Abstract

It is demonstrated by numerical simulations that the CS2-filled dual-core fiber coupler with appropriate parameters can provide single-mode operation, normal dispersions, low loss and high nonlinearities in 1550-nm and 2000-nm wavelength windows, which can contribute to a saturable absorber (SA) with short fiber length and low power needed for nonlinearity-induced saturation. The effects of stimulated Raman scattering (SRS) play a key role in the process of nonlinearity-induced saturation. The numerical results indicate that the SAs can be employed in the mode-locking fiber lasers with self-similar (SS) or dissipative soliton (DS) operations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (2)

Y. Tang, A. Chong, and F. W. Wise, “Generation of 8 nJ pulses from a normal-dispersion thulium fiber laser,” Opt. Lett. 40(10), 2361–2364 (2015).
[Crossref] [PubMed]

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (2)

2012 (1)

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]

2011 (1)

2010 (2)

2009 (1)

2008 (2)

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25(2), 140–148 (2008).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).
[Crossref]

2007 (1)

2006 (1)

2005 (2)

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005).
[Crossref]

J. R. Buckley, F. W. Wise, F. Ö. Ilday, and T. Sosnowski, “Femtosecond fiber lasers with pulse energies above 10 nJ,” Opt. Lett. 30(14), 1888–1890 (2005).
[Crossref] [PubMed]

2004 (2)

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]

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]

2003 (2)

F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 µm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003).
[Crossref] [PubMed]

A. Samoc, “Dispersion of refractive properties of solvents: Chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. 94(9), 6167–6174 (2003).
[Crossref]

1995 (2)

1992 (2)

1988 (1)

1984 (1)

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18(10), 1580–1583 (1982).
[Crossref]

1972 (1)

Buckley, J. R.

J. R. Buckley, F. W. Wise, F. Ö. Ilday, and T. Sosnowski, “Femtosecond fiber lasers with pulse energies above 10 nJ,” Opt. Lett. 30(14), 1888–1890 (2005).
[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]

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]

F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 µm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003).
[Crossref] [PubMed]

Chen, C.-J.

Chiang, K. S.

Chichkov, N. B.

Chong, A.

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]

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]

F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 µm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003).
[Crossref] [PubMed]

Doran, N. J.

Doty, S. L.

Fleming, J. W.

Fork, R. L.

Giessen, H.

Haus, J. W.

Hausmann, K.

Huang, C.

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

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]

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.

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18(10), 1580–1583 (1982).
[Crossref]

Kieu, K.

Kracht, D.

Lim, H.

Limpert, J.

Liu, A. Q.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005).
[Crossref]

Mafi, A.

Menyuk, C. R.

Morgner, U.

Nazemosadat, E.

Neumann, J.

Oh, Y.

Oktem, B.

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

Ortaç, B.

Pricking, S.

Renninger, W. H.

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]

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25(2), 140–148 (2008).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).
[Crossref]

Samoc, A.

A. Samoc, “Dispersion of refractive properties of solvents: Chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. 94(9), 6167–6174 (2003).
[Crossref]

Schreiber, T.

Shang, W.

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

Snyder, A. W.

Sosnowski, T.

Tang, D. Y.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005).
[Crossref]

Tang, Y.

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

Y. Tang, A. Chong, and F. W. Wise, “Generation of 8 nJ pulses from a normal-dispersion thulium fiber laser,” Opt. Lett. 40(10), 2361–2364 (2015).
[Crossref] [PubMed]

Teipel, J.

Tünnermann, A.

Ülgüdür, C.

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

Vieweg, M.

Wai, P. K. A.

Walton, D. T.

Wandt, D.

Wang, C.

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

Winful, H. G.

Wise, F. W.

Y. Tang, A. Chong, and F. W. Wise, “Generation of 8 nJ pulses from a normal-dispersion thulium fiber laser,” Opt. Lett. 40(10), 2361–2364 (2015).
[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]

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]

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25(2), 140–148 (2008).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).
[Crossref]

J. R. Buckley, F. W. Wise, F. Ö. Ilday, and T. Sosnowski, “Femtosecond fiber lasers with pulse energies above 10 nJ,” Opt. Lett. 30(14), 1888–1890 (2005).
[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]

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]

F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 µm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003).
[Crossref] [PubMed]

Wood, D.

