Abstract

Cavity ring-up spectroscopy (CRUS) provides an advanced technique to sense ultrafast phenomena, but there is no thorough discussion on its theory. Here we give a detailed theoretical analysis of CRUS with and without modal coupling, and present exact analytical expressions for the normalized transmission, which are very simple under certain reasonable conditions. Our results provide a solid theoretical basis for the applications of CRUS.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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  1. G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).
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    [Crossref] [PubMed]
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    [Crossref]
  4. G. Wang, M. Zhao, Y. Qin, Z. Yin, X. Jiang, and M. Xiao, “Demonstration of an ultra-low-threshold phonon laser with coupled microtoroid resonators in vacuum,” Photon. Res. 5, 73–76 (2017).
    [Crossref]
  5. J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
    [Crossref]
  6. Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
    [Crossref] [PubMed]
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  8. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
    [Crossref] [PubMed]
  9. L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
    [Crossref] [PubMed]
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    [Crossref]
  11. E. Kim, M. D. Baaskea, and F. Vollmer, “Towards next-generation label-free biosensors: recent advances in whispering gallery mode sensors,” Lab Chip 17, 1190–1205 (2017).
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    [Crossref]
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    [Crossref] [PubMed]
  14. Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
    [Crossref]
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    [Crossref]

2017 (3)

G. Wang, M. Zhao, Y. Qin, Z. Yin, X. Jiang, and M. Xiao, “Demonstration of an ultra-low-threshold phonon laser with coupled microtoroid resonators in vacuum,” Photon. Res. 5, 73–76 (2017).
[Crossref]

E. Kim, M. D. Baaskea, and F. Vollmer, “Towards next-generation label-free biosensors: recent advances in whispering gallery mode sensors,” Lab Chip 17, 1190–1205 (2017).
[Crossref] [PubMed]

Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29, 1604920 (2017).
[Crossref]

2016 (2)

Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

2015 (1)

S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015).
[Crossref] [PubMed]

2014 (1)

J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
[Crossref]

2013 (1)

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

2011 (1)

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

2010 (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. Photon. 4, 46–49 (2010).
[Crossref]

2008 (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]

2007 (1)

2006 (1)

Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[Crossref] [PubMed]

2002 (1)

1997 (1)

Arazi, L.

S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015).
[Crossref] [PubMed]

Arnold, S.

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

Baaskea, M. D.

E. Kim, M. D. Baaskea, and F. Vollmer, “Towards next-generation label-free biosensors: recent advances in whispering gallery mode sensors,” Lab Chip 17, 1190–1205 (2017).
[Crossref] [PubMed]

Birks, T. A.

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. Photon. 4, 46–49 (2010).
[Crossref]

Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

Cheung, G.

Chormaic, S. N.

Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
[Crossref]

Clements, W. R.

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Cook, A. K.

Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[Crossref] [PubMed]

Dayan, B

S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015).
[Crossref] [PubMed]

Dong, C.-H.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

Dumeige, Y.

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

Féron, P.

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

Ferrari, M.

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

Gong, Q.

Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29, 1604920 (2017).
[Crossref]

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Guo, G.-C.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[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. Photon. 4, 46–49 (2010).
[Crossref]

Jacques, F.

Jiang, X.

Jiang, X. F.

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Kasumie, S.

Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
[Crossref]

Kim, E.

E. Kim, M. D. Baaskea, and F. Vollmer, “Towards next-generation label-free biosensors: recent advances in whispering gallery mode sensors,” Lab Chip 17, 1190–1205 (2017).
[Crossref] [PubMed]

Kippenberg, T. J.

Knight, J. C.

Li, B. B.

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Li, J.

J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
[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. Photon. 4, 46–49 (2010).
[Crossref]

Lovsky, Y.

S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015).
[Crossref] [PubMed]

Madugani, R.

Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
[Crossref]

Nunzi Conti, G.

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

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. Photon. 4, 46–49 (2010).
[Crossref]

Park, Y.-S.

Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[Crossref] [PubMed]

Qin, Y.

Righini, G. C.

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

Ristic, D.

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

Rosenblum, S.

