Abstract

We present comprehensive case studies on trapping of light in plasmonic waveguides, including the metal-insulator-metal (MIM) and insulator-metal-insulator (IMI) waveguides. Due to the geometrical symmetry, the guided modes are classified into the anti-symmetric and symmetric modes. For the lossless case, where the relative electric permittivity of metal (εm) and dielectric (εd) are purely real, we define ρ as ρ = -εm/εd. It is shown that trapping of light occurs in the following cases: the anti-symmetric mode in the MIM waveguide with 1 < ρ < 1.28, the symmetric mode in the MIM waveguide with ρ ≪ 1, and the symmetric mode in the IMI waveguide with ρ < 1. The physical interpretation reveals that these conditions are closely connected with the field distributions in the core and the cladding. Various mode properties such as the number of supported modes and the core width for the mode cut off are also presented.

© 2010 Optical Society of America

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    [CrossRef]
  34. J. Takahara and T. Kobayashi, "Low-dimensional optical waves and nano-optical circuits," Optics and Photonics News 6, 54-59 (2004).
    [CrossRef]
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    [CrossRef]
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2009 (3)

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ""Rainbow" trapping and releasing at telecommunication wavelengths," Phys. Rev. Lett. 102, 056801 (2009).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning," Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

E. Feigenbaum and M. Orenstein, "Backward propagating slow light in inverted plasmonic taper," Opt. Express 17, 2465-2469 (2009).
[CrossRef] [PubMed]

2008 (16)

E. Verhagen, A. Polman, and L. Kuipers, "Nanofocusing in laterally tapered plasmonic waveguides," Opt. Express 16, 45-57 (2008).
[CrossRef] [PubMed]

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

D. M. Beggs, T. P. White, L. O'Faolain, and T. F. Krauss, "Ultracompact and low-power optical switch based on silicon photonic crystals," Opt. Lett. 33, 147-149 (2008).
[CrossRef] [PubMed]

J. Park, H. Kim, I.-M. Lee, S. Kim, J. Jung, and B. Lee, "Resonant tunneling of surface plasmon polariton in the plasmonic nano-cavity," Opt. Express 16, 16903-16915 (2008).
[CrossRef] [PubMed]

M. Notomi and H. Taniyama, "On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning," Opt. Express 16, 18657-18666 (2008).
[CrossRef]

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, "Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries," Opt. Express 16, 19001-19017 (2008).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear plasmonic slot waveguides," Opt. Express 16, 21209-21214 (2008).
[CrossRef] [PubMed]

J. Park and B. Lee, "An approximate formula of the effective refractive index of the metal-insulator-metal surface plasmon polariton waveguide in the infrared region," Jpn. J. Appl. Phys. 47, 8449-8451 (2008).
[CrossRef]

T. F. Krauss, "Why do we need slow light?," Nature Photon. 2, 448-450 (2008).
[CrossRef]

L. Thevenaz, "Slow and fast light in optical fibres," Nature Photon. 2, 474-481 (2008).
[CrossRef]

K. Y. Kim, "Tunneling-induced temporary light trapping in negative-index-clad slab waveguide," Jpn. J. Appl. Phys. 47, 4843-4845 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

T. Baba, "Slow light in photonic crystals," Nature Photon. 2, 465-473 (2008).
[CrossRef]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, "From waveguiding to spatial localization of THz waves within a plasmonic metallic grating," IEEE J. Sel. Top. Quantum Electron. 14, 486-490 (2008).
[CrossRef]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, "Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures," Phys. Rev. Lett. 101, 169903 (2008).
[CrossRef]

2007 (6)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "'Trapped rainbow' storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2007).
[CrossRef]

T. Baba and D. Mori, "Slow light engineering in photonic crystals," J. Phys. D 40, 2659-2665 (2007).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316, 430-432 (2007).
[CrossRef] [PubMed]

