Abstract

A completely analytical approach to analysis of energy-scalable ultrashort-pulse oscillators operating in both normal- and anomalous-dispersion regimes is developed. The theory, based on the approximated solutions of the generalized complex nonlinear Ginzburg-Landau equation allows the problem to be reduced to a purely algebraic model, so that the oscillator characteristics are easy to trace and are completely characterized by only two parameters defining the so-called master diagram of the pulse energy scalability. The proposed theory covers all types of energy-scalable oscillators: all-normal-dispersion fiber, chirped-pulse and thin-disk solid-state ones and is validated by numerical simulations.

© 2010 OSA

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  4. J. Neuhaus, D. Bauer, J. Zhang, A. Killi, J. Kleinbauer, M. Kumkar, S. Weiler, M. Guina, D. H. Sutter, and Th. Dekorsy, “Subpicosecond thin-disk laser oscillator with pulse energies of up to 25.9 microjoules by use of an active multipass geometry,” Opt. Express 16, 20530–20539 (2008).
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  5. G. Palmer, M. Schultze, M. Siegel, M. Emons, U. Bünting, and U. Morgner, “Passively mode-locked Yb:KLu(WO4)2 thin-disk oscillator operated in the positive and negative dispersion regime,” Opt. Lett. 33, 1608–1610 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  39. A. K. Komarov and K. P. Komarov, “Pulse splitting in a passive mode-locked laser,” Optics Commun. 183, 265–270 (2000).
    [CrossRef]
  40. S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
    [CrossRef]
  41. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18, 1732–1741 (2001).
    [CrossRef]
  42. O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007).
    [CrossRef]
  43. A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140–148 (2008).
    [CrossRef]
  44. S. Lefrançois, K. Kieu, Y. Deng, J. D. Kafka, and F. W. Wise, “Scaling of dissipative soliton fiber lasers to megawatt peak powers by use of large-area photonic crystal fiber,” Opt. Lett. 35, 1569–1571 (2010).
    [CrossRef] [PubMed]
  45. M. Siegel, G. Palmer, M. Emons, M. Schutze, A. Ruehl, and U. Morgner, “Pulsing dynamics in Ytterbium based chirped-pulse oscillators,” Opt. Express 16, 14314–14320 (2008).
    [CrossRef] [PubMed]

2010 (3)

2009 (4)

B. G. Bale, S. Boscolo, and S. K. Turitsyn, “Dissipative dispersion-managed solitons in mode-locked lasers,” Opt. Lett. 34, 3286–3288 (2009).
[CrossRef] [PubMed]

V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009).
[CrossRef]

V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: A unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[CrossRef]

2008 (12)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008).
[CrossRef]

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

T. Südmeyer, S. V. Marchese, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Photonics 2, 599–604 (2008).
[CrossRef]

S. V. Marchese, C. R. E. Baer, A. G. Engqvist, S. Hashimoto, D. J. H. C. Maas, M. Golling, T. Südmeyer, and U. Keller, “Femtosecond thin disk laser oscillator with pulse energy beyond the 10-microjoule level,” Opt. Express 16, 6397–6407 (2008).
[CrossRef] [PubMed]

J. Neuhaus, D. Bauer, J. Zhang, A. Killi, J. Kleinbauer, M. Kumkar, S. Weiler, M. Guina, D. H. Sutter, and Th. Dekorsy, “Subpicosecond thin-disk laser oscillator with pulse energies of up to 25.9 microjoules by use of an active multipass geometry,” Opt. Express 16, 20530–20539 (2008).
[CrossRef] [PubMed]

G. Palmer, M. Schultze, M. Siegel, M. Emons, U. Bünting, and U. Morgner, “Passively mode-locked Yb:KLu(WO4)2 thin-disk oscillator operated in the positive and negative dispersion regime,” Opt. Lett. 33, 1608–1610 (2008).
[CrossRef] [PubMed]

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).
[CrossRef]

B. G. Bale, J. N. Kutz, A. Chong, W. H. Renninger, and F. W. Wise, “Spectral filtering for high-energy mode-locking in normal dispersion fiber lasers,” J. Opt. Soc. Am. B 25, 1763–1770 (2008).
[CrossRef]

