Abstract

We theoretically study the plasmonic modes in metal-multi-insulator-metal (MMIM) waveguides. Two types of symmetric MMIM structures consisting of three insulators are investigated thoroughly. The effective refractive index, energy confinement, propagation length, and figure of merit are given in terms of geometric parameters. Due to the step index modulation, these properties of MMIM structures differ from the metal-insulator-metal (MIM) structure. Compared with the corresponding MIM structures, MMIM structures can possess either better energy confinement or larger propagation length, which depends on the geometric parameters and the index distribution. Propagation length of up to 103 µm and a figure of merit of up to 104 are observed for MMIM structure with core thickness of several hundred nanometers.

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2011 (4)

X. Zhu, J. Zhang, J. Xu, and D. Yu, “Vertical plasmonic resonant nanocavities,” Nano Lett. 11(3), 1117–1121 (2011).
[CrossRef] [PubMed]

Y. A. Akimov and H. S. Chu, “Plasmon coupling effect on propagation of surface plasmon polaritons at a continuous metal/dielectric interface,” Phys. Rev. B 83(16), 165412 (2011).
[CrossRef]

S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011).
[CrossRef]

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[CrossRef] [PubMed]

2010 (8)

2009 (4)

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009).
[CrossRef] [PubMed]

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]

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

2008 (3)

J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,” Opt. Express 16(1), 413–425 (2008).
[CrossRef] [PubMed]

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

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78(4), 045425 (2008).
[CrossRef]

2007 (5)

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[CrossRef]

Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
[CrossRef]

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

N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett. 32(21), 3086–3088 (2007).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

2006 (6)

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006).
[CrossRef]

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. B 73(3), 035407 (2006).
[CrossRef]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006).
[CrossRef] [PubMed]

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[CrossRef] [PubMed]

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (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,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

2005 (4)

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (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]

D. F. P. Pile and D. K. Gramotnev, “Nanoscale fabry-prot interferometer using channel plasmon-polaritons in triangular metallic grooves,” Appl. Phys. Lett. 86(16), 161101 (2005).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005).
[CrossRef]

2004 (2)

2000 (1)

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]

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter 44(24), 13556–13572 (1991).
[CrossRef] [PubMed]

1981 (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
[CrossRef]

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

Akimov, Y. A.

Y. A. Akimov and H. S. Chu, “Plasmon coupling effect on propagation of surface plasmon polaritons at a continuous metal/dielectric interface,” Phys. Rev. B 83(16), 165412 (2011).
[CrossRef]

Almeida, V. R.

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. B 73(3), 035407 (2006).
[CrossRef]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005).
[CrossRef]

Barrios, C. A.

Bartal, G.

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009).
[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]

Berini, P.

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]

Bouhelier, A.

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[CrossRef]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[CrossRef]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (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 resonators,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Brongersma, M. L.

Buchholz, D. B.

Catrysse, P. B.

Chang, R. P.

Chang, Y.-J.

Y.-J. Chang and G.-Y. Lo, “A narrow band metal-multi-insulator-metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. 22(9), 634–636 (2010).
[CrossRef]

Y.-J. Chang, “Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating,” Opt. Express 18(12), 13258–13270 (2010).
[CrossRef] [PubMed]

Chen, Z.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

Chu, H. S.

Y. A. Akimov and H. S. Chu, “Plasmon coupling effect on propagation of surface plasmon polaritons at a continuous metal/dielectric interface,” Phys. Rev. B 83(16), 165412 (2011).
[CrossRef]

Colas des Francs, G.

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[CrossRef]

Dai, D.

Dal Negro, L.

de Vries, T.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

de Waardt, H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

Dereux, A.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[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]

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

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(9), 1928–1932 (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. B 73(3), 035407 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005).
[CrossRef]

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

Economou, E. N.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

Eijkemans, T. J.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

Fan, S.

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005).
[CrossRef]

Feng, N.-N.

Geluk, E. J.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

Genov, D. A.

