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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 27434–27440
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Vector solitons with locked and precessing states of polarization

Sergey V. Sergeyev, Chengbo Mou, Aleksey Rozhin, and Sergei K. Turitsyn  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 27434-27440 (2012)
http://dx.doi.org/10.1364/OE.20.027434


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Abstract

We demonstrate experimentally new families of vector solitons with locked and precessing states of polarization for fundamental and multipulse soliton operations in a carbon nanotube mode-locked fiber laser with anomalous dispersion laser cavity.

© 2012 OSA

1. Introduction

In this work we report to the best of our knowledge a first complete experimental characterization of new families of vector solitons in a carbon nanotube mode-locked fiber laser with anomalous dispersion laser cavity. Experimental data has been collected using an in-line polarimeter. By tuning an in-cavity polarization controller (POC) and POC for the pump laser (Fig. 1
Fig. 1 Experimental set-up
), we have found a new type of vector solitons with locked and precessing SOPs for fundamental soliton and multipulsing operations on a time scale of 40-40000 round-trips. The observed polarization attractors might be a key to the future enabling technologies of secure communications [1

1. G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88(9), 097903 (2002). [CrossRef] [PubMed]

], trapping and manipulation of atoms and nanoparticles [2

2. L. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10(1), 268–273 (2010). [CrossRef] [PubMed]

, 3

3. M. Spanner, K. M. Davitt, and M. Y. Ivanova, “Stability of angular confinement and rotational acceleration of a diatomic molecule in an optical centrifuge,” J. Chem. Phys. 115(18), 8403–8410 (2001). [CrossRef]

] and vectorial control of magnetization [4

4. N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata-Gonokami, “The vectorial control of magnetization by light,” Nat Commun 2, 362 (2011). [CrossRef] [PubMed]

].

2. Experimental set-up and results

The experimental setup is shown in Fig. 1. The ring cavity fiber laser with a total length of 7.83m comprises 2 m of high concentration erbium doped fiber (LIEKKITM Er80-8/125) and single mode (SM) fiber with anomalous dispersion (group velocity dispersion (GVD) parameter for erbium fiber β2,EDF = −19.26 fs2 /mm,), polarization controllers (POCs), wavelength division multiplexing (WDM) coupler, optical isolator (OIS) to provide unidirectional lasing, saturable absorber (polymer film with carbon nanotubes (CNT)), and output coupler. The CNT mode-locker is embedded between two standard fiber connector ferrules and index matching gel is applied to minimize the transmission loss. The cavity is pumped via 980/1550 WDM by a 976 nm laser diode (LD) with a maximum current of about 355 mA which provides 170 mW of optical power. With the help of a 90:10 coupler 90% of the intracavity power was directed out of the cavity. Output lasing has been analyzed with help of an auto-correlator (Pulsecheck), oscilloscope (Tektronix), optical spectrum analyzer (ANDO AQ6317B) and in-line polarimeter (Thorlabs, IPM5300).

S0=|u|2+|v|2,S1=|u|2|v|2,S=22|u||v|cosΔφ,S=32|u||v|sinΔφ,si=SiS12+S22+S32,DOP=S12+S22+S32S0,(i=1,2,3).
(1)

In the experiment, pump current has been changed from 209 mA to 355 mA and the in-cavity polarization and pump polarization controller have been tuned to obtain the polarization attractors shown in Figs. 2
Fig. 2 Polarization locked vector soliton. (a) output optical spectrum, (b) single pulse train, (c) measured auto-correlation trace. Polarization dynamics in the time frame of 40-40 000 round trips (1 µs – 1 ms) in terms of (d) optical power of orthogonally polarized modes Ix (solid line) and Iy (dashed line), total power I = Ix + Iy (dotted line), (e) phase difference and degree of polarization, and (f) Stokes parameters at Poincaré sphere. Parameters: pump current Ip = 209, mA, period T = 38.9 ns, pulse width Tp = 455 fs, output power I = 0.15 mW, phase difference Δφπand DOP = 61%.
-5. In view of auto-correlator sensitivity to the input SOP, all auto-correlation traces have been averaged over 16 samples.

