Abstract

Metal-insulator-metal (MIM) surface plasmon polaritons (SPPs) waveguides with side-coupled resonators have been widely studied through various approaches. However, few methods are both physically transparent and complete. Here, an analytical approach, which is based on the Green’s function method, is developed in order to investigate electromagnetic wave transmission across SPPs MIM waveguide networks. The proposed method is applied in order to model different MIM-waveguide geometries with weakly-coupled side stubs, comparing to the geometries with strongly-coupled stubs. The weak coupling between the backbone and stubs is taken into account by the electromagnetic field leakage at metal-insulator interface. Analytical expressions for transmittance in cases of single stub and cavity are obtained straightforwardly. Our method shows excellent computational efficiency in contrast with solving Maxwell equations numerically.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2017 (4)

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

2016 (11)

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

2015 (5)

G. Lai, R. Liang, Y. Zhang, Z. Bian, L. Yi, G. Zhan, and R. Zhao, “Double plasmonic nanodisks design for electromagnetically induced transparency and slow light,” Opt. Express 23(5), 6554–6561 (2015).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

L. Y. He, T. J. Wang, Y. P. Gao, C. Cao, and C. Wang, “Discerning electromagnetically induced transparency from Autler-Townes splitting in plasmonic waveguide and coupled resonators system,” Opt. Express 23(18), 23817–23826 (2015).
[Crossref] [PubMed]

2014 (4)

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

K. Wen, Y. Hu, L. Chen, J. Zhou, L. Lei, and Z. Guo, “Design of an Optical Power and Wavelength Splitter Based on Subwavelength Waveguides,” J. Lightwave Technol. 32(17), 3020–3026 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

2013 (2)

A. Sellier, T. V. Teperik, and A. de Lustrac, “Resonant circuit model for efficient metamaterial absorber,” Opt. Express 21(Suppl 6), A997–A1006 (2013).
[Crossref] [PubMed]

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

2012 (2)

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

2011 (3)

2010 (2)

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[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(6), 6191–6204 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (1)

2006 (1)

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

2004 (1)

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Abouti, O. E.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Agrawal, G. P.

Akjouj, A.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Bayati, M. S.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Bian, Z.

Bonyadi Ram, S.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Cao, C.

Cao, G.

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

Chen, L.

Chen, X.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

Chen, Z.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

Cheng, B. H.

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Cherbi, L.

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

de Lustrac, A.

Djafarirouhani, B.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Djafari-Rouhani, B.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Dobrzynski, L.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Duan, L.

Ebadi, S. M.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

El Boudouti, E. H.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Elbialy, S.

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Elboudouti, E.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Fang, G.

Fikri, R.

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

Fukui, M.

Gao, Y. P.

Gong, Y.

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Guo, Z.

Gwon, H. R.

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[Crossref]

Haddouche, I.

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

Haraguchi, M.

Hattori, H. T.

He, J.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

He, L. Y.

He, Z.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Helmy, A. S.

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

Hu, F.

Hu, Y.

Huang, X.

Huang, X. G.

Jamili, M.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Jin, X.

Kaatuzian, H.

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

Lai, G.

Lan, Y. C.

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Lang, P.

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

Lee, S. H.

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[Crossref]

Lei, L.

Li, B.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Li, C.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Li, G.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Li, H.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Li, X.

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

Liang, R.

Lin, X.

Liu, J.

Liu, S.

Liu, X.

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[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(4), 3513–3518 (2011).
[Crossref] [PubMed]

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Lu, H.

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[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(4), 3513–3518 (2011).
[Crossref] [PubMed]

Lu, W.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Mansouri-Birjandi, M. A.

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

Mao, D.

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[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(4), 3513–3518 (2011).
[Crossref] [PubMed]

Matsuzaki, Y.

Nakagaki, M.

Noual, A.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Okamoto, T.

Ouyang, M.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Pannipitiya, A.

Paul, S.

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

Pennec, Y.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Poursajadi, S. M.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Premaratne, M.

Rakhshani, M. R.

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

Ray, M.

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

Rukhlenko, I. D.

Salim, M.

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

Samra, A.

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Sellier, A.

Song, J.

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

Su, R.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Su, X.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Swillam, M. A.

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

Taheri, A. N.

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

Tang, B.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Tao, J.

Teperik, T. V.

Vasseur, J.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
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Vigneron, J. P.

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

Wang, C.

