Abstract

Properties of a coupled optoelectronic microwave oscillator are studied theoretically. We find analytical expressions for the steady-state values of the pulse duration, chirp, optical power, and the power of the microwave signal generated in the systems as the function of the parameters of the optical and microwave loops. The analytical expressions are compared with the results of the numerical simulations as well as the experimental data.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245-262 (1965).
    [CrossRef]
  2. D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser-Part I: Theory,” IEEE J. Quantum Electron. 6, 694-708 (1970).
    [CrossRef]
  3. J. T. Darrow and R. K. Jain, “Active mode locking of broad band continuous wave lasers,” IEEE J. Quantum Electron. 27, 1048-1060 (1991).
    [CrossRef]
  4. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1996).
  5. G. H. C. New, “The generation of ultrashort laser pulses,” Rep. Prog. Phys. 46, 877-971 (1983).
    [CrossRef]
  6. L. A. Jiang, M. E. Grein, and E. P. Ippen, “Quantum-limited noise performance of a mode-locked laser diode,” Opt. Lett. 27, 49-51 (2002).
    [CrossRef]
  7. M. E. Grein, L. A. Jiang, H. A. Haus, E. P. Ippen, C. McNeilage, J. H. Searls, and R. S. Windeler, “Observation of quantum-limited timing jitter in an active, harmonically mode-locked fiber laser,” Opt. Lett. 27, 957-959 (2002).
    [CrossRef]
  8. F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, “Characterization of the noise and correlations in harmonically mode-locked lasers,” J. Opt. Soc. Am. B 19, 2609-2621 (2002).
    [CrossRef]
  9. X. S. Yao and L. Maleki, “Dual microwave and optical oscillator,” Opt. Lett. 22, 1867-1869 (1997).
    [CrossRef]
  10. X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73-78 (2000).
    [CrossRef]
  11. N. Yu, E. Salik, and L. Maleki, “Ultralow-noise mode-locked laser with coupled optoelectronic oscillator configuration,” Opt. Lett. 30, 1231-1233 (2005).
    [CrossRef] [PubMed]
  12. D. Eliyahu and L. Maleki, “Modulation response (S21) of the coupled opto-electronic oscillator,” Proceedings of the 2005 IEEE International Frequency Control Symposium and Exposition (IEEE, 2005), pp. 850-856.
    [CrossRef]
  13. E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444-446 (2007).
    [CrossRef]
  14. G. R. Huggett, “Mode-locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
    [CrossRef]
  15. T. S. Kinsel, “A stabilized mode-locked Nd:YAG laser using electronic feedback,” IEEE J. Quantum Electron. 9, 38 (1973).
    [CrossRef]
  16. K. Y. Lau and A. Yariv, “Self-sustained picosecond pulse generation in a GaAlAs laser at an electrically tunable repetition rate by optoelectronic feedback,” Appl. Phys. Lett. 45, 124-126 (1984).
    [CrossRef]
  17. M. Nakazawa, T. Nakashima, and M. Tokuda, “An optoelectronic self-oscillatory circuit with an optical fiber delayed feedback and its injection locking technique,” J. Lightwave Technol. 2, 719-730 (1984).
    [CrossRef]
  18. D. E. Spence, J. M. Evans, W. E. Sleat, and W. Sibbett, “Regeneratively initiated self-mode-locked Ti:Sapphire laser,” Opt. Lett. 16, 1762-1764 (1991).
    [CrossRef] [PubMed]
  19. J. D. Kafka, M. L. Watts, and J. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti: Sapphire laser,” IEEE J. Quantum Electron. 28, 2151-2161 (1992).
    [CrossRef]
  20. M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarization-maintaining erbium fiber laser,” Electron. Lett. 30, 1603-1605 (1994).
    [CrossRef]
  21. M. Nakazawa, E. Yoshida, E. Yamada, and Y. Kimura, “A repetition rate stabilized and tunable, regeneratively mode-locked fiber laser using an offset-locking technique,” Jpn. J. Appl. Phys., Part 2 35, L691-L694 (1996).
    [CrossRef]
  22. M. Nakazawa, E. Yoshida, and K. Tamura, “Ideal phase-locked-loop (PLL) operation of a 10 GHz erbium-doped fiber laser using regenerative mode-locking as an optical voltage controlled oscillator,” Electron. Lett. 33, 1318-1319 (1997).
    [CrossRef]
  23. B. Bakhshi, P. A. Andrekson, and X. Zhang, “10 GHz modelocked, dispersion-managed and polarization-maintaining erbium fiber ring laser with variable output coupling,” Electron. Lett. 34, 884-885 (1998).
    [CrossRef]
  24. K. K. Gupta, D. Novak, and H.-F. Liu, “Noise characterization of a regeneratively mode-locked fiber ring laser,” IEEE J. Quantum Electron. 36, 70-78 (2000).
    [CrossRef]
  25. K. K. Gupta and N. Onodera, “Regenerative mode locking via superposition of higher-order cavity modes in composite cavity fiber lasers,” Opt. Lett. 30, 2221-2223 (2005).
    [CrossRef] [PubMed]
  26. L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
    [CrossRef]
  27. N. G. Usechak and G. P. Agrawal, “Rate-equation approach for frequency-modulation mode locking using the moment method,” J. Opt. Soc. Am. B 22, 2570-2580 (2005).
    [CrossRef]
  28. N. Usechak and G. Agrawal, “Semi-analytic technique for analyzing mode-locked lasers,” Opt. Express 13, 2075-2081 (2005).
    [CrossRef] [PubMed]
  29. F. Rana, R. J. Ram, and H. A. Haus, “Quantum noise of actively mode-locked lasers with dispersion and amplitude/phase modulation,” IEEE J. Quantum Electron. 40, 41-56 (2004).
    [CrossRef]
  30. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
    [CrossRef]
  31. A. B. Matsko, D. Eliyahu, P. Koonath, D. Seidel, and L. Maleki are preparing a manuscript to be called “Theory of coupled optoelectronic microwave oscillator II: noise properties.”
  32. H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. 22, 325-331 (1986).
    [CrossRef]
  33. G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297-2305 (1989).
    [CrossRef]
  34. D. Eliyahu, R. A. Salvatore, and A. Yariv, “Effect of noise on the power spectrum of passively mode-locked lasers,” J. Opt. Soc. Am. B 14, 167-174 (1997).
    [CrossRef]
  35. M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458-1470 (2004).
    [CrossRef]
  36. H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, “Theory of soliton stability in asynchronous modelocking,” J. Lightwave Technol. 14, 622-627 (1996).
    [CrossRef]
  37. S. Longhi and P. Laporta, “Time-domain analysis of frequency modulation laser oscillation,” Appl. Phys. Lett. 73, 720-722 (1998).
    [CrossRef]