With pump current increased to 306 mA, multi-pulsing was observed (Fig. 3
Fig. 3 Vector soliton with slowly evolving state of polarization for two-pulse operation. (a) output optical spectrum, (b) single pulse train, (c) measured auto-correlation trace. Polarization dynamics in the time frame of 40-40 000 round trips (1 µs – 1 ms) in terms of (d) optical power of orthogonally polarized modes Ix (solid line) and Iy (dashed line), total power I = Ix + Iy (dotted line), (e) phase difference and degree of polarization, and (f) Stokes parameters at Poincaré sphere. Parameters: pump current Ip = 306, mA, period T = 38.9 ns, pulse width Tp = 247 fs, output power I≈0.55 mW.
). The multi-pulsing arises as a result of interplay between the laser cavities’ bandwidth constraints and the energy quantization associated with the resulting mode-locked pulses [17

17. F. Li, P. K. A. Wai, and J. N. Kutz, “Geometrical description of the onset of multipulsing in mode-locked laser cavities,” J. Opt. Soc. Am. B 27(10), 2068–2077 (2010). [CrossRef]

]. The mode-locked pulse has increasing peak power and spectral bandwidth with increased pump power. However, the increase in the mode-locked spectral bandwidth is limited by the gain bandwidth of the cavity. To overcome this constraint with further increasing the pump power, a single pulse is split into two pulses per round trip with energy divided between two pulses and spectral bandwidths within the gain bandwidth window. As a result, double pulsing with the period T = 38.9 ns, pulse width Tp = 247 fs, and output power I≈0.55 mW has been observed (Fig. 3(a-d)). Anti-phase dynamics of oscillations for two cross polarized SOPs results in cw operation for the total output power (Fig. 3(d)). DOP oscillations around a low value of 12% indicate the presence of SOP oscillations faster than the polarimeter resolution time of 1 µs (Fig. 3(e)). The trace of the fast oscillations can be found in Fig. 3(e) as fast phase difference jumps and so the resulting polarization attractor at the Poincaré sphere comprises a polyline winding around a circle (Fig. 3(f)). This attractor is located close to the left circularly polarized SOP which is an eigenstate for isotropic laser along with the right circularly polarized SOP and all linearly polarized SOPs (equator at the Poincaré sphere) [5

5. S. V. Sergeyev, “Spontaneous Light Polarization Symmetry Breaking for an anisotropic ring cavity dye laser,” Phys. Rev. A 59(5), 3909–3917 (1999). [CrossRef]

].

Further pump power current increase up to 320 mA results in five-pulse soliton dynamics with with period T = 38.9 ns, pulse width Tp = 292 fs, output power I≈0.65 mW (Fig. 4(a-d)
Fig. 4 Vector soliton with slowly evolving state of polarization for five-pulse operation. (a) output optical spectrum, (b) single pulse train, (c) measured auto-correlation trace. Polarization dynamics in the time frame of 40-40 000 round trips (1 µs – 1 ms) in terms of (d) optical power of orthogonally polarized modes Ix (solid line) and Iy (dashed line), total power I = Ix + Iy (dotted line), (e) phase difference and degree of polarization, and (f) Stokes parameters at Poincaré sphere. Parameters: pump current Ip = 320 mA, period T = 38.9 ns, pulse width Tp = 292 fs, output power I≈0.65 mW.
). As follows from Fig. 4(c), output laser SOP is changing fast and so averaging over 16 samples is not enough to obtain smooth fundamental soliton shape of the autocorrelation trace. Anti-phase dynamics of oscillations for two cross polarized SOPs results in weak periodic oscillations of the total output power (Fig. 4(d)). As compared to Fig. 3, DOP is oscillating around the higher value of 30% that also indicates the presence of SOP oscillations faster than polarimeter resolution time of 1 µs (Fig. 4(e)). The trace of the fast oscillations can be also found in the phase difference dynamics shown in Fig. 4(e). As a result of the fast phase jumps between cross polarized SOPs and slow SOP precessing, the polarization attractor at the Poincaré sphere comprises a polyline with an outline in the form of a circle (Fig. 4(f)). This attractor is located close to the equator at the Poincaré sphere which is an eigenstate for an isotropic laser [5

5. S. V. Sergeyev, “Spontaneous Light Polarization Symmetry Breaking for an anisotropic ring cavity dye laser,” Phys. Rev. A 59(5), 3909–3917 (1999). [CrossRef]

].

