Abstract

We propose the use of an intra-cavity Mach Zehnder interferometer (MZI), for increasing the repetition rate at which carrier-envelope phase-locked pulses are generated in passively mode-locked fiber lasers. The attractive feature of the proposed scheme is that light escaping through the open output ports of the MZI can be used as a monitor signal feeding a servo loop that allows multiple pulses to co-exist in the cavity, while rigidly controlling their separation. The proposed scheme enables in principle a significant increase in the pulse-rate with no deterioration in the properties of the generated pulses.

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  1. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
    [CrossRef] [PubMed]
  2. R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett., 85, 2264–2267 (2000).
    [CrossRef] [PubMed]
  3. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 013902 (2008)
    [CrossRef] [PubMed]
  4. S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
    [CrossRef] [PubMed]
  5. S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306, 1318–1324 (2004).
    [CrossRef] [PubMed]
  6. J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
    [CrossRef]
  7. S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B. 27, B51–B62 (2010).
    [CrossRef]
  8. A. Bartels, D. Heinecke, and S. A. Diddams, “10-GHz self-referenced optical frequency comb,” Science, vol.  326, p. 681 (2009).
    [CrossRef] [PubMed]
  9. D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” Proc. 2005 Int. Freq. Control Symp. and Precise Time and Time Interval Sys. Mtg. IEEE: Piscataway, NJ, 2005, 481–487.
  10. I. Hartl, A. Romann, and M. E. Fermann, “Passively mode locked GHz femtosecond Yb-fiber laser using an intra-cavity martinez compressor,” Proc. Conf. Lasers and Electro-Optics2011, Optical Society of America, paper CMD3.
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    [CrossRef] [PubMed]
  13. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B 24, 1756–1770 (2007).
    [CrossRef]
  14. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited lasers and amplifiers to high average power,” Opt. Express 16, 13240–13266 (2008).
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    [CrossRef]
  16. E. Yoshida, Y. Kimura, and M. Nakazawa, “Laser diode-pumped femtosecond Erbium doped fiber laser with a sub-ring cavity for repitition rate control,” Appl. Phys. Lett. 60, 932–934 (1992).
    [CrossRef]
  17. G.T. Harvey and L.F. Mollenauer, “Harmonically mode-locked fiber ring laser with an internal Fabry-Perot stabilizer for soliton transmission,” Opt. Lett. 18, 107–109 (1993).
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    [CrossRef]
  19. Y. Parkhomenko, M. Horowitz, C. R. Menyuk, and T. F. Carruthers, “Theoretical study of an actively mode-locked fiber laser stabilized by an intra-cavity Fabry-Perot etalon: Linear regime,” J. Opt. Soc. Am. B. 24, 1793–1802 (2007).
    [CrossRef]
  20. F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, “Harmonically modelocked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources,” J. Opt. A: Pure Appl. Opt. 11, 1–23 (2009)
    [CrossRef]
  21. R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solitons in erbium fibre laser,” Electron. Lett. 27, 1257–1258 (1991).
    [CrossRef]
  22. J. Schröder, S. Coen, F. Vanholsbeeck, and T. Sylvestre, “Passively modelocked fiber Raman laser with 100 GHz repetition rate,” Opt. Lett. 31, 3489–3491 (2006).
    [CrossRef] [PubMed]
  23. D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
    [CrossRef]
  24. A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, “Acoustic effect in passively mode-locked fiber ring lasers,” Opt. Lett. 20, 907–909 (1996).
    [CrossRef]
  25. B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgenson, “Phase-locked, erbium-fiber-laser based frequency comb in the near infrared,” Opt. Lett., vol.  29, 250–252 (2004).
    [CrossRef] [PubMed]
  26. P. Pal, W. H. Knox, I. Hartl, and M. E. Fermann, “Self referenced Yb-fiber-laser frequency comb using a dispersion micromanaged tapered holey fiber,” Opt. Express 15, 12161–12166 (2007).
    [CrossRef] [PubMed]
  27. E. Baumann, F. R. Giorgetta, J. W. Nicholson, W. C. Swann, I. Coddington, and N. R. Newbury, “High-performance, vibration-immune, fiber-laser comb, Opt. Lett.,  34, 638–640 (2009).
    [CrossRef] [PubMed]
  28. J. Lim, K. Knabe, K. A. Tillman, W. Neely, Y. Wang, R. Amezcua-Correa, F. Couny, P. S. Light, F. Benabid, J. C. Knight, K. L. Corwin, J. W. Nicholson, and B. R. Washburn, “A phase-stabilized nanotube fiber laser frequency comb,” Opt. Express 17, 14115–14120 (2009).
    [CrossRef] [PubMed]
  29. S. K. Sheem, “Optical fiber interferometers with [3 × 3] directional couplers: Analysis,” J. Ap. Phys. 52, 3865–3872 (1981).
    [CrossRef]
  30. R. W. C. Vance and J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc. Opto-Electron. 141, 231–241 (1994).
    [CrossRef]
  31. At the time of this writing, companies that produce 3 × 3 fiber couplers include the Shenzhen Technology Company and Rayscience Optoelectronic Innovation.
  32. R. G. Priest, “Analysis of fiber interferometer utilizing 3× 3 fiber coupler,” Trans. Micro. Theory Tech. MTT-30, 1589–1591 (1982).
    [CrossRef]
  33. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking, J. Opt. Soc. Amer. E,  8, 2068–2076 (1991).
    [CrossRef]
  34. C. Antonelli, J. Chen, and F. Kartner, “Intracavity pulse dynamics and stability for passively mode-locked lasers,” Opt. Express 15, 5919–5924 (2007).
    [CrossRef] [PubMed]
  35. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–995 (1993).
    [CrossRef]
  36. F. M. Gardner, Phaselock techniques3rd ed. (Wiley-Interscience, 2005).
    [CrossRef]
  37. S. T. Cundiff, J. Ye, and J. L. Hall, “Optical Frequency Synthesis Based on Mode-Locked Lasers,” Review of Scientific Instruments,  72, 3749–3771, (2001).
    [CrossRef]

