Abstract

A metal-insulator-metal (MIM) waveguide can support two plasmonic modes. Efficient conversion between the two modes can be achieved by reshaping of both phase and power density distributions of the guided mode. The converters are designed with the assistance of transformation optics. We propose two practical configurations for mode conversion, which only consist of homogeneous materials yielded from linear coordinate transformations. The functionalities of the converters are demonstrated by full wave simulations. Without consideration of transmission loss, conversion efficiency of as high as 95% can be realized.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2012 (3)

2011 (4)

2010 (3)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett.10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Z. Han, A. Y. Elezzabi, and V. Van, “Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform,” Opt. Lett.35(4), 502–504 (2010).
[CrossRef] [PubMed]

2009 (2)

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009).
[CrossRef]

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett.95(1), 013504 (2009).
[CrossRef]

2006 (4)

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,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407 (2006).
[CrossRef]

L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett.31(14), 2133–2135 (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(23), 3447–3449 (2006).
[CrossRef] [PubMed]

2005 (3)

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(26), 261114 (2005).
[CrossRef]

J.-C. Weeber, M. U. Gonzalez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett.87(22), 221101 (2005).
[CrossRef]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408(3-4), 131–314 (2005).
[CrossRef]

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter44(24), 13556–13572 (1991).
[CrossRef] [PubMed]

1983 (1)

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407 (2006).
[CrossRef]

Barbastathis, G.

Bartal, G.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett.10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Baudrion, A.-L.

J.-C. Weeber, M. U. Gonzalez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett.87(22), 221101 (2005).
[CrossRef]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[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, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Chen, L.

Chen, X.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Dereux, A.

J.-C. Weeber, M. U. Gonzalez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett.87(22), 221101 (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,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407 (2006).
[CrossRef]

Dong, J.-W.

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,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Elezzabi, A. Y.

Feng, Y.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009).
[CrossRef]

Fukui, M.

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(26), 261114 (2005).
[CrossRef]

Garcia-Vidal, F. J.

Gonzalez, M. U.

J.-C. Weeber, M. U. Gonzalez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett.87(22), 221101 (2005).
[CrossRef]

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[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(26), 261114 (2005).
[CrossRef]

Han, T.

Han, Z.

Hao, Y.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009).
[CrossRef]

Haraguchi, M.

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(26), 261114 (2005).
[CrossRef]

Jiang, K.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Jiang, T.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009).
[CrossRef]

Kim, M. S.

Krasavin, A. V.

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,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Lee, M.-H.

Lipson, M.

Liu, L.

Liu, Y.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett.10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Luo, Y.

J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structures,” Opt. Express19(9), 8625–8631 (2011).
[CrossRef] [PubMed]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408(3-4), 131–314 (2005).
[CrossRef]

G. I. Stegeman, R. F. Wallis, and A. A. Maradudin, “Excitation of surface polaritons by end-fire coupling,” Opt. Lett.8(7), 386–388 (1983).
[CrossRef] [PubMed]

Martin-Moreno, L.

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(26), 261114 (2005).
[CrossRef]

Moreno, E.

Mortensen, N. A.

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter44(24), 13556–13572 (1991).
[CrossRef] [PubMed]

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(26), 261114 (2005).
[CrossRef]

Okamoto, 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(26), 261114 (2005).
[CrossRef]

Park, H.-R.

Park, J.-M.

Pendry, J. B.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Pile, D. F. P.

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(26), 261114 (2005).
[CrossRef]

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407 (2006).
[CrossRef]

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter44(24), 13556–13572 (1991).
[CrossRef] [PubMed]

Premaratne, M.

Qiu, C.-W.

Qiu, M.

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett.95(1), 013504 (2009).
[CrossRef]

Rodrigo, S. G.

Rukhlenko, I. D.

Shakya, J.

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408(3-4), 131–314 (2005).
[CrossRef]

Stegeman, G. I.

