Abstract

A surface plasmon polaritons (SPPs) refractive index sensor which consists of two metal-insulator-metal (MIM) waveguides coupled to each other by a ring resonator is proposed. The transmission properties are numerically simulated by finite element method. The sensing characteristics of such structure are systematically analyzed by investigating the transmission spectrum. The results indicate that there exist three resonance peaks in the transmission spectrum, and all of which have a linear relationship with the refractive index of the material under sensing. Through the optimization of structural parameters, we achieve a theoretical value of the refractive index sensitivity as high as 3460nmRIU−1. Furthermore, this structure can also be used as a temperature sensor with temperature sensitivity of 1.36nm/°C. This work paves the way toward sensitive nanometer scale refractive index sensor and temperature sensor for design and application.

© 2014 Optical Society of America

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  1. V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
    [Crossref]
  2. D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
    [Crossref]
  3. F. Fan, S. Chen, X. H. Wang, and S. J. Chang, “Tunable nonreciprocal terahertz transmission and enhancement based on metal/magneto-optic plasmonic lens,” Opt. Express 21(7), 8614–8621 (2013).
    [Crossref] [PubMed]
  4. A. Dolatabady, N. Granpayeh, and V. F. Nezhad, “A nanoscale refractive index sensor in two dimensional plasmonic waveguide with nanodisk resonator,” Opt. Commun. 300, 265–268 (2013).
    [Crossref]
  5. V. E. Bochenkov, M. Frederiksen, and D. S. Sutherland, “Enhanced refractive index sensitivity of elevated short-range ordered nanohole arrays in optically thin plasmonic Au films,” Opt. Express 21(12), 14763–14770 (2013).
    [Crossref] [PubMed]
  6. J. Zhu, J. J. Li, and J. W. Zhao, “Improve the refractive index sensitivity of coaxial-cable type gold nanostructure: the effect of dielectric polarization from the separate layer,” J. Nanopart. Res. 15(6), 1721 (2013).
    [Crossref]
  7. S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
    [Crossref]
  8. A. Sun and Z. S. Wu, “A Hybrid LPG/CFBG for Highly Sensitive Refractive Index Measurements,” Sensors (Basel) 12(12), 7318–7325 (2012).
    [Crossref] [PubMed]
  9. Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
    [Crossref] [PubMed]
  10. B. Gallinet and O. J. F. Martin, “Refractive index sensing with subradiant modes: A framework to reduce losses in plasmonic nanostructures,” ACS Nano 7(8), 6978–6987 (2013).
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  11. M. X. Ren, C. P. Pan, Q. Q. Li, W. Cai, X. Z. Zhang, Q. Wu, S. Fan, and J. Xu, “Isotropic spiral plasmonic metamaterial for sensing large refractive index change,” Opt. Lett. 38(16), 3133–3136 (2013).
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  17. K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
    [Crossref]
  18. T. S. Wu, L. Wang, and Z. Wang, “A photonic crystal fiber temperature sensor based on Signac interferometer structure,” Chin. J. Lasers 39(11), 1114002 (2012).
    [Crossref]
  19. L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
    [Crossref]
  20. M. M. Luo, Y. G. Liu, Z. Wang, T. T. Han, Z. Wu, J. Guo, and W. Huang, “Twin-resonance-coupling and high sensitivity sensing characteristics of a selectively fluid-filled microstructured optical fiber,” Opt. Express 21(25), 30911–30917 (2013).
    [Crossref] [PubMed]
  21. C. Nicolaou, W. T. Lau, R. Gad, H. Akhavan, R. Schilling, and O. Levi, “Enhanced detection limit by dark mode perturbation in 2D photonic crystal slab refractive index sensors,” Opt. Express 21(25), 31698–31712 (2013).
    [Crossref] [PubMed]
  22. S. C. Warren-Smith and T. M. Monro, “Exposed core microstructured optical fiber Bragg gratings: refractive index sensing,” Opt. Express 22(2), 1480–1489 (2014).
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  23. D. K. C. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009).
    [Crossref] [PubMed]
  24. J. J. Chen, C. W. Sun, and Q. H. Gong, “Fano resonances in a single defect nanocavity coupled with a plasmonic waveguide,” Opt. Lett. 39(1), 52–55 (2014).
    [Crossref] [PubMed]
  25. B. W. You, J. Y. Lu, T. A. Liu, and J. L. Peng, “Hybrid terahertz plasmonic waveguide for sensing applications,” Opt. Express 21(18), 21087–21096 (2013).
    [Crossref] [PubMed]
  26. E. A. Velichko and A. I. Nosich, “Refractive-index sensitivities of hybrid surface-plasmon resonances for a core-shell circular silver nanotube sensor,” Opt. Lett. 38(23), 4978–4981 (2013).
    [Crossref] [PubMed]
  27. J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
    [Crossref]
  28. X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33(23), 2874–2876 (2008).
    [Crossref] [PubMed]
  29. 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(26), 24096–24101 (2009).
    [Crossref] [PubMed]