2010 (1)

S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B. 27, B51–B62 (2010).
[CrossRef]

2009 (5)

2008 (4)

2007 (5)

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
[CrossRef] [PubMed]

N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs,” J. Opt. Soc. Am. B 24, 1756–1770 (2007).
[CrossRef]

Y. Parkhomenko, M. Horowitz, C. R. Menyuk, and T. F. Carruthers, “Theoretical study of an actively mode-locked fiber laser stabilized by an intra-cavity Fabry-Perot etalon: Linear regime,” J. Opt. Soc. Am. B. 24, 1793–1802 (2007).
[CrossRef]

P. Pal, W. H. Knox, I. Hartl, and M. E. Fermann, “Self referenced Yb-fiber-laser frequency comb using a dispersion micromanaged tapered holey fiber,” Opt. Express 15, 12161–12166 (2007).
[CrossRef] [PubMed]

C. Antonelli, J. Chen, and F. Kartner, “Intracavity pulse dynamics and stability for passively mode-locked lasers,” Opt. Express 15, 5919–5924 (2007).
[CrossRef] [PubMed]

2006 (2)

J. Schröder, S. Coen, F. Vanholsbeeck, and T. Sylvestre, “Passively modelocked fiber Raman laser with 100 GHz repetition rate,” Opt. Lett. 31, 3489–3491 (2006).
[CrossRef] [PubMed]

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

2005 (2)

D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” Proc. 2005 Int. Freq. Control Symp. and Precise Time and Time Interval Sys. Mtg. IEEE: Piscataway, NJ, 2005, 481–487.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

2004 (2)

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgenson, “Phase-locked, erbium-fiber-laser based frequency comb in the near infrared,” Opt. Lett., vol.  29, 250–252 (2004).
[CrossRef] [PubMed]

2002 (1)

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

2001 (1)

S. T. Cundiff, J. Ye, and J. L. Hall, “Optical Frequency Synthesis Based on Mode-Locked Lasers,” Review of Scientific Instruments,  72, 3749–3771, (2001).
[CrossRef]