2007 (1)

E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444-446 (2007).
[CrossRef]

2005 (4)

2004 (2)

F. Rana, R. J. Ram, and H. A. Haus, “Quantum noise of actively mode-locked lasers with dispersion and amplitude/phase modulation,” IEEE J. Quantum Electron. 40, 41-56 (2004).
[CrossRef]

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

2002 (3)

2000 (2)

X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73-78 (2000).
[CrossRef]

K. K. Gupta, D. Novak, and H.-F. Liu, “Noise characterization of a regeneratively mode-locked fiber ring laser,” IEEE J. Quantum Electron. 36, 70-78 (2000).
[CrossRef]

1998 (2)

B. Bakhshi, P. A. Andrekson, and X. Zhang, “10 GHz modelocked, dispersion-managed and polarization-maintaining erbium fiber ring laser with variable output coupling,” Electron. Lett. 34, 884-885 (1998).
[CrossRef]

S. Longhi and P. Laporta, “Time-domain analysis of frequency modulation laser oscillation,” Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

1997 (3)

D. Eliyahu, R. A. Salvatore, and A. Yariv, “Effect of noise on the power spectrum of passively mode-locked lasers,” J. Opt. Soc. Am. B 14, 167-174 (1997).
[CrossRef]

X. S. Yao and L. Maleki, “Dual microwave and optical oscillator,” Opt. Lett. 22, 1867-1869 (1997).
[CrossRef]

M. Nakazawa, E. Yoshida, and K. Tamura, “Ideal phase-locked-loop (PLL) operation of a 10 GHz erbium-doped fiber laser using regenerative mode-locking as an optical voltage controlled oscillator,” Electron. Lett. 33, 1318-1319 (1997).
[CrossRef]

1996 (2)

M. Nakazawa, E. Yoshida, E. Yamada, and Y. Kimura, “A repetition rate stabilized and tunable, regeneratively mode-locked fiber laser using an offset-locking technique,” Jpn. J. Appl. Phys., Part 2 35, L691-L694 (1996).
[CrossRef]

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, “Theory of soliton stability in asynchronous modelocking,” J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

1994 (1)

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarization-maintaining erbium fiber laser,” Electron. Lett. 30, 1603-1605 (1994).
[CrossRef]

1993 (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

1992 (2)

J. D. Kafka, M. L. Watts, and J. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti: Sapphire laser,” IEEE J. Quantum Electron. 28, 2151-2161 (1992).
[CrossRef]

L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
[CrossRef]

1991 (2)

J. T. Darrow and R. K. Jain, “Active mode locking of broad band continuous wave lasers,” IEEE J. Quantum Electron. 27, 1048-1060 (1991).
[CrossRef]

D. E. Spence, J. M. Evans, W. E. Sleat, and W. Sibbett, “Regeneratively initiated self-mode-locked Ti:Sapphire laser,” Opt. Lett. 16, 1762-1764 (1991).
[CrossRef] [PubMed]

1989 (1)

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297-2305 (1989).
[CrossRef]

1986 (1)

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. 22, 325-331 (1986).
[CrossRef]

