Abstract

We present the design and theoretical demonstration of a microstructured optical fiber (MOF) for multichannel sensing applications based on the Fano resonance among the different whispering-gallery modes (WGMs) propagating in the MOF. The proposed MOF consists of a number of capillary channels with different diameters inside a tubular frame. When the phases of the WGMs in the capillary channels and the frame are matched, the Fano resonance will occur and the resonant peaks can be observed in the output spectrum of the tubular frame resonator. Sensing signals from the individual channels can be detected by measuring the central wavelengths of the corresponding Fano resonant peaks. To demonstrate the practicality, we study a dual-channel MOF for bio-sensing applications, i.e., detecting the refractive index variation in biological samples. In the analysis, we have shown that channel 1 and 2 achieve a sensitivity of 29.0557 nm/RIU (refractive index unit) and 22.9160 nm/RIU in the TE mode; and 16.0694 nm/RIU and 13.3181 nm/RIU in the TM mode respectively, when the refractive index of the biological samples varies between 1.330 and 1.345. The new MOF can be a compact, flexible, and low-cost solution for a variety of applications including multichannel bio/chemical sensing, multi-microcavity laser, and tunable photonics devices.

© 2017 Optical Society of America

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  1. M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
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    [Crossref]
  4. V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
    [Crossref] [PubMed]
  5. Y. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
    [Crossref] [PubMed]
  6. F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
    [Crossref]
  7. C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
    [Crossref] [PubMed]
  8. Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
    [Crossref] [PubMed]
  9. M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
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  10. I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31(9), 1319–1321 (2006).
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  12. P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
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  13. Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
    [Crossref]
  14. M. Luo, Y. G. Liu, Z. Wang, 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]
  15. G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28(12), 989–991 (2003).
    [Crossref] [PubMed]
  16. W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
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  17. A. Cavanna, F. Just, X. Jiang, G. Leuchs, M. V. Chekhova, P. J. Russell, and N. Y. Joly, “Hybrid photonic-crystal fiber for single-mode phase matched generation of third harmonic and photon triplets,” Optica 3(9), 952–955 (2016).
    [Crossref]
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    [Crossref] [PubMed]
  19. J. Du, Y. Liu, Z. Wang, B. Zou, B. Liu, and X. Dong, “Electrically tunable Sagnac filter based on a photonic bandgap fiber with liquid crystal infused,” Opt. Lett. 33(19), 2215–2217 (2008).
    [Crossref] [PubMed]
  20. Q. Liu, S. Li, J. Li, C. Dou, X. Wang, G. Wang, and M. Shi, “Tunable fiber polarization filter by filling different index liquids and gold wire into photonic crystal fiber,” J. Lightwave Technol. 34(10), 2484–2490 (2016).
    [Crossref]
  21. W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
    [Crossref] [PubMed]
  22. A. Mahmood, V. Kavungal, S. S. Ahmed, G. Farrell, and Y. Semenova, “Magnetic-field sensor based on whispering-gallery modes in a photonic crystal fiber infiltrated with magnetic fluid,” Opt. Lett. 40(21), 4983–4986 (2015).
    [Crossref] [PubMed]
  23. D. D. Smith, H. Chang, and K. A. Fuller, “Whispering-gallery mode splitting in coupled microresonators,” J. Opt. Soc. Am. B 20(9), 1967–1974 (2003).
    [Crossref]
  24. T. Ling and L. J. Guo, “A unique resonance mode observed in a prism-coupled micro-tube resonator sensor with superior index sensitivity,” Opt. Express 15(25), 17424–17432 (2007).
    [Crossref] [PubMed]
  25. B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
    [Crossref] [PubMed]
  26. J. Liao, X. Wu, L. Liu, and L. Xu, “Fano resonance and improved sensing performance in a spectral-simplified optofluidic micro-bubble resonator by introducing selective modal losses,” Opt. Express 24(8), 8574–8580 (2016).
    [Crossref] [PubMed]
  27. S. I. Shopova, Y. Sun, A. T. Rosenberger, and X. Fan, “Highly sensitive tuning of coupled optical ring resonators by microfluidics,” Microfluid. Nanofluidics 6(3), 425–429 (2009).
    [Crossref]
  28. A. N. Kolyadin, A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. G. Plotnichenko, and E. M. Dianov, “Light transmission in negative curvature hollow core fiber in extremely high material loss region,” Opt. Express 21(8), 9514–9519 (2013).
    [Crossref] [PubMed]
  29. F. Yu and J. C. Knight, “Spectral attenuation limits of silica hollow core negative curvature fiber,” Opt. Express 21(18), 21466–21471 (2013).
    [Crossref] [PubMed]
  30. F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31(24), 3574–3576 (2006).
    [Crossref] [PubMed]

