Abstract

Determination of the intrinsic quality factor of a loaded whispering gallery mode microcavity can be important for many applications where the coupling conditions cannot be tuned. We propose a single-scan technique based on a Stokes parameters analysis to extract the intrinsic quality factor and therefore determine the coupling regime. We propose a simple model for this analysis and present experimental measurements, which are in very good agreement with the model.

© 2011 OSA

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References

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    [CrossRef] [PubMed]
  2. J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
    [CrossRef]
  3. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
    [CrossRef]
  4. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
    [CrossRef]
  5. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
    [CrossRef] [PubMed]
  6. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  20. M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
    [CrossRef]

2009 (2)

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

C. R. Locke, D. Stuart, E. N. Ivanov, and A. N. Luiten, “A simple technique for accurate and complete characterisation of a Fabry-Perot cavity,” Opt. Express 17, 21935–21943 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (4)

P. Bianucci, C. R. Fietz, J. W. Robertson, G. Shvets, and C.-K. Shih, “Whispering gallery mode microresonators as polarization converters,” Opt. Lett. 32, 2224–2226 (2007).
[CrossRef] [PubMed]

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

M. Sumetsky, “Optimization of optical ring resonator devices for sensing applications,” Opt. Lett. 32, 2577–2579 (2007).
[CrossRef] [PubMed]

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

2005 (1)

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

2004 (3)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef] [PubMed]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
[CrossRef]

2003 (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]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

2000 (1)

1998 (1)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

1996 (2)

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef] [PubMed]

Armani, D. K.

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]

Arnold, S.

Baigent, K. G.

Bandy, D. K.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef] [PubMed]

Bianucci, P.

Carmon, T.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Chen, D.-R.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Connolly, J.

Dale, E.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Dumeige, Y.

Féron, P.

Fietz, C. R.

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Ghisa, L.

Goldstein, D.

D. Goldstein, Polarized Light (Marcel Dekker, Inc., New York, 2003), 2nd ed.
[CrossRef]

Gorodetsky, M. L.

Gray, M. B.

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Griffel, G.

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

He, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Hossein-Zadeh, M.

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

Humphrey, M. J.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Ilchenko, V. S.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Ito, T.

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
[CrossRef]

Ivanov, E. N.

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

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, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Kokubun, Y.

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
[CrossRef]

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Li, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef] [PubMed]

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Locke, C. R.

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Luiten, A. N.

Maleki, L.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

Matsko, A. B.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

McClelland, D. E.

Morris, N.

Nguyên, T. K. N.

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef] [PubMed]

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Robertson, J. W.

Rokhsari, H.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Rosenberger, A. T.

M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. 271, 124–131 (2007).
[CrossRef]

Savchenkov, A. A.

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

Scherer, A.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Serpengüzel, A.

Shih, C.-K.

Shvets, G.

Slagmolen, B. J. J.

Slusher, R. E.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Spillane, S. M.

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, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Stuart, D.

Sumetsky, M.

Taskent, D.

Tavernier, H.

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Trebaol, S.

Vahala, K. J.

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

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, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Vollmer, F.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

Xiao, Y. F.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Yang, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Zhu, J.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

M. Hossein-Zadeh and K. J. Vahala, “Importance of intrinsic-Q in microring-based optical filters and dispersion-compensation devices,” IEEE Photonics Technol. Lett. 19, 1045–1047 (2007).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

T. Ito and Y. Kokubun, “Nondestructive measurement of propagation loss and coupling efficiency in microring resonator filters using filter responses,” Jpn. J. Appl. Phys. 43, 1002–1005 (2004).
[CrossRef]

Nat. Methods (1)

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

Nat. Photonics (1)

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[CrossRef]

Nature (2)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef] [PubMed]

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]

Opt. Commun. (1)

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Opt. Express (1)

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Phys. Rev. A (1)

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Figures (5)

Fig. 1
Fig. 1

Lossless coupling scheme between a single mode waveguide and an optical resonator. Polarization states axx⃗ + ayey⃗ and bxx⃗ + byy⃗ are entering and exiting the coupling region respectively. An amplitude transmission coefficient of |T| and a phase difference of θ are added by the cavity. The power fraction κ2 = 1 – t2 coupled to the cavity and the losses α characterize the waveguide-resonator system.

Fig. 3
Fig. 3

Experimental setup used for S0 and S2 parameters measurements: TLS - Tunable laser source, OSC - Oscilloscope, TRIG - Trigger signal, CAV - Microcavity, TAP - Tapered fiber, C - 50:50 non-polarizing beamsplitter cube, CO1, CO2 et CO3 - Collimators, HWP1 et HWP2 - λ/2 wave plates, POL - Polarizer, L1 et L2 - Lens, D1 et D2 - Detectors. A SMF-28 fiber is used up to CO3.

Fig. 2
Fig. 2

Calculated spectra of the Stokes parameters across a resonance for the undercoupled regime 2(a)–2(b) (Q0 = 1 × 106 and Qc = 3 × 106), the critically coupled regime 2(c)–2(d) (Q0 = Qc = 1×106) and the overcoupled regime 2(e)–2(f) (Q0 = 1×106 and Qc = 3×105). The case where ϕ = 0 and ϕ = −π/5 are shown on the left and right side respectively. The black and red dots represent the extrema of the S3 spectra and the position of the FWHM values of S0 respectively. ax and ay are set to 1 / 2.

