Abstract

Due to ultra high quality factor (106 − 109), axisymmetric optical microcavities are popular platforms for biosensing applications. It has been recently demonstrated that a microcavity biosensor can track a biodetection event as a function of its quality factor by using phase shift cavity ring down spectroscopy (PS-CRDS). However, to achieve maximum sensitivity, it is necessary to optimize the microcavity parameters for a given sensing application. Here, we introduce an improved finite element model which allows us to determine the optimized geometry for the PS-CRDS sensor. The improved model not only provides fast and accurate determination of quality factors but also determines the tunneling distance of axisymmetric resonators. The improved model is validated numerically, analytically, and experimentally.

© 2013 OSA

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  1. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).
  2. X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
    [CrossRef]
  3. L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” California Institute of Technology, US Patent 6,490,039, (2002).
  4. J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).
  5. A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-q microcavities,” Opt. Lett. 31, 1896–1898 (2006).
    [CrossRef] [PubMed]
  6. J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express 16, 13158–13167 (2008).
    [CrossRef] [PubMed]
  7. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012).
    [CrossRef] [PubMed]
  8. C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311, (1994).
    [CrossRef] [PubMed]
  9. P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
    [CrossRef] [PubMed]
  10. V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, (2005).
    [CrossRef] [PubMed]
  11. M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express 17, 19160 (2009).
    [CrossRef]
  12. O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. 21, 1263–1274 (2005).
    [CrossRef]
  13. M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).
  14. M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. 33, 227–233, (1985).
    [CrossRef]
  15. R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).
  16. M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. 55, 1209–1218 (2007).
    [CrossRef]
  17. S. M. Spillane, “Fiber-coupled ultra-high-Q microresonators for nonlinear and quantum optics,” PhD. thesis, CALTECH (2004)
  18. M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
    [CrossRef] [PubMed]
  19. https://sites.google.com/site/axisymmetricmarkoxborrow/home
  20. M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature 488, 353–356 (2012).
    [CrossRef] [PubMed]
  21. C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett. 100, 013305 (2012).
    [CrossRef]
  22. T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
    [CrossRef]
  23. Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
    [CrossRef]
  24. M. I. Cheema and A. G. Kirk, “Implementation of the perfectly matched layer to determine the quality factor of axisymmetric resonators in COMSOL,” in COMSOL conference,Boston, Oct 8 2010.
  25. A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO, October24, 2010.
  26. F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373, (1997).
    [CrossRef]
  27. V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys. B54, 184–187, (1992).
  28. L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, USA, 1969) pp. 298.
  29. J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67033806 (2003).
    [CrossRef]
  30. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A 10, 2, (1993).
    [CrossRef]
  31. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
    [CrossRef] [PubMed]
  32. S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
    [CrossRef]
  33. M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993).
    [CrossRef]
  34. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer, 1997).
  35. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157–162, (2003).
    [CrossRef]
  36. A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
    [CrossRef]

2012 (4)

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).

M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012).
[CrossRef] [PubMed]

M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature 488, 353–356 (2012).
[CrossRef] [PubMed]

C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett. 100, 013305 (2012).
[CrossRef]

2011 (1)

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

2010 (3)

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO, October24, 2010.

M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
[CrossRef] [PubMed]

2009 (3)

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express 17, 19160 (2009).
[CrossRef]

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

2008 (2)

2007 (1)

M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. 55, 1209–1218 (2007).
[CrossRef]

2006 (1)

2005 (3)

V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, (2005).
[CrossRef] [PubMed]

O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. 21, 1263–1274 (2005).
[CrossRef]

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
[CrossRef]

2003 (3)

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157–162, (2003).
[CrossRef]

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67033806 (2003).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

2002 (1)

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).

2000 (1)

P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
[CrossRef] [PubMed]

1997 (1)

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373, (1997).
[CrossRef]

1994 (2)

C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311, (1994).
[CrossRef] [PubMed]

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

1993 (2)

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A 10, 2, (1993).
[CrossRef]

M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993).
[CrossRef]

1992 (1)

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys. B54, 184–187, (1992).

