Abstract

The master equation of mode locking written in operator notation is supplemented with noise-source terms that conserve commutator brackets. The noise sources are associated with the reservoirs responsible for loss and gain. The output of a mode-locked laser with the least possible quantum noise is determined.

© 2000 Optical Society of America

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  1. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [CrossRef]
  2. H. A. Haus, “Measurement of amplitude noise in optical cavity masers,” Appl. Phys. Lett. 6, 85–87 (1965).
    [CrossRef]
  3. S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
    [CrossRef] [PubMed]
  4. F. Haake and M. Lewenstein, “Adiabatic expansion for the single-mode laser,” Phys. Rev. A 27, 1013–1021 (1983).
    [CrossRef]
  5. H. Haken, “Quantum fluctuations in the laser,” in Frontiers in Quantum Optics, E. Pike and S. Sarkas, eds. (Hilger, London, 1986), pp. 355–398.
  6. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, New York, 1992).
  7. L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
    [CrossRef]
  8. J. L. Vey and W. Elasser, “Semiclassical description of noise and generation of amplitude squeezed states with vertical cavity surface emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 1299–1304 (1997).
    [CrossRef]
  9. A. Jann and Y. Ben-Aryeh, “Quantum noise reduction in semiconductor lasers,” J. Opt. Soc. Am. B 13, 761–767 (1996).
    [CrossRef]
  10. Z. Yifu and X. Min, “Amplitude squeezing and a transition from lasing with inversion to lasing without inversion in a four level laser,” Phys. Rev. A 48, 3895–3899 (1993).
    [CrossRef]
  11. J. Arnaud, “Amplitude squeezing from spectral-hole burning: a semiclassical theory,” Phys. Rev. A 48, 2235–2245 (1993).
    [CrossRef] [PubMed]
  12. A. N. Oraevsky, “Quantum fluctuations and formation of coherency in lasers,” J. Opt. Soc. Am. B 5, 933–945 (1988).
    [CrossRef]
  13. F. Jeremie, J. Vey, and P. Gallion, “Optical corpuscular theory of semiconductor laser intensity noise and intensity squeezed light generation,” J. Opt. Soc. Am. B 14, 250–257 (1996).
    [CrossRef]
  14. M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
    [CrossRef] [PubMed]
  15. C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
    [CrossRef] [PubMed]
  16. M. O. Scully, G. Sussmann, and C. Benkert, “Quantum noise reduction via maser memory effects: theory and applications,” Phys. Rev. Lett. 60, 1014–1017 (1988).
    [CrossRef] [PubMed]
  17. H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1992).
    [CrossRef]
  18. H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599–1611 (1970).
    [CrossRef]
  19. H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
    [CrossRef]
  20. Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
    [CrossRef] [PubMed]
  21. D. J. Jones, K. Hall, H. A. Haus, and E. P. Ippen, “Asynchronous pulse-modulated optical fiber-ring buffer,” Opt. Lett. 23, 177–179 (1998).
    [CrossRef]
  22. H. A. Haus, “A theory of forced modelocking,” IEEE J. Quantum Electron. 11, 323–330 (1975).
    [CrossRef]
  23. Y. Chen, F. X. Kärtner, U. Morgner, S. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999).
    [CrossRef]
  24. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 368–392 (1990).
    [CrossRef]
  25. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
    [CrossRef] [PubMed]
  26. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
    [CrossRef] [PubMed]
  27. P. L. Hagelstein, “Application of a photon configuration-space model to soliton propagation in a fiber,” Phys. Rev. A 54, 2426–2438 (1996).
    [CrossRef] [PubMed]
  28. I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
    [CrossRef] [PubMed]
  29. S. Carter and P. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
    [CrossRef] [PubMed]
  30. J. M. Fini, P. Hagelstein, and H. A. Haus, “Agreement of stochastic soliton formalism with second quantized and configuration-space models,” Phys. Rev. A 57, 4842–4585 (1998).
    [CrossRef]
  31. J. P. Gordon, “Dispersive perturbations of the nonlinear Schrödinger equations,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  32. H. A. Haus, W. Wong, and F. I. Khatri, “Continuum generation by perturbation of solitons,” J. Opt. Soc. Am. B 14, 304–313 (1997).
    [CrossRef]
  33. H. A. Haus and C. Yu, “Soliton squeezing and the continuum,” J. Opt. Soc. Am. B 17, 618–628 (2000).
    [CrossRef]
  34. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  35. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
    [CrossRef]
  36. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect” Opt. Lett. 17, 31–33 (1992).
    [CrossRef] [PubMed]
  37. S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996).
    [CrossRef]
  38. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
    [CrossRef]

