Abstract

We report on the experimental observation of efficient all-resonant three-wave mixing using high-Q whispering-gallery modes. The modes were excited in a millimeter size toroidal cavity fabricated from LiNbO3. We implemented a low-noise resonant electro-optic modulator based on this wave mixing process. We observe an efficient modulation of light with coherent microwave pumping at 9 GHz with applied power of approximately 10 mW. Used as a receiver, the modulator allows us to detect nanowatt microwave radiation. Preliminary results with a 33-GHz modulator prototype are also reported. We present a theoretical interpretation of the experimental results and discuss possible applications of the device.

© 2003 Optical Society of America

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  7. F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
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    [CrossRef]
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    [CrossRef]
  30. D. A. Cohen and A. F. J. Levi, “Microphotonic components for a mm-wave receiver,” Solid-State Electron. 45, 495-505 (2001).
    [CrossRef]
  31. D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “High-Q microphotonic electro-optic modulator,” Solid-State Electron. 45, 1577-1589 (2001).
    [CrossRef]
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    [CrossRef]
  38. R. Graham and H. Haken, “The quantum fluctuations of the optical parametric oscillator,” Z. Phys. 210, 276–302 (1968).
    [CrossRef]

2001

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, “Microtorus: a high-finesse microcavity with whispering-gallery modes,” Opt. Lett. 26, 256-258 (2001).
[CrossRef]

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 141-144 (2001).
[CrossRef]

S. Blair and Y. Chen, “Resonant-enhanced evanescent-wave fluorescence biosensing with cylindrical optical cavities,” Appl. Opt. 40, 570-582 (2001).
[CrossRef]

R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742-5747 (2001).
[CrossRef]

D. A. Cohen and A. F. J. Levi, “Microphotonic millimetre-wave receiver architecture,” Electron. Lett. 37, 37-39 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “Microphotonic modulator for microwave receiver,” Electron. Lett. 37, 300-301 (2001).
[CrossRef]

D. A. Cohen and A. F. J. Levi, “Microphotonic components for a mm-wave receiver,” Solid-State Electron. 45, 495-505 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “High-Q microphotonic electro-optic modulator,” Solid-State Electron. 45, 1577-1589 (2001).
[CrossRef]

2000

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

1999

M. Pelton and Y. Yamamoto, “Ultralow threshold laser using a single quantum dot and a microsphere cavity,” Phys. Rev. A 59, 2418-2421 (1999).
[CrossRef]

A. N. Oraevsky, M. O. Scully, T. V. Sarkisyan, and D. K. Bandy, “Using whispering gallery modes in semiconductor microdevices,” Laser Phys. 9, 990-1003 (1999).

J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24, 847-849 (1999).
[CrossRef]

1998

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247-249 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

1997

F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
[CrossRef]

J. Popp, M. H. Fields, and R. K. Chang, “Q switching by saturable absorption in microdroplets: elastic scattering and laser emission,” Opt. Lett. 22, 1296-1298 (1997).
[CrossRef]

1996

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

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

1994

1993

L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroshe, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

1992

1991

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

1990

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open system with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

1989

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

1984

1968

R. Graham and H. Haken, “The quantum fluctuations of the optical parametric oscillator,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

1961

W. H. Louisell, A. Yariv, and A. E. Siegmann, “Quantum fluctuations and noise in parametric processes,” Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

An, K. Y.

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Bandy, D. K.

A. N. Oraevsky, M. O. Scully, T. V. Sarkisyan, and D. K. Bandy, “Using whispering gallery modes in semiconductor microdevices,” Laser Phys. 9, 990-1003 (1999).

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open system with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Blair, S.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Blom, F. C.

F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
[CrossRef]

Boyd, R. W.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

Brune, M.

L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroshe, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

Cai, M.

Campillo, A. J.

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

Chang, J. S.

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

Chang, R. K.

Chen, Y.

Choi, Y. S.

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

Cohen, D. A.

D. A. Cohen and A. F. J. Levi, “Microphotonic millimetre-wave receiver architecture,” Electron. Lett. 37, 37-39 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “Microphotonic modulator for microwave receiver,” Electron. Lett. 37, 300-301 (2001).
[CrossRef]

D. A. Cohen and A. F. J. Levi, “Microphotonic components for a mm-wave receiver,” Solid-State Electron. 45, 495-505 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “High-Q microphotonic electro-optic modulator,” Solid-State Electron. 45, 1577-1589 (2001).
[CrossRef]

Collot, L.

