Abstract

We analyze normal mode splitting in a pair of vertically coupled microdisk resonators. A full vectorial finite-element model is used to find the eigenfrequencies of the symmetric and antisymmetric composite modes as a function of coupling distance. We find that the coupled microdisks can compete with the best Fabry–Perot resonators in displacement sensing. We also show how we configured FreeFem++ for the sphere eigenvalue problem.

© 2012 Optical Society of America

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  1. R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
    [CrossRef]
  2. A. N. Oraevsky, “Whispering gallery waves,” Quantum Electron. 32, 377–400 (2002).
    [CrossRef]
  3. M. L. Gorodetsky, Optical Microresonators with Gigantic Quality Factor (in Russian) (Fizmatlit, 2011), http://www.ozon.ru/context/detail/id/6210477/ .
  4. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
    [CrossRef]
  5. R. T. Wang and G. J. Dick, “Cryocooled sapphire oscillator with ultrahigh stability,” IEEE Trans. Instrum. Meas. 48, 528–531 (1999).
    [CrossRef]
  6. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–12305 (2007).
    [CrossRef]
  7. A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering–gallery modes—part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
    [CrossRef]
  8. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [CrossRef]
  9. J. Moore, M. Tomes, T. Carmon, and M. Jarrahi, “Continuous-wave ultraviolet emission through fourth-harmonic generation in a whispering-gallery resonator,” Opt. Express 19, 24139–24146 (2011).
    [CrossRef]
  10. A. B. Matsko, A. A. Savchenkov, N. Yu, and L. Maleki, “Whispering-gallery-mode resonators as frequency references. I. Fundamental limitations,” J. Opt. Soc. Am. B 24, 1324–1335 (2007).
    [CrossRef]
  11. 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]
  12. F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. 105, 20701–20704 (2008).
    [CrossRef]
  13. 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. 108, 5976–5979 (2011).
    [CrossRef]
  14. I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express 20, 6604–6609 (2012).
    [CrossRef]
  15. M. Hossein-Zadeh and A. F. J. Levi, “Ring resonator-based photonic microwave receiver modulator with picowatt sensitivity,” IET Optoelectron. 5, 36–39 (2011).
    [CrossRef]
  16. W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
    [CrossRef]
  17. V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering–gallery modes: a basis for coordinate meter,” Opt. Commun. 107, 41–48 (1994).
    [CrossRef]
  18. V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, “On the ultimate sensitivity in coordinate measurements,” Phys. Lett. A 179, 244–248 (1993).
    [CrossRef]
  19. W. P. Huang, “Coupled–mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  20. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two–level system,” Phys. Rev. Lett. 104, 083901 (2010).
    [CrossRef]
  21. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
    [CrossRef]
  22. M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
    [CrossRef]
  23. J. P. Webb, “The finite–element method for finding modes of dielectric–loaded cavities,” IEEE Trans. Microwave Theor. mtt-33, 635–639 (1985).
    [CrossRef]
  24. M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theor. 55, 1209–1218 (2007).
    [CrossRef]
  25. S. H. Wong and Z. J. Cendes, “Combined finite element–modal solution of three-dimensional eddy current problems,” IEEE Trans. Magn. 24, 2685–2687 (1988).
    [CrossRef]
  26. O. Pironneau, F. Hecht, A. Le Hyaric, and J. Morice, “FreeFem++,” http://www.freefem.org/ .
  27. http://www.caam.rice.edu/software/ARPACK/ .
  28. http://www.cise.ufl.edu/research/sparse/umfpack/ .
  29. M. A. Arain and G. Mueller, “Optical layout and parameters for the advanced LIGO cavities,” LIGO-T0900043-10 (2009).
  30. V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (Chicago, 1985).
  31. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17363–365 (1992).
    [CrossRef]
  32. M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B 21, 697–705 (2004).
    [CrossRef]
  33. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorical finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express 14, 11128–11141 (2006).
    [CrossRef]
  34. D. B. Thompson, D. A. Keating, E. Guler, K. Ichimura, M. E. Williams, and K. A. Fuller, “Separation-sensitive measurements of morphology dependent resonances in coupled fluorescent microspheres,” Opt. Express 18, 19209–19218 (2010).
    [CrossRef]
  35. B. Wu, Y. Liu, Z. Dai, and S. Liu, “Stable narrow linewidth Er-doped fiber laser at 1550 nm,” Microw. Opt. Technol. Lett. 49, 1453–1456 (2007).
    [CrossRef]
  36. M. L. Povinelli, S. G. Johnson, M. Lonar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering gallery-mode resonators,” Opt. Express 13, 8286–8295 (2005).
    [CrossRef]
  37. FreeFem++ example is available, http://arxiv.org/abs/1208.4320.

