Abstract

We present a simple method to determine simultaneously the main characteristics of passive or active high-Q optical resonators. The method is based on cavity ringdown spectroscopy, where the probe wavelength is rapidly swept across the resonance. It has already been shown that this technique allows the loaded cavity lifetime of passive resonators to be obtained. We show that we can also infer the coupling regime for passive resonators and the resonant gain for active resonators. The method is tested on Er3+ doped fiber resonators and also applied to determine the intrinsic and external Q-factors of an MgF2 whispering gallery mode resonator.

© 2008 Optical Society of America

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    [CrossRef]

2008

Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Féron, “Measurement of the dispersion induced by a slow-light system based on coupled active resonator induced transparency,” Phys. Rev. A 78, 013818 (2008).
[CrossRef]

2007

2006

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[CrossRef]

2005

2004

S. Minin, M. R. Fisher, and S. L. Chuang, “Current-controlled group delay using a semiconductor Fabry-Perot amplifier,” Appl. Phys. Lett. 84, 3238-3240 (2004).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef] [PubMed]

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

2003

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. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272-274 (2003).
[CrossRef] [PubMed]

2002

2001

J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator-fiber configuration: application to wavelength-selective switching, modulation, amplification, and oscillation,” Opt. Lett. 26, 1236-1238 (2001).
[CrossRef]

G. Stewart, K. Atherton, H. Yu, and B. Culshaw, “An investigation of an optical fibre amplifier loop for intra-cavity and ring-down cavity loss measurements,” Meas. Sci. Technol. 12, 843-849 (2001).
[CrossRef]

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321-322 (2001).
[CrossRef]

2000

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect,” Phys. Lett. A 271, 314-318 (2000).
[CrossRef]

Y. He and B. J. Orr, “Ringdown and cavity-enhanced absorption spectroscopy using a continuous-wave tunable diode laser and a rapidly swept optical cavity,” Chem. Phys. Lett. 319, 131-137 (2000).
[CrossRef]

M. C. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74-77 (2000).
[CrossRef] [PubMed]

B. J. J. Slagmolen, M. B. Gray, K. G. Baigent, and D. E. McClelland, “Phase-sensitive reflection technique for characterization of a Fabry-Perot interferometer,” Appl. Opt. 39, 3638-3643 (2000).
[CrossRef]

1999

1997

1993

R. Loudon, M. Harris, and T. J. Shepherd, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681-701 (1993).
[CrossRef] [PubMed]

1992

J. T. Kringlebotn, P. R. Morkel, C. N. Pannell, D. N. Payne, and R. I. Lamimg, “Amplified fibre delay line with 27000 recirculations,” Electron. Lett. 28, 201-202 (1992).
[CrossRef]

G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363-365 (1992).
[CrossRef] [PubMed]

1991

G. S. Pandian and F. E. Seraji, “Optical pulse response of a fiber ring resonator,” Proc. IEE 138, 235-239 (1991).

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51-57 (1991).
[CrossRef]

1988

1986

1982

1966

H. J. Schmitt and H. Zimmer, “Fast sweep measurements of relaxation times in superconducting cavities,” IEEE Trans. Microwave Theory Tech. MTT-14, 206-207 (1966).
[CrossRef]

Aoki, T.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[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.

Atherton, K.

G. Stewart, K. Atherton, H. Yu, and B. Culshaw, “An investigation of an optical fibre amplifier loop for intra-cavity and ring-down cavity loss measurements,” Meas. Sci. Technol. 12, 843-849 (2001).
[CrossRef]

Baigent, K. G.

Barsuglia, M.

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect,” Phys. Lett. A 271, 314-318 (2000).
[CrossRef]

Bennett, R. G. T.

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51-57 (1991).
[CrossRef]

Bondu, F.

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect,” Phys. Lett. A 271, 314-318 (2000).
[CrossRef]

Bowen, W. P.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Boyd, R. W.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Bretenaker, F.

Byer, R. L.

Cai, M. C.

