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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 27045–27050
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Sideband-controllable mode-locking fiber laser based on chirped fiber Bragg gratings

Dongdong Han and Xueming Liu  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 27045-27050 (2012)
http://dx.doi.org/10.1364/OE.20.027045


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Abstract

We have proposed a sideband-controllable fiber soliton laser by means of chirped fiber Bragg gratings (CFBGs). Each side of the spectral sidebands of laser could be removed by using a CFBG with proper dispersion. Numerical simulations have well reproduced the experimental observations. The numerical and experimental investigations show that the generation of the unilateral sidebands is attributed to the CFBG-induced spectral filtering effect. Our work provides an effective way to manage conventional solitons with spectral sidebands.

© 2012 OSA

1. Introduction

With the compact cavity design, low cost, environmentally stable performance, and the capability of generating short pulses, passively mode-locked fiber lasers have attracted a great deal of research interests recently [1

1. X. Liu, “Interaction and motion of solitons in passively-mode-locked fiber lasers,” Phys. Rev. A 84(5), 053828 (2011). [CrossRef]

5

5. X. Liu, “Dissipative soliton evolution in ultra-large normal-cavity-dispersion fiber lasers,” Opt. Express 17(12), 9549–9557 (2009). [CrossRef] [PubMed]

]. Mode-lockers such as semiconductor saturable absorption mirrors [6

6. Y. Deng, M. Koch, F. Lu, G. Wicks, and W. Knox, “Colliding-pulse passive harmonic mode-locking in a femtosecond Yb-doped fiber laser with a semiconductor saturable absorber,” Opt. Express 12(16), 3872–3877 (2004). [CrossRef] [PubMed]

], carbon nanotubes [7

7. 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]

], graphenes [8

8. H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17(20), 17630–17635 (2009). [CrossRef] [PubMed]

], nonlinear optical loop mirrors [9

9. L. Yun and X. Liu, “Generation and Propagation of Bound-State Pulses in a Passively Mode-Locked Figure-Eight Laser,” IEEE Photon. J. 4(2), 512–519 (2012). [CrossRef]

], and nonlinear polarization rotation (NPR) techniques [10

10. X. Liu, “Mechanism of high-energy pulse generation without wave breaking in mode-locked fiber lasers,” Phys. Rev. A 82(5), 053808 (2010). [CrossRef]

, 11

11. L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal-dispersion fiber lasers,” Laser Phys. Lett. 7(1), 63–67 (2010). [CrossRef]

] have been used to achieve ultrashort pulses. Depending on the net dispersion of laser cavity, various pulses including conventional solitons (CSs) [12

12. X. Liu, “Soliton formation and evolution in passively-mode-locked lasers with ultralong anomalous-dispersion fibers,” Phys. Rev. A 84(2), 023835 (2011). [CrossRef]

], stretched pulses [13

13. K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, and J. G. Fujimoto, “Unidirectional ring resonators for self-starting passively mode-locked lasers,” Opt. Lett. 18(3), 220–222 (1993). [CrossRef] [PubMed]

], self-similar pulses [14

14. F. Ö. 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]

], and dissipative solitons [15

15. X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A 81(2), 023811 (2010). [CrossRef]

, 16

16. X. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A 82(6), 063834 (2010). [CrossRef]

] have been observed experimentally. According to the soliton theory, CSs are originated from the balance between group velocity dispersion and fiber nonlinearity [12

12. X. Liu, “Soliton formation and evolution in passively-mode-locked lasers with ultralong anomalous-dispersion fibers,” Phys. Rev. A 84(2), 023835 (2011). [CrossRef]

].

Once a CS is formed, discrete sidebands acting like spikes will generate and distribute on the both sides of the spectrum. The positions of the sidebands on the soliton spectrum are determined by the cavity dispersion, cavity length, and pulse duration [17

17. G. P. Agrawal, “Applications of Nonlinear Fiber Optics,” Fourth ed., (Academic Press, Boston, 2007).

, 18

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

]. So far, several techniques have been proposed to control the generation of the sidebands. Noske et al. have inserted a Fabry-Perot filter into the cavity to avoid the generation of soliton sidebands [19

19. D. U. Noske and J. R. Taylor, “Spectral and temporal stabilization of a diode-pumped ytterbium-erbium fiber soliton laser,” Electron. Lett. 29(25), 2200–2201 (1993). [CrossRef]

]. The sidebands could also be suppressed by rotating the polarization controllers (PCs) in cavities [20