D. Mori, S. Kubo, H. Sasaki, and T. Baba, "Experimental demonstration of wideband dispersion compensated slow light by a chirped photonic crystal directional coupler," Opt. Express 15, 5264-5270 (2007).
[CrossRef] [PubMed]

T. Kawasaki, D. Mori, and T. Baba, "Experimental observation of slow light in photonic crystal coupled waveguides," Opt. Express 15, 10274-10281 (2007).
[CrossRef] [PubMed]

2006 (4)

F. Kusunoki, T. Yotsuya, and J. Takahara, "Confinement and guiding of two-dimensional optical waves by low-refractive-index cores," Opt. Express 14, 5651-5656 (2006).
[CrossRef] [PubMed]

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] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization," Phys. Rev. B 73, 035407 (2006).
[CrossRef]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104 (2006).
[CrossRef]

2005 (1)

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, 075405 (2005).
[CrossRef]

2004 (2)

M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

J. Takahara and T. Kobayashi, "Low-dimensional optical waves and nano-optical circuits," Optics and Photonics News 6, 54-59 (2004).
[CrossRef]

2002 (1)

J. E. Heebner, R. W. Boyd, and Q. H. Park, "Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide," Phys. Rev. E 65, 036619 (2002).
[CrossRef]

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, 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 44, 13556-13572 (1991).
[CrossRef]

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).
[CrossRef]

Atwater, H. A.

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, "Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries," Opt. Express 16, 19001-19017 (2008).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316, 430-432 (2007).
[CrossRef] [PubMed]

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] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization," Phys. Rev. B 73, 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, 075405 (2005).
[CrossRef]

Baba, T.

Bartal, G.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

Bartoli, F. J.

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ""Rainbow" trapping and releasing at telecommunication wavelengths," Phys. Rev. Lett. 102, 056801 (2009).
[CrossRef] [PubMed]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, "From waveguiding to spatial localization of THz waves within a plasmonic metallic grating," IEEE J. Sel. Top. Quantum Electron. 14, 486-490 (2008).
[CrossRef]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, "Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures," Phys. Rev. Lett. 101, 169903 (2008).
[CrossRef]

Beggs, D. M.

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, 10484-10503 (2000).
[CrossRef]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "'Trapped rainbow' storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Boyd, R. W.

J. E. Heebner, R. W. Boyd, and Q. H. Park, "Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide," Phys. Rev. E 65, 036619 (2002).
[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).
[CrossRef]

Davoyan, A. R.

Ding, Y. J.

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ""Rainbow" trapping and releasing at telecommunication wavelengths," Phys. Rev. Lett. 102, 056801 (2009).
[CrossRef] [PubMed]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, "Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures," Phys. Rev. Lett. 101, 169903 (2008).
[CrossRef]

Ding, Y. J. J.

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, "From waveguiding to spatial localization of THz waves within a plasmonic metallic grating," IEEE J. Sel. Top. Quantum Electron. 14, 486-490 (2008).
[CrossRef]

Dionne, J. A.

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, "Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries," Opt. Express 16, 19001-19017 (2008).
[CrossRef]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316, 430-432 (2007).
[CrossRef] [PubMed]

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] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization," Phys. Rev. B 73, 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, 075405 (2005).
[CrossRef]

Feigenbaum, E.

Fu, Z.

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, "Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures," Phys. Rev. Lett. 101, 169903 (2008).
[CrossRef]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, "From waveguiding to spatial localization of THz waves within a plasmonic metallic grating," IEEE J. Sel. Top. Quantum Electron. 14, 486-490 (2008).
[CrossRef]

Gan, Q. Q.

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ""Rainbow" trapping and releasing at telecommunication wavelengths," Phys. Rev. Lett. 102, 056801 (2009).
[CrossRef] [PubMed]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, "From waveguiding to spatial localization of THz waves within a plasmonic metallic grating," IEEE J. Sel. Top. Quantum Electron. 14, 486-490 (2008).
[CrossRef]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, "Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures," Phys. Rev. Lett. 101, 169903 (2008).
[CrossRef]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

Heebner, J. E.

