Abstract

It has been recently demonstrated that a metallic surface with periodic grooves can support a laterally-confined surface wave called spoof plasmon polaritons (SSPPs). Here we propose a SSPPs waveguide drilled with L-shaped grooves which can support SSPPs efficiently. Dispersion relations based on the modal expansion method (MEM) are derived and discussed. Under the deep subwavelength condition, a concise formula for the dispersion relations is obtained. Our results show that the dispersion relations are sensitive to the transversal depths. The L-shaped groove is equivalent to a deeper rectangular groove, but more compact than the straight one. As an example of the applications, the rainbow-trapping effect is realized by changing the transversal depths of the L-shaped grooves.

© 2016 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  22. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
    [Crossref] [PubMed]
  23. X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
    [Crossref]
  24. H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
    [Crossref] [PubMed]
  25. M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
    [Crossref] [PubMed]
  26. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
    [Crossref] [PubMed]
  27. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
    [Crossref]
  28. 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. 100(25), 256803 (2008).
    [Crossref] [PubMed]
  29. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
    [Crossref] [PubMed]
  30. Y. Yang, X. Shen, P. Zhao, H. C. Zhang, and T. J. Cui, “Trapping surface plasmon polaritons on ultrathin corrugated metallic strips in microwave frequencies,” Opt. Express 23(6), 7031–7037 (2015).
    [Crossref] [PubMed]

2016 (1)

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

2015 (2)

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Y. Yang, X. Shen, P. Zhao, H. C. Zhang, and T. J. Cui, “Trapping surface plasmon polaritons on ultrathin corrugated metallic strips in microwave frequencies,” Opt. Express 23(6), 7031–7037 (2015).
[Crossref] [PubMed]

2014 (1)

M. H. Shih, “Plasmonics: Small and fast plasmonic modulator,” Nat. Photonics 8(3), 171–172 (2014).
[Crossref]

2013 (3)

X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013).
[Crossref] [PubMed]

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

2012 (3)

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

2011 (3)

2010 (3)

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref] [PubMed]

2009 (3)

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref] [PubMed]

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

2008 (4)

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

L. Shen, X. Chen, and T.-J. Yang, “Terahertz surface plasmon polaritons on periodically corrugated metal surfaces,” Opt. Express 16(5), 3326–3333 (2008).
[Crossref] [PubMed]

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

2007 (1)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

2005 (2)

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
[Crossref]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

2004 (2)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

2001 (1)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Atwater, H. A.

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

Bartoli, F. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

Brongersma, M. L.

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Chen, X.

Cheng, X. C.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Cui, T. J.

Y. Yang, X. Shen, P. Zhao, H. C. Zhang, and T. J. Cui, “Trapping surface plasmon polaritons on ultrathin corrugated metallic strips in microwave frequencies,” Opt. Express 23(6), 7031–7037 (2015).
[Crossref] [PubMed]

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

De Vries, T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Ding, Y. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Du, C. L.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Feng, Y.

Fernandez-Dominguez, A. I.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref] [PubMed]

Fernández-Domínguez, A. I.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref] [PubMed]

Fernández-García, R.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Fu, Q. H.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Fu, Z.

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Gan, D. C.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Gan, Q.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Gao, X.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Gao, Y.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Garcia-Vidal, F. J.

D. Martin-Cano, O. Quevedo-Teruel, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Waveguided spoof surface plasmons with deep-subwavelength lateral confinement,” Opt. Lett. 36(23), 4635–4637 (2011).
[Crossref] [PubMed]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

García-Vidal, F. J.

Giannini, V.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Gong, Y.

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Hess, O.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Hu, H.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Huang, J. X.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Ji, D.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Jiang, T.

X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013).
[Crossref] [PubMed]

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Kee, C.-S.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Khoe, G. D.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Khurgin, J. B.

Kim, J.-E.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Kim, K.-J.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Kim, S.-H.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Leijtens, X. J.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Li, L.M.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Liu, C.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Liu, J.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Liu, K.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Liu, S.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Liu, X.

Liu, X. M.

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

Lu, H.

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

H. Lu, X. Liu, L. Wang, Y. Gong, and D. Mao, “Ultrafast all-optical switching in nanoplasmonic waveguide with Kerr nonlinear resonator,” Opt. Express 19(4), 2910–2915 (2011).
[Crossref] [PubMed]

Luo, C. R.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Luo, W.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Luo, X. G.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Lyandres, O.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Ma, H. F.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Maier, S. A.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Mao, D.

Martin-Cano, D.

Martin-Moreno, L.

Martín-Moreno, L.

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

Moreno, E.

Nesterov, M. L.

Oei, Y. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Oh, S. S.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Park, H. Y.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Pendry, J. B.

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

Polman, A.

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

Quevedo-Teruel, O.

Ran, L. X.

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Roschuk, T.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Shalaev, V. M.

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Shen, L.

Shen, L. F.

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Shen, X.

Shen, X. P.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Shi, J. H

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Shih, M. H.

M. H. Shih, “Plasmonics: Small and fast plasmonic modulator,” Nat. Photonics 8(3), 171–172 (2014).
[Crossref]

Smalbrugge, B.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Smit, M. K.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Song, K.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Sonnefraud, Y.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Tian, L.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Tomlinson, L. A.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

Van Duyne, R. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Vezenov, D.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Wagner, K.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Wang, C. T.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Wang, G. X.

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

Wang, L.

Wood, J. J.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

Xu, T.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Yang, T.-J.

Yang, Y.

Zeng, X.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Zhang, H. C.

Zhang, X. F.

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Zhao, J.

X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013).
[Crossref] [PubMed]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Zhao, P.

Zhao, X. P.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Zhao, Y. H.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Zhou, K.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Zhu, B.