1984 (2)

K. Y. Lau and A. Yariv, “Self-sustained picosecond pulse generation in a GaAlAs laser at an electrically tunable repetition rate by optoelectronic feedback,” Appl. Phys. Lett. 45, 124-126 (1984).
[CrossRef]

M. Nakazawa, T. Nakashima, and M. Tokuda, “An optoelectronic self-oscillatory circuit with an optical fiber delayed feedback and its injection locking technique,” J. Lightwave Technol. 2, 719-730 (1984).
[CrossRef]

1983 (1)

G. H. C. New, “The generation of ultrashort laser pulses,” Rep. Prog. Phys. 46, 877-971 (1983).
[CrossRef]

1973 (1)

T. S. Kinsel, “A stabilized mode-locked Nd:YAG laser using electronic feedback,” IEEE J. Quantum Electron. 9, 38 (1973).
[CrossRef]

1970 (1)

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser-Part I: Theory,” IEEE J. Quantum Electron. 6, 694-708 (1970).
[CrossRef]

1968 (1)

G. R. Huggett, “Mode-locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
[CrossRef]

1965 (1)

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245-262 (1965).
[CrossRef]

Agrawal, G.

Agrawal, G. P.

N. G. Usechak and G. P. Agrawal, “Rate-equation approach for frequency-modulation mode locking using the moment method,” J. Opt. Soc. Am. B 22, 2570-2580 (2005).
[CrossRef]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297-2305 (1989).
[CrossRef]

Ahmed, Z.

L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
[CrossRef]

Andrekson, P. A.

B. Bakhshi, P. A. Andrekson, and X. Zhang, “10 GHz modelocked, dispersion-managed and polarization-maintaining erbium fiber ring laser with variable output coupling,” Electron. Lett. 34, 884-885 (1998).
[CrossRef]

Bakhshi, B.

B. Bakhshi, P. A. Andrekson, and X. Zhang, “10 GHz modelocked, dispersion-managed and polarization-maintaining erbium fiber ring laser with variable output coupling,” Electron. Lett. 34, 884-885 (1998).
[CrossRef]

Chen, Y.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

Darrow, J. T.

J. T. Darrow and R. K. Jain, “Active mode locking of broad band continuous wave lasers,” IEEE J. Quantum Electron. 27, 1048-1060 (1991).
[CrossRef]

Davis, L.

Eliyahu, D.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “Effect of noise on the power spectrum of passively mode-locked lasers,” J. Opt. Soc. Am. B 14, 167-174 (1997).
[CrossRef]

A. B. Matsko, D. Eliyahu, P. Koonath, D. Seidel, and L. Maleki are preparing a manuscript to be called “Theory of coupled optoelectronic microwave oscillator II: noise properties.”

D. Eliyahu and L. Maleki, “Modulation response (S21) of the coupled opto-electronic oscillator,” Proceedings of the 2005 IEEE International Frequency Control Symposium and Exposition (IEEE, 2005), pp. 850-856.
[CrossRef]

Evans, J. M.

Grein, M. E.

Gupta, K. K.

K. K. Gupta and N. Onodera, “Regenerative mode locking via superposition of higher-order cavity modes in composite cavity fiber lasers,” Opt. Lett. 30, 2221-2223 (2005).
[CrossRef] [PubMed]

K. K. Gupta, D. Novak, and H.-F. Liu, “Noise characterization of a regeneratively mode-locked fiber ring laser,” IEEE J. Quantum Electron. 36, 70-78 (2000).
[CrossRef]

Harris, S. E.

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245-262 (1965).
[CrossRef]

Haus, H. A.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

F. Rana, R. J. Ram, and H. A. Haus, “Quantum noise of actively mode-locked lasers with dispersion and amplitude/phase modulation,” IEEE J. Quantum Electron. 40, 41-56 (2004).
[CrossRef]

F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, “Characterization of the noise and correlations in harmonically mode-locked lasers,” J. Opt. Soc. Am. B 19, 2609-2621 (2002).
[CrossRef]

M. E. Grein, L. A. Jiang, H. A. Haus, E. P. Ippen, C. McNeilage, J. H. Searls, and R. S. Windeler, “Observation of quantum-limited timing jitter in an active, harmonically mode-locked fiber laser,” Opt. Lett. 27, 957-959 (2002).
[CrossRef]

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, “Theory of soliton stability in asynchronous modelocking,” J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. 22, 325-331 (1986).
[CrossRef]

Huggett, G. R.

G. R. Huggett, “Mode-locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
[CrossRef]

Ippen, E. P.

Jain, R. K.

J. T. Darrow and R. K. Jain, “Active mode locking of broad band continuous wave lasers,” IEEE J. Quantum Electron. 27, 1048-1060 (1991).
[CrossRef]

Jiang, L. A.

Jones, D. J.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, “Theory of soliton stability in asynchronous modelocking,” J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

Kafka, J. D.

