Abstract

With the combining effects of the fiber birefringence induced round-trip phase variation and the gain profile reshaping induced spectral filtering in the Erbium-doped fiber laser (EDFL) cavity, the mechanism corresponding to the central wavelength tunability of the EDFL passively mode-locked by nonlinear polarization rotation is explored. Bending the intracavity fiber induces the refractive index difference between orthogonal axes, which enables the dual-band central wavelength shift of 2.9 nm at 1570 nm region and up to 10.2 nm at 1600 nm region. The difference between the wavelength shifts at two bands is attributed to the gain dispersion decided by the gain spectral curvature of the EDFA, and the spacing between two switchable bands is provided by the birefringence induced variation on phase delay which causes transmittance variation. In addition, the central wavelength shift can also be controlled by varying the pumping geometry. At 1570 nm regime, an offset of up to 5.9 nm between the central wavelengths obtained under solely forward or backward pumping condition is observed, whereas the bidirectional pumping scheme effectively compensates the gain spectral reshaping effects to minimize the central wavelength shift. In contrast, the wavelength offset shrinks to only 1.1 nm when mode-locking at 1600 nm under single-sided pumping, as the gain profile strongly depends on the spatial distribution of the excited erbium ions under different pumping schemes. Except the birefringence variation and the gain spectral filtering phenomena, the gain-saturation mechanism induced refractive index change and its influence to the dual-band central wavelength tunability are also observed and analyzed.

© 2014 Optical Society of America

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    [Crossref]
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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] [PubMed]

2013 (2)

Y.-H. Lin, Y.-C. Chi, and G.-R. Lin, “Nanoscale charcoal powder induced saturable absorption and mode-locking of a low-gain erbium-doped fiber-ring laser,” Laser Phys. Lett. 10(5), 055105 (2013).
[Crossref]

Y.-H. Lin, J.-Y. Lo, W.-H. Tseng, C.-I. Wu, and G.-R. Lin, “Self-amplitude and self-phase modulation of the charcoal mode-locked erbium-doped fiber lasers,” Opt. Express 21(21), 25184–25196 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (2)

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98(7), 073106 (2011).
[Crossref]

G.-R. Lin and Y.-C. Lin, “Directly exfoliated and imprinted graphite nano-particle saturable absorber for passive mode-locking erbium-doped fiber laser,” Laser Phys. Lett. 8(12), 880–886 (2011).
[Crossref]

2010 (1)

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

2009 (2)

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Z. Han, B. Qiaoliang, T. Dingyuan, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

2008 (1)

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

2007 (1)

2004 (1)

2000 (1)

1994 (1)

1992 (3)

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28(24), 2226–2228 (1992).
[Crossref]

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

H. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (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(10), 2068–2076 (1991).
[Crossref]

G. P. Agrawal, “Effect of gain dispersion on ultrashort pulse amplification in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27(6), 1843–1849 (1991).
[Crossref]

1989 (1)

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, “Effect of gain dispersion on ultrashort pulse amplification in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27(6), 1843–1849 (1991).
[Crossref]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Bao, Q.

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Chen, Y.

Cheng, W.-H.

Chi, Y.-C.

Y.-H. Lin, Y.-C. Chi, and G.-R. Lin, “Nanoscale charcoal powder induced saturable absorption and mode-locking of a low-gain erbium-doped fiber-ring laser,” Laser Phys. Lett. 10(5), 055105 (2013).
[Crossref]

Demokan, M. S.

Dingyuan, T.

Z. Han, B. Qiaoliang, T. Dingyuan, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Farrell, G.

Ferrari, A. C.

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98(7), 073106 (2011).
[Crossref]

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Fontana, F.

Franco, P.

Fujimoto, J. G.

H. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8(10), 2068–2076 (1991).
[Crossref]

Han, Z.

Z. Han, B. Qiaoliang, T. Dingyuan, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Hasan, T.

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98(7), 073106 (2011).
[Crossref]

Haus, H.

H. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

Haus, H. A.

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28(24), 2226–2228 (1992).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8(10), 2068–2076 (1991).
[Crossref]

Hennrich, F.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Huang, P. L.

Huang, S.-H.

Ippen, E. P.

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28(24), 2226–2228 (1992).
[Crossref]

H. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8(10), 2068–2076 (1991).
[Crossref]

Jablonski, M.

