Abstract

This paper theoretically analyzes a hollow cylindrical whispering gallery mode resonator with radially inhomogeneous cladding. We propose an index profile of n(r) = b/r to enhance field penetration towards the resonator core. With such index profile, externally coupled evanescent wave can easily penetrate the resonator cladding without any potential barrier.

© 2012 OSA

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References

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    [CrossRef]
  3. A. A. Savchenkov, W. Liang, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Narrowband tunable photonic notch filter,” Opt. Lett.34(9), 1318–1320 (2009).
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    [CrossRef]
  10. S. I. Shopova, H. Zhou, X. Fan, and P. Zhang, “Optofluidic ring resonator based dye laser,” Appl. Phys. Lett.90(22), 221101 (2007).
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    [CrossRef]
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    [CrossRef]
  18. I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
    [CrossRef]
  19. I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett.31(9), 1319–1321 (2006).
    [CrossRef] [PubMed]
  20. I. M. White, H. Zhu, J. D. Suter, X. Fan, and M. Zourob, “Label-free detection with the liquid core optical ring resonator sensing platform,” Methods Mol. Biol.503, 139–165 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
  22. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A20(1), 157–162 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]

2012

F. Zhao, T. Zhan, G. Huang, Y. Mei, and X. Hu, “Liquid sensing capability of rolled-up tubular optical microcavities: a theoretical study,” Lab Chip12(19), 3798–3802 (2012).
[CrossRef] [PubMed]

2011

R. Yang, A. P. Yun, Y. X. Zhang, and X. Y. Pu, “Quantum theory of whispering gallery modes in a cylindrical optical microcavity,” Optik (Stuttg.)122(10), 900–909 (2011).
[CrossRef]

N. Lin, L. Jiang, S. M. Wang, H. Xiao, Y. F. Lu, and H. Tsai, “Thermostable refractive index sensors based on whispering gallery modes in a microsphere coated with poly(methyl methacrylate),” Appl. Opt.50(7), 992–998 (2011).
[CrossRef] [PubMed]

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

2010

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: Analytical and rigorous full wave solution,” Phys. Rev. B82(5), 054204 (2010).
[CrossRef]

2009

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, “Electrically tunable liquid crystal optical microresonators,” Nat. Photonics3(10), 595–600 (2009).
[CrossRef]

A. A. Savchenkov, W. Liang, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Narrowband tunable photonic notch filter,” Opt. Lett.34(9), 1318–1320 (2009).
[CrossRef] [PubMed]

I. M. White, H. Zhu, J. D. Suter, X. Fan, and M. Zourob, “Label-free detection with the liquid core optical ring resonator sensing platform,” Methods Mol. Biol.503, 139–165 (2009).
[CrossRef] [PubMed]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett.95(4), 041106 (2009).
[CrossRef]

2007

2006

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron.12(1), 3–14 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron.12(1), 15–32 (2006).
[CrossRef]

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
[CrossRef]

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

2003

2000

1992

L. G. Guimaraes and H. M. Nussenzveig, “Theory of Mie Resonances and Ripple Fluctuations,” Opt. Commun.89(5-6), 363–369 (1992).
[CrossRef]

Andrés, M. V.

Arnold, S.

Cai, M.

Chan, C. T.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: Analytical and rigorous full wave solution,” Phys. Rev. B82(5), 054204 (2010).
[CrossRef]

Chen, H. Y.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: Analytical and rigorous full wave solution,” Phys. Rev. B82(5), 054204 (2010).
[CrossRef]

Díez, A.

Dulashko, Y.

Fan, X.

I. M. White, H. Zhu, J. D. Suter, X. Fan, and M. Zourob, “Label-free detection with the liquid core optical ring resonator sensing platform,” Methods Mol. Biol.503, 139–165 (2009).
[CrossRef] [PubMed]

J. D. Suter, I. M. White, H. Zhu, and X. Fan, “Thermal characterization of liquid core optical ring resonator sensors,” Appl. Opt.46(3), 389–396 (2007).
[CrossRef] [PubMed]

M. Sumetsky, R. S. Windeler, Y. Dulashko, and X. Fan, “Optical liquid ring resonator sensor,” Opt. Express15(22), 14376–14381 (2007).
[CrossRef] [PubMed]

S. I. Shopova, H. Zhou, X. Fan, and P. Zhang, “Optofluidic ring resonator based dye laser,” Appl. Phys. Lett.90(22), 221101 (2007).
[CrossRef]

I. M. White, J. Gohring, Y. Sun, G. Yang, S. Lacey, and X. Fan, “Versatile waveguide-coupled optofluidic devices based on liquid core optical ring resonators,” Appl. Phys. Lett.91(24), 241104 (2007).
[CrossRef] [PubMed]

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

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
[CrossRef]

Gimeno, B.