L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroshe, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

Domokos, P.

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

Driessen, A.

F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
[CrossRef]

Driver, H. S. T.

Dubreuil, N.

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Dupriez, P.

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

Eversole, J. D.

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

Feron, P.

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

Fields, M. H.

Funabashi, M.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Gorodetsky, M. L.

Graham, R.

R. Graham and H. Haken, “The quantum fluctuations of the optical parametric oscillator,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

Haken, H.

R. Graham and H. Haken, “The quantum fluctuations of the optical parametric oscillator,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

Hare, J.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 141-144 (2001).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

Haroche, S.

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

Haroshe, S.

L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroshe, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

Heebner, J. E.

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open system with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Hoekstra, H. J. W. M.

F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
[CrossRef]

Hosaya, K.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Hossein-Zadeh, M.

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “High-Q microphotonic electro-optic modulator,” Solid-State Electron. 45, 1577-1589 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “Microphotonic modulator for microwave receiver,” Electron. Lett. 37, 300-301 (2001).
[CrossRef]

Hutcheon, R. J.

Ilchenko, V. S.

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, “Microtorus: a high-finesse microcavity with whispering-gallery modes,” Opt. Lett. 26, 256-258 (2001).
[CrossRef]

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 141-144 (2001).
[CrossRef]

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247-249 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[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]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

Inoue, A.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Inoue, T.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Kimble, H. J.

Knight, J. C.

Kuwabara, T.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open system with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Lee, J. H.

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

Lee, S. B.

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

Lefevre, V.

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

Lefevre-Seguin, V.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 141-144 (2001).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroshe, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open system with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Levi, A. F. J.

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “High-Q microphotonic electro-optic modulator,” Solid-State Electron. 45, 1577-1589 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “Microphotonic modulator for microwave receiver,” Electron. Lett. 37, 300-301 (2001).
[CrossRef]

D. A. Cohen and A. F. J. Levi, “Microphotonic components for a mm-wave receiver,” Solid-State Electron. 45, 495-505 (2001).
[CrossRef]

D. A. Cohen and A. F. J. Levi, “Microphotonic millimetre-wave receiver architecture,” Electron. Lett. 37, 37-39 (2001).
[CrossRef]

Lin, H. B.

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

Lissillour, F.

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

Long, M. B.

Long, R.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 141-144 (2001).
[CrossRef]

Louisell, W. H.

W. H. Louisell, A. Yariv, and A. E. Siegmann, “Quantum fluctuations and noise in parametric processes,” Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

Mabuchi, H.

Maleki, L.

Maruhashi, K.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Moon, H. J.

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

Nagai, H.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Ohata, K.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Oraevsky, A. N.

A. N. Oraevsky, M. O. Scully, T. V. Sarkisyan, and D. K. Bandy, “Using whispering gallery modes in semiconductor microdevices,” Laser Phys. 9, 990-1003 (1999).

Painter, O.

Pelton, M.

M. Pelton and Y. Yamamoto, “Ultralow threshold laser using a single quantum dot and a microsphere cavity,” Phys. Rev. A 59, 2418-2421 (1999).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Pompa, T. J. A.

F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
[CrossRef]

Popp, J.

Poulain, M.

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

Raimond, J. M.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

Raimond, J.-M.

L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroshe, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

Robertson, G. N.

Roch, J. F.

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

Sandoghdar, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

Sarkisyan, T. V.

A. N. Oraevsky, M. O. Scully, T. V. Sarkisyan, and D. K. Bandy, “Using whispering gallery modes in semiconductor microdevices,” Laser Phys. 9, 990-1003 (1999).

Savchenkov, A. A.

Scully, M. O.

A. N. Oraevsky, M. O. Scully, T. V. Sarkisyan, and D. K. Bandy, “Using whispering gallery modes in semiconductor microdevices,” Laser Phys. 9, 990-1003 (1999).

Sercel, P. C.

Shinozaki, S.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Siegmann, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegmann, “Quantum fluctuations and noise in parametric processes,” Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

Stephan, G. M.

