Abstract

A plasmonic mode-selection filter based on end-coupled circular split-ring resonators is proposed and demonstrated. In contrast to regular ring resonators, which can support only the integer modes, extra noninteger resonance modes will be excited by placing metallic nanowalls in the proposed circular ring resonators. Since all the modes are highly sensitive to the position of the nanowall, the associated modes are effectively excited or suppressed by manipulating the position. This proposed scheme offers great flexibility to design the transmission spectrum with expected modes. Moreover, each integer or noninteger resonance mode with high transmittance can be selected individually by cascading two proposed split-ring resonators that share an intersection of transmission peaks. The corresponding spectra and the propagation characteristics are numerically investigated by using the finite-difference time-domain method.

© 2014 Optical Society of America

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  1. C. Zhang, R. Wang, Y. Wang, S. Zhu, C. Min, and X. C. Yuan, “Phase-stepping technique for highly sensitive microscopic surface plasmon resonance biosensor,” Appl. Opt. 53, 836–840 (2014).
    [CrossRef]
  2. H. Fan and P. Berini, “Thermo-optic characterization of long-range surface-plasmon devices in Cytop,” Appl. Opt. 52, 162–170 (2013).
    [CrossRef]
  3. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
    [CrossRef]
  4. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125409 (2004).
    [CrossRef]
  5. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
    [CrossRef]
  6. X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometric sizes,” Opt. Lett. 33, 2874–2976 (2008).
    [CrossRef]
  7. C. T. Wang, C. L. Du, Y. G. Lv, and X. G. Luo, “Surface electromagnetic wave excitation and diffraction by subwavelength slit with periodically patterned metallic grooves,” Opt. Express 14, 5671–5681 (2006).
    [CrossRef]
  8. K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, Y. H. Guo, and X. G. Luo, “Spectral characteristics of plasmonic metal-insulator-metal waveguides with a tilted groove,” IEEE Photon. J. 4, 1794–1800 (2012).
    [CrossRef]
  9. Q. Zhang, X. G. Huang, X. S. Lin, J. Tao, and X. P. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express 17, 7549–7554 (2009).
    [CrossRef]
  10. J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
    [CrossRef]
  11. F. F. Hu, H. X. Yi, and Z. P. Zhou, “Wavelength demultiplexing structure based on arrayed plasmonic slot cavities,” Opt. Lett. 36, 1500–1502 (2011).
    [CrossRef]
  12. K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Wavelength demultiplexing structure based on plasmonic metal-insulator-metal waveguide,” J. Opt. 14, 075001 (2012).
    [CrossRef]
  13. K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Design of plasmonic comb-like filters using loop-based resonators,” Plasmonics 8, 1017–1022 (2013).
    [CrossRef]
  14. X. Zou, M. Li, W. Pan, L. Yan, J. Azaña, and J. Yao, “All-fiber optical filter with an ultra-narrow and rectangular spectral response,” Opt. Lett. 38, 3096–3098 (2013).
    [CrossRef]
  15. Y. Yang, H. T. Dai, and X. W. Sun, “Split ring aperture for optical magnetic field enhancement by radially polarized beam,” Opt. Express 21, 6845–6850 (2013).
    [CrossRef]
  16. S. S. Walavalkar, P. Latawiec, A. Homyk, and A. Scherer, “Scalable method for the fabrication and testing of glass filled, three-dimensionally sculpted extraordinary transmission apertures,” Nano Lett. 14, 311–317 (2014).
    [CrossRef]
  17. P. A. Brandão and S. B. Cavalcanti, “Optical spin-to-orbital plasmonic angular momentum conversion in subwavelength apertures,” Opt. Lett. 38, 920–922 (2013).
    [CrossRef]
  18. 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, 508–511 (2006).
    [CrossRef]