2016 (6)

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

A. Cavanna, F. Just, X. Jiang, G. Leuchs, M. V. Chekhova, P. J. Russell, and N. Y. Joly, “Hybrid photonic-crystal fiber for single-mode phase matched generation of third harmonic and photon triplets,” Optica 3(9), 952–955 (2016).
[Crossref]

Q. Liu, S. Li, J. Li, C. Dou, X. Wang, G. Wang, and M. Shi, “Tunable fiber polarization filter by filling different index liquids and gold wire into photonic crystal fiber,” J. Lightwave Technol. 34(10), 2484–2490 (2016).
[Crossref]

J. Liao, X. Wu, L. Liu, and L. Xu, “Fano resonance and improved sensing performance in a spectral-simplified optofluidic micro-bubble resonator by introducing selective modal losses,” Opt. Express 24(8), 8574–8580 (2016).
[Crossref] [PubMed]

2015 (4)

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

A. Mahmood, V. Kavungal, S. S. Ahmed, G. Farrell, and Y. Semenova, “Magnetic-field sensor based on whispering-gallery modes in a photonic crystal fiber infiltrated with magnetic fluid,” Opt. Lett. 40(21), 4983–4986 (2015).
[Crossref] [PubMed]

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

2014 (2)

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

2013 (4)

2010 (2)

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. S. Russell, “Emergence of geometrical optical nonlinearities in photonic crystal fiber nanowires,” Phys. Rev. Lett. 105(9), 093904 (2010).
[Crossref] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

2009 (1)

S. I. Shopova, Y. Sun, A. T. Rosenberger, and X. Fan, “Highly sensitive tuning of coupled optical ring resonators by microfluidics,” Microfluid. Nanofluidics 6(3), 425–429 (2009).
[Crossref]

2008 (1)

2007 (2)

Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
[Crossref]

T. Ling and L. J. Guo, “A unique resonance mode observed in a prism-coupled micro-tube resonator sensor with superior index sensitivity,” Opt. Express 15(25), 17424–17432 (2007).
[Crossref] [PubMed]

2006 (2)

2003 (6)

J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
[Crossref] [PubMed]

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28(12), 989–991 (2003).
[Crossref] [PubMed]

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

D. D. Smith, H. Chang, and K. A. Fuller, “Whispering-gallery mode splitting in coupled microresonators,” J. Opt. Soc. Am. B 20(9), 1967–1974 (2003).
[Crossref]

Ahmed, S. S.

Aksyuk, V.

Y. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref] [PubMed]

Asano, M.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

Benabid, F.

Biancalana, F.

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. S. Russell, “Emergence of geometrical optical nonlinearities in photonic crystal fiber nanowires,” Phys. Rev. Lett. 105(9), 093904 (2010).
[Crossref] [PubMed]

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Biriukov, A. S.

Bo, F.

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

Brasch, V.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

Cavanna, A.

Chang, H.

Chekhova, M. V.

Chen, W.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

Couny, F.

Davanço, M.

Y. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref] [PubMed]

Dianov, E. M.

Dong, C. H.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Dong, X.

Dou, C.