2000 (1)

1999 (1)

1998 (2)

D. J. Jones, K. Hall, H. A. Haus, and E. P. Ippen, “Asynchronous pulse-modulated optical fiber-ring buffer,” Opt. Lett. 23, 177–179 (1998).
[CrossRef]

J. M. Fini, P. Hagelstein, and H. A. Haus, “Agreement of stochastic soliton formalism with second quantized and configuration-space models,” Phys. Rev. A 57, 4842–4585 (1998).
[CrossRef]

1997 (2)

1996 (5)

1993 (5)

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

Z. Yifu and X. Min, “Amplitude squeezing and a transition from lasing with inversion to lasing without inversion in a four level laser,” Phys. Rev. A 48, 3895–3899 (1993).
[CrossRef]

J. Arnaud, “Amplitude squeezing from spectral-hole burning: a semiclassical theory,” Phys. Rev. A 48, 2235–2245 (1993).
[CrossRef] [PubMed]

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

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

1992 (4)

1991 (1)

S. Carter and P. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
[CrossRef] [PubMed]

1990 (3)

Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 368–392 (1990).
[CrossRef]

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
[CrossRef] [PubMed]

1989 (2)

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

1988 (2)

M. O. Scully, G. Sussmann, and C. Benkert, “Quantum noise reduction via maser memory effects: theory and applications,” Phys. Rev. Lett. 60, 1014–1017 (1988).
[CrossRef] [PubMed]

A. N. Oraevsky, “Quantum fluctuations and formation of coherency in lasers,” J. Opt. Soc. Am. B 5, 933–945 (1988).
[CrossRef]

1987 (1)

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

1986 (1)

1984 (1)

H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
[CrossRef]

1983 (1)

F. Haake and M. Lewenstein, “Adiabatic expansion for the single-mode laser,” Phys. Rev. A 27, 1013–1021 (1983).
[CrossRef]

1975 (1)

H. A. Haus, “A theory of forced modelocking,” IEEE J. Quantum Electron. 11, 323–330 (1975).
[CrossRef]

1970 (1)

H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599–1611 (1970).
[CrossRef]

1965 (1)

H. A. Haus, “Measurement of amplitude noise in optical cavity masers,” Appl. Phys. Lett. 6, 85–87 (1965).
[CrossRef]

1963 (1)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Arnaud, J.

J. Arnaud, “Amplitude squeezing from spectral-hole burning: a semiclassical theory,” Phys. Rev. A 48, 2235–2245 (1993).
[CrossRef] [PubMed]

Ben-Aryeh, Y.

Benkert, C.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
[CrossRef] [PubMed]

M. O. Scully, G. Sussmann, and C. Benkert, “Quantum noise reduction via maser memory effects: theory and applications,” Phys. Rev. Lett. 60, 1014–1017 (1988).
[CrossRef] [PubMed]

Bergou, J.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
[CrossRef] [PubMed]

Carter, S.

S. Carter and P. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
[CrossRef] [PubMed]

Chen, Y.

Chiao, R. Y.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

Cho, S.

Davidovich, L.

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[CrossRef]

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
[CrossRef] [PubMed]

Deutsch, I. H.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

Drummond, P.

S. Carter and P. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
[CrossRef] [PubMed]

Elasser, W.

Fabre, C.

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Fini, J. M.

J. M. Fini, P. Hagelstein, and H. A. Haus, “Agreement of stochastic soliton formalism with second quantized and configuration-space models,” Phys. Rev. A 57, 4842–4585 (1998).
[CrossRef]

Fujimoto, J. G.