  19. H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, and H. Yao, “Surface plasmon polariton propagation and combination in Y-shaped metallic channels,” Opt. Express 13, 10795–10800 (2005).
    [CrossRef]
  20. T. W. Lee and S. Gray, “Subwavelength light bending by metal slit structures,” Opt. Express 13, 9652–9659 (2005).
    [CrossRef]
  21. R. A. Wahsheh, Z. Lu, and M. A. G. Abushagur, “Nanoplasmonic couplers and splitters,” Opt. Express 17, 19033–19040 (2009).
    [CrossRef]
  22. K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “A four-port plasmonic quasi-circulator based on metal-insulator-metal waveguides,” Opt. Express 20, 28025–28032 (2012).
    [CrossRef]
  23. M. A. Bavil, Z. Zhou, and Q. Deng, “Active unidirectional propagation of surface plasmons at subwavelength slits,” Opt. Express 21, 17066–17076 (2013).
    [CrossRef]
  24. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010).
    [CrossRef]
  25. J. Tao, X. G. Huang, X. S. Lin, Q. Zhang, and X. P. Jin, “A narrow-band subwavelength plasmonic waveguide filter with asymmetrical multiple-teeth-shaped structure,” Opt. Express 17, 13989–13994 (2009).
    [CrossRef]
  26. Z. H. Han, E. Forsberg, and S. L. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
    [CrossRef]
  27. B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
    [CrossRef]
  28. X. Luo, X. H. Zou, X. F. Li, Z. Zhou, W. Pan, L. S. Yan, and K. H. Wen, “High-uniformity multichannel plasmonic filter using linearly lengthened insulators in metal-insulator-metal waveguide,” Opt. Lett. 38, 1585–1587 (2013).
    [CrossRef]
  29. G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19, 3513–3518 (2011).
    [CrossRef]
  30. H. Lu, X. Liu, D. Mao, L. Wang, and Y. Gong, “Tunable band-pass plasmonic waveguide filters with nanodisk resonators,” Opt. Express 18, 17922–17927 (2010).
    [CrossRef]
  31. Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, H. Li, and X. G. Luo, “A plasmonic splitter based on slot cavity,” Opt. Express 19, 13831–13838 (2011).
    [CrossRef]
  32. A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
    [CrossRef]
  33. T. B. Wang, X. W. Wen, C. P. Yin, and H. Z. Wang, “The transmission characteristics of surface plasmon polaritons in ring resonator,” Opt. Express 17, 24096–24101 (2009).
    [CrossRef]
  34. T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide ring resonators,” Opt. Express 17, 2968–2975 (2009).
    [CrossRef]
  35. B. Yun, G. Hu, and Y. Cui, “Theoretical analysis of a nanoscale plasmonic filter based on a rectangular metal-insulator-metal waveguide,” J. Phys. D 43, 385102 (2010).
    [CrossRef]
  36. F. S. Ma and C. Lee, “Optical nanofilters based on meta-atom side-coupled plasmonics metal-insulator-metal waveguides,” J. Lightwave Technol. 31, 2876–2880 (2013).
    [CrossRef]
  37. I. Zand, M. S. Abrishamian, and P. Berini, “Highly tunable nanoscale metal-insulator-metal split ring core ring resonators (SRCRRs),” Opt. Express 21, 79–86 (2013).
    [CrossRef]
  38. J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
    [CrossRef]
  39. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  40. X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).
  41. Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, and X. G. Luo, “Electromagnetically induced transparency (EIT)-like transmission in side-coupled complementary split-ring resonators,” Opt. Express 20, 24348–24355 (2012).
    [CrossRef]

2014

S. S. Walavalkar, P. Latawiec, A. Homyk, and A. Scherer, “Scalable method for the fabrication and testing of glass filled, three-dimensionally sculpted extraordinary transmission apertures,” Nano Lett. 14, 311–317 (2014).
[CrossRef]

C. Zhang, R. Wang, Y. Wang, S. Zhu, C. Min, and X. C. Yuan, “Phase-stepping technique for highly sensitive microscopic surface plasmon resonance biosensor,” Appl. Opt. 53, 836–840 (2014).
[CrossRef]

