Abstract

We study the influence of noise in ultrashort-pulse trains in terms of the concepts of optical coherence theory. Specifically, we consider the effects of temporal partial coherence and the timing jitter, including their possible statistical dependence. Analytical expressions for the optical power spectrum, average intensity, mutual coherence function, and the cross-spectral density function of different partially coherent pulse trains affected by stationary noise are given. We show that the new frequencies appearing due to the presence of noise are always fully uncorrelated, unless they are separated by an integral multiple of the repetition rate of the pulses in the train. We also find that spikes appear in the optical power spectrum when the noise and timing jitter are correlated.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
    [CrossRef]
  2. G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2000).
  3. M. E. Fermann, A. Galvanauskas, and G. Sucha, eds., Ultrafast Lasers: Technology and Applications (Dekker, 2003).
  4. H. A. Haus and A. Mecozzi, "Noise of mode-locked lasers," IEEE J. Quantum Electron. 29, 983-996 (1993).
    [CrossRef]
  5. V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007 (2007).
    [CrossRef]
  6. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Structures for additive pulse mode locking," J. Opt. Soc. Am. B 8, 2068-2076 (1991).
    [CrossRef]
  7. 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]
  8. A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998).
    [CrossRef]
  9. D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B 39, 201-217 (1986).
    [CrossRef]
  10. M. Guina, N. Xiang, A. Vainionpää, O. G. Okhotnikov, T. Sajavaara, and J. Keinonen, "Self-starting stretched-pulse fiber laser mode locked and stabilized with slow and fast semiconductor saturable absorbers," Opt. Lett. 26, 1809-1811 (2001).
    [CrossRef]
  11. J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
    [CrossRef]
  12. H. Tsuchida, "Timing noise measurement of 320 GHz optical pulses using an improved optoelectronic mixer," Opt. Lett. 31, 628-630 (2006).
    [CrossRef] [PubMed]
  13. D. Eliyahu, R. A. Salvatore, and A. Yariv, "Noise characterization of a pulse train generated by actively mode-locked lasers," J. Opt. Soc. Am. B 13, 1619-1626 (1996).
    [CrossRef]
  14. S. T. Cundiff and J. Ye, "Colloquium: femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
    [CrossRef]
  15. A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
    [CrossRef] [PubMed]
  16. S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D 35, R43-R59 (2002).
    [CrossRef]
  17. F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).
  18. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Noise-induced linewidth in frequency combs," Opt. Lett. 31, 1875-1877 (2006).
    [CrossRef] [PubMed]
  19. D. W. Rush, G. L. Burdge, and P. T. Ho, "The linewidth of a modelocked semiconductor laser caused by spontaneous emission: experimental comparison to single-mode operation," IEEE J. Quantum Electron. QE-22, 2088-2091 (1986).
    [CrossRef]
  20. F. K. Fatemi, J. W. Lou, and T. F. Carruthers, "Frequency comb linewidth of an actively mode-locked fiber laser," Opt. Lett. 29, 944-946 (2004).
    [CrossRef] [PubMed]
  21. K. Haneda, M. Yoshida, M. Nakazawa, H. Yokohama, and Y. Ogawa, "Linewidth and relative intensity noise measurements of longitudinal modes in ultrahigh-speed mode-locked laser dioses," Opt. Lett. 30, 1000-1002 (2005).
    [CrossRef] [PubMed]
  22. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
    [CrossRef]
  23. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
    [CrossRef] [PubMed]
  24. Q. Lin, L. G. Wang, and S. Y. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
    [CrossRef]
  25. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, "Energy spectrum of a nonstationary ensemble of pulses," Opt. Lett. 29, 394-396 (2004).
    [CrossRef] [PubMed]
  26. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
    [CrossRef]
  27. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent plane-wave-type pulses," Opt. Lett. 30, 2973-2975 (2005).
    [CrossRef] [PubMed]
  28. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  29. M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153-171 (1997).
    [CrossRef]
  30. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
    [CrossRef]
  31. P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. QE-21, 1806-1813 (1985).
    [CrossRef]
  32. Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-666 (2006).
    [CrossRef]

2007 (1)

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007 (2007).
[CrossRef]

2006 (3)

2005 (4)

K. Haneda, M. Yoshida, M. Nakazawa, H. Yokohama, and Y. Ogawa, "Linewidth and relative intensity noise measurements of longitudinal modes in ultrahigh-speed mode-locked laser dioses," Opt. Lett. 30, 1000-1002 (2005).
[CrossRef] [PubMed]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent plane-wave-type pulses," Opt. Lett. 30, 2973-2975 (2005).
[CrossRef] [PubMed]