Fig. 4
Fig. 4

Micrography of the resonator and waveguide. A 1.2 μm diameter tapered optical fiber is brought near to a toroidal silica microtoroid resonator using a piezoeletric stage.

Fig. 5
Fig. 5

Experimental and calculated S0 and S2 parameters obtained for three coupling conditions giving (a) Q 0 ( e ) = ( 2.273 ± 0.017 ) × 10 6 and Q c ( e ) = ( 1.58 ± 0.12 ) × 10 8 (b) Q 0 ( e ) = ( 2.333 ± 0.010 ) × 10 6 and Q c ( e ) = ( 3.81 ± 0.08 ) × 10 7 and (c) Q 0 ( e ) = ( 2.311 ± 0.009 ) × 10 6 and Q c ( e ) = ( 5.11 ± 0.03 ) × 10 6.

Tables (1)

Tables Icon

Table 1 Extracted parameters from S 0 N ( e ) (red curve) and S 2 N ( e ) (green curve) presented in Fig. 5(a)–5(c).

Equations (18)

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[ b x b y ] = [ 1 0 0 | T | e i θ ] [ a x a y e i ϕ ] .
| T | 2 = t 2 + e 2 α L 2 t e α L cos ( β 0 L ) 1 + t 2 e 2 α L 2 t e α L cos ( β 0 L )
θ = tan 1 ( [ t 2 1 ] e α L sin ( β 0 L ) t ( 1 + e 2 α L ) e α L [ 1 + t 2 ] cos ( β 0 L ) )
S 0 = | b x | 2 + | b y | 2 = a x 2 + | T | 2 a y 2 S 1 = | b x | 2 | b y | 2 = a x 2 | T | 2 a y 2 S 2 = 2 | b x | | b y | cos ( δ ) = 2 a x a y | T | cos ( θ + ϕ ) S 3 = 2 | b x | | b y | sin ( δ ) = 2 a x a y | T | sin ( θ + ϕ )
S 2 = 2 a x a y cos ( ϕ ) [ t ( 1 + e 2 α L ) e α L ( 1 + t 2 ) cos ( β 0 L ) ] + sin ( ϕ ) e α L ( 1 t 2 ) sin ( β 0 L ) 1 + t 2 e 2 α L 2 t e α L cos ( β 0 L ) .
β 0 L ( 1 ) = arctan ( ( 1 t 2 e 2 α L ) sin ( ϕ ) cos ( ϕ ) ( 1 + t 2 e 2 α L ) + 2 t e α L )
β 0 L ( 2 ) = arctan ( ( 1 t 2 e 2 α L ) sin ( ϕ ) cos ( ϕ ) ( 1 + t 2 e 2 α L ) 2 t e α L ) + π
S 2 N , max = S 2 max 2 a x a y = t ( 1 e 2 α L ) cos ( ϕ ) + e α L ( 1 t 2 ) 1 t 2 e 2 α L
S 2 N , min = S 2 min 2 a x a y = t ( 1 e 2 α L ) cos ( ϕ ) e α L ( 1 t 2 ) 1 t 2 e 2 α L
Δ S 2 N = Δ S 2 2 a x a y = 2 e α L ( 1 t 2 ) 1 t 2 e 2 α L 2 Q 0 β 0 L Q 0 + Q c β 0 L
Q T = λ Δ λ = Q 0 Q c Q 0 + Q c .
Q 0 ( e ) = 4 a x a y Q T 4 a x a y Δ S 2 = 2 Q T 1 Δ S 2 N and Q c ( e ) = 4 a x a y Q T Δ S 2 = 2 Q T Δ S 2 N
Δ Q 0 Q 0 = 1 4 a x a y Q T ( 4 a x a y Δ S 2 ) Q 0 , Δ Q c Q c = 1 4 a x a y Q T Q c Δ S 2 .
I ( α p , ϕ l ) = 1 2 [ S 0 + S 1 cos ( 2 α p ) + S 2 cos ( ϕ l ) sin ( 2 α p ) + S 3 sin ( ϕ l ) sin ( 2 α p ) ] .
S 0 = I ( 0 ° , 0 ° ) + I ( 90 ° , 0 ° ) S 1 = I ( 0 ° , 0 ° ) I ( 90 ° , 0 ° ) S 2 = 2 I ( 45 ° , 0 ° ) I ( 0 ° , 0 ° ) I ( 90 ° , 0 ° ) = 2 I ( 45 ° , 0 ° ) S 0 S 3 = 2 I ( 45 ° , 90 ° ) I ( 0 ° , 0 ° ) I ( 90 ° , 0 ° ) = 2 I ( 45 ° , 90 ° ) S 0 .
I 1 = | T 45 | 2 a x 2 + | T 45 | 2 | T | 2 a y 2 | T 45 | 2 ( a x 2 + | T | 2 a y 2 ) I 2 p x 2 | R 45 | 2 2 [ a x 2 + | T | 2 a y 2 + 2 a x a y | T | cos ( θ + ϕ ) ] .
S 0 N ( e ) = I 1 | T 45 | 2 P tot = I 1 I 1 off S 2 N ( e ) = 2 I 2 p x 2 | R 45 | 2 I 1 | T 45 | 2 2 P tot f ( 1 f )
P tot = a x 2 + a y 2 = I 1 off | T 45 | 2 a x = ( 1 f ) P tot and a y = f P tot .

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