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

Gong, Y.

Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
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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. Photonics 4(2), 83–91 (2010).
[CrossRef]

D. F. P. Pile and D. K. Gramotnev, “Nanoscale fabry-prot interferometer using channel plasmon-polaritons in triangular metallic grooves,” Appl. Phys. Lett. 86(16), 161101 (2005).
[CrossRef]

Grandidier, J.

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[CrossRef]

He, S.

Hill, M. T.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

Holmgaard, T.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
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T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
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Jang, J. I.

Karouta, F.

Ketterson, J. B.

Kim, H.

Krasavin, A. V.

A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett. 35(13), 2118–2120 (2010).
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
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A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78(4), 045425 (2008).
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H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).
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Kwon, M.-S.

Kwon, S.-H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
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Kwong, D. L.

S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011).
[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(7083), 508–511 (2006).
[CrossRef] [PubMed]

Lee, B.

Lee, Y.-H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
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Leong, E. S. P.

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(9), 1928–1932 (2006).
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Li, Q.

Liow, T. Y.

S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011).
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Lipson, M.

Lo, G. Q.

S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011).
[CrossRef]

Lo, G.-Y.

Y.-J. Chang and G.-Y. Lo, “A narrow band metal-multi-insulator-metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. 22(9), 634–636 (2010).
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S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006).
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Markey, L.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[CrossRef]

Massenot, S.

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[CrossRef]

Miyazaki, H. T.

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).
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Mu, W.

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter 44(24), 13556–13572 (1991).
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Ning, C.-Z.

Notzel, R.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
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Nötzel, R.

Oei, Y.-S.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
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M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
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Oulton, R. F.

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009).
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R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
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Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
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D. F. P. Pile and D. K. Gramotnev, “Nanoscale fabry-prot interferometer using channel plasmon-polaritons in triangular metallic grooves,” Appl. Phys. Lett. 86(16), 161101 (2005).
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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. B 73(3), 035407 (2006).
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J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005).
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B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter 44(24), 13556–13572 (1991).
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Y. Song, J. Wang, Q. Li, M. Yan, and M. Qiu, “Broadband coupler between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 18(12), 13173–13179 (2010).
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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).
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D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
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Smalbrugge, B.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
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Smit, M. K.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
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Song, Y.

Sorger, V. J.

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009).
[CrossRef] [PubMed]

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

Sukharev, M.

Sun, M.

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. B 73(3), 035407 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005).
[CrossRef]

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]

Turkiewicz, J. P.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
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M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

Veronis, G.

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (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. Matter 44(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,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Vuckovic, J.

Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
[CrossRef]

Wang, J.

Weeber, J.-C.

J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B 76(11), 113405 (2007).
[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]

Xu, J.

X. Zhu, J. Zhang, J. Xu, and D. Yu, “Vertical plasmonic resonant nanocavities,” Nano Lett. 11(3), 1117–1121 (2011).
[CrossRef] [PubMed]

Xu, Q.

Yan, M.

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]

Yao, J.

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009).
[CrossRef] [PubMed]

Yu, D.

X. Zhu, J. Zhang, J. Xu, and D. Yu, “Vertical plasmonic resonant nanocavities,” Nano Lett. 11(3), 1117–1121 (2011).
[CrossRef] [PubMed]

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]

Zayats, A. V.

A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett. 35(13), 2118–2120 (2010).
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78(4), 045425 (2008).
[CrossRef]

Zhang, J.

X. Zhu, J. Zhang, J. Xu, and D. Yu, “Vertical plasmonic resonant nanocavities,” Nano Lett. 11(3), 1117–1121 (2011).
[CrossRef] [PubMed]

Zhang, X.

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009).
[CrossRef] [PubMed]

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

Zhu, S.

S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011).
[CrossRef]

Zhu, X.

X. Zhu, J. Zhang, J. Xu, and D. Yu, “Vertical plasmonic resonant nanocavities,” Nano Lett. 11(3), 1117–1121 (2011).
[CrossRef] [PubMed]

Zhu, Y.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

Zia, R.