As follows from Figs. 2-5, phase difference dynamics indicate the presence of coherent coupling between cross-polarized SOPs through the gain sharing and pump and in-cavity polarization controllers similar to the polarization dynamics of single- and multi-mode lasers without a saturable absorber [5

5. S. V. Sergeyev, “Spontaneous Light Polarization Symmetry Breaking for an anisotropic ring cavity dye laser,” Phys. Rev. A 59(5), 3909–3917 (1999). [CrossRef]

10

10. S. Sergeyev, K. O’Mahoney, S. Popov, and A. T. Friberg, “Coherence and anticoherence resonance in high-concentration erbium-doped fiber laser,” Opt. Lett. 35(22), 3736–3738 (2010). [CrossRef] [PubMed]

]. It is well known from the theory of nonlinear coupled oscillators that weak coupling leads to a complex behavior, and increasing the coupling leads to stabilization of the behavior, i.e. coupled attractors approach a stable steady state [27

27. D. G. Aronson, G. B. Ermentrout, and N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41(3), 403–449 (1990). [CrossRef]

]. The coupling is determined by pump power, amplitude and phase anisotropy of the cavity caused by polarization controllers. Complex polarization attractors in Figs. 3 and 4 are the result of a weak coupling caused by isotropic cavity. Stabilization takes place with an increased amplitude and phase anisotropy in the cavity and leads to the more simple attractors in the form of the fixed point and the limit cycle (Figs. 2 and 5).

4. Conclusions

Using an in-line polarimeter for erbium doped fiber laser passively mode locked with carbon nanotubes, we demonstrated for the first time new types of vector solitons with locked and slowly evolving states of polarization on a time scale of 40-40000 round-trips for fundamental soliton and multipulsing operations. The obtained results can find a practical implementation in secure communications [1

1. G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88(9), 097903 (2002). [CrossRef] [PubMed]

], trapping and manipulation of atoms and nanoparticles [2

2. L. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10(1), 268–273 (2010). [CrossRef] [PubMed]

, 3

3. M. Spanner, K. M. Davitt, and M. Y. Ivanova, “Stability of angular confinement and rotational acceleration of a diatomic molecule in an optical centrifuge,” J. Chem. Phys. 115(18), 8403–8410 (2001). [CrossRef]

] and vectorial control of magnetization [4

4. N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata-Gonokami, “The vectorial control of magnetization by light,” Nat Commun 2, 362 (2011). [CrossRef] [PubMed]

].

Acknowledgment

S. Sergeyev acknowledges financial support from the European Union program FP7 PEOPLE-2009-IEF (grant 253297). The work was also supported by the European Research Council, the Leverhulme Trust, FP7 Marie Curie project IRSES, and the grant N11.519.11.6038 of the Ministry of Education and Science of the Russian Federation.

References and links

1.

G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88(9), 097903 (2002). [CrossRef] [PubMed]

2.

L. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10(1), 268–273 (2010). [CrossRef] [PubMed]

3.

M. Spanner, K. M. Davitt, and M. Y. Ivanova, “Stability of angular confinement and rotational acceleration of a diatomic molecule in an optical centrifuge,” J. Chem. Phys. 115(18), 8403–8410 (2001). [CrossRef]

4.

N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata-Gonokami, “The vectorial control of magnetization by light,” Nat Commun 2, 362 (2011). [CrossRef] [PubMed]

5.

