Abstract

We have investigated the dynamical behavior of various passively mode-locked solid-state lasers by measuring how a modulation of the pump power affects the output power. We show theoretically and experimentally how the damping of the relaxation oscillations is reduced and finally becomes zero when the pump power is reduced so that the threshold for Q-switched mode locking is approached. For the first time to our knowledge, this method provides important information on the stability of mode locking above the Q-switched mode-locking threshold. It is applicable to lasers that are mode locked with slow-saturable absorbers. The results helped to explain the cause of unexpectedly low Q-switching thresholds in two cases. Also we obtain some useful spectroscopic information.

© 2004 Optical Society of America

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  1. U. Keller, “Recent developments in compact ultrafast lasers,” Nature 424, 831–838 (2003).
    [CrossRef] [PubMed]
  2. U. Keller, D. A. B. Miller, G. D. Boyd, T. H. Chiu, J. F. Ferguson, and M. T. Asom, “Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an antiresonant semiconductor Fabry-Perot saturable absorber,” Opt. Lett. 17, 505–507 (1992).
    [CrossRef] [PubMed]
  3. U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
    [CrossRef]
  4. U. Keller, T. H. Chiu, and J. F. Ferguson, “Self-starting and self-Q-switching dynamics of a passively mode-locked Nd: YLF and Nd:YAG laser,” Opt. Lett. 18, 217–219 (1993).
    [CrossRef]
  5. F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
    [CrossRef]
  6. C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, “Q-switching stability limits of continuous-wave passive mode locking,” J. Opt. Soc. Am. B 16, 46–56 (1999).
    [CrossRef]
  7. G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
    [CrossRef]
  8. L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
    [CrossRef]
  9. G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
    [CrossRef]
  10. S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).
  11. D. Burns, M. Hetterich, A. I. Ferguson, E. Bente, M. D. Dawson, J. I. Davies, and S. W. Bland, “High-average-power (>20 W) Nd:YVO4 lasers mode locked by strain-compensated saturable Bragg reflectors,” J. Opt. Soc. Am. B 17, 919–926 (2000).
    [CrossRef]
  12. L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
    [CrossRef]
  13. L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
    [CrossRef]
  14. R. Fluck, G. Zhang, U. Keller, K. J. Weingarten, and M. Moser, “Diode-pumped passively mode-locked 1.3 μm Nd:YVO4 and Nd:YLF lasers by use of semiconductor saturable absorbers,” Opt. Lett. 21, 1378–1380 (1996).
    [CrossRef] [PubMed]
  15. A. L. Harmer, A. Linz, and D. R. Gabbe, “Fluorescence of Nd3+ in lithium yttrium fluoride,” J. Phys. Chem. Solids 30, 1483–1491 (1969).
    [CrossRef]
  16. R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
    [CrossRef]
  17. S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
    [CrossRef]
  18. E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
    [CrossRef]
  19. A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
    [CrossRef]
  20. T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, “Suppression of Q-switched mode locking and breakup into multiple pulses by inverse saturable absorption,” Appl. Phys. B 70, 41–49 (2000).
    [CrossRef]

2003 (3)

U. Keller, “Recent developments in compact ultrafast lasers,” Nature 424, 831–838 (2003).
[CrossRef] [PubMed]

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

2002 (2)

L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
[CrossRef]

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

2001 (1)

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

2000 (2)

D. Burns, M. Hetterich, A. I. Ferguson, E. Bente, M. D. Dawson, J. I. Davies, and S. W. Bland, “High-average-power (>20 W) Nd:YVO4 lasers mode locked by strain-compensated saturable Bragg reflectors,” J. Opt. Soc. Am. B 17, 919–926 (2000).
[CrossRef]

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, “Suppression of Q-switched mode locking and breakup into multiple pulses by inverse saturable absorption,” Appl. Phys. B 70, 41–49 (2000).
[CrossRef]

1999 (3)

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, “Q-switching stability limits of continuous-wave passive mode locking,” J. Opt. Soc. Am. B 16, 46–56 (1999).
[CrossRef]

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

1998 (2)

S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
[CrossRef]

1996 (2)

R. Fluck, G. Zhang, U. Keller, K. J. Weingarten, and M. Moser, “Diode-pumped passively mode-locked 1.3 μm Nd:YVO4 and Nd:YLF lasers by use of semiconductor saturable absorbers,” Opt. Lett. 21, 1378–1380 (1996).
[CrossRef] [PubMed]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

1995 (1)

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

1993 (1)

1992 (1)

1986 (1)

A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
[CrossRef]

1969 (1)

A. L. Harmer, A. Linz, and D. R. Gabbe, “Fluorescence of Nd3+ in lithium yttrium fluoride,” J. Phys. Chem. Solids 30, 1483–1491 (1969).
[CrossRef]

Asom, M. T.