Appl. Phys. Lett. (7)

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]

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005).
[CrossRef]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

D. F. P. Pile and D. K. Gramotnev, “Nanoscale fabry-prot interferometer using channel plasmon-polaritons in triangular metallic grooves,” Appl. Phys. Lett. 86(16), 161101 (2005).
[CrossRef]

Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
[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]

S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y.-J. Chang and G.-Y. Lo, “A narrow band metal-multi-insulator-metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. 22(9), 634–636 (2010).
[CrossRef]

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

Nano Lett. (3)

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006).
[CrossRef] [PubMed]

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009).
[CrossRef] [PubMed]

X. Zhu, J. Zhang, J. Xu, and D. Yu, “Vertical plasmonic resonant nanocavities,” Nano Lett. 11(3), 1117–1121 (2011).
[CrossRef] [PubMed]

Nat. Photonics (3)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[CrossRef]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
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Nature (1)

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Figures (8)

Fig. 1
Fig. 1

Geometries and notations of a general MMIM structure with three insulators (a), and two symmetric candidates, MHLHM (b) and MLHLM (c). In (b) and (c), the letters H and L denote the high and low refractive index layers, respectively.

Fig. 2
Fig. 2

Cutoff properties and Re(neff) of the eigenmodes in MHLHM [(a), (b), (c)] and MLHLM [(d), (e), (f)] in terms of the geometric parameters (d1 and d2). (a), (d) Number of solutions of the characteristic equation. The top and right axes are labeled by the optical thicknesses of the first and second insulator layers measured in λ0, respectively. (b), (e) Re(neff) as a function of d1 with d2 fixed at 0 (black), 50 (blue), 100 (green), 200 (red) nm. (c), (f) Re(neff) as a function of d2 with d1 = 0 (black), 50 (blue), 100 (green), 200 (red) nm. In (b), (c), (e), (f), the curves for the ab, sb, and TM1 modes are plotted in solid, dashed, and dotted lines, respectively. In (b), the gray lines started from the cutoff points of the sb and TM1 lines are extrapolated from the connected ones. The curves of the ab modes in (c) and (e) intersect at points C (8.48, 3.66) and C1 (4.77, 3.48), respectively, which correspond with the solid orange lines in (b) and (f), respectively. The curves of the sb modes in (f) intersect at point C2 (193.3, 1.452), which corresponds with the dashed orange line in (e).

Fig. 3
Fig. 3

Dependence of the critical values dC, dC1, and dC2 on (a) the L layer refractive index nL with nH = 3.48, and (b) the H layer refractive index nH with nL = 1.44. The gray dashed lines show the asymptotes for the curves of dC and dC2; the red dashed line in (a) shows the asymptote for the curve of dC1.

Fig. 4
Fig. 4

Typical field profiles of Re(Ez) for the ab and sb modes in the (a) MHLHM and (b), (c) MLHLM structures. Solid lines: ab modes; dashed lines: sb modes. In (a), d1 = d2 = d3 = 120 nm. The black lines show Re(Ez) for the MIM structure with core refractive index equal to nH and core thickness equal to 3d1. In (b), d1 = d2 = d3 = 150 nm. In (c), d1 = d2 = d3 = 1 (blue), 4.77 (red), 20 (green) nm. The scale of x axis varies for different curves.

Fig. 5
Fig. 5

Energy proportions in the metal, H, and L layers of (a) the ab and sb modes in MHLHM with respect to dH, (b) the ab mode in MLHLM with respect to dH, and (c) the sb mode in MLHLM with respect to dL. Solid lines: ab mode; dashed lines: sb mode. In (a), (b), dL is fixed at 50 (blue), 100 (green), and 200 (red) nm. In (c), dH is fixed at 50 (blue), 100 (green), and 200 (red) nm.