J. E. Heebner, R. W. Boyd, and Q. H. Park, "Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide," Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Hermann, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "'Trapped rainbow' storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Jamois, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Jung, J.

Kawasaki, T.

Kim, H.

Kim, K. Y.

K. Y. Kim, "Tunneling-induced temporary light trapping in negative-index-clad slab waveguide," Jpn. J. Appl. Phys. 47, 4843-4845 (2008).
[CrossRef]

Kim, S.

Kivshar, Y. S.

Klaedtke, A.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Kobayashi, T.

J. Takahara and T. Kobayashi, "Low-dimensional optical waves and nano-optical circuits," Optics and Photonics News 6, 54-59 (2004).
[CrossRef]

Krauss, T. F.

Kubo, S.

Kuipers, L.

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning," Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

Kusunoki, F.

Lee, B.

Lee, I.-M.

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316, 430-432 (2007).
[CrossRef] [PubMed]

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] [PubMed]

Liu, Y. M.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Liu, Z. W.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Mori, D.

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric-constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning," Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

M. Notomi and H. Taniyama, "On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning," Opt. Express 16, 18657-18666 (2008).
[CrossRef]

O'Faolain, L.

Orenstein, M.

Park, J.

Park, Q. H.

J. E. Heebner, R. W. Boyd, and Q. H. Park, "Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide," Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Polman, A.

E. Verhagen, A. Polman, and L. Kuipers, "Nanofocusing in laterally tapered plasmonic waveguides," Opt. Express 16, 45-57 (2008).
[CrossRef] [PubMed]

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, "Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries," Opt. Express 16, 19001-19017 (2008).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization," Phys. Rev. B 73, 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, 075405 (2005).
[CrossRef]

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric-constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Sasaki, H.

Shadrivov, I. V.

Shalaev, V. M.

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2007).
[CrossRef]

Stacy, A. M.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

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).
[CrossRef]

Stockman, M. I.

M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

Sun, C.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization," Phys. Rev. B 73, 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, 075405 (2005).
[CrossRef]

Takahara, J.

F. Kusunoki, T. Yotsuya, and J. Takahara, "Confinement and guiding of two-dimensional optical waves by low-refractive-index cores," Opt. Express 14, 5651-5656 (2006).
[CrossRef] [PubMed]

J. Takahara and T. Kobayashi, "Low-dimensional optical waves and nano-optical circuits," Optics and Photonics News 6, 54-59 (2004).
[CrossRef]

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).
[CrossRef]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning," Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning," Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

M. Notomi and H. Taniyama, "On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning," Opt. Express 16, 18657-18666 (2008).
[CrossRef]

Thevenaz, L.

L. Thevenaz, "Slow and fast light in optical fibres," Nature Photon. 2, 474-481 (2008).
[CrossRef]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "'Trapped rainbow' storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

Verhagen, E.

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric-constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Wang, Y.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

White, T. P.

Yao, J.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Yotsuya, T.

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

Zhang, X.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, "From waveguiding to spatial localization of THz waves within a plasmonic metallic grating," IEEE J. Sel. Top. Quantum Electron. 14, 486-490 (2008).
[CrossRef]

J. Phys. D (1)

T. Baba and D. Mori, "Slow light engineering in photonic crystals," J. Phys. D 40, 2659-2665 (2007).
[CrossRef]

Jpn. J. Appl. Phys. (2)

J. Park and B. Lee, "An approximate formula of the effective refractive index of the metal-insulator-metal surface plasmon polariton waveguide in the infrared region," Jpn. J. Appl. Phys. 47, 8449-8451 (2008).
[CrossRef]

K. Y. Kim, "Tunneling-induced temporary light trapping in negative-index-clad slab waveguide," Jpn. J. Appl. Phys. 47, 4843-4845 (2008).
[CrossRef]