Gallion, P.

Garrison, J. C.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, “Two-photon bound states: the diphoton bullet in dispersive self-focusing media,” Phys. Rev. A 47, 3330–3336 (1993).
[CrossRef] [PubMed]

Giacobino, E.

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Gordon, J. P.

Haake, F.

F. Haake and M. Lewenstein, “Adiabatic expansion for the single-mode laser,” Phys. Rev. A 27, 1013–1021 (1983).
[CrossRef]

Hagelstein, P.

J. M. Fini, P. Hagelstein, and H. A. Haus, “Agreement of stochastic soliton formalism with second quantized and configuration-space models,” Phys. Rev. A 57, 4842–4585 (1998).
[CrossRef]

Hagelstein, P. L.

P. L. Hagelstein, “Application of a photon configuration-space model to soliton propagation in a fiber,” Phys. Rev. A 54, 2426–2438 (1996).
[CrossRef] [PubMed]

Hall, K.

Hasegawa, A.

Haus, H. A.

H. A. Haus and C. Yu, “Soliton squeezing and the continuum,” J. Opt. Soc. Am. B 17, 618–628 (2000).
[CrossRef]

Y. Chen, F. X. Kärtner, U. Morgner, S. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999).
[CrossRef]

J. M. Fini, P. Hagelstein, and H. A. Haus, “Agreement of stochastic soliton formalism with second quantized and configuration-space models,” Phys. Rev. A 57, 4842–4585 (1998).
[CrossRef]

D. J. Jones, K. Hall, H. A. Haus, and E. P. Ippen, “Asynchronous pulse-modulated optical fiber-ring buffer,” Opt. Lett. 23, 177–179 (1998).
[CrossRef]

H. A. Haus, W. Wong, and F. I. Khatri, “Continuum generation by perturbation of solitons,” J. Opt. Soc. Am. B 14, 304–313 (1997).
[CrossRef]

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996).
[CrossRef]

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

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
[CrossRef]

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1992).
[CrossRef]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 368–392 (1990).
[CrossRef]

Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[CrossRef] [PubMed]

H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
[CrossRef]

H. A. Haus, “A theory of forced modelocking,” IEEE J. Quantum Electron. 11, 323–330 (1975).
[CrossRef]

H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599–1611 (1970).
[CrossRef]

H. A. Haus, “Measurement of amplitude noise in optical cavity masers,” Appl. Phys. Lett. 6, 85–87 (1965).
[CrossRef]

Hillery, M.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
[CrossRef] [PubMed]

Ippen, E. P.

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Jann, A.

Jeremie, F.

Jones, D. J.

Kärtner, F. X.

Khatri, F. I.

Kodama, Y.

Kolobov, M. I.

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

Lai, Y.

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
[CrossRef]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 368–392 (1990).
[CrossRef]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

Lewenstein, M.

F. Haake and M. Lewenstein, “Adiabatic expansion for the single-mode laser,” Phys. Rev. A 27, 1013–1021 (1983).
[CrossRef]

Machida, S.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Mecozzi, A.

Min, X.

Z. Yifu and X. Min, “Amplitude squeezing and a transition from lasing with inversion to lasing without inversion in a four level laser,” Phys. Rev. A 48, 3895–3899 (1993).
[CrossRef]

Moores, J. D.

Morgner, U.

Mullen, J. A.

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1992).
[CrossRef]

Namiki, S.

Oraevsky, A. N.

Orszag, M.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
[CrossRef] [PubMed]

Scully, M. O.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, “Role of pumping statistics in laser dynamics: quantum Langevin approach,” Phys. Rev. A 41, 2756–2765 (1990).
[CrossRef] [PubMed]

M. O. Scully, G. Sussmann, and C. Benkert, “Quantum noise reduction via maser memory effects: theory and applications,” Phys. Rev. Lett. 60, 1014–1017 (1988).
[CrossRef] [PubMed]

Sussmann, G.