Sun, H.

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407 (2006).
[CrossRef]

Tang, X.

Tian, J.

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett.95(1), 013504 (2009).
[CrossRef]

Van, V.

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(26), 261114 (2005).
[CrossRef]

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter44(24), 13556–13572 (1991).
[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,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Wallis, R. F.

Weeber, J.-C.

J.-C. Weeber, M. U. Gonzalez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett.87(22), 221101 (2005).
[CrossRef]

Xu, H.

Xu, X.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009).
[CrossRef]

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(26), 261114 (2005).
[CrossRef]

Yan, W.

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett.95(1), 013504 (2009).
[CrossRef]

Yu, S.

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett.95(1), 013504 (2009).
[CrossRef]

Yu, T.

Zayats, A. V.

Zentgraf, T.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett.10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Zhang, B.

Zhang, J.

J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structures,” Opt. Express19(9), 8625–8631 (2011).
[CrossRef] [PubMed]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Zhang, S.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structures,” Opt. Express19(9), 8625–8631 (2011).
[CrossRef] [PubMed]

Zhang, X.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett.10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Zhao, J.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009).
[CrossRef]

Zhu, W.

Zouhdi, S.

Appl. Phys. Lett. (4)

J.-C. Weeber, M. U. Gonzalez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett.87(22), 221101 (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(26), 261114 (2005).
[CrossRef]

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett.95(1), 013504 (2009).
[CrossRef]

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nano Lett. (1)

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett.10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Nat Commun (1)

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[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,” Nature440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408(3-4), 131–314 (2005).
[CrossRef]

Phys. Rev. B (1)

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407 (2006).
[CrossRef]

Phys. Rev. B Condens. Matter (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter44(24), 13556–13572 (1991).
[CrossRef] [PubMed]

Other (2)

U. Leonhardt and T. G. Philbin, Transformation Optics and the Geometry of Light (Elsevier Science Bv, Amsterdam, 2009), vol. 53, pp. 69–152.

E. D. Palik, Handbook of optical constants of solids (Academic Press, London, 1985).

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Figures (5)

Fig. 1
Fig. 1

Basic properties of the plasmonic eigenmodes (the ab and sb modes) in MIM. (a) Schematic and notations of MIM waveguide; (b) Cutoff properties; (c) Typical field profiles (red lines) and phase distributions [arg(Ez), blue lines] of Ez at z = 0; (d) Typical profiles of power density, where the powers of the two modes are normalized equally. In (b)-(d), solid and dashed lines correspond to the ab and sb modes, respectively. In (c) and (d), the insulator thickness is 300 nm.

Fig. 2
Fig. 2

Mode conversion by only phase reshaping. (a) Physical and virtual spaces of converter region. The horizontal and vertical grids in both the spaces show constant x' and constant z', respectively. (b) Power flows of the ab and sb component, P(a) and P(s), and the reflected power flow as functions of the length difference between the virtual and physical spaces, (l'l), at z = l + 600 nm. Here l is fixed at 500 nm, d = 300 nm, and incident power is assumed to unit. (c) Snapshot of Ez field when (l'l) = 230 nm. White line shows where P(a) and P(s) are extracted in (b).

Fig. 3
Fig. 3

Mode conversion by reshaping both phase and power density. (a) and (d) show two divisions of the physical and virtual spaces. In each panels, the horizontal and vertical grids show contours of x' and z', respectively. (b) and (e) show the power flows of transmitted ab and sb components, P(a) and P(s), as well as the reflected power, as functions of (l'l), corresponding with (a) and (d) respectively. l = 500 nm, d = 300 nm and d1 = 75 nm. The incident power flow is unit. The power flows are extracted at z = l + 500 nm. (c) and (f) show the snapshots of Ez fields at the maximum conversion efficiencies of (a) and (d) respectively. In (c) and (f), (l'l) is equal to 226 and 220 nm respectively. White lines show where the data of power flow are extracted.