Nano Lett. (1)

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] [PubMed]

Nature (2)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature 455, 376-U332 (2008).
[CrossRef] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "'Trapped rainbow' storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Nature Photon. (4)

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2007).
[CrossRef]

T. F. Krauss, "Why do we need slow light?," Nature Photon. 2, 448-450 (2008).
[CrossRef]

L. Thevenaz, "Slow and fast light in optical fibres," Nature Photon. 2, 474-481 (2008).
[CrossRef]

T. Baba, "Slow light in photonic crystals," Nature Photon. 2, 465-473 (2008).
[CrossRef]

Opt. Express (10)

F. Kusunoki, T. Yotsuya, and J. Takahara, "Confinement and guiding of two-dimensional optical waves by low-refractive-index cores," Opt. Express 14, 5651-5656 (2006).
[CrossRef] [PubMed]

D. Mori, S. Kubo, H. Sasaki, and T. Baba, "Experimental demonstration of wideband dispersion compensated slow light by a chirped photonic crystal directional coupler," Opt. Express 15, 5264-5270 (2007).
[CrossRef] [PubMed]

T. Kawasaki, D. Mori, and T. Baba, "Experimental observation of slow light in photonic crystal coupled waveguides," Opt. Express 15, 10274-10281 (2007).
[CrossRef] [PubMed]

E. Verhagen, A. Polman, and L. Kuipers, "Nanofocusing in laterally tapered plasmonic waveguides," Opt. Express 16, 45-57 (2008).
[CrossRef] [PubMed]

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

J. Park, H. Kim, I.-M. Lee, S. Kim, J. Jung, and B. Lee, "Resonant tunneling of surface plasmon polariton in the plasmonic nano-cavity," Opt. Express 16, 16903-16915 (2008).
[CrossRef] [PubMed]

M. Notomi and H. Taniyama, "On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning," Opt. Express 16, 18657-18666 (2008).
[CrossRef]

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, "Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries," Opt. Express 16, 19001-19017 (2008).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear plasmonic slot waveguides," Opt. Express 16, 21209-21214 (2008).
[CrossRef] [PubMed]

E. Feigenbaum and M. Orenstein, "Backward propagating slow light in inverted plasmonic taper," Opt. Express 17, 2465-2469 (2009).
[CrossRef] [PubMed]

Opt. Lett. (1)

Optics and Photonics News (1)

J. Takahara and T. Kobayashi, "Low-dimensional optical waves and nano-optical circuits," Optics and Photonics News 6, 54-59 (2004).
[CrossRef]

Phys. Rev. B (6)

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

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).
[CrossRef]

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric-constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization," Phys. Rev. B 73, 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, 075405 (2005).
[CrossRef]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, "Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability," Phys. Rev. B 73, 085104 (2006).
[CrossRef]

Phys. Rev. E (1)

J. E. Heebner, R. W. Boyd, and Q. H. Park, "Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide," Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Phys. Rev. Lett. (4)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning," Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, "Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures," Phys. Rev. Lett. 101, 169903 (2008).
[CrossRef]

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ""Rainbow" trapping and releasing at telecommunication wavelengths," Phys. Rev. Lett. 102, 056801 (2009).
[CrossRef] [PubMed]

M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

Science (2)

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316, 430-432 (2007).
[CrossRef] [PubMed]

Other (1)

E. Feigenbaum, N. Kaminski, and M. Orenstein, "Negative group velocity: Is it a backward wave or fast light?," arXiv:0807.4915 (2008).

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Figures (14)

Fig. 1.
Fig. 1.

Schematic diagram of the MIM waveguide

Fig. 2.
Fig. 2.