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

2004 (3)

2003 (3)

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

Q. Lin, L. G. Wang, and S. Y. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

S. T. Cundiff and J. Ye, "Colloquium: femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

2002 (2)

S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D 35, R43-R59 (2002).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

2001 (1)

2000 (2)

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

1998 (1)

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998).
[CrossRef]

1997 (2)

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]

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153-171 (1997).
[CrossRef]

1996 (1)

1993 (1)

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

1991 (1)

1986 (2)

D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B 39, 201-217 (1986).
[CrossRef]

D. W. Rush, G. L. Burdge, and P. T. Ho, "The linewidth of a modelocked semiconductor laser caused by spontaneous emission: experimental comparison to single-mode operation," IEEE J. Quantum Electron. QE-22, 2088-2091 (1986).
[CrossRef]

1985 (1)

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. QE-21, 1806-1813 (1985).
[CrossRef]

Ablowitz, M. J.

Agrawal, G. P.

S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, "Energy spectrum of a nonstationary ensemble of pulses," Opt. Lett. 29, 394-396 (2004).
[CrossRef] [PubMed]

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
[CrossRef]

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2000).

Andres, P.

Apolonski, A.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Bertolotti, M.

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153-171 (1997).
[CrossRef]

Burdge, G. L.

D. W. Rush, G. L. Burdge, and P. T. Ho, "The linewidth of a modelocked semiconductor laser caused by spontaneous emission: experimental comparison to single-mode operation," IEEE J. Quantum Electron. QE-22, 2088-2091 (1986).
[CrossRef]

Carruthers, T. F.

Cundiff, S. T.

M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Noise-induced linewidth in frequency combs," Opt. Lett. 31, 1875-1877 (2006).
[CrossRef] [PubMed]

S. T. Cundiff and J. Ye, "Colloquium: femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D 35, R43-R59 (2002).
[CrossRef]

Delfyett, P. J.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

Eliyahu, D.

Ell, R.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

Fanto, M. L.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

Fatemi, F. K.

Fermann, M. E.

M. E. Fermann, A. Galvanauskas, and G. Sucha, eds., Ultrafast Lasers: Technology and Applications (Dekker, 2003).

Ferrari, A.

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153-171 (1997).
[CrossRef]

Friberg, A. T.

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007 (2007).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Fujimoto, J. G.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

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

Galvanauskas, A.

M. E. Fermann, A. Galvanauskas, and G. Sucha, eds., Ultrafast Lasers: Technology and Applications (Dekker, 2003).

Guina, M.

Haneda, K.

Hansch, T. W.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Haus, H. A.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

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

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

Hayduk, M. J.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

Ho, P. T.

D. W. Rush, G. L. Burdge, and P. T. Ho, "The linewidth of a modelocked semiconductor laser caused by spontaneous emission: experimental comparison to single-mode operation," IEEE J. Quantum Electron. QE-22, 2088-2091 (1986).
[CrossRef]

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. QE-21, 1806-1813 (1985).
[CrossRef]

Holzwarth, R.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Ilan, B.

Ippen, E. P.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

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

Jiang, Z.

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-666 (2006).
[CrossRef]

Kärtner, F. X.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

Keinonen, J.

Krausz, F.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998).
[CrossRef]

Lajunen, H.

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007 (2007).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

Lancis, J.

Leaird, D. E.

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-666 (2006).
[CrossRef]

Lin, Q.

Q. Lin, L. G. Wang, and S. Y. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

Lou, J. W.

Malowicki, J. E.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Mecozzi, A.

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

Morgner, U.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

Nakazawa, M.

Ogawa, Y.

Okhotnikov, O. G.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Ponomarenko, S. A.

Poppe, A.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998).
[CrossRef]

Rush, D. W.

D. W. Rush, G. L. Burdge, and P. T. Ho, "The linewidth of a modelocked semiconductor laser caused by spontaneous emission: experimental comparison to single-mode operation," IEEE J. Quantum Electron. QE-22, 2088-2091 (1986).
[CrossRef]

Sajavaara, T.

Salvatore, R. A.

Schibli, T.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

Sereda, L.

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153-171 (1997).
[CrossRef]

Silvestre, E.

Spielmann, C.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998).
[CrossRef]

Sucha, G.

M. E. Fermann, A. Galvanauskas, and G. Sucha, eds., Ultrafast Lasers: Technology and Applications (Dekker, 2003).

Tempea, G.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Tervo, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

Torres-Company, V.

