Abstract

Fast saturable absorbers (FSAs) play a critical role in stabilizing many passively modelocked lasers. The most commonly used averaged model to study these lasers is the Haus modelocking equation (HME) that includes a third-order nonlinear FSA. However, it predicts a narrow region of stability that is inconsistent with experiments. To better replicate the laser physics, averaged laser models that include FSAs with higher-than-third-order nonlinearities have been introduced. Here, we compare three common FSA models to each other and to the HME using the recently-developed boundary tracking algorithms. The three FSA models are the cubic-quintic model, the sinusoidal model, and the algebraic model. We find that all three models predict the existence of a stable high-energy solution that is not present in the HME and have a much larger stable operating region. We also find that all three models predict qualitatively similar stability diagrams. We conclude that averaged laser models that include FSAs with higher-than-third-order nonlinearity should be used when studying the stability of passively modelocked lasers.

© 2016 Optical Society of America

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    [Crossref] [PubMed]
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2016 (1)

C. R. Menyuk and S. Wang, “Spectral methods for determining the stability and noise performance of passively modelocked lasers,” Nanophotonics 5, 332–350 (2016).
[Crossref]

2015 (2)

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

C. Bao, W. Chang, C. Yang, N. Akhmediev, and S. T. Cundiff, “Observation of coexisting dissipative solitons in a mode-locked fiber laser,” Phys. Rev. Lett. 115, 253903 (2015).
[Crossref]

2014 (2)

2011 (1)

2010 (1)

2009 (1)

2008 (2)

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

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

2007 (2)

2006 (2)

2005 (5)

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
[Crossref]

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[Crossref]

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quant. Electron. 41, 1388–1402 (2005).
[Crossref]

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quantum Electron. 41, 1388–1402 (2005).
[Crossref]

S. T. Cundiff and J. Ye, “Phase stabilization of mode-locked lasers,” J. Mod. Opt. 52, 201–219 (2005).
[Crossref]

2004 (3)

B. R. Washburn, R. W. Fox, N. R. Newbury, J. W. Nicholson, K. Feder, P. S. Westbrook, and C. G. Jørgensen, “Fiber-laser-based frequency comb with a tunable repetition rate,” Opt. Express 12, 4999–5004 (2004).
[Crossref] [PubMed]

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[Crossref]

R. Paschotta, “Noise of mode-locked lasers (part i): numerical model,” Appl. Phys. B 79, 153–162 (2004).
[Crossref]

2003 (1)

T. Fortier, D. Jones, J. Ye, and S. Cundiff, “Highly phase stable mode-locked lasers,” IEEE J. Sel. Topics Quantum Electron. 9, 1002–1010 (2003).
[Crossref]

2002 (2)

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
[Crossref]

2000 (2)

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quant. Electron. 6, 1173–1185 (2000).
[Crossref]

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

1998 (2)

1996 (2)

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[Crossref]

F. X. Kärtner, I. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[Crossref]

1995 (1)

1994 (2)

I. N. Duling, C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Operation of a nonlinear loop mirror in a laser cavity,” IEEE J. Quant. Electron. 30, 194–199 (1994).
[Crossref]

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Stability of passively mode-locked fiber lasers with fast saturable absorption,” Opt. Lett. 19, 198–200 (1994).
[Crossref] [PubMed]

1993 (1)

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

1992 (4)

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[Crossref]

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17, 417–419 (1992).
[Crossref] [PubMed]

M. Hofer, M. Ober, F. Haberl, and M. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 21, 1981–1983 (1992).
[Crossref]

1991 (2)

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]

D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas, “320 fs soliton generation with passively mode-locked erbium fibre laser,” Electron. Lett. 27, 730–732 (1991).
[Crossref]

1990 (2)

B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg-Landau equation,” Phys. Rev. A 42, 6009–6014 (1990).
[Crossref] [PubMed]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[Crossref] [PubMed]

1987 (1)

B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D 29, 155–172 (1987).
[Crossref]

1978 (1)

M. Gupta, B. Som, and B. Dasgupta, “Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile,” Phys. Lett. A 69, 172–174 (1978).
[Crossref]

1975 (3)

H. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quant. Electron. 11, 736–746 (1975).
[Crossref]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

Ablowitz, M. J.

Afanasjev, V. V.

Akhmediev, N.

C. Bao, W. Chang, C. Yang, N. Akhmediev, and S. T. Cundiff, “Observation of coexisting dissipative solitons in a mode-locked fiber laser,” Phys. Rev. Lett. 115, 253903 (2015).
[Crossref]

Akhmediev, N. N.