Graphical method for the anti-symmetric plasmonic mode in the MIM waveguide. (a) The case ρ > 1. (εd ,ρ) = (1.0,4.0) and ak 0 = 1.2ξ. The definition of ξ is given in Eq. (14). It is observed that, if ak 0 > ξ, then the anti-symmetric plasmonic mode is supported in the MIM waveguide. (b) The case ρ < 1. (εd ,ρ) = (1.0,0.8) and ak 0 = 0.8ξ. The anti-symmetric plasmonic mode exists in the MIM waveguide when ak 0 < ξ.

Fig. 3.
Fig. 3.

Graphical method for the anti-symmetric photonic mode in the MIM waveguide. (a) The case ρ > V. (εd ,ρ) = (1.0,2.0), resulting in ζ < ξ. The definition of ζ is given in Eq. (15). ak 0 = (ζ + ξ)/2 . It is shown that, if ζ < ak 0 < ξ, then the anti-symmetric photonic mode is supported in the MIM waveguide. (b) The case ρ < V. (εd ,ρ) = (1.0,1.1), leading to ζ > ξ . ak 0 = (ζ + ξ)/2. The anti-symmetric photonic mode exists in the MIM waveguide when ξ < ak 0 < ζ.

Fig. 4.
Fig. 4.

(a)-(c) Dependence of the effective refractive index neff as a function of the reduced core width ak 0 for the anti-symmetric mode in the MIM waveguide. (d)-(f) Effect of ak 0 on the normalized optical power flow Pnorm . The definition of Pnorm is given in Appendix A. (a) and (d) show the results for the case ρ < 1 with (εd ,ρ) = (1.0,0.8). (b) and (e) depict the results for the case 1 < ρ < 1.28 with (εd , ρ) = (1.0,1.1). (c) and (f) illustrate the results for the case ρ > 1.28 with (εd , ρ) (1.0,2.0). It is seen that, in the case 1 < ρ < 1.28 the mode degeneracy occurs at a certain point ak 0 < ξ. This corresponds to the Pnorm = 0 (See Fig. 4(e)) i.e., trapping of light.

Fig. 5.
Fig. 5.

(a) Graphical method for the anti-symmetric plasmonic mode in the MIM waveguide. The case 1 < ρ < 1.28. (εd , ρ) = (1.0,1.1). ak 0 = 0.98ξ. Two solutions are observed. (b) The graph of ρU cothU - √U 2 + V 2 for (εd ,ρ) = (1.0,1.1) . The arrow shows the tendency of decreasing ak 0 i.e., ak 0 = ζ, ξ, 0.98ξ, 0.938ξ(= hc ), and 0.91ξ.

Fig. 6.
Fig. 6.

Dependence of (a) |neff | and (b) Pnorm on εd . (εd ,ak 0) = (-1.1,0.712) (c) ω - β dispersion relation. (εd ,2a,ωP ) = (1.0,90nm,6.83 × 1015). The abscissa is normalized with the bulk plasma wavenumber βP , while the ordinate is normalized with the bulk plasma frequency ωP . (d) The dependence of Pnorm on ω. The parameters are the same as those in (c).

Fig. 7.
Fig. 7.

Graphical method for the symmetric plasmonic mode in the MIM waveguide. (a) The case ρ>1. (εd ,ρ) = (1.0,1.8) and ak 0 = 0.8 . The symmetric plasmonic mode is always supported in the MIM waveguide regardless of ak 0. (b) The case ρ<1 . (εd ,p) = (1.0,0.3) and ak 0 = 0.3 . No symmetric plasmonic mode exists regardless of ak 0.

Fig. 8.
Fig. 8.

Graphical method for the symmetric photonic mode in the MIM waveguide. (a) The case with usual ρ. (εd ,ρ = (1.0,2.0) and ak 0 = 1.2χ. The definition of χ is given in Eq. (18). If W at the point A is smaller than that at the point B, it is guaranteed that the symmetric plasmonic mode is supported in the MIM waveguide. (b) The case ρ≪1. (εd ,ρ) = (1.0,0.01) . ak 0 =0.99χ. Two solutions are observed. As ak 0 decreases, the radius V of the circle W = √-J 2 + V 2 also decreases and two solutions come closer to each other. It is expected that, at a certain value of ak 0, two solutions would degenerate into one, resulting in trapped light.