M. C. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74-77 (2000).
[CrossRef] [PubMed]

Cavalier, F.

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect,” Phys. Lett. A 271, 314-318 (2000).
[CrossRef]

Chang, H.

Chenevier, M.

Chodorow, M.

Choi, J. M.

Chuang, S. L.

S. Minin, M. R. Fisher, and S. L. Chuang, “Current-controlled group delay using a semiconductor Fabry-Perot amplifier,” Appl. Phys. Lett. 84, 3238-3240 (2004).
[CrossRef]

Crosignani, B.

Culshaw, B.

G. Stewart, K. Atherton, H. Yu, and B. Culshaw, “An investigation of an optical fibre amplifier loop for intra-cavity and ring-down cavity loss measurements,” Meas. Sci. Technol. 12, 843-849 (2001).
[CrossRef]

Dayan, B.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Di Porto, P.

Dumeige, Y.

Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Féron, “Measurement of the dispersion induced by a slow-light system based on coupled active resonator induced transparency,” Phys. Rev. A 78, 013818 (2008).
[CrossRef]

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[CrossRef]

Féron, P.

Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Féron, “Measurement of the dispersion induced by a slow-light system based on coupled active resonator induced transparency,” Phys. Rev. A 78, 013818 (2008).
[CrossRef]

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[CrossRef]

Fisher, M. R.

S. Minin, M. R. Fisher, and S. L. Chuang, “Current-controlled group delay using a semiconductor Fabry-Perot amplifier,” Appl. Phys. Lett. 84, 3238-3240 (2004).
[CrossRef]

Ghisa, L.

Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Féron, “Measurement of the dispersion induced by a slow-light system based on coupled active resonator induced transparency,” Phys. Rev. A 78, 013818 (2008).
[CrossRef]

Giles, I. P.

Gray, M. B.

Gustafson, E. K.

Hahn, J. W.

Harris, M.

R. Loudon, M. Harris, and T. J. Shepherd, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681-701 (1993).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

He, Y.

Y. He and B. J. Orr, “Continuous-wave cavity ringdown absorption spectroscopy with a swept-frequency laser: rapid spectral sensing of gas-phase molecules,” Appl. Opt. 44, 6752-6761 (2005).
[CrossRef] [PubMed]

Y. He and B. J. Orr, “Ringdown and cavity-enhanced absorption spectroscopy using a continuous-wave tunable diode laser and a rapidly swept optical cavity,” Chem. Phys. Lett. 319, 131-137 (2000).
[CrossRef]

Heebner, J. E.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Heitmann, H.

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect,” Phys. Lett. A 271, 314-318 (2000).
[CrossRef]

Holler, S.

Husman, M. E.

Ilchenko, V. S.

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] [PubMed]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef] [PubMed]

Ioannidis, Z. K.

Jackson, D. J.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Jin, Z.

Kachanov, A.

Khoshsima, M.

Kim, J. W.

Kimble, H. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363-365 (1992).
[CrossRef] [PubMed]

Kippenberg, T. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (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]

Kringlebotn, J. T.

J. T. Kringlebotn, P. R. Morkel, C. N. Pannell, D. N. Payne, and R. I. Lamimg, “Amplified fibre delay line with 27000 recirculations,” Electron. Lett. 28, 201-202 (1992).
[CrossRef]

Kuramochi, E.

Lalezari, R.

Lamimg, R. I.

J. T. Kringlebotn, P. R. Morkel, C. N. Pannell, D. N. Payne, and R. I. Lamimg, “Amplified fibre delay line with 27000 recirculations,” Electron. Lett. 28, 201-202 (1992).
[CrossRef]

Lawrence, M. J.

Le Floch, A.

Lee, H.-W.

Lee, J. Y.

Lee, R. K.

Li, Z.

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51-57 (1991).
[CrossRef]

Loudon, R.

R. Loudon, M. Harris, and T. J. Shepherd, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681-701 (1993).
[CrossRef] [PubMed]

Ma, H.

Maleki, L.

Man, N.