M. Siegel, G. Palmer, M. Emons, M. Schutze, A. Ruehl, and U. Morgner, “Pulsing dynamics in Ytterbium based chirped-pulse oscillators,” Opt. Express 16, 14314–14320 (2008).
[CrossRef] [PubMed]

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140–148 (2008).
[CrossRef]

2007 (2)

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

O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007).
[CrossRef]

2006 (4)

Ch. Jirauschek and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B 23, 1776–1784 (2006).
[CrossRef]

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtoseond fiber laser,” Opt. Express 14, 10095–10100 (2006).
[CrossRef] [PubMed]

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B: Lasers Opt. 83, 503–510(2006).
[CrossRef]

2005 (4)

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

2004 (1)

R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B: Lasers Opt. 79, 163–173 (2004).

2003 (2)

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003).
[CrossRef]

S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
[CrossRef]

2001 (2)

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18, 1732–1741 (2001).
[CrossRef]

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear equations in optics,” Pramana J. Phys. 57, 917–936 (2001).
[CrossRef]

2000 (1)

A. K. Komarov and K. P. Komarov, “Pulse splitting in a passive mode-locked laser,” Optics Commun. 183, 265–270 (2000).
[CrossRef]

1998 (1)

S. Ch. Cerda, S. B. Cavalvanti, and J. M. Hickmann, “A variational approach of nonlinear dissipative pulse propagation,” Eur. Phys. J. D 1, 313–316 (1998).
[CrossRef]

1995 (1)

1992 (1)

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

1985 (1)

1975 (1)

H. A. Haus, “A theory of fast saturable absorber modelocking,” J. Appl. Phys. 46, 3049 (1975).
[CrossRef]

Akhmediev, N.

Ph. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonance as a guideline for high-energy pulse laser oscillators,” J. Opt. Soc. Am. B 27, 2336–2341 (2010).
[CrossRef]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[CrossRef]

W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev and A. AnkiewiczSolitons: nonlinear pulses and beams (Chapman&Hall, London, 1997).

Anderson, D.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear equations in optics,” Pramana J. Phys. 57, 917–936 (2001).
[CrossRef]

Ankiewicz, A.

Ph. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonance as a guideline for high-energy pulse laser oscillators,” J. Opt. Soc. Am. B 27, 2336–2341 (2010).
[CrossRef]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[CrossRef]

W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008).
[CrossRef]

N. N. Akhmediev and A. AnkiewiczSolitons: nonlinear pulses and beams (Chapman&Hall, London, 1997).

Antonelli, C.

Apolonski, A.

V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: A unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B: Lasers Opt. 83, 503–510(2006).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

Baer, C. R. E.

Bale, B. G.

Bauer, D.

Berntson, A.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear equations in optics,” Pramana J. Phys. 57, 917–936 (2001).
[CrossRef]

Boscolo, S.

Brabec, Th.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Buckley, J.

Bulanov, S. V.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Bünting, U.

Caputo, J. G.

Cavalvanti, S. B.

S. Ch. Cerda, S. B. Cavalvanti, and J. M. Hickmann, “A variational approach of nonlinear dissipative pulse propagation,” Eur. Phys. J. D 1, 313–316 (1998).
[CrossRef]

Cerda, S. Ch.

S. Ch. Cerda, S. B. Cavalvanti, and J. M. Hickmann, “A variational approach of nonlinear dissipative pulse propagation,” Eur. Phys. J. D 1, 313–316 (1998).
[CrossRef]

Chang, W.

Ph. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonance as a guideline for high-energy pulse laser oscillators,” J. Opt. Soc. Am. B 27, 2336–2341 (2010).
[CrossRef]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[CrossRef]

W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008).
[CrossRef]

Chen, J.

Chernykh, A.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B: Lasers Opt. 83, 503–510(2006).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

Chong, A.

Curley, P. F.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Dekorsy, Th.

Deng, Y.

Dombi, P.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

Durr, M.

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

Emons, M.

Engqvist, A. G.

Fermann, M. E.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Fernandez, A.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

Flytzanis, N.

Gingras, G.

T. Südmeyer, S. V. Marchese, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Photonics 2, 599–604 (2008).
[CrossRef]

Glick, Y.

O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007).
[CrossRef]

Gohle, Ch.

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

Golling, M.

Graf, R.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

Grelu, Ph.

Guelachvili, G.

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

Guina, M.

