Multiple-soliton in spatiotemporal mode-locked multimode fiber lasers

Yihang Ding; Xiaosheng Xiao; Pan Wang; Changxi Yang

doi:10.1364/OE.27.011435

Multiple-soliton in spatiotemporal mode-locked multimode fiber lasers

Yihang Ding, Xiaosheng Xiao, Pan Wang, and Changxi Yang

Optics Express
Vol. 27,
Issue 8,
pp. 11435-11446
(2019)
•https://doi.org/10.1364/OE.27.011435

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Yihang Ding, Xiaosheng Xiao, Pan Wang, and Changxi Yang, "Multiple-soliton in spatiotemporal mode-locked multimode fiber lasers," Opt. Express 27, 11435-11446 (2019)

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Related Topics
Table of Contents Category
- Lasers, Optical Amplifiers, and Laser Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: February 7, 2019
- Revised Manuscript: March 28, 2019
- Manuscript Accepted: March 28, 2019
- Published: April 9, 2019

Accessible

Open Access

Abstract

Experimental observations of spatiotemporal mode-locked multiple-soliton, including harmonic mode locking and multiple pulses, in multimode fiber (MMF) lasers are reported. Numerical simulations are conducted to investigate the nonlinear dynamics of multi-pulsing. The influences of cavity parameters on the spatiotemporal outputs are analyzed by simulations, which agree with the experimental observations qualitatively. This work would contribute to understanding the complex spatiotemporal nonlinear dynamics in MMF lasers.

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References

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|
Author
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Publication

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]
D. Gloge, “Optical fibers for communication,” Appl. Opt. 13(2), 249–254 (1974).
[Crossref] [PubMed]
W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref] [PubMed]
Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]
K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]
L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]
U. Teğin and B. Ortaç, “Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers,” IEEE Photonics Technol. Lett. 29(24), 2195–2198 (2017).
[Crossref]
O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]
L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]
Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref] [PubMed]
A. Mafi, “Pulse propagation in a short nonlinear graded-index multimode optical fiber,” J. Lightwave Technol. 30(17), 2803–2811 (2012).
[Crossref]
M. Conforti, C. Mas Arabi, A. Mussot, and A. Kudlinski, “Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers,” Opt. Lett. 42(19), 4004–4007 (2017).
[Crossref] [PubMed]
K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]
K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]
G. Lopez-Galmiche, Z. Sanjabi Eznaveh, M. A. Eftekhar, J. Antonio Lopez, L. G. Wright, F. Wise, D. Christodoulides, and R. Amezcua Correa, “Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm,” Opt. Lett. 41(11), 2553–2556 (2016).
[Crossref] [PubMed]
R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]
L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]
P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
[Crossref]
A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
[Crossref]
P. Wang, X. Xiao, H. Zhao, and C. Yang, “Observation of duration-tunable soliton explosion in passively mode-locked fiber laser,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]
H. Zhang, D. Y. Tang, L. M. Zhao, and R. J. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express 18(5), 4428–4433 (2010).
[Crossref] [PubMed]
D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]
Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]
A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
[Crossref]
Z.-C. Luo, M. Liu, H. Liu, X.-W. Zheng, A.-P. Luo, C.-J. Zhao, H. Zhang, S.-C. Wen, and W.-C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based topological insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
[Crossref] [PubMed]
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).
[Crossref] [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).
[Crossref] [PubMed]
X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]
Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]
Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]
P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
[Crossref]
H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
[Crossref] [PubMed]
E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
[Crossref] [PubMed]
L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]
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).
[Crossref] [PubMed]
J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
[Crossref]
M. Olivier and M. Piché, “Origin of the bound states of pulses in the stretched-pulse fiber laser,” Opt. Express 17(2), 405–418 (2009).
[Crossref] [PubMed]

2019 (1)

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]

2018 (6)

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
[Crossref]

H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
[Crossref] [PubMed]

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

2017 (9)

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

M. Conforti, C. Mas Arabi, A. Mussot, and A. Kudlinski, “Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers,” Opt. Lett. 42(19), 4004–4007 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

U. Teğin and B. Ortaç, “Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers,” IEEE Photonics Technol. Lett. 29(24), 2195–2198 (2017).
[Crossref]

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

P. Wang, X. Xiao, H. Zhao, and C. Yang, “Observation of duration-tunable soliton explosion in passively mode-locked fiber laser,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