M. O. Scully, G. Sussmann, and C. Benkert, “Quantum noise reduction via maser memory effects: theory and applications,” Phys. Rev. Lett. 60, 1014–1017 (1988).
[CrossRef] [PubMed]

Vey, J.

Vey, J. L.

Wong, W.

Yamamoto, Y.

Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984).
[CrossRef]

Yifu, Z.

Z. Yifu and X. Min, “Amplitude squeezing and a transition from lasing with inversion to lasing without inversion in a four level laser,” Phys. Rev. A 48, 3895–3899 (1993).
[CrossRef]

Yu, C.

Yu, C. X.

Appl. Phys. Lett. (1)

H. A. Haus, “Measurement of amplitude noise in optical cavity masers,” Appl. Phys. Lett. 6, 85–87 (1965).
[CrossRef]

IEEE J. Quantum Electron. (2)

H. A. Haus, “A theory of forced modelocking,” IEEE J. Quantum Electron. 11, 323–330 (1975).
[CrossRef]

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

J. Opt. Soc. Am. B (11)

H. A. Haus and C. Yu, “Soliton squeezing and the continuum,” J. Opt. Soc. Am. B 17, 618–628 (2000).
[CrossRef]

A. N. Oraevsky, “Quantum fluctuations and formation of coherency in lasers,” J. Opt. Soc. Am. B 5, 933–945 (1988).
[CrossRef]

J. P. Gordon, “Dispersive perturbations of the nonlinear Schrödinger equations,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[CrossRef]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
[CrossRef]

A. Jann and Y. Ben-Aryeh, “Quantum noise reduction in semiconductor lasers,” J. Opt. Soc. Am. B 13, 761–767 (1996).
[CrossRef]

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996).
[CrossRef]

H. A. Haus, W. Wong, and F. I. Khatri, “Continuum generation by perturbation of solitons,” J. Opt. Soc. Am. B 14, 304–313 (1997).
[CrossRef]

J. L. Vey and W. Elasser, “Semiclassical description of noise and generation of amplitude squeezed states with vertical cavity surface emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 1299–1304 (1997).
[CrossRef]

Y. Chen, F. X. Kärtner, U. Morgner, S. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999).
[CrossRef]

F. Jeremie, J. Vey, and P. Gallion, “Optical corpuscular theory of semiconductor laser intensity noise and intensity squeezed light generation,” J. Opt. Soc. Am. B 14, 250–257 (1996).
[CrossRef]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 368–392 (1990).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (2)

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1992).
[CrossRef]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Phys. Rev. A (12)

M. I. Kolobov, L. Davidovich, E. Giacobino, and C. Fabre, “Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources,” Phys. Rev. A 47, 1431–1446 (1993).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of a laser resonator containing a gain element and a loss element, (b) fiber ring resonator.

Fig. 2
Fig. 2

Two ways of producing additive-pulse mode locking: (a) with a nonlinear Mach–Zehnder interferometer, (b) by nonlinear rotation of polarization.

Fig. 3
Fig. 3

Power spectra of (a) amplitude normalized to shot noise and (b) frequency fluctuations and time evolutions of (c) the phase and (d) the position for several filter bandwidths. ξ=257 fs, γ/δ=0.015, TR=24 ns, TR D=0.0384 ps2, τe=20, τo=100. Filter bandwidths: Δλf=15 nm (solid curves), Δλf=35 nm (dotted curves), Δλf=25 nm (dashed–dotted curves), Δλf=40 nm (dashed curves).

Fig. 4
Fig. 4

Power spectra of (a) amplitude normalized to shot noise and (b) frequency fluctuations and time evolutions of (c) the phase and (d) the position for several SAM’s. ξ=257 fs, Ωf=35 nm, TR=24 ns, TRD=0.0384 ps2, τe=20, τo=100. γ/δ=0.01 (solid curves), γ/δ=0.015 (dotted curves), γ/δ=0.02 (dashed–dotted curves), γ/δ=0.028 (dashed curves).