J. D. Kafka, M. L. Watts, and J. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti: Sapphire laser,” IEEE J. Quantum Electron. 28, 2151-2161 (1992).
[CrossRef]

Kimura, Y.

M. Nakazawa, E. Yoshida, E. Yamada, and Y. Kimura, “A repetition rate stabilized and tunable, regeneratively mode-locked fiber laser using an offset-locking technique,” Jpn. J. Appl. Phys., Part 2 35, L691-L694 (1996).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarization-maintaining erbium fiber laser,” Electron. Lett. 30, 1603-1605 (1994).
[CrossRef]

Kinsel, T. S.

T. S. Kinsel, “A stabilized mode-locked Nd:YAG laser using electronic feedback,” IEEE J. Quantum Electron. 9, 38 (1973).
[CrossRef]

Koonath, P.

A. B. Matsko, D. Eliyahu, P. Koonath, D. Seidel, and L. Maleki are preparing a manuscript to be called “Theory of coupled optoelectronic microwave oscillator II: noise properties.”

Kuizenga, D. J.

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser-Part I: Theory,” IEEE J. Quantum Electron. 6, 694-708 (1970).
[CrossRef]

Laporta, P.

S. Longhi and P. Laporta, “Time-domain analysis of frequency modulation laser oscillation,” Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

Lau, K. Y.

K. Y. Lau and A. Yariv, “Self-sustained picosecond pulse generation in a GaAlAs laser at an electrically tunable repetition rate by optoelectronic feedback,” Appl. Phys. Lett. 45, 124-126 (1984).
[CrossRef]

Lee, H. L. T.

Liu, H. -F.

K. K. Gupta, D. Novak, and H.-F. Liu, “Noise characterization of a regeneratively mode-locked fiber ring laser,” IEEE J. Quantum Electron. 36, 70-78 (2000).
[CrossRef]

Longhi, S.

S. Longhi and P. Laporta, “Time-domain analysis of frequency modulation laser oscillation,” Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

Lowery, A. J.

L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
[CrossRef]

Maleki, L.

E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444-446 (2007).
[CrossRef]

N. Yu, E. Salik, and L. Maleki, “Ultralow-noise mode-locked laser with coupled optoelectronic oscillator configuration,” Opt. Lett. 30, 1231-1233 (2005).
[CrossRef] [PubMed]

X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73-78 (2000).
[CrossRef]

X. S. Yao and L. Maleki, “Dual microwave and optical oscillator,” Opt. Lett. 22, 1867-1869 (1997).
[CrossRef]

D. Eliyahu and L. Maleki, “Modulation response (S21) of the coupled opto-electronic oscillator,” Proceedings of the 2005 IEEE International Frequency Control Symposium and Exposition (IEEE, 2005), pp. 850-856.
[CrossRef]

A. B. Matsko, D. Eliyahu, P. Koonath, D. Seidel, and L. Maleki are preparing a manuscript to be called “Theory of coupled optoelectronic microwave oscillator II: noise properties.”

Matsko, A. B.

A. B. Matsko, D. Eliyahu, P. Koonath, D. Seidel, and L. Maleki are preparing a manuscript to be called “Theory of coupled optoelectronic microwave oscillator II: noise properties.”

McDuff, O. P.

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245-262 (1965).
[CrossRef]

McNeilage, C.

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

Nakashima, T.

M. Nakazawa, T. Nakashima, and M. Tokuda, “An optoelectronic self-oscillatory circuit with an optical fiber delayed feedback and its injection locking technique,” J. Lightwave Technol. 2, 719-730 (1984).
[CrossRef]

Nakazawa, M.

M. Nakazawa, E. Yoshida, and K. Tamura, “Ideal phase-locked-loop (PLL) operation of a 10 GHz erbium-doped fiber laser using regenerative mode-locking as an optical voltage controlled oscillator,” Electron. Lett. 33, 1318-1319 (1997).
[CrossRef]

M. Nakazawa, E. Yoshida, E. Yamada, and Y. Kimura, “A repetition rate stabilized and tunable, regeneratively mode-locked fiber laser using an offset-locking technique,” Jpn. J. Appl. Phys., Part 2 35, L691-L694 (1996).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarization-maintaining erbium fiber laser,” Electron. Lett. 30, 1603-1605 (1994).
[CrossRef]

M. Nakazawa, T. Nakashima, and M. Tokuda, “An optoelectronic self-oscillatory circuit with an optical fiber delayed feedback and its injection locking technique,” J. Lightwave Technol. 2, 719-730 (1984).
[CrossRef]

New, G. H. C.

G. H. C. New, “The generation of ultrashort laser pulses,” Rep. Prog. Phys. 46, 877-971 (1983).
[CrossRef]

Novak, D.

K. K. Gupta, D. Novak, and H.-F. Liu, “Noise characterization of a regeneratively mode-locked fiber ring laser,” IEEE J. Quantum Electron. 36, 70-78 (2000).
[CrossRef]

Olsson, N. A.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297-2305 (1989).
[CrossRef]

Onodera, N.