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007 (2007).
[CrossRef]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent plane-wave-type pulses," Opt. Lett. 30, 2973-2975 (2005).
[CrossRef] [PubMed]

Tsuchida, H.

Turunen, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Udem, T.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Vahimaa, P.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Vainionpää, A.

von der Linde, D.

D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B 39, 201-217 (1986).
[CrossRef]

Wang, L. G.

Q. Lin, L. G. Wang, and S. Y. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

Weiner, A. M.

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-666 (2006).
[CrossRef]

Wolf, E.

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Xiang, N.

Xu, L.

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998).
[CrossRef]

Yariv, A.

Ye, J.

S. T. Cundiff and J. Ye, "Colloquium: femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

Yokohama, H.

Yoshida, M.

Zhu, S. Y.

Q. Lin, L. G. Wang, and S. Y. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

Appl. Phys. B (1)

D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B 39, 201-217 (1986).
[CrossRef]

IEEE J. Quantum Electron. (4)

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

D. W. Rush, G. L. Burdge, and P. T. Ho, "The linewidth of a modelocked semiconductor laser caused by spontaneous emission: experimental comparison to single-mode operation," IEEE J. Quantum Electron. QE-22, 2088-2091 (1986).
[CrossRef]

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. QE-21, 1806-1813 (1985).
[CrossRef]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-666 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007 (2007).
[CrossRef]

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

J. Phys. D (1)

S. T. Cundiff, "Phase stabilization of ultrashort optical pulses," J. Phys. D 35, R43-R59 (2002).
[CrossRef]

JEOS A: Pure Appl. Opt. (1)

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153-171 (1997).
[CrossRef]

Opt. Commun. (3)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Q. Lin, L. G. Wang, and S. Y. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Phys. Rev. Lett. (1)

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hansch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

S. T. Cundiff and J. Ye, "Colloquium: femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

Top. Appl. Phys. (1)

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," Top. Appl. Phys. 95, 73-136 (2004).

Other (4)

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2000).

M. E. Fermann, A. Galvanauskas, and G. Sucha, eds., Ultrafast Lasers: Technology and Applications (Dekker, 2003).

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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 (7)

Fig. 1
Fig. 1

(a) Absolute value of the mutual coherence function, Γ ( t 1 , t 2 ) , of a train of Gaussian pulses of 10 ps rms intensity width, 15 GHz repetition rate, and ω 0 T Δ Φ C E = 0 , affected by multiplicative noise. The coherence time of the Gaussian-correlated noise Γ N ( τ ) is taken as t c n = 100 ps . Note that the average intensity of the train can be seen in the diagonal. (b) Degree of coherence γ ( t 1 , t 2 ) of the same pulse train.

Fig. 2
Fig. 2

(a) Spectra of partially coherent trains of Gaussian pulses of 3 ps rms intensity width, 25 GHz repetition rate, and ω 0 T Δ Φ C E = 0 . The coherence times of the Gaussian noise Γ N ( τ ) are t c n = 200 [solid (blue) curve], 100 [dashed (red) curve], and 50 ps [dashed-dotted (green) line]. (b) Close-up of one of the comb lines.

Fig. 3
Fig. 3

(a) Absolute value of the mutual coherence function, Γ ( t 1 , t 2 ) , of a train of Gaussian pulses of 10 ps rms intensity width, 15 GHz repetition rate, and ω 0 T Δ Φ C E = 0 , affected by uncorrelated multiplicative noise and timing jitter. The coherence times of the Gaussian noise and jitter correlation, Γ N ( τ ) and Γ J ( τ ) , are chosen to be t c n = 100 ps and t c j = 20 ps , respectively. The coefficient of the strength of the timing jitter is α = 0.01 . The average intensity of the pulse train can be seen in the diagonal. (b) Degree of coherence γ ( t 1 , t 2 ) of the same pulse train.

Fig. 4
Fig. 4

Spectra of trains of Gaussian pulses of 3 ps rms intensity width, 25 GHz repetition rate, and ω 0 T Δ Φ C E = 0 , affected by uncorrelated multiplicative noise and timing jitter. The coherence times of the Gaussian correlation functions Γ N ( τ ) and Γ J ( τ ) are t c n = 100 ps for the noise, and t c j = 40 [solid (blue) curve], 20 [dashed (red) curve], and 10 ps [dashed-dotted (green) curve] for the jitter. The coefficient specifying the strength of the timing jitter is α = 0.01 .