Fig. 5
Fig. 5

Power spectra of (a) amplitude normalized to shot noise and (b) frequency fluctuations and time evolutions of (c) the phase and (d) the position for several dispersions. ξ=257 fs, γ/δ=0.015, Ωf=35 nm, TR=24 ns, τe=20, τo=100. TR D=0.0384 ps2 (solid curves), TR D=0.0288 ps2 (dotted curves), TR D=0.0192 ps2 (dashed–dotted curves), TR D=0.0096 ps2 (dashed curves).

Equations (135)

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ddtAˆν=-iων+1τo-1τg+1τeAˆν+Nˆoν+Nˆgν+2/τeSˆ,
ddtAˆν=-iων+1τoAˆν+Nˆoν.
ddt[Aˆν, Aˆμ]=-2τo[Aˆν, Aˆμ]+{[Aˆν, Nˆoμ]+[Nˆoν, Aˆμ]}.
[Aˆν, Aˆμ]=δνμ.
ddt[Aˆν, Aˆμ]=-2τo[Aˆν, Aˆμ]+Δt2{[Nˆoν, Nˆoμ]+[Nˆoν, Nˆoμ]}=0.
[Nˆoν(t), Nˆoμ(t)]=2τoδ(t-t)δνμ,
[Sˆν(t), Sˆμ(t)]=δ(t-t)δνμ,
[Nˆgν(t), Nˆgμ(t)]=-2τgδ(t-t)δνμ.
Rˆν=-Sˆν+2/τeAˆν.
Aˆν(ω)=2/τeSˆν(ω)+Nˆν(ω)-i(ω-ων)+1/τe+1/τo-1/τg,
[Sˆν(ω), Sˆμ(ω)]=12πδ(ω-ω)δνμ,
[Nˆoν(ω), Nˆoμ(ω)]=12π2τoδ(ω-ω)δνμ,
[Nˆgν(ω), Nˆgμ(ω)]=-12π2τgδ(ω-ω)δνμ.
[Aˆν(ω), Aˆμ(ω)]=(2/τe)[Sˆ(ω), Sˆ(ω)]+[Nˆν(ω), Nˆμ(ω)][-i(ω-ων)+1/τe+1/τo-1/τg][i(ω-ωμ)+1/τe+1/τo-1/τg]=2/τe+2/τo-2/τg[(ω-ων)2+(1/τe+1/τo-1/τg)2]12πδ(ω-ω)δνμ.
Rˆν(ω)=-Sˆν(ω)+2/τeAˆν(ω)=i(ω-ων)+1/τe-1/τo-1/τg-i(ω-ων)+1/τe+1/τo-1/τgSˆν(ω)+2τeNˆν(ω)-i(ω-ων)+1/τe+1/τo-1/τg.
[Rˆν(ω), Rˆμ(ω)]
=12πδ(ω-ω)×δνμ(ω-ων)2+(-1/τe+1/τo-1/τg)2(ω-ων)2+(1/τe+1/τo-1/τg)2×2τe2/τo-2τg(ω-ων)2+(1/τe+1/τo-1/τg)2
=12πδ(ω-ω)δνμ.
 dω  dω[Aˆν(ω), Aˆμ(ω)]= dω  dω(2/τe)[Sˆν(ω), Sˆμ(ω)]+[Nˆν(ω), Nˆμ(ω)][-i(ω-ων)+1/τe+1/τo-1/τg][i(ω-ωμ)+1/τe+1/τo-1/τg]=12π dω2/τe+2/τo-2/τg[(ω-ων)2+(1/τe+1/τo-1/τg)2]δνμ=δνμ.