K. K. Gupta and N. Onodera, “Regenerative mode locking via superposition of higher-order cavity modes in composite cavity fiber lasers,” Opt. Lett. 30, 2221-2223 (2005).
[CrossRef] [PubMed]

L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
[CrossRef]

Pieterse, J. J.

J. D. Kafka, M. L. Watts, and J. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti: Sapphire laser,” IEEE J. Quantum Electron. 28, 2151-2161 (1992).
[CrossRef]

Ram, R. J.

F. Rana, R. J. Ram, and H. A. Haus, “Quantum noise of actively mode-locked lasers with dispersion and amplitude/phase modulation,” IEEE J. Quantum Electron. 40, 41-56 (2004).
[CrossRef]

F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, “Characterization of the noise and correlations in harmonically mode-locked lasers,” J. Opt. Soc. Am. B 19, 2609-2621 (2002).
[CrossRef]

Rana, F.

F. Rana, R. J. Ram, and H. A. Haus, “Quantum noise of actively mode-locked lasers with dispersion and amplitude/phase modulation,” IEEE J. Quantum Electron. 40, 41-56 (2004).
[CrossRef]

F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, “Characterization of the noise and correlations in harmonically mode-locked lasers,” J. Opt. Soc. Am. B 19, 2609-2621 (2002).
[CrossRef]

Salik, E.

E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444-446 (2007).
[CrossRef]

N. Yu, E. Salik, and L. Maleki, “Ultralow-noise mode-locked laser with coupled optoelectronic oscillator configuration,” Opt. Lett. 30, 1231-1233 (2005).
[CrossRef] [PubMed]

Salvatore, R. A.

Searls, J. H.

Seidel, D.

A. B. Matsko, D. Eliyahu, P. Koonath, D. Seidel, and L. Maleki are preparing a manuscript to be called “Theory of coupled optoelectronic microwave oscillator II: noise properties.”

Sibbett, W.

Siegman, A. E.

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser-Part I: Theory,” IEEE J. Quantum Electron. 6, 694-708 (1970).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1996).

Silberberg, Y.

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. 22, 325-331 (1986).
[CrossRef]

Sleat, W. E.

Spence, D. E.

Tamura, K.

M. Nakazawa, E. Yoshida, and K. Tamura, “Ideal phase-locked-loop (PLL) operation of a 10 GHz erbium-doped fiber laser using regenerative mode-locking as an optical voltage controlled oscillator,” Electron. Lett. 33, 1318-1319 (1997).
[CrossRef]

Tokuda, M.

M. Nakazawa, T. Nakashima, and M. Tokuda, “An optoelectronic self-oscillatory circuit with an optical fiber delayed feedback and its injection locking technique,” J. Lightwave Technol. 2, 719-730 (1984).
[CrossRef]

Tucker, R. S.

L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
[CrossRef]

Usechak, N.

Usechak, N. G.

Watts, M. L.

J. D. Kafka, M. L. Watts, and J. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti: Sapphire laser,” IEEE J. Quantum Electron. 28, 2151-2161 (1992).
[CrossRef]

Windeler, R. S.

Wong, W. S.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, “Theory of soliton stability in asynchronous modelocking,” J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

Yamada, E.

M. Nakazawa, E. Yoshida, E. Yamada, and Y. Kimura, “A repetition rate stabilized and tunable, regeneratively mode-locked fiber laser using an offset-locking technique,” Jpn. J. Appl. Phys., Part 2 35, L691-L694 (1996).
[CrossRef]

Yao, X. S.

Yariv, A.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “Effect of noise on the power spectrum of passively mode-locked lasers,” J. Opt. Soc. Am. B 14, 167-174 (1997).
[CrossRef]

K. Y. Lau and A. Yariv, “Self-sustained picosecond pulse generation in a GaAlAs laser at an electrically tunable repetition rate by optoelectronic feedback,” Appl. Phys. Lett. 45, 124-126 (1984).
[CrossRef]

Yoshida, E.

M. Nakazawa, E. Yoshida, and K. Tamura, “Ideal phase-locked-loop (PLL) operation of a 10 GHz erbium-doped fiber laser using regenerative mode-locking as an optical voltage controlled oscillator,” Electron. Lett. 33, 1318-1319 (1997).
[CrossRef]

M. Nakazawa, E. Yoshida, E. Yamada, and Y. Kimura, “A repetition rate stabilized and tunable, regeneratively mode-locked fiber laser using an offset-locking technique,” Jpn. J. Appl. Phys., Part 2 35, L691-L694 (1996).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarization-maintaining erbium fiber laser,” Electron. Lett. 30, 1603-1605 (1994).
[CrossRef]

Yu, N.

E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444-446 (2007).
[CrossRef]

N. Yu, E. Salik, and L. Maleki, “Ultralow-noise mode-locked laser with coupled optoelectronic oscillator configuration,” Opt. Lett. 30, 1231-1233 (2005).
[CrossRef] [PubMed]

Zhai, L.