Fig. 9.
Fig. 9.

(a)-(c) Dependence of the effective refractive index neff as a function of the reduced core width ak 0 for the symmetric mode in the MIM waveguide. (d)-(f) Effect of ak 0 on the normalized optical power flow Pnorm . (a) and (d) show the results for the case ρ ≪ 1 with (εd ,ρ) = (1.0,0.01) . (b) and (e) depict the results for the case ρ<1 with (εd ,ρ) = (1.0,0.2) . (c) and (f) illustrate the results for the case p>1 with (εd ,ρ) = (1.0,1.5) . Note that the mode degeneracy occurs at a certain point ak 0<χ in the case ρ ≪ 1, leading to trapping of light.

Fig. 10.
Fig. 10.

Schematic diagram of the IMI waveguide

Fig. 11.
Fig. 11.

Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (εd ,ρ) = (1.0,1.8) and ak 0 = 0.8 . There is one intersection regardless of the change in ak 0, indicating that one symmetric plasmonic mode is always allowed in this case. (b) The case ρ < 1 . (εd ,ρ) = (1.0,0.5) and ak 0 = 0.1 . For ak 0 below a certain value, there are two intersections. As ak 0 increases, two intersections come closer to each other, and at the certain value, they degenerate into single intersection.

Fig. 12.
Fig. 12.

(a)-(b) Dependence of the effective refractive index |neff | as a function of the reduced core width ak 0 for the symmetric mode in the IMI waveguide. (c)-(d) Effect of ak 0 on the normalized optical power flow Pnorm . (a) and (c) show the results for the case ρ < 1 with (εd ,ρ) = (1.0,0.5 ) . It is noteworthy that the mode degeneracy occurs at a certain point in the case ρ<1 . (b) and (d) depict the results for the case ρ>1 with (εd , ρ) = (1.0,2.0).

Fig. 13.
Fig. 13.

Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (εd ,ρ) = (1.0,1.8) and ak 0 = 0.8 . The anti-symmetric plasmonic mode is supported in the IMI waveguide regardless of the core width. (b) The case ρ < 1 . (εd , ρ) = (1.0,0.5) and ak 0 = 2.0 . No anti-symmetric mode is guided in the IMI waveguide.

Fig. 14.
Fig. 14.

(a) Dependence of the effective refractive index |neff | as a function of the reduced core width ak 0 for the anti-symmetric mode in the IMI waveguide for the case ρ > 1. (b) Effect of ak 0 on the normalized optical power flow Pnorm . (εd ,ρ) = (1.0,2.0) in common. Regardless of ak 0, a plasmonic positive mode always exists.

Tables (5)

Tables Icon

Table 1. Conditions for existence of the anti-symmetric mode in the MIM waveguide. ξ and ζ are defined in Eqs. (14) and (15), respectively.

Tables Icon

Table 2. Supplemented conditions for existence of the anti-symmetric mode in the MIM waveguide

Tables Icon

Table 3. Conditions for existence of the symmetric mode in the MIM waveguide. χ is defined in Eq. (18).

Tables Icon

Table 4. Conditions for existence of the symmetric mode in the IMI waveguide

Tables Icon

Table 5. Conditions for existence of the anti-symmetric mode in the IMI waveguide