Du, J.

Efimov, A.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Fan, X.

S. I. Shopova, Y. Sun, A. T. Rosenberger, and X. Fan, “Highly sensitive tuning of coupled optical ring resonators by microfluidics,” Microfluid. Nanofluidics 6(3), 425–429 (2009).
[Crossref]

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31(9), 1319–1321 (2006).
[Crossref] [PubMed]

Farrell, G.

Foreman, M. R.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

Foster, M. A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Fu, W.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Fuller, K. A.

Gaeta, A. L.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Geiselmann, M.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Gorodetsky, M. L.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

Gu, C.

Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
[Crossref]

Guo, G. C.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Guo, J.

Guo, L. J.

Han, T.

Herr, T.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

Huang, W.

Ikuta, R.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

Imoto, N.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

Jiang, X.

Joly, N. Y.

Just, F.

Kavungal, V.

Kippenberg, T. J.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

Knight, J. C.

F. Yu and J. C. Knight, “Spectral attenuation limits of silica hollow core negative curvature fiber,” Opt. Express 21(18), 21466–21471 (2013).
[Crossref] [PubMed]

J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
[Crossref] [PubMed]

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Kolyadin, A. N.

Kosolapov, A. F.

Kuhlmey, B.

Leuchs, G.

Levy, J. S.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Li, J.

Li, S.

Li, Y.

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

Liao, J.

Light, P. S.

Lihachev, G.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

Lin, W.

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

Ling, T.

Lipson, M.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Liu, B.

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

J. Du, Y. Liu, Z. Wang, B. Zou, B. Liu, and X. Dong, “Electrically tunable Sagnac filter based on a photonic bandgap fiber with liquid crystal infused,” Opt. Lett. 33(19), 2215–2217 (2008).
[Crossref] [PubMed]

Liu, L.

Liu, Q.

Liu, Y.

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

Y. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref] [PubMed]

J. Du, Y. Liu, Z. Wang, B. Zou, B. Liu, and X. Dong, “Electrically tunable Sagnac filter based on a photonic bandgap fiber with liquid crystal infused,” Opt. Lett. 33(19), 2215–2217 (2008).
[Crossref] [PubMed]

Liu, Y. G.

Liu, Y.-X.

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

Luo, M.

Mahmood, A.

McPhedran, R.

Monifi, F.

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

Nori, F.

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

Omenetto, F. G.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Oveys, H.

Ozdemir, S. K.

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

Özdemir, S. K.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

Peng, B.

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

Pfeiffer, M. H. P.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

Plotnichenko, V. G.

Pryamikov, A. D.

Pu, S.

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

Reeves, W. H.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Renversez, G.

Rosenberger, A. T.

S. I. Shopova, Y. Sun, A. T. Rosenberger, and X. Fan, “Highly sensitive tuning of coupled optical ring resonators by microfluidics,” Microfluid. Nanofluidics 6(3), 425–429 (2009).
[Crossref]

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

Russell, P. J.

Russell, P. S.

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. S. Russell, “Emergence of geometrical optical nonlinearities in photonic crystal fiber nanowires,” Phys. Rev. Lett. 105(9), 093904 (2010).
[Crossref] [PubMed]

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Schmidt, M. A.

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. S. Russell, “Emergence of geometrical optical nonlinearities in photonic crystal fiber nanowires,” Phys. Rev. Lett. 105(9), 093904 (2010).
[Crossref] [PubMed]

Seballos, L.

Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
[Crossref]

Semenova, Y.

Shen, Z.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Shi, C.

Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
[Crossref]

Shi, L.

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

Shi, M.

Shopova, S. I.

S. I. Shopova, Y. Sun, A. T. Rosenberger, and X. Fan, “Highly sensitive tuning of coupled optical ring resonators by microfluidics,” Microfluid. Nanofluidics 6(3), 425–429 (2009).
[Crossref]

Skryabin, D. V.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Smith, D. D.