L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
[CrossRef]

Zhang, X.

B. Bakhshi, P. A. Andrekson, and X. Zhang, “10 GHz modelocked, dispersion-managed and polarization-maintaining erbium fiber ring laser with variable output coupling,” Electron. Lett. 34, 884-885 (1998).
[CrossRef]

Appl. Phys. Lett. (3)

G. R. Huggett, “Mode-locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
[CrossRef]

K. Y. Lau and A. Yariv, “Self-sustained picosecond pulse generation in a GaAlAs laser at an electrically tunable repetition rate by optoelectronic feedback,” Appl. Phys. Lett. 45, 124-126 (1984).
[CrossRef]

S. Longhi and P. Laporta, “Time-domain analysis of frequency modulation laser oscillation,” Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

Electron. Lett. (4)

L. Zhai, A. J. Lowery, Z. Ahmed, N. Onodera, and R. S. Tucker, “Locking bandwidth of mode-locked semiconductor lasers,” Electron. Lett. 28, 545-546 (1992).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively modelocked polarization-maintaining erbium fiber laser,” Electron. Lett. 30, 1603-1605 (1994).
[CrossRef]

M. Nakazawa, E. Yoshida, and K. Tamura, “Ideal phase-locked-loop (PLL) operation of a 10 GHz erbium-doped fiber laser using regenerative mode-locking as an optical voltage controlled oscillator,” Electron. Lett. 33, 1318-1319 (1997).
[CrossRef]

B. Bakhshi, P. A. Andrekson, and X. Zhang, “10 GHz modelocked, dispersion-managed and polarization-maintaining erbium fiber ring laser with variable output coupling,” Electron. Lett. 34, 884-885 (1998).
[CrossRef]

IEEE J. Quantum Electron. (11)

K. K. Gupta, D. Novak, and H.-F. Liu, “Noise characterization of a regeneratively mode-locked fiber ring laser,” IEEE J. Quantum Electron. 36, 70-78 (2000).
[CrossRef]

J. D. Kafka, M. L. Watts, and J. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti: Sapphire laser,” IEEE J. Quantum Electron. 28, 2151-2161 (1992).
[CrossRef]

T. S. Kinsel, “A stabilized mode-locked Nd:YAG laser using electronic feedback,” IEEE J. Quantum Electron. 9, 38 (1973).
[CrossRef]

S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. 1, 245-262 (1965).
[CrossRef]

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser-Part I: Theory,” IEEE J. Quantum Electron. 6, 694-708 (1970).
[CrossRef]

J. T. Darrow and R. K. Jain, “Active mode locking of broad band continuous wave lasers,” IEEE J. Quantum Electron. 27, 1048-1060 (1991).
[CrossRef]

F. Rana, R. J. Ram, and H. A. Haus, “Quantum noise of actively mode-locked lasers with dispersion and amplitude/phase modulation,” IEEE J. Quantum Electron. 40, 41-56 (2004).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. 22, 325-331 (1986).
[CrossRef]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297-2305 (1989).
[CrossRef]

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

E. Salik, N. Yu, and L. Maleki, “An ultralow phase noise coupled optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19, 444-446 (2007).
[CrossRef]

J. Lightwave Technol. (3)

M. Nakazawa, T. Nakashima, and M. Tokuda, “An optoelectronic self-oscillatory circuit with an optical fiber delayed feedback and its injection locking technique,” J. Lightwave Technol. 2, 719-730 (1984).
[CrossRef]

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, “Theory of soliton stability in asynchronous modelocking,” J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73-78 (2000).
[CrossRef]

J. Opt. Soc. Am. B (3)

Jpn. J. Appl. Phys., Part 2 (1)

M. Nakazawa, E. Yoshida, E. Yamada, and Y. Kimura, “A repetition rate stabilized and tunable, regeneratively mode-locked fiber laser using an offset-locking technique,” Jpn. J. Appl. Phys., Part 2 35, L691-L694 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Rep. Prog. Phys. (1)

G. H. C. New, “The generation of ultrashort laser pulses,” Rep. Prog. Phys. 46, 877-971 (1983).
[CrossRef]

Other (3)

D. Eliyahu and L. Maleki, “Modulation response (S21) of the coupled opto-electronic oscillator,” Proceedings of the 2005 IEEE International Frequency Control Symposium and Exposition (IEEE, 2005), pp. 850-856.
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1996).

A. B. Matsko, D. Eliyahu, P. Koonath, D. Seidel, and L. Maleki are preparing a manuscript to be called “Theory of coupled optoelectronic microwave oscillator II: noise properties.”