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect,” Phys. Lett. A 271, 314-318 (2000).
[CrossRef]

Matone, L.

L. Matone, M. Barsuglia, F. Bondu, F. Cavalier, H. Heitmann, and N. Man, “Finesse and mirror speed measurement for a suspended Fabry-Perot cavity using the ringing effect,” Phys. Lett. A 271, 314-318 (2000).
[CrossRef]

Matsko, A. B.

McClelland, D. E.

Minin, S.

S. Minin, M. R. Fisher, and S. L. Chuang, “Current-controlled group delay using a semiconductor Fabry-Perot amplifier,” Appl. Phys. Lett. 84, 3238-3240 (2004).
[CrossRef]

Mohageg, M.

Morkel, P. R.

J. T. Kringlebotn, P. R. Morkel, C. N. Pannell, D. N. Payne, and R. I. Lamimg, “Amplified fibre delay line with 27000 recirculations,” Electron. Lett. 28, 201-202 (1992).
[CrossRef]

Morville, J.

Nguyen, T. K. N.

Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Féron, “Measurement of the dispersion induced by a slow-light system based on coupled active resonator induced transparency,” Phys. Rev. A 78, 013818 (2008).
[CrossRef]

Notomi, M.

Orr, B. J.

Y. He and B. J. Orr, “Continuous-wave cavity ringdown absorption spectroscopy with a swept-frequency laser: rapid spectral sensing of gas-phase molecules,” Appl. Opt. 44, 6752-6761 (2005).
[CrossRef] [PubMed]

Y. He and B. J. Orr, “Ringdown and cavity-enhanced absorption spectroscopy using a continuous-wave tunable diode laser and a rapidly swept optical cavity,” Chem. Phys. Lett. 319, 131-137 (2000).
[CrossRef]

Painter, O.

M. C. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74-77 (2000).
[CrossRef] [PubMed]

Pandian, G. S.

G. S. Pandian and F. E. Seraji, “Optical pulse response of a fiber ring resonator,” Proc. IEE 138, 235-239 (1991).

Pannell, C. N.

J. T. Kringlebotn, P. R. Morkel, C. N. Pannell, D. N. Payne, and R. I. Lamimg, “Amplified fibre delay line with 27000 recirculations,” Electron. Lett. 28, 201-202 (1992).
[CrossRef]

Parkins, A. S.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Payne, D. N.

J. T. Kringlebotn, P. R. Morkel, C. N. Pannell, D. N. Payne, and R. I. Lamimg, “Amplified fibre delay line with 27000 recirculations,” Electron. Lett. 28, 201-202 (1992).
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Smith, D. D.

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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).
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Trebaol, S.

Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Féron, “Measurement of the dispersion induced by a slow-light system based on coupled active resonator induced transparency,” Phys. Rev. A 78, 013818 (2008).
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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
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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. C. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74-77 (2000).
[CrossRef] [PubMed]

Vallet, M.

Vollmer, F.

Wilcut, E.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Willke, B.

Wong, V.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Yariv, A.

Ying, G.

Yoo, Y. S.

Yu, H.

G. Stewart, K. Atherton, H. Yu, and B. Culshaw, “An investigation of an optical fibre amplifier loop for intra-cavity and ring-down cavity loss measurements,” Meas. Sci. Technol. 12, 843-849 (2001).
[CrossRef]

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[CrossRef]

IEEE J. Quantum Electron.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

H. J. Schmitt and H. Zimmer, “Fast sweep measurements of relaxation times in superconducting cavities,” IEEE Trans. Microwave Theory Tech. MTT-14, 206-207 (1966).
[CrossRef]

J. Opt. Soc. Am. B

Meas. Sci. Technol.

G. Stewart, K. Atherton, H. Yu, and B. Culshaw, “An investigation of an optical fibre amplifier loop for intra-cavity and ring-down cavity loss measurements,” Meas. Sci. Technol. 12, 843-849 (2001).
[CrossRef]

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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671-674 (2006).
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Opt. Commun.