Fig. 5
Fig. 5

(a) Absolute value of the mutual coherence function, Γ ( t 1 , t 2 ) , of a train of Gaussian pulses of 10 ps rms intensity width, 15 GHz repetition rate, and ω 0 T Δ Φ C E = 0 , affected by correlated multiplicative noise and timing jitter. The coherence times of the Gaussian noise, jitter, and noise–jitter correlations, Γ N ( τ ) , Γ J ( τ ) , and Γ N J ( τ ) , are taken to be t c n = 100 ps , t c j = 20 ps , and t c n j = 80 ps , respectively. The coefficients determining the strengths of the timing jitter and the noise–jitter correlation are α = 0.01 and β = 0.05 . Note that the average intensity of the train can be seen in the diagonal. (b) Degree of coherence γ ( t 1 , t 2 ) of the same pulse train.

Fig. 6
Fig. 6

Average intensities of single pulses from the trains considered in Figs. 1, 3, 5. The solid (blue) curve represents the case of multiplicative noise (Fig. 1), the dashed (red) curve uncorrelated noise and timing jitter (Fig. 3), and the dashed–dotted (green) curve correlated noise and timing jitter (Fig. 5).

Fig. 7
Fig. 7

Right-hand-side part of the spectra of trains of Gaussian pulses of 3 ps rms intensity width, 25 GHz repetition rate, and ω 0 T Δ Φ C E = 0 , influenced by multiplicative noise [solid (blue) curve], uncorrelated noise and timing jitter [dashed (red) curve], and correlated noise and timing jitter [dashed–dotted (green) curve]. The coherence times of the Gaussian noise, jitter, and noise–jitter correlations, Γ N ( τ ) , Γ J ( τ ) , and Γ N J ( τ ) , are chosen to be t c n = 100 ps , t c j = 20 ps , and t c n j = 80 ps , respectively. The coefficients specifying the strengths of the timing jitter and the noise–jitter correlation are α = 0.01 and β = 0.005 .

Equations (40)

