Abstract

Using the example of whispering gallery mode (WGM) electro-optic modulator (EOM) we show that the majority of phase EOMs, particularly the resonant types, introduce additional quantum noise to the modulated light. The noise power grows quadratically with the optical power and results from the unavoidable spontaneous emission process originating from the strongly nondegenerate parametric interaction. This latter process is the physical basis for modulation.

© 2007 Optical Society of America

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  1. E. I. Gordon and J. D. Rigden, "Fabry-Perot electro-optic modulator," Bell System Tech. J. 42, 155-179 (1963).
  2. R. C. Alferness, "Waveguide electro-optic modulators," IEEE Trans. Microwave Theor. and Techniques 30, 1121-1137 (1982).
    [CrossRef]
  3. K. P. Ho and J. M. Kahn, "Optical frequency comb generator using phase modulation in amplified circulating loop", IEEE Photon. Technol. Lett. 5, 721-725 (1993).
    [CrossRef]
  4. T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).
  5. I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," IEEE Photon. Technol. Lett. 14, 795-797 (2002).
    [CrossRef]
  6. M. Kato, K. Fujiura, and T. Kurihara, "Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator," Electron. Lett. 40, 299-301 (2001).
    [CrossRef]
  7. N. Benter, R. P. Bertram, E. Soergel, K. Buse, D. Apitz, L. B. Jacobsen, and P. M. Johansen, "Large-area Fabry-Perot modulator based on electro-optic polymers," Appl. Opt. 44, 6235-6239 (2005).
    [CrossRef] [PubMed]
  8. M. Kato, K. Fujiura, T. Kurihara, "Generation of a superstable Lorentzian pulse train with a high repetition frequency based on a Fabry-Perot resonator integrated with an electro-optic phase modulator," Appl. Opt. 44, 1263-1269 (2005).
    [CrossRef] [PubMed]
  9. V. S. Ilchenko, X. S. Yao, and L. Maleki, "Microsphere integration in active and passive photonics devices," Proc. SPIE 3930, 154-162 (2000).
    [CrossRef]
  10. V. S. Ilchenko and L. Maleki, "Novel whispering-gallery resonators for lasers, modulators, and sensors," Proc. SPIE 4270, 120-130 (2001).
    [CrossRef]
  11. D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, "Microphotonic modulator for microwave receiver," Electron. Lett. 37, 300-301 (2001).
    [CrossRef]
  12. 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]
  13. D. A. Cohen and A. F. J. Levi, "Microphotonic millimetr-wave receiver architecture," Electron. Lett. 37, 37-39 (2001).
    [CrossRef]
  14. D. A. Cohen and A. F. J. Levi, "Microphotonic components for a mm-wave receiver," Solid State Electron. 45, 495-505 (2001).
    [CrossRef]
  15. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
    [CrossRef]
  16. L. Maleki, A. F. J. Levi, S. Yao, and V. Ilchenko, "Light modulation in whispering-gallery-mode resonators," US papent 6,473,218 (2002).
  17. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002).
    [CrossRef]
  18. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Whispering gallery mode electro-optic modulator and photonic microwave receiver," J. Opt. Soc. Am. B 20, 333-342 (2003).
    [CrossRef]
  19. M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004).
    [CrossRef]
  20. M. Hossein-Zadeh, and A. F. J. Levi, "Self-homodyne RF-optical LiNbO3 microdisk receiver," Solid State Electron. 49, 1428-1434 (2005).
    [CrossRef]
  21. H. Tazawa and W. H. Steier, "Linearity of ring resonator-based electrooptic polymer modulator," Electron. Lett. 41, 12971298 (2005).
    [CrossRef]
  22. M. Hossein-Zadeh, and A. F. J. Levi, "14.6-GHz LiNbO3 microdisk photonic self-homodyne RF receiver," IEEE Trans. Microwave Theory Tech. 54, 821-831 (2006).
    [CrossRef]
  23. H. Tazawa, W. H. Steier "Analysis of ring resonator-based traveling-wave modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
    [CrossRef]
  24. H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
    [CrossRef]
  25. S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007).
    [CrossRef]
  26. B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
    [CrossRef]
  27. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002).
    [CrossRef]
  28. E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
    [CrossRef]
  29. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett A 287, 331-338 (2001).
    [CrossRef]
  30. V. B. Braginsky and S. P. Vyatchanin, "Low quantum noise tranquilizer for Fabry-Perot interferometer", Phys. Lett A 293, 228-234 (2002).
    [CrossRef]
  31. B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
    [CrossRef]
  32. 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]
  33. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005).
    [CrossRef] [PubMed]
  34. T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
    [CrossRef]
  35. E. DAmbrosio and W. Kells, "Considerations on parametric instability in Fabry Perot interferometer," Phys. Lett. A 299, 326-330 (2002).
    [CrossRef]

