Abstract

Period multiplication is a bifurcation phenomenon, which depicts a typical roadmap from periodic steady-state to chaos. There has been particular interest in studying period multiplication properties in mode-locked lasers, but the mechanism behind is still not well-studied. Here, we experimentally demonstrate a figure-of-9 fiber laser which can generate period-multiplied pulses with long-term stability. Further theoretical insight is obtained by numerical simulations, which clearly shows that the ratio between the distances of each bifurcation point is approaching the Feigenbaum constant. We believe that the laser design presented here could be a stable and reliable ultrafast optical platform to study bifurcation and chaos, which also has potential applications in optical frequency comb and all-optical modulation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]

2019 (3)

J. Peng, S. Boscolo, Z. Zhao, and H. Zeng, “Breathing dissipative solitons in mode-locked fiber lasers,” Sci. Adv. 5(11), eaax1110 (2019).
[Crossref]

H. J. Chen, Y. J. Tan, J. G. Long, W. C. Chen, W. Y. Hong, H. Cui, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dynamical diversity of pulsating solitons in a fiber laser,” Opt. Express 27(20), 28507–28522 (2019).
[Crossref]

J. Zhou, W. Pan, X. Fu, L. Zhang, and Y. Feng, “Environmentally-stable 50-fs pulse generation directly from an Er:fiber oscillator,” Opt. Fiber Technol. 52, 101963 (2019).
[Crossref]

2018 (3)

2017 (2)

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

2016 (4)

M. Lezius, T. Wilken, C. Deutsch, M. Giunta, O. Mandel, A. Thaller, V. Schkolnik, M. Schiemangk, A. Dinkelaker, and A. Kohfeldt, “Space-borne frequency comb metrology,” Optica 3, 1381–1387 (2016).
[Crossref]

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

J. Zhou and X. Gu, “32-nJ 615-fs stable dissipative soliton ring cavity fiber laser with Raman scattering,” IEEE Photonics Technol. Lett. 28(4), 453–456 (2016).
[Crossref]

N. Kuse, J. Jiang, C. C. Lee, T. R. Schibli, and M. E. Fermann, “All polarization-maintaining Er fiber-based optical frequency combs with nonlinear amplifying loop mirror,” Opt. Express 24(3), 3095–3102 (2016).
[Crossref]

2015 (1)

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

2013 (3)

2010 (1)

S. Chouli and P. Grelu, “Soliton rains in a fiber laser: An experimental study,” Phys. Rev. A 81(6), 063829 (2010).
[Crossref]

2009 (1)

2008 (1)

L. Zhao, D. Tang, X. Wu, and H. Zhang, “Period-doubling of gain-guided solitons in fiber lasers of large net normal dispersion,” Opt. Commun. 281(13), 3557–3560 (2008).
[Crossref]

2006 (1)

2004 (3)

L. Zhao, D. Tang, F. Lin, and B. Zhao, “Observation of period-doubling bifurcations in a femtosecond fiber soliton laser with dispersion management cavity,” Opt. Express 12(19), 4573–4578 (2004).
[Crossref]

B. Zhao, D. Y. Tang, L. M. Zhao, P. Shum, and H. Y. Tam, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004).
[Crossref]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70(6), 066612 (2004).
[Crossref]

2001 (1)

D. Tang, W. 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).
[Crossref]

1988 (1)

1979 (1)

M. J. Feigenbaum, “The universal metric properties of nonlinear transformations,” J. Stat. Phys. 21(6), 669–706 (1979).
[Crossref]

1978 (1)

M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19(1), 25–52 (1978).
[Crossref]

Akhmediev, N.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70(6), 066612 (2004).
[Crossref]

Amrani, F.

Backus, S.

Bai, D.

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Bao, C.

Boscolo, S.

J. Peng, S. Boscolo, Z. Zhao, and H. Zeng, “Breathing dissipative solitons in mode-locked fiber lasers,” Sci. Adv. 5(11), eaax1110 (2019).
[Crossref]

Brunel, M.

Chédot, C.

Chen, H. J.

Chen, W. C.

Chen, X.

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Cheng, T.

L. Zhao, D. Tang, T. Cheng, H. Tam, and C. Lu, “Period-doubling of dispersion-managed soliton in Erbium-doped fiber lasers at around zero dispersion,” in OSA/FIO FTuR4 (2007).

Chouli, S.

S. Chouli and P. Grelu, “Soliton rains in a fiber laser: An experimental study,” Phys. Rev. A 81(6), 063829 (2010).
[Crossref]

Cleff, C.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Cui, H.

Deutsch, C.

Ding, J.

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Dinkelaker, A.

Dobner, S.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Doran, N. J.