2000 (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

1996 (1)

1994 (1)

R. W. C. Vance and J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc. Opto-Electron. 141, 231–241 (1994).
[CrossRef]

1993 (2)

1992 (1)

E. Yoshida, Y. Kimura, and M. Nakazawa, “Laser diode-pumped femtosecond Erbium doped fiber laser with a sub-ring cavity for repitition rate control,” Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

1991 (2)

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solitons in erbium fibre laser,” Electron. Lett. 27, 1257–1258 (1991).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking, J. Opt. Soc. Amer. E,  8, 2068–2076 (1991).
[CrossRef]

1982 (1)

R. G. Priest, “Analysis of fiber interferometer utilizing 3× 3 fiber coupler,” Trans. Micro. Theory Tech. MTT-30, 1589–1591 (1982).
[CrossRef]

1981 (1)

S. K. Sheem, “Optical fiber interferometers with [3 × 3] directional couplers: Analysis,” J. Ap. Phys. 52, 3865–3872 (1981).
[CrossRef]

Amezcua-Correa, R.

Antonelli, C.

Bartels, A.

A. Bartels, D. Heinecke, and S. A. Diddams, “10-GHz self-referenced optical frequency comb,” Science, vol.  326, p. 681 (2009).
[CrossRef] [PubMed]

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

Barty, C. P. J.

Baumann, E.

Beach, R. J.

Benabid, F.

Bergquist, J. C.

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

Blondel, M.

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

Braje, D.

Carruthers, T. F.

Y. Parkhomenko, M. Horowitz, C. R. Menyuk, and T. F. Carruthers, “Theoretical study of an actively mode-locked fiber laser stabilized by an intra-cavity Fabry-Perot etalon: Linear regime,” J. Opt. Soc. Am. B. 24, 1793–1802 (2007).
[CrossRef]

Chen, J.

Coddington, I.

E. Baumann, F. R. Giorgetta, J. W. Nicholson, W. C. Swann, I. Coddington, and N. R. Newbury, “High-performance, vibration-immune, fiber-laser comb, Opt. Lett.,  34, 638–640 (2009).
[CrossRef] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 013902 (2008)
[CrossRef] [PubMed]

Coen, S.

Corwin, K. L.

Couny, F.

Cundiff, S. T.

S. T. Cundiff, J. Ye, and J. L. Hall, “Optical Frequency Synthesis Based on Mode-Locked Lasers,” Review of Scientific Instruments,  72, 3749–3771, (2001).
[CrossRef]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Davey, R. P.

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solitons in erbium fibre laser,” Electron. Lett. 27, 1257–1258 (1991).
[CrossRef]

Dawson, J. W.

Delfyett, P. J.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, “Harmonically modelocked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources,” J. Opt. A: Pure Appl. Opt. 11, 1–23 (2009)
[CrossRef]

Deparis, O.

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

Diddams, S. A.

S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B. 27, B51–B62 (2010).
[CrossRef]

S. A. Diddams, M. Kirchner, T. Fortier, D. Braje, A. M. Weiner, and L. Hollberg, “Improved signal-to-noise ratio of 10 GHz microwave signals generated with a mode-filtered femtosecond laser frequency comb,” Opt. Express 17, 3331–3340 (2009).
[CrossRef] [PubMed]

A. Bartels, D. Heinecke, and S. A. Diddams, “10-GHz self-referenced optical frequency comb,” Science, vol.  326, p. 681 (2009).
[CrossRef] [PubMed]

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgenson, “Phase-locked, erbium-fiber-laser based frequency comb in the near infrared,” Opt. Lett., vol.  29, 250–252 (2004).
[CrossRef] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Emplit, P.

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

Fendel, P.

Ferguson, A. I.