2012

I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express 20, 6604–6609 (2012).
[CrossRef]

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]

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

2011

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. 108, 5976–5979 (2011).
[CrossRef]

M. Hossein-Zadeh and A. F. J. Levi, “Ring resonator-based photonic microwave receiver modulator with picowatt sensitivity,” IET Optoelectron. 5, 36–39 (2011).
[CrossRef]

J. Moore, M. Tomes, T. Carmon, and M. Jarrahi, “Continuous-wave ultraviolet emission through fourth-harmonic generation in a whispering-gallery resonator,” Opt. Express 19, 24139–24146 (2011).
[CrossRef]

2010

2008

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. 105, 20701–20704 (2008).
[CrossRef]

2007

A. B. Matsko, A. A. Savchenkov, N. Yu, and L. Maleki, “Whispering-gallery-mode resonators as frequency references. I. Fundamental limitations,” J. Opt. Soc. Am. B 24, 1324–1335 (2007).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–12305 (2007).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
[CrossRef]

B. Wu, Y. Liu, Z. Dai, and S. Liu, “Stable narrow linewidth Er-doped fiber laser at 1550 nm,” Microw. Opt. Technol. Lett. 49, 1453–1456 (2007).
[CrossRef]

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

2006

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
[CrossRef]

K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorical finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express 14, 11128–11141 (2006).
[CrossRef]

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

2005

2004

2003

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

2002

A. N. Oraevsky, “Whispering gallery waves,” Quantum Electron. 32, 377–400 (2002).
[CrossRef]

1999

R. T. Wang and G. J. Dick, “Cryocooled sapphire oscillator with ultrahigh stability,” IEEE Trans. Instrum. Meas. 48, 528–531 (1999).
[CrossRef]

1994

W. P. Huang, “Coupled–mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering–gallery modes: a basis for coordinate meter,” Opt. Commun. 107, 41–48 (1994).
[CrossRef]

1993

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, “On the ultimate sensitivity in coordinate measurements,” Phys. Lett. A 179, 244–248 (1993).
[CrossRef]

1992

1988

S. H. Wong and Z. J. Cendes, “Combined finite element–modal solution of three-dimensional eddy current problems,” IEEE Trans. Magn. 24, 2685–2687 (1988).
[CrossRef]

1985

J. P. Webb, “The finite–element method for finding modes of dielectric–loaded cavities,” IEEE Trans. Microwave Theor. mtt-33, 635–639 (1985).
[CrossRef]

1939

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[CrossRef]

Arain, M. A.

M. A. Arain and G. Mueller, “Optical layout and parameters for the advanced LIGO cavities,” LIGO-T0900043-10 (2009).

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]

Arnold, S.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. 105, 20701–20704 (2008).
[CrossRef]

Baumgartel, L.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, “On the ultimate sensitivity in coordinate measurements,” Phys. Lett. A 179, 244–248 (1993).
[CrossRef]

V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (Chicago, 1985).

Capasso, F.

Carmon, T.

Cendes, Z. J.