2014 (3)

2013 (16)

B. W. You, J. Y. Lu, T. A. Liu, and J. L. Peng, “Hybrid terahertz plasmonic waveguide for sensing applications,” Opt. Express 21(18), 21087–21096 (2013).
[Crossref] [PubMed]

E. A. Velichko and A. I. Nosich, “Refractive-index sensitivities of hybrid surface-plasmon resonances for a core-shell circular silver nanotube sensor,” Opt. Lett. 38(23), 4978–4981 (2013).
[Crossref] [PubMed]

M. M. Luo, Y. G. Liu, Z. Wang, T. T. Han, Z. Wu, J. Guo, and W. Huang, “Twin-resonance-coupling and high sensitivity sensing characteristics of a selectively fluid-filled microstructured optical fiber,” Opt. Express 21(25), 30911–30917 (2013).
[Crossref] [PubMed]

C. Nicolaou, W. T. Lau, R. Gad, H. Akhavan, R. Schilling, and O. Levi, “Enhanced detection limit by dark mode perturbation in 2D photonic crystal slab refractive index sensors,” Opt. Express 21(25), 31698–31712 (2013).
[Crossref] [PubMed]

J. H. Zhou, X. P. Xu, W. B. Han, D. Mu, H. Song, Y. Meng, X. Leng, J. Yang, X. Di, and Q. Chang, “Fano resonance of nanoparticles embedded in Fabry-Perot cavities,” Opt. Express 21(10), 12159–12164 (2013).
[Crossref] [PubMed]

T. Cao and L. Zhang, “Enhancement of Fano resonance in metal/dielectric/metal metamaterials at optical regime,” Opt. Express 21(16), 19228–19239 (2013).
[Crossref] [PubMed]

X. B. Kang, H. D. Li, J. Ding, and Z. G. Wang, “Fano resonance and step-like transmission via guide-mode resonance structure,” Opt. Lett. 38(5), 715–717 (2013).
[Crossref] [PubMed]

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

F. Fan, S. Chen, X. H. Wang, and S. J. Chang, “Tunable nonreciprocal terahertz transmission and enhancement based on metal/magneto-optic plasmonic lens,” Opt. Express 21(7), 8614–8621 (2013).
[Crossref] [PubMed]

A. Dolatabady, N. Granpayeh, and V. F. Nezhad, “A nanoscale refractive index sensor in two dimensional plasmonic waveguide with nanodisk resonator,” Opt. Commun. 300, 265–268 (2013).
[Crossref]

V. E. Bochenkov, M. Frederiksen, and D. S. Sutherland, “Enhanced refractive index sensitivity of elevated short-range ordered nanohole arrays in optically thin plasmonic Au films,” Opt. Express 21(12), 14763–14770 (2013).
[Crossref] [PubMed]