limT1T dtAˆ(t)Aˆ(t)
=limT1T dt  dω  dω exp[i(ω-ω)t]Aˆ(ω)Aˆ(ω)=limT2πT dω  dωδ(ω-ω)Aˆ(ω)Aˆ(ω)
=limT2πT dωAˆ(ω)Aˆ(ω).
Sˆν(ω)Sˆμ(ω)=Nˆoν(ω)Nˆoμ(ω)=0,
Nˆgν(ω)Nˆgμ(ω)=12π2τgδ(ω-ω)δμν.
limT2πT dωAˆν(ω)Aˆν(ω)
=12π dω2/τg(ω-ων)2+(1/τe+1/τo-1/τg)2
=1/τg(1/τe+1/τo-1/τg).
ων limT2πT dωRˆν(ω)Rˆν(ω)
=ων2/(τeτg)(1/τe+1/τo-1/τg).
Nˆg=Nˆu+Nˆl
[Nˆg, Nˆg]=-2τu-2τlδ(t-t)=-2τgδ(t-t),
NuNu.
2/τu2/τu-2/τl=nunu-neχ,
1τfν=1τfoν2ΔΩ2Ωf2=1τfovg2ν2Δβ2Ωf2,
aˆ(β)=(L/2π)Aˆν.
[aˆ(β), aˆ(β)]=L(2π)2[Aˆm, Aˆn]=L(2π)2δmn12πδ(β-β),
taˆ(δβ, t)=-1τo+1τe-1τgaˆ(δβ, t)+1τfovg2Ωf2δβ2aˆ(δβ, t)+nˆ(δβ, t)+2/τesˆ(δβ, t),
nˆ(δβ)=limLL2π(Nˆo+Nˆf+Nˆg).
[nˆf(δβ, t), nˆf(δβ, t)]=2τfovg2Ωf2δβ212π×δ(δβ-δβ)δ(t-t).
[Sˆ(δβ, t), Sˆ(δβ, t)]=12πδ(δβ-δβ)δ(t-t).
aˆ(x)=-Δβ/2Δβ/2 dδβ exp(iδβx)aˆ(δβ),
[aˆ(δβ), aˆ(δβ)]=12πδ(δβ-δβ)
[aˆ(x), aˆ(x)]
=-Δβ/2Δβ/2 dδβ -Δβ/2Δβ/2 dδβ[aˆ(δβ), aˆ(δβ)]×exp[i(δβx-δβx)]=12π-Δβ/2Δβ/2 dδβ -Δβ/2Δβ/2 dδβδ(δβ-δβ)×exp[i(δβx-δβx)]=12π-Δβ/2Δβ/2 dβ exp[iδβ(x-x)]=Δβ2πsin[Δβ(x-x)/2]Δβ(x-x)/2=N(x-x).
aˆ(δβ)=12π- dxaˆ(x)exp(-iδβx).
sˆ(x)=0 dδβ exp(iδβx)sˆ(δβ)=ν2πLL2πSˆν exp[i(βν-βo)x].
[sˆ(x),sˆ(x)]
=exp[-iβo(x-x)]1Lν,μ[Sˆν,Sˆμ]exp[i(βνx-βμx)]=exp[-iβo(x-x)]1Lν,μ δνμ exp[i(βνx-βμx)]×δ(t-t)δ(x-x)δ(t-t).
rˆ(x)=-sˆ(x)+2/τeaˆ(x).
taˆ(x, t)=-1τe+1τo-1τgaˆ(x, t)+vg2Ωf2τfo2x2aˆ(x, t)+nˆ(x, t)+2/τesˆ(x, t).
[nˆo(x, t), nˆo(x, t)]=2τoδ(x-x)δ(t-t),
[nˆg(x, t), nˆg(x, t)]=-2τgδ(x-x)δ(t-t).
taˆ=iD2x2aˆ+iδaˆaˆaˆ,
Hˆ=Daˆxaˆx-δ2aˆaˆaˆaˆ.
taˆ=-1τaˆ+1τfovg2Ωf2+iD2x2aˆ+iδaˆaˆaˆ+nˆ+2τe sˆ,
1τ1τe+1τo-1τg.
absorberaction(-la+γaˆaˆ)aˆSˆaˆ.
aˆ(x)=ao(x)+Δaˆ(x).