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

ε m = 1 ω p 2 ω ( ω + i γ ) ,
ρ = ε m ε d = ε m ε d .
κ m ε m + κ d ε d coth ( κ d a ) = 0 ( the anti symmetric plasmonic mode ) ,
κ m ε m + κ d ε d cot ( κ d a ) = 0 ( the anti symmetric photonic mode ) ,
κ m ε m + κ d ε d tanh ( κ d a ) = 0 ( the symmetric plasmonic mode ) ,
κ m ε m + κ d ε d tan ( κ d a ) = 0 ( the symmetric photonic mode ) .
κ m 2 + β 2 = ε m k 0 2 ,
κ d 2 + β 2 = ε d k 0 2 ,
κ d 2 + β 2 = ε d k 0 2 .
W = a κ m ,
U = a κ d ,
J = a k d ,
V = a k 0 ε d ε m .
ξ = ρ ε d ε m = ε m ε d ( ε d ε m ) .
ζ = 1 ε d cot 1 ε m ρ ε d = 1 ε d cot 1 ε d ε m .
ρ c = tan ρ c ( 1 + ρ c ) .
σ = π 2 ε d .
χ = 1 ε d tan 1 ( ε d ε m ) .
κ d ε d + κ m ε m tanh ( κ m a ) = 0 ( the symmetric plasmonic mode ) ,
κ d ε d + κ m ε m coth ( κ m a ) = 0 ( the anti symmetric plasmonic mode ) .
H y ( x , z ; t ) = { A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 1 sinh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a )
E x ( x , z ; t ) = { ( β ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ( β ω ε 0 ε d ) B 1 sinh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) ( β ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) .
E z ( x , z ; t ) = { ( κ m j ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ( κ d j ω ε 0 ε d ) B 1 cosh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) ( κ m j ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a )
A 1 = B 1 sinh ( κ d a ) ,
κ m ε m A 1 = κ d ε d B 1 cosh ( κ d a ) ,
κ m ε m = κ d ε d coth ( κ d a ) .
P z ( x , z ; t ) = { ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) exp [ 2 κ m ( x a ) ] ( x a ) ( β B 1 2 ω ε 0 ) ( 1 ε d ) sinh 2 ( κ d x ) ( a x a ) ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) exp [ 2 κ m ( x a ) ] ( x a ) .
P metal = 2 a { ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) exp [ 2 κ m ( x a ) ] } d x
= 2 [ ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) ] 0 exp [ 2 κ m x ] d x
= ( β B 1 2 ω ε 0 ) ( sinh 2 ( κ d a ) ε m κ m ) ,
P dielectric = 2 a a ( β B 1 2 ω ε 0 ) ( 1 ε d ) sinh 2 ( κ d x ) d x
= 2 ( β B 1 2 ω ε 0 ) ( 1 ε d ) 0 a sinh 2 ( κ d x ) d x
= 2 ( β B 1 2 ω ε 0 ) ( 1 ε d ) [ sinh ( 2 κ d x ) 4 κ d x 2 ] 0 a
= ( β B 1 2 ω ε 0 ) ( 1 ε d κ d ) [ sinh ( 2 κ d a ) 2 a κ d ] .
P norm = sinh ( 2 κ d a ) / 2 a κ d ε d κ d + sinh 2 ( κ d a ) ε m κ m sinh ( 2 κ d a ) / 2 a κ d ε d κ d sinh 2 ( κ d a ) ε m κ m .
H y ( x , z ; t ) = { A 2 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 2 sin ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 2 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 3 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 3 cosh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 3 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 4 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 4 cos ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 4 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 5 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 5 sinh ( κ m x ) exp [ j ( β z ω t ) ] ( a x a ) A 5 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 6 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 6 cosh ( κ m x ) exp [ j ( β z ω t ) ] ( a x a ) A 6 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
n eff 2 ε m ε m + n eff 2 ε d ε d coth ( a k 0 n eff 2 ε d ) = 0 .
n eff k 0 ε m κ m + n eff k 0 ε d κ d coth ( a k 0 n eff 2 ε d ) κ d ε d k 0 csch 2 ( a k 0 n eff 2 ε d ) ( d ( a k 0 ) d n eff κ d k 0 + a k 0 2 n eff κ d ) = 0 ,
d ( a k 0 ) d n eff = n eff ε d k 0 3 κ d 2 [ sinh ( 2 κ d a ) / 2 a κ d ε d κ d + sinh 2 ( κ d a ) ε m κ m ] .

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