J. Zhu, J. J. Li, and J. W. Zhao, “Improve the refractive index sensitivity of coaxial-cable type gold nanostructure: the effect of dielectric polarization from the separate layer,” J. Nanopart. Res. 15(6), 1721 (2013).
[Crossref]

S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
[Crossref]

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

B. Gallinet and O. J. F. Martin, “Refractive index sensing with subradiant modes: A framework to reduce losses in plasmonic nanostructures,” ACS Nano 7(8), 6978–6987 (2013).
[Crossref] [PubMed]

M. X. Ren, C. P. Pan, Q. Q. Li, W. Cai, X. Z. Zhang, Q. Wu, S. Fan, and J. Xu, “Isotropic spiral plasmonic metamaterial for sensing large refractive index change,” Opt. Lett. 38(16), 3133–3136 (2013).
[Crossref] [PubMed]

2012 (4)

A. Sun and Z. S. Wu, “A Hybrid LPG/CFBG for Highly Sensitive Refractive Index Measurements,” Sensors (Basel) 12(12), 7318–7325 (2012).
[Crossref] [PubMed]

T. S. Wu, L. Wang, and Z. Wang, “A photonic crystal fiber temperature sensor based on Signac interferometer structure,” Chin. J. Lasers 39(11), 1114002 (2012).
[Crossref]

J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
[Crossref]

Y. H. Fu, J. B. Zhang, Y. F. Yu, and B. Luk’yanchuk, “Generating and Manipulating Higher Order Fano Resonances in Dual-Disk Ring Plasmonic Nanostructures,” ACS Nano 6(6), 5130–5137 (2012).
[Crossref] [PubMed]

2010 (2)

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

2009 (3)

2008 (1)

Akhavan, H.

Armelles, G.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Bochenkov, V. E.

Borghs, G.

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

Bratschitsch, R.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Cai, W.

Cao, T.

Cebollada, A.

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

Chang, Q.

Chang, S. J.

Chen, J. J.

Chen, S.

Di, X.

Ding, J.

Dolatabady, A.

A. Dolatabady, N. Granpayeh, and V. F. Nezhad, “A nanoscale refractive index sensor in two dimensional plasmonic waveguide with nanodisk resonator,” Opt. Commun. 300, 265–268 (2013).
[Crossref]

Dorpe, P. V.

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

Eggleton, B. J.

Fan, F.

Fan, S.

Frederiksen, M.

Fu, Y. H.

Y. H. Fu, J. B. Zhang, Y. F. Yu, and B. Luk’yanchuk, “Generating and Manipulating Higher Order Fano Resonances in Dual-Disk Ring Plasmonic Nanostructures,” ACS Nano 6(6), 5130–5137 (2012).
[Crossref] [PubMed]

Gad, R.

Gallinet, B.

B. Gallinet and O. J. F. Martin, “Refractive index sensing with subradiant modes: A framework to reduce losses in plasmonic nanostructures,” ACS Nano 7(8), 6978–6987 (2013).
[Crossref] [PubMed]

Garcia-Martin, A.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

Garcia-Martin, J.-M.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

García-Martín, A.

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Gong, Q. H.

González, M. U.

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

González-Díaz, J. B.

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Granpayeh, N.

A. Dolatabady, N. Granpayeh, and V. F. Nezhad, “A nanoscale refractive index sensor in two dimensional plasmonic waveguide with nanodisk resonator,” Opt. Commun. 300, 265–268 (2013).
[Crossref]

Guo, J.

Guzatov, D.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

Han, T. T.

Han, W. B.

Hao, F.

F. Hao, P. Nordlander, Y. Sonnefraud, P. van Dorpe, and S. A. Maier, “Tunability of Subradiant Dipolar and Fano-Type Plasmon Resonances in Metallic Ring/Disk Cavities: Implications for Nanoscale Optical Sensing,” ACS Nano 3(3), 643–652 (2009).
[Crossref] [PubMed]

Huang, W.