Gohring, J.

I. M. White, J. Gohring, Y. Sun, G. Yang, S. Lacey, and X. Fan, “Versatile waveguide-coupled optofluidic devices based on liquid core optical ring resonators,” Appl. Phys. Lett.91(24), 241104 (2007).
[CrossRef] [PubMed]

Guimaraes, L. G.

L. G. Guimaraes and H. M. Nussenzveig, “Theory of Mie Resonances and Ripple Fluctuations,” Opt. Commun.89(5-6), 363–369 (1992).
[CrossRef]

Guo, L. J.

Holler, S.

Hu, X.

F. Zhao, T. Zhan, G. Huang, Y. Mei, and X. Hu, “Liquid sensing capability of rolled-up tubular optical microcavities: a theoretical study,” Lab Chip12(19), 3798–3802 (2012).
[CrossRef] [PubMed]

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

Huang, G.

F. Zhao, T. Zhan, G. Huang, Y. Mei, and X. Hu, “Liquid sensing capability of rolled-up tubular optical microcavities: a theoretical study,” Lab Chip12(19), 3798–3802 (2012).
[CrossRef] [PubMed]

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

Humar, M.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, “Electrically tunable liquid crystal optical microresonators,” Nat. Photonics3(10), 595–600 (2009).
[CrossRef]

Ilchenko, V. S.

A. A. Savchenkov, W. Liang, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Narrowband tunable photonic notch filter,” Opt. Lett.34(9), 1318–1320 (2009).
[CrossRef] [PubMed]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron.12(1), 3–14 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron.12(1), 15–32 (2006).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A20(1), 157–162 (2003).
[CrossRef] [PubMed]

Jiang, L.

Khoshsima, M.

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett.95(4), 041106 (2009).
[CrossRef]

Lacey, S.

I. M. White, J. Gohring, Y. Sun, G. Yang, S. Lacey, and X. Fan, “Versatile waveguide-coupled optofluidic devices based on liquid core optical ring resonators,” Appl. Phys. Lett.91(24), 241104 (2007).
[CrossRef] [PubMed]

Li, L.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: Analytical and rigorous full wave solution,” Phys. Rev. B82(5), 054204 (2010).
[CrossRef]

Liang, W.

Lin, N.

Lin, Z.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: Analytical and rigorous full wave solution,” Phys. Rev. B82(5), 054204 (2010).
[CrossRef]

Ling, T.

Liu, S.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: Analytical and rigorous full wave solution,” Phys. Rev. B82(5), 054204 (2010).
[CrossRef]

Lu, Y. F.

Maleki, L.

Matsko, A. B.

A. A. Savchenkov, W. Liang, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Narrowband tunable photonic notch filter,” Opt. Lett.34(9), 1318–1320 (2009).
[CrossRef] [PubMed]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron.12(1), 3–14 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron.12(1), 15–32 (2006).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A20(1), 157–162 (2003).
[CrossRef] [PubMed]

Mei, Y.

F. Zhao, T. Zhan, G. Huang, Y. Mei, and X. Hu, “Liquid sensing capability of rolled-up tubular optical microcavities: a theoretical study,” Lab Chip12(19), 3798–3802 (2012).
[CrossRef] [PubMed]

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

Musevic, I.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, “Electrically tunable liquid crystal optical microresonators,” Nat. Photonics3(10), 595–600 (2009).
[CrossRef]

Narimanov, E. E.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett.95(4), 041106 (2009).
[CrossRef]

Nussenzveig, H. M.

L. G. Guimaraes and H. M. Nussenzveig, “Theory of Mie Resonances and Ripple Fluctuations,” Opt. Commun.89(5-6), 363–369 (1992).
[CrossRef]

Oveys, H.

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

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
[CrossRef]

Painter, O.

Pajk, S.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, “Electrically tunable liquid crystal optical microresonators,” Nat. Photonics3(10), 595–600 (2009).
[CrossRef]

Pu, X. Y.

R. Yang, A. P. Yun, Y. X. Zhang, and X. Y. Pu, “Quantum theory of whispering gallery modes in a cylindrical optical microcavity,” Optik (Stuttg.)122(10), 900–909 (2011).
[CrossRef]

Ravnik, M.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, “Electrically tunable liquid crystal optical microresonators,” Nat. Photonics3(10), 595–600 (2009).
[CrossRef]

Savchenkov, A. A.