Hänsch, Th. W.

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

Hashimoto, S.

Haus, H. A.

Herrmann, M.

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

Hickmann, J. M.

S. Ch. Cerda, S. B. Cavalvanti, and J. M. Hickmann, “A variational approach of nonlinear dissipative pulse propagation,” Eur. Phys. J. D 1, 313–316 (1998).
[CrossRef]

Hofer, M.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Holzwarth, R.

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

Jirauschek, Ch.

Kafka, J. D.

Kalashnikov, V. L.

V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009).
[CrossRef]

V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: A unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B: Lasers Opt. 83, 503–510(2006).
[CrossRef]

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003).
[CrossRef]

S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
[CrossRef]

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18, 1732–1741 (2001).
[CrossRef]

V. L. Kalashnikov, “The unified theory of chirped-pulse oscillators,” Proc. SPIE Nonlinear Opt. Appl. III, Vol. 7354, Mario Bertolotti, Ed., p. 73540T (also arXiv:0903.5396 [physics.optics]) (2009).

Kärtner, F. X.

Katz, O.

O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007).
[CrossRef]

Keller, U.

Kieu, K.

Kieu, Kh.

Killi, A.

Kleinbauer, J.

Komarov, A. K.

A. K. Komarov and K. P. Komarov, “Pulse splitting in a passive mode-locked laser,” Optics Commun. 183, 265–270 (2000).
[CrossRef]

Komarov, K. P.

A. K. Komarov and K. P. Komarov, “Pulse splitting in a passive mode-locked laser,” Optics Commun. 183, 265–270 (2000).
[CrossRef]

Krausz, F.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Kumkar, M.

Kutz, J. N.

Lefrançois, S.

Lisak, M.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear equations in optics,” Pramana J. Phys. 57, 917–936 (2001).
[CrossRef]

Liu, Y.

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

Maas, D. J. H. C.

Mandon, J.

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

Marchese, S. V.

Morgner, U.

Moshammer, R.

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

Mourou, G. A.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Nafcha, Y.

O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007).
[CrossRef]

Naumov, S.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
[CrossRef]

Neuhaus, J.

Ober, M. H.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Oughstun, K. E.

K. E. Oughstun, Electromagmetic and Optical Pulse Propagation 1 (Springer, NY, 2006).

Palmer, G.

Paschotta, R.

R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B: Lasers Opt. 79, 163–173 (2004).

Picqué, N.

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

Podivilov, E.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B: Lasers Opt. 83, 503–510(2006).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

Rauschenberger, J.

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

Renninger, W.

Renninger, W. H.

Rudenko, A.

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

Ruehl, A.

Schmidt, A. J.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Schuessler, H. A.

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

Schultze, M.

Schutze, M.

Siegel, M.

Silberberg, Y.

Sintov, Y.

O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007).
[CrossRef]

Sørensen, M. P.

Sorokin, E.

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003).
[CrossRef]

S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
[CrossRef]

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18, 1732–1741 (2001).
[CrossRef]

Sorokina, I. T.

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003).
[CrossRef]

S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
[CrossRef]

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18, 1732–1741 (2001).
[CrossRef]

Soto-Crespo, J. M.

Ph. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonance as a guideline for high-energy pulse laser oscillators,” J. Opt. Soc. Am. B 27, 2336–2341 (2010).
[CrossRef]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[CrossRef]

W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008).
[CrossRef]

Spielmann, Ch.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Südmeyer, T.

Sutter, D. H.

Tajima, T.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Tempea, G.

S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
[CrossRef]

Tschuch, S.

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

Turitsyn, S. K.

Udem, Th.

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

Ullrich, J.

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

Weiler, S.

Wintner, E.

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Wise, F.

Wise, F. W.

Witzel, B.

T. Südmeyer, S. V. Marchese, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Photonics 2, 599–604 (2008).
[CrossRef]

Zhang, J.

Appl. Phys. B: Lasers Opt. (3)

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B: Lasers Opt. 83, 503–510(2006).
[CrossRef]

R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B: Lasers Opt. 79, 163–173 (2004).