Doubek, R.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Drummond, P. D.

D. Tang, W. 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).
[Crossref]

Feigenbaum, M. J.

M. J. Feigenbaum, “The universal metric properties of nonlinear transformations,” J. Stat. Phys. 21(6), 669–706 (1979).
[Crossref]

M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19(1), 25–52 (1978).
[Crossref]

Feng, Y.

J. Zhou, W. Pan, X. Fu, L. Zhang, and Y. Feng, “Environmentally-stable 50-fs pulse generation directly from an Er:fiber oscillator,” Opt. Fiber Technol. 52, 101963 (2019).
[Crossref]

J. Zhou, W. Pan, X. Gu, L. Zhang, and Y. Feng, “Dissipative-soliton generation with nonlinear-polarization-evolution in a polarization maintaining fiber,” Opt. Express 26(4), 4166–4171 (2018).
[Crossref]

J. Zhou, W. Pan, and Y. Feng, “Generation of ultra-stable period-multiplied pulsing in a figure-of-9 fiber cavity,” in OSA/FIO, JW4A.42 (2019)

Fermann, M. E.

Fischer, M.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Fu, S.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Fu, W.

Fu, X.

J. Zhou, W. Pan, X. Fu, L. Zhang, and Y. Feng, “Environmentally-stable 50-fs pulse generation directly from an Er:fiber oscillator,” Opt. Fiber Technol. 52, 101963 (2019).
[Crossref]

Giunta, M.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

M. Lezius, T. Wilken, C. Deutsch, M. Giunta, O. Mandel, A. Thaller, V. Schkolnik, M. Schiemangk, A. Dinkelaker, and A. Kohfeldt, “Space-borne frequency comb metrology,” Optica 3, 1381–1387 (2016).
[Crossref]

Grapinet, M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70(6), 066612 (2004).
[Crossref]

Grelu, P.

S. Chouli and P. Grelu, “Soliton rains in a fiber laser: An experimental study,” Phys. Rev. A 81(6), 063829 (2010).
[Crossref]

F. Amrani, A. Haboucha, M. Salhi, H. Leblond, A. Komarov, P. Grelu, and F. Sanchez, “Passively mode-locked erbium-doped double-clad fiber laser operating at the 322nd harmonic,” Opt. Lett. 34(14), 2120–2122 (2009).
[Crossref]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70(6), 066612 (2004).
[Crossref]

Gu, X.

J. Zhou, W. Pan, X. Gu, L. Zhang, and Y. Feng, “Dissipative-soliton generation with nonlinear-polarization-evolution in a polarization maintaining fiber,” Opt. Express 26(4), 4166–4171 (2018).
[Crossref]

J. Zhou and X. Gu, “32-nJ 615-fs stable dissipative soliton ring cavity fiber laser with Raman scattering,” IEEE Photonics Technol. Lett. 28(4), 453–456 (2016).
[Crossref]

Haboucha, A.

Hänsel, W.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Hartl, I.

M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013).
[Crossref]

Hideur, A.

Holzwarth, R.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Hong, W. Y.

Hoogland, H.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Ilday, FÖ

Jiang, J.

Jiang, K.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Kohfeldt, A.

Komarov, A.

Kuse, N.

Leblond, H.

Lee, C. C.

Lezius, M.

Li, H.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Li, W.

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Liang, Z.

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Limpert, J.

Lin, F.

Liu, D.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Liu, H.

Liu, M.

Long, J. G.

Lu, C.

L. Zhao, D. Tang, T. Cheng, H. Tam, and C. Lu, “Period-doubling of dispersion-managed soliton in Erbium-doped fiber lasers at around zero dispersion,” in OSA/FIO FTuR4 (2007).

Lu, Q.

Luo, A.

Luo, A. P.

Luo, Z.

Luo, Z. C.

Ma, J.

Ma, T.

Man, W.

D. Tang, W. 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).
[Crossref]

Mandel, O.

Mao, Q.

Mayer, P.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Ortaç, B.

Pan, W.

J. Zhou, W. Pan, X. Fu, L. Zhang, and Y. Feng, “Environmentally-stable 50-fs pulse generation directly from an Er:fiber oscillator,” Opt. Fiber Technol. 52, 101963 (2019).
[Crossref]

J. Zhou, W. Pan, X. Gu, L. Zhang, and Y. Feng, “Dissipative-soliton generation with nonlinear-polarization-evolution in a polarization maintaining fiber,” Opt. Express 26(4), 4166–4171 (2018).
[Crossref]

J. Zhou, W. Pan, and Y. Feng, “Generation of ultra-stable period-multiplied pulsing in a figure-of-9 fiber cavity,” in OSA/FIO, JW4A.42 (2019)

Peng, J.