2007 (2)

S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007).
[CrossRef]

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

2006 (4)

M. Hossein-Zadeh, and A. F. J. Levi, "14.6-GHz LiNbO3 microdisk photonic self-homodyne RF receiver," IEEE Trans. Microwave Theory Tech. 54, 821-831 (2006).
[CrossRef]

H. Tazawa, W. H. Steier "Analysis of ring resonator-based traveling-wave modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
[CrossRef]

2005 (6)

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]

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005).
[CrossRef] [PubMed]

M. Hossein-Zadeh, and A. F. J. Levi, "Self-homodyne RF-optical LiNbO3 microdisk receiver," Solid State Electron. 49, 1428-1434 (2005).
[CrossRef]

H. Tazawa and W. H. Steier, "Linearity of ring resonator-based electrooptic polymer modulator," Electron. Lett. 41, 12971298 (2005).
[CrossRef]

N. Benter, R. P. Bertram, E. Soergel, K. Buse, D. Apitz, L. B. Jacobsen, and P. M. Johansen, "Large-area Fabry-Perot modulator based on electro-optic polymers," Appl. Opt. 44, 6235-6239 (2005).
[CrossRef] [PubMed]

M. Kato, K. Fujiura, T. Kurihara, "Generation of a superstable Lorentzian pulse train with a high repetition frequency based on a Fabry-Perot resonator integrated with an electro-optic phase modulator," Appl. Opt. 44, 1263-1269 (2005).
[CrossRef] [PubMed]

2004 (2)

M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004).
[CrossRef]

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
[CrossRef]

2003 (1)

2002 (6)

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002).
[CrossRef]

I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," IEEE Photon. Technol. Lett. 14, 795-797 (2002).
[CrossRef]

E. DAmbrosio and W. Kells, "Considerations on parametric instability in Fabry Perot interferometer," Phys. Lett. A 299, 326-330 (2002).
[CrossRef]

V. B. Braginsky and S. P. Vyatchanin, "Low quantum noise tranquilizer for Fabry-Perot interferometer", Phys. Lett A 293, 228-234 (2002).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002).
[CrossRef]

2001 (8)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett A 287, 331-338 (2001).
[CrossRef]

M. Kato, K. Fujiura, and T. Kurihara, "Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator," Electron. Lett. 40, 299-301 (2001).
[CrossRef]

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).

V. S. Ilchenko and L. Maleki, "Novel whispering-gallery resonators for lasers, modulators, and sensors," Proc. SPIE 4270, 120-130 (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, 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 and A. F. J. Levi, "Microphotonic millimetr-wave receiver architecture," Electron. Lett. 37, 37-39 (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]

2000 (1)

V. S. Ilchenko, X. S. Yao, and L. Maleki, "Microsphere integration in active and passive photonics devices," Proc. SPIE 3930, 154-162 (2000).
[CrossRef]

1993 (1)

K. P. Ho and J. M. Kahn, "Optical frequency comb generator using phase modulation in amplified circulating loop", IEEE Photon. Technol. Lett. 5, 721-725 (1993).
[CrossRef]

1982 (1)

R. C. Alferness, "Waveguide electro-optic modulators," IEEE Trans. Microwave Theor. and Techniques 30, 1121-1137 (1982).
[CrossRef]

1969 (1)

E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
[CrossRef]

1963 (1)

E. I. Gordon and J. D. Rigden, "Fabry-Perot electro-optic modulator," Bell System Tech. J. 42, 155-179 (1963).

Alferness, R. C.