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[Crossref]

Bao, C.

C. Bao, W. Chang, C. Yang, N. Akhmediev, and S. T. Cundiff, “Observation of coexisting dissipative solitons in a mode-locked fiber laser,” Phys. Rev. Lett. 115, 253903 (2015).
[Crossref]

Benedick, A.

Beyatli, E.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Birge, J.

Brunel, M.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Buckley, J.

Canbaz, F.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Chang, W.

C. Bao, W. Chang, C. Yang, N. Akhmediev, and S. T. Cundiff, “Observation of coexisting dissipative solitons in a mode-locked fiber laser,” Phys. Rev. Lett. 115, 253903 (2015).
[Crossref]

Chartier, T.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Chen, C. J.

I. N. Duling, C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Operation of a nonlinear loop mirror in a laser cavity,” IEEE J. Quant. Electron. 30, 194–199 (1994).
[Crossref]

Chen, C.-J.

Chen, L. J.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Chong, A.

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

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

Cihan, C.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Crespo, H. M.

Cundiff, S.

T. Fortier, D. Jones, J. Ye, and S. Cundiff, “Highly phase stable mode-locked lasers,” IEEE J. Sel. Topics Quantum Electron. 9, 1002–1010 (2003).
[Crossref]

Cundiff, S. T.

C. Bao, W. Chang, C. Yang, N. Akhmediev, and S. T. Cundiff, “Observation of coexisting dissipative solitons in a mode-locked fiber laser,” Phys. Rev. Lett. 115, 253903 (2015).
[Crossref]

J. K. Wahlstrand, J. T. Willits, T. R. Schibli, C. R. Menyuk, and S. T. Cundiff, “Quantitative measurement of timing and phase dynamics in a mode-locked laser,” Opt. Lett. 32, 3426–3428 (2007).
[Crossref] [PubMed]

C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. R. Schibli, and S. T. Cundiff, “Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling,” Opt. Express 15, 6677–6689 (2007).
[Crossref] [PubMed]

S. T. Cundiff and J. Ye, “Phase stabilization of mode-locked lasers,” J. Mod. Opt. 52, 201–219 (2005).
[Crossref]

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

Dasgupta, B.

M. Gupta, B. Som, and B. Dasgupta, “Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile,” Phys. Lett. A 69, 172–174 (1978).
[Crossref]

Demirbas, U.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Diddams, S. A.

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

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

Docherty, A.

Drazin, P. G.

P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge Computer Science Texts (Cambridge University Press, 1989).
[Crossref]

Duling, I. N.

I. N. Duling, C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Operation of a nonlinear loop mirror in a laser cavity,” IEEE J. Quant. Electron. 30, 194–199 (1994).
[Crossref]

Dunlop, A.

Erbert, G.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Feder, K.

Fermann, M.

M. Hofer, M. Ober, F. Haberl, and M. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Firth, W.

Fortier, T.

T. Fortier, D. Jones, J. Ye, and S. Cundiff, “Highly phase stable mode-locked lasers,” IEEE J. Sel. Topics Quantum Electron. 9, 1002–1010 (2003).
[Crossref]

Fox, R. W.

Frantzeskakis, D. J.

Fujimoto, J. G.

Grelu, P.

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

Gupta, M.

M. Gupta, B. Som, and B. Dasgupta, “Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile,” Phys. Lett. A 69, 172–174 (1978).
[Crossref]

Haberl, F.

M. Hofer, M. Ober, F. Haberl, and M. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Hall, J. L.

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

Haus, H.

H. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quant. Electron. 11, 736–746 (1975).
[Crossref]

Haus, H. A.

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quant. 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]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

Hideur, A.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Hofer, M.

M. Hofer, M. Ober, F. Haberl, and M. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Horikis, T. P.

Ippen, E. P.

Johnson, R. S.

P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge Computer Science Texts (Cambridge University Press, 1989).
[Crossref]

Jones, D.

T. Fortier, D. Jones, J. Ye, and S. Cundiff, “Highly phase stable mode-locked lasers,” IEEE J. Sel. Topics Quantum Electron. 9, 1002–1010 (2003).
[Crossref]

Jones, D. J.

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

Jørgensen, C. G.

Jung, I.

F. X. Kärtner, I. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[Crossref]

Kapitula, T.

Kärtner, F. X.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

M. Y. Sander, J. Birge, A. Benedick, H. M. Crespo, and F. X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26, 743–749 (2009).
[Crossref]

F. X. Kärtner, I. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[Crossref]

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[Crossref] [PubMed]

Keller, U.