S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015).
[Crossref] [PubMed]

Shao, L.

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Shen, Z.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

Soria, S.

G. C. Righini, Y. Dumeige, P. Féron, M. Ferrari, G. Nunzi Conti, D. Ristic, and S. Soria, “Whispering gallery mode microresonators: fundamentals and applications,” Riv. Nuovo Cimento Soc. Ital. Fis. 34, 435–488 (2011).

Spillane, S. M.

Sun, F.-W.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

Vahala, K. J.

Vollmer, F.

E. Kim, M. D. Baaskea, and F. Vollmer, “Towards next-generation label-free biosensors: recent advances in whispering gallery mode sensors,” Lab Chip 17, 1190–1205 (2017).
[Crossref] [PubMed]

S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015).
[Crossref] [PubMed]

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

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

Wang, G.

Wang, H.

Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[Crossref] [PubMed]

Wang, W.

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Ward, J. M.

Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
[Crossref]

Wu, Y.

J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
[Crossref]

Xiao, M.

Xiao, Y. F.

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Xiao, Y.-F.

Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29, 1604920 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

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. Photon. 4, 46–49 (2010).
[Crossref]

Yang, L.

Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29, 1604920 (2017).
[Crossref]

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. Photon. 4, 46–49 (2010).
[Crossref]

Yang, Y.

Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
[Crossref]

Yin, Z.

Yu, R.

J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
[Crossref]

Yu, X. C.

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Yu, X.-C.

Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29, 1604920 (2017).
[Crossref]

Zhang, D.

J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
[Crossref]

Zhang, S.

J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
[Crossref]

Zhang, Y.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

Zhao, M.

Zhi, Y.

Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29, 1604920 (2017).
[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. Photon. 4, 46–49 (2010).
[Crossref]

Zou, C.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

Zou, X.-B.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. photon. 10, 657–661 (2016).
[Crossref]

Adv. Mater. (2)

L. Shao, X. F. Jiang, X. C. Yu, B. B. Li, W. R. Clements, F. Vollmer, W. Wang, Y. F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mater. 25, 5616–5620 (2013).
[Crossref] [PubMed]

Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, “Single nanoparticle detection using optical microcavities,” Adv. Mater. 29, 1604920 (2017).
[Crossref]

Appl. Phys. B (1)

Y. Yang, R. Madugani, S. Kasumie, J. M. Ward, and S. N. Chormaic, “Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator,” Appl. Phys. B 122, 291 (2016).
[Crossref]

Lab Chip (1)

E. Kim, M. D. Baaskea, and F. Vollmer, “Towards next-generation label-free biosensors: recent advances in whispering gallery mode sensors,” Lab Chip 17, 1190–1205 (2017).
[Crossref] [PubMed]

Nano Lett. (1)

Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006).
[Crossref] [PubMed]

Nat. Commun. (1)

S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015).
[Crossref] [PubMed]

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).
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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. Photon. 4, 46–49 (2010).
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J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90, 053832 (2014).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1 Schematic illustration of a WGM microresonator coupled to a fiber taper (right) and a step-like modulation function f(t) (left).
Fig. 2
Fig. 2 Comparison between the exact T(t) in Eq. (23) and the approximate T(t) in Eq. (27) with κe/2π = 4.5 MHz, κ/2π = 5 MHz, β/2π = 10 MHz and δ/2π = 200 MHz. For exact T(t), α+ and α are calculated from f (t) = t/tr(0 ≤ ttr) with tr = 0.1 ns. For approximate T(t), α = 1 is used (see the section 4). Their difference, i.e., the exact T(t) minus the approximate T(t), is shown on the bottom.