S. H. Wong and Z. J. Cendes, “Combined finite element–modal solution of three-dimensional eddy current problems,” IEEE Trans. Magn. 24, 2685–2687 (1988).
[CrossRef]

Chen, T.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

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. 108, 5976–5979 (2011).
[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]

Dai, Z.

B. Wu, Y. Liu, Z. Dai, and S. Liu, “Stable narrow linewidth Er-doped fiber laser at 1550 nm,” Microw. Opt. Technol. Lett. 49, 1453–1456 (2007).
[CrossRef]

Dick, G. J.

R. T. Wang and G. J. Dick, “Cryocooled sapphire oscillator with ultrahigh stability,” IEEE Trans. Instrum. Meas. 48, 528–531 (1999).
[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. 108, 5976–5979 (2011).
[CrossRef]

Fomin, A. E.

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
[CrossRef]

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. 108, 5976–5979 (2011).
[CrossRef]

Fuller, K. A.

Gorodetsky, M. L.

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
[CrossRef]

M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B 21, 697–705 (2004).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering–gallery modes: a basis for coordinate meter,” Opt. Commun. 107, 41–48 (1994).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, “On the ultimate sensitivity in coordinate measurements,” Phys. Lett. A 179, 244–248 (1993).
[CrossRef]

M. L. Gorodetsky, Optical Microresonators with Gigantic Quality Factor (in Russian) (Fizmatlit, 2011), http://www.ozon.ru/context/detail/id/6210477/ .

Grudinin, I. S.

Guler, E.

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. 108, 5976–5979 (2011).
[CrossRef]

Hossein-Zadeh, M.

M. Hossein-Zadeh and A. F. J. Levi, “Ring resonator-based photonic microwave receiver modulator with picowatt sensitivity,” IET Optoelectron. 5, 36–39 (2011).
[CrossRef]

Huang, W. P.

Ibanescu, M.

Ichimura, K.

Ilchenko, V. S.

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
[CrossRef]

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

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering–gallery modes: a basis for coordinate meter,” Opt. Commun. 107, 41–48 (1994).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, “On the ultimate sensitivity in coordinate measurements,” Phys. Lett. A 179, 244–248 (1993).
[CrossRef]

Jarrahi, M.

Jeon, S.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kakihara, K.

Keating, D. A.

Keng, D.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. 105, 20701–20704 (2008).
[CrossRef]

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. 108, 5976–5979 (2011).
[CrossRef]

Kimble, H. J.

Kippenberg, T. J.

Kono, N.

Koshiba, M.

Lalezari, R.

Lee, H.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

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. 108, 5976–5979 (2011).
[CrossRef]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two–level system,” Phys. Rev. Lett. 104, 083901 (2010).
[CrossRef]

Levi, A. F. J.

M. Hossein-Zadeh and A. F. J. Levi, “Ring resonator-based photonic microwave receiver modulator with picowatt sensitivity,” IET Optoelectron. 5, 36–39 (2011).
[CrossRef]

Li, J.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

Liang, W.

Liu, S.

B. Wu, Y. Liu, Z. Dai, and S. Liu, “Stable narrow linewidth Er-doped fiber laser at 1550 nm,” Microw. Opt. Technol. Lett. 49, 1453–1456 (2007).
[CrossRef]

Liu, Y.

B. Wu, Y. Liu, Z. Dai, and S. Liu, “Stable narrow linewidth Er-doped fiber laser at 1550 nm,” Microw. Opt. Technol. Lett. 49, 1453–1456 (2007).
[CrossRef]

Lonar, M.

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. 108, 5976–5979 (2011).
[CrossRef]

Maleki, L.

Matsko, A. B.

Mitrofanov, V. P.

V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (Chicago, 1985).

Moore, J.

Mueller, G.

M. A. Arain and G. Mueller, “Optical layout and parameters for the advanced LIGO cavities,” LIGO-T0900043-10 (2009).

Oraevsky, A. N.

A. N. Oraevsky, “Whispering gallery waves,” Quantum Electron. 32, 377–400 (2002).
[CrossRef]

Oxborrow, M.