1985 (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. 33, 227–233, (1985).
[CrossRef]

Adibi, A.

Alford, N. M.

M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature 488, 353–356 (2012).
[CrossRef] [PubMed]

Arbabi, A.

A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO, October24, 2010.

Arbenz, P.

O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. 21, 1263–1274 (2005).
[CrossRef]

Armani, A. M.

C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett. 100, 013305 (2012).
[CrossRef]

M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012).
[CrossRef] [PubMed]

A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-q microcavities,” Opt. Lett. 31, 1896–1898 (2006).
[CrossRef] [PubMed]

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
[CrossRef]

Armani, D. K.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

Barnes, J.

Bearman, G. H.

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).

Belle, S.

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

Breeze, J. D.

M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature 488, 353–356 (2012).
[CrossRef] [PubMed]

Buck, J. R.

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67033806 (2003).
[CrossRef]

Burger, S.

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

Carmon, T.

Carver, B.

Cheema, M. I.

M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012).
[CrossRef] [PubMed]

M. I. Cheema and A. G. Kirk, “Implementation of the perfectly matched layer to determine the quality factor of axisymmetric resonators in COMSOL,” in COMSOL conference,Boston, Oct 8 2010.

Chen, T.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Chew, W. C.

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373, (1997).
[CrossRef]

Chinellato, O.

O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. 21, 1263–1274 (2005).
[CrossRef]

Choi, H. S.

C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett. 100, 013305 (2012).
[CrossRef]

Datsyuk, V. V.

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys. B54, 184–187, (1992).

Dick, G. J.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Dong, C. H.

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

Emmerson, G.

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

Fan, X.

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
[CrossRef]

Flagan, R. C.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Fraser, J. M.

Fraser, S. E.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Gagliardi, G.

Gallignani, M.

M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993).
[CrossRef]

Garrigues, S.

M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993).
[CrossRef]

Gil, L. M.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Goddard, L. L.

A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO, October24, 2010.

Gong, Q.

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

Guardia, M.

M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993).
[CrossRef]

Han, Z. F.

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

Hayat, A. A.

Hayata, K.

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. 33, 227–233, (1985).
[CrossRef]

Hellmann, R.

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

Herchak, S.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Hetterich, M.

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

Ilchenko, V. S.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157–162, (2003).
[CrossRef]

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” California Institute of Technology, US Patent 6,490,039, (2002).

Johnson, B. R.

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A 10, 2, (1993).
[CrossRef]

Kalt, H.

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

Kang, Y. M.

A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO, October24, 2010.

Karl, M.

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

Kettner, B.

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

Kim, J. H.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Kimble, H. J.

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67033806 (2003).
[CrossRef]

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

Kirk, A. G.

M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012).
[CrossRef] [PubMed]

M. I. Cheema and A. G. Kirk, “Implementation of the perfectly matched layer to determine the quality factor of axisymmetric resonators in COMSOL,” in COMSOL conference,Boston, Oct 8 2010.

Koshiba, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. 33, 227–233, (1985).
[CrossRef]

Kossakovski, D.

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).

Krausz, F.

C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311, (1994).
[CrossRef] [PubMed]

Lee, H.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Li, B. B.

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

Li, Q.

Li, Y.

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

Loock, H. P.

Lu, T.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Maleki, L.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157–162, (2003).
[CrossRef]

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” California Institute of Technology, US Patent 6,490,039, (2002).

Matsko, A. B.

Mehrabani, S.

Min, B.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
[CrossRef]

Nadeau, J. L.

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).

Narimanov, E. E.

V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, (2005).
[CrossRef] [PubMed]

Osegueda, R. A.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Oxborrow, M.

M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature 488, 353–356 (2012).
[CrossRef] [PubMed]

M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. 55, 1209–1218 (2007).
[CrossRef]

Pereyra, P.

P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
[CrossRef] [PubMed]

Peter, Y. A.

Pierluissi, J. H.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Podolskiy, V. A.

V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, (2005).
[CrossRef] [PubMed]

Quarteroni, A.

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer, 1997).

Revilla, A.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Santiago, D. G.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Savchenkov, A. A.