S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, “Self-starting five optical cycle pulse generation in Cr4+:YAG laser,” Appl. Phys. B: Lasers Opt. 76, 1–11 (2003).
[CrossRef]

Eur. Phys. J. D (1)

S. Ch. Cerda, S. B. Cavalvanti, and J. M. Hickmann, “A variational approach of nonlinear dissipative pulse propagation,” Eur. Phys. J. D 1, 313–316 (1998).
[CrossRef]

IEEE J. Quantum Electron. (2)

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003).
[CrossRef]

F. Krausz, M. E. Fermann, Th. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

J. Appl. Phys. (1)

H. A. Haus, “A theory of fast saturable absorber modelocking,” J. Appl. Phys. 46, 3049 (1975).
[CrossRef]

J. Opt. Soc. Am. B (8)

JETP Lett. (1)

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

Nat. Photonics (1)

T. Südmeyer, S. V. Marchese, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Photonics 2, 599–604 (2008).
[CrossRef]

Nature (1)

Ch. Gohle, Th. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and Th. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
[CrossRef] [PubMed]

New J. Phys. (3)

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators. Theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005).
[CrossRef]

E. Sorokin, V. L. Kalashnikov, J. Mandon, G. Guelachvili, N. Picqué, and I. T. Sorokina, “Cr4+:YAG chirped-pulse oscillator,” New J. Phys. 10, 083022 (2008).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (4)

Optics Commun. (2)

O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Optics Commun. 269, 156–165 (2007).
[CrossRef]

A. K. Komarov and K. P. Komarov, “Pulse splitting in a passive mode-locked laser,” Optics Commun. 183, 265–270 (2000).
[CrossRef]

Phys. Rev. A (4)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008).
[CrossRef]

V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: A unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in the anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[CrossRef]

Phys. Rev. E (1)

V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Liu, S. Tschuch, A. Rudenko, M. Durr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001-1–4 (2008).
[CrossRef]

Pramana J. Phys. (1)

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear equations in optics,” Pramana J. Phys. 57, 917–936 (2001).
[CrossRef]

Rev. Mod. Phys. (1)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[CrossRef]

Other (5)

N. N. Akhmediev and A. AnkiewiczSolitons: nonlinear pulses and beams (Chapman&Hall, London, 1997).

The Wolfram Mathematica 7 notebook is accessible at http://info.tuwien.ac.at/kalashnikov/variational.html

K. E. Oughstun, Electromagmetic and Optical Pulse Propagation 1 (Springer, NY, 2006).

N. N. Akhmediev and A. Ankiewicz (Eds.), Dissipative Solitons: From Optics to Biology and Medicine. (Springer-Verlag, Berlin, Heidelberg, 2008).

V. L. Kalashnikov, “The unified theory of chirped-pulse oscillators,” Proc. SPIE Nonlinear Opt. Appl. III, Vol. 7354, Mario Bertolotti, Ed., p. 73540T (also arXiv:0903.5396 [physics.optics]) (2009).

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Figures (6)

Fig. 1
Fig. 1

Maps for two thin-disk Yb:YAG oscillators: an airless one (a) and air-filled one (b). Ĥ blocks correspond to the delay lines with the dispersion compensators, Ĝ is the gain head, is the SESAM.

Fig. 2
Fig. 2

Thresholds (cτΣγ/βζ vs. E) of the pulse stability (solid curves) and the pulse widths TFWHM at these thresholds (dashed curves) in the ADR. The different colors correspond to μ=0.002 (black), 0.0025 (green), 0.004 (red), 0.005 (magenta), and 0.008 (blue). Analytical thresholds correspond to the lines without symbols, numerical ones correspond to the curves with symbols. The dimensional values of the GDD and the output pulse energies at the extreme points are shown for oscillator types a and b. For an oscillator of the b′-type, the intracavity energies are shown. The pulses are stable below the thresholds shown.

Fig. 3
Fig. 3

Thresholds (cτ Σγ/βζvs. E) of the pulse stability (solid curves) and the pulse widths TFWHM at these thresholds (dashed curves) in the ADR. The different colors correspond to μ=0.002 (black), 0.008 (blue), and 0.016 (green). Symbols demonstrate the operational points in the vicinity of stability threshold for the different oscillators (corresponding references and output energies are inscribed). Dotted curves correspond to the criterion c/μ=1 [38].