Song, B.

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

Srinivasan, K.

Y. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref] [PubMed]

Stark, S.

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. S. Russell, “Emergence of geometrical optical nonlinearities in photonic crystal fiber nanowires,” Phys. Rev. Lett. 105(9), 093904 (2010).
[Crossref] [PubMed]

Sun, Y.

S. I. Shopova, Y. Sun, A. T. Rosenberger, and X. Fan, “Highly sensitive tuning of coupled optical ring resonators by microfluidics,” Microfluid. Nanofluidics 6(3), 425–429 (2009).
[Crossref]

Swaim, J. D.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

Takeuchi, Y.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

Taylor, A. J.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

Tran, T. X.

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. S. Russell, “Emergence of geometrical optical nonlinearities in photonic crystal fiber nanowires,” Phys. Rev. Lett. 105(9), 093904 (2010).
[Crossref] [PubMed]

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

Vollmer, F.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

Wang, G.

Wang, X.

Wang, Z.

White, I. M.

Wu, X.

Wu, Z.

Xu, L.

Xu, X.

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

Yamamoto, T.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

Yang, C.

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

Yang, L.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

Yu, F.

Zhang, H.

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

Zhang, J.

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

Zhang, J. Z.

Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
[Crossref]

Zhang, X.

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
[Crossref]

Zhang, Y. L.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Zhao, P.

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

Zou, B.

Zou, C. L.

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Adv. Opt. Photonics 7(2), 168–240 (2015).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

Y. Zhang, C. Shi, C. Gu, L. Seballos, and J. Z. Zhang, “Liquid core photonic crystal fiber sensor based on surface enhanced Raman scattering,” Appl. Phys. Lett. 90(19), 193504 (2007).
[Crossref]

J. Lightwave Technol. (1)

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

Lab Chip (1)

Y. Liu, L. Shi, X. Xu, P. Zhao, Z. Wang, S. Pu, and X. Zhang, “All-optical tuning of a magnetic-fluid-filled optofluidic ring resonator,” Lab Chip 14(16), 3004–3010 (2014).
[Crossref] [PubMed]

Laser Photonics Rev. (1)

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, “Observation of optomechanical coupling in a microbottle resonator,” Laser Photonics Rev. 10(4), 603–611 (2016).
[Crossref]

Microfluid. Nanofluidics (1)

S. I. Shopova, Y. Sun, A. T. Rosenberger, and X. Fan, “Highly sensitive tuning of coupled optical ring resonators by microfluidics,” Microfluid. Nanofluidics 6(3), 425–429 (2009).
[Crossref]

Nat. Commun. (2)

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not Electromagnetically Induced Transparency in Whispering-Gallery Microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Nat. Photonics (2)

F. Monifi, J. Zhang, S. K. Ozdemir, B. Peng, Y.-X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10(6), 399–405 (2016).
[Crossref]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Nature (3)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref] [PubMed]

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. S. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424(6948), 511–515 (2003).
[Crossref] [PubMed]

J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (5)

Optica (1)

Phys. Rev. Lett. (2)

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. S. Russell, “Emergence of geometrical optical nonlinearities in photonic crystal fiber nanowires,” Phys. Rev. Lett. 105(9), 093904 (2010).
[Crossref] [PubMed]

Y. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref] [PubMed]

Sci. Rep. (1)

W. Lin, H. Zhang, B. Liu, B. Song, Y. Li, C. Yang, and Y. Liu, “Laser-tuned whispering gallery modes in a solid-core microstructured optical fibre integrated with magnetic fluids,” Sci. Rep. 5, 17791 (2015).
[Crossref] [PubMed]

Science (2)

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351(6271), 357–360 (2016).
[Crossref] [PubMed]

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 Cross-sections of the proposed MOF

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Fig. 2
Fig. 2 (a) Mode coupling in the dual-channel MOF; (b) an equivalent model for (a)

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Fig. 3
Fig. 3 Transmittance of channel 1 (red), channel 2 (green), frame (blue), and the output spectrum (black) of the dual-channel WGM resonator system under the conditions of: (a) mode splitting (b) Fano resonance

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Fig. 4
Fig. 4 Output spectrum of the TE mode with different media in the channels: (a) both channels filled with water; (b) channel 1: bio-samples with different RI and channel 2: water; and (c) channel 1: water and channel 2: bio-samples with different RI. The color changes from red to blue indicating Fano resonance shifts towards longer wavelengths with increasing RI.