Fig. 6
Fig. 6

(a)-(d) Effective mode width (Aeff) of the ab [(a), (c)] and sb [(b), (d)] modes in MHLHM [(a), (b)] and MLHLM [(c), (d)]. The maximum energy density (Wmax) can appear at different positions within the core. The regions 1, 2, …, 5 are divided according to the positions of Wmax, by the gray lines. (e)-(i) Typical energy density profiles of the supported modes in MHLHM [(e), (f)] and MLHLM [(g)-(h)] corresponding with the regions 1, 2, …, 5 in (a)-(d). Solid lines: ab modes; dashed lines: sb modes. In (e)-(i), d1 = d2 = 50, 120, 20, 100, 180 nm, respectively. Aeff of both MHLHM and MLHLM is less than that of M-L-M structure with the same core thickness, except for the bottom right region bounded by the white lines in (c).

Fig. 7
Fig. 7

Propagation length (Lp) of the ab [(a), (c)] and sb [(b), (d)] modes of MHLHM [(a), (b)] and MLHLM [(c), (d)] in terms of d1 and d2. In (a), at the left (or right) side of the white line, Lp of the ab mode is greater (or less) than that of the corresponding M-H-M structure. In (c), at the left (or right) side of the white line, Lp of the ab mode is less (or greater) than that of the corresponding M-L-M structure.

Fig. 8
Fig. 8

Figure of merit (FOM) of the ab [(a), (c)] and sb [(b), (d)] modes of MHLHM [(a), (b)] and MLHLM [(c), (d)] in terms of d1 and d2.

Equations (13)

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ε(x)={ ε 0 = ε m 1 , for x a 0 =0, ε p = n p 2 , for a p1 <x a p ,p=1,2,3, ε 4 = ε m 2 , for x> a 3 .
E=( E x ,0, E z ) e i(βzωt) , H= 1 μ 0 c (0, H y ,0) e i(βzωt) ,
0=( N 01 N 23 + N 12 N 34 ) T 1 T 2 T 3 +( N 12 + N 01 N 24 ) T 1 T 2 +( N 13 + N 01 N 34 ) T 1 T 3 +( N 23 + N 02 N 34 ) T 2 T 3 +( N 01 + N 14 ) T 1 +( N 02 + N 24 ) T 2 +( N 03 + N 34 ) T 3 +(1+ N 04 ),
W= 1 2 { Re[ d(ω ε 0 ε) dω ]E E * + μ 0 H H * }.
A eff = 1 W max W(x)dx ,
E x ={ iA n eff k 0 U 0 e U 0 x , x0 i n eff k 0 U p [ A p1 cosh( U p x)+ A p2 sinh( U p x) ], a p1 x a p ,(p=1,2,3) i A 4 n eff k 0 U 4 e U 4 (x a 3 ) , x a 3 ;
E z ={ A e U 0 x , x0 A p1 sinh( U p x)+ A p2 cosh( U p x), a p1 x a p ,(p=1,2,3) A 4 e U 4 (x a 3 ) , x a 3 ;
H y ={ iA ε 0 k 0 U 0 e U 0 x , x0 i ε p k 0 U p [ A p1 cosh( U p x)+ A p2 sinh( U p x) ], a p1 x a p ,(p=1,2,3). i A 4 ε 4 k 0 U 4 e U 4 (x a 3 ) , x a 3
[ A 11 A 12 ]=[ N 01 1 ]A,
[ A 21 A 22 ]=[ S 12 C 12 C 12 S 12 ][ S 11 C 11 N 12 C 11 N 12 S 11 ][ A 11 A 12 ],
[ A 31 A 32 ]=[ S 23 C 23 C 23 S 23 ][ S 22 C 22 N 23 C 22 N 23 S 22 ][ A 21 A 22 ],
[ A 4 A 4 ]=[ S 33 C 33 N 34 C 33 N 34 S 33 ][ A 31 A 32 ].
f r (p) { n r (p) ; n i (p1) } = 0, f i (p) { n r (p) ; n i (p) } = 0,

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