R. C. Alferness, "Waveguide electro-optic modulators," IEEE Trans. Microwave Theor. and Techniques 30, 1121-1137 (1982).
[CrossRef]

Apitz, D.

Benter, N.

Bertram, R. P.

Bortnik, B.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

Braginsky, V. B.

V. B. Braginsky and S. P. Vyatchanin, "Low quantum noise tranquilizer for Fabry-Perot interferometer", Phys. Lett A 293, 228-234 (2002).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett A 287, 331-338 (2001).
[CrossRef]

Buse, K.

Carmon, T.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005).
[CrossRef] [PubMed]

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]

Cohen, D. A.

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, 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 and A. F. J. Levi, "Microphotonic millimetr-wave receiver architecture," Electron. Lett. 37, 37-39 (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]

Corbitt, T.

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
[CrossRef]

Dalton, L. R.

DAmbrosio, E.

E. DAmbrosio and W. Kells, "Considerations on parametric instability in Fabry Perot interferometer," Phys. Lett. A 299, 326-330 (2002).
[CrossRef]

Davies, R. J.

M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004).
[CrossRef]

Dunayevskiy, I.

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

Fetterman, H. R.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

Fujiura, K.

M. Kato, K. Fujiura, T. Kurihara, "Generation of a superstable Lorentzian pulse train with a high repetition frequency based on a Fabry-Perot resonator integrated with an electro-optic phase modulator," Appl. Opt. 44, 1263-1269 (2005).
[CrossRef] [PubMed]

M. Kato, K. Fujiura, and T. Kurihara, "Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator," Electron. Lett. 40, 299-301 (2001).
[CrossRef]

Gheorma, I. L.

I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," IEEE Photon. Technol. Lett. 14, 795-797 (2002).
[CrossRef]

Gordon, E. I.

E. I. Gordon and J. D. Rigden, "Fabry-Perot electro-optic modulator," Bell System Tech. J. 42, 155-179 (1963).

Gray, M. B.

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
[CrossRef]

Higuma, K.

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).

Ho, K. P.

K. P. Ho and J. M. Kahn, "Optical frequency comb generator using phase modulation in amplified circulating loop", IEEE Photon. Technol. Lett. 5, 721-725 (1993).
[CrossRef]

Hossein-Zadeh, M.

M. Hossein-Zadeh, and A. F. J. Levi, "14.6-GHz LiNbO3 microdisk photonic self-homodyne RF receiver," IEEE Trans. Microwave Theory Tech. 54, 821-831 (2006).
[CrossRef]

M. Hossein-Zadeh, and A. F. J. Levi, "Self-homodyne RF-optical LiNbO3 microdisk receiver," Solid State Electron. 49, 1428-1434 (2005).
[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]

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

Hum, S. V.

M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004).
[CrossRef]

Hung, Y.-C.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

Ilchenko, V. S.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Whispering gallery mode electro-optic modulator and photonic microwave receiver," J. Opt. Soc. Am. B 20, 333-342 (2003).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002).
[CrossRef]

V. S. Ilchenko and L. Maleki, "Novel whispering-gallery resonators for lasers, modulators, and sensors," Proc. SPIE 4270, 120-130 (2001).
[CrossRef]

V. S. Ilchenko, X. S. Yao, and L. Maleki, "Microsphere integration in active and passive photonics devices," Proc. SPIE 3930, 154-162 (2000).
[CrossRef]

Innerhofer, E.

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
[CrossRef]

Izutsu, M.

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).

Jacobsen, L. B.

Jen, A. K.-Y.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

Johansen, P. M.

Kahn, J. M.

K. P. Ho and J. M. Kahn, "Optical frequency comb generator using phase modulation in amplified circulating loop", IEEE Photon. Technol. Lett. 5, 721-725 (1993).
[CrossRef]

Kato, M.