Equations (34)

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d E cw ( t ) d t = ( j w c κ ) E cw ( t ) + 2 κ e E in ( t ) ,
E in ( t ) = f ( t ) s e j w l t ,
E cw ( t ) = e ( j w c κ ) t 0 t e ( j w c + κ ) t 2 κ e E in ( t ) d t .
E cw ( t ) = s 2 κ e e ( j w c κ ) t 0 t e ( j δ + κ ) t f ( t ) d t ,
E cw ( t ) = s 2 κ e j δ + κ e ( j w c κ ) t { e ( j δ + κ ) t f ( t ) 0 t e ( j δ + κ ) t d f ( t ) d t d t } .
α = 0 t r e ( j δ + κ ) ( t t r ) d f ( t ) d t d t ,
E cw ( t ) = s 2 κ e j δ + κ e ( j w c κ ) t { e ( j δ + κ ) t α e ( j δ + κ ) t r }
= 2 κ e j δ + κ { 1 α e ( j δ κ ) ( t t r ) } E in ( t ) , t t r ,
E out ( t ) = E in ( t ) + 2 κ e E cw ( t ) .
E out ( t ) = { A e j θ A B e j θ B e ( j δ κ ) ( t t r ) } E in ( t ) , t t r ,
A e j θ A = 1 + 2 κ e j δ + κ , B e j θ B = α 2 κ e j δ + κ .
T ( t ) = A 2 + B 2 e 2 κ ( t t r ) 2 A B e κ ( t t r ) cos { δ ( t t r ) + θ B θ A } , t t r ,
E out ( t ) { 1 2 κ e α δ j e ( j δ κ ) ( t t r ) } E in ( t ) , t t r ,
T ( t ) 1 4 κ e δ | α | e κ ( t t r ) sin { δ ( t t r ) + θ α } , t t r ,
d E cw ( t ) d t = ( j w c κ ) E cw ( t ) + j β E ccw ( t ) + 2 κ e E in ( t ) ,
d E ccw ( t ) d t = ( j w c κ ) E ccw ( t ) + j β E cw ( t ) ,
E + ( t ) = E cw ( t ) + E ccw ( t ) , E ( t ) = E cw ( t ) E ccw ( t ) .
d E + ( t ) d t = { j ( w c β ) κ } E + ( t ) + 2 k e E in ( t ) ,
d E ( t ) d t = { j ( w c + β ) κ } E ( t ) + 2 k e E in ( t ) ,
E + ( t ) = 2 κ e j δ + + κ { 1 α + e ( j δ + κ ) ( t t r ) } E in ( t ) , t t r ,
E ( t ) = 2 κ e j δ + κ { 1 α e ( j δ κ ) ( t t r ) } E in ( t ) , t t r ,
α + = 0 t r e ( j δ + + κ ) ( t t r ) d f ( t ) d t d t , α = 0 t r e ( j δ + κ ) ( t t r ) d f ( t ) d t d t .
T ( t ) = | C κ e j δ + + κ α + e ( j δ + κ ) ( t t r ) κ e j δ + κ α e ( j δ κ ) ( t t r ) | 2 , t t r ,
C = 1 + κ e j δ + + κ + κ e j δ + κ .
κ e j δ + + κ κ e j δ + κ κ e δ j .
T ( t ) | 1 κ e α δ j e ( j δ + κ ) ( t t r ) κ e α δ j e ( j δ κ ) ( t t r ) | 2
1 4 κ e δ | α | e κ ( t t r ) cos [ β ( t t r ) ] sin [ δ ( t t r ) + θ α ] , t t r ,
| α | < 0 t r | e ( j δ + κ ) ( t t r ) d f ( t ) d t | d t < 0 t r d f ( t ) d t d t = 1 ,
α 0 t r d f ( t ) d t d t = f ( t r ) f ( 0 ) = 1 ,
α 0 t r { 1 + ( j δ + κ ) ( t t r ) } d f ( t ) d t d t = 1 γ ( j δ + κ ) t r .
γ = 1 t r 0 t r ( t t r ) d f ( t ) d t d t = 1 t r 0 t r f ( t ) d t .
α = 0 t r e ( j δ + κ ) ( t t r ) 1 t r d t = 1 ( j δ + κ ) t r { 1 e ( j δ κ ) t r } .
α = π 2 t r 0 t r e ( j δ + κ ) ( t t r ) cos ( π t 2 t r ) d t
= π 4 1 ( j δ + κ ) t r { j e ( j δ κ ) t r } π 4 1 ( j δ + κ ) t r { j + e ( j δ κ ) t r } ,

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