Fig. 4
Fig. 4

Mode conversion in MIM waveguides with real metal. (a) Ez snapshot with d = 300 nm, d1 = 80 nm, l = 500 nm and (l'l) = 224 nm. (b) Ez snapshot with d = 200 nm, d1 = 34 nm, l = 500 nm and (l'l) = 210 nm.

Fig. 5
Fig. 5

Mode conversion in metal slot waveguides. (a) and (b) show Ez snapshots using the configurations shown in Figs. 3(a) and 3(d) respectively. Inset of (a) shows the cross section of the slot waveguide.

Equations (16)

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E=( E x ,0, E z ) e i( k 0 n eff zωt) , H=(0, H y ,0) e i( k 0 n eff zωt) ,
a b : tanh dU 2 = ε 1 W ε m U ; s b : tanh dU 2 = ε m U ε 1 W ,
ε= ε 1 A A T detA , μ= μ 1 A A T detA ,
x= x , y= y , z=(l/ l ) z ,
ε z,x =diag( ε 1 , ε 1 ( l /l) 2 ).
x= x ,y= y ,z= l l z ; x= d 1 d/2 x d/2 d 1 l z + d 2 d 1 ,y= y ,z= l l z ; x= d 1 d/2 x + d/2 d 1 l z + d 2 d 1 ,y= y ,z= z ;
ε z,x = ε 1 diag(1, A 2 ) ε z,x = ε 1 [ C 2 A B 1 C 2 A B 1 C 2 A 2 ( B 1 2 C 2 +1) ] ε z,x = ε 1 [ C 2 B 2 C 2 B 2 C 2 B 2 2 C 2 +1 ]
x= d 1 d/2 x ,y= y ,z= l l z ; x= x d/2 d 1 l z ,y= y ,z= l l z ; x= x + d/2 d 1 l z ,y= y ,z= z ; x= d 1 d/2 x +d2 d 1 ,y= y ,z= z .
ε z,x = ε 1 diag( C 2 , A 2 ) ε z,x = ε 1 [ 1 A B 1 A B 1 A 2 ( B 1 2 +1) ] ε z,x = ε 1 [ 1 B 2 B 2 B 2 2 +1 ] ε z,x = ε 1 diag( C 2 ,1)
E x ={ iA n eff k 0 W e Wx , x<0 iA n eff k 0 U [ ε m U ε 1 W coshUx+sinhUx ], 0<x<d imA n eff k 0 W e W(xd) , x>d
E z ={ A e Wx , x<0 A[ ε m U ε 1 W sinhUx+coshUx ], 0<x<d mA e W(xd) , x>d
H y ={ iA ε m k 0 μ 0 cW e Wx , x<0 iA ε 1 k 0 μ 0 cU [ ε m U ε 1 W coshUx+sinhUx ], 0<x<d imA ε m k 0 μ 0 cW e W(xd) , x>d
A (a) = 1 2 [ E z | x=0 E z | x=d ], A (s) = 1 2 [ E z | x=0 + E z | x=d ].
P(z)=Re + 1 2 E x H y * dx e 2Im( n eff ) k 0 z .
+ 1 2 E x H y * dx = | A | 2 ε m * n eff k 0 2 μ 0 c|W | 2 1 W+ W * + 1 2 | A | 2 ε 1 n eff k 0 2 μ 0 c|U | 2 × { 1 U+ U * [ R+ R * 2 cosh(U+ U * )d+ R R * +1 2 sinh(U+ U * )d R+ R * 2 ] + 1 U U * [ R * R 2 cosh(U U * )d+ R R * 1 2 sinh(U U * )d R * R 2 ] }
p c (z,x)= 1 2 Re[ E x (a) H y (s)* e i( n eff (a) n eff (s)* ) k 0 z + E x (s) H y (a)* e i( n eff (s) n eff (a)* ) k 0 z ]

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