Abstract

Through thorough numerical simulations, we investigated the molecular and polarization properties of the vector soliton molecules in an anomalous-dispersion fiber laser for the first time to our best knowledge. The molecular properties of the fast and slow modes of the vector soliton molecule can have different evolution characteristics on the interaction plane. The polarization dynamics of the leading and trailing vector pulses of the vector soliton molecule can have different evolution dynamics on the polarization Poincare sphere. The balance between gain and loss, the coupling between the orthogonal components and the interaction between the leading and trailing pulses result in various physical pictures of the vector soliton molecules in fiber lasers. Our results enrich the vector soliton dynamics in fiber lasers and have potential in optical signal processing and polarization division multiplexing for optical communications.

© 2017 Optical Society of America

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References

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    [PubMed]

2017 (4)

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[PubMed]

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[PubMed]

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Y. Du, X. Shu, and P. Cheng, “Numerical simulations of fast-axis instability of vector solitons in mode-locked fiber lasers,” Opt. Express 25(2), 1131–1141 (2017).
[PubMed]

2016 (2)

2015 (1)

2013 (2)

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

C. Mou, S. V. Sergeyev, A. G. Rozhin, and S. K. Turitsyn, “Bound state vector solitons with locked and precessing states of polarization,” Opt. Express 21(22), 26868–26875 (2013).
[PubMed]

2012 (2)

P. Serena, N. Rossi, and A. Bononi, “PDM-iRZ-QPSK vs. PS-QPSK at 100 Gbit/s over dispersion-managed links,” Opt. Express 20(7), 7895–7900 (2012).
[PubMed]

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

2010 (3)

2009 (1)

A. Zaviyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80(4), 043829 (2009).

2008 (2)

H. Zhang, D. Y. Tang, L. M. Zhao, and N. Xiang, “Coherent energy exchange between components of a vector soliton in fiber lasers,” Opt. Express 16(17), 12618–12623 (2008).
[PubMed]

A. Haus, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78(6), 063817 (2008).

2007 (1)

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1 Pt 2), 016613 (2007).
[PubMed]

2006 (2)

M. Grapinet and P. Grelu, “Vibrating soliton pairs in a mode-locked laser cavity,” Opt. Lett. 31(14), 2115–2117 (2006).
[PubMed]

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4 Pt 2), 046605 (2006).
[PubMed]

2005 (1)

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016616 (2005).
[PubMed]

2003 (3)

2002 (1)

2001 (2)

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64(3), 033814 (2001).

D. Y. Tang, B. Zhao, D. Y. Shen, and C. Lu, “Bound-soliton fiber laser,” Phys. Rev. A 66(3), 033806 (2001).

1999 (3)

1998 (1)

1997 (2)

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the Complex Ginzburg-Landau Equations,” Phys. Rev. Lett. 79(21), 4047–4051 (1997).

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78(17), 3290–3293 (1997).

1993 (1)

B. A. Malomed, “Bound state of envelop solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(4), 2875–2880 (1993).

1991 (1)

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation,” Phys. Rev. A 44(10), 6954–6957 (1991).
[PubMed]

1989 (1)

1983 (1)

Akhmediev, N.

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

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1 Pt 2), 016613 (2007).
[PubMed]

J. M. Soto-Crespo, N. Akhmediev, P. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28(19), 1757–1759 (2003).
[PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Multisoliton regime of pulse generation by lasers passively mode-locked with a slow saturable absorber,” J. Opt. Soc. Am. B 16(4), 674–677 (1999).

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15(2), 515–523 (1998).

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the Complex Ginzburg-Landau Equations,” Phys. Rev. Lett. 79(21), 4047–4051 (1997).

Akhmediev, N. N.

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. 80(20), 3988–3991 (1999).

Andral, U.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[PubMed]

Ankiewicz, A.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15(2), 515–523 (1998).

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the Complex Ginzburg-Landau Equations,” Phys. Rev. Lett. 79(21), 4047–4051 (1997).

Barad, Y.

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78(17), 3290–3293 (1997).

Batshon, H. G.

Béal, J.

Belhache, F.

Bergman, K.

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. 80(20), 3988–3991 (1999).

Bohm, M.

A. Haus, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78(6), 063817 (2008).

Bononi, A.

Chan, C. C.

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4 Pt 2), 046605 (2006).
[PubMed]

Cheng, J.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Cheng, P.

Collings, B. C.

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. 80(20), 3988–3991 (1999).

Cundiff, S. T.

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. 80(20), 3988–3991 (1999).

Devine, N.

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1 Pt 2), 016613 (2007).
[PubMed]

Djordjevic, I.

