Abstract

Different pulse-shaping mechanisms were investigated experimentally and numerically in passively mode-locked thulium-doped fiber lasers. Conventional solitons were demonstrated in a passively semiconductor saturable absorber mirror mode-locked anomalous dispersion thulium-doped fiber laser. With normal dispersion fiber and spectral filter added in cavity, pulse-shaping processes were theoretically analyzed in the presence of dispersion map and dissipation in thulium-doped fiber lasers. The existence of parabolic pulse as nonlinear attraction was proved and distinct pulse intensity profiles evolution from Gaussian shape to parabolic shape was proposed in dissipative dispersion-managed thulium-doped fiber lasers.

© 2015 Optical Society of America

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References

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2014 (1)

2013 (4)

I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013).
[Crossref]

H. Xia, H. Li, X. Zhang, S. Zhang, X. Tang, and Y. Liu, “Characteristics of dissipative solitons in an all-fiber thulium-doped fiber ring laser,” Opt. Eng. 52(5), 054201 (2013).
[Crossref]

Q. Q. Wang, T. Chen, M. S. Li, B. T. Zhang, Y. F. Lu, and K. P. Chen, “All-fiber ultrafast thulium-doped fiber ring laser with dissipative soliton and noise-like output in normal dispersion by single-wall carbon nanotubes,” Appl. Phys. Lett. 103(1), 011103 (2013).
[Crossref]

P. Wan, L. M. Yang, and J. Liu, “High power 2 µm femtosecond fiber laser,” Opt. Express 21(18), 21374–21379 (2013).
[Crossref] [PubMed]

2012 (4)

A. A. Krylov, M. A. Chernyshova, D. S. Chernykh, A. K. Senatorov, I. M. Tupitsyn, P. G. Kryukov, and E. M. Dianov, “High-power thulium-doped fibre laser with intracavity dispersion management,” Quantum Electron. 42(5), 427–431 (2012).
[Crossref]

J. Liu, Q. Wang, and P. Wang, “High average power picosecond pulse generation from a thulium-doped all-fiber MOPA system,” Opt. Express 20(20), 22442–22447 (2012).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

S. K. Turitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

2011 (1)

2010 (2)

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton–similariton fibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

2009 (4)

2008 (3)

2007 (1)

2005 (1)

N. M. Fried, “Thulium fiber laser lithotripsy: An in vitro analysis of stone fragmentation using a modulated 110-watt Thulium fiber laser at 1.94 microm,” Lasers Surg. Med. 37(1), 53–58 (2005).
[Crossref] [PubMed]

2004 (1)

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

2000 (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26 Pt 1), 6010–6013 (2000).
[Crossref] [PubMed]

1997 (1)

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65(2), 277–294 (1997).
[Crossref]

1995 (1)

L. Nelson, E. Ippen, and H. Haus, “Broadly tunable sub-500 fs pulses from an additive-pulse mode-locked thulium-doped fiber ring laser,” Appl. Phys. Lett. 67(1), 19–21 (1995).
[Crossref]

1994 (1)

M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30(6), 1469–1477 (1994).
[Crossref]

1992 (2)

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, “Self-starting passively mode-locked fiber ring soliton laser exploiting nonlinear polarization rotation,” Electron. Lett. 28(15), 1391–1393 (1992).
[Crossref]

D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking in nonlinear optical fibers,” J. Opt. Soc. Am. B 9(8), 1358–1361 (1992).
[Crossref]

1989 (1)

1985 (1)

Anderson, D.

Bale, B. G.

S. K. Turitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

B. G. Bale, S. Boscolo, and S. K. Turitsyn, “Dissipative dispersion-managed solitons in mode-locked lasers,” Opt. Lett. 34(21), 3286–3288 (2009).
[Crossref] [PubMed]

Boscolo, S.

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

B. G. Bale, S. Boscolo, and S. K. Turitsyn, “Dissipative dispersion-managed solitons in mode-locked lasers,” Opt. Lett. 34(21), 3286–3288 (2009).
[Crossref] [PubMed]

Buckley, J. R.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Budni, P. A.

Chen, K. P.

Q. Q. Wang, T. Chen, M. S. Li, B. T. Zhang, Y. F. Lu, and K. P. Chen, “All-fiber ultrafast thulium-doped fiber ring laser with dissipative soliton and noise-like output in normal dispersion by single-wall carbon nanotubes,” Appl. Phys. Lett. 103(1), 011103 (2013).
[Crossref]

Chen, T.