Xu, J.

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

Yang, N.

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

Zhang, R.

Zhao, B.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005).
[Crossref]

Zhao, L. M.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18(10), 1580–1583 (1982).
[Crossref]

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

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]

J. Appl. Phys. (1)

A. Samoc, “Dispersion of refractive properties of solvents: Chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. 94(9), 6167–6174 (2003).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

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

Opt. Express (4)

Opt. Lett. (10)

Y. Tang, A. Chong, and F. W. Wise, “Generation of 8 nJ pulses from a normal-dispersion thulium fiber laser,” Opt. Lett. 40(10), 2361–2364 (2015).
[Crossref] [PubMed]

E. Nazemosadat and A. Mafi, “Nonlinear switching in a two-concentric-core chalcogenide glass optical fiber for passively mode-locking a fiber laser,” Opt. Lett. 39(16), 4675–4678 (2014).
[Crossref] [PubMed]

K. S. Chiang, “Intermodal dispersion in two-core optical fibers,” Opt. Lett. 20(9), 997–999 (1995).
[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]

N. B. Chichkov, K. Hausmann, D. Wandt, U. Morgner, J. Neumann, and D. Kracht, “High-power dissipative solitons from an all-normal dispersion erbium fiber oscillator,” Opt. Lett. 35(16), 2807–2809 (2010).
[Crossref] [PubMed]

N. J. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13(1), 56–58 (1988).
[Crossref] [PubMed]

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17(6), 417–419 (1992).
[Crossref] [PubMed]

H. G. Winful and D. T. Walton, “Passive mode locking through nonlinear coupling in a dual-core fiber laser,” Opt. Lett. 17(23), 1688–1690 (1992).
[Crossref] [PubMed]

F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 µm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003).
[Crossref] [PubMed]

J. R. Buckley, F. W. Wise, F. Ö. Ilday, and T. Sosnowski, “Femtosecond fiber lasers with pulse energies above 10 nJ,” Opt. Lett. 30(14), 1888–1890 (2005).
[Crossref] [PubMed]

Phys. Rev. A (2)

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005).
[Crossref]

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).
[Crossref]

Phys. Rev. Lett. (2)

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]

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]

Sci. Rep. (1)

C. Huang, C. Wang, W. Shang, N. Yang, Y. Tang, and J. Xu, “Developing high energy dissipative soliton fiber lasers at 2 micron,” Sci. Rep. 5(1), 13680 (2015).
[Crossref] [PubMed]

Other (1)

G. P. Agrawal, Applications of Nonlinear Fiber Optics, Second ed. (Elsevier, 2009).

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Figures (24)