20. Z. Luo, A. Luo, W. Xu, C. Song, Y. Gao, and W. Chen, “Sideband controllable soliton all-fiber ring laser passively mode-locked by nonlinear polarization rotation,” Laser Phys. Lett. 6(8), 582–585 (2009). [CrossRef]

]. However, both sides of spectral sidebands are suppressed or enhanced simultaneously. In addition, Katz et al. have reported the phenomenon of the unilateral sidebands when they utilizing chirped fiber Bragg grating (CFBG) for dispersion compensating in ytterbium (Yb3+) doped fiber laser [21

21. O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Opt. Commun. 269(1), 156–165 (2007). [CrossRef]

]. On the other hand, the net-normal fiber laser can deliver pulses without the sideband spectrum [4

4. L. R. Wang, X. M. Liu, Y. K. Gong, D. Mao, and H. Feng, “Ultra-broadband high-energy pulse generation and evolution in a compact erbium-doped all-fiber laser,” Laser Phys. Lett. 8(5), 376–381 (2011). [CrossRef]

, 5

5. X. Liu, “Dissipative soliton evolution in ultra-large normal-cavity-dispersion fiber lasers,” Opt. Express 17(12), 9549–9557 (2009). [CrossRef] [PubMed]

].

In this paper, we have proposed a sideband-controllable fiber soliton laser by means of CFBGs. On the case of no CFBG, the sidebands exist on both sides of the soliton spectrum. However, each side of the sidebands could be completely suppressed whereas the other side is retained by inserting a CFBG with suitable dispersion. Numerical simulations well confirm the experimental results. The theoretical and experimental investigations infer that the CFBG-induced spectral filtering effect contributes on the unilateral sidebands on the optical spectrum. Our work provides an effective way to control spectral sidebands of CSs.

2. Experimental setup

The proposed fiber laser system is shown schematically in Fig. 1
Fig. 1 Schematic diagram of the experimental fiber laser cavity.
. A 1480-nm Raman fiber laser is coupled into the laser with 1480/1550 nm wavelength-division-multiplexer (WDM). A 15-m-long erbium-doped fiber (EDF) with the dispersion parameter D of −9 ps/nm/km contributes to the gain media. An optical circulator (OC) is used to incorporate the CFBG into the cavity. Two CFBGs operating at 1550 nm with the 3-dB bandwidths of 7 and 12 nm are respectively inserted into the cavity. The dispersion parameters for the two CFBG are 5.7 ps/nm and 3.3 ps/nm, respectively. A polarization-sensitive isolator (PS-ISO) together with two PCs forms the NPR technique to implement mode locking operation. A 30% fiber coupler is used to output the signal. The other fibers are standard single-mode fiber (SMF) with the total length of ~39 m and D of 17 ps/nm/km. An optical spectrum analyzer, an autocorrelator, a 6-GHz oscilloscope, a radio-frequency (RF) analyzer and a 10-GHz photodetector are employed to monitor the laser output simultaneously.

3. Experimental results and analyses

With an appropriate PC state, self-started mode locking could be easily achieved when the pump power P is beyond a threshold value. Figure 2(a)
Fig. 2 (a) Optical spectrum, (b) autocorrelation trace, (c) oscilloscope trace, and (d) RF spectrum of CS in the fiber laser without CFBG.
shows the output spectrum of CSs at P = 100 mW when no CFBG is used. One can see that the spectral sidebands are symmetrically distributed at both sides of the spectrum. The central wavelength and 3-dB bandwidth of the CS are 1550 nm and 2.6 nm, respectively. The autocorrelation trace of the soliton is shown in Fig. 2(b). If a sech2 profile is assumed for fitting, the pulse duration is estimated as ~0.977 ps. The time-bandwidth product (TBP) is calculated as ~0.317, indicating that the pulse is nearly chirp free. The corresponding pulse train in Fig. 2(c) shows that the laser operates at multi-pulse mode-locking state. The slight nonuniformity of the measured pulse height is caused by the limit of the electronic detection system. The RF spectrum in Fig. 2(d) shows that signal/noise ratio is higher than 60 dB, indicating that stable mode locking is achieved. The fundamental frequency is ~3.81 MHz, corresponding to the total cavity length is 54 m.