Wang, G.

Wang, L.

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Wang, T. J.

Wang, Y.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Wang, Z.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

Wen, K.

Wu, H. H.

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Wu, J.

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

Xu, H.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Yang, H.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

Yi, H.

Yi, L.

Yousif, B.

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Zafar, R.

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

Zhan, G.

Zhan, S.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Zhang, J. X. J.

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

Zhang, Q.

Zhang, R.

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

Zhang, X.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Zhang, Y.

Zhao, H.

Zhao, R.

Zhao, W.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Zhao, Z.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Zheng, M.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Zhou, J.

Zhou, X.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Zhou, Z.

Zhu, Q.

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

Appl. Phys. B (1)

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (2)

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

EPL (1)

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

IEEE Photonic. Tech. L (4)

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

J. Appl. Phys. (1)

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

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

J. Phys. D Appl. Phys. (1)

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

Mater. Trans. (1)

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[Crossref]

Opt. Commun. (4)

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Opt. Express (8)

A. Sellier, T. V. Teperik, and A. de Lustrac, “Resonant circuit model for efficient metamaterial absorber,” Opt. Express 21(Suppl 6), A997–A1006 (2013).
[Crossref] [PubMed]

Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express 16(21), 16314–16325 (2008).
[Crossref] [PubMed]

L. Y. He, T. J. Wang, Y. P. Gao, C. Cao, and C. Wang, “Discerning electromagnetically induced transparency from Autler-Townes splitting in plasmonic waveguide and coupled resonators system,” Opt. Express 23(18), 23817–23826 (2015).
[Crossref] [PubMed]

G. Lai, R. Liang, Y. Zhang, Z. Bian, L. Yi, G. Zhan, and R. Zhao, “Double plasmonic nanodisks design for electromagnetically induced transparency and slow light,” Opt. Express 23(5), 6554–6561 (2015).
[Crossref] [PubMed]

J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009).
[Crossref] [PubMed]

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(4), 3513–3518 (2011).
[Crossref] [PubMed]

J. Tao, X. G. Huang, X. Lin, Q. Zhang, and X. Jin, “A narrow-band subwavelength plasmonic waveguide filter with asymmetrical multiple-teeth-shaped structure,” Opt. Express 17(16), 13989–13994 (2009).
[Crossref] [PubMed]

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(6), 6191–6204 (2010).
[Crossref] [PubMed]

Opt. Lett. (1)

Opt. Quantum Electron. (3)

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Phys. Rev. A (1)

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

Plasmonics (3)

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

Proc. SPIE 6343, Photonics North (1)

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

RSC Advances (1)

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Sci. Rep. (1)

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Surf. Sci. Rep. (1)

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Other (3)

S. A. Maier, Plasmonics: fundamentals and applications (Springer Science & Business Media 2007).

K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, Berlin, 2008).

S. Wu and D. Wu, “Adjusting Spectrum of MIM Optical Filters by Stub Inclination,” in Frontiers in Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTh2A.157.

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Figures (6)

Fig. 1
Fig. 1 Schematic illustrations of (a) infinite MIM waveguide (b) infinite metallic waveguide, and (c) semi-infinite MIM waveguide.
Fig. 2
Fig. 2 Schematic of the type of MIM waveguide networks system. (a) Input and output waveguides (b) short-circuit waveguide (c) open-circuit structure (d) unconnected side-coupled cavity.
Fig. 3
Fig. 3 Schematic diagrams of three typical constructions. (a) Arbitrary open-circuit stubs model (b) arbitrary side-couped cavities model (c) short circuit junctions’ networks.
Fig. 4
Fig. 4 (a) Schematic of single stub structure. (b) Transmission spectra for single side-coupled waveguide structure with L = 400 nm , d = 300 nm and h = w = 50 nm . (c) Magnetic field at the transmitted-dip wavelength λ = 1820 nm . The inset shows the FDTD simulation result for comparison.
Fig. 5
Fig. 5 (a) Schematic of single cavity structure. (b) Magnetic field distribution at λ = 1412 nm with s = 30 nm . Inset image: The magnetic-field distribution by the FDTD simulation. (c) Transmission spectrums for single cavity with various gap distances s = 10 nm , 20 nm , 30 nm , respectively. Simulation data (red dashed), conventional model without length modification (black dashed), and the proposed method results (blue solid) are demonstrated.
Fig. 6
Fig. 6 (a) Schematic of MIM waveguide with four serial stubs structure. (b)Transmission spectra for serial stubs structure with   L = d = 400 nm and h = w = 50 nm .