Fig. 4
Fig. 4

Thresholds (|c| ≡ τΣγ/|β|ζ vs. E) of the pulse stability. The different colors correspond to μ=0.002 (black), 0.005 (magenta), and 0.008 (blue); bγ/ζ=2×10−4 (black and blue), 4×10−3 (magenta). Analytical thresholds correspond to the lines without symbols, numerical ones correspond to the curves with symbols. The dimensional values of the GDD and the output pulse energies in the extreme points are shown for oscillator types a and b. The pulses are stable below the thresholds shown.

Fig. 5
Fig. 5

Thresholds (|c| ≡ τΣγ/|β|ζ vs. E) of the pulse stability. The different colors correspond to bγ/ζ=0.001 (black), 0.1 (red), and 10 (blue); μ=0.008. Analytical thresholds from [20] correspond to the curves (the CDS are stable in the direction of the arrows marked in regular type). Analytical thresholds from the variational model correspond to the symbols (the CDS are stable in the direction of the arrows marked in italics).

Fig. 6
Fig. 6

Thresholds (|c| ≡ τΣγ/|β|ζ for the solid-state oscillators and |c| ≡ τγ/|β|ζ for the fiber ones vs. E/b) of the pulse stability. The different colors correspond to bγ/ζ =33 and μ=0.5 (red), b =0.3 and μ=0.02 (blue), b =7×105 and μ=0.025 (black). Analytical thresholds from [20] correspond to the curves (the CDS is stable in the direction of the arrow marked in regular type). Analytical threshold from the variational model is shown by crosses (the CDS is stable in the direction of the arrows marked in italics). Symbols demonstrate the operational points in the vicinity of the stability threshold for the different oscillators (corresponding references and output energies are given).

Tables (2)

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Table 1 Main abbreviations and symbols

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Table 2 Simulation parameters (per round-trip) for the oscillator maps shown in Fig. 1 (a) and (b). Map b′ is geometrically identical to the b one, but has other parameters.

Equations (14)

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Ldt f d d z Ldt f = 2 Q a f dt ,
L = 1 2 { i [ a * a t a a * t ] + [ β a t a * t γ | a | 4 ] } .
a ( z , t ) = A ( z ) exp [ i ϕ ( z ) ] sech [ t T ( z ) ] 1 + i ψ ( z ) ,
Q i [ + ρ 1 + σ | a | 2 d t ( 1 + π 2 t 2 ) + μ ζ | a | 2 1 + ζ | a | 2 ] a .
ϕ z = b 2 A 2 , T = 1 A c ,
μ ( 2 + ln ( 1 + A 2 A 1 + A 2 1 + A 2 + A 1 + A 2 ) A 1 + A 2 ) + 2 Ξ 2 c A 2 3 = 0 ,
ϕ z = b 24 A 2 12 b A 4 + ψ ( 12 A 2 Ξ + μ π 2 32 c A 4 ) b 2 c ψ . T = 2 A ( 1 + ψ 2 ) ( 2 c ψ b ) c ( ( 12 A 2 Ξ + μ π 2 ) ψ 4 b A 4 ) , ψ = 4 b A ( A 1 + A 2 ( c A 2 3 ( μ + Ξ ) ) + 3 μ arctanh [ A 1 + A 2 ] ) c ( 1 + A 2 ( 12 A 2 ( 2 μ Ξ ) μ π 2 ) ) 24 A arctanh [ A 1 + A 2 ] .
L ^ [ A ] = A ( t ) exp [ κ μ 1 + ζ | A ( t ) | 2 ]
G ^ [ A ] = i β g 2 2 t 2 A ( t ) + i γ g | A ( t ) | 2 A ( t ) + ( ρ Ω g 1 + N σ | A ( t ) | 2 d t t exp [ Ω g ( t t ) ] A ( t ) d t ) + s ( t ) .
s ( t ) s * ( t ) = 2 Σ θ h ν δ t δ ( t t )
A z + 1 ( t ) = [ H ^ H ^ G ^ H ^ G ^ L ^ G ^ H ^ G ^ H ^ H ^ ] A z ( t ) , A z + 1 ( t ) = [ L ^ H ^ G ^ H ^ L ^ ] A z ( t ) ,
ρ k + δ k = P T r ρ max + ( ρ k P T r ρ max ) exp [ Δ t T r ] ,
c E ζ / τ Σ ,
E ζ 2 c 2 / γ τ Σ ϒ ,

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