Then, a CFBG with the reflection bandwidth of about 7 nm is inserted into the cavity, as shown in Fig. 1. At appropriate PC states, self-started mode locking could also be achieved. Figure 3
Fig. 3 (a) Spectrum and (b) autocorrelation trace of left unilateral sidebands CS pulse with 7-nm CFBG.
shows a typical state of mode-locked pulses. We can observe from Fig. 3(a) that the right side of the spectrum is very smooth and the sidebands are completely suppressed. However, the sidebands on the left side are still maintained. The central wavelength locates at 1552.3 nm, which is close to the right side of the CFBG reflection spectrum (i.e., 1553.5 nm). The 3-dB width of the spectrum and the pulse duration are estimated as 0.4 nm and 7.83 ps, respectively. The TBP is given by ~0.389, which indicates that the pulse is almost chirp-free. Here, the pulse duration is much larger than that of Fig. 2(b), which can be attributed to the large dispersion induced by the CFBG. The formation of the unilateral sidebands could be understood as follows. In the cavity without the CFBG, the interference between the soliton and dispersive waves forms the sidebands on both sides of the spectrum. Once the CFBG is added into the cavity, it works as a spectral filter that passes through certain optical wavelength bands. When the mode-locking wavelength is close to right end of the CFBG reflection spectrum, the right sidebands will be suppressed due to the strong spectral filtering effect. However, Kelly sidebands on left side are observed because the spectrum almost is unaffected by the filtering effect.

In our experiment, the sidebands on the left side could also be completely suppressed while that of right side is maintained by using a CFBG with the bandwidth of 12 nm. The corresponding spectrum and autocorrelation trace are shown in Fig. 4
Fig. 4 (a) Spectrum and (b) autocorrelation trace of right unilateral sidebands CS pulse with 12-nm CFBG.
. The central wavelength locates at 1544.3 nm, which is close to the left side of the CFBG reflection spectrum (i.e., 1544 nm). The 3-dB width of the spectrum and the pulse duration are estimated as 0.6 nm and 5.4 ps, respectively. The formation of unilateral sidebands on the right side can be explained similarly. We note that the two mode-locked operations have different operating wavelengths. Soliton only can be formed at certain wavelengths when the cavity anomalous dispersion and positive nonlinearity reach dynamic balance [12

12. X. Liu, “Soliton formation and evolution in passively-mode-locked lasers with ultralong anomalous-dispersion fibers,” Phys. Rev. A 84(2), 023835 (2011). [CrossRef]

]. In Fig. 3, the balance between the two effects is achieved and mode-locked operation is established at the longer wavelength of the 7-nm CFBG. However, when the 12-nm CFBG is inserted in the cavity, the cavity dispersion is changed evidently. Then, at the shorter wavelength of CFBG, the balance between dispersion and nonlinearity can be achieved and CSs are formed in cavity. The mechanism is very different from the Ref [21

21. O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Opt. Commun. 269(1), 156–165 (2007). [CrossRef]

], where the Yb3+ spectral gain characteristics play a key role. As a result, each side of the sidebands could be completely suppressed in our experiment.

4. Simulation results

Equation (1) in the numerical model is solved with a split-step Fourier method [15

15. X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A 81(2), 023811 (2010). [CrossRef]

, 23

23. X. Liu and B. Lee, “A fast method for nonlinear Schrödinger equation,” IEEE Photon. Technol. Lett. 15(11), 1549–1551 (2003). [CrossRef]

]. For the modeling, we use the following parameters in our simulations to match the experimental conditions. g0 = 6 dB/m, Ωg = 25 nm. γ = 3 W−1km−1, β2 = 11.5 ps2/km for EDF and γ = 1.3 W−1km−1, β2 = −21.6 ps2/km for SMF. The lengths of the EDF and SMF are 15 and 39 m, respectively. Each CFBG is written on a 15-mm-length SMF. The dispersion parameters for the 7-nm and 12-nm CFBG are 5.7 ps/nm and 3.3 ps/nm, respectively. The above parameters are obtained from the data sheets of fiber products.

The simulation starts from an arbitrary signal, and after a finite number of circulations it converges into a stable solution with appropriate parameters. Without adding CFBG, we can observe that the sidebands are symmetrically distributed on the both sides of the spectrum, as depicted by the black curve in Fig. 5(a)
Fig. 5 Optical spectra (black curves) and pulse profiles (blue curves) (a) without CFBG, (b) with 7-nm CFBG, and (c) with 12-nm CFBG.
. The blue curve shows the corresponding pulse profile. By taking CFBGs effects into account, unilateral sidebands on the left and right sides of the spectrum have been obtained. Figure 5(b) and Fig. 5(c) show the typical states of output pulses by using the 7-nm and 12-nm CFBGs, respectively. Because the mode-locked wavelength is close to the edge of the CFBG reflection spectrum, the sidebands are cut off and the corresponding spectrum exhibit smooth profile on one side. Numerical simulations confirm that the unilateral sidebands on the spectrum are caused by the CFBG-induced filtering effect.