F. X. Kärtner, I. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[Crossref]

Kodama, Y.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 21, 1981–1983 (1992).
[Crossref]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[Crossref]

Komarov, A.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[Crossref]

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
[Crossref]

Kutz, J. N.

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Review 48, 629–678 (2006).
[Crossref]

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[Crossref]

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
[Crossref]

Laming, R. I.

D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas, “320 fs soliton generation with passively mode-locked erbium fibre laser,” Electron. Lett. 27, 730–732 (1991).
[Crossref]

Leblond, H.

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
[Crossref]

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[Crossref]

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Leitenstorfer, A.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Malomed, B. A.

B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg-Landau equation,” Phys. Rev. A 42, 6009–6014 (1990).
[Crossref] [PubMed]

B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D 29, 155–172 (1987).
[Crossref]

Marks, B. S.

S. Wang, A. Docherty, B. S. Marks, and C. R. Menyuk, “Boundary tracking algorithms for determining the stability of mode-locked pulses,” J. Opt. Soc. Am. B 31, 2914–2930 (2014).
[Crossref]

S. Wang, B. S. Marks, and C. R. Menyuk, “Nonlinear stabilization of high-energy and broadband pulses in passively modelocking models,” (2016), arXiv:1607.03162.

Matsas, V. J.

D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas, “320 fs soliton generation with passively mode-locked erbium fibre laser,” Electron. Lett. 27, 730–732 (1991).
[Crossref]

Mecozzi, A.

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

Menyuk, C.

S. Wang and C. Menyuk, “Computational methods for determining the stability of pulses in passively modelocked laser systems,” in 2013 IEEE Photonics Conference, (2013), pp. 392–393.

Menyuk, C. R.

Nepomnyashchy, A. A.

B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg-Landau equation,” Phys. Rev. A 42, 6009–6014 (1990).
[Crossref] [PubMed]

Newbury, N. R.

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quantum Electron. 41, 1388–1402 (2005).
[Crossref]

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quant. Electron. 41, 1388–1402 (2005).
[Crossref]

B. R. Washburn, R. W. Fox, N. R. Newbury, J. W. Nicholson, K. Feder, P. S. Westbrook, and C. G. Jørgensen, “Fiber-laser-based frequency comb with a tunable repetition rate,” Opt. Express 12, 4999–5004 (2004).
[Crossref] [PubMed]

Nicholson, J. W.

Nixon, S. D.

Ober, M.

M. Hofer, M. Ober, F. Haberl, and M. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Paschotta, R.

R. Paschotta, “Noise of mode-locked lasers (part i): numerical model,” Appl. Phys. B 79, 153–162 (2004).
[Crossref]

Payne, D. N.

D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas, “320 fs soliton generation with passively mode-locked erbium fibre laser,” Electron. Lett. 27, 730–732 (1991).
[Crossref]

Phillips, M. W.

D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas, “320 fs soliton generation with passively mode-locked erbium fibre laser,” Electron. Lett. 27, 730–732 (1991).
[Crossref]

Ranka, J. K.

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

Renninger, W.

Renninger, W. H.

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

Richardson, D. J.

D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas, “320 fs soliton generation with passively mode-locked erbium fibre laser,” Electron. Lett. 27, 730–732 (1991).
[Crossref]

Romagnoli, M.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 21, 1981–1983 (1992).
[Crossref]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[Crossref]

Salhi, M.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Sanchez, F.

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
[Crossref]

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[Crossref]

Sanchez, F. m. c.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Sander, M. Y.

Sandstede, B.

Schibli, T. R.

Sennaroglu, A.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Smith, R. P.

Som, B.

M. Gupta, B. Som, and B. Dasgupta, “Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile,” Phys. Lett. A 69, 172–174 (1978).
[Crossref]

Soto-Crespo, J. M.

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[Crossref]

Stentz, A.

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

Sumpf, B.

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

Wabnitz, S.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 21, 1981–1983 (1992).
[Crossref]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[Crossref]

Wahlstrand, J. K.

Wai, P. K. A.

Wang, S.

C. R. Menyuk and S. Wang, “Spectral methods for determining the stability and noise performance of passively modelocked lasers,” Nanophotonics 5, 332–350 (2016).
[Crossref]

S. Wang, A. Docherty, B. S. Marks, and C. R. Menyuk, “Boundary tracking algorithms for determining the stability of mode-locked pulses,” J. Opt. Soc. Am. B 31, 2914–2930 (2014).
[Crossref]

S. Wang and C. Menyuk, “Computational methods for determining the stability of pulses in passively modelocked laser systems,” in 2013 IEEE Photonics Conference, (2013), pp. 392–393.