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).
[Crossref] [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).
[Crossref] [PubMed]

2016 (5)

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref] [PubMed]

G. Lopez-Galmiche, Z. Sanjabi Eznaveh, M. A. Eftekhar, J. Antonio Lopez, L. G. Wright, F. Wise, D. Christodoulides, and R. Amezcua Correa, “Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm,” Opt. Lett. 41(11), 2553–2556 (2016).
[Crossref] [PubMed]

2015 (2)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
[Crossref]

2013 (3)

Z.-C. Luo, M. Liu, H. Liu, X.-W. Zheng, A.-P. Luo, C.-J. Zhao, H. Zhang, S.-C. Wen, and W.-C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based topological insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
[Crossref] [PubMed]

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref] [PubMed]

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

2012 (3)

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

A. Mafi, “Pulse propagation in a short nonlinear graded-index multimode optical fiber,” J. Lightwave Technol. 30(17), 2803–2811 (2012).
[Crossref]

Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]

2011 (1)

E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
[Crossref] [PubMed]

2010 (1)

H. Zhang, D. Y. Tang, L. M. Zhao, and R. J. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express 18(5), 4428–4433 (2010).
[Crossref] [PubMed]

2009 (1)

M. Olivier and M. Piché, “Origin of the bound states of pulses in the stretched-pulse fiber laser,” Opt. Express 17(2), 405–418 (2009).
[Crossref] [PubMed]

2007 (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

2003 (1)

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).
[Crossref] [PubMed]

1998 (1)

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
[Crossref]

1992 (1)

A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
[Crossref]

1974 (1)

D. Gloge, “Optical fibers for communication,” Appl. Opt. 13(2), 249–254 (1974).
[Crossref] [PubMed]

Akhmediev, N.

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

Amezcua Correa, R.

Andral, U.

Antonio Lopez, J.

Barthelemy, A.

Barthélémy, A.

Béal, J.

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).
[Crossref] [PubMed]

Bendahmane, A.

Bergman, K.

Billet, C.

Broderick, N. G. R.

A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
[Crossref]

Caravaca-Aguirre, A. M.

Christodoulides, D.

Christodoulides, D. N.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

Coillet, A.

Collings, B. C.

Conforti, M.

Couderc, V.

Cui, Y.

X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]

Ding, E.

Du, J.

Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]

Dudley, J. M.

Dupiol, R.

Eftekhar, M. A.

Erkintalo, M.

A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
[Crossref]

Fabert, M.

Fini, J.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Genty, G.

Gloge, D.

D. Gloge, “Optical fibers for communication,” Appl. Opt. 13(2), 249–254 (1974).
[Crossref] [PubMed]

Grelu, P.

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

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).
[Crossref] [PubMed]

Grudinin, A.

A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
[Crossref]

Hao, Y.

Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]

Herink, G.

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Kibler, B.

Knize, R. J.

H. Zhang, D. Y. Tang, L. M. Zhao, and R. J. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express 18(5), 4428–4433 (2010).
[Crossref] [PubMed]

Knox, W. H.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Krupa, K.

Kudlinski, A.

Kurtz, F.

Kutz, J. N.

Lefrancois, S.

Li, B.

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

Li, H.

Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]

Li, M.-J.

Li, X.

Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]

Liu, H.

Liu, M.

Liu, X.

X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]

Liu, Z.

Lopez-Galmiche, G.

Luo, A.-P.

Luo, Z.-C.

Lushnikov, P. M.

Mafi, A.

A. Mafi, “Pulse propagation in a short nonlinear graded-index multimode optical fiber,” J. Lightwave Technol. 30(17), 2803–2811 (2012).
[Crossref]

Mas Arabi, C.

Meng, Y.

Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]

Merolla, J.-M.

Millot, G.

Modotto, D.

Mussot, A.

Närhi, M.

Nelson, L.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Nithyanandan, K.

Nolan, D. A.

Olivier, M.

M. Olivier and M. Piché, “Origin of the bound states of pulses in the stretched-pulse fiber laser,” Opt. Express 17(2), 405–418 (2009).
[Crossref] [PubMed]

Ortaç, B.