Fig. 1
Fig. 1 The cross section of the CS2-filled dual-core silica fiber, where D and d represent the diameter of the hollow core and core-to-core spacing, respectively. n1 and n2 are the refractive indices for CS2 and silica. The right part shows n1 [16] and n2 [17] as functions of wavelength.
Fig. 2
Fig. 2 The properties of CS2-filled single-core silica fiber with different core diameters, including (a) the normalized frequency V, (b) loss, (c) group velocity dispersion (GVD) D(λ) and (d) nonlinear parameter γ(λ) of the foundamental mode (FM) in the fiber.
Fig. 3
Fig. 3 The coupling length Lc as a function of wavelength for the CS2-filled two-core fibers with D = 1.8 μm (a) and D = 2.2 μm (b).
Fig. 4
Fig. 4 The sketch of the SA based on one segment of CS2-filled two-core fiber, where Lc is the coupling length at the center wavelength.
Fig. 5
Fig. 5 The output powers P1(Lc) (blue) and P2(Lc) (red) as functions of wavelengths in the CW–input case when P1(0) = 20 W (a), 50 W (b), 100 W (c), 300 W (d), 600 W (e) and 1000 W (f), respectively. D = 1.8 μm, d = 5.5 μm, and Lc = 64.5 mm at the 1550 nm operating wavelength.
Fig. 6
Fig. 6 The output powers P1(Lc) (blue) and P2(Lc) (red) as functions of wavelengths in the CW–input case when P1(0) = 20 W (a), 50 W (b), 100 W (c), 300 W (d), 600 W (e) and 1000 W (f), respectively. D = 2.2 μm, d = 6.5 μm, and Lc = 22.3 mm at the 2000 nm operating wavelength.
Fig. 7
Fig. 7 The curves of transmission Tr as a function of peak power P0 (a), output pulse |A1(Lc,T)|2 (b), output spectra |A1(Lc, λ)|2 (c) and |A2(Lc, λ)|2 (d) with P0 = 104 W for different TFWHM of input pulses, where D = 1.8 μm, d = 3 μm, and Lc = 0.41 mm at center wavelength of initial pulse, i.e., λ0 = 1550 nm.
Fig. 8
Fig. 8 The curves of transmission Tr as a function of peak power P0 (a), output pulse |A1(Lc,T)|2 (b), output spectra |A1(Lc, λ)|2 (c) and |A2(Lc, λ)|2 (d) with P0 = 6000 W for different TFWHM of input pulses, where D = 1.8 μm, d = 5 μm, and Lc = 23.7 mm at center wavelength of initial pulse, i.e., λ0 = 1550 nm.
Fig. 9
Fig. 9 The curves of transmission Tr as a function of peak power P0 (a), output pulse |A1(Lc,T)|2 (b), output spectra |A1(Lc, λ)|2 (c) and |A2(Lc, λ)|2 (d) with P0 = 104 W for different TFWHM of input pulses, where D = 2.2 μm, d = 4 μm, and Lc = 0.68 mm at center wavelength of initial pulse, i.e., λ0 = 2000 nm.
Fig. 10
Fig. 10 The curves of transmission Tr as a function of peak power P0 (a), output pulse |A1(Lc,T)|2 (b), output spectra |A1(Lc, λ)|2 (c) and |A2(Lc, λ)|2 (d) with P0 = 6000 W for different TFWHM of input pulses, where D = 2.2 μm, d = 6 μm, and Lc = 15 mm at center wavelength of initial pulse, i.e., λ0 = 2000 nm.
Fig. 11
Fig. 11 The curves of transmission Tr as a function of peak power P0 (a) and output pulse |A1(Lc,T)|2 with P0 = 10000 W (b) for different TFWHM of input pulses, where D = 1.8 μm, d = 5 μm, Lc = 23.7 mm, and λ0 = 1550 nm. The dashed curves represent the case of neglecting the effect of SRS.
Fig. 12
Fig. 12 The curves of transmission Tr as a function of peak power P0 (a) and output pulse |A1(Lc,T)|2 with P0 = 15000 W (b) for different TFWHM of input pulses, where D = 2.2 μm, d = 6 μm, Lc = 15 mm and λ0 = 2000 nm. The dashed curves represent the case of neglecting the effect of SRS.
Fig. 13
Fig. 13 The curves of transmission Tr as a function of peak power P0 (a) and output pulse |A1(Lc,T)|2 with P0 = 10000 W (b) for different TFWHM of input pulses, where D = 1.8 μm, d = 5 μm, Lc = 23.7 mm, and λ0 = 1550 nm. The dashed curves represent the case of neglecting the self-steepening effect.
Fig. 14
Fig. 14 The curves of transmission Tr as a function of peak power P0 (a) and output pulse |A1(Lc,T)|2 with P0 = 15000 W (b) for different TFWHM of input pulses, where D = 2.2 μm, d = 6 μm, Lc = 15 mm and λ0 = 2000 nm. The dashed curves represent the case of neglecting the self-steepening effect.
Fig. 15
Fig. 15 The maximum transmission Trmax (a)(b) and the corresponding peak power P0I (c)(d) of the input pulse as functions of core-to-core spacing d and temporal width TFWHM of the input pulse, where D = 1.8 μm, λ0 = 1550 nm (a)(c), and D = 2.2 μm, λ0 = 2000 nm (b)(d).
Fig. 16
Fig. 16 Simple schematics of the Er-doped fiber laser in two configurations.
Fig. 17
Fig. 17 Simple schematics of the Tm-doped fiber laser in two configurations.
Fig. 18
Fig. 18 Transient evolutions of pulses at the output end of the Er fibers in the temporal domain from white noise to the steady state for the type-A (a) and type-B schematics (b), where Δωf is taken to be 7 nm of wavelength bandwidth. D = 1.8 μm, d = 5.5 μm, Lc = 64.3 mm at λ0 = 1550 nm for the SA. The other parameters can be seen in Tab. 1.
Fig. 19
Fig. 19 The evolutions of pulses along the ring cavity in temporal (a)(c) and spectral (b)(d) domains for the type-A (a)(b) and type-B schematics (c)(d), where Δωf is taken to be 7 nm of wavelength bandwidth. For the SA, D = 1.8 μm, d = 5.5 μm, Lc = 64.3 mm at λ0 = 1550 nm. The other parameters can be seen in Tab. 1.
Fig. 20
Fig. 20 Evolution of the kurtosis with respect to the position in the ring cavity (a)(c), and the temporal shapes (blue) with frequency chirps compared with expected parabolic pulses (red) at the output end of Er fiber (b)(d) for the type-A (a)(b) and type-B schematics (c)(d), where Δωf is taken to be 7 nm of wavelength bandwidth. For the SA, D = 1.8 μm, d = 5.5 μm, Lc = 64.3 mm at λ0 = 1550 nm. The other parameters can be seen in Tab. 1.
Fig. 21
Fig. 21 The temporal shapes with frequency chirps (a)(b) and spectra (c)(d) of the output pulses with different values of d in the type-A (a)(c) and type-B (b)(d) fiber lasers, where Δωf is taken to be 7 nm of wavelength bandwidth. For the SA, D = 1.8 μm, and λ0 = 1550 nm. The other parameters can be seen in Tab. 1.
Fig. 22
Fig. 22 The evolutions of pulses along the ring cavity in temporal (a)(c) and spectral (b)(d) domains for the type-A (a)(b) and type-B (c)(d) schematics, where Δωf is taken to be 15 nm of wavelength bandwidth. For the SA, D = 2.2 μm, d = 6.5 μm, Lc = 50.1 mm at λ0 = 1920 nm. The other parameters can be seen in Tab. 2.
Fig. 23
Fig. 23 The temporal shapes (blue) with frequency chirps compared with expected parabolic pulses (red) at the output end of Tm fiber (a)(b) and spectra (c)(d) for the type-A (a)(c) and type-B (b)(d) schematics, where Δωf is taken to be 15 nm of wavelength bandwidth. For the SA, D = 2.2 μm, d = 6.5 μm, Lc = 50.1 mm at λ0 = 1920 nm. The other parameters can be seen in Tab. 2.
Fig. 24
Fig. 24 The temporal shapes with frequency chirps (a)(b) and spectra (c)(d) of the output pulses with different values of d in the type-A (a)(c) and type-B (b)(d) schematics, where Δωf is taken to be 15 nm of wavelength bandwidth. For the SA, D = 2.2 μm, and λ0 = 1920 nm. The other parameters can be seen in Table 2.