Aus der Au, J.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Bente, E.

Bland, S. W.

Boyd, G. D.

Braun, B.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Brovelli, L. R.

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Brownell, M.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Burns, D.

Calasso, I.

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Chai, B. H. T.

L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
[CrossRef]

Chiu, T. H.

Crosby, P. A.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Davies, J. I.

Dawson, M. D.

Ferguson, A. I.

Ferguson, J. F.

Fluck, R.

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

R. Fluck, G. Zhang, U. Keller, K. J. Weingarten, and M. Moser, “Diode-pumped passively mode-locked 1.3 μm Nd:YVO4 and Nd:YLF lasers by use of semiconductor saturable absorbers,” Opt. Lett. 21, 1378–1380 (1996).
[CrossRef] [PubMed]

Fornasiero, L.

L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
[CrossRef]

Gabbe, D. R.

A. L. Harmer, A. Linz, and D. R. Gabbe, “Fluorescence of Nd3+ in lithium yttrium fluoride,” J. Phys. Chem. Solids 30, 1483–1491 (1969).
[CrossRef]

Gallmann, L.

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

Geronimo, G. D.

S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

Golding, P. S.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Grange, R.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Haiml, M.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Harder, C.

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

Harmer, A. L.

A. L. Harmer, A. Linz, and D. R. Gabbe, “Fluorescence of Nd3+ in lithium yttrium fluoride,” J. Phys. Chem. Solids 30, 1483–1491 (1969).
[CrossRef]

Hetterich, M.

Hönninger, C.

C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, “Q-switching stability limits of continuous-wave passive mode locking,” J. Opt. Soc. Am. B 16, 46–56 (1999).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Huber, G.

L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
[CrossRef]

Ippen, E. P.

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, “Suppression of Q-switched mode locking and breakup into multiple pulses by inverse saturable absorption,” Appl. Phys. B 70, 41–49 (2000).
[CrossRef]

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

Jensen, T.

L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
[CrossRef]

Ji, W.

A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
[CrossRef]

Joschko, M.

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

Jung, I. D.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Kamp, M.

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Kar, A. K.

A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
[CrossRef]

Kärtner, F. X.

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, “Suppression of Q-switched mode locking and breakup into multiple pulses by inverse saturable absorption,” Appl. Phys. B 70, 41–49 (2000).
[CrossRef]

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Keller, U.

U. Keller, “Recent developments in compact ultrafast lasers,” Nature 424, 831–838 (2003).
[CrossRef] [PubMed]

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
[CrossRef]

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, “Q-switching stability limits of continuous-wave passive mode locking,” J. Opt. Soc. Am. B 16, 46–56 (1999).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

R. Fluck, G. Zhang, U. Keller, K. J. Weingarten, and M. Moser, “Diode-pumped passively mode-locked 1.3 μm Nd:YVO4 and Nd:YLF lasers by use of semiconductor saturable absorbers,” Opt. Lett. 21, 1378–1380 (1996).
[CrossRef] [PubMed]

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

U. Keller, T. H. Chiu, and J. F. Ferguson, “Self-starting and self-Q-switching dynamics of a passively mode-locked Nd: YLF and Nd:YAG laser,” Opt. Lett. 18, 217–219 (1993).
[CrossRef]

U. Keller, D. A. B. Miller, G. D. Boyd, T. H. Chiu, J. F. Ferguson, and M. T. Asom, “Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an antiresonant semiconductor Fabry-Perot saturable absorber,” Opt. Lett. 17, 505–507 (1992).
[CrossRef] [PubMed]

A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
[CrossRef]

Kilburn, I. J.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Klimov, I.

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

Kolodziejski, L. A.

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

Koontz, E. M.

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

Kopf, D.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Krainer, L.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
[CrossRef]

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

Kück, S.

L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
[CrossRef]

Langlois, P.

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

Laporta, P.

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

Lecomte, S.

L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
[CrossRef]

Linz, A.

A. L. Harmer, A. Linz, and D. R. Gabbe, “Fluorescence of Nd3+ in lithium yttrium fluoride,” J. Phys. Chem. Solids 30, 1483–1491 (1969).
[CrossRef]

Matuschek, N.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Miller, D. A. B.

Morier-Genoud, F.

Moser, M.

Paschotta, R.

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
[CrossRef]

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, “Q-switching stability limits of continuous-wave passive mode locking,” J. Opt. Soc. Am. B 16, 46–56 (1999).
[CrossRef]

Schibli, T. R.

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, “Suppression of Q-switched mode locking and breakup into multiple pulses by inverse saturable absorption,” Appl. Phys. B 70, 41–49 (2000).
[CrossRef]

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

Smith, S. D.

A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
[CrossRef]

Spühler, G. J.

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

Svelto, O.