Seidel, D.

Sercel, P. C.

Shopova, S. I.

S. I. Shopova, H. Zhou, X. Fan, and P. Zhang, “Optofluidic ring resonator based dye laser,” Appl. Phys. Lett.90(22), 221101 (2007).
[CrossRef]

Smith, T. L.

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
[CrossRef]

Sumetsky, M.

Sun, Y.

I. M. White, J. Gohring, Y. Sun, G. Yang, S. Lacey, and X. Fan, “Versatile waveguide-coupled optofluidic devices based on liquid core optical ring resonators,” Appl. Phys. Lett.91(24), 241104 (2007).
[CrossRef] [PubMed]

Suter, J. D.

I. M. White, H. Zhu, J. D. Suter, X. Fan, and M. Zourob, “Label-free detection with the liquid core optical ring resonator sensing platform,” Methods Mol. Biol.503, 139–165 (2009).
[CrossRef] [PubMed]

J. D. Suter, I. M. White, H. Zhu, and X. Fan, “Thermal characterization of liquid core optical ring resonator sensors,” Appl. Opt.46(3), 389–396 (2007).
[CrossRef] [PubMed]

Teraoka, I.

Tsai, H.

Vahala, K. J.

Vollmer, F.

Wang, S. M.

White, I. M.

I. M. White, H. Zhu, J. D. Suter, X. Fan, and M. Zourob, “Label-free detection with the liquid core optical ring resonator sensing platform,” Methods Mol. Biol.503, 139–165 (2009).
[CrossRef] [PubMed]

J. D. Suter, I. M. White, H. Zhu, and X. Fan, “Thermal characterization of liquid core optical ring resonator sensors,” Appl. Opt.46(3), 389–396 (2007).
[CrossRef] [PubMed]

I. M. White, J. Gohring, Y. Sun, G. Yang, S. Lacey, and X. Fan, “Versatile waveguide-coupled optofluidic devices based on liquid core optical ring resonators,” Appl. Phys. Lett.91(24), 241104 (2007).
[CrossRef] [PubMed]

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

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
[CrossRef]

Windeler, R. S.

Xiao, H.

Xiong, Z.

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

Xu, C.

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

Yang, G.

I. M. White, J. Gohring, Y. Sun, G. Yang, S. Lacey, and X. Fan, “Versatile waveguide-coupled optofluidic devices based on liquid core optical ring resonators,” Appl. Phys. Lett.91(24), 241104 (2007).
[CrossRef] [PubMed]

Yang, R.

R. Yang, A. P. Yun, Y. X. Zhang, and X. Y. Pu, “Quantum theory of whispering gallery modes in a cylindrical optical microcavity,” Optik (Stuttg.)122(10), 900–909 (2011).
[CrossRef]

Yun, A. P.

R. Yang, A. P. Yun, Y. X. Zhang, and X. Y. Pu, “Quantum theory of whispering gallery modes in a cylindrical optical microcavity,” Optik (Stuttg.)122(10), 900–909 (2011).
[CrossRef]

Zamora, V.

Zhan, T.

F. Zhao, T. Zhan, G. Huang, Y. Mei, and X. Hu, “Liquid sensing capability of rolled-up tubular optical microcavities: a theoretical study,” Lab Chip12(19), 3798–3802 (2012).
[CrossRef] [PubMed]

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

Zhang, J.

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
[CrossRef]

Zhang, P.

S. I. Shopova, H. Zhou, X. Fan, and P. Zhang, “Optofluidic ring resonator based dye laser,” Appl. Phys. Lett.90(22), 221101 (2007).
[CrossRef]

Zhang, Y. X.

R. Yang, A. P. Yun, Y. X. Zhang, and X. Y. Pu, “Quantum theory of whispering gallery modes in a cylindrical optical microcavity,” Optik (Stuttg.)122(10), 900–909 (2011).
[CrossRef]

Zhao, F.

F. Zhao, T. Zhan, G. Huang, Y. Mei, and X. Hu, “Liquid sensing capability of rolled-up tubular optical microcavities: a theoretical study,” Lab Chip12(19), 3798–3802 (2012).
[CrossRef] [PubMed]

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

Zhou, H.

S. I. Shopova, H. Zhou, X. Fan, and P. Zhang, “Optofluidic ring resonator based dye laser,” Appl. Phys. Lett.90(22), 221101 (2007).
[CrossRef]

Zhu, H.