Huang, X. G.

J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
[Crossref]

X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33(23), 2874–2876 (2008).
[Crossref] [PubMed]

Jauho, A. P.

S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
[Crossref]

Jiang, R. B.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Jin, C.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Jin, S.

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Jin, X. P.

J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
[Crossref]

Kang, X. B.

Kuhlmey, B. T.

Lagae, L.

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

Lau, W. T.

Leitenstorfer, A.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Leng, X.

Levi, O.

Li, H. D.

Li, J. J.

J. Zhu, J. J. Li, and J. W. Zhao, “Improve the refractive index sensitivity of coaxial-cable type gold nanostructure: the effect of dielectric polarization from the separate layer,” J. Nanopart. Res. 15(6), 1721 (2013).
[Crossref]

Li, Q. Q.

Lin, X. S.

Liu, M. X.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Liu, T. A.

Liu, T. R.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Liu, Y. G.

Lodewijks, K.

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

Lu, J. Y.

Luk’yanchuk, B.

Y. H. Fu, J. B. Zhang, Y. F. Yu, and B. Luk’yanchuk, “Generating and Manipulating Higher Order Fano Resonances in Dual-Disk Ring Plasmonic Nanostructures,” ACS Nano 6(6), 5130–5137 (2012).
[Crossref] [PubMed]

Luo, M. M.

Maier, S. A.

F. Hao, P. Nordlander, Y. Sonnefraud, P. van Dorpe, and S. A. Maier, “Tunability of Subradiant Dipolar and Fano-Type Plasmon Resonances in Metallic Ring/Disk Cavities: Implications for Nanoscale Optical Sensing,” ACS Nano 3(3), 643–652 (2009).
[Crossref] [PubMed]

Martin, O. J. F.

B. Gallinet and O. J. F. Martin, “Refractive index sensing with subradiant modes: A framework to reduce losses in plasmonic nanostructures,” ACS Nano 7(8), 6978–6987 (2013).
[Crossref] [PubMed]

Martín-Becerra, D.

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Mei, X.

J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
[Crossref]

Meng, Y.

Monro, T. M.

Mortensen, N. A.

S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
[Crossref]

Mu, D.

Nezhad, V. F.

A. Dolatabady, N. Granpayeh, and V. F. Nezhad, “A nanoscale refractive index sensor in two dimensional plasmonic waveguide with nanodisk resonator,” Opt. Commun. 300, 265–268 (2013).
[Crossref]

Nicolaou, C.

Nordlander, P.

F. Hao, P. Nordlander, Y. Sonnefraud, P. van Dorpe, and S. A. Maier, “Tunability of Subradiant Dipolar and Fano-Type Plasmon Resonances in Metallic Ring/Disk Cavities: Implications for Nanoscale Optical Sensing,” ACS Nano 3(3), 643–652 (2009).
[Crossref] [PubMed]

Nosich, A. I.

Pan, C. P.

Peng, J. L.

Qi, L.

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Raza, S.

S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
[Crossref]

Ren, M. X.

Roy, W. V.

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

Ryken, J.

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

Schilling, R.

Shen, Y.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Song, H.

Sonnefraud, Y.

F. Hao, P. Nordlander, Y. Sonnefraud, P. van Dorpe, and S. A. Maier, “Tunability of Subradiant Dipolar and Fano-Type Plasmon Resonances in Metallic Ring/Disk Cavities: Implications for Nanoscale Optical Sensing,” ACS Nano 3(3), 643–652 (2009).
[Crossref] [PubMed]

Sun, A.

A. Sun and Z. S. Wu, “A Hybrid LPG/CFBG for Highly Sensitive Refractive Index Measurements,” Sensors (Basel) 12(12), 7318–7325 (2012).
[Crossref] [PubMed]

Sun, C. W.

Sutherland, D. S.