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

Γ ( t 1 , t 2 ) = U * ( t 1 ) U ( t 2 ) ,
γ ( t 1 , t 2 ) = Γ ( t 1 , t 2 ) I ¯ ( t 1 ) I ¯ ( t 2 ) ,
I ¯ ( t ) = Γ ( t , t ) = U ( t ) 2 .
W ( ω 1 , ω 2 ) = U ̃ * ( ω 1 ) U ̃ ( ω 2 ) ,
W ( ω 1 , ω 2 ) = 1 ( 2 π ) 2 Γ ( t 1 , t 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 ,
Γ ( t 1 , t 2 ) = W ( ω 1 , ω 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d ω 1 d ω 2 .
S ( ω ) = W ( ω , ω ) = U ̃ ( ω ) 2 .
I ( t 1 ) I ( t 2 ) = I ( t 1 ) I ( t 2 ) + Γ ( t 1 , t 2 ) 2 ,
Δ I ( t 1 ) Δ I ( t 2 ) I ¯ ( t 1 ) I ¯ ( t 2 ) = γ ( t 1 , t 2 ) 2 .
U ( t ) = exp ( i ω 0 t ) N ( t ) n = L L ψ ( t n T ) exp [ i ( n ω 0 T n Δ Φ CE ) ] .
Γ a ( t 1 , t 2 ) = Γ N ( τ ) M * ( t 1 ) M ( t 2 ) exp ( i ω 0 τ ) ,
M ( t ) = n = L L ψ ( t n T ) exp [ i ( n ω 0 T n Δ Φ CE ) ] .
I ¯ ( t ) = I a ( t ) = Γ a ( t , t ) = n = L L ψ ( t n T ) exp [ i ( n ω 0 T n Δ Φ CE ) ] 2 .
W a ( ω 1 , ω 2 ) = W N ( ω ) M ̃ * ( Ω 1 ω ) M ̃ ( Ω 2 ω ) d ω ,
W N ( ω ) = 1 2 π Γ N ( τ ) exp ( i ω τ ) d τ .
M ̃ ( ω ) = ψ ̃ ( ω ) n = L L exp ( i ω n T ) exp [ i ( n ω 0 T n Δ Φ CE ) ] ,
W a ( ω 1 , ω 2 ) = n = m = W N ( Ω 1 2 π n T Δ Φ CE T + ω 0 ) ψ ̃ * ( 2 π n T + Δ Φ CE T ω 0 ) ψ ̃ ( Ω 2 Ω 1 + 2 π n T + Δ Φ CE T ω 0 ) δ [ Ω 2 Ω 1 + 2 π ( n + m ) T ] .
S a ( ω ) = W N ( Ω ) M ̃ ( Ω ) 2 ,
S a ( ω ) = n = W N ( Ω 2 π n T Δ Φ C E T + ω 0 ) ψ ̃ ( 2 π n T + Δ Φ C E T ω 0 ) 2 .
U ( t ) = exp ( i ω 0 t ) N ( t ) n = L L ψ [ t n T J ( t ) T ] exp [ i ( n ω 0 T n Δ Φ C E ) ] ,
ψ [ t n T T J ( t ) ] ψ ( t n T ) T J ( t ) ψ ̇ ( t n T ) ,
U ( t ) = N ( t ) exp ( i ω 0 t ) [ M ( t ) T J ( t ) M ̇ ( t ) ] .
Γ b ( t 1 , t 2 ) = Γ a ( t 1 , t 2 ) + Γ j 1 ( t 1 , t 2 ) ,
Γ j 1 ( t 1 , t 2 ) = T 2 Γ N ( τ ) Γ J ( τ ) M ̇ * ( t 1 ) M ̇ ( t 2 ) exp ( i ω 0 τ ) ,
I b ( t ) = I a ( t ) + I j 1 ( t ) ,
S b ( ω ) = S a ( Ω ) + S j 1 ( Ω ) ,
S j 1 ( Ω ) = ( T Ω ) 2 M ̃ ( Ω ) 2 W J ( Ω ) W N ( Ω ) .
Γ c ( t 1 , t 2 ) = Γ b ( t 1 , t 2 ) + Γ j 2 ( t 1 , t 2 ) ,
Γ j 2 ( t 1 , t 2 ) = T 2 M ̇ * ( t 1 ) M ̇ ( t 2 ) [ Γ N J ( τ ) Γ N J * ( τ ) + Γ N J ( 0 ) 2 ] exp ( i ω 0 τ ) .
I c ( t ) = I b ( t ) + I j 2 ( t ) ,
S c ( ω ) = S b ( ω ) + S j 2 ( ω ) ,
S j 2 ( ω ) = ( T Ω ) 2 M ̃ ( Ω ) 2 { δ ( Ω ) Γ N J ( 0 ) 2 + W N J ( Ω ) W N J * ( Ω ) } ,
Γ ( t 1 , t 2 ) = M * ( t 1 ) M ( t 2 ) Γ N ( τ ) T M * ( t 1 ) M ̇ ( t 2 ) N * ( t 1 ) N ( t 2 ) J ( t 2 ) T M ̇ * ( t 1 ) M ( t 2 ) N * ( t 1 ) N ( t 2 ) J ( t 1 ) + T 2 M ̇ * ( t 1 ) M ̇ * ( t 2 ) N * ( t 1 ) N ( t 2 ) J ( t 1 ) J ( t 2 ) ,
N * ( t 1 ) N ( t 2 ) J ( t 2 ) = 0 ,
N * ( t 1 ) N ( t 2 ) J ( t 1 ) = 0 ,
N * ( t 1 ) N ( t 2 ) J ( t 1 ) J ( t 2 ) = N * ( t 1 ) N ( t 2 ) J ( t 1 ) J ( t 2 ) + N * ( t 1 ) J ( t 2 ) J ( t 1 ) N ( t 2 ) + N * ( t 1 ) J ( t 1 ) J ( t 2 ) N ( t 2 ) = Γ N ( τ ) Γ J ( τ ) + Γ N J ( τ ) Γ N J * ( τ ) + Γ N J ( 0 ) 2 ,
Γ N J ( t 1 , t 2 ) = ( N * ( t 1 ) N ( t 2 ) N * ( t 1 ) J ( t 2 ) J ( t 1 ) N ( t 2 ) J ( t 1 ) J ( t 2 ) ) = ( Γ N ( τ ) Γ N J ( τ ) Γ J N ( τ ) Γ J ( τ ) ) .
Γ N J ( τ ) = Γ J N * ( τ ) .
[ f 1 * ( t 1 ) f 1 ( t 2 ) Γ N ( τ ) + f 1 * ( t 1 ) f 2 ( t 2 ) Γ N J ( τ ) + f 2 * ( t 1 ) f 1 ( t 2 ) Γ J N ( τ ) + f 2 * ( t 1 ) f 2 ( t 2 ) Γ J ( τ ) ] d t 1 d t 2 0
[ W N ( ω ) W J ( ω ) ] 1 2 W N J ( ω ) .

Metrics