Drummond, P. D.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64(3), 033814 (2001).

Du, Y.

Egorov, O.

B. Ortac, A. Zaviyalov, C. K. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35(10), 1578–1580 (2010).
[PubMed]

A. Zaviyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80(4), 043829 (2009).

Fu, S.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Gordon, J. P.

Grapinet, M.

Grelu, P.

Gutty, F.

Han, M.

Han, X. X.

X. M. Liu, X. X. Han, and X. K. Yao, “Discrete bisoliton fiber laser,” Sci. Rep. 6, 34414 (2016).
[PubMed]

Hartwig, H.

A. Haus, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78(6), 063817 (2008).

Haus, A.

A. Haus, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78(6), 063817 (2008).

Herink, G.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[PubMed]

Iliew, R.

B. Ortac, A. Zaviyalov, C. K. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35(10), 1578–1580 (2010).
[PubMed]

A. Zaviyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80(4), 043829 (2009).

Jalali, B.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[PubMed]

Knox, W. H.

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. 80(20), 3988–3991 (1999).

Krupa, K.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[PubMed]

Kurtz, F.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[PubMed]

Lederer, F.

B. Ortac, A. Zaviyalov, C. K. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35(10), 1578–1580 (2010).
[PubMed]

A. Zaviyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80(4), 043829 (2009).

Li, L.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Li, X.

Limpert, J.

Liu, B.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Liu, D.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Liu, X. M.

X. M. Liu, X. X. Han, and X. K. Yao, “Discrete bisoliton fiber laser,” Sci. Rep. 6, 34414 (2016).
[PubMed]

X. M. Liu, “Dynamic evolution of temporal dissipative-soliton molecule in a large normal path-averaged dispersion fiber laser,” Phys. Rev. A 82(6), 063834 (2010).

Lu, C.

D. Y. Tang, B. Zhao, D. Y. Shen, and C. Lu, “Bound-soliton fiber laser,” Phys. Rev. A 66(3), 033806 (2001).

Luo, Y.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Malomed, B. A.

B. A. Malomed, “Bound state of envelop solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(4), 2875–2880 (1993).

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation,” Phys. Rev. A 44(10), 6954–6957 (1991).
[PubMed]

Man, W. S.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64(3), 033814 (2001).

Mikhailov, V.

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

Mitschke, F.

A. Haus, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78(6), 063817 (2008).

Mollenauer, L. F.

Mou, C.

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

C. Mou, S. V. Sergeyev, A. G. Rozhin, and S. K. Turitsyn, “Bound state vector solitons with locked and precessing states of polarization,” Opt. Express 21(22), 26868–26875 (2013).
[PubMed]

Nielsen, C. K.

Nithyanandan, K.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[PubMed]

Ortac, B.

Rabin, B.

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

Romagnoli, M.

Ropers, C.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[PubMed]

Rossi, N.

Rozhin, A.

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

Rozhin, A. G.

Serena, P.

Sergeyev, S. V.

C. Mou, S. V. Sergeyev, A. G. Rozhin, and S. K. Turitsyn, “Bound state vector solitons with locked and precessing states of polarization,” Opt. Express 21(22), 26868–26875 (2013).
[PubMed]

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

Shen, D. Y.

Shu, X.

Silberberg, Y.

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78(17), 3290–3293 (1997).

Smith, K.

Socci, L.

Solli, D. R.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[PubMed]

Song, Y. F.

Soto-Crespo, J.

Soto-Crespo, J. M.

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1 Pt 2), 016613 (2007).
[PubMed]

P. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Relative phase-locking of pulses in a passively mode-locked fiber laser,” J. Opt. Soc. Am. B 20(5), 863–870 (2003).

J. M. Soto-Crespo, N. Akhmediev, P. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28(19), 1757–1759 (2003).
[PubMed]

P. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Phase-locked soliton pairs in a stretched-pulse fiber laser,” Opt. Lett. 27(11), 966–968 (2002).
[PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Multisoliton regime of pulse generation by lasers passively mode-locked with a slow saturable absorber,” J. Opt. Soc. Am. B 16(4), 674–677 (1999).

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. 80(20), 3988–3991 (1999).

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15(2), 515–523 (1998).

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the Complex Ginzburg-Landau Equations,” Phys. Rev. Lett. 79(21), 4047–4051 (1997).

Sun, Q.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Tam, H. Y.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016616 (2005).
[PubMed]

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64(3), 033814 (2001).

Tang, D.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Tang, D. Y.