Tao, J.

J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
[Crossref]

Tao, Y. T.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Temnov, V. V.

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

Thomay, T.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Toscano, G.

S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
[Crossref]

van Dorpe, P.

F. Hao, P. Nordlander, Y. Sonnefraud, P. van Dorpe, and S. A. Maier, “Tunability of Subradiant Dipolar and Fano-Type Plasmon Resonances in Metallic Ring/Disk Cavities: Implications for Nanoscale Optical Sensing,” ACS Nano 3(3), 643–652 (2009).
[Crossref] [PubMed]

Velichko, E. A.

Wang, H. Z.

Wang, J.

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Wang, L.

T. S. Wu, L. Wang, and Z. Wang, “A photonic crystal fiber temperature sensor based on Signac interferometer structure,” Chin. J. Lasers 39(11), 1114002 (2012).
[Crossref]

Wang, T. B.

Wang, X.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Wang, X. H.

Wang, Z.

Wang, Z. G.

Warren-Smith, S. C.

Wen, X. W.

Woggon, U.

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

Wu, D. K. C.

Wu, Q.

Wu, T. S.

T. S. Wu, L. Wang, and Z. Wang, “A photonic crystal fiber temperature sensor based on Signac interferometer structure,” Chin. J. Lasers 39(11), 1114002 (2012).
[Crossref]

Wu, Z.

Wu, Z. S.

A. Sun and Z. S. Wu, “A Hybrid LPG/CFBG for Highly Sensitive Refractive Index Measurements,” Sensors (Basel) 12(12), 7318–7325 (2012).
[Crossref] [PubMed]

Wubs, M.

S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
[Crossref]

Xiao, G.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Xu, J.

Xu, X. P.

Yang, J.

Ye, M. P.

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Yin, C. P.

You, B. W.

Yu, Y. F.

Y. H. Fu, J. B. Zhang, Y. F. Yu, and B. Luk’yanchuk, “Generating and Manipulating Higher Order Fano Resonances in Dual-Disk Ring Plasmonic Nanostructures,” ACS Nano 6(6), 5130–5137 (2012).
[Crossref] [PubMed]

Yuan, J. Y.

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Zhang, J. B.

Y. H. Fu, J. B. Zhang, Y. F. Yu, and B. Luk’yanchuk, “Generating and Manipulating Higher Order Fano Resonances in Dual-Disk Ring Plasmonic Nanostructures,” ACS Nano 6(6), 5130–5137 (2012).
[Crossref] [PubMed]

Zhang, L.

Zhang, X. Z.

Zhang, Z.

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Zhao, C. L.

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Zhao, J. W.

J. Zhu, J. J. Li, and J. W. Zhao, “Improve the refractive index sensitivity of coaxial-cable type gold nanostructure: the effect of dielectric polarization from the separate layer,” J. Nanopart. Res. 15(6), 1721 (2013).
[Crossref]

Zhou, J. H.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

J. H. Zhou, X. P. Xu, W. B. Han, D. Mu, H. Song, Y. Meng, X. Leng, J. Yang, X. Di, and Q. Chang, “Fano resonance of nanoparticles embedded in Fabry-Perot cavities,” Opt. Express 21(10), 12159–12164 (2013).
[Crossref] [PubMed]

Zhou, Z. K.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Zhu, J.

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

J. Zhu, J. J. Li, and J. W. Zhao, “Improve the refractive index sensitivity of coaxial-cable type gold nanostructure: the effect of dielectric polarization from the separate layer,” J. Nanopart. Res. 15(6), 1721 (2013).
[Crossref]

Zhu, J. H.