Q. Q. Wang, T. Chen, M. S. Li, B. T. Zhang, Y. F. Lu, and K. P. Chen, “All-fiber ultrafast thulium-doped fiber ring laser with dissipative soliton and noise-like output in normal dispersion by single-wall carbon nanotubes,” Appl. Phys. Lett. 103(1), 011103 (2013).
[Crossref]

Chernykh, D. S.

A. A. Krylov, M. A. Chernyshova, D. S. Chernykh, A. K. Senatorov, I. M. Tupitsyn, P. G. Kryukov, and E. M. Dianov, “High-power thulium-doped fibre laser with intracavity dispersion management,” Quantum Electron. 42(5), 427–431 (2012).
[Crossref]

Chernyshova, M. A.

A. A. Krylov, M. A. Chernyshova, D. S. Chernykh, A. K. Senatorov, I. M. Tupitsyn, P. G. Kryukov, and E. M. Dianov, “High-power thulium-doped fibre laser with intracavity dispersion management,” Quantum Electron. 42(5), 427–431 (2012).
[Crossref]

Chicklis, E. P.

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

K. Kieu, W. H. Renninger, A. Chong, and F. W. Wise, “Sub-100 fs pulses at watt-level powers from a dissipative-soliton fiber laser,” Opt. Lett. 34(5), 593–595 (2009).
[Crossref] [PubMed]

Clark, W. G.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Creeden, D.

Dennis, M. L.

M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30(6), 1469–1477 (1994).
[Crossref]

Desaix, M.

Dianov, E. M.

A. A. Krylov, M. A. Chernyshova, D. S. Chernykh, A. K. Senatorov, I. M. Tupitsyn, P. G. Kryukov, and E. M. Dianov, “High-power thulium-doped fibre laser with intracavity dispersion management,” Quantum Electron. 42(5), 427–431 (2012).
[Crossref]

Dudley, J. M.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26 Pt 1), 6010–6013 (2000).
[Crossref] [PubMed]

Duling, I. N.

M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30(6), 1469–1477 (1994).
[Crossref]

Engelbrecht, M.

Fedoruk, M. P.

S. K. Turitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26 Pt 1), 6010–6013 (2000).
[Crossref] [PubMed]

Finot, C.

Fried, N. M.

N. M. Fried, “Thulium fiber laser lithotripsy: An in vitro analysis of stone fragmentation using a modulated 110-watt Thulium fiber laser at 1.94 microm,” Lasers Surg. Med. 37(1), 53–58 (2005).
[Crossref] [PubMed]

Fuchs, F.

Gaida, C.

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

Gebhardt, M.

Gordon, J. P.

Gumenyuk, R.

Harvey, J. D.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26 Pt 1), 6010–6013 (2000).
[Crossref] [PubMed]

Haus, H.

L. Nelson, E. Ippen, and H. Haus, “Broadly tunable sub-500 fs pulses from an additive-pulse mode-locked thulium-doped fiber ring laser,” Appl. Phys. Lett. 67(1), 19–21 (1995).
[Crossref]

Haus, H. A.

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65(2), 277–294 (1997).
[Crossref]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6(6), 1159–1166 (1989).
[Crossref]

Haxsen, F.

Ilday, F. O.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Ilday, F. Ö.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton–similariton fibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

Ippen, E.

L. Nelson, E. Ippen, and H. Haus, “Broadly tunable sub-500 fs pulses from an additive-pulse mode-locked thulium-doped fiber ring laser,” Appl. Phys. Lett. 67(1), 19–21 (1995).
[Crossref]

Ippen, E. P.

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65(2), 277–294 (1997).
[Crossref]

Jansen, F.

Jauregui, C.

Jiang, M.

Johnson, A. M.

Jones, D. J.

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65(2), 277–294 (1997).
[Crossref]

Ketteridge, P. A.

Kieu, K.

K. Kieu, W. H. Renninger, A. Chong, and F. W. Wise, “Sub-100 fs pulses at watt-level powers from a dissipative-soliton fiber laser,” Opt. Lett. 34(5), 593–595 (2009).
[Crossref] [PubMed]

K. Kieu and F. W. Wise, “Soliton thulium-doped fiber laser with carbon nanotube saturable absorber,” IEEE Photon. Technol. Lett. 21(3), 128–130 (2009).
[Crossref] [PubMed]

Kracht, D.

Kruglov, V. I.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26 Pt 1), 6010–6013 (2000).
[Crossref] [PubMed]

Krylov, A. A.