Compared to Fig. 5(a), the pulse duration and the spectral width in Figs. 5(b) and 5(c) have been significantly increased and decreased, respectively. The increase of pulse duration attributes to the large dispersion induced by the additional CFBGs. Our results may provide a principle for managing the pulse duration in fiber laser. The numerical results have fully confirmed the experimental observations.

5. Conclusions

In this paper, we have proposed a sideband-controllable fiber soliton laser by using CFBGs. The typical CS with symmetrical spectral sidebands is observed when no CFBG is inserted into the cavity. However, unilateral sidebands on either the right or the left side of the spectrum are generated by using a CFBG with suitable dispersion. The formation of the unilateral sidebands could be attributed to the CFBG-induced spectral filtering effect. Numerical simulations have well reproduced the experimental results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 11204368, 10874239 and 10604066. Corresponding author (X. Liu). Tel.: + 862988881560; fax: + 862988887603; electronic mail: liuxueming72@yahoo.com and liuxm@opt.ac.cn.

References and links

1.

X. Liu, “Interaction and motion of solitons in passively-mode-locked fiber lasers,” Phys. Rev. A 84(5), 053828 (2011). [CrossRef]

2.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012). [CrossRef]

3.

D. Mao, X. M. Liu, L. R. Wang, X. H. Hu, and H. Lu, “Partially polarized wave-breaking-free dissipative soliton with super-broad spectrum in a mode-locked fiber laser,” Laser Phys. Lett. 8(2), 134–138 (2011). [CrossRef]

4.

L. R. Wang, X. M. Liu, Y. K. Gong, D. Mao, and H. Feng, “Ultra-broadband high-energy pulse generation and evolution in a compact erbium-doped all-fiber laser,” Laser Phys. Lett. 8(5), 376–381 (2011). [CrossRef]

5.

X. Liu, “Dissipative soliton evolution in ultra-large normal-cavity-dispersion fiber lasers,” Opt. Express 17(12), 9549–9557 (2009). [CrossRef] [PubMed]

6.

Y. Deng, M. Koch, F. Lu, G. Wicks, and W. Knox, “Colliding-pulse passive harmonic mode-locking in a femtosecond Yb-doped fiber laser with a semiconductor saturable absorber,” Opt. Express 12(16), 3872–3877 (2004). [CrossRef] [PubMed]

7.

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]

8.

H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17(20), 17630–17635 (2009). [CrossRef] [PubMed]

9.

L. Yun and X. Liu, “Generation and Propagation of Bound-State Pulses in a Passively Mode-Locked Figure-Eight Laser,” IEEE Photon. J. 4(2), 512–519 (2012). [CrossRef]

10.

X. Liu, “Mechanism of high-energy pulse generation without wave breaking in mode-locked fiber lasers,” Phys. Rev. A 82(5), 053808 (2010). [CrossRef]

11.

L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal-dispersion fiber lasers,” Laser Phys. Lett. 7(1), 63–67 (2010). [CrossRef]

12.

X. Liu, “Soliton formation and evolution in passively-mode-locked lasers with ultralong anomalous-dispersion fibers,” Phys. Rev. A 84(2), 023835 (2011). [CrossRef]

13.

K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, and J. G. Fujimoto, “Unidirectional ring resonators for self-starting passively mode-locked lasers,” Opt. Lett. 18(3), 220–222 (1993). [CrossRef] [PubMed]

14.

F. Ö. 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]

15.

X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A 81(2), 023811 (2010). [CrossRef]

16.

X. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A 82(6), 063834 (2010). [CrossRef]

17.

G. P. Agrawal, “Applications of Nonlinear Fiber Optics,” Fourth ed., (Academic Press, Boston, 2007).

18.

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

19.

D. U. Noske and J. R. Taylor, “Spectral and temporal stabilization of a diode-pumped ytterbium-erbium fiber soliton laser,” Electron. Lett. 29(25), 2200–2201 (1993). [CrossRef]

20.

Z. Luo, A. Luo, W. Xu, C. Song, Y. Gao, and W. Chen, “Sideband controllable soliton all-fiber ring laser passively mode-locked by nonlinear polarization rotation,” Laser Phys. Lett. 6(8), 582–585 (2009). [CrossRef]

21.

O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Opt. Commun. 269(1), 156–165 (2007). [CrossRef]

22.

X. Liu, “Numerical and experimental investigation of dissipative solitons in passively mode-locked fiber lasers with large net-normal-dispersion and high nonlinearity,” Opt. Express 17(25), 22401–22416 (2009). [CrossRef] [PubMed]

23.