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solitons in erbium fibre laser,” Electron. Lett. 27, 1257–1258 (1991).
[CrossRef]

Fermann, M. E.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

P. Pal, W. H. Knox, I. Hartl, and M. E. Fermann, “Self referenced Yb-fiber-laser frequency comb using a dispersion micromanaged tapered holey fiber,” Opt. Express 15, 12161–12166 (2007).
[CrossRef] [PubMed]

I. Hartl, A. Romann, and M. E. Fermann, “Passively mode locked GHz femtosecond Yb-fiber laser using an intra-cavity martinez compressor,” Proc. Conf. Lasers and Electro-Optics2011, Optical Society of America, paper CMD3.

Foreman, S. M.

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
[CrossRef] [PubMed]

Fortier, T.

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking, J. Opt. Soc. Amer. E,  8, 2068–2076 (1991).
[CrossRef]

Gardner, F. M.

F. M. Gardner, Phaselock techniques3rd ed. (Wiley-Interscience, 2005).
[CrossRef]

Gee, S.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, “Harmonically modelocked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources,” J. Opt. A: Pure Appl. Opt. 11, 1–23 (2009)
[CrossRef]

Giorgetta, F. R.

Golovchenko, E. A.

Haelterman, M.

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

Hall, J. L.

S. T. Cundiff, J. Ye, and J. L. Hall, “Optical Frequency Synthesis Based on Mode-Locked Lasers,” Review of Scientific Instruments,  72, 3749–3771, (2001).
[CrossRef]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Hänsch, T. W.

Hartl, I.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

P. Pal, W. H. Knox, I. Hartl, and M. E. Fermann, “Self referenced Yb-fiber-laser frequency comb using a dispersion micromanaged tapered holey fiber,” Opt. Express 15, 12161–12166 (2007).
[CrossRef] [PubMed]

I. Hartl, A. Romann, and M. E. Fermann, “Passively mode locked GHz femtosecond Yb-fiber laser using an intra-cavity martinez compressor,” Proc. Conf. Lasers and Electro-Optics2011, Optical Society of America, paper CMD3.

Harvey, G.T.

Hati, A.

D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” Proc. 2005 Int. Freq. Control Symp. and Precise Time and Time Interval Sys. Mtg. IEEE: Piscataway, NJ, 2005, 481–487.

Haus, H. A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–995 (1993).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking, J. Opt. Soc. Amer. E,  8, 2068–2076 (1991).
[CrossRef]

Heebner, J. E.

Heinecke, D.

A. Bartels, D. Heinecke, and S. A. Diddams, “10-GHz self-referenced optical frequency comb,” Science, vol.  326, p. 681 (2009).
[CrossRef] [PubMed]

Hollberg, L.

S. A. Diddams, M. Kirchner, T. Fortier, D. Braje, A. M. Weiner, and L. Hollberg, “Improved signal-to-noise ratio of 10 GHz microwave signals generated with a mode-filtered femtosecond laser frequency comb,” Opt. Express 17, 3331–3340 (2009).
[CrossRef] [PubMed]

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

Holman, K. W.

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
[CrossRef] [PubMed]

Holzwarth, R.

Horowitz, M.

Y. Parkhomenko, M. Horowitz, C. R. Menyuk, and T. F. Carruthers, “Theoretical study of an actively mode-locked fiber laser stabilized by an intra-cavity Fabry-Perot etalon: Linear regime,” J. Opt. Soc. Am. B. 24, 1793–1802 (2007).
[CrossRef]

Howe, D. A.

D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” Proc. 2005 Int. Freq. Control Symp. and Precise Time and Time Interval Sys. Mtg. IEEE: Piscataway, NJ, 2005, 481–487.

Hudson, D. D.

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
[CrossRef] [PubMed]

Ippen, E. P.

Ivanov, E. N.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

Jefferts, S. R.

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

Jones, D. J.

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
[CrossRef] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Jørgenson, C. G.

Kartner, F.

Kärtner, F. X.

Kimura, Y.

E. Yoshida, Y. Kimura, and M. Nakazawa, “Laser diode-pumped femtosecond Erbium doped fiber laser with a sub-ring cavity for repitition rate control,” Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

Kirchner, M.

Kiyan, R.

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

Knabe, K.

Knight, J. C.

Knox, W. H.

Langford, N.

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solitons in erbium fibre laser,” Electron. Lett. 27, 1257–1258 (1991).
[CrossRef]

Light, P. S.