J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
[Crossref]

ACS Nano (3)

B. Gallinet and O. J. F. Martin, “Refractive index sensing with subradiant modes: A framework to reduce losses in plasmonic nanostructures,” ACS Nano 7(8), 6978–6987 (2013).
[Crossref] [PubMed]

F. Hao, P. Nordlander, Y. Sonnefraud, P. van Dorpe, and S. A. Maier, “Tunability of Subradiant Dipolar and Fano-Type Plasmon Resonances in Metallic Ring/Disk Cavities: Implications for Nanoscale Optical Sensing,” ACS Nano 3(3), 643–652 (2009).
[Crossref] [PubMed]

Y. H. Fu, J. B. Zhang, Y. F. Yu, and B. Luk’yanchuk, “Generating and Manipulating Higher Order Fano Resonances in Dual-Disk Ring Plasmonic Nanostructures,” ACS Nano 6(6), 5130–5137 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

D. Martín-Becerra, J. B. González-Díaz, V. V. Temnov, A. Cebollada, G. Armelles, T. Thomay, A. Leitenstorfer, R. Bratschitsch, A. García-Martín, and M. U. González, “Enhancement of the magnetic modulation of surface plasmon polaritons in Au/Co/Au films,” Appl. Phys. Lett. 97(18), 183114 (2010).
[Crossref]

Chin. J. Lasers (1)

T. S. Wu, L. Wang, and Z. Wang, “A photonic crystal fiber temperature sensor based on Signac interferometer structure,” Chin. J. Lasers 39(11), 1114002 (2012).
[Crossref]

J. Nanopart. Res. (1)

J. Zhu, J. J. Li, and J. W. Zhao, “Improve the refractive index sensitivity of coaxial-cable type gold nanostructure: the effect of dielectric polarization from the separate layer,” J. Nanopart. Res. 15(6), 1721 (2013).
[Crossref]

Nat. Commun. (1)

Y. Shen, J. H. Zhou, T. R. Liu, Y. T. Tao, R. B. Jiang, M. X. Liu, G. Xiao, J. Zhu, Z. K. Zhou, X. Wang, C. Jin, and J. Wang, “Plasmonic gold mushroom arrays with refractive index sensing figures of merit approaching the theoretical limit,” Nat. Commun. 4, 2381 (2013).
[Crossref] [PubMed]

Nat. Photonics (1)

V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Photonics 4(2), 107–111 (2010).
[Crossref]

Opt. Commun. (2)

J. H. Zhu, X. G. Huang, J. Tao, X. P. Jin, and X. Mei, “Nanometeric plasmonic refractive index senor,” Opt. Commun. 285(13-14), 3242–3245 (2012).
[Crossref]

A. Dolatabady, N. Granpayeh, and V. F. Nezhad, “A nanoscale refractive index sensor in two dimensional plasmonic waveguide with nanodisk resonator,” Opt. Commun. 300, 265–268 (2013).
[Crossref]

Opt. Express (9)

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(26), 24096–24101 (2009).
[Crossref] [PubMed]

F. Fan, S. Chen, X. H. Wang, and S. J. Chang, “Tunable nonreciprocal terahertz transmission and enhancement based on metal/magneto-optic plasmonic lens,” Opt. Express 21(7), 8614–8621 (2013).
[Crossref] [PubMed]

J. H. Zhou, X. P. Xu, W. B. Han, D. Mu, H. Song, Y. Meng, X. Leng, J. Yang, X. Di, and Q. Chang, “Fano resonance of nanoparticles embedded in Fabry-Perot cavities,” Opt. Express 21(10), 12159–12164 (2013).
[Crossref] [PubMed]

V. E. Bochenkov, M. Frederiksen, and D. S. Sutherland, “Enhanced refractive index sensitivity of elevated short-range ordered nanohole arrays in optically thin plasmonic Au films,” Opt. Express 21(12), 14763–14770 (2013).
[Crossref] [PubMed]

T. Cao and L. Zhang, “Enhancement of Fano resonance in metal/dielectric/metal metamaterials at optical regime,” Opt. Express 21(16), 19228–19239 (2013).
[Crossref] [PubMed]