[Δaˆ(x), Δaˆ(x)]=δ(x-x).
signalresponse=(-la+γ|ao|2)ao
[(-la+2γ|ao|2)Δaˆ+γao2Δaˆ]+nˆa
PΔaˆ+QΔaˆ+nˆa.
(1+P)Δaˆ+QΔaˆ+nˆa.
[nˆa(x), nˆa(x)]=-{P+P*+|P|2-|Q|2}×δ(x-x)δ(t-t).
[nˆa(x), nˆa(x)]=-2Pδ(x-x)δ(t-t)=2(la-2γ|ao|2)δ(x-x)δ(t-t).
taˆ=-1τaˆ+vg2Ωf2τfo2+iD2x2aˆ+iδaˆaˆaˆ+Sˆ(aˆ,aˆ)aˆ+nˆ,
1τ=1τe+1τo-1τg+la
tao=-1τao+vg2Ωf2τfo+iD2t2ao+(γ+iδ)|ao|2ao.
ao(x, t)=Ao exp(iθ)sechxξ(1+iβ) expi|Ao|22t.
taˆ(x, t)=iD2x2aˆ(x, t)+iδaˆ(x, t)aˆ(x, t)aˆ(x, t).
ao(t, x)=Ao sechx-xo-2Dpotξ×expiδAo22t-Dpo2t+pox+θ,
Aoξ=2D/δ.
no= dx|ao2(t, x)|=2Ao2ξ.
ao(t, x)=Ao sechxξexpiδAo22t.
 dxvg2Ωf2τfao*2x2ao+c.c.=-43Ao2vg2Ωf2τfoξ2.
4ξ/τ+43vg2Ωf2τfoξ-83γξAo2=0.
vg2Ωf2τfoξ2=2γAo2.
tΔaˆ=-1τΔaˆ+1τfovg2Ωf2+iD2x2Δaˆ+2(γ+iδ)|ao2|Δaˆ+(γ+iδ)ao2Δaˆ+nˆ
Δaˆ=Δaˆsol+Δaˆcont.
Δaˆsol=[ΔAˆ1(t)f1(x)+ΔAˆ2(t)f2(x)+ΔXˆ(t)fx(x)+ΔPˆ(t)fp(x)]exp(iδAo2t/2),
f1(x)=1ξ1-xξtanhxξsechxξ.
f2(x)=iξsech(x/ξ).
fX(x)=1ξtanh(x/ξ)sech(x/ξ),
fP(x)=iξxξsech(x/ξ).
Re  dxf̲Q*(x)f̲R(x)=δQR,
f̲1(x)=1ξsech(x/ξ),
f̲2(x)=iξ1-xξtanh(x/ξ)sech(x/ξ),
f̲X(x)=1ξxξsech(x/ξ),
f̲P(x)=iξtanh(x/ξ)sech(x/ξ).
tΔaˆ=iD2x2Δaˆ+2iδ|ao|2Δaˆ+iδao2Δaˆ.
ddtΔAˆ1=0,
ddtΔAˆ2=-µΔA1,
ddtΔXˆ=-µΔPˆ,
ddtΔPˆ=0,
ΔAˆ1(0)=12 dxf̲1*(x)Δaˆ(0, x)+h.c.= df̲1*(x)Δaˆ1(0, x).
[ΔAˆ1(0),ΔAˆ2(0)]
= dx|f̲1*(x)| dx|f̲2*(x)|[Δaˆ1(0, x)Δaˆ2(0, x)]=i2 dx|f̲2*(x)| dx|f̲2*(x)|δ(x-x)=i/2.
[ΔXˆ(0), ΔPˆ(0)]=i/2.
[Δxˆ,noΔpˆ]=i.
ddtΔAˆ1=-1τ1ΔAˆ+Nˆ1,
ddtΔAˆ2=µΔAˆ1+Nˆ2,
ddtΔXˆ=-µΔPˆ+NˆX,
ddtΔPˆ=-1τPΔPˆ+NˆP,
NˆQ=12 dxf̲Q*(x)nˆ(t, x)+h.c.,Q=1, 2, X, P.
 df̲1*(x)1τfovg2Ωf22x2f1(x)ΔAˆ1
=1τfovg2Ωf2 dx2f̲1*(x)x2f1(x)ΔAˆ1.