2013

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Design of plasmonic comb-like filters using loop-based resonators,” Plasmonics 8, 1017–1022 (2013).
[CrossRef]

I. Zand, M. S. Abrishamian, and P. Berini, “Highly tunable nanoscale metal-insulator-metal split ring core ring resonators (SRCRRs),” Opt. Express 21, 79–86 (2013).
[CrossRef]

H. Fan and P. Berini, “Thermo-optic characterization of long-range surface-plasmon devices in Cytop,” Appl. Opt. 52, 162–170 (2013).
[CrossRef]

P. A. Brandão and S. B. Cavalcanti, “Optical spin-to-orbital plasmonic angular momentum conversion in subwavelength apertures,” Opt. Lett. 38, 920–922 (2013).
[CrossRef]

Y. Yang, H. T. Dai, and X. W. Sun, “Split ring aperture for optical magnetic field enhancement by radially polarized beam,” Opt. Express 21, 6845–6850 (2013).
[CrossRef]

X. Luo, X. H. Zou, X. F. Li, Z. Zhou, W. Pan, L. S. Yan, and K. H. Wen, “High-uniformity multichannel plasmonic filter using linearly lengthened insulators in metal-insulator-metal waveguide,” Opt. Lett. 38, 1585–1587 (2013).
[CrossRef]

M. A. Bavil, Z. Zhou, and Q. Deng, “Active unidirectional propagation of surface plasmons at subwavelength slits,” Opt. Express 21, 17066–17076 (2013).
[CrossRef]

X. Zou, M. Li, W. Pan, L. Yan, J. Azaña, and J. Yao, “All-fiber optical filter with an ultra-narrow and rectangular spectral response,” Opt. Lett. 38, 3096–3098 (2013).
[CrossRef]

F. S. Ma and C. Lee, “Optical nanofilters based on meta-atom side-coupled plasmonics metal-insulator-metal waveguides,” J. Lightwave Technol. 31, 2876–2880 (2013).
[CrossRef]

2012

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Wavelength demultiplexing structure based on plasmonic metal-insulator-metal waveguide,” J. Opt. 14, 075001 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, Y. H. Guo, and X. G. Luo, “Spectral characteristics of plasmonic metal-insulator-metal waveguides with a tilted groove,” IEEE Photon. J. 4, 1794–1800 (2012).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, and X. G. Luo, “Electromagnetically induced transparency (EIT)-like transmission in side-coupled complementary split-ring resonators,” Opt. Express 20, 24348–24355 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “A four-port plasmonic quasi-circulator based on metal-insulator-metal waveguides,” Opt. Express 20, 28025–28032 (2012).
[CrossRef]

2011

2010

2009

2008

2007

Z. H. Han, E. Forsberg, and S. L. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[CrossRef]

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
[CrossRef]

J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
[CrossRef]

2006

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, 508–511 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

C. T. Wang, C. L. Du, Y. G. Lv, and X. G. Luo, “Surface electromagnetic wave excitation and diffraction by subwavelength slit with periodically patterned metallic grooves,” Opt. Express 14, 5671–5681 (2006).
[CrossRef]

2005

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[CrossRef]

T. W. Lee and S. Gray, “Subwavelength light bending by metal slit structures,” Opt. Express 13, 9652–9659 (2005).
[CrossRef]

H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, and H. Yao, “Surface plasmon polariton propagation and combination in Y-shaped metallic channels,” Opt. Express 13, 10795–10800 (2005).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

2004

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125409 (2004).
[CrossRef]

2003

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

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Abrishamian, M. S.

Abushagur, M. A. G.

Agrawal, G. P.

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

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

Azaña, J.

Bavil, M. A.

Berini, P.

Bouhelier, A.

J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
[CrossRef]

Bozhevolnyi, S. I.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide ring resonators,” Opt. Express 17, 2968–2975 (2009).
[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, 508–511 (2006).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Brandão, P. A.

Cavalcanti, S. B.