Y. F. Song, H. Zhang, L. M. Zhao, D. Y. Shen, and D. Y. Tang, “Coexistence and interaction of vector and bound vector solitons in a dispersion-managed fiber laser mode locked by graphene,” Opt. Express 24(2), 1814–1822 (2016).
[PubMed]

H. Zhang, D. Y. Tang, L. M. Zhao, and N. Xiang, “Coherent energy exchange between components of a vector soliton in fiber lasers,” Opt. Express 16(17), 12618–12623 (2008).
[PubMed]

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4 Pt 2), 046605 (2006).
[PubMed]

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016616 (2005).
[PubMed]

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64(3), 033814 (2001).

D. Y. Tang, B. Zhao, D. Y. Shen, and C. Lu, “Bound-soliton fiber laser,” Phys. Rev. A 66(3), 033806 (2001).

Tchofo-Dinda, P.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[PubMed]

Tsatourian, V.

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

Tünnermann, A.

Turitsyn, S. K.

C. Mou, S. V. Sergeyev, A. G. Rozhin, and S. K. Turitsyn, “Bound state vector solitons with locked and precessing states of polarization,” Opt. Express 21(22), 26868–26875 (2013).
[PubMed]

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

Wang, T.

Westbrook, P. S.

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

Wu, J.

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4 Pt 2), 046605 (2006).
[PubMed]

Xiang, N.

Xu, L.

Yang, H.

Yao, X. K.

X. M. Liu, X. X. Han, and X. K. Yao, “Discrete bisoliton fiber laser,” Sci. Rep. 6, 34414 (2016).
[PubMed]

Yuan, T.

Zaviyalov, A.

B. Ortac, A. Zaviyalov, C. K. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35(10), 1578–1580 (2010).
[PubMed]

A. Zaviyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80(4), 043829 (2009).

Zhang, H.

Zhang, S.

Zhao, B.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016616 (2005).
[PubMed]

D. Y. Tang, B. Zhao, D. Y. Shen, and C. Lu, “Bound-soliton fiber laser,” Phys. Rev. A 66(3), 033806 (2001).

Zhao, L.

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

Zhao, L. M.

Y. F. Song, H. Zhang, L. M. Zhao, D. Y. Shen, and D. Y. Tang, “Coexistence and interaction of vector and bound vector solitons in a dispersion-managed fiber laser mode locked by graphene,” Opt. Express 24(2), 1814–1822 (2016).
[PubMed]

H. Zhang, D. Y. Tang, L. M. Zhao, and N. Xiang, “Coherent energy exchange between components of a vector soliton in fiber lasers,” Opt. Express 16(17), 12618–12623 (2008).
[PubMed]

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4 Pt 2), 046605 (2006).
[PubMed]

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016616 (2005).
[PubMed]

J. Opt. Soc. Am. B (4)

Nat. Photonics (1)

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

Opt. Express (8)

C. Mou, S. V. Sergeyev, A. G. Rozhin, and S. K. Turitsyn, “Bound state vector solitons with locked and precessing states of polarization,” Opt. Express 21(22), 26868–26875 (2013).
[PubMed]

H. G. Batshon, I. Djordjevic, L. Xu, and T. Wang, “Modified hybrid subcarrier/amplitude/ phase/polarization LDPC-coded modulation for 400 Gb/s optical transmission and beyond,” Opt. Express 18(13), 14108–14113 (2010).
[PubMed]

P. Serena, N. Rossi, and A. Bononi, “PDM-iRZ-QPSK vs. PS-QPSK at 100 Gbit/s over dispersion-managed links,” Opt. Express 20(7), 7895–7900 (2012).
[PubMed]

H. Zhang, D. Y. Tang, L. M. Zhao, and N. Xiang, “Coherent energy exchange between components of a vector soliton in fiber lasers,” Opt. Express 16(17), 12618–12623 (2008).
[PubMed]

M. Han, S. Zhang, X. Li, H. Zhang, H. Yang, and T. Yuan, “Polarization dynamic patterns of vector solitons in a graphene mode-locked fiber laser,” Opt. Express 23(3), 2424–2435 (2015).
[PubMed]

Y. F. Song, H. Zhang, L. M. Zhao, D. Y. Shen, and D. Y. Tang, “Coexistence and interaction of vector and bound vector solitons in a dispersion-managed fiber laser mode locked by graphene,” Opt. Express 24(2), 1814–1822 (2016).
[PubMed]

Y. Du, X. Shu, and P. Cheng, “Numerical simulations of fast-axis instability of vector solitons in mode-locked fiber lasers,” Opt. Express 25(2), 1131–1141 (2017).
[PubMed]

P. Grelu, J. Béal, and J. Soto-Crespo, “Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime,” Opt. Express 11(18), 2238–2243 (2003).
[PubMed]

Opt. Lett. (6)

Phys. Rev. A (6)

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64(3), 033814 (2001).