Appl. Phys. Lett. (4)

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

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

Nat. Mater. (2)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

M. H. Shih, “Plasmonics: Small and fast plasmonic modulator,” Nat. Photonics 8(3), 171–172 (2014).
[Crossref]

Nature (3)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. B (1)

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Phys. Rev. Lett. (1)

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Prog. Electromagn. Res. M (1)

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Sci. Rep. (2)

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Science (2)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Small (1)

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Other (1)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

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Figures (6)

Fig. 1
Fig. 1 (a) Front view of the waveguide drilled with periodic L-shaped grooves. (b) Three-dimensional perspective of a unit cell of the waveguide. (c) Schematic for the partition of the waveguide unit.
Fig. 2
Fig. 2 Dispersion relations of SSPPs supported by waveguides with different hl. d = 50 μm, a = 0.2d, ht = 0 and x0 = 0.1d. The curves and symbols represent the analytical and simulated results, respectively. The solid, dash and dash-dot-dot curves correspond to the fundamental, the 1st and 2nd order modes, respectively.
Fig. 3
Fig. 3 (a) Evolution of dispersion relations with the transversal depth ht. (b) - (d) Ex distributions of the SSPPs supported by the waveguide with ht = 0.6d at β=π/d . (b) The fundamental mode. (c) The 1st order mode. (d) The 2nd order mode. d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.
Fig. 4
Fig. 4 Evolution of the equivalent length hequ as fuctions of ht at f = 0.55 THz, 1.8 THz and 2.95 THz.
Fig. 5
Fig. 5 (a) Changing of c/ v g with ht at f = 0.65 THz, 0.62 THz and 0.6 THz. The purple curve denotes waveguides with straight grooves whose depths equate to hl + ht. The blue, green and red curves correspond to waveguides with L-shaped grooves whose longitudinal and transversal depths are hl and ht, respectively. (b) Simulated field distribution on the slow-wave structure with straight grooves at f = 0.65 THz. (c) – (e) Simulated field distributions on the slow-wave structure with L-shaped grooves at f = 0.65 THz, 0.62 THz and 0.6 THz. (f) One-dimensional |E| distributions along x direction 1 μm above the waveguide surfaces. It shares the same legend with (a). d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.
Fig. 6
Fig. 6 (a) Dispersion relations for different waveguides. (b)-(e) Waveguide units with zero, one, two and three transversal grooves, respectively. (f) Simulated field distribution on the slow-wave structure at 0.535 THz. (g) Simulated field distribution on the slow-wave structure at 0.515 THz. d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.

Equations (18)

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

H y I ( x,z )= n= + A n (1) e q n (1) z e i β n x ,
E x I ( x,z )= n= + q n (1) i k 0 A n (1) e q n (1) z e i β n x .
H y II ( x,z )= m=0 ψ m (2) ( x )[ A m (2) e i q m (2) z + B m (2) e i q m (2) (z+ h l a) ]
E x II ( x,z )= m=0 q m (2) k 0 ψ m (2) ( x )[ A m (2) e i q m (2) z B m (2) e i q m (2) (z+ h l a) ] ,
ψ m ( 2 ) (x)={ 0, if 0<x< x 0 1 γ m a cos[ mπ a ( x x 0 ) ], if x 0 x x 0 +a 0, if x 0 +a<x<d .
H y III ( x,z )= m=0 ψ m (3) (x)[ A m (3) e i q m (3) (z+ h l a) + B m (3) e i q m (3) (z+ h l ) ] ,
E x III ( x,z )= m=0 q m (3) i k 0 ψ m (3) (x)[ A m (3) e i q m (3) (z+ h l a) B m (3) e i q m (3) (z+ h l ) ] ,
ψ m ( 3 ) (x)={ 0, if 0<x< x 0 1 γ m ( h t +a) cos[ mπ h t +a ( x x 0 ) ], if x 0 x x 0 +a+ h t 0, if x 0 +a+ h t <x<d .
( m'=0 W m'm + g m )[ A m (2) B m (2) e i q m (2) ( h l a) ]= f m [ A m (2) e i q m (2) ( h l a) B m (2) ],
f m [ A m (2) B m (2) e i q m (2) ( h l a) ]=( g m + m'=0 T mm' (23) l m' T m'm (32) )[ A m (2) e i q m (2) ( h l a) B m (2) ],
| [W]+[g] [f] [f] [g]+[ T (23) ][l][ T (32) ] |=0.
β 2 k 0 2 k 0 = a d ( a h t +a )tan[ k 0 ( h l a) ]+tan( k 0 a ) ( a h t +a )tan[ k 0 ( h l a) ]tan( k 0 a ) .
β 2 k 0 2 k 0 = a d tan( k 0 h l ),
β 2 k 0 2 k 0 = a d tan( k 0 h equ ).
tan( k 0 h equ )= ( a h t +a )tan[ k 0 ( h l a) ]+tan( k 0 a ) ( a h t +a )tan[ k 0 ( h l a) ]tan( k 0 a ) .
β 2 k 0 2 k 0 = a d tan[ k 0 ( h l + h t ) ].
v g =c { P 2 +1 + aω d P P 2 +1 a c a h t +a se c 2 ( ωa c )[ 1+ tan 2 ( ω h l c ωa c ) ]+ h l a c se c 2 ( ω h l c ωa c )[ ( a h t +a ) 2 + tan 2 ( ωa c ) ] [ a h t +a tan( k 0 h l ωa c ) )tan( k 0 a ) ] 2 } 1
P= a d ( a h t +a )tan[ k 0 ( h l a) ]+tan( k 0 a) ( a h t +a )tan[ k 0 ( h l a) ]tan( k 0 a) .

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