S. Wang, B. S. Marks, and C. R. Menyuk, “Nonlinear stabilization of high-energy and broadband pulses in passively modelocking models,” (2016), arXiv:1607.03162.

Washburn, B. R.

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quantum Electron. 41, 1388–1402 (2005).
[Crossref]

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quant. Electron. 41, 1388–1402 (2005).
[Crossref]

B. R. Washburn, R. W. Fox, N. R. Newbury, J. W. Nicholson, K. Feder, P. S. Westbrook, and C. G. Jørgensen, “Fiber-laser-based frequency comb with a tunable repetition rate,” Opt. Express 12, 4999–5004 (2004).
[Crossref] [PubMed]

Westbrook, P. S.

Willits, J.

Willits, J. T.

Windeler, R. S.

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

Wise, F.

Wise, F. W.

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

Wright, E.

Yang, C.

C. Bao, W. Chang, C. Yang, N. Akhmediev, and S. T. Cundiff, “Observation of coexisting dissipative solitons in a mode-locked fiber laser,” Phys. Rev. Lett. 115, 253903 (2015).
[Crossref]

Ye, J.

S. T. Cundiff and J. Ye, “Phase stabilization of mode-locked lasers,” J. Mod. Opt. 52, 201–219 (2005).
[Crossref]

T. Fortier, D. Jones, J. Ye, and S. Cundiff, “Highly phase stable mode-locked lasers,” IEEE J. Sel. Topics Quantum Electron. 9, 1002–1010 (2003).
[Crossref]

Appl. Phys. B (1)

R. Paschotta, “Noise of mode-locked lasers (part i): numerical model,” Appl. Phys. B 79, 153–162 (2004).
[Crossref]

Electron. Lett. (3)

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 21, 1981–1983 (1992).
[Crossref]

D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas, “320 fs soliton generation with passively mode-locked erbium fibre laser,” Electron. Lett. 27, 730–732 (1991).
[Crossref]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[Crossref]

IEEE J Quantum Electron. (1)

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

IEEE J. Quant. Electron. (3)

I. N. Duling, C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Operation of a nonlinear loop mirror in a laser cavity,” IEEE J. Quant. Electron. 30, 194–199 (1994).
[Crossref]

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quant. Electron. 41, 1388–1402 (2005).
[Crossref]

H. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quant. Electron. 11, 736–746 (1975).
[Crossref]

IEEE J. Quantum Electron. (2)

M. Hofer, M. Ober, F. Haberl, and M. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

N. R. Newbury and B. R. Washburn, “Theory of the frequency comb output from a femtosecond fiber laser,” IEEE J. Quantum Electron. 41, 1388–1402 (2005).
[Crossref]

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

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quant. Electron. 6, 1173–1185 (2000).
[Crossref]

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

F. X. Kärtner, I. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[Crossref]

C. Cihan, E. Beyatli, F. Canbaz, L. J. Chen, B. Sumpf, G. Erbert, A. Leitenstorfer, F. X. Kärtner, A. Sennaroglu, and U. Demirbas, “Gain-matched output couplers for efficient Kerr-lens mode-locking of low-cost and high-peak power cr:lisaf lasers,” IEEE J. Sel. Top. Quantum Electron. 21, 94–105 (2015).
[Crossref]

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

T. Fortier, D. Jones, J. Ye, and S. Cundiff, “Highly phase stable mode-locked lasers,” IEEE J. Sel. Topics Quantum Electron. 9, 1002–1010 (2003).
[Crossref]

Indiana Univ. Math. J. (1)

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[Crossref]

J. Appl. Phys. (2)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[Crossref]

J. Mod. Opt. (1)

S. T. Cundiff and J. Ye, “Phase stabilization of mode-locked lasers,” J. Mod. Opt. 52, 201–219 (2005).
[Crossref]

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

Nanophotonics (1)

C. R. Menyuk and S. Wang, “Spectral methods for determining the stability and noise performance of passively modelocked lasers,” Nanophotonics 5, 332–350 (2016).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Lett. A (2)

M. Gupta, B. Som, and B. Dasgupta, “Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile,” Phys. Lett. A 69, 172–174 (1978).
[Crossref]

N. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

Phys. Rev. A (5)