Chen, Z.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Cui, Y.

B. Yun, G. Hu, and Y. Cui, “Theoretical analysis of a nanoscale plasmonic filter based on a rectangular metal-insulator-metal waveguide,” J. Phys. D 43, 385102 (2010).
[CrossRef]

Dai, H. T.

Deng, Q.

Dereux, A.

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

J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
[CrossRef]

Des, F. G.

J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
[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, 508–511 (2006).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Du, C.

Du, C. L.

Duan, L.

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, 508–511 (2006).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Fan, H.

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125409 (2004).
[CrossRef]

Forsberg, E.

Z. H. Han, E. Forsberg, and S. L. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[CrossRef]

Gao, H.

Gong, Y.

Gray, S.

Guo, Y. H.

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Design of plasmonic comb-like filters using loop-based resonators,” Plasmonics 8, 1017–1022 (2013).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, and X. G. Luo, “Electromagnetically induced transparency (EIT)-like transmission in side-coupled complementary split-ring resonators,” Opt. Express 20, 24348–24355 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, Y. H. Guo, and X. G. Luo, “Spectral characteristics of plasmonic metal-insulator-metal waveguides with a tilted groove,” IEEE Photon. J. 4, 1794–1800 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Wavelength demultiplexing structure based on plasmonic metal-insulator-metal waveguide,” J. Opt. 14, 075001 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “A four-port plasmonic quasi-circulator based on metal-insulator-metal waveguides,” Opt. Express 20, 28025–28032 (2012).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, H. Li, and X. G. Luo, “A plasmonic splitter based on slot cavity,” Opt. Express 19, 13831–13838 (2011).
[CrossRef]

Guo, Z.

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Design of plasmonic comb-like filters using loop-based resonators,” Plasmonics 8, 1017–1022 (2013).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, and X. G. Luo, “Electromagnetically induced transparency (EIT)-like transmission in side-coupled complementary split-ring resonators,” Opt. Express 20, 24348–24355 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “A four-port plasmonic quasi-circulator based on metal-insulator-metal waveguides,” Opt. Express 20, 28025–28032 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, Y. H. Guo, and X. G. Luo, “Spectral characteristics of plasmonic metal-insulator-metal waveguides with a tilted groove,” IEEE Photon. J. 4, 1794–1800 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Wavelength demultiplexing structure based on plasmonic metal-insulator-metal waveguide,” J. Opt. 14, 075001 (2012).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, H. Li, and X. G. Luo, “A plasmonic splitter based on slot cavity,” Opt. Express 19, 13831–13838 (2011).
[CrossRef]

Han, Z. H.

Z. H. Han, E. Forsberg, and S. L. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[CrossRef]

Harel, E.

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

Hattori, H. T.

He, S. L.

Z. H. Han, E. Forsberg, and S. L. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[CrossRef]

Holmgaard, T.

Homyk, A.

S. S. Walavalkar, P. Latawiec, A. Homyk, and A. Scherer, “Scalable method for the fabrication and testing of glass filled, three-dimensionally sculpted extraordinary transmission apertures,” Nano Lett. 14, 311–317 (2014).
[CrossRef]

Hosseini, A.

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
[CrossRef]

Hu, F. F.

Hu, G.

B. Yun, G. Hu, and Y. Cui, “Theoretical analysis of a nanoscale plasmonic filter based on a rectangular metal-insulator-metal waveguide,” J. Phys. D 43, 385102 (2010).
[CrossRef]

Huang, X. G.

Jin, X. P.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kik, P. G.

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

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[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, 508–511 (2006).
[CrossRef]

Latawiec, P.

S. S. Walavalkar, P. Latawiec, A. Homyk, and A. Scherer, “Scalable method for the fabrication and testing of glass filled, three-dimensionally sculpted extraordinary transmission apertures,” Nano Lett. 14, 311–317 (2014).
[CrossRef]

Lee, C.

Lee, T. W.

Li, H.

Li, M.

Li, W.

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).