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

Painter, O.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two–level system,” Phys. Rev. Lett. 104, 083901 (2010).
[CrossRef]

Panov, V. I.

V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (Chicago, 1985).

Povinelli, M. L.

Rempe, G.

Richtmyer, R. D.

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[CrossRef]

Saitoh, K.

Savchenkov, A. A.

Seidel, D.

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]

Smythe, E. J.

Thompson, D. B.

Thompson, R. J.

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. 108, 5976–5979 (2011).
[CrossRef]

Vahala, K. J.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two–level system,” Phys. Rev. Lett. 104, 083901 (2010).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–12305 (2007).
[CrossRef]

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

Vollmer, F.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. 105, 20701–20704 (2008).
[CrossRef]

Vyatchanin, S. P.

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering–gallery modes: a basis for coordinate meter,” Opt. Commun. 107, 41–48 (1994).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, “On the ultimate sensitivity in coordinate measurements,” Phys. Lett. A 179, 244–248 (1993).
[CrossRef]

Wang, R. T.

R. T. Wang and G. J. Dick, “Cryocooled sapphire oscillator with ultrahigh stability,” IEEE Trans. Instrum. Meas. 48, 528–531 (1999).
[CrossRef]

Webb, J. P.

J. P. Webb, “The finite–element method for finding modes of dielectric–loaded cavities,” IEEE Trans. Microwave Theor. mtt-33, 635–639 (1985).
[CrossRef]

Williams, M. E.

Wong, S. H.

S. H. Wong and Z. J. Cendes, “Combined finite element–modal solution of three-dimensional eddy current problems,” IEEE Trans. Magn. 24, 2685–2687 (1988).
[CrossRef]

Wu, B.

B. Wu, Y. Liu, Z. Dai, and S. Liu, “Stable narrow linewidth Er-doped fiber laser at 1550 nm,” Microw. Opt. Technol. Lett. 49, 1453–1456 (2007).
[CrossRef]

Yang, K. Y.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012).
[CrossRef]

Yu, N.

Appl. Phys. Lett.

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]

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, 3–14 (2006).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Vertical coupling of two silica microdisks with refractive index n = 1.46 , thickness 2 μm, wedge angle 45°, radius 180 μm, air gap 0.5 μm, and the mode orbital index M = 588 corresponding to optical wavelength of about 1.56 μm. The amplitude of the magnetic field vector is shown with lines of equal value. From outside, the lines are H max / 1000 , H max / 100 , H max / 10 , and H max / 2 . Approximate electric field vector directions are shown with arrows.

Fig. 2.
Fig. 2.

Mode splitting in vertically coupled microdisks. NMS of silica microtoroids coupled side by side is shown for comparison (adapted from [20]). Error bars are similar in size to point markers and are not shown.

Fig. 3.
Fig. 3.

Adapted mesh in the mode localization region. Inset: FEM solution for a sphere— T M 1000 , 1000 , 1 mode. Iso-value lines are equally spaced, scale for the mesh and the solution is the same.

Fig. 4.
Fig. 4.

Convergence of TE composite mode frequencies with increasing number of mesh elements. Vertical axis shows optical frequency offset by 192 THz. Air gap 2.5 μm, R = 1750 μm , M = 10 4 .

Tables (2)

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Table 1. Displacement Sensitivities of Coupled Microdisks for Optical Frequency around 193 THz (Wavelength 1.56 μm)a

Tables Icon

Table 2. Computed Errors of y = k a for a Spherea, M = 10 3 , n = 1.46

Equations (2)

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Δ x = d Q h ν W τ = 2.2 × 10 - 11 h ν W τ .
y ( P n h M ( 1 ) ( y ) j M - 1 ( n y ) - j M ( n y ) h M - 1 ( 1 ) ( y ) ) + M ( 1 - P ) ( h M ( 1 ) ( y ) j M ( n y ) ) = 0 .

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