M. Kato, K. Fujiura, T. Kurihara, "Generation of a superstable Lorentzian pulse train with a high repetition frequency based on a Fabry-Perot resonator integrated with an electro-optic phase modulator," Appl. Opt. 44, 1263-1269 (2005).
[CrossRef] [PubMed]

M. Kato, K. Fujiura, and T. Kurihara, "Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator," Electron. Lett. 40, 299-301 (2001).
[CrossRef]

Kawanishi, T.

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).

Kells, W.

E. DAmbrosio and W. Kells, "Considerations on parametric instability in Fabry Perot interferometer," Phys. Lett. A 299, 326-330 (2002).
[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]

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005).
[CrossRef] [PubMed]

Kuo, Y. H.

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

Kurihara, T.

M. Kato, K. Fujiura, T. Kurihara, "Generation of a superstable Lorentzian pulse train with a high repetition frequency based on a Fabry-Perot resonator integrated with an electro-optic phase modulator," Appl. Opt. 44, 1263-1269 (2005).
[CrossRef] [PubMed]

M. Kato, K. Fujiura, and T. Kurihara, "Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator," Electron. Lett. 40, 299-301 (2001).
[CrossRef]

Levi, A. F. J.

M. Hossein-Zadeh, and A. F. J. Levi, "14.6-GHz LiNbO3 microdisk photonic self-homodyne RF receiver," IEEE Trans. Microwave Theory Tech. 54, 821-831 (2006).
[CrossRef]

M. Hossein-Zadeh, and A. F. J. Levi, "Self-homodyne RF-optical LiNbO3 microdisk receiver," Solid State Electron. 49, 1428-1434 (2005).
[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, "Microphotonic modulator for microwave receiver," Electron. Lett. 37, 300-301 (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]

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

Li, S.

S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007).
[CrossRef]

Luo, J. D.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

Maleki, L.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Whispering gallery mode electro-optic modulator and photonic microwave receiver," J. Opt. Soc. Am. B 20, 333-342 (2003).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002).
[CrossRef]

V. S. Ilchenko and L. Maleki, "Novel whispering-gallery resonators for lasers, modulators, and sensors," Proc. SPIE 4270, 120-130 (2001).
[CrossRef]

V. S. Ilchenko, X. S. Yao, and L. Maleki, "Microsphere integration in active and passive photonics devices," Proc. SPIE 3930, 154-162 (2000).
[CrossRef]

Matsko, A. B.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Whispering gallery mode electro-optic modulator and photonic microwave receiver," J. Opt. Soc. Am. B 20, 333-342 (2003).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002).
[CrossRef]

Matsuo, Y.

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).

Mavalvala, N.

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
[CrossRef]

McClelland, D. E.

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
[CrossRef]

Mishkin, E. A.

E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
[CrossRef]

Mow-Lowry, C. M.

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
[CrossRef]

Oikawa, S.

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).

Okoniewski, M.

M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004).
[CrossRef]

Osgood, R. M.

I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," IEEE Photon. Technol. Lett. 14, 795-797 (2002).
[CrossRef]

Ottaway, D.

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
[CrossRef]

Pelc, J.

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
[CrossRef]

Rabiei, P.

Rigden, J. D.

E. I. Gordon and J. D. Rigden, "Fabry-Perot electro-optic modulator," Bell System Tech. J. 42, 155-179 (1963).

Rokhsari, H.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005).
[CrossRef] [PubMed]

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]

Savchenkov, A. A.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Whispering gallery mode electro-optic modulator and photonic microwave receiver," J. Opt. Soc. Am. B 20, 333-342 (2003).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002).
[CrossRef]

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]

Seo, B.-J.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

Sheard, B. S.

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
[CrossRef]

Soergel, E.

Steier, W. H.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

H. Tazawa, W. H. Steier "Analysis of ring resonator-based traveling-wave modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

H. Tazawa and W. H. Steier, "Linearity of ring resonator-based electrooptic polymer modulator," Electron. Lett. 41, 12971298 (2005).
[CrossRef]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

Strigin, S. E.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett A 287, 331-338 (2001).
[CrossRef]

Tazawa, H.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

H. Tazawa, W. H. Steier "Analysis of ring resonator-based traveling-wave modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

H. Tazawa and W. H. Steier, "Linearity of ring resonator-based electrooptic polymer modulator," Electron. Lett. 41, 12971298 (2005).
[CrossRef]

Vahala, K. J.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005).
[CrossRef] [PubMed]

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]

Vyatchanin, S. P.

