Abstract

This study uses a full vector pseudospectral scheme in the frequency domain to investigate the mode characteristics of surface plasmon-polariton (SPP) waveguides. The wave equations solved in this study are based on the transverse magnetic field components, and thus the spurious modes are removed due to the constraint of divergence-free magnetic vector. The waveguide dimension dependences on the mode confinement and propagation length of the dielectric-loaded surface plasmon-polariton waveguide (DLSPPW) are extensively studied and characterized. The numerical results of the DLSPPW show that the proposed scheme is highly efficient and yields accurate complex effective indices while requiring much less memory than the commonly used finite element method. This study also analyzes the propagation characteristics and figures of merit of an inverted metal slot waveguide (IMSW) in detail. The IMSW achieves a propagation loss an order of magnitude lower than nanoparticle chains with comparable degrees of lateral confinement.

© 2010 OSA

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    [CrossRef] [PubMed]

2010

2009

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009).
[CrossRef] [PubMed]

G. H. Yuan, P. Wang, Y. H. Lu, and H. Ming, “Multimode interference splitter based on dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 17(15), 12594–12600 (2009).
[CrossRef] [PubMed]

C. C. Huang, “Improved pseudospectral mode solver by prolate spheroidal wave functions for optical waveguides with step-index,” J. Lightwave Technol. 27(5), 597–605 (2009).
[CrossRef]

K. Y. Jung, F. L. Teixeira, and R. M. Reano, “Surface plasmon coplanar waveguides: mode characteristics and mode conversion losses,” IEEE Photon. Technol. Lett. 21(10), 630–632 (2009).
[CrossRef]

M. Fujii, J. Leuthold, and W. Freude, “Dispersion relation and loss of subwavelength confined mode of metal-dielectric-gap optical waveguides,” IEEE Photon. Technol. Lett. 21(6), 362–364 (2009).
[CrossRef]

K. Tanaka, T. T. Minh, and M. Tanaka, “Analysis of propagation characteristics in the surface plasmon polariton gap waveguides by method of lines,” Opt. Express 17(2), 1078–1092 (2009).
[CrossRef] [PubMed]

J. Grandidier, G. C. des Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

G. Colas des Francs, J. Grandidier, S. Massenot, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Integrated plasmonic waveguides: a mode solver based on density of states formulation,” Phys. Rev. B 80(11), 115419 (2009).
[CrossRef]

Y. Binfeng, H. Guohua, and C. Yiping, “Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides,” Opt. Express 17(5), 3610–3618 (2009).
[CrossRef] [PubMed]

D. X. Dai and S. L. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009).
[CrossRef] [PubMed]

2008

A. Hosseini, A. Nieuwoudt, and Y. Massoud, “On the design of dielectric strip plasmonic structures for subwavelength waveguiding applications,” IEEE Trans. NanoTechnol. 7(2), 189–196 (2008).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

T. T. Minh, K. Tanaka, and M. Tanaka, “Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines,” Opt. Express 16(13), 9378–9390 (2008).
[CrossRef] [PubMed]

S. Massenot, J. C. Weeber, A. Bouhelier, G. Colas des Francs, J. Grandidier, L. Markey, and A. Dereux, “Differential method for modeling dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 16(22), 17599–17608 (2008).
[CrossRef] [PubMed]

C. C. Huang, “Simulation of optical waveguides by novel full-vectorial pseudospectral-based imaginary-distance beam propagation method,” Opt. Express 16(22), 17915–17934 (2008).
[CrossRef] [PubMed]

J. Grandidier, S. Massenot, G. C. Francs, A. Bouhelier, J. C. Weeber, L. Markey, A. Dereux, J. Renger, M. U. González, and R. Quidant, “Dielectric-loaded surface plasmon polariton waveguides: figures of merit and mode characterization by image and Fourier plane leakage microscopy,” Phys. Rev. B 78(24), 245419 (2008).
[CrossRef]

2007

2006

2005

J. Shibayama, T. Yamazaki, J. Yamauchi, and H. Nakano, “Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam propagation method,” J. Lightwave Technol. 23(3), 1533–1539 (2005).
[CrossRef]

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (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(4), 046802 (2005).
[CrossRef] [PubMed]

G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30(24), 3359–3361 (2005).
[CrossRef]

2004

D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004).
[CrossRef] [PubMed]

S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron. 40(3), 325–329 (2004).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82(5), 668–670 (2003).
[CrossRef]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[CrossRef] [PubMed]

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21(10), 2284–2296 (2003).
[CrossRef]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[CrossRef] [PubMed]

2000

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000).
[CrossRef]

P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express 7(10), 329–335 (2000).
[CrossRef] [PubMed]

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000).
[CrossRef]

1997

1996

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996).
[CrossRef]

1986

A. J. Kobelansky and J. P. Webb, “Eliminating spurious modes in finite-element waveguide problems by using divergence-free fields,” Electron. Lett. 22(11), 569–570 (1986).
[CrossRef]

1984

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microw. Theory Tech. 32(8), 922–928 (1984).
[CrossRef]

Al-Bader, S. J.