Tables (2)

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Table 1 The Fiber Parameters in Er-Doped Fiber Lasers [6]

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Table 2 The Fiber Parameters in Tm-Doped Fiber Lasers [26]

Equations (11)

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L C = π 2κ
A 1 z + α 2 A 1 + κ 1 A 2 T k2 i k+1 k! β k A 1 T k =iκ A 2 +iγ(1+ 1 ω 0 T )( A 1 (z,T) 0 R( t )| A 1 (z,T t ) | 2 d t )
A 2 z + α 2 A 2 + κ 1 A 1 T + k2 i k+1 k! β k A 2 T k =iκ A 1 +iγ(1+ i ω 0 T )( A 2 (z,T) 0 R( t )| A 2 (z,T t ) | 2 d t )
d A 1 dz =iκ A 2 +iγ | A 1 | 2 A 1
d A 2 dz =iκ A 1 +iγ | A 2 | 2 A 2
P 1 (z)= | A 1 (z) | 2 =0.5 P total [ 1+cn(2κz|m) ]; m=( P total / P c ) 2
P total = P 1 (z)+ P 2 (z); P 2 (z)= | A 2 (z) | 2
P c =4κ/γ
T r = E out E in = 0 | A 1 ( L c ,T) | 2 dT 0 | A 1 (0,T) | 2 dT
A 1 z g 2 A 1 k2 i k+1 k! β k A 1 T k =iγ( A 1 (z,T) 0 R( t )| A 1 (z,T t ) | 2 d t )
g= g 0 1+ E pulse / E sat + (ω ω 0 ) 2 /Δ ω g 2

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