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

A possible scheme of the coupled opto-electronic oscillator. The optical and RF loops are shown by solid black and red dashed lines, respectively. Both loops include amplifiers and filters. We assume that the optical loop contains a semiconductor optical amplifier (SOA) with amplitude gain g and a filter with △ FWHM. The RF loop has an amplifier with amplitude gain G and a filter with Δ RF FWHM. Gains of amplifiers are high enough to support the stable mode-locked oscillation in the optical loop. The oscillation starts from vacuum and thermal fluctuations in the loops. The optical loop is physically closed. The microwave loop has an optical insert. A part of the light circulating in the optical loop is taken out using an optical beam splitter (amplitude splitting coefficient r 1 ) and demodulated on the fast photodiode (PD) generating an RF signal with carrier frequency equivalent to the optical pulse repetition rate. The amplified and filtered RF signal is fed into an electro-optic amplitude modulator (EOM). The phase of the RF signal is tuned via the phase rotator ϕ.

Fig. 2
Fig. 2

Temporal shape of an optical pulse in a COEO obtained by simulations (a) and observed experimentally (b). The knee in the pulse is due to the slow carrier dynamics in the SOA. The pulses have nearly Gaussian shape if the fast gain is considered.

Fig. 3
Fig. 3

Obtained by simulations (a) and experimentally observed (b) optical spectrum of COEO generating pulses shown in Fig. 2.

Equations (63)