B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg-Landau equation,” Phys. Rev. A 42, 6009–6014 (1990).
[Crossref] [PubMed]

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. m. c. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
[Crossref]

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

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[Crossref] [PubMed]

Phys. Rev. E (1)

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[Crossref]

Phys. Rev. Lett. (1)

C. Bao, W. Chang, C. Yang, N. Akhmediev, and S. T. Cundiff, “Observation of coexisting dissipative solitons in a mode-locked fiber laser,” Phys. Rev. Lett. 115, 253903 (2015).
[Crossref]

Physica D (1)

B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D 29, 155–172 (1987).
[Crossref]

Science (1)

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

SIAM Review (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Review 48, 629–678 (2006).
[Crossref]

Other (3)

S. Wang and C. Menyuk, “Computational methods for determining the stability of pulses in passively modelocked laser systems,” in 2013 IEEE Photonics Conference, (2013), pp. 392–393.

S. Wang, B. S. Marks, and C. R. Menyuk, “Nonlinear stabilization of high-energy and broadband pulses in passively modelocking models,” (2016), arXiv:1607.03162.

P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge Computer Science Texts (Cambridge University Press, 1989).
[Crossref]

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Figures (10)

Figure 1
Figure 1 Comparison of the nonlinear gain for the three models of fast saturable absorption given in Eq. (10).
Figure 2
Figure 2 The stability regions of the GHME with a cubic-quintic saturable absorber fsa,cq(|u|). The stability boundaries are marked by three curves, C1, C2, and C3. This figure reproduces Fig. 16 of ref. [21]
Figure 3
Figure 3 For the GHME with the cubic-quintic model of Eq. (10a), the pulse profiles of the low-amplitude solution (LAS) and the high-amplitude solution (HAS) with (σ, δ) = (0.004, 0.036).
Figure 4
Figure 4 Stability boundaries for models with the three different absorbers; black (cubic-quintic), blue (algebraic), red (sinusoidal).
Figure 5
Figure 5 Variation of the profile parameters of the low-amplitude solution (LAS) when σ = 0.004. The black curves represent the pulse profiles of the LAS with the cubic-quintic model, and, similarly, the blue curves represent those of the algebraic model, and the red curves represent those of the sinusoidal model.
Figure 6
Figure 6 With σ = 0.004, the variation of the profile parameters of the high-amplitude solution (HAS): sub-figures (a) and (b) show the profiles of the cubic-quintic model and the sinusoidal model, while sub-figures (c) and (d) show the algebraic model. The black curves represent the pulse profiles with the cubic-quintic model, and, similarly, the blue curves represent the algebraic model, and the red curves represent the sinusoidal model.
Figure 7
Figure 7 The stability boundary of the GHME with the cubic-quintic model and the sinusoidal model at large values of δ as σ varies. The unstable region of both models lies above each curve. The pulse solution of the GHME with the algebraic model is always stable when δ increases, as we prove in Appendix B.
Figure 8
Figure 8 Comparison of the computational stationary pulses with the experimental pulses using parameters in (a) set 1 [39,40], and (b) set 2 [41].
Figure 9
Figure 9 Illustrations of the linearized spectrum of the eigenvalues of (a) the nonlinear Schrödinger equation and (b) the generalized Haus modelocking equation.
Figure 10
Figure 10 As the nonlinear gain coefficient δ increases, for the GHME in Eq. (18), the variations of (a) the peak amplitude of the HAS A0 and (b) the amplitude value λa.

Tables (3)

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Table 1 Normalized values of parameters.

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Table 2 Instability mechanisms of the GHME shown in Fig. 2, where LAS represents the low-amplitude solution, and HAS represents the high-amplitude solution.

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Table 3 Values of parameters we use in validating the experimental results.

Equations (31)