S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron. 40(3), 325–329 (2004).
[CrossRef]

Andersen, T. B.

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[CrossRef] [PubMed]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[CrossRef] [PubMed]

Aussenegg, F. R.

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104–1 (2006).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Berini, P.

Binfeng, Y.

Bouhelier, A.

J. Grandidier, G. C. des Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009).
[CrossRef] [PubMed]

G. Colas des Francs, J. Grandidier, S. Massenot, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Integrated plasmonic waveguides: a mode solver based on density of states formulation,” Phys. Rev. B 80(11), 115419 (2009).
[CrossRef]

J. Grandidier, S. Massenot, G. C. Francs, A. Bouhelier, J. C. Weeber, L. Markey, A. Dereux, J. Renger, M. U. González, and R. Quidant, “Dielectric-loaded surface plasmon polariton waveguides: figures of merit and mode characterization by image and Fourier plane leakage microscopy,” Phys. Rev. B 78(24), 245419 (2008).
[CrossRef]

S. Massenot, J. C. Weeber, A. Bouhelier, G. Colas des Francs, J. Grandidier, L. Markey, and A. Dereux, “Differential method for modeling dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 16(22), 17599–17608 (2008).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup-Hansen, L. Markey, and A. Dereux, “Thermo-optic control of dielectric-loaded plasmonic waveguide components,” Opt. Express 18(2), 1207–1216 (2010).
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009).
[CrossRef] [PubMed]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

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(23), 3447–3449 (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(4), 046802 (2005).
[CrossRef] [PubMed]

T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82(5), 668–670 (2003).
[CrossRef]

Chang, S. H.

Chen, L.

Chen, Z.

Chiu, T. C.

Colas des Francs, G.

G. Colas des Francs, J. Grandidier, S. Massenot, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Integrated plasmonic waveguides: a mode solver based on density of states formulation,” Phys. Rev. B 80(11), 115419 (2009).
[CrossRef]

S. Massenot, J. C. Weeber, A. Bouhelier, G. Colas des Francs, J. Grandidier, L. Markey, and A. Dereux, “Differential method for modeling dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 16(22), 17599–17608 (2008).
[CrossRef] [PubMed]

Conway, J. A.

Dai, D. X.

Davies, J. B.

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microw. Theory Tech. 32(8), 922–928 (1984).
[CrossRef]

Dereux, A.

J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup-Hansen, L. Markey, and A. Dereux, “Thermo-optic control of dielectric-loaded plasmonic waveguide components,” Opt. Express 18(2), 1207–1216 (2010).
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009).
[CrossRef] [PubMed]

G. Colas des Francs, J. Grandidier, S. Massenot, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Integrated plasmonic waveguides: a mode solver based on density of states formulation,” Phys. Rev. B 80(11), 115419 (2009).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

(a) The cross section of DLSPPW with refractive indices of core np , gold film ng , substrate nd , and air na .(b) the division of computational domain for the DLSPPW structure.

Fig. 2
Fig. 2

(a) The computational time and (b) the memory capacity versus the number of unknown of the proposed scheme.

Fig. 3
Fig. 3

The relative errors of (a) the real parts of neff and (b) the propagation lengths Lc (μm) of the proposed scheme at t = 0.6μm versus different width t. The reference values are obtained by the FEM [17].

Fig. 4
Fig. 4

The relative errors of (a) the real parts of neff and (b) the propagation lengths Lc of the proposed scheme at w = 0.5μm versus different thicknesses t. The reference values are obtained by the FEM [17].

Fig. 5
Fig. 5

(a) The real parts of effective index and (b) the propagation lengths of DLSPPW versus the ridge width w for several thicknesses t.

Fig. 6
Fig. 6

(a) The lateral mode extents and (b) the confinement areas of DLSPPW versus the ridge width w for several thicknesses t.

Fig. 7
Fig. 7

The mode intensity profiles |Hx |2 of DLSPPW at w = 0.8μm for (a) t = 0.6μm (b) t = 0.4μm (c) t = 0.3μm.

Fig. 8
Fig. 8

The mode intensity profiles |Hx |2 of DLSPPW at w = 0.2μm for (a) t = 0.6μm (b) t = 0.4μm (c) t = 0.3μm.