Lim, J.

Love, J. D.

R. W. C. Vance and J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc. Opto-Electron. 141, 231–241 (1994).
[CrossRef]

Mansuripur, M.

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

Marcinkevicius, M.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

Martin, M. J.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

McFerran, J. J.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–995 (1993).
[CrossRef]

Mégret, P.

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

Menyuk, C. R.

Y. Parkhomenko, M. Horowitz, C. R. Menyuk, and T. F. Carruthers, “Theoretical study of an actively mode-locked fiber laser stabilized by an intra-cavity Fabry-Perot etalon: Linear regime,” J. Opt. Soc. Am. B. 24, 1793–1802 (2007).
[CrossRef]

A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, “Acoustic effect in passively mode-locked fiber ring lasers,” Opt. Lett. 20, 907–909 (1996).
[CrossRef]

Messerly, M. J.

Mollenauer, L.F.

Moloney, J. V.

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

Nakazawa, M.

E. Yoshida, Y. Kimura, and M. Nakazawa, “Laser diode-pumped femtosecond Erbium doped fiber laser with a sub-ring cavity for repitition rate control,” Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

Neely, W.

Newbury, N. R.

Nicholson, J. W.

Oates, C. W.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

Ozharar, S.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, “Harmonically modelocked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources,” J. Opt. A: Pure Appl. Opt. 11, 1–23 (2009)
[CrossRef]

Pal, P.

Panasenko, D.

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

Parkhomenko, Y.

Y. Parkhomenko, M. Horowitz, C. R. Menyuk, and T. F. Carruthers, “Theoretical study of an actively mode-locked fiber laser stabilized by an intra-cavity Fabry-Perot etalon: Linear regime,” J. Opt. Soc. Am. B. 24, 1793–1802 (2007).
[CrossRef]

Pax, P. H.

Peyghambarian, N.

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

Pilipetskii, A. N.

Polynkin, A.

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

Polynkin, P.

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

Pottiez, O.

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

Priest, R. G.

R. G. Priest, “Analysis of fiber interferometer utilizing 3× 3 fiber coupler,” Trans. Micro. Theory Tech. MTT-30, 1589–1591 (1982).
[CrossRef]

Quinlan, F.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, “Harmonically modelocked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources,” J. Opt. A: Pure Appl. Opt. 11, 1–23 (2009)
[CrossRef]

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Romann, A.

I. Hartl, A. Romann, and M. E. Fermann, “Passively mode locked GHz femtosecond Yb-fiber laser using an intra-cavity martinez compressor,” Proc. Conf. Lasers and Electro-Optics2011, Optical Society of America, paper CMD3.

Russell, P. St. J.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett., 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Schibli, T. R.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

Schröder, J.

Sheem, S. K.

S. K. Sheem, “Optical fiber interferometers with [3 × 3] directional couplers: Analysis,” J. Ap. Phys. 52, 3865–3872 (1981).
[CrossRef]

Shverdin, M. Y.

Sickler, J. W.

Siders, C. W.

Sridharan, A. K.

Stappaerts, E. A.

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Swann, W. C.

Sylvestre, T.

Tillman, K. A.

Udem, Th.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett., 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Vance, R. W. C.

R. W. C. Vance and J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc. Opto-Electron. 141, 231–241 (1994).
[CrossRef]

Vanholsbeeck, F.

Wadsworth, W. J.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett., 85, 2264–2267 (2000).
[CrossRef] [PubMed]

Wang, Y.

Washburn, B. R.

Weiner, A. M.

Wilken, T.

Wilpers, G.

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Yan, M. F.

Ye, J.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
[CrossRef] [PubMed]

S. T. Cundiff, J. Ye, and J. L. Hall, “Optical Frequency Synthesis Based on Mode-Locked Lasers,” Review of Scientific Instruments,  72, 3749–3771, (2001).
[CrossRef]

Yoshida, E.