Equations (55)

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( 2 x 2 + 2 z 2 + k x 2 + k z 2 ) H y ( x , z ) = 0 ,
( k x 2 + 2 x 2 ) u ( x ) = 0.
Z k x ( k x 2 + 2 x 2 ) G ( x , x ' ) = δ ( x x ' ) ,
G ( x , x ' ) = e i k x | x x ' | 2 i Z .
tanh ( k sp 2 ε d k 0 2 d 2 ) = ( ε d k sp 2 ε m μ m k 0 2 ε m k sp 2 ε d μ d k 0 2 ) ,
k sp k 0 ε d 2 ε d ε d ε m k 0 ε d d .
k sp ' = k 0 ε m ,
Z ' = k sp ' d ' ω ε m .
[ g ( M M ) ] 1 = [ i Z Z cot ( k sp L ' ) Z sin ( k sp L ' ) Z sin ( k sp L ' ) i Z ' Z cot ( k sp L ' ) ] ,
g ( M M ) ( I ( M M ) + A ( M M ) ) = G ( M M ) i n D = [ D j ] ,
[ g j ( M M ) ] 1 = Δ j ( M M ) [ G j ( M M ) ] 1 ,
Δ j ( M M ) = I j ( M M ) + A j ( M M ) ,
A j ( x ' ' , x ' ) = V j ( x ) G j ( x , x ' ) d x ,
V j ( x ) = δ ( x ) n + δ ( x L j ) n ,
[ g j ( M M ) ] 1 = ( Z j cot ( k sp L j ) Z j sin ( k sp L j ) Z j sin ( k sp L j ) Z j cot ( k sp L j ) ) .
V j ( x ) = δ ( x ) n ,
[ g k ( M M ) ] 1 = ( Z k cot ( k sp L k ' ) Z k sin ( k sp L k ' ) Z k sin ( k sp L k ' ) Z k cot ( k sp L k ' ) ) ,
u ( D ) = U ( D ) U ( M ) [ G ( M M ) ] 1 G ( M D ) + U ( M ) [ G ( M M ) ] 1 g ( M M ) [ G ( M M ) ] 1 G ( M D ) .
u ( x ) = 2 i Z in sin ( α j d j ) { [ g ( 0 , x 1 ) g ( 0 , x 2 ) ] [ sin [ α j ( d j x ) ] sin ( α j x ) ] } , x | x 1 x 2 | = d j ,
u 0 1 ( x ) = u a ( x ) + u b ( x ) + u c ( x ) + u d ( x ) , ( 0 < x < d ) ,
u a ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 0 ) G 0 1 1 ( 0 , 0 ) G 0 1 ( 0 , x ) ,
u b ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 0 ) G 0 1 1 ( 0 , 1 ) G 0 1 ( 1 , x ) ,
u c ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 1 ) G 0 1 1 ( 1 , 1 ) G 0 1 ( 1 , x ) ,
u d ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 1 ) G 0 1 1 ( 1 , 0 ) G 0 1 ( 0 , x ) .
H I y ( x , z ) = u ( x ) e k z m ( z + d 2 ) ( z < d 2 ) ,
H II y ( x , z ) = u ( x ) ( 2 e k z d d 2 e k z d d + 1 cos h ( k z d z ) ) ( d 2 < z < d 2 ) ,
H III y ( x , z ) = u ( x ) e k z m ( z d 2 ) ( z > d 2 ) ,
H [ metallicWG ] y ( x , z ) = u ( x ) ( 2 e k z m d 2 e k z m d + 1 cos h ( k z m z ) ) .
T ( 1 n ) = | 2 i Z 1 g ( 1 , n ) | 2 ,
R 1 = | 1 2 i Z 1 g ( 1 , 1 ) | 2 .
T 1 , 2 , 3 n = | 2 i Z 1 A 1 e i φ 1 g ( 1 , n ) + 2 i Z 2 A 2 e i φ 2 g ( 2 , n ) + 2 i Z 3 A 3 e i φ 3 g ( 3 , n ) + | 2 ,