B. W. You, J. Y. Lu, T. A. Liu, and J. L. Peng, “Hybrid terahertz plasmonic waveguide for sensing applications,” Opt. Express 21(18), 21087–21096 (2013).
[Crossref] [PubMed]

M. M. Luo, Y. G. Liu, Z. Wang, T. T. Han, Z. Wu, J. Guo, and W. Huang, “Twin-resonance-coupling and high sensitivity sensing characteristics of a selectively fluid-filled microstructured optical fiber,” Opt. Express 21(25), 30911–30917 (2013).
[Crossref] [PubMed]

C. Nicolaou, W. T. Lau, R. Gad, H. Akhavan, R. Schilling, and O. Levi, “Enhanced detection limit by dark mode perturbation in 2D photonic crystal slab refractive index sensors,” Opt. Express 21(25), 31698–31712 (2013).
[Crossref] [PubMed]

S. C. Warren-Smith and T. M. Monro, “Exposed core microstructured optical fiber Bragg gratings: refractive index sensing,” Opt. Express 22(2), 1480–1489 (2014).
[Crossref] [PubMed]

Opt. Lett. (6)

Plasmonics (2)

S. Raza, G. Toscano, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Refractive-Index Sensing with Ultrathin Plasmonic Nanotubes,” Plasmonics 8(2), 193–199 (2013).
[Crossref]

K. Lodewijks, J. Ryken, W. V. Roy, G. Borghs, L. Lagae, and P. V. Dorpe, “Tuning the Fano Resonance Between Localized and Propagating Surface Plasmon Resonances for Refractive Index Sensing Applications,” Plasmonics 8(3), 1379–1385 (2013).
[Crossref]

Sens. Actuators B Chem. (1)

L. Qi, C. L. Zhao, J. Y. Yuan, M. P. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193, 185–189 (2014).
[Crossref]

Sensors (Basel) (1)

A. Sun and Z. S. Wu, “A Hybrid LPG/CFBG for Highly Sensitive Refractive Index Measurements,” Sensors (Basel) 12(12), 7318–7325 (2012).
[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 Real part of the effective index versus the refractive index n of insulator in a slit MIM SPPs waveguide structure for different incident wavelengths.

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Fig. 2
Fig. 2 Structure schematic of two slits MIM SPPs waveguide with a ring resonator. (a) Three-dimensional structure. (b) Two-dimensional structure.

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Fig. 3
Fig. 3 (a) The transmission spectrum of the sensor structure. The contour profiles of field Hz of the device at different wavelengths of (b) λ = 2000nm, (c) λ = 1488nm, (d) λ = 760nm.

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Fig. 4
Fig. 4 The transmission spectrum of the structure for different refractive indices with d = 50nm, w = 10nm, and r = 170nm.

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Fig. 5
Fig. 5 The three peaks of transmission spectrum versus the refractive index n of the material under sensing.

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Fig. 6
Fig. 6 The drift of the transmission peaks versus the temperature.

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Fig. 7
Fig. 7 The peaks of the transmission spectra versus the refractive index n with r = 150nm, r = 160nm, and r = 170nm.

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Fig. 8
Fig. 8 The peaks of the transmission spectra versus the refractive index with d = 30nm, d = 50nm, and d = 70nm.

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Fig. 9
Fig. 9 Transmission spectra of the structure for different radius and different refractive index n

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Equations (6)

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

ε i n k z 2 + ε m k z 1 coth ( i k z 1 ω / 2 ) = 0 ,
k z 1 2 = ε i n k 0 2 − β 2 ,
k z 2 2 = ε m k 0 2 − β 2 ,
ε m ( ω ) = ε ∞ − ω p 2 ω 2 + i ω γ ,
J n ' ( k r a ) J n ' ( k r i ) − N n ' ( k r a ) N n ' ( k r i ) = 0 ,
n = 1.36048 − 3.94 × 10 − 4 ( T − T 0 ) ,

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