Li, X. F.

Lin, X.

Lin, X. S.

Liu, X.

Lu, H.

Lu, Z.

Luo, B.

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Design of plasmonic comb-like filters using loop-based resonators,” Plasmonics 8, 1017–1022 (2013).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, and X. G. Luo, “Electromagnetically induced transparency (EIT)-like transmission in side-coupled complementary split-ring resonators,” Opt. Express 20, 24348–24355 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “A four-port plasmonic quasi-circulator based on metal-insulator-metal waveguides,” Opt. Express 20, 28025–28032 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, Y. H. Guo, and X. G. Luo, “Spectral characteristics of plasmonic metal-insulator-metal waveguides with a tilted groove,” IEEE Photon. J. 4, 1794–1800 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Wavelength demultiplexing structure based on plasmonic metal-insulator-metal waveguide,” J. Opt. 14, 075001 (2012).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, H. Li, and X. G. Luo, “A plasmonic splitter based on slot cavity,” Opt. Express 19, 13831–13838 (2011).
[CrossRef]

Luo, X.

Luo, X. G.

Lv, Y.

Lv, Y. G.

Ma, F. S.

Maier, S. A.

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

Mao, D.

Markey, L.

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

J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
[CrossRef]

Massoud, Y.

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
[CrossRef]

Meltzer, S.

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

Min, C.

Pan, W.

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Design of plasmonic comb-like filters using loop-based resonators,” Plasmonics 8, 1017–1022 (2013).
[CrossRef]

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).

X. Zou, M. Li, W. Pan, L. Yan, J. Azaña, and J. Yao, “All-fiber optical filter with an ultra-narrow and rectangular spectral response,” Opt. Lett. 38, 3096–3098 (2013).
[CrossRef]

X. Luo, X. H. Zou, X. F. Li, Z. Zhou, W. Pan, L. S. Yan, and K. H. Wen, “High-uniformity multichannel plasmonic filter using linearly lengthened insulators in metal-insulator-metal waveguide,” Opt. Lett. 38, 1585–1587 (2013).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, Y. H. Guo, and X. G. Luo, “Spectral characteristics of plasmonic metal-insulator-metal waveguides with a tilted groove,” IEEE Photon. J. 4, 1794–1800 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Wavelength demultiplexing structure based on plasmonic metal-insulator-metal waveguide,” J. Opt. 14, 075001 (2012).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “A four-port plasmonic quasi-circulator based on metal-insulator-metal waveguides,” Opt. Express 20, 28025–28032 (2012).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, and X. G. Luo, “Electromagnetically induced transparency (EIT)-like transmission in side-coupled complementary split-ring resonators,” Opt. Express 20, 24348–24355 (2012).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, H. Li, and X. G. Luo, “A plasmonic splitter based on slot cavity,” Opt. Express 19, 13831–13838 (2011).
[CrossRef]

Pannipitiya, A.

Premaratne, M.

Requicha, A. G.

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

Rukhlenko, I. D.

Scherer, A.

S. S. Walavalkar, P. Latawiec, A. Homyk, and A. Scherer, “Scalable method for the fabrication and testing of glass filled, three-dimensionally sculpted extraordinary transmission apertures,” Nano Lett. 14, 311–317 (2014).
[CrossRef]

Shi, H.

Sun, X. W.

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Tao, J.

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, 508–511 (2006).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Wahsheh, R. A.

Walavalkar, S. S.

S. S. Walavalkar, P. Latawiec, A. Homyk, and A. Scherer, “Scalable method for the fabrication and testing of glass filled, three-dimensionally sculpted extraordinary transmission apertures,” Nano Lett. 14, 311–317 (2014).
[CrossRef]

Wang, B.

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[CrossRef]

Wang, C.

Wang, C. T.

Wang, G.

Wang, G. P.

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[CrossRef]

Wang, H. Z.

Wang, L.

Wang, R.

Wang, T. B.

Wang, Y.

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125409 (2004).
[CrossRef]

Weeber, J. C.