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u z = [ i ϕ l 2 + g ( | u | ) 2 ( 1 + 1 2 ω g 2 2 t 2 ) i β 2 2 t 2 + i γ | u | 2 ] u + f sa ( | u | ) u ,
g ( | u | ) = g 0 1 + P av ( | u | ) / P sat ,
f sa ( | u | ) = δ | u | 2 .
u z | ab = f ab ( | u | ) u = f 0 u 1 + | u ( t ) | 2 / P ab ,
f sa ( | u | ) = [ f ab ( | u | ) f 0 ] = f 0 | u | 2 / P ab 1 + | u | 2 / P ab .
f sa ( | u | ) = f 0 P ab | u ( t ) | 2 f 0 P ab 2 | u ( t ) | 4 .
f sa ( | u | ) = δ | u ( t ) | 2 σ | u ( t ) | 4 ,
f ab ( | u | ) = f 0 + f 1 cos ( μ | u | 2 ν ) ,
f ab ( | u | ) = f 0 + f 1 cos ν + μ f 1 ( sin ν ) | u | 2 ( μ 2 f 1 / 2 ) ( cos ν ) | u | 4 .
δ = f 0 P ab = μ f 1 sin ν , σ = f 0 P ab 2 = μ 2 f 1 cos ν 2 ,
f sa , cq ( | u | ) = δ | u | 2 σ | u | 4 ,
f sa , al ( | u | ) = δ | u | 2 1 + ( σ / δ ) | u | 2 ,
f sa , sn ( | u | ) = δ 2 2 σ [ 2 sin ( 2 σ δ | u | 2 + π 4 ) 1 ] ,
u h ( t ) = A h sech ( 1 + i β h ) ( t / t h ) ,
u 0 ( t ) = | u 0 ( t ) | exp [ θ 0 ( t ) ] ,
b 0 = Im t u * [ d u / d t ] d t t 2 | u | 2 d t .
u z = i ϕ u + i 2 2 u t 2 + i | u | 2 u ,
u 0 ( t ) = sech ( t ) , ϕ 0 = 1 / 2 .
Δ u z = i ϕ 0 Δ u + i 2 2 Δ u t 2 + ( 2 i sech 2 t ) Δ u ( i sech 2 t ) Δ u ¯ Δ u ¯ z = i ϕ 0 Δ u ¯ i 2 2 Δ u ¯ t 2 ( 2 i sech 2 t ) Δ u ¯ ( i sech 2 t ) Δ u ,
Δ u z = ( u 0 ) Δ u = λ Δ u ,
u z = [ i ϕ l 2 + g ( | u | ) 2 ( 1 + 1 2 ω g 2 2 t 2 ) i β 2 2 t 2 + i γ | u | 2 + δ | u | 2 1 + σ / δ | u | 2 ] u ,
u 0 = A 0 sech ( γ β A 0 t ) , ϕ 0 = γ 2 A 0 2
u z = i ϕ u i β 2 2 u t 2 + i γ | u | 2 u .
f sa , al ( | u | ) = δ | u | 2 1 + ( σ / δ ) | u | 2 δ 2 σ .
u * u z = [ i ϕ l 2 + g | u | 2 ] | u | 2 + [ g | u | 4 ω g 2 i β 2 ] u * 2 u t 2 + i γ | u | 4 + δ | u | 4 1 + σ / δ | u | 2 .
u u * z = [ i ϕ l 2 + g ( | u | ) 2 ] | u | 2 + [ g ( | u | ) 4 ω g 2 + i β 2 ] u 2 u * t 2 i γ | u | 4 + δ | u | 4 1 + σ / δ | u | 2 .
| u | 2 z = [ g ( | u | ) l ] | u | 2 + g ( | u | ) 4 ω g 2 ( u * 2 u t 2 + u 2 u * t 2 ) + i β 2 ( u 2 u * t 2 u * 2 u t 2 ) + 2 δ | u | 4 1 + σ / δ | u | 2 ,
w z = [ g ( | u | ) l ] w g ( | u | ) 2 ω g 2 T / 2 T / 2 | u t | 2 d t + T / 2 T / 2 2 δ | u | 4 1 + σ / δ | u | 2 d t .
d A 0 d z = f ( A 0 ) = g 0 E sat E sat + 2 A 0 β / γ ( A 0 + γ A 0 3 6 β ω 2 ) l A 0 + 2 δ 2 A 0 σ δ 3 σ 3 / 2 δ + σ A 0 2 log ( δ + σ A 0 2 + σ A 0 δ + σ A 0 2 σ A 0 ) .
d Δ A d z d f ( A ) d A | A = A 0 Δ A = λ a Δ A ,
λ a = g 0 E sat E sat + 2 A 0 β / γ ( 1 + γ A 0 2 2 β ω g 2 ) 2 g 0 E sat β / γ ( E sat + 2 A 0 β / γ ) 2 ( A 0 + γ A 0 3 6 β ω g 2 ) l + 2 δ 2 σ + δ 3 A 0 σ ( δ + σ A 0 2 ) 3 / 2 log ( δ + σ A 0 2 + σ A 0 δ + σ A 0 2 σ A 0 ) 2 δ 3 σ ( δ + σ A 0 2 ) .

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