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

Streed, E. W.

Takimoto, Y.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

Treussart, F.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

Tzeng, H. M.

Vahala, K. J.

van Dijk, D. R.

F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
[CrossRef]

Vernooy, D. W.

von Klitzing, W.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 141-144 (2001).
[CrossRef]

Wall, K. F.

Yamamoto, Y.

M. Pelton and Y. Yamamoto, “Ultralow threshold laser using a single quantum dot and a microsphere cavity,” Phys. Rev. A 59, 2418-2421 (1999).
[CrossRef]

Yao, X. S.

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegmann, “Quantum fluctuations and noise in parametric processes,” Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

Young, K.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open system with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

F. C. Blom, D. R. van Dijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Pompa, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747-749 (1997).
[CrossRef]

Electron. Lett.

F. Lissillour, P. Feron, N. Dubreuil, P. Dupriez, M. Poulain, and G. M. Stephan, “Erbium-doped microspherical lasers at 1.56 μm,” Electron. Lett. 36, 1382-1384 (2000).
[CrossRef]

D. A. Cohen and A. F. J. Levi, “Microphotonic millimetre-wave receiver architecture,” Electron. Lett. 37, 37-39 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “Microphotonic modulator for microwave receiver,” Electron. Lett. 37, 300-301 (2001).
[CrossRef]

Eur. Phys. J. D

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235-238 (1998).
[CrossRef]

Europhys. Lett.

L. Collot, V. Lefevre-Seguin, M. Brune, J.-M. Raimond, and S. Haroshe, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327-334 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

K. Ohata, T. Inoue, M. Funabashi, A. Inoue, Y. Takimoto, T. Kuwabara, S. Shinozaki, K. Maruhashi, K. Hosaya, and H. Nagai, “Sixty-GHz-bang ultra-miniature monolithic T/R modules for multimedia wireless communication systems,” IEEE Trans. Microwave Theory Tech. 44, 2354-2360 (1996).
[CrossRef]

J. Korean Phys. Soc.

Y. S. Choi, H. J. Moon, K. Y. An, S. B. Lee, J. H. Lee, and J. S. Chang, “Ultrahigh-Q microsphere dye laser based on evanescent-wave coupling,” J. Korean Phys. Soc. 39, 928-931 (2001).

J. Lumin.

F. Treussart, V. S. Ilchenko, J. F. Roch, P. Domokos, J. Hare, V. Lefevre, J. M. Raimond, and S. Haroche, “Whispering gallery mode microlaser at liquid helium temperature,” J. Lumin. 76, 670-673 (1998).
[CrossRef]

Laser Phys.

A. N. Oraevsky, M. O. Scully, T. V. Sarkisyan, and D. K. Bandy, “Using whispering gallery modes in semiconductor microdevices,” Laser Phys. 9, 990-1003 (1999).

New J. Phys.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, “Tunable whispering gallery modes for spectroscopy and CQED experiments,” New J. Phys. 3, 141-144 (2001).
[CrossRef]

Opt. Lett.

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, “Microtorus: a high-finesse microcavity with whispering-gallery modes,” Opt. Lett. 26, 256-258 (2001).
[CrossRef]

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

H. Mabuchi and H. J. Kimble, “Atom galleries for whispering atoms—binding atoms in stable orbits around an optical resonator,” Opt. Lett. 19, 749-751 (1994).
[CrossRef] [PubMed]

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

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247-249 (1998).
[CrossRef]

J. Popp, M. H. Fields, and R. K. Chang, “Q switching by saturable absorption in microdroplets: elastic scattering and laser emission,” Opt. Lett. 22, 1296-1298 (1997).
[CrossRef]

J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24, 847-849 (1999).
[CrossRef]

H. M. Tzeng, K. F. Wall, M. B. Long, and R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499-501 (1984).
[CrossRef] [PubMed]

J. C. Knight, H. S. T. Driver, R. J. Hutcheon, and G. N. Robertson, “Core-resonance capillary-fiber whispering-gallery-mode laser,” Opt. Lett. 17, 1280-1282 (1992).
[CrossRef] [PubMed]

Phys. Lett. A

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering gallery modes,” Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

Phys. Rev.