J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
[CrossRef]

Wen, K. H.

Wen, X. W.

Yan, L.

X. Zou, M. Li, W. Pan, L. Yan, J. Azaña, and J. Yao, “All-fiber optical filter with an ultra-narrow and rectangular spectral response,” Opt. Lett. 38, 3096–3098 (2013).
[CrossRef]

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).

Yan, L. S.

Yang, Y.

Yao, H.

Yao, J.

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).

X. Zou, M. Li, W. Pan, L. Yan, J. Azaña, and J. Yao, “All-fiber optical filter with an ultra-narrow and rectangular spectral response,” Opt. Lett. 38, 3096–3098 (2013).
[CrossRef]

Yi, H. X.

Yin, C. P.

Yuan, X. C.

Yun, B.

B. Yun, G. Hu, and Y. Cui, “Theoretical analysis of a nanoscale plasmonic filter based on a rectangular metal-insulator-metal waveguide,” J. Phys. D 43, 385102 (2010).
[CrossRef]

Zand, I.

Zhang, C.

Zhang, Q.

Zhou, Z.

Zhou, Z. P.

Zhu, S.

Zou, X.

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).

X. Zou, M. Li, W. Pan, L. Yan, J. Azaña, and J. Yao, “All-fiber optical filter with an ultra-narrow and rectangular spectral response,” Opt. Lett. 38, 3096–3098 (2013).
[CrossRef]

Zou, X. H.

Appl. Opt.

Appl. Phys. Lett.

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[CrossRef]

A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90, 181102 (2007).
[CrossRef]

IEEE Photon. J.

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, Y. H. Guo, and X. G. Luo, “Spectral characteristics of plasmonic metal-insulator-metal waveguides with a tilted groove,” IEEE Photon. J. 4, 1794–1800 (2012).
[CrossRef]

IEEE Photon. Technol. Lett.

Z. H. Han, E. Forsberg, and S. L. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

X. Zou, W. Li, W. Pan, L. Yan, and J. Yao, “Photonic-assisted microwave channelizer with improved channel characteristics based on spectrum-controlled stimulated Brillouin scattering,” IEEE Trans. Microw. Theory Tech. 61, 3470–3478 (2013).

J. Lightwave Technol.

J. Opt.

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Wavelength demultiplexing structure based on plasmonic metal-insulator-metal waveguide,” J. Opt. 14, 075001 (2012).
[CrossRef]

J. Phys. D

B. Yun, G. Hu, and Y. Cui, “Theoretical analysis of a nanoscale plasmonic filter based on a rectangular metal-insulator-metal waveguide,” J. Phys. D 43, 385102 (2010).
[CrossRef]

Nano Lett.

J. C. Weeber, A. Bouhelier, F. G. Des, L. Markey, and A. Dereux, “Submicrometer in-plane integrated surface plasmon cavities,” Nano Lett. 7, 1352–1359 (2007).
[CrossRef]

S. S. Walavalkar, P. Latawiec, A. Homyk, and A. Scherer, “Scalable method for the fabrication and testing of glass filled, three-dimensionally sculpted extraordinary transmission apertures,” Nano Lett. 14, 311–317 (2014).
[CrossRef]

Nat. Mater.

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

Nature

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, 508–511 (2006).
[CrossRef]

Opt. Express

H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, and H. Yao, “Surface plasmon polariton propagation and combination in Y-shaped metallic channels,” Opt. Express 13, 10795–10800 (2005).
[CrossRef]

T. W. Lee and S. Gray, “Subwavelength light bending by metal slit structures,” Opt. Express 13, 9652–9659 (2005).
[CrossRef]

R. A. Wahsheh, Z. Lu, and M. A. G. Abushagur, “Nanoplasmonic couplers and splitters,” Opt. Express 17, 19033–19040 (2009).
[CrossRef]

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “A four-port plasmonic quasi-circulator based on metal-insulator-metal waveguides,” Opt. Express 20, 28025–28032 (2012).
[CrossRef]