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Fig. 5
Fig. 5 Responses of the Fano resonant peaks as a function of sample RI in the TE mode: (a) channel 1 filled with bio-samples and channel 2 filled with water; and (b) channel 1 filled with water and channel 2 filled with bio-samples. For (a) and (b), the corresponding transmission of channel 1 and 2 is plotted in the middle and bottom rows respectively.

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Fig. 6
Fig. 6 Output spectrum of the TM mode with different RI in the channels: (a) both channels filled with water; (b) channel 1: bio-samples with different RI and channel 2: water; and (c) channel 1: water and channel 2: bio-samples with different RI. The color changes from red to blue indicating Fano resonance shifts towards longer wavelengths with increasing RI.

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Fig. 7
Fig. 7 Responses of the Fano resonant peaks as a function of the sample RI for the TM mode: (a) channel 1 filled with bio-samples and channel 2 filled with water; and (b) channel 1 filled with water and channel 2 filled with bio-samples. For (a) and (b), the corresponding transmission of channel 1 and 2 is plotted in the middle and bottom rows respectively.

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Fig. 8
Fig. 8 (a) Sensitivity as a function of the channel radius with a wall thickness (t) of 1.5 μm, 2 μm and 2.5μm; (b) sensitivity as a function of wall thickness when the channel radius is 40 μm.

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Tables (2)

Tables Icon

Table 1 Design parameters of the MOF.

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Table 2 Expressions of Leff.

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Equations (10)

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

Ψz(r)={ A J m ( k 0 n 1 r) B J m ( k 0 n 2 r)+C N m ( k 0 n 2 r) D H m (1) ( k 0 n 3 r) r> R 1 R 1 ≤r≤ R 2 r> R 2
N J ' m ( k 0 n 1 R 1 ) J m ( k 0 n 1 R 1 ) = (B / C)J ' m ( k 0 n 2 R 1 )+ N ' m ( k 0 n 2 R 1 ) (B / C) J m ( k 0 n 2 R 1 )+ N m ( k 0 n 2 R 1 ) ; N={ n 1 / n 2 ; TM n 2 / n 1 ; TE
B C = M H m (1) '( k 0 n 3 R 2 ) N m ( k 0 n 2 R 2 )− H m (1) ( k 0 n 3 R 2 ) N m '( k 0 n 2 R 2 ) H m (1) ( k 0 n 3 R 2 )J ' m ( k 0 n 2 R 2 )−M H m (1) '( k 0 n 3 R 2 ) J m ( k 0 n 2 R 2 ) ; M={ n 3 / n 2 ; TM n 2 / n 3 ; TE
t out = t w + | κ | 2 α αt− e iφ
[ E out E f2 ]=[ t w κ 0 − κ 0 * t ][ E in E f1 ]
E f1 =( α f3 e −i φ f3 ) t ch,1 ( α f2 e −i φ f2 ) t ch,2 ( α f1 e −i φ f1 ) E f2 = α f e −i φ f t ch,1 t ch,2 E f2
E f2 = κ 0 ∗ α f t t ch,1 t ch,2 e −i φ f −1 E in
E out =( t w + | κ 0 | 2 α f t ch,1 t ch,2 α f t t ch,1 t ch,2 − e i φ f ) E in
t out = E out E in = t w + α f t ch1 t ch2 | κ 0 | 2 α f t t ch1 t ch2 − e i φ f
T out = | t out | 2

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