Scheurich, S.

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

Schmidt, F.

M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

Shi, C.

C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett. 100, 013305 (2012).
[CrossRef]

Shopoua, S. I.

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
[CrossRef]

So, S.

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

Soltani, M.

Sparrow, I.J.G.

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

Spielmann, C.

C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311, (1994).
[CrossRef] [PubMed]

Spillane, S. M.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

S. M. Spillane, “Fiber-coupled ultra-high-Q microresonators for nonlinear and quantum optics,” PhD. thesis, CALTECH (2004)

Stingl, A.

C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311, (1994).
[CrossRef] [PubMed]

Streiff, M.

O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. 21, 1263–1274 (2005).
[CrossRef]

Sun, Y.

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
[CrossRef]

Suter, J. D.

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
[CrossRef]

Suzuki, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. 33, 227–233, (1985).
[CrossRef]

Szipöcs, R.

C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311, (1994).
[CrossRef] [PubMed]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373, (1997).
[CrossRef]

Tian, Z.

Tomes, M.

Vahala, K.

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Vahala, K. J.

M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express 17, 19160 (2009).
[CrossRef]

A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-q microcavities,” Opt. Lett. 31, 1896–1898 (2006).
[CrossRef] [PubMed]

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

Valli, A.

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer, 1997).

Villalva, G. J.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Vollmer, F.

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).

Wang, R. T.

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Weinstein, L. A.

L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, USA, 1969) pp. 298.

White, I. M.

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
[CrossRef]

Wilson, M. W. B.

Witzig, A.

O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. 21, 1263–1274 (2005).
[CrossRef]

Xiao, Y. F.

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

Yam, S.

Yang, L.

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).

Yastrubshak, O.

Yegnanarayanan, S.

Zhu, H.

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
[CrossRef]

Zou, C. L.

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

10th annual review of progress in applied computational electromagnetics (1)

R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics 1, 159–170, (1994).

Analyst (1)

M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993).
[CrossRef]

Analytica Chimica Acta (1)

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008)
[CrossRef]

Appl. Phys. (1)

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys. B54, 184–187, (1992).

Appl. Phys. Lett. (2)

C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett. 100, 013305 (2012).
[CrossRef]

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O, Appl. Phys. Lett. 87, (2005).
[CrossRef]

FIO (1)

A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO, October24, 2010.

Future Gener. Comp. Sy. (1)

O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. 21, 1263–1274 (2005).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373, (1997).
[CrossRef]

IEEE Trans. on Microwave Theory Tech. (2)

M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. 55, 1209–1218 (2007).
[CrossRef]

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. 33, 227–233, (1985).
[CrossRef]

J. Opt. Soc. Am. A (2)

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A 10, 2, (1993).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157–162, (2003).
[CrossRef]

Nanophotonics (1)

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).

Nature (2)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature 488, 353–356 (2012).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-q microcavities,” Opt. Lett. 31, 1896–1898 (2006).
[CrossRef] [PubMed]

V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, (2005).
[CrossRef] [PubMed]

Phys. Rev. A (1)

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67033806 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902, (2010).
[CrossRef]

C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311, (1994).
[CrossRef] [PubMed]

P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. (1)

T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011).
[CrossRef]

Proc. SPIE (2)

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009)
[CrossRef]

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 72 (2002).

Other (6)

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” California Institute of Technology, US Patent 6,490,039, (2002).

S. M. Spillane, “Fiber-coupled ultra-high-Q microresonators for nonlinear and quantum optics,” PhD. thesis, CALTECH (2004)

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer, 1997).

L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, USA, 1969) pp. 298.

https://sites.google.com/site/axisymmetricmarkoxborrow/home

M. I. Cheema and A. G. Kirk, “Implementation of the perfectly matched layer to determine the quality factor of axisymmetric resonators in COMSOL,” in COMSOL conference,Boston, Oct 8 2010.

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Figures (7)

Fig. 1
Fig. 1

a) (False color) Logarithmic intensity of the fundamental TE mode of a silica microsphere in air. b) Normalized field intensity along the dashed line shown in (a). Dotted curve shows potential, normalized to k2. Tunneling distance is indicated by t.