ξ2  dx2f̲1*(x)x2f1(x)
= dy[1-2 sech2(y)][1-y tanh(y)]sech2(y)=-1.
3γAo2  df̲1*(x)sech2(x)f1(x)ΔAˆ1
=3γAo2ΔAˆ1  dy[1-y tanh(y)]sech4(x)
=3γAo2ΔAˆ1.
1τ1=1τ+vg2Ωf2τfoξ2-3γAo2.
vg2Ωf2τfoξ2>72γAo2.
 dxf̲P*(x)1τfovg2Ωf22x2fP(x)ΔPˆ=1τfovg2Ωf2ξ2ΔPˆ  dy tanh(y)sech(y)2y2[y sech(y)]=-1τfovg2Ωf2ξ2ΔPˆ-1τPΔPˆ.
NˆQ2= dx|f̲Q(x)|nˆ(j)(x, t) dx|f̲Q(x)|nˆ(j)(x, t)=NQδ(t-t),
NˆP2=1(2π)2 dβ|f̲P(β, t)|nˆf(j)(β)× dβ|f̲P(β, t)|nˆf(j)(β)=NPδ(t-t).
ΔQˆ(t)=0t dthQ(t-t)NˆQ(t)+ΔQ(0).
(1/2){ΔPˆ(t)ΔPˆ(t)+ΔPˆ(t)ΔPˆ(t)}
=0t dthP(t-t)0t dthP(t-t)NˆP(t)NˆP(t)+ΔPˆ(0)ΔPˆ(0)hP(t)hP(t)=NPτp2{exp[-(t-t)/τP]-exp[-(t+t)/τp]}+ΔPˆ2(0)exp[-(t+t)/τp]t>t,=NPτp2{exp[-(t-t)/τp]-exp[-(t+t)/τp]}+ΔPˆ2(0)exp[-(t+t)/τp]t<t.
(1/2){ΔPˆ(t)ΔPˆ(t)+ΔPˆ(t)ΔPˆ(t)}
=NPτP2exp(-|t-t|/τP)
|ΔXˆ2(t)|=0t dthX(t-t)0t dthX(t-t)×NˆX(t)NˆX(t)+ΔXˆ2(0)hX2(t)+μ2 0t dt 0t dt(1/2)×{ΔPˆ(t)ΔP(t)+ΔPˆ(t)ΔPˆ(t)}.
ΔXˆ2(t)=NXt+ΔXˆ2(0)+μ2NPτP3tτP-12[1-exp(-2t/τP)]+ΔPˆ2(0)1-exp(-t/τP)1/τP2.
[ΔXˆ(t)-ΔXˆ(t)]2
=ΔXˆ2(t)+ΔXˆ2(t)-ΔXˆ(t)ΔXˆ(t)-ΔXˆ(t)ΔX(t).
limt,t[ΔXˆ(t)-ΔXˆ(t)]2
=NX|t-t|+μ2NPτP2|t-t|.
Δrˆ(x)=-sˆ(x)+2/τeΔaˆ(x).
ΔrˆQ=-sˆQ+2/τeΔQˆ,
[ΔrˆQ,ΔrˆR]=[sˆQ,sˆR]-2τe [sˆQ,ΔRˆ]-2τe [ΔQˆ,sˆR]+2τe[ΔQˆ,ΔRˆ]=i2-2τe [sˆQ,ΔRˆ]-2τe [ΔQˆ,sˆR]+2τei2.
[ΔsˆQ,ΔsˆR]=[ΔQˆ,ΔRˆ]=i/2.
[ΔQˆ,sˆR]=-t dt  dxf̲Q*(x)2τe sˆ(1)(t, x), dxf̲R*(x)sˆ(2)(t, x)=i22τe -t dt dxf̲Q*(x) dxf̲R*(x)× δ(x-x)δ(t-t)=i22τe -t dxf̲Q*(x)fR*(x)δ(t-t)=i22τe -t dtδ(t-t)=i22τe12.
 dxf̲Q*(x)f̲R*(x)=1.
ΔrˆQ2=sˆQ2-2τe sˆQΔQˆ-2τe ΔQˆsˆQ+2τeΔQ2.
ΔrˆQ2=sˆQ2+2τeΔQˆ2,

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