S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

Taccheo, S.

S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

Teisset, C. Y.

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

Thoen, E. R.

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, “Suppression of Q-switched mode locking and breakup into multiple pulses by inverse saturable absorption,” Appl. Phys. B 70, 41–49 (2000).
[CrossRef]

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

Walker, A. C.

A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
[CrossRef]

Weingarten, K. J.

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
[CrossRef]

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

R. Fluck, G. Zhang, U. Keller, K. J. Weingarten, and M. Moser, “Diode-pumped passively mode-locked 1.3 μm Nd:YVO4 and Nd:YLF lasers by use of semiconductor saturable absorbers,” Opt. Lett. 21, 1378–1380 (1996).
[CrossRef] [PubMed]

Zeller, S. C.

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

Zhang, G.

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

R. Fluck, G. Zhang, U. Keller, K. J. Weingarten, and M. Moser, “Diode-pumped passively mode-locked 1.3 μm Nd:YVO4 and Nd:YLF lasers by use of semiconductor saturable absorbers,” Opt. Lett. 21, 1378–1380 (1996).
[CrossRef] [PubMed]

Appl. Phys. B (5)

S. C. Zeller, L. Krainer, G. J. Spühler, K. J. Weingarten, R. Paschotta, and U. Keller, “Passively mode-locked 40-GHz Er:Yb:glass laser,” Appl. Phys. B 76, 1181–1182 (2003).

L. Fornasiero, S. Kück, T. Jensen, G. Huber, and B. H. T. Chai, “Excited state absorption and stimulated emission of Nd3+ in crystals. Part 2. YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B 67, 549–553 (1998).
[CrossRef]

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, “Suppression of Q-switched mode locking and breakup into multiple pulses by inverse saturable absorption,” Appl. Phys. B 70, 41–49 (2000).
[CrossRef]

Appl. Phys. Lett. (2)

E. R. Thoen, E. M. Koontz, M. Joschko, P. Langlois, T. R. Schibli, F. X. Kärtner, E. P. Ippen, and L. A. Kolodziejski, “Two-photon absorption in semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 74, 3927–3929 (1999).
[CrossRef]

A. C. Walker, A. K. Kar, W. Ji, U. Keller, and S. D. Smith, “All-optical power limiting of CO2 laser pulses using cascaded optical bistable elements,” Appl. Phys. Lett. 48, 683–685 (1986).
[CrossRef]

Electron. Lett. (3)

G. J. Spühler, L. Gallmann, R. Fluck, G. Zhang, L. R. Brovelli, C. Harder, P. Laporta, and U. Keller, “Passively model-locked diode-pumped erbium-ytterbium glass laser using a semiconductor saturable absorber mirror,” Electron. Lett. 35, 567–568 (1999).
[CrossRef]

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38, 225–227 (2002).
[CrossRef]

G. J. Spühler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Novel multi-wavelength source with 25-GHz channel spacing tunable over the C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. Krainer, R. Paschotta, S. Lecomte, M. Moser, K. J. Weingarten, and U. Keller, “Compact Nd:YVO4 lasers with pulse repetition rates up to 160 GHz,” IEEE J. Quantum Electron. 38, 1331–1338 (2002).
[CrossRef]

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

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Sel. Top. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

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

J. Phys. Chem. Solids (1)

A. L. Harmer, A. Linz, and D. R. Gabbe, “Fluorescence of Nd3+ in lithium yttrium fluoride,” J. Phys. Chem. Solids 30, 1483–1491 (1969).
[CrossRef]

Nature (1)

U. Keller, “Recent developments in compact ultrafast lasers,” Nature 424, 831–838 (2003).
[CrossRef] [PubMed]

Opt. Eng. (1)

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid-state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995).
[CrossRef]

Opt. Lett. (3)

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Figures (11)

Fig. 1
Fig. 1

(a) Modulus and (b) phase of a typical transfer function. The transfer function is completely characterized by the resonance frequency (frequency where the modulus of the transfer function has a peak), the peak width, and the peak height.

Fig. 2
Fig. 2

Nd:YVO4 laser cavity. The beam radius in the crystal is designed to be 80 µm; the beam radius on the SESAM is 140 µm.

Fig. 3
Fig. 3

Setup used for transfer-function measurements.

Fig. 4
Fig. 4

(a), (b) Transfer functions of the cw and (c), (d) the mode-locked Nd:YVO4 laser at the highest and lowest measured intracavity power. For clarity, the other curves are not shown here. (e)–(h) Parameters fres and ξ as functions of the intracavity power for the cw and the mode-locked laser.

Fig. 5
Fig. 5

Nd:YLF laser cavity. The beam radii in the crystal are designed to be 45 µm and 67 µm, in sagittal and horizontal direction, respectively, while the beam radius on the SESAM is 90 µm.