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

A ( T , t ) = [ E π τ ] 1 / 2 [ exp ( t ξ 2 τ ) 2 ] 1 + i q e i Ω ( t ξ ) ,
E = T m / 2 T m / 2 | A ( T , t ) | 2 d t ,
T R A T + i 2 ( β 2 Σ + i Ω f 2 ) 2 A t 2 1 6 β 3 Σ 3 A t 3 = i γ Σ | A | 2 A + 1 2 ( g Σ α Σ ) A A Δ A M 2 ω m 2 ( t t m ) 2 ,
g Σ = g 0 Σ ( 1 + P ave P sat ) 1 = g 0 Σ ( 1 + E E sat ) 1 .
Δ A M = π V π ρ G r α m w R P PDsat E / E PDsat 1 + E / E PDsat ,
t m ( T + T m w d ) = ξ ( T ) ,
g Σ α Σ 1 + q 2 2 τ 2 Ω f 2 1 2 Δ A M ω m 2 τ 2 = 0 ,
[ β 2 Σ τ 2 q τ 2 Ω f 2 ] ( 1 + q 2 ) + E γ Σ 2 π τ q Δ A M ω m 2 τ 2 = 0 ,
q β 2 Σ τ + 1 q 2 2 τ Ω f 2 1 2 Δ A M ω m 2 τ 3 = 0.
1 + q 2 2 τ 2 Ω f 2 1 ,
Δ A M ω m 2 τ 2 1.
E = E sat ( g 0 Σ α Σ 1 ) .
τ 4 = 1 Δ A M ω m 2 Ω f 2 { 2 ( 1 + 1 β 2 Σ 2 Ω f 4 ) × [ 1 + β 2 Σ 2 Ω f 4 ( 1 + E γ Σ τ 2 π β 2 Σ ) 1 ] E γ Σ τ 2 π β 2 Σ } ,
q = 1 β 2 Σ Ω f 2 [ 1 + β 2 Σ 2 Ω f 4 ( 1 + E γ Σ τ 2 π β 2 Σ ) 1 ] ,
τ 4 = 1 Δ A M ω m 2 Ω f 2 ( 1 1 8 π Ω f 4 E 2 γ Σ 2 τ 2 ) ,
τ = 1 2 [ ( ( 1 8 π Ω f 2 E 2 γ Σ 2 ω m 2 Δ A M ) 2 + 4 Δ A M ω m 2 Ω f 2 ) 1 / 2 1 8 π Ω f 2 E 2 γ Σ 2 ω m 2 Δ A M ] 1 / 2 ,
q = Ω f 2 E γ Σ τ 2 2 π .
T R d ξ d T = ( β 2 Σ q Ω f 2 ) Ω Δ A M ω m 2 τ 2 ( ξ t m ) ,
T R d Ω d T = 1 + q 2 Ω f 2 τ 2 Ω + Δ A M ω m 2 q ( ξ t m ) ,
T m w d d t m d T = ξ t m ,
λ 0 = 0 ,
λ ± = 1 2 { 1 T m w d + ( 1 + q 2 Ω f 2 τ 2 + Δ A M ω m 2 τ 2 ) 1 T R ± [ 1 T m w d + ( 1 + q 2 Ω f 2 τ 2 + Δ A M ω m 2 τ 2 ) 1 T R ] 2 4 ( β 2 Σ Ω f 2 q ) Δ A M ω m 2 q Ω f 2 T R 2 } .
Ω = 0 ,     ξ t m = 0 ,
1 T + = 1 T m w d + Δ A M ω m 2 τ 2 1 T R ,
1 T = 1 + q 2 Ω f 2 τ 2 1 T R .
g t = g 0 g τ c g τ c P P sat ,
1 V g ϕ T = 1 2 α g ,
T A M = α A M 2 [ 1 + cos ( π V V π + θ 0 ) ] ,
T R A T + i 2 ( β 2 Σ + i Ω f 2 ) 2 A t 2 = i γ Σ | A | 2 A + 1 2 ( g Σ α Σ ) A A Δ A M 2 sin 2 [ ω m ( t t m ( T ) ) ] .
T R d ξ d T = ( β 2 Σ q Ω f 2 ) Ω 1 2 e ω m 2 τ 2 Δ A M ω m τ 2   sin [ 2 ω m ( ξ t m ) ] ,
T R d Ω d T = 1 + q 2 Ω f 2 τ 2 Ω Δ A M Ω ( 1 e ω m 2 τ 2   cos [ 2 ω m ( ξ t m ) ] ) + 1 2 e ω m 2 τ 2 Δ A M ω m q   sin [ 2 ω m ( ξ t m ) ] ,
t m ( T + T mwd ) = ξ ( T ) ,
t m = t m 0 + δ t m T T R .
ξ = t m 0 + δ t m T + T m w d T R ,
ξ t m = δ t m T m w d T R
δ = Δ A M ω m 2 τ 2 T m w d T R × [ q ( β 2 Σ Ω f 2 q ) 1 + q 2 + Δ A M Ω f 2 τ 2 ( 1 cos [ δ ] ) + 1 ] sin [ δ ] ,
δ = 2 ω m δ t m T m w d T R ,
δ t m < 1 2 Δ A M ω m τ 2 ,
δ ω RFml < Δ A M ω m 2 τ 2 2 T R .
j = R r | A ( T , t ) | 2 ,
j = R r T m / 2 T m / 2 | A ( T , t ) | 2 exp [ i 2 π ( t 1 t ) T m ] d t 1 T m + c .c . ,
j r R P ave 1 + P ave / P PDsat cos [ ω m ( t ξ ( T ) ) ] .
V ( T , t ) = | V ( T ) | cos [ ω m ( t + t m ( T ) ) ] .
| V ( T + T m w 1 ) | cos [ ω m ( t t m ( T + T m w 1 ) + T m w 1 ) ] = ρ G r α m w R P ave 1 + P ave / P PDsat [ 1 2 Δ RF Δ RF 2 i [ ω m ( 1 ξ / T ) ω RF ] e i ω m ( t ξ ) i ϕ + c .c . ] ,
ω m T m w 1 + arg [ Δ RF Δ RF 2 i ( ω m ω RF ) ] = 2 π l + ϕ ,
t m ( T + T m w d ) = ξ ( T ) ,
| V ( T + T m w d ) | = ρ G r α m w R P ave 1 + P ave / P PDsat ,
Δ A M 2 ω m 2 ( t t m ) 2 .
E ( T ) = | A ( T , t ) | 2 d t ,
ξ ( T ) = 1 E t | A ( T , t ) | 2 d t ,
Ω ( T ) = i 2 E [ A A t A A t ] d t ,
q ( T ) = i E ( t ξ ) [ A A t A A t ] d t ,
τ 2 ( T ) = 2 E ( t ξ ) 2 | A ( T , t ) | 2 d t .
d E d T = [ A A T + A A T ] d t ,
d ξ d T = 1 E d E d T ξ + 1 E t [ A A T + A A T ] d t = 1 E ( t ξ ) [ A A T + A A T ] d t ,
d Ω d T = 1 E d E d T Ω i 2 E [ T ( A A t ) T ( A A t ) ] d t ,
d q d T = 1 E d E d T q + i E ( t ξ ) [ T ( A A t ) T ( A A t ) ] d t ,
τ d τ d T = 1 2 E d E d T τ 2 + 1 E ( t ξ ) 2 [ A A T + A A T ] d t .
T R d E d T = ( g Σ α Σ ) E 1 2 Ω f 2 τ 2 [ 1 + q 2 + 2 Ω 2 τ 2 ] E Δ A M ω m 2 2 [ τ 2 + 2 ( t m ξ ) 2 ] E ,
T R d ξ d T = ( β 2 Σ q Ω f 2 ) Ω + β 3 Σ 4 τ 2 [ 1 + q 2 + 2 Ω 2 τ 2 ] Δ A M ω m 2 τ 2 ( ξ t m ) ,
T R d Ω d T = 1 Ω f 2 τ 2 ( 1 + q 2 ) Ω + Δ A M ω m 2 q ( ξ t m ) ,
T R d q d T = β 2 Σ q / Ω f 2 τ 2 [ 1 + q 2 + 2 Ω 2 τ 2 ] + E γ Σ 2 π τ β 3 Σ Ω 2 τ 2 [ 3 + 3 q 2 + 2 Ω 2 τ 2 ] + Δ A M ω m 2 τ 2 [ 2 Ω ( ξ t m ) q ] ,
T R d τ d T = q τ [ β 2 Σ β 3 Σ Ω ] + 1 2 Ω f 2 τ ( 1 q 2 ) Δ A M ω m 2 τ 3 2 .

Metrics