I. M. White, H. Zhu, J. D. Suter, X. Fan, and M. Zourob, “Label-free detection with the liquid core optical ring resonator sensing platform,” Methods Mol. Biol.503, 139–165 (2009).
[CrossRef] [PubMed]

J. D. Suter, I. M. White, H. Zhu, and X. Fan, “Thermal characterization of liquid core optical ring resonator sensors,” Appl. Opt.46(3), 389–396 (2007).
[CrossRef] [PubMed]

Zi, J.

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: Analytical and rigorous full wave solution,” Phys. Rev. B82(5), 054204 (2010).
[CrossRef]

Zourob, M.

I. M. White, H. Zhu, J. D. Suter, X. Fan, and M. Zourob, “Label-free detection with the liquid core optical ring resonator sensing platform,” Methods Mol. Biol.503, 139–165 (2009).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

T. Zhan, C. Xu, F. Zhao, Z. Xiong, X. Hu, G. Huang, Y. Mei, and J. Zi, “Optical resonances in tubular microcavities with subwavelength wall thicknesses,” Appl. Phys. Lett.99(21), 211104 (2011).
[CrossRef]

S. I. Shopova, H. Zhou, X. Fan, and P. Zhang, “Optofluidic ring resonator based dye laser,” Appl. Phys. Lett.90(22), 221101 (2007).
[CrossRef]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett.95(4), 041106 (2009).
[CrossRef]

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” Appl. Phys. Lett.89(19), 191106 (2006).
[CrossRef]

I. M. White, J. Gohring, Y. Sun, G. Yang, S. Lacey, and X. Fan, “Versatile waveguide-coupled optofluidic devices based on liquid core optical ring resonators,” Appl. Phys. Lett.91(24), 241104 (2007).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron.12(1), 3–14 (2006).
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Figures (5)

Fig. 1
Fig. 1

(a) A typical capillary tube based cylindrical WGM refractometer; (b) cross section of a step index cavity; (c) graded index cavity with index profile b/r in the cladding

Fig. 2
Fig. 2

Field distribution in the cavity. (a) For lower q orders, field is strictly confined inside the core and fast decays in the cladding; while for higher q orders, field in cladding exhibits sinusoidal behavior across the cladding and leaks an evanescent tail to the outside for external coupling. (b) When full resonance happens in the cladding, cladding retains more field energy; (c) partial resonances in cladding results in strong resonances in core. n1 = 3, n3 = 1, b = 15 RIU-μm, R1 = 10μm, R2 = 14µm, m = 110.

Fig. 3
Fig. 3

2D FDTD simulation results. (a) and (d) show transmission spectra obtained from step index and graded index cavity respectively. (b), (c), (e) and (f) show field distribution at 4 different resonance frequencies.

Fig. 4
Fig. 4

Field intensity distribution of first three TM WGMs (m = 350, l = 1, 2 and 3) for step index (top) and graded index (bottom) claddings respectively. Independent of cladding thickness and l order, graded index always allows field to penetrate the cladding and extend evanescent tail into the core. n1 = 1.33, n3 = 1, b = 1.45 × 40 RIU-μm (for graded index), n2 = 1.45 (for step index), R1 = 36μm, R2 = 40µm.

Fig. 5
Fig. 5

Effective potential Ueff for a cylindrical tube cavity with air surrounded and high index material inside. Graded index designs flatten the potential barriers in the cladding, and n2(r)~r−1 is the critical changing rate.

Equations (8)

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2 H+ ε ε ××H= k 2 n 2 (r)H
2 E+(E ε ε )= k 2 n 2 (r)E
d 2 H z (r) d r 2 +( 1 r 2 n(r) dn(r) dr ) d dr H z (r)+( k 2 n 2 (r) m 2 r 2 ) H z (r)=0
d 2 d r 2 E z (r)+ 1 r d dr E z (r)+( k 2 n 2 (r) m 2 r 2 ) E z (r)=0
E z (r)= E 1 (r)= a 1 J m ( n 1 kr)
E z (r)= E 3 (r)= a 3 H m (1) ( n 3 kr)
E z (r)= E 2 (r)= a 21 cos( ln(r)Δ )+ a 22 sin( ln(r)Δ )
J m (k n 1 R 1 ) H m (1) '(k n 3 R 2 ) J m '(k n 1 R 1 ) H m (1) (k n 3 R 2 ) tan(ln( R 2 )Δ)tan(ln( R 1 )Δ)= R 1 n 1 R 2 n 3

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