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51-57 (1991).
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Opt. Express

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[CrossRef] [PubMed]

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[CrossRef]

Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Féron, “Measurement of the dispersion induced by a slow-light system based on coupled active resonator induced transparency,” Phys. Rev. A 78, 013818 (2008).
[CrossRef]

Phys. Rev. Lett.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
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Figures (14)

Fig. 1
Fig. 1

Optical resonator coupled to a single access line. The input and output fields are s in and s out , u is the mode amplitude, τ 0 is the intrinsic photon lifetime, and τ e is the coupling photon lifetime. The inset represents the amplitude coupling coefficient κ.

Fig. 2
Fig. 2

Transmission at resonance T ( δ = 0 ) in decibels as a function of 1 τ 0 for the different regimes: undercoupling, critical coupling, overcoupling, and selective amplification. We also highlight the transparency of the resonator and the laser threshold.

Fig. 3
Fig. 3

Transmission T as a function of the normalized frequency detuning for different coupling regimes: (i) τ 0 = τ e 4 overcouping, (ii) τ 0 = τ e critical coupling, (iii) τ 0 = 3 τ e undercoupling, and (iv) τ 0 = 4 τ e selective amplification. We also indicate the FWHM 2 δ 1 2 of the transmission spectrum for the critical coupling and amplification regimes.

Fig. 4
Fig. 4

Transmission as a function of t τ in the case of a critically coupled resonator for different sweeping speeds. (a) V ̃ S = 0.0075 V ̃ 0 , (b) V ̃ S = 0.3 V ̃ 0 , (c) V ̃ S = 3 V ̃ 0 , and (d) V ̃ S = 30 V ̃ 0 , where V ̃ 0 = 2 ( π τ 2 ) .

Fig. 5
Fig. 5

(a) Transmission in the stationary case as a function of the normalized detuning in the case of τ 0 = 3 τ e and τ e = 3 τ 0 . The two curves are exactly superimposed. (b) Transmission as a function of time in the identical two cases with V ̃ S = 2.25 V ̃ 0 . Note that the two responses are easily distinguishable.

Fig. 6
Fig. 6

Transmission as a function of t τ in the case of selective amplification ( τ 0 = 2.5 τ e ) for different sweeping speeds. (a) V ̃ S = 0.0075 V ̃ 0 , (b) V ̃ S = 0.3 V ̃ 0 , (c) V ̃ S = 3 V ̃ 0 , and (d) V ̃ S = 30 V ̃ 0 , where V ̃ 0 = 2 ( π τ 2 ) .

Fig. 7
Fig. 7

Experimental setup used to test fiber resonators. The resonator consists of SMF 28 fiber with a spliced 50 cm long section of Er 3 + fiber pumped with a 980 nm laser diode using wavelength multiplexers M. C, tunable coupler; I, optical isolator; PC, polarization controller; D, fast detector. The probe is a tunable 1550 nm laser diode (bandwidth 150 kHz ) whose central frequency is linearly swept with a controllable period. P 0 is the optical power of the input signal. The fiber resonator is immersed in a water bath to limit thermal fluctuations.

Fig. 8
Fig. 8

Experimental (black curves) and theoretical (gray curves) results using the fiber resonator, from the top image to the bottom one the pumping rate increases (a) 3.8, (b) 4.5, and (c) 5.2 mW . In the three experiments the coupling coefficient between the ring and the coupler C is kept constant and P 0 = 2 mW . From the fast scanning measurements ( V ̃ S = 5.2 ± 0.2 MHz μ s ) we deduced (a) undercoupling, (b) critical coupling, and (c) overcoupling. The slow scanning values are fitted using the same parameters apart from V S ( V ̃ S = 0.40 ± 0.02 MHz μ s using the FSR). For the critical coupling we measured 2 δ 1 2 = 1.02 MHz in the stationary case, which gives a finesse F = c ( 2 δ 1 2 n L ) = 80.6 .