Knize, R. J.

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Kuo, H.-H.

Li, L.-J.

Lin, G.-R.

Y.-H. Lin, J.-Y. Lo, W.-H. Tseng, C.-I. Wu, and G.-R. Lin, “Self-amplitude and self-phase modulation of the charcoal mode-locked erbium-doped fiber lasers,” Opt. Express 21(21), 25184–25196 (2013).
[Crossref] [PubMed]

Y.-H. Lin, Y.-C. Chi, and G.-R. Lin, “Nanoscale charcoal powder induced saturable absorption and mode-locking of a low-gain erbium-doped fiber-ring laser,” Laser Phys. Lett. 10(5), 055105 (2013).
[Crossref]

Y.-H. Lin and G.-R. Lin, “Free-standing nano-scale graphite saturable absorber for passively mode-locked erbium doped fiber ring laser,” Laser Phys. Lett. 9(5), 398–404 (2012).
[Crossref]

P. L. Huang, S.-C. Lin, C.-Y. Yeh, H.-H. Kuo, S.-H. Huang, G.-R. Lin, L.-J. Li, C.-Y. Su, and W.-H. Cheng, “Stable mode-locked fiber laser based on CVD fabricated graphene saturable absorber,” Opt. Express 20(3), 2460–2465 (2012).
[Crossref] [PubMed]

G.-R. Lin and Y.-C. Lin, “Directly exfoliated and imprinted graphite nano-particle saturable absorber for passive mode-locking erbium-doped fiber laser,” Laser Phys. Lett. 8(12), 880–886 (2011).
[Crossref]

Lin, S.-C.

Lin, Y.-C.

G.-R. Lin and Y.-C. Lin, “Directly exfoliated and imprinted graphite nano-particle saturable absorber for passive mode-locking erbium-doped fiber laser,” Laser Phys. Lett. 8(12), 880–886 (2011).
[Crossref]

Lin, Y.-H.

Y.-H. Lin, Y.-C. Chi, and G.-R. Lin, “Nanoscale charcoal powder induced saturable absorption and mode-locking of a low-gain erbium-doped fiber-ring laser,” Laser Phys. Lett. 10(5), 055105 (2013).
[Crossref]

Y.-H. Lin, J.-Y. Lo, W.-H. Tseng, C.-I. Wu, and G.-R. Lin, “Self-amplitude and self-phase modulation of the charcoal mode-locked erbium-doped fiber lasers,” Opt. Express 21(21), 25184–25196 (2013).
[Crossref] [PubMed]

Y.-H. Lin and G.-R. Lin, “Free-standing nano-scale graphite saturable absorber for passively mode-locked erbium doped fiber ring laser,” Laser Phys. Lett. 9(5), 398–404 (2012).
[Crossref]

Lo, J.-Y.

Loh, K.

Z. Han, B. Qiaoliang, T. Dingyuan, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Loh, K. P.

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Lu, S.

Man, W. S.

Matsas, V. J.

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

Midrio, M.

Milne, W. I.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Newson, T. P.

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

Ni, Z.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Olsson, N. A.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Payne, D. N.

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

Popa, D.

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98(7), 073106 (2011).
[Crossref]

Qiaoliang, B.

Z. Han, B. Qiaoliang, T. Dingyuan, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Rajan, G.

Richardson, D. J.

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

Romagnoli, M.

Rozhin, A. G.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Scardaci, V.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Set, S. Y.

Shen, Z. X.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Su, C.-Y.

Sun, Z.

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98(7), 073106 (2011).
[Crossref]

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Tam, H. Y.

Tamura, K.

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28(24), 2226–2228 (1992).
[Crossref]

Tanaka, Y.

Tang, D.

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Tang, D. Y.

Torrisi, F.

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98(7), 073106 (2011).
[Crossref]

Tozzato, A.

Tseng, W.-H.

Wai, P. K. A.

Wang, F.

D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. 98(7), 073106 (2011).
[Crossref]

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Wang, P.

Wang, Q.

Wang, Y.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Wang, Z.

Wen, S.

White, I. H.

F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008).
[Crossref] [PubMed]

Wu, C.-I.

Yaguchi, H.

Yan, Y.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Yeh, C.-Y.

Zhang, H.