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Phys. Rev. A (Coll. Park) (1)

Phys. Rev. Lett. (3)

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Science (2)

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Fig. 1 Experimental setup. (a) The MMF laser (SPDM, short pass dichroic mirror; L₁ and L₂, lens; OM4, passive GRIN multimode fiber; HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; BS, beam splitter; ISO, isolator; SF, spectral filter; M1-3, mirrors); (b) Measurement setup. Method 1: spectral filtering; Method 2: spatial sampling (BS: beam splitter; OSA: optical spectrum analyzer; PD, photodetector).

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Fig. 2 Second-order harmonic mode locking state. (a) Pulse train measured by oscilloscope; (b) RF spectrum of pulse train with a resolution bandwidth of 1 Hz (inset: RF spectrum over a 1 GHz span with a resolution bandwidth of 2 kHz); (c) Autocorrelation trace of output pulses; (d) Spectra of the whole output (solid curve) and the spectral-filtered outputs (dashed curves), with the corresponding beam profiles shown in (e)-(h); (e) Beam profile of the whole output, with black circles represented the spatial sampling positions (one sampler fixed at the center, and another sampler moving along the direction of arrow); (f)-(h) Beam profiles corresponding to different spectrum regions by tunable spectral-filtering of the output; (i) Pulse intensities and pulse intervals of the pulse trains sampled by the moving sampler at different position (the intensity corresponding to the fixed sampler is 53); (j) pulse train sampled by the fixed sampler; (k) a typical pulse train sampled by the moving sampler; (l) and (n) are the corresponding optical and RF spectra of (j) and (k), respectively; (m) RF spectra of the spectral-filtered outputs corresponding to (f)-(h).

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Fig. 3 Multiple pulses state. (a) Pulse train measured by oscilloscope; (b) Optical spectrum (inset: beam profile); (c) Pulse intensities and pulse intervals of the pulse trains sampled at different position.

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Fig. 4 Soliton molecule state. (a) Pulse train measured by oscilloscope; (b) Optical spectrum (inset: beam profile); (c) Autocorrelation traces of different spatial samplings, S1 and S2.

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Fig. 5 Mode-resolved simulation results for soliton molecule in the MMF laser. E_sat = 2 nJ, small signal gain g₀ = 15 m⁻¹ and P_sat of the SA is 180 W. (a) Mode-resolved pulse evolution with roundtrip (inset: beam profile); (b) Pulse evolution with roundtrip; (c) Intracavity pulse evolution in the last roundtrip; (d) Temporal output of the last roundtrip; (e) Pulse energy evolution with roundtrip. In (a), the initial pulses intensities are enlarged 200 times for clarity. In mode-resolved figures, the modes are color-coded. Among those modes, mode 2 is the highest, modes 1, 3 and 4 are almost overlapped, modes 5 and 6 are almost overlapped. OC: output coupler, SA: saturable absorber, BF: bandpass filter, SF: spatial filter.

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Fig. 6 Evolution of multiple pulses with different initial conditions. Small signal gain g₀ = 20 m⁻¹ and P_sat of the SA is 180 W. (a) Evolution with two unequal pulse input, and (b) is the corresponding mode-resolved steady output; (c) Evolution with random noise input, and (d) is the corresponding steady output.

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Fig. 7 Mode-resolved output intensities and the corresponding beam profiles with various coupling coefficients (CCs) operating in multiple pulses state. CC I = [4,2,2,1,1,1], CC II = [2,4,2,2,1,1], CC III = [1,1,1,2,2,4], and other parameters are the same as Fig. 6.

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Fig. 8 The intensities of modes 2 and 3, normalized to that of mode 1, of the steady output versus (a) small signal gain g₀ and (b) P_sat of the SA. Regimes in different colors represent different operation states of the STML MMF laser.

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Equations (1)

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\begin{array}{l} \partial_{z} A_{p} (z, t) = i (β_{0}^{(p)} - ℜ [β_{0}^{(0)}]) A_{p} - (β_{1}^{(p)} - ℜ [β_{1}^{(0)}]) \frac{\partial A_{p}}{\partial t} - i \frac{β_{2}^{(p)}}{2} \partial_{t}^{2} A_{p} \\ i \frac{n_{2} ω_{0}}{c} \sum_{l, m, n} {(1 - f_{R}) S_{p l m n}^{K} A_{l} A_{m} A_{n}^{*} + f_{R} A_{l} S_{p l m n}^{R} \int_{- \infty}^{t} d τ A_{m} (z, t - τ) A_{n}^{*} (z, t - τ) h_{R} (τ)} \end{array}