E. Yoshida, Y. Kimura, and M. Nakazawa, “Laser diode-pumped femtosecond Erbium doped fiber laser with a sub-ring cavity for repitition rate control,” Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

Yost, D. C.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

E. Yoshida, Y. Kimura, and M. Nakazawa, “Laser diode-pumped femtosecond Erbium doped fiber laser with a sub-ring cavity for repitition rate control,” Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

Electron. Lett. (2)

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solitons in erbium fibre laser,” Electron. Lett. 27, 1257–1258 (1991).
[CrossRef]

J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett. 41, 650–651 (2005).
[CrossRef]

IEE Proc. Opto-Electron. (1)

R. W. C. Vance and J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc. Opto-Electron. 141, 231–241 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

O. Pottiez, O. Deparis, R. Kiyan, M. Haelterman, P. Emplit, P. Mégret, and M. Blondel, “Supermode noise of harmonically mode-locked erbium fiber lasers with composite cavity,” IEEE J. Quantum Electron.,  38, 252–259 (2002).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–995 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

D. Panasenko, P. Polynkin, A. Polynkin, J. V. Moloney, M. Mansuripur, and N. Peyghambarian, “Er-Yb femtosecond ring fiber oscillator with 1.1-W average power and GHz repetition rates,” IEEE Photon. Technol. Lett. 18, 853–855 (2006).
[CrossRef]

J. Ap. Phys. (1)

S. K. Sheem, “Optical fiber interferometers with [3 × 3] directional couplers: Analysis,” J. Ap. Phys. 52, 3865–3872 (1981).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, “Harmonically modelocked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources,” J. Opt. A: Pure Appl. Opt. 11, 1–23 (2009)
[CrossRef]

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

J. Opt. Soc. Am. B. (2)

Y. Parkhomenko, M. Horowitz, C. R. Menyuk, and T. F. Carruthers, “Theoretical study of an actively mode-locked fiber laser stabilized by an intra-cavity Fabry-Perot etalon: Linear regime,” J. Opt. Soc. Am. B. 24, 1793–1802 (2007).
[CrossRef]

S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B. 27, B51–B62 (2010).
[CrossRef]

J. Opt. Soc. Amer. E (1)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking, J. Opt. Soc. Amer. E,  8, 2068–2076 (1991).
[CrossRef]

Nature Photon. (1)

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, M. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nature Photon. 2, 355–359 (2008).
[CrossRef]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 013902 (2008)
[CrossRef] [PubMed]

Proc. 2005 Int. Freq. Control Symp. and Precise Time and Time Interval Sys. Mtg. IEEE: Piscataway, NJ (1)

D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” Proc. 2005 Int. Freq. Control Symp. and Precise Time and Time Interval Sys. Mtg. IEEE: Piscataway, NJ, 2005, 481–487.

Rev. Sci. Instrum. (1)

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007).
[CrossRef] [PubMed]

Review of Scientific Instruments (1)

S. T. Cundiff, J. Ye, and J. L. Hall, “Optical Frequency Synthesis Based on Mode-Locked Lasers,” Review of Scientific Instruments,  72, 3749–3771, (2001).
[CrossRef]

Science (3)

S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of time and frequency at the outset of the 21st century,” Science 306, 1318–1324 (2004).
[CrossRef] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

A. Bartels, D. Heinecke, and S. A. Diddams, “10-GHz self-referenced optical frequency comb,” Science, vol.  326, p. 681 (2009).
[CrossRef] [PubMed]

Trans. Micro. Theory Tech. (1)

R. G. Priest, “Analysis of fiber interferometer utilizing 3× 3 fiber coupler,” Trans. Micro. Theory Tech. MTT-30, 1589–1591 (1982).
[CrossRef]

Other (4)

F. M. Gardner, Phaselock techniques3rd ed. (Wiley-Interscience, 2005).
[CrossRef]

I. Hartl, A. Romann, and M. E. Fermann, “Passively mode locked GHz femtosecond Yb-fiber laser using an intra-cavity martinez compressor,” Proc. Conf. Lasers and Electro-Optics2011, Optical Society of America, paper CMD3.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett., 85, 2264–2267 (2000).
[CrossRef] [PubMed]

At the time of this writing, companies that produce 3 × 3 fiber couplers include the Shenzhen Technology Company and Rayscience Optoelectronic Innovation.