S. V. Sergeyev, “Spontaneous Light Polarization Symmetry Breaking for an anisotropic ring cavity dye laser,” Phys. Rev. A 59(5), 3909–3917 (1999). [CrossRef]

6.

H. Zeghlache and A. Boulnois, “Polarization instability in lasers. I. Model and steady states of neodymium-doped fiber lasers,” Phys. Rev. A 52(5), 4229–4242 (1995). [CrossRef] [PubMed]

7.

R. Leners and G. Stéphan, “Rate equation analysis of a multimode bipolarization Nd3+ doped fibre laser,” Quantum Semiclass. Opt. 7(5), 757–794 (1995). [CrossRef]

8.

Yu. V. Loiko, A. M. Kul’minskii, and A. P. Voitovich, “Impact of the vectorial degrees of freedom on the nonlinear behavior of class B lasers,” Opt. Commun. 210(1-2), 121–148 (2002). [CrossRef]

9.

G. D. Van Wiggeren and R. Roy, “High-speed fiber-optic polarization analyzer: measurements of the polarization dynamics of an erbium-doped fiber ring laser,” Opt. Commun. 164(1-3), 107–120 (1999). [CrossRef]

10.

S. Sergeyev, K. O’Mahoney, S. Popov, and A. T. Friberg, “Coherence and anticoherence resonance in high-concentration erbium-doped fiber laser,” Opt. Lett. 35(22), 3736–3738 (2010). [CrossRef] [PubMed]

11.

J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sanchez-Mondragon, “Vector soliton fiber lasers,” Opt. Lett. 24(6), 376–378 (1999). [CrossRef] [PubMed]

12.

A. Martinez, M. Omura, M. Takiguchi, B. Xu, T. Kuga, T. Ishigure, and S. Yamashita, “Multi-solitons in a dispersion managed fiber laser using a carbon nanotube-coated taper fiber,” in Conference on Nonlinear Photonics (NP), Technical Digest (CD) (Optical Society of America, 2012) paper JT5A.29.

13.

Y. S. Fedotov, S. M. Kobtsev, R. N. Arif, A. G. Rozhin, C. Mou, and S. K. Turitsyn, “Spectrum-, pulsewidth-, and wavelength-switchable all-fiber mode-locked Yb laser with fiber based birefringent filter,” Opt. Express 20(16), 17797–17805 (2012). [CrossRef] [PubMed]

14.

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]

15.

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

16.

J. M. Soto-Crespo and N. Akhmediev, “Soliton as Strange Attractor: Nonlinear Synchronization and Chaos,” Phys. Rev. Lett. 95(2), 024101 (2005). [CrossRef] [PubMed]

17.

F. Li, P. K. A. Wai, and J. N. Kutz, “Geometrical description of the onset of multipulsing in mode-locked laser cavities,” J. Opt. Soc. Am. B 27(10), 2068–2077 (2010). [CrossRef]

18.

S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of polarization-locked vector solitons in an optical fiber,” Phys. Rev. Lett. 82(20), 3988–3991 (1999). [CrossRef]

19.

B. C. Collings, S. T. Cundiff, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B 17(3), 354–365 (2000). [CrossRef]

20.

H. Zhang, D. Y. Tang, L. M. Zhao, and H. Y. Tam, “Induced solitons formed by cross-polarization coupling in a birefringent cavity fiber laser,” Opt. Lett. 33(20), 2317–2319 (2008). [CrossRef] [PubMed]

21.

L. M. Zhao, D. Y. Tang, X. Wu, H. Zhang, and H. Y. Tam, “Coexistence of polarization-locked and polarization-rotating vector solitons in a fiber laser with SESAM,” Opt. Lett. 34(20), 3059–3061 (2009). [CrossRef] [PubMed]

22.

L. M. Zhao, D. Y. Tang, H. Zhang, and X. Wu, “Polarization rotation locking of vector solitons in a fiber ring laser,” Opt. Express 16(14), 10053–10058 (2008). [CrossRef] [PubMed]

23.