W. H. Louisell, A. Yariv, and A. E. Siegmann, “Quantum fluctuations and noise in parametric processes,” Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Phys. Rev. A

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open system with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

M. Pelton and Y. Yamamoto, “Ultralow threshold laser using a single quantum dot and a microsphere cavity,” Phys. Rev. A 59, 2418-2421 (1999).
[CrossRef]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett.

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437-440 (1991).
[CrossRef] [PubMed]

Solid-State Electron.

D. A. Cohen and A. F. J. Levi, “Microphotonic components for a mm-wave receiver,” Solid-State Electron. 45, 495-505 (2001).
[CrossRef]

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “High-Q microphotonic electro-optic modulator,” Solid-State Electron. 45, 1577-1589 (2001).
[CrossRef]

Z. Phys.

R. Graham and H. Haken, “The quantum fluctuations of the optical parametric oscillator,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

Other

M. Abramowitz and I. A. Stegun, eds., Handbook on Mathematical Functions (Dover, New York, 1970).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, New York, 1994).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

V. S. Ilchenko, X. S. Yao, and L. Maleki, “Microsphere integration in active and passive photonics devices,” in Laser Resonators III; A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE 3930, 154-162 (2000).

Toroidal cavity shape is essential here to obtain a large free spectral range (frequency splitting between neighboring cavity modes).

V. S. Ilchenko and L. Maleki, “Novel whispering-gallery resonators for lasers, modulators, and sensors,” in Laser Resonators IV, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE 4270, 120-130 (2001).

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Figures (12)

Fig. 1
Fig. 1

Experimental setup: (1) LiNbO3 optical cavity, (2) microwave resonator, (3) microwave feeding strip line, and (4) diamond coupling prism. Inset: geometric characteristics of the nonlinear optical cavity.

Fig. 2
Fig. 2

Reflection spectrum of the nonlinear optical cavity. The free spectral range is 9.155 GHz. The mode quality factor is approximately 5×106.

Fig. 3
Fig. 3

Demodulated microwave power versus frequency of the microwave pumping. The zero level corresponds to the saturation power. The curve for the 2.5-nW pump was taken with 20-s averaging. This explains the lowering of the background noise level. However, even without the averaging we can see the nanowatt signals.

Fig. 4
Fig. 4

Optical etalon transmission (frequency spectrum of the modulated signal) versus etalon frequency scanning. Zero frequency corresponds to the carrier frequency of pumping the system light.

Fig. 5
Fig. 5

Normalized demodulated microwave power versus power of the microwave pump. The absolute value of the demodulated microwave signal is approximately 30 dB less than the input microwave power.

Fig. 6
Fig. 6

Normalized power of optical harmonics generated in the modulator versus power of the microwave pump. The unity power corresponds to the maximum power of the first harmonic measured in our experiment. The curve is taken with optical etalon transmission dependence (Fig. 4).

Fig. 7
Fig. 7

Whispering-gallery-mode response curves for various powers of the microwave pump.

Fig. 8
Fig. 8

Top: whispering-gallery-mode resonance. Bottom: demodulated microwave power versus detuning of the pump light from the whispering-gallery-mode resonance. No signal is found for the resonant tuning.

Fig. 9
Fig. 9

Normalized demodulated microwave power versus frequency of the pump microwave field for the Ka-band modulation frequency.

Fig. 10
Fig. 10

Top: theoretically evaluated whispering-gallery-mode resonance. Bottom: theoretically evaluated demodulated microwave power versus detuning of the pump light from the whispering-gallery-mode resonance. No signal is found for the resonant tuning.

Fig. 11
Fig. 11

Evaluated frequency spectrum of the modulated signal. Zero frequency corresponds to the carrier frequency of the pumping light.

Fig. 12
Fig. 12

Evaluated normalized power of optical harmonics generated in the modulator versus power of the microwave pump.