C. Zhao, Y. Zou, Y. Chen, Z. Wang, S. Lu, H. Zhang, S. Wen, and D. Y. Tang, “Wavelength-tunable picosecond soliton fiber laser with Topological Insulator: Bi2Se3 as a mode locker,” Opt. Express 20(25), 27888–27895 (2012).
[Crossref] [PubMed]

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Zhao, C.

Zhao, L.

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[Crossref]

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Figures (9)

Fig. 1
Fig. 1

The experimental setup of the NPRML-EDFL for investigating the central wavelength offset due to (a) the bended intracavity SMF circle and (b) the different pumping geometries.

Fig. 2
Fig. 2

The experimental setup for measuring the gain and ASE spectra of the EDFA.

Fig. 3
Fig. 3

(a) The gain spectrum and (b) the ASE spectrum of the EDFA in the experimental system.

Fig. 4
Fig. 4

The spectra and autocorrelation traces of the soliton pulses generated from the EDFL passively mode-locked at two wavelength regions of (a) 1570 nm and (b) 1600 nm, which are obtained at different conditions by bending the intracavity bended SMF circle with varying radius from 3 cm (left column) to 1 cm (right column).

Fig. 5
Fig. 5

The polarization state of the circulated pulse versus the fast and slow axes of the fiber in the NPRML-EDFL system; f and s, the fast and slow axis of the fiber; f’ and s’, the fast and slow axis of the fiber after the rotation; P, the polarization direction of the polarizer; A, the polarization direction of the analyzer.

Fig. 6
Fig. 6

The linear transmittance versus the total phase delay between the two polarization components with different Δθ values when assuming θ1 = π/4 and θ2 = θ1 + Δθ.

Fig. 7
Fig. 7

(a) Linear transmittance of the EDFL versus the phase delay between two orthogonal-polarization components changed by bending the intracavity fiber. (b) Optical spectrum corresponding to the intracavity fiber possessing the different bending radius at (a) 1570 nm regime and (b) 1600 nm regime.

Fig. 8
Fig. 8

Soliton spectra and autocorrelation traces obtained from the passively mode-locked EDFL at (a) 1570 nm and (b) 1600 nm regimes under different pumping conditions.

Fig. 9
Fig. 9

(a) The gain spectrum and the central wavelength of the passively mode-locked EDFL under different pumping geometry conditions. (b)(c) The optical spectra of the passively mode-locked EDFL at 1570 nm regime and 1600 nm regime under different pumping geometries.

Tables (2)

Tables Icon

Table 1 Parameters of Passively Mode-locked EDFL Soliton Pulses at (a) 1570 nm and (b) 1600 nm Regimes Obtained by Bending Intracavity SMF Circle at Different Radii

Tables Icon

Table 2 Parameters of Soliton Pulse Corresponding to Different Pump Geometry Obtained by Passively Mode-locked EDFL at (a) 1570 nm and (b) 1600nm Regimes

Equations (10)

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| T | 2 = sin 2 θ 1 sin 2 θ 2 + cos 2 θ 1 cos 2 θ 2 + 1 2 sin2 θ 1 sin2 θ 2 cos(Δϕ),
Δϕ=Δ ϕ PC +Δ ϕ LB +Δ ϕ NL ,
Δ ϕ LB =2π(1 δλ λ s ) L L b =2π(1 δλ λ s ) L λ s Δ n eff ,
Δ ϕ NL =2γLPcos(2 θ 1 )/3,
A z + 1 v g A t + i 2 β 2 2 A t 2 = 1 2 (1iα)gA 1 2 α int A,
g(ω) g p ( ω 0 )+g'(ω ω 0 )+ 1 2 g" (ω ω 0 ) 2 = g p [ 1+ 1 Ω g 2 (ω ω 0 ) 2 ].
gA= g p (1+ 1 Ω g 2 2 t 2 )A.
g p t = g o g p τ c g p | A | 2 E sat ,
g p (t)= g 0 exp[ t ( | A(z,t) | 2 / E sat )dt ],
A z + 1 v g A t + i 2 β 2 2 A t 2 + 1 2 α int A= 1 2 (1iα)( A+ 1 Ω g 2 2 A t 2 )[ g 0 exp( 1 E sat t | A(z,t) | 2 dt ) ].

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