Fig. 6
Fig. 6

(a), (b) Transfer functions of the cw and (c), (d) the mode-locked Nd:YLF laser. (e)–(h) Parameters fres and ξ as functions of the intracavity power for the cw and the mode-locked laser.

Fig. 7
Fig. 7

61-MHz Er:Yb:glass laser cavity. The sagittal and the tangential beam radius in the crystal are calculated to be 26 µm and 40 µm, respectively. The beam radius on the SESAM is 67 µm.

Fig. 8
Fig. 8

(a)–(d) Modulus and phase of measured transfer functions of 61-MHz mode-locked Er:Yb:glass laser at two different intracavity powers. The low-frequency part was fitted with χYbEr(f) (solid curves). (e), (f) By removing χYbEr(f), we find the transfer functions χEr(f) of the erbium ions alone (see text for explanation). (g), (h) fres and ξ of the measured erbium transfer functions as functions of the intracavity power. The solid curves show ξ(P) and fres(P), taking into account additional nonlinear loss in SESAM. The dashed curves [not distinguishable from the solid curve in (g)] are plotted with the additional nonlinear loss reduced to the amount expected from TPA in the SESAM (see also Fig. 9). The ξ(P) corresponding to the dotted curve in Fig. 9 is not within the plotting area anymore.

Fig. 9
Fig. 9

Measured and fitted reflectivity curve of the SESAM used in the 61-MHz Er:Yb:glass laser. The measurement was done using 2.6-ps pulses. The rollover at high pulse fluences is clearly visible. The dotted curve shows the same reflectivity curve without the additional nonlinear loss (see the text); the dashed curve shows the reflectivity curve, including the expected amount of two-photon absorption.

Fig. 10
Fig. 10

Cavity of the 10-GHz Er:Yb:glass laser.

Fig. 11
Fig. 11

Measured transfer functions of the 10-GHz mode-locked Er:Yb:glass laser. The low-frequency part was fitted with χYbEr(f) (solid curves) allowing for an estimate of the energy-transfer coupling constant between the ytterbium and erbium ions.

Equations (37)

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

dPdt=g-l-qP(EP)TRP,
dgdt=g0-gτL+η(g)PPEsat,L-PEsat,Lg,
qP(EP)=ΔRS[1-exp(-S)],S=EPEsat,A.
ddt δPδg=αβ-γ-δPδg+ζ0δPP,
α=-P qPEP(EP),β=PTR,γ=gEsat,L,
=1τL+PEsat,L+PPEsat,L ηg(g),ζ=η(g)Esat,L.
fro=12π βγ-α-14(-α)21/2
τro=2-α.
χ(ω)δPoutδPP=Tocβζ-ω2+(-α)iω+βγ-α.
fres=12π βγ-α-12(-α)21/2.
ξf1/22-fres2=34π2(-α)βγ-α-14(-α)21/2=3π froτro
χˆ=12π Tocβζ(-α)βγ-α-14(-α)21/2=14πTocβζ τrofro.
|χ(f)|=χˆξ3f4-2 fres2f2+13ξ2+fres41/2,
arg χ(f)=-arctan f2fres4+13ξ21/2-fres21/2fres4+13ξ21/2-f2.
fres12π βγ=12π PgTREsat,L1/2.
frofres,
τro=3π froξ3π fresξ.
τP1.1Δfg gΔR,
dN2dtYbEr=CN1M2,
dM2dt=κ(M1)×PPVLhνP-M2τYb-CN1M2,
dPdt=g-l-qP(EP)TRP,
dgdt=g0-gτL-PEsat,Lg+νLνPC EYbEsat,L σemσtotN-Esat,LVLhνLg,
dEYbdt=κ(EYb)PP-EYbτYb-CEYbσemσtotN-Esat,LVLhνLg.
ddt δPδgδE=αβ0-γ-m230m32-m33δPδgδE+κ(E)00δPP,
=1τL+PEsat,L+CEYbVLhνP,
m23=CEsat,L νLνP σemσtotN-Esat,LVLhνLg,
m32=CEYb Esat,LVLhνL,
m33=1τYb+CσemσtotN-Esat,LVLhνLg-PP dκdE(EYb).
χ(ω)=δPoutδPP=δEYbδPP×δPoutδEYb=χYbEr(ω)×χEr(ω),
χYbEr(ω)=Tocβκm23m33βγ-α 11+i ωm33 ,
χEr(ω)=βγ-α-ω2+(-α)iω+βγ-α.
>αandm33>0.
fro=12π βγ-α-14(-α)21/2.
χYbEr(f)=A1+i ffco 
fco=m332π=12π 1τYb+CN1-PP dκdE(E)12π 1τYb+CN1,
N1=σemσtotN-Esat,LVLhνLg,
qnew(EP)=a×EP.

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