V. B. Braginsky and S. P. Vyatchanin, "Low quantum noise tranquilizer for Fabry-Perot interferometer", Phys. Lett A 293, 228-234 (2002).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett A 287, 331-338 (2001).
[CrossRef]

Walls, D. F.

E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
[CrossRef]

Weldon, M. A. J.

M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004).
[CrossRef]

Whitcomb, S. E.

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
[CrossRef]

Yao, X. S.

V. S. Ilchenko, X. S. Yao, and L. Maleki, "Microsphere integration in active and passive photonics devices," Proc. SPIE 3930, 154-162 (2000).
[CrossRef]

Yi, F.

S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007).
[CrossRef]

Zhang, C.

Zhang, X. M.

S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007).
[CrossRef]

Zheng, S. L.

S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007).
[CrossRef]

Appl. Opt. (2)

Bell System Tech. J. (1)

E. I. Gordon and J. D. Rigden, "Fabry-Perot electro-optic modulator," Bell System Tech. J. 42, 155-179 (1963).

Electron. Lett. (5)

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).

M. Kato, K. Fujiura, and T. Kurihara, "Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator," Electron. Lett. 40, 299-301 (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 millimetr-wave receiver architecture," Electron. Lett. 37, 37-39 (2001).
[CrossRef]

H. Tazawa and W. H. Steier, "Linearity of ring resonator-based electrooptic polymer modulator," Electron. Lett. 41, 12971298 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

H. Tazawa, W. H. Steier "Analysis of ring resonator-based traveling-wave modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002).
[CrossRef]

M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004).
[CrossRef]

K. P. Ho and J. M. Kahn, "Optical frequency comb generator using phase modulation in amplified circulating loop", IEEE Photon. Technol. Lett. 5, 721-725 (1993).
[CrossRef]

I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," IEEE Photon. Technol. Lett. 14, 795-797 (2002).
[CrossRef]

IEEE Trans. Microwave Theor. and Techniques (1)

R. C. Alferness, "Waveguide electro-optic modulators," IEEE Trans. Microwave Theor. and Techniques 30, 1121-1137 (1982).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Hossein-Zadeh, and A. F. J. Levi, "14.6-GHz LiNbO3 microdisk photonic self-homodyne RF receiver," IEEE Trans. Microwave Theory Tech. 54, 821-831 (2006).
[CrossRef]

J. Lightwave Technol. (1)

J. Lightwave Technology (1)

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006).
[CrossRef]

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

Microwave Opt. Technol. Lett. (1)

S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007).
[CrossRef]

Opt. Express (1)

Phys. Lett A (2)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett A 287, 331-338 (2001).
[CrossRef]

V. B. Braginsky and S. P. Vyatchanin, "Low quantum noise tranquilizer for Fabry-Perot interferometer", Phys. Lett A 293, 228-234 (2002).
[CrossRef]

Phys. Lett. A (1)

E. DAmbrosio and W. Kells, "Considerations on parametric instability in Fabry Perot interferometer," Phys. Lett. A 299, 326-330 (2002).
[CrossRef]

Phys. Rev. (1)

E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
[CrossRef]

Phys. Rev. A (3)

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
[CrossRef]

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002).
[CrossRef]

Phys. Rev. Lett. (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]

Proc. SPIE (2)

V. S. Ilchenko, X. S. Yao, and L. Maleki, "Microsphere integration in active and passive photonics devices," Proc. SPIE 3930, 154-162 (2000).
[CrossRef]

V. S. Ilchenko and L. Maleki, "Novel whispering-gallery resonators for lasers, modulators, and sensors," Proc. SPIE 4270, 120-130 (2001).
[CrossRef]

Solid State Electron. (3)

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]

M. Hossein-Zadeh, and A. F. J. Levi, "Self-homodyne RF-optical LiNbO3 microdisk receiver," Solid State Electron. 49, 1428-1434 (2005).
[CrossRef]

Other (1)

L. Maleki, A. F. J. Levi, S. Yao, and V. Ilchenko, "Light modulation in whispering-gallery-mode resonators," US papent 6,473,218 (2002).