Equations (58)

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

nklmqa=tlq-ξ(n2-1)1/2,
nk˜mqa=Tmq-ξ(n2-1)1/2,
k22(l-m)+1a2(1-ε2)1/2 m.
klmqk˜mq+k22k˜mq.
nklmqa=tlq-ξ(n2-1)1/2+2(l-m)+121-(1-ε2)1/2(1-ε2)1/2.
ωl+1,mq-ωlmqωlmq-ωl-1,mq=cna [1+0.62l-2/3+O(l-5/3)],
ωl,m+1,q-ωlmqωlmq-ωl,m-1,q=-cna1-(1-ε2)1/2(1-ε2)1/2.
E1(t)=Ein(t)T-E2(t)(1-T)1/2,
Eout(t)=-Ein(t)(1-T)1/2-E2(t)T.
E2(t)=-E1(t-τ),
E˜2(t)-E˜1(t)+τE˜˙1(t).
1-(1-T)1/2T2.
E˜˙1(t)+T2τ E˜1(t)=Ein(t)τexp(iωt)Tτ.
E˙1(t)+(iω+γ)E1(t)=F(t).
Eout(t)=-Ein(t)+E1(t)T.
Hˆ=H^0+Vˆ.
H^0=ωa^aˆ+ω-b^-b^-+ω+b^+b^++ωcc^cˆ,
Vˆ=g(b^-c^aˆ+b^+cˆaˆ)+adjoint,
g=4πωa χ(2)2πωccVc1/21VVdV ΨaΨbΨc
a^˙=-iωaˆ-ig*(b^-cˆ+c^b^+),
b^˙-=-iω-b^--igc^aˆ,
b^˙+=-iω+b^+-igcˆaˆ,
c^˙=-iωccˆ-igb^-aˆ-ig*a^b^+.
A˙=-ΓAA-ig*(B-C+CB+)+FA,
B˙-=-ΓB-B--igCA,
B˙+=-ΓB+B+-igCA,
C˙=-ΓCC-igB-A-ig*AB++FM,
ΓA=i(ω-ω0)+γ,
ΓB=i(ω-ω0±ωM)+γ,
ΓC=i(ωc-ωM)+γM,
A=-i g*ΓA (B-C+CB+)+FAΓA,
CFMΓC.
B-=-igFAFM*ΓB+ΓCΓB-ΓB+ΓA|ΓC|2+|g|2|FM|2(ΓB++ΓB-),
B+=-igFAFMΓB-ΓC*ΓB-ΓB+ΓA|ΓC|2+|g|2|FM|2(ΓB++ΓB-).
A=ΓB-ΓB+|ΓC|2FAΓB-ΓB+ΓA|ΓC|2+|g|2|FM|2(ΓB++ΓB-).
Eout-
=-Ein2igγFM*ΓB+ΓCΓB-ΓB+ΓA|ΓC|2+|g|2|FM|2(ΓB++ΓB-),
Eout+
=-Ein2igγFMΓB-ΓC*ΓB-ΓB+ΓA|ΓC|2+|g|2|FM|2(ΓB++ΓB-),
Eout
=Ein
×ΓB-ΓB+(ΓA-2γ)|ΓC|2+|g|2|FM|2(ΓB++ΓB-)ΓB-ΓB+ΓA|ΓC|2+|g|2x|FM|2(ΓB++ΓB-).
W±Win=2S1+2S22,
W0Win=1-2S21+2S22,
S=4|g|Qω0WMQMωM21/2
S=Q 16πχ(2)a2πWMQMcωMVC1/21VVdVΨaΨbΨc.
I|Eout+Eout-exp(iωMt)+Eout+exp(-iωMt)|2.
IM(EoutEout*+Eout*Eout+)exp(-iωMt)+c.c.,
Vˆ=gn=-(a^n-1c^a^n+a^n+1cˆa^n)+adjoint,
A˙n=-γAn-ig(An-1C+C*An+1)+FAδn,0,
C˙=-γMC-ign=-(An-1*An+An*An+1)+FM,
γAn+ig(An-1C+C*An+1)=FAδn,0,
γMC+ign=-(An-1*An+An*An+1)=FM.
A(t)=n=-Anexp(-iωMnt).
A(t)=FAγMγγM+2ig|FM|cos(ωMt+ϕM).
An=12π02π/ωMA(t)exp(iωMnt)dt.
Eout(t)=1-2iScos(ωMt+ϕM)1+2iScos(ωMt+ϕM) Ein(t)=Ein(t)exp{-2i arctan[2Scos(ωMt+ϕM)]}.

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