J. H. Wong, K. Wu, H. H. Liu, Ch. Ouyang, H. Wang, Sh. Aditya, P. Shum, S. Fu, E. J. R. Kelleher, A. Chernov, and E. D. Obraztsova, “Vector solitons in a laser passively mode-locked by single-wall carbon nanotubes,” Opt. Commun. 284(7), 2007–2011 (2011). [CrossRef]

24.

D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu, “Observation of high-order polarization-locked vector solitons in a fiber laser,” Phys. Rev. Lett. 101(15), 153904 (2008). [CrossRef] [PubMed]

25.

C. Mou, S. Sergeyev, A. Rozhin, and S. Turistyn, “All-fiber polarization locked vector soliton laser using carbon nanotubes,” Opt. Lett. 36(19), 3831–3833 (2011). [CrossRef] [PubMed]

26.

C. Mou, S. Sergeyev, A. Rozhin, and S. K. Turitsyn, “Vector Solitons with Slowly Precessing States of Polarization,” in Conference on Nonlinear Photonics (NP), Technical Digest (CD) (Optical Society of America, 2012) paper NTu2D.

27.

D. G. Aronson, G. B. Ermentrout, and N. Kopell, “Amplitude response of coupled oscillators,” Physica D 41(3), 403–449 (1990). [CrossRef]

OCIS Codes
(140.4050) Lasers and laser optics : Mode-locked lasers
(250.5530) Optoelectronics : Pulse propagation and temporal solitons
(060.3510) Fiber optics and optical communications : Lasers, fiber

ToC Category:
Polarization Control and Vector Solitons

History
Original Manuscript: September 12, 2012
Revised Manuscript: October 10, 2012
Manuscript Accepted: October 10, 2012
Published: November 19, 2012

Virtual Issues
Nonlinear Photonics (2012) Optics Express

Citation
Sergey V. Sergeyev, Chengbo Mou, Aleksey Rozhin, and Sergei K. Turitsyn, "Vector solitons with locked and precessing states of polarization," Opt. Express 20, 27434-27440 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27434