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Equations (57)

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

H ̂ 0 = h ¯ ω a ̂ a ̂ + h ¯ ω b ̂ b ̂ + h ¯ ω + b ̂ + b ̂ + + h ¯ ω c c ̂ c ̂ ,
V ̂ = h ¯ g ( b ̂ c ̂ a ̂ + b ̂ + c ̂ a ̂ ) + adjoint ,
a ̂ ˙ = i ω a ̂ i g * ( b ̂ c ̂ + c ̂ b ̂ + ) ,
b ̂ ˙ = i ω b ̂ i g c ̂ a ̂ ,
b ̂ ˙ + = i ω + b ̂ + i g c ̂ a ̂ ,
c ̂ ˙ = i ω c c ̂ i g b ̂ a ̂ i g * a ̂ b ̂ + .
n ̂ b + + n ̂ b + n ̂ a = N ̂ ,
n ̂ b n ̂ b + n ̂ c = N ̂ S ,
S ̂ + S ̂ + + S ̂ + S ̂ + ( n ̂ b n ̂ b + ) 2 = N ̂ C ,
S ̂ + C ̂ + C ̂ S ̂ + = N ̂ I ;
C ̂ ˙ = i g S ̂ +
S ̂ ˙ + = i g [ n ̂ c + N ̂ S ] C ̂ ,
C ̂ ̈ + g 2 [ n ̂ c ( t ) + N ̂ S ] C ̂ = 0 .
n ̂ ̈ c + g 2 [ n ̂ c 2 + n ̂ c N ̂ C N ̂ S 2 + N ̂ S ] = 0 .
n ̂ c 6 N ̂ sin 2 [ g t ( N ̂ 3 ) 1 / 4 ] .
n ̂ c | t 0 N ̂ g 2 t 2 .
b ̂ ( t ) = b ̂ ( 0 ) + i g * a * ( c ̂ ( 0 ) t + c ̃ t 2 / 2 ) ,
b ̂ + ( t ) = b ̂ + ( 0 ) i g a ( c ̂ ( 0 ) t + c ̃ t 2 / 2 ) ,
c ̂ = c ̂ ( 0 ) + c ˜ t , c ˜ = i ( g a b ̂ ( 0 ) + g * a * b ̂ + ( 0 ) ) .
n ̂ b | t 0 N ̂ g 2 t 2 + N ̂ 2 g 4 t 4 / 4 ,
n ̂ b + t 0 N ̂ 2 g 4 t 4 / 4 .
A ̂ ˙ = γ A ̂ i g * ( B ̂ C ̂ + C ̂ B ̂ + ) + F ̂ A ,
B ̂ ˙ = γ B ̂ i g C ̂ A ̂ + F ̂ B ,
B ̂ ˙ + = γ B ̂ + i g C ̂ A ̂ + F ̂ B + ,
C ̂ ˙ = γ M C ̂ i g B ̂ A ̂ i g * A ̂ B ̂ + + F ̂ C .
F ̂ A = 2 P γ h ¯ ω ,
F ̂ C = 2 P m w γ M h ¯ ω c ,
F ̂ A ( t ) F ̂ A ( t ) = 2 γ δ ( t t ) ,
F ̂ B ± ( t ) F ̂ B ± ( t ) = 2 γ δ ( t t ) ,
F ̂ C ( t ) F ̂ C ( t ) = 2 γ M ( n ¯ t h + 1 ) δ ( t t ) ,
F ̂ C ( t ) F ̂ C ( t ) = 2 γ M n ¯ t h δ ( t t ) ,
F ̂ B ± = f ̂ B ± ( ω ˜ ) e i ω ˜ t d ω ˜ 2 π ,
F ̂ C = F ̂ C + f ̂ C ( ω ˜ ) e i ω ˜ t d ω ˜ 2 π ,
B ̂ ± = δ B ̂ ± ( ω ˜ ) e i ω ˜ t d ω ˜ 2 π ,
C ̂ = C + δ C ̂ ( ω ˜ ) e i ω ˜ t d ω ˜ 2 π .
