 2.
Fig. 2.

Comparison between propagation constants calculated by the MoL and those calculated by the VIE method [26]. The open and solid circles show the results obtained by the MoL and those obtained by the VIE method as given in ref. [26], respectively. The dependences of normalized phase and attenuation constants on the narrow-gap width ax are shown in (a) and (b) respectively, while their dependences on the narrow-gap depth ay are shown in (c) and (d), respectively. The relative permittivity of the metal and space inside the waveguide are given by ε 2/ε 0= −13.2 − j1.08 (Au) and ε 1/ε 0 =1, respectively.

Fig. 3.
Fig. 3.

Comparison between field distributions calculated by the MoL and those by the VIE method [26] for the case of ax × ay = 101 nm×101 nm. Distributions shown in (a), (b) and (c) represent the Re[Ex (x,y)], Re[Ey (x,y)] and Re[Ez (x,y)] calculated by the MoL respectively, while those shown in (c), (d) and (e) represent Re[Ex (x,y), Re[Ey (x,y)] and Re[Ez (x,y)] calculated by the VIE method, respectively. The intensity scale is arbitrary.

Fig. 4.
Fig. 4.

The dependences of the normalized phase constant β and the propagation length L on the wide-gap height b y are shown respectively in (a) and (b) for five values of the wide-gap width b x. The size of narrow-gap region is fixed to a x × a y = 20 nm×20 nm. (ε 2/ε 0= −13.2 −j1.08, ε 1/ε 0 =1).

Fig. 5.
Fig. 5.

The dependences of the normalized phase constant β and the propagation length L on the wide-gap width b x are shown in (a) and (b), respectively, for three values of the cross section of the narrow-gap region a x × a y. The wide-gap height b y is fixed to a large value (b y=600 nm). The smallest value of b x corresponds to a waveguide that consists of two plates; i.e., a x =b x (ε 0/ε 0= −13.2 − j1.08, ε1/ε 0 =1).

Fig. 6.
Fig. 6.

The dependences of the normalized phase constant β and the propagation length L on the narrow-gap depth a y are shown in (a) and (b), respectively, for three values of the narrow-gap width a x. The results for the case of SPP between two plates are indicated by the straight lines (ε 2/ε 0= −13.2 − j1.08, ε 1/ε 0 =1).

Fig. 7.
Fig. 7.

The dependences of the normalized phase constant β and the propagation length L on the gap width a x are respectively shown in (a) and (b) for five values of the slab thicknesses a y. The results for the case when one slab is removed (i.e., a x =∞) are indicated by the straight lines (ε 2/ε 0= -13.2 − j1.08, ε 1/ε 0 =1).

Fig. 8.
Fig. 8.

The dependences of the normalized phase constant β and the propagation length on the gap width a x of the slab-plate structure SPGW shown in the inset are shown in (a) and (b) respectively, with the slab thickness a y as a parameter. The results for the edge mode are indicated by the straight lines. The open squares represent the results for the slab-slab structures shown in Fig. 7 (ε 2/ε 0= -13.2 − j1.08, ε 1/ε 0 =1).

Fig. 9.
Fig. 9.

Typical distributions of the electric field components (a) Re[E x(x,y)], (b) Re[E y(x,y)] and (c) Re[E z(x,y)] of the slab-plate structure for a x=101 nm and a y=101 nm.

Fig. 10.
Fig. 10.

The dependences of the normalized phase constant β and the propagation length L on the narrow-gap width a x are shown in (a) and (b) respectively, for five values of the staggered parameter S (see the inset). The slab thickness is fixed to a y=101 nm (ε 2/ε 0= -13.2 − j1.08, ε 1/ε 0 =1).

Fig. 11.
Fig. 11.

Typical distributions of the main electric field component Re[E x(x,y)] of the staggered slab-slab structure of SPGW for (a) S=101 nm, (b) S=60 nm, (c) S=40 nm and (d) S=20 nm. The narrow-gap width a x and slab thickness a y are fixed to a x=40 nm and a y=101 nm, respectively.

Fig. 12.
Fig. 12.

Typical distributions of the main electric field component Re[E x(x,y)] of the staggered slab-slab structure of SPGW for (a) S=202 nm and (b) S=20 nm. The narrow-gap width a x and slab thickness a y are fixed to a x=40 nm and a y=202 nm, respectively

Equations (4)

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E=εr1(x) ××Πejk0×Πh
η0H=jk0×Πe+××Πh
Πe,h=k02 Ψe,h ejkzz ix
[Ȳ(kz)]Ē=0,

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