Fig. 2
Fig. 2

Modeling results at 1550nm. WGM Quality factors for various silica sphere diameters of fundamental TE mode. OUB and OLB represent upper and lower bounds that are calculated by using Oxborrow’s model [16].

Fig. 3
Fig. 3

Modeling results at 1550nm. Tunneling distance of a fundamental TE mode as a function of the microcavity geometries. Inset of Fig. 3(b) shows the cross section of a microtoroidal cavity.

Fig. 4
Fig. 4

Comparison between the modeling and the experimental results. WGM Quality factors of fundamental TM mode for various silica microtoroidal cavity diameters immersed in ethanol. Minor diameter d of each cavity is slightly different and is around 5μm ± 1μm. The experimental quality factors are determined by Lorentz curve fitting of the resonant peaks (Q = λλ) where λ ≈ 1530nm.

Fig. 5
Fig. 5

Convergence plot for the quality factor determination of a microsphere.

Fig. 6
Fig. 6

Modeling results show the existence of an optimum geometry. Change in quality factor (ΔQ) of fundamental TM mode as a function of major diameter (D) of a microtoroidal cavity (minor diameter(d) = 6μm) immersed in water. ΔQ shows the difference between Qtotal for water and that measured when small refractive index change (δn = 10−3) is introduced into the water. (a)Modeling results for λ = 633nm (b)Modeling results for λ = 1530nm.

Fig. 7
Fig. 7

Modeling results. Various quality factors (Eq. (22)) for fundamental TM mode involved in a microtoroidal cavity immersed in water as a function of D (d = 6μm). In the optimum region the Qwgm is close to Qtotal.(a) Modeling results at λ = 633nm. (b) Modeling results at λ = 1530nm. Qsilica ≈ 1011 (Silica has very low absorption at 1530nm) and is omitted in the figure to reduce the range in y-axis.

Equations (22)

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

V ( × H ˜ * ) ε 1 ( × H ) α ( H ˜ * ) ( H ) + c 2 H ˜ * 2 H t 2 d V ) = 0
ε ¯ = ε Λ ¯ , μ ¯ = μ Λ ¯
Λ ¯ = ( r ˜ r ) ( s z s r ) r ^ + ( r r ˜ ) ( s z s r ) ϕ ^ + ( r ˜ r ) ( s r s z ) z ^
s r = { n medium 0 r r pml n medium j G ( r r pml t rpml ) N r > r pml
s z = { n medium j G ( z lpml z t lpml ) N z < z lpml n medium z lpml z z upml n medium j G ( z z upml t upml ) N z > z upml
r ˜ = { r 0 r r pml r j G ( ( r r pml ) N + 1 ( N + 1 ) t pml N ) r > r pml
r pml 6 t
z ( u , l ) pml | 5.5 w z |
V ( ( × H ˜ * ) ε ¯ 1 ( × H ) α ( × H ˜ * ) ( H ) + c 2 H ˜ * μ ¯ 2 H t 2 ) d V = 0
Q wgm = ( f r ) 2 ( f r )
Q wgm = 1 2 ( m + 1 2 ) p 1 2 M ( p 2 1 ) 1 2 e 2 T m , m 1
p 2 = ε sphere ε medium
T m = ( m + 1 2 ) ( m l tanh ( m l )
m l = cosh 1 ( p ( 1 1 m + 1 2 ( t q 0 β + p 1 2 M ( p 2 1 ) ) ) 1 )
β = ( 1 2 ( m + 1 2 ) ) 1 3
M = { 0 For TE 1 For TM
p 1 2 M j m ' [ p k 0 a ] j m [ p k 0 a ] = h m ' [ k 0 a ] h m [ k 0 a ]
d 2 Ψ r d r 2 + V r Ψ r = E Ψ r
V r = k 2 ( 1 p r 2 ) + m ( m + 1 ) r 2
t = m ( m + 1 ) k a
t = m ( m + 1 ) k D + d 2
Q total = 1 Q wgm 1 + Q surroundings 1 + Q material 1 + Q coupling 1

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