X. Liu and B. Lee, “A fast method for nonlinear Schrödinger equation,” IEEE Photon. Technol. Lett. 15(11), 1549–1551 (2003). [CrossRef]

OCIS Codes
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(140.3500) Lasers and laser optics : Lasers, erbium
(140.3510) Lasers and laser optics : Lasers, fiber

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: August 22, 2012
Revised Manuscript: November 7, 2012
Manuscript Accepted: November 9, 2012
Published: November 15, 2012

Citation
Dongdong Han and Xueming Liu, "Sideband-controllable mode-locking fiber laser based on chirped fiber Bragg gratings," Opt. Express 20, 27045-27050 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27045


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References

  1. X. Liu, “Interaction and motion of solitons in passively-mode-locked fiber lasers,” Phys. Rev. A84(5), 053828 (2011). [CrossRef]
  2. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics6(2), 84–92 (2012). [CrossRef]
  3. D. Mao, X. M. Liu, L. R. Wang, X. H. Hu, and H. Lu, “Partially polarized wave-breaking-free dissipative soliton with super-broad spectrum in a mode-locked fiber laser,” Laser Phys. Lett.8(2), 134–138 (2011). [CrossRef]
  4. L. R. Wang, X. M. Liu, Y. K. Gong, D. Mao, and H. Feng, “Ultra-broadband high-energy pulse generation and evolution in a compact erbium-doped all-fiber laser,” Laser Phys. Lett.8(5), 376–381 (2011). [CrossRef]
  5. X. Liu, “Dissipative soliton evolution in ultra-large normal-cavity-dispersion fiber lasers,” Opt. Express17(12), 9549–9557 (2009). [CrossRef] [PubMed]
  6. Y. Deng, M. Koch, F. Lu, G. Wicks, and W. Knox, “Colliding-pulse passive harmonic mode-locking in a femtosecond Yb-doped fiber laser with a semiconductor saturable absorber,” Opt. Express12(16), 3872–3877 (2004). [CrossRef] [PubMed]
  7. 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]
  8. H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express17(20), 17630–17635 (2009). [CrossRef] [PubMed]
  9. L. Yun and X. Liu, “Generation and Propagation of Bound-State Pulses in a Passively Mode-Locked Figure-Eight Laser,” IEEE Photon. J.4(2), 512–519 (2012). [CrossRef]
  10. X. Liu, “Mechanism of high-energy pulse generation without wave breaking in mode-locked fiber lasers,” Phys. Rev. A82(5), 053808 (2010). [CrossRef]
  11. L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal-dispersion fiber lasers,” Laser Phys. Lett.7(1), 63–67 (2010). [CrossRef]
  12. X. Liu, “Soliton formation and evolution in passively-mode-locked lasers with ultralong anomalous-dispersion fibers,” Phys. Rev. A84(2), 023835 (2011). [CrossRef]
  13. K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, and J. G. Fujimoto, “Unidirectional ring resonators for self-starting passively mode-locked lasers,” Opt. Lett.18(3), 220–222 (1993). [CrossRef] [PubMed]
  14. F. Ö. 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]
  15. X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A81(2), 023811 (2010). [CrossRef]
  16. X. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A82(6), 063834 (2010). [CrossRef]
  17. G. P. Agrawal, “Applications of Nonlinear Fiber Optics,” Fourth ed., (Academic Press, Boston, 2007).
  18. M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber laser,” IEEE J. Quantum Electron.30(6), 1469–1477 (1994). [CrossRef]
  19. D. U. Noske and J. R. Taylor, “Spectral and temporal stabilization of a diode-pumped ytterbium-erbium fiber soliton laser,” Electron. Lett.29(25), 2200–2201 (1993). [CrossRef]
  20. Z. Luo, A. Luo, W. Xu, C. Song, Y. Gao, and W. Chen, “Sideband controllable soliton all-fiber ring laser passively mode-locked by nonlinear polarization rotation,” Laser Phys. Lett.6(8), 582–585 (2009). [CrossRef]
  21. O. Katz, Y. Sintov, Y. Nafcha, and Y. Glick, “Passively mode-locked ytterbium fiber laser utilizing chirped-fiber-Bragg-gratings for dispersion control,” Opt. Commun.269(1), 156–165 (2007). [CrossRef]
  22. X. Liu, “Numerical and experimental investigation of dissipative solitons in passively mode-locked fiber lasers with large net-normal-dispersion and high nonlinearity,” Opt. Express17(25), 22401–22416 (2009). [CrossRef] [PubMed]
  23. X. Liu and B. Lee, “A fast method for nonlinear Schrödinger equation,” IEEE Photon. Technol. Lett.15(11), 1549–1551 (2003). [CrossRef]

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