J. Peng, S. Boscolo, Z. Zhao, and H. Zeng, “Breathing dissipative solitons in mode-locked fiber lasers,” Sci. Adv. 5(11), eaax1110 (2019).
[Crossref]

Salhi, M.

Sanchez, F.

Schibli, T. R.

Schiemangk, M.

Schkolnik, V.

Schmid, S.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Shen, D.

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Shen, X.

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Shi, L.

Shum, P.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

B. Zhao, D. Y. Tang, L. M. Zhao, P. Shum, and H. Y. Tam, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004).
[Crossref]

Sidorenko, P.

Song, J.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Song, Y.

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Soto-Crespo, J. M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70(6), 066612 (2004).
[Crossref]

Steinmetz, T.

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

Tam, H.

L. Zhao, D. Tang, T. Cheng, H. Tam, and C. Lu, “Period-doubling of dispersion-managed soliton in Erbium-doped fiber lasers at around zero dispersion,” in OSA/FIO FTuR4 (2007).

Tam, H. Y.

B. Zhao, D. Y. Tang, L. M. Zhao, P. Shum, and H. Y. Tam, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004).
[Crossref]

D. Tang, W. 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).
[Crossref]

Tan, Y. J.

Tang, D.

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

L. Zhao, D. Tang, X. Wu, and H. Zhang, “Period-doubling of gain-guided solitons in fiber lasers of large net normal dispersion,” Opt. Commun. 281(13), 3557–3560 (2008).
[Crossref]

L. Zhao, D. Tang, F. Lin, and B. Zhao, “Observation of period-doubling bifurcations in a femtosecond fiber soliton laser with dispersion management cavity,” Opt. Express 12(19), 4573–4578 (2004).
[Crossref]

D. Tang, W. 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).
[Crossref]

L. Zhao, D. Tang, T. Cheng, H. Tam, and C. Lu, “Period-doubling of dispersion-managed soliton in Erbium-doped fiber lasers at around zero dispersion,” in OSA/FIO FTuR4 (2007).

Tang, D. Y.

B. Zhao, D. Y. Tang, L. M. Zhao, P. Shum, and H. Y. Tam, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004).
[Crossref]

Tang, M.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Thaller, A.

Tünnermann, A.

Wen, S.

Wilken, T.

Wise, F. W.

Wood, D.

Wright, L. G.

Wu, H.

Wu, X.

L. Zhao, D. Tang, X. Wu, and H. Zhang, “Period-doubling of gain-guided solitons in fiber lasers of large net normal dispersion,” Opt. Commun. 281(13), 3557–3560 (2008).
[Crossref]

Wu, Z.

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Xiao, X.

Xu, W.

Xu, W. C.

Yang, C.

Yang, K.

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Zeng, H.

J. Peng, S. Boscolo, Z. Zhao, and H. Zeng, “Breathing dissipative solitons in mode-locked fiber lasers,” Sci. Adv. 5(11), eaax1110 (2019).
[Crossref]

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Zhang, H.

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Z. Luo, M. Liu, H. Liu, X. Zheng, A. Luo, C. Zhao, H. Zhang, S. Wen, and W. 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]

L. Zhao, D. Tang, X. Wu, and H. Zhang, “Period-doubling of gain-guided solitons in fiber lasers of large net normal dispersion,” Opt. Commun. 281(13), 3557–3560 (2008).
[Crossref]

Zhang, L.

J. Zhou, W. Pan, X. Fu, L. Zhang, and Y. Feng, “Environmentally-stable 50-fs pulse generation directly from an Er:fiber oscillator,” Opt. Fiber Technol. 52, 101963 (2019).
[Crossref]

J. Zhou, W. Pan, X. Gu, L. Zhang, and Y. Feng, “Dissipative-soliton generation with nonlinear-polarization-evolution in a polarization maintaining fiber,” Opt. Express 26(4), 4166–4171 (2018).
[Crossref]

Zhang, Q.

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Zhao, B.

B. Zhao, D. Y. Tang, L. M. Zhao, P. Shum, and H. Y. Tam, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004).
[Crossref]

L. Zhao, D. Tang, F. Lin, and B. Zhao, “Observation of period-doubling bifurcations in a femtosecond fiber soliton laser with dispersion management cavity,” Opt. Express 12(19), 4573–4578 (2004).
[Crossref]

Zhao, C.

Zhao, J.

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

Zhao, L.