Fig. 9
Fig. 9

The mode intensity profiles |Hx |2 of DLSPPW at w = 0.4μm and t = 0.3μm for (a) λ = 1.55μm (b) λ = 1.22μm (c) λ = 0.893μm.

Fig. 10
Fig. 10

The cross section of the IMSW with refractive indices of core nSi , gold cladding nAu , substrate nSiO2 , and silica cladding nSiO2 .

Fig. 11
Fig. 11

(a) The real parts of effective index and (b) the propagation lengths of the IMSW versus the core width w for several different core thicknesses t.

Fig. 12
Fig. 12

The mode intensity profiles |Hy |2 of the IMSW at t = 250nm for (a) w = 150nm (b) w = 100nm (c) w = 50nm (d) w = 10nm.

Fig. 13
Fig. 13

The mode intensity profiles |Hy |2 of the IMSW at w = 150nm for (a) t = 230nm (b) t = 130nm (c) t = 50nm (d) t = 10nm.

Fig. 14
Fig. 14

(a) The real parts of effective index and (b) the propagation lengths of the IMSW versus the core thickness t for three different core widths of w.

Fig. 15
Fig. 15

(a) Mode beam height in vertical direction versus the core thickness t for three different core width w and (b) the figure of merit (FOM) versus the core width w for several core thicknesses t of the IMSW.

Fig. 16
Fig. 16

The mode intensity profiles |Hy |2 of the IMSW at the optimum FOMs of w = 30nm for (a) t = 250nm (b) t = 150nm (c) t = 50nm, and that at (d) w = 20nm and t = 10nm.

Fig. 17
Fig. 17

The mode intensity profiles |Hy |2 of the IMSW at t = 250nm and w = 150nm for (a) λ = 1.22μm (b) λ = 0.893μm.

Tables (4)

Tables Icon

Table 1 The convergence of the real parts of effective index calculated by the proposed scheme for different widths w of ridge at t = 0.6μm versus the terms of basis functions.

Tables Icon

Table 2 The convergence of the propagation lengths L c(μm) calculated by the proposed scheme for different widths w of ridge at t = 0.6μm versus the terms of basis functions.

Tables Icon

Table 3 Comparisons of the real parts of effective index obtained by the Finite Element Method (FEM), the differential method (DM), and the proposed scheme (this work) for different widths w of ridge at t = 0.6μm.

Tables Icon

Table 4 Comparisons of the propagation lengths L c (μm) obtained by Finite Element Method (FEM), the differential method (DM), and the proposed scheme (this work) for different widths w of ridge at t = 0.6μm.

Equations (14)

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2 H + k 0 2 n 2 H + n 2 n 2 × ( × H ) = 0
2 H s x 2 + 2 H s y 2 + k 0 2 ( n 2 n e f f 2 ) H s = 0 ,     ( s = x , y ) .
n y + 2 H x y | y n y 2 H x y | y + = ( n y + 2 n y 2 ) H y x
H y y | y + = H y y | y
n x + 2 H y x | x n x 2 H y x | x + = ( n x + 2 n x 2 ) H x y
H x x | x + = H x x | x
H s r ( x , y ) = p = 0 n x q = 0 n y θ p r ( x ) ψ q r ( y ) H s , p q r ,     ( s = x , y )
[ P r 0 0 P r ] [ H x r H y r ] = ( k 0 n e f f ) 2 [ H x r H y r ] ,
P r H s r = [ 2 H s r x 2 + 2 H s r y 2 + k 0 2 n r 2 H s , p q r ] | x = x i , y = y j , ( s = x , y )
= i = 1 n x 1 j = 1 n y 1 [ p = 0 n x q = 0 n y { θ p r ( 2 ) ( x ) ψ q r ( y ) + θ p r ( x ) ψ q r ( 2 ) ( y ) + k 0 2 n r 2 ( x , y ) θ p r ( x ) ψ q r ( y ) } ] | x = x i , y = y j [ H s , p q r ]
[ Q 1 0 0 0 0 Q 2 0 0 0 0 0 0 0 0 Q s ] [ H 1 H 2 H s ] = ( k 0 n e f f ) 2 [ H 1 H 2 H s ]
Q r = [ P r 0 0 P r ] , Η r = [ H x r H y r ] ,     ( r = 1 , 2 , 3... m ) .
θ p ( x ) = ( 1 ) p + 1 ( 1 x 2 ) T v ' ( x ) c p n 2 ( x x p ) , c p = { 2 , if p =0,   N 1, if 1 p N -1
θ p ( α x ) = x L v ( α x ) α ( x x p ) [ x L v ' ( α x ) ] | x = x p e α ( x x p ) / 2 ,

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