Fig. 9
Fig. 9

Fast and slow scanning experimental results (black curves) and fitting curves (gray curves) for two combinations of coupling coefficient and pumping rate (4.5 and 7 mW ) and an input power P 0 = 1.5 mW . These two parameters are chosen to switch the role played by τ 0 and τ e . (a) Undercoupling regime, τ 0 = 404 , τ e = 958 , and τ = 284 ns . (b) Overcoupling regime, τ 0 = 895 , τ e = 321 , and τ = 236 ns . The slow scanning (stationary regime) provides almost the same transmission spectrum.

Fig. 10
Fig. 10

Description of the Mg F 2 WGM resonator with its coupling fiber taper. The minimal diameter of the taper is about 3 μ m . D = 5.2 mm , e = 0.7 mm , and the spherical polished part has a diameter ρ = 60 μ m .

Fig. 11
Fig. 11

Experimental results with the Mg F 2 WGM resonator (black curve). The theoretical fit (gray curve) gives Q 0 = 3.3 × 10 8 , Q e = 2.5 × 10 9 , and V ̃ S = 3.3 MHz μ s .

Fig. 12
Fig. 12

Experimental (black curves) and theoretical (gray curves) results with the fiber resonator for two pumping rates and an input power P 0 = 0.8 mW allowing switching between (a) undercoupling regime (pump power of 5.7 mW ) and (b) selective amplification (pump power of 8 mW ); (a) τ 0 = 530 , τ e = 594 ns and (b) τ 0 = 2686 ns , τ e = 510 ns .

Fig. 13
Fig. 13

Fast and slow scanning experimental results (black curves) and fitting curves (gray curves) for two pumping rates (11.0 and 14.6 mW ) and an input power P 0 = 1.5 mW in the selective amplification regime: (a) τ 0 = 1017 ns , τ e = 196 ns , then T ( 0 ) = 2.2 in a frequency bandwidth 2 δ 1 2 = 1.3 MHz and (b) τ 0 = 470 ns , τ e = 190 ns , then T ( 0 ) = 5.5 in a frequency bandwidth 2 δ 1 2 = 1.0 MHz .

Fig. 14
Fig. 14

Experimental results for P 0 = 1.5 mW with the fiber resonator under high pumping ( 22.1 mW ) . The theoretical fit (gray curve) gives τ 0 = 476 ns , τ e = 287 ns , then T ( 0 ) = 16.3 (or T ( 0 ) = 12.1 dB ) in a frequency bandwidth 2 δ 1 2 = 440 kHz .

Equations (16)

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

1 τ = 1 τ 0 + 1 τ e ,
κ 2 = 2 τ L τ e ,
a 2 = 1 2 τ L τ 0 .
d u d t = ( j ω 0 1 τ ) u ( t ) + 2 τ e s in ( t ) .
s out ( t ) = s in ( t ) + 2 τ e u ( t ) .
T = s out s in 2 = s out s 0 2 .
T ( δ ) = ( 1 τ e 1 τ 0 ) 2 + 4 π 2 δ 2 ( 1 τ e + 1 τ 0 ) 2 + 4 π 2 δ 2 .
Q = ν 0 2 δ 1 2 ,
T ( 0 ) = ( τ e τ 0 τ e + τ 0 ) 2 .
F = Δ ν 2 δ 1 2 = π κ 2 .
ω ( t ) = ω i + Ω 2 T S t ,
d φ d t = ω i + Ω T S t .
u ( t ) = 2 τ e s 0 exp ( j ω 0 t t τ ) × [ f ( t ) f ( 0 ) + 1 j ( ω i ω 0 ) + 1 τ ] ,
f ( t ) = 0 t exp [ j φ ( t ) + ( 1 τ j ω 0 ) t ] d t ,
f ( t ) = j π 2 V S exp [ j ( 2 π δ i j τ ) 2 2 V S ] × erf ( j τ 2 π δ i V S t 2 j V S ) ,
σ 2 ( τ 0 , τ e , V S ) = i = 1 N [ T mes , i T theo , i ( τ 0 , τ e , V S ) ] 2 ,

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