M. A. Bavil, Z. Zhou, and Q. Deng, “Active unidirectional propagation of surface plasmons at subwavelength slits,” Opt. Express 21, 17066–17076 (2013).
[CrossRef]

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010).
[CrossRef]

J. Tao, X. G. Huang, X. S. Lin, Q. Zhang, and X. P. Jin, “A narrow-band subwavelength plasmonic waveguide filter with asymmetrical multiple-teeth-shaped structure,” Opt. Express 17, 13989–13994 (2009).
[CrossRef]

T. B. Wang, X. W. Wen, C. P. Yin, and H. Z. Wang, “The transmission characteristics of surface plasmon polaritons in ring resonator,” Opt. Express 17, 24096–24101 (2009).
[CrossRef]

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

G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19, 3513–3518 (2011).
[CrossRef]

H. Lu, X. Liu, D. Mao, L. Wang, and Y. Gong, “Tunable band-pass plasmonic waveguide filters with nanodisk resonators,” Opt. Express 18, 17922–17927 (2010).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, H. Li, and X. G. Luo, “A plasmonic splitter based on slot cavity,” Opt. Express 19, 13831–13838 (2011).
[CrossRef]

Q. Zhang, X. G. Huang, X. S. Lin, J. Tao, and X. P. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express 17, 7549–7554 (2009).
[CrossRef]

C. T. Wang, C. L. Du, Y. G. Lv, and X. G. Luo, “Surface electromagnetic wave excitation and diffraction by subwavelength slit with periodically patterned metallic grooves,” Opt. Express 14, 5671–5681 (2006).
[CrossRef]

Y. Yang, H. T. Dai, and X. W. Sun, “Split ring aperture for optical magnetic field enhancement by radially polarized beam,” Opt. Express 21, 6845–6850 (2013).
[CrossRef]

I. Zand, M. S. Abrishamian, and P. Berini, “Highly tunable nanoscale metal-insulator-metal split ring core ring resonators (SRCRRs),” Opt. Express 21, 79–86 (2013).
[CrossRef]

Y. H. Guo, L. S. Yan, W. Pan, B. Luo, K. H. Wen, Z. Guo, and X. G. Luo, “Electromagnetically induced transparency (EIT)-like transmission in side-coupled complementary split-ring resonators,” Opt. Express 20, 24348–24355 (2012).
[CrossRef]

Opt. Lett.

Phys. Rev. B

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125409 (2004).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Plasmonics

K. H. Wen, L. S. Yan, W. Pan, B. Luo, Z. Guo, and Y. H. Guo, “Design of plasmonic comb-like filters using loop-based resonators,” Plasmonics 8, 1017–1022 (2013).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of the proposed plasmonic filter based on the SRR structure (a) including a metallic nanowall, (b) without defect, and (c)–(i) including a nanowall with angles from 0° to 90°.

Fig. 2.
Fig. 2.

Transmission spectra of the circular ring (a) without defect, and (b)–(h) with a nanowall whose angle φ increases from 0° to 90°. The insets are the schematic of the SRR and the magnetic-field distributions at the resonance wavelengths.

Fig. 3.
Fig. 3.

Transmission spectra of the SRR with dual nanowalls in a line. The insets illustrate the schematic of the SRR and the magnetic-field distributions at the resonance wavelengths.

Fig. 4.
Fig. 4.

Transmission spectra of the separated modes by cascading two SRRs: (a) TM1, (b) TM1.5, (c) TM2, and (d) TM2.5. The insets are the schematics of the SRR and the magnetic-field distributions at the resonance wavelengths.

Tables (2)

Tables Icon

Table 1. Transmittances of Resonant Modes with Different Positions of Nanowalls

Tables Icon

Table 2. Transmittances and Wavelengths for Each Mode

Equations (2)

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

kRe(neff)Leff+Δθ=2mπ,m=1,2,3,,
εikm+εmkitanh(jkiW/2)=0.

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