A. A. Krylov, M. A. Chernyshova, D. S. Chernykh, A. K. Senatorov, I. M. Tupitsyn, P. G. Kryukov, and E. M. Dianov, “High-power thulium-doped fibre laser with intracavity dispersion management,” Quantum Electron. 42(5), 427–431 (2012).
[Crossref]

Kryukov, P. G.

A. A. Krylov, M. A. Chernyshova, D. S. Chernykh, A. K. Senatorov, I. M. Tupitsyn, P. G. Kryukov, and E. M. Dianov, “High-power thulium-doped fibre laser with intracavity dispersion management,” Quantum Electron. 42(5), 427–431 (2012).
[Crossref]

Kutz, J. N.

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

Li, H.

H. Xia, H. Li, X. Zhang, S. Zhang, X. Tang, and Y. Liu, “Characteristics of dissipative solitons in an all-fiber thulium-doped fiber ring laser,” Opt. Eng. 52(5), 054201 (2013).
[Crossref]

Li, M. S.

Q. Q. Wang, T. Chen, M. S. Li, B. T. Zhang, Y. F. Lu, and K. P. Chen, “All-fiber ultrafast thulium-doped fiber ring laser with dissipative soliton and noise-like output in normal dispersion by single-wall carbon nanotubes,” Appl. Phys. Lett. 103(1), 011103 (2013).
[Crossref]

Limpert, J.

Lisak, M.

Liu, J.

Liu, Y.

H. Xia, H. Li, X. Zhang, S. Zhang, X. Tang, and Y. Liu, “Characteristics of dissipative solitons in an all-fiber thulium-doped fiber ring laser,” Opt. Eng. 52(5), 054201 (2013).
[Crossref]

Lu, Y. F.

Q. Q. Wang, T. Chen, M. S. Li, B. T. Zhang, Y. F. Lu, and K. P. Chen, “All-fiber ultrafast thulium-doped fiber ring laser with dissipative soliton and noise-like output in normal dispersion by single-wall carbon nanotubes,” Appl. Phys. Lett. 103(1), 011103 (2013).
[Crossref]

Matsas, V. J.

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, “Self-starting passively mode-locked fiber ring soliton laser exploiting nonlinear polarization rotation,” Electron. Lett. 28(15), 1391–1393 (1992).
[Crossref]

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[Crossref]

McCarthy, J. C.

Morgner, U.

Nelson, L.

L. Nelson, E. Ippen, and H. Haus, “Broadly tunable sub-500 fs pulses from an additive-pulse mode-locked thulium-doped fiber ring laser,” Appl. Phys. Lett. 67(1), 19–21 (1995).
[Crossref]

Nelson, L. E.

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65(2), 277–294 (1997).
[Crossref]

Newson, T. P.

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, “Self-starting passively mode-locked fiber ring soliton laser exploiting nonlinear polarization rotation,” Electron. Lett. 28(15), 1391–1393 (1992).
[Crossref]

Okhotnikov, O. G.

Oktem, B.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton–similariton fibre laser,” Nat. Photonics 4(5), 307–311 (2010).
[Crossref]

Payne, D. N.

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, “Self-starting passively mode-locked fiber ring soliton laser exploiting nonlinear polarization rotation,” Electron. Lett. 28(15), 1391–1393 (1992).
[Crossref]

Petropoulos, P.

Pollak, T. M.

Provost, L.

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H. Xia, H. Li, X. Zhang, S. Zhang, X. Tang, and Y. Liu, “Characteristics of dissipative solitons in an all-fiber thulium-doped fiber ring laser,” Opt. Eng. 52(5), 054201 (2013).
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Appl. Phys. B (1)

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65(2), 277–294 (1997).
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V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, “Self-starting passively mode-locked fiber ring soliton laser exploiting nonlinear polarization rotation,” Electron. Lett. 28(15), 1391–1393 (1992).
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IEEE Photon. Technol. Lett. (1)

K. Kieu and F. W. Wise, “Soliton thulium-doped fiber laser with carbon nanotube saturable absorber,” IEEE Photon. Technol. Lett. 21(3), 128–130 (2009).
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J. Opt. Soc. Am. B (2)

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Opt. Eng. (1)

H. Xia, H. Li, X. Zhang, S. Zhang, X. Tang, and Y. Liu, “Characteristics of dissipative solitons in an all-fiber thulium-doped fiber ring laser,” Opt. Eng. 52(5), 054201 (2013).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rep. (1)