R k = | 1 2 i Z k A k g ( k , k ) + 2 i Z 1 A 1 e i φ 1 g ( 1 , k ) + 2 i Z 2 A 2 e i φ 2 g ( 2 , k ) + | 2 ,
g n 1 = { i Z 0 + φ right ( 1 ) + φ s ( 1 ) , n = 1 ; φ left ( j ) + φ s ( j ) + i Z 0 , n = j ; φ left ( n ) + φ right ( n ) + φ s ( n ) , n = 2 , 3 , , j 1.
g n ' 1 = i Z n ' + φ s ( n ) , n = 1 , 2 , 3 , j .
g ( p , q ) 1 = Z 0 sin ( k s p Δ ( p , q ) ) , if | p q | = 1.
g ( n , n ' ) 1 = Z n sin [ k sp ( d j + δ metal ) ] .
g n 1 = { i Z 0 + φ right ( 1 ) + φ gap ( 1 ) , n = 1 ; φ left ( j ) + φ gap ( j ) + i Z 0 , n = j ; φ left ( n ) + φ right ( n ) + φ gap ( n ) , n = 2 , 3 , , j 1.
g C n 1 = φ gap ( n ) + φ cavity ( n ) ,
g C n ' 1 = i Z n ' + φ cavity ( n ) ,
g ( p , q ) 1 = Z 0 sin ( k sp Δ ( p , q ) ) , if | p q | 1.
g ( n , C n ) 1 = Z n sin ( k sp s n ) .
g ( C n , C n ' ) 1 = Z n sin ( k sp d n ' ) .
g ( 1 , 1 ) 1 = i Z L 1 + i Z w 1 Z L 1 cot ( k sp Δ w 1 ) Z w 1 cot ( k sp Δ L 1 ) ,
g ( 1 , q ) 1 = i Z L 1 + i Z w q Z L 1 cot ( k sp Δ w q ) Z w q cot ( k sp Δ L 1 ) ,
g ( p , q ) 1 = i Z L p + i Z w q Z L p cot ( k sp Δ w q ) Z w q cot ( k sp Δ L p ) ,
g ( p , 1 ) 1 = i Z L p + i Z w 1 Z L p cot ( k sp Δ w 1 ) Z w 1 cot ( k sp Δ L p ) ,
g ( a , 1 ) 1 = i Z L a Z w 1 cot ( k sp Δ L a 1 ) Z w 1 cot ( k sp Δ L a + 1 ) Z L a cot ( k sp Δ w 1 ) ,
g ( 1 , b ) 1 = i Z w b Z L 1 cot ( k sp Δ w b 1 ) Z L 1 cot ( k sp Δ w b + 1 ) Z w b cot ( k sp Δ L 1 ) ,
g ( c , q ) 1 = i Z L c Z w q cot ( k sp Δ L c 1 ) Z w q cot ( k sp Δ L c + 1 ) Z L c cot ( k sp Δ w q ) ,
g ( p , d ) 1 = i Z w d Z L p cot ( k sp Δ L d 1 ) Z L p cot ( k sp Δ L d + 1 ) Z w d cot ( k sp Δ L p ) ,
g ( m , n ) 1 = Z w n cot ( k sp Δ L m 1 ) Z w n cot ( k sp Δ L m + 1 ) Z L m cot ( k sp Δ w n 1 ) Z L m cot ( k sp Δ w n + 1 ) ,
g [ ( r 1 , s 1 ) , ( r 2 , s 2 ) ] 1 = { Z L ( r 1 ) sin ( k sp Δ L [ min ( s 1 , s 2 ) ] ) , if r 1 = r 2 , and | s 1 s 2 | =1; Z w ( s 1 ) sin ( k sp Δ L [ min ( r 1 , r 2 ) ] ) , if s 1 = s 2 , and | r 1 r 2 | =1; 0 , else .
ε m ( ω ) = ε ω p 2 ω 2 i ω γ .
T stub = | 2 Z ' + i Z cot ( k sp ( d + δ metal ) ) i Z Z ' cot ( k sp ( d + δ metal ) ) e 2 i k sp L | 2 .
T cavity = | 4 i Z sin ( α ) sin 2 ( β ) ( Z + i Z ' cot ( β ) ) i Z cot ( α ) cot ( β ) Z ' cot ( α ) i ( 2 Z 2 + ( Z ' ) 2 ) [ sin ( α + 2 β ) sin ( α ) ] 3 Z Z ' [ cos ( α + 2 β ) cos ( α ) ] e 2 i k sp L | 2 ,

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