D. Y. Tang, B. Zhao, D. Y. Shen, and C. Lu, “Bound-soliton fiber laser,” Phys. Rev. A 66(3), 033806 (2001).

A. Haus, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78(6), 063817 (2008).

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation,” Phys. Rev. A 44(10), 6954–6957 (1991).
[PubMed]

A. Zaviyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80(4), 043829 (2009).

X. M. Liu, “Dynamic evolution of temporal dissipative-soliton molecule in a large normal path-averaged dispersion fiber laser,” Phys. Rev. A 82(6), 063834 (2010).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016616 (2005).
[PubMed]

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1 Pt 2), 016613 (2007).
[PubMed]

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4 Pt 2), 046605 (2006).
[PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

B. A. Malomed, “Bound state of envelop solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(4), 2875–2880 (1993).

Phys. Rev. Lett. (4)

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the Complex Ginzburg-Landau Equations,” Phys. Rev. Lett. 79(21), 4047–4051 (1997).

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[PubMed]

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78(17), 3290–3293 (1997).

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. 80(20), 3988–3991 (1999).

Sci. Rep. (3)

Y. Luo, J. Cheng, B. Liu, Q. Sun, L. Li, S. Fu, D. Tang, L. Zhao, and D. Liu, “Group-Velocity-locked vector soliton molecules in fiber lasers,” Sci. Rep. 7(1), 2369 (2017).
[PubMed]

V. Tsatourian, S. V. Sergeyev, C. Mou, A. Rozhin, V. Mikhailov, B. Rabin, P. S. Westbrook, and S. K. Turitsyn, “Polarisation Dynamics of Vector Soliton Molecules in Mode Locked Fibre Laser,” Sci. Rep. 3, 3154 (2013).
[PubMed]

X. M. Liu, X. X. Han, and X. K. Yao, “Discrete bisoliton fiber laser,” Sci. Rep. 6, 34414 (2016).
[PubMed]

Science (1)

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[PubMed]

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Figures (12)