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Figures (5)

Fig. 1
Fig. 1

Schematic illustration of the proposed multi-pulse carrier-envelope phase-locked laser.

Fig. 2
Fig. 2

The steady-state waveform for several values of the MZI time and phase offsets from their ideal values. The solid curves show the computational solution of the full CGLE, while the dashed curves show the approximate solution that we obtain by using a chirped hyperbolic secant waveform with the modified parameters given in Eq. (10). The powers are normalized to the peak power of the steady-state pulse that corresponds to a perfectly-matched MZI.

Fig. 3
Fig. 3

Buildup of stable pulses from noise. The MZI mismatch parameters are ΔT = 0.2τ and Δθ = 0.3 rad. The blue, green and red curves correspond to the waveform after 100, 300 and 1200 roundtrips, respectively. The case of n = 1200 is indistinguishable from the steady state solution. The powers are normalized to the peak power of the steady-state pulse that corresponds to a perfectly-matched MZI.

Fig. 4
Fig. 4

The value of the monitor signal mθ as a function of the phase mismatch Δθ and time mismatch ΔT parameters.

Fig. 5
Fig. 5

Evolution of the control signals (a) and the offsets in the laser parameters (b) after a large perturbation. The offsets in the laser parameters ΔTR, ΔθR, and Δθm are highly correlated to the control signals ΔTs, Δθs, and mθ, respectively.

Equations (16)

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

d A 1 dz = ia A 2 + ib A 3 , d A 2 dz = ia A 1 + ia A 3 , d A 3 dz = ib A 1 + ia A 3 ,
k a ( t k T s ) exp ( i θ s ) ,
a ˜ ± ( ω ) = 1 3 { 1 r 2 ] + r exp [ i ( ϕ s + η s ω ± 2 π / 3 ) ] } a ˜ ( ω ) .
p ± = 1 2 π W d ω | a ˜ ± ( ω ) | 2 = 1 3 + r 1 r 2 π W 3 d ω cos ( ϕ s + η s ω ± 2 π / 3 ) | a ˜ ( ω ) | 2 1 3 2 r 1 r 2 3 [ 1 2 ± 3 2 ( ϕ s + η s ω ) ] ,
ω = 1 2 π W d ω ω | a ˜ ( ω ) | 2
m θ = p p + 2 r 1 r 2 ϕ s ,
a ˜ c ( ω ) = 1 3 { 1 r 2 + r exp [ i ( ϕ s + η s ω ) ] } a ˜ ( ω ) A [ 1 + ρ 2 i ( ϕ s + η s ω ) 1 2 ρ 2 ( ϕ s + η s ω ) 2 ] a ˜ ( ω ) ,
A = 1 3 ( 1 r 2 + r ) , ρ 2 = r 1 r 2 + r .
T R a T + Δ t a t = ( g l ) a i 2 ( β D + i β g ) 2 a t 2 + i ( γ K i γ a ) | a | 2 a + i θ R a ,
l = l + 1 2 ρ 2 ( 1 ρ 2 ) β g ϕ s 2 β g + ρ 2 ( 1 ρ 2 ) η s 2 ,
β g = β g + ρ 2 ( 1 ρ 2 ) η s 2 ,
θ R = θ R 2 β D [ ρ 2 ( 1 ρ 2 ) η s ϕ s β g + ρ 2 ( 1 ρ 2 ) η s 2 ] 2 + ρ 2 ϕ s β g β g + ρ 2 ( 1 ρ 2 ) η s 2 + R ϕ s R θ M ,
Δ t = β D ρ 2 ( 1 ρ ) 2 η s ϕ s β g + ρ 2 ( 1 ρ 2 ) η s 2 + ρ 2 η s .
Δ ω = ρ 2 ( 1 ρ 2 ) η s ϕ s β g + ρ 2 ( 1 ρ 2 ) η s 2 .
θ R = Δ θ NL = N θ s ,
T R + Δ t = N T s ,

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