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References

  1. G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett.88(9), 097903 (2002). [CrossRef] [PubMed]
  2. L. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett.10(1), 268–273 (2010). [CrossRef] [PubMed]
  3. M. Spanner, K. M. Davitt, and M. Y. Ivanova, “Stability of angular confinement and rotational acceleration of a diatomic molecule in an optical centrifuge,” J. Chem. Phys.115(18), 8403–8410 (2001). [CrossRef]
  4. N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata-Gonokami, “The vectorial control of magnetization by light,” Nat Commun2, 362 (2011). [CrossRef] [PubMed]
  5. S. V. Sergeyev, “Spontaneous Light Polarization Symmetry Breaking for an anisotropic ring cavity dye laser,” Phys. Rev. A59(5), 3909–3917 (1999). [CrossRef]
  6. H. Zeghlache and A. Boulnois, “Polarization instability in lasers. I. Model and steady states of neodymium-doped fiber lasers,” Phys. Rev. A52(5), 4229–4242 (1995). [CrossRef] [PubMed]
  7. R. Leners and G. Stéphan, “Rate equation analysis of a multimode bipolarization Nd3+ doped fibre laser,” Quantum Semiclass. Opt.7(5), 757–794 (1995). [CrossRef]
  8. Yu. V. Loiko, A. M. Kul’minskii, and A. P. Voitovich, “Impact of the vectorial degrees of freedom on the nonlinear behavior of class B lasers,” Opt. Commun.210(1-2), 121–148 (2002). [CrossRef]
  9. G. D. Van Wiggeren and R. Roy, “High-speed fiber-optic polarization analyzer: measurements of the polarization dynamics of an erbium-doped fiber ring laser,” Opt. Commun.164(1-3), 107–120 (1999). [CrossRef]
  10. S. Sergeyev, K. O’Mahoney, S. Popov, and A. T. Friberg, “Coherence and anticoherence resonance in high-concentration erbium-doped fiber laser,” Opt. Lett.35(22), 3736–3738 (2010). [CrossRef] [PubMed]
  11. J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sanchez-Mondragon, “Vector soliton fiber lasers,” Opt. Lett.24(6), 376–378 (1999). [CrossRef] [PubMed]
  12. A. Martinez, M. Omura, M. Takiguchi, B. Xu, T. Kuga, T. Ishigure, and S. Yamashita, “Multi-solitons in a dispersion managed fiber laser using a carbon nanotube-coated taper fiber,” in Conference on Nonlinear Photonics (NP), Technical Digest (CD) (Optical Society of America, 2012) paper JT5A.29.
  13. Y. S. Fedotov, S. M. Kobtsev, R. N. Arif, A. G. Rozhin, C. Mou, and S. K. Turitsyn, “Spectrum-, pulsewidth-, and wavelength-switchable all-fiber mode-locked Yb laser with fiber based birefringent filter,” Opt. Express20(16), 17797–17805 (2012). [CrossRef] [PubMed]
  14. B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A81(3), 033828 (2010). [CrossRef]
  15. Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics6(2), 84–92 (2012). [CrossRef]
  16. J. M. Soto-Crespo and N. Akhmediev, “Soliton as Strange Attractor: Nonlinear Synchronization and Chaos,” Phys. Rev. Lett.95(2), 024101 (2005). [CrossRef] [PubMed]
  17. F. Li, P. K. A. Wai, and J. N. Kutz, “Geometrical description of the onset of multipulsing in mode-locked laser cavities,” J. Opt. Soc. Am. B27(10), 2068–2077 (2010). [CrossRef]
  18. S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of polarization-locked vector solitons in an optical fiber,” Phys. Rev. Lett.82(20), 3988–3991 (1999). [CrossRef]
  19. B. C. Collings, S. T. Cundiff, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B17(3), 354–365 (2000). [CrossRef]
  20. H. Zhang, D. Y. Tang, L. M. Zhao, and H. Y. Tam, “Induced solitons formed by cross-polarization coupling in a birefringent cavity fiber laser,” Opt. Lett.33(20), 2317–2319 (2008). [CrossRef] [PubMed]
  21. L. M. Zhao, D. Y. Tang, X. Wu, H. Zhang, and H. Y. Tam, “Coexistence of polarization-locked and polarization-rotating vector solitons in a fiber laser with SESAM,” Opt. Lett.34(20), 3059–3061 (2009). [CrossRef] [PubMed]
  22. L. M. Zhao, D. Y. Tang, H. Zhang, and X. Wu, “Polarization rotation locking of vector solitons in a fiber ring laser,” Opt. Express16(14), 10053–10058 (2008). [CrossRef] [PubMed]
  23. J. H. Wong, K. Wu, H. H. Liu, Ch. Ouyang, H. Wang, Sh. Aditya, P. Shum, S. Fu, E. J. R. Kelleher, A. Chernov, and E. D. Obraztsova, “Vector solitons in a laser passively mode-locked by single-wall carbon nanotubes,” Opt. Commun.284(7), 2007–2011 (2011). [CrossRef]
  24. D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu, “Observation of high-order polarization-locked vector solitons in a fiber laser,” Phys. Rev. Lett.101(15), 153904 (2008). [CrossRef] [PubMed]
  25. C. Mou, S. Sergeyev, A. Rozhin, and S. Turistyn, “All-fiber polarization locked vector soliton laser using carbon nanotubes,” Opt. Lett.36(19), 3831–3833 (2011). [CrossRef] [PubMed]
  26. C. Mou, S. Sergeyev, A. Rozhin, and S. K. Turitsyn, “Vector Solitons with Slowly Precessing States of Polarization,” in Conference on Nonlinear Photonics (NP), Technical Digest (CD) (Optical Society of America, 2012) paper NTu2D.
  27. D. G. Aronson, G. B. Ermentrout, and N. Kopell, “Amplitude response of coupled oscillators,” Physica D41(3), 403–449 (1990). [CrossRef]

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