Fig. 1
Fig. 1 Schematic of the fiber laser in our simulation.
Fig. 2
Fig. 2 Soliton molecule evolution of the vector soliton molecule, peak separations (blue solid), phase difference (red solid) between leading and trailing pulses, interaction evolution traces of the slow (pink solid) and fast (black solid) soliton molecules: (a) slow soliton molecule with 800fs separation, (b) fast soliton molecule with 800fs separation, (c) interaction traces of the slow and fast soliton molecules with 800fs separation and 2000 rounds, (d) slow soliton molecule with 7.9ps separation, (e) fast soliton molecule with 7.9ps separation, (f) interaction traces of the slow and fast soliton molecules with 7.9ps separation and 2000 rounds, (g) slow soliton molecule with 17ps separation, (h) fast soliton molecule with 17ps separation, (i) interaction traces of the slow and fast soliton molecules with 17ps separation and 1000 rounds.
Fig. 3
Fig. 3 Pulse interaction of the vector soliton molecule under beat lengths of 5m and 7m, peak separations (blue solid), phase difference between leading and trailing pulses (red solid), interaction traces of the slow soliton molecule (pink solid) and fast(black solid) soliton molecule: (a) soliton molecule of slow mode with 5m beat length, (b) soliton molecule of fast mode with 5m beat length, (c) interaction traces of the soliton molecules of slow and fast modes with 5m beat length and 1000 rounds, (d) soliton molecule of slow mode with 7m beat length, (e) soliton molecule of fast mode with 7m beat length, (f) interaction traces of the soliton molecules of slow and fast modes with 7m beat length and 1000 rounds.
Fig. 4
Fig. 4 Pulse interaction of the vector soliton molecule under 10m beat length, peak separations (blue solid), phase difference between leading and trailing pulses (red solid), interaction traces of the soliton molecule on the slow (pink solid) and fast (black solid) axes: (a) slow soliton molecule with 2.1ps separation, (b) fast soliton molecule with 2.1ps separation, (c) interaction traces of the soliton molecules with 2.1ps separation and 2000 rounds, (d) slow soliton molecule with 2.48ps separation, (e) fast soliton molecule with 2.48ps separation, (f) interaction traces of the soliton molecules of slow and fast modes with 2.48ps separation and 1000 rounds, (g) slow soliton molecule with 10.3ps separation, (h) fast soliton molecule with 10.3ps separation, (i) interaction traces of the slow and fast soliton molecules with 10.3ps separation and 1000 rounds.
Fig. 5
Fig. 5 (a)Intensity profile of the slow soliton molecule and the amplitude ratio between slow and fast modes across the time slot of the soliton molecule, (b)Intensity profile of the slow soliton molecule and the phase difference between slow and fast modes across the time slot of the soliton molecule, (c) polarization trajectories of vector L and vector T within their time slot of full width of quarter maximum.
Fig. 6
Fig. 6 Polarization dynamics of the leading and trailing pulses with beat length of 2m and peak separation of 790fs: (a) the amplitude evolutions of the four components of the vector soliton molecule, (b) phase difference between the two polarization components of the leading and trailing pulses, (c) polarization trajectories on the normalized polarization Poincare sphere, (d) spectra of the soliton molecule of the slow(pink solid) and fast mode(black solid), (e) pulse intensity of the soliton molecule of the slow mode, (f) pulse intensity of the soliton molecule of the fast mode.
Fig. 7
Fig. 7 Polarization dynamics of the vector L (blue solid) and vector T (red solid) with different beat lengths and peak separations: (a) 5m beat length with 820fs separation, (b) 5m beat length with 7.95ps separation, (c) 7m beat length with 800fs separation, (d) 7m beat length with 7.95ps separation.
Fig. 8
Fig. 8 Phase difference between the slow and fast components of vector L (blue solid) and vector T (red dash) and intensity difference between slow and fast components of vector L (pink solid) and vector T (black dash), the insert is the enlargement portion to show the difference between the vector L and T: (a) and (e) 5m beat length with ~820fs separation, (b) and (f) 5m beat length with ~7.95ps separation, (c) and (g) 7m beat length with ~800fs separation, (d) and (h) 7m beat length with ~7.95ps separation.
Fig. 9
Fig. 9 Polarization trajectories of vector L (blue solid) and vector T (red solid) under different temporal separations: (a) ~1.94ps, (b) ~2.49ps, (c) 4.89ps, (d) 10.37ps.
Fig. 10
Fig. 10 Detailed evolution of the vector soliton molecules under beat length of 10m. Polarization phase difference between the orthogonal components of vector L and vector T with different separation: (a) 1.94ps separation, (b) 2.49ps separation, (c) 4.89ps separation, (d) 10.37ps separation. Intensity of leading and trailing pulses of slow mode and fast mode: (e) 1.94ps separation, (f) 2.49ps separation, (g) 4.89ps separation, (h)10.37ps separation. Intensity profiles of the vector soliton molecules with different separation: (i) 1.94ps separation, (j) 2.49ps separation, (k) 4.89ps separation, (l) 10.37ps separation. The inset in Fig. 10(c) is the enlargement portion of the vertical axis.
Fig. 11
Fig. 11 Intra-cavity dynamics of the vector soliton molecule with beat length of 2m and 790fs temporal separation: (a) amplitude evolution of slow (blue solid) and fast (red solid) modes and phase difference between the slow and fast components of vector L (pink solid), (b) amplitude evolution of trailing pulses of slow (blue solid) and fast (red solid) modes and phase difference between the slow and fast components of vector T (pink solid), (c) intensity difference (blue dashed) and phase difference between the leading and trailing pulses of the slow soliton molecule, (d) intensity difference (blue dashed) and phase difference between the leading and trailing pulses of the fast soliton molecule.
Fig. 12
Fig. 12 Intra-cavity evolutions of the amplitudes of the slow component (blue solid), fast component (red solid) and phase difference (pink solid) between the slow and fast components under beat length of 10m: (a) vector L with separation of 4.89ps, (b) vector T with separation of 4.89ps, (c) total spectrum at the end of the cavity with separation of 4.89ps, (d) vector L with separation of 10.37ps, (e)vector T with separation of 10.37ps, (f) total spectrum at the end of the cavity with separation of 10.37ps.

Equations (4)

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u z = i β u δ u t i β 2 2 2 u t 2 + i γ ( | u | 2 + 2 3 | v | 2 ) u + i γ 3 v 2 u + g 2 u ( 1 + 1 Ω 2 2 t 2 ) v z = i β v + δ v t i β 2 2 2 v t 2 + i γ ( | v | 2 + 2 3 | u | 2 ) v + i γ 3 u 2 v + g 2 v ( 1 + 1 Ω 2 2 t 2 )
g = 2 * exp ( ( | u | 2 + | v | 2 ) d t / E s )
T = 1 0.4 / ( 1 + P / 500 )
S 1 = | u | 2 | v | 2 , S 2 = 2 | u | | v | cos φ , S 3 = 2 | u | | v | sin φ , s i = S i / S 1 2 + S 2 2 + S 3 2

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