S. K. Turitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

Phys. Rev. A (1)

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

Phys. Rev. Lett. (2)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26 Pt 1), 6010–6013 (2000).
[Crossref] [PubMed]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Quantum Electron. (2)

A. A. Krylov, M. A. Chernyshova, D. S. Chernykh, A. K. Senatorov, I. M. Tupitsyn, P. G. Kryukov, and E. M. Dianov, “High-power thulium-doped fibre laser with intracavity dispersion management,” Quantum Electron. 42(5), 427–431 (2012).
[Crossref]

I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013).
[Crossref]

Other (2)

H. H. Li, J. Liu, Z. C. Cheng, J. Xu, F. Z. Tan, and P. Wang, “Characteristics of pulses in passively mode-locked thulium-doped fiber laser,” in Advanced Solid State Lasers (Optical Society of America, 2014), paper ATh2A–34.

C. Rudy, M. Digonnet, R. Byer, S. Jiang, and Q. Wang, “Thulium-doped Germanosilicate Mode-locked Fiber Lasers,” in Lasers, Sources, and Related Photonic Devices, OSA Technical Digest (CD) (Optical Society of America, 2012), paper FTh4A.4.

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Figures (12)

Fig. 1
Fig. 1 Schematic setup of passively mode-locked thulium-doped fiber laser. SESAM, semiconductor saturable absorber mirror; WDM, 1550/2000 nm wavelength division multiplexer coupler; HR, high-reflectivity broadband mirror.
Fig. 2
Fig. 2 (a) Stable pulse train of the conventional soliton thulium-doped fiber laser. (b) Average output power with the increase of incident pump power.
Fig. 3
Fig. 3 (a) Optical spectrum (b) Autocorrelation trace at maximum average output power of the conventional soliton in thulium-doped fiber laser. Dashed curve: simulated fit.
Fig. 4
Fig. 4 Schematic of the dispersion-managed thulium-doped fiber laser. TDF, thulium-doped fiber; SA, saturable absorber; OC, output coupler; SMF, single mode fiber; NDF, normal dispersion fiber.
Fig. 5
Fig. 5 Evolution in the temporal domain from white noise to steady solution.
Fig. 6
Fig. 6 (a) Pulse shape. Inset: dechirped pulse; Blue curve: frequency chirp; Red dashed curve: Gaussian fit. (b) Optical spectrum of the dissipative dispersion-managed solitons.
Fig. 7
Fig. 7 The dissipative dispersion-managed solitons: the intra-cavity evolutions of pulse duration and spectral bandwidth along the cavity position.
Fig. 8
Fig. 8 (a) Pulse shape. Red dashed curve: Gaussian fit; Blue curve: frequency chirp. (b) Optical spectrum of the conservative dispersion-managed solitons. Red dashed curve: Gaussian fit.
Fig. 9
Fig. 9 The conservative dispersion-managed solitons: the intra-cavity evolutions of pulse duration and spectral bandwidth along the cavity position.
Fig. 10
Fig. 10 Schematic of the thulium-doped fiber ring laser. TDF, thulium-doped fiber; SMF, single mode fiber; NDF, normal dispersion fiber; SA, saturable absorber; Filter, 30 nm Gaussian filter.
Fig. 11
Fig. 11 (a) Pulse shape and (b) optical spectrum after the NDF. Inset: frequency chirp; Blue dashed curve: parabolic fit. (c) Pulse shape and (d) optical spectrum after the Gaussian filter. Red dotted curve: Gaussian fit.
Fig. 12
Fig. 12 The parabolic pulses: the intra-cavity evolutions of pulse duration and spectral bandwidth along the cavity position.

Tables (2)

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Table 1 The parameters of the numerical model

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Table 2 The parameters of the numerical model

Equations (6)

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i A z β 2 2 2 A t 2 +γ( A(z,t) R(τ) | A(z,tτ) | 2 dτ )=ig(z)A+ ig(z) Δ Ω g 2 2 A t 2 i α 2 A
g(z)= g 0 1+ E(z) / E Sat
Δ Ω g =2πc Δ λ g / λ 0 2
R(t)=(1 f R )δ(t)+ f R h R (t)
h R (t)= τ 1 2 + τ 2 2 τ 1 τ 2 2 exp(t/ τ 2 )sin(t/ τ 1 )
A out = A in [ R unsat +ΔR ΔR (1+ | A(t) | 2 / P sat ) ]

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