δ B ̂ ( ω ) = ( 1 + g 2 A 2 Γ Γ M ) f ̂ B ( ω ˜ ) Γ + g 2 A 2 Γ Γ M f ̂ B + ( ω ˜ ) Γ i g A Γ f ̂ C ( ω ˜ ) Γ M ,
δ B ̂ + ( ω ˜ ) = ( 1 g 2 A 2 Γ Γ M ) f ̂ B + ( ω ˜ ) Γ g 2 A 2 Γ Γ M f ̂ B ( ω ˜ ) Γ i g A Γ f ̂ C ( ω ˜ ) Γ M ,
n ̂ B = ( 2 γ + γ M ) γ M 2 γ 2 [ g 2 A 2 γ M ( γ M + γ ) ] 2 + g 2 A 2 γ ( γ M + γ ) ( n ¯ t h + 1 ) ,
n ̂ B + = ( 2 γ + γ M ) γ M 2 γ 2 [ g 2 A 2 γ M ( γ M + γ ) ] 2 + g 2 A 2 γ ( γ M + γ ) n ¯ t h ,
n ̂ C = n ¯ th + g 2 A 2 γ M ( γ M + γ ) .
A F ̂ A γ , C F ̂ C γ M , B igA γ C * , B + igA γ C .
E out E in , E out = E in 2 ig γ C * , E out + = E in 2 ig γ C .
e ̂ out ( ω ˜ ) = ( Γ * Γ + 2 γ g 2 A 2 Γ 2 Γ M ) e ̂ in ( ω ˜ ) + 2 γ g 2 A 2 Γ 2 Γ M e ̂ in + ( ω ˜ ) 2 igA γ γ M Γ Γ M e ̂ C ( ω ˜ ) ,
e ̂ out + ( ω ˜ ) = ( Γ * Γ 2 γ g 2 A 2 Γ 2 Γ M ) e ̂ in+ ( ω ˜ ) 2 γ g 2 A 2 Γ 2 Γ M e ̂ in ( ω ˜ ) 2 igA γ γ M Γ Γ M e ̂ C ( ω ˜ ) .
P out + = P in 4 g 2 γ 2 2 ( P mw + Δ P mw ) h ¯ ω c γ M .
P mw = P in mw + S ( ω ˜ ) d ω ˜ 2 π = P in mw + γ γ M γ + γ M k B T 2 + P g 2 ( γ + γ M ) 2 ω c ω ( 1 + γ M 2 γ ) ,
S ( ω ˜ ) = k B T γ 2 γ M 2 ( γ 2 + ω ˜ 2 ) ( γ M 2 + ω ˜ 2 ) + P ω c ω 2 g 2 γ 2 γ M ( γ 2 + ω ˜ 2 ) 2 ( γ M 2 + ω ˜ 2 ) ,
Δ P ( ω ˜ ) = h ¯ ω c [ h ¯ ω γ 2 P γ γ M 4 g 2 + 1 2 2 P h ¯ ω γ 4 g 2 γ γ M γ 4 γ M 2 Γ 4 Γ M 2 ] + 2 k B T γ 2 γ M 2 Γ 2 Γ M 2 .
Δ P ( ω ˜ ) min = 2 h ¯ ω c + 2 k B T ,
2 P opt h ¯ ω γ 4 g 2 γ γ M = 2 .
m = P out + P out = 8 g 2 γ 2 P mw h ¯ ω c γ M .
P opt = 2 P mw m ω ω c P mw sat ω ω c .
Δ P ( ω ˜ ) P mw 2 h ¯ ω P out + + 2 k B T P mw .
P mw sat h ¯ ω 2 ω c 2 16 g 2 Q 2 Q m .
g = ω L h ¯ 2 M ω M .
P th = γ 4 M ω M 2 L 2 Q Q M ,
n ̂ M = n ¯ th + P P th

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