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

L. Zhao, D. Tang, X. Wu, and H. Zhang, “Period-doubling of gain-guided solitons in fiber lasers of large net normal dispersion,” Opt. Commun. 281(13), 3557–3560 (2008).
[Crossref]

L. Zhao, D. Tang, F. Lin, and B. Zhao, “Observation of period-doubling bifurcations in a femtosecond fiber soliton laser with dispersion management cavity,” Opt. Express 12(19), 4573–4578 (2004).
[Crossref]

L. Zhao, D. Tang, T. Cheng, H. Tam, and C. Lu, “Period-doubling of dispersion-managed soliton in Erbium-doped fiber lasers at around zero dispersion,” in OSA/FIO FTuR4 (2007).

Zhao, L. M.

B. Zhao, D. Y. Tang, L. M. Zhao, P. Shum, and H. Y. Tam, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004).
[Crossref]

Zhao, Z.

J. Peng, S. Boscolo, Z. Zhao, and H. Zeng, “Breathing dissipative solitons in mode-locked fiber lasers,” Sci. Adv. 5(11), eaax1110 (2019).
[Crossref]

Zheng, X.

Zhou, J.

J. Zhou, W. Pan, X. Fu, L. Zhang, and Y. Feng, “Environmentally-stable 50-fs pulse generation directly from an Er:fiber oscillator,” Opt. Fiber Technol. 52, 101963 (2019).
[Crossref]

J. Zhou, W. Pan, X. Gu, L. Zhang, and Y. Feng, “Dissipative-soliton generation with nonlinear-polarization-evolution in a polarization maintaining fiber,” Opt. Express 26(4), 4166–4171 (2018).
[Crossref]

J. Zhou and X. Gu, “32-nJ 615-fs stable dissipative soliton ring cavity fiber laser with Raman scattering,” IEEE Photonics Technol. Lett. 28(4), 453–456 (2016).
[Crossref]

J. Zhou, W. Pan, and Y. Feng, “Generation of ultra-stable period-multiplied pulsing in a figure-of-9 fiber cavity,” in OSA/FIO, JW4A.42 (2019)

Appl. Phys. B (1)

W. Hänsel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 123(1), 41 (2017).
[Crossref]

IEEE Photonics J. (1)

Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

IEEE Photonics Technol. Lett. (2)

J. Zhao, W. Li, D. Bai, X. Shen, K. Yang, X. Chen, J. Ding, and H. Zeng, “Self-referenced f-to-2f beat note of a period-doubling mode-locked Yb-fiber laser,” IEEE Photonics Technol. Lett. 27(5), 459–461 (2015).
[Crossref]

J. Zhou and X. Gu, “32-nJ 615-fs stable dissipative soliton ring cavity fiber laser with Raman scattering,” IEEE Photonics Technol. Lett. 28(4), 453–456 (2016).
[Crossref]

J. Stat. Phys. (2)

M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19(1), 25–52 (1978).
[Crossref]

M. J. Feigenbaum, “The universal metric properties of nonlinear transformations,” J. Stat. Phys. 21(6), 669–706 (1979).
[Crossref]

Nat. Photonics (1)

M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013).
[Crossref]

Opt. Commun. (1)

L. Zhao, D. Tang, X. Wu, and H. Zhang, “Period-doubling of gain-guided solitons in fiber lasers of large net normal dispersion,” Opt. Commun. 281(13), 3557–3560 (2008).
[Crossref]

Opt. Express (7)

Opt. Fiber Technol. (1)

J. Zhou, W. Pan, X. Fu, L. Zhang, and Y. Feng, “Environmentally-stable 50-fs pulse generation directly from an Er:fiber oscillator,” Opt. Fiber Technol. 52, 101963 (2019).
[Crossref]

Opt. Lett. (4)

Optica (1)

Phys. Rev. A (3)

B. Zhao, D. Y. Tang, L. M. Zhao, P. Shum, and H. Y. Tam, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004).
[Crossref]

S. Chouli and P. Grelu, “Soliton rains in a fiber laser: An experimental study,” Phys. Rev. A 81(6), 063829 (2010).
[Crossref]

D. Tang, W. 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).
[Crossref]

Phys. Rev. E (1)

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70(6), 066612 (2004).
[Crossref]

Sci. Adv. (1)

J. Peng, S. Boscolo, Z. Zhao, and H. Zeng, “Breathing dissipative solitons in mode-locked fiber lasers,” Sci. Adv. 5(11), eaax1110 (2019).
[Crossref]

Sci. Rep. (1)

Z. Wu, S. Fu, K. Jiang, J. Song, H. Li, M. Tang, P. Shum, and D. Liu, “Switchable thulium-doped fiber laser from polarization rotation vector to scalar soliton,” Sci. Rep. 6(1), 34844 (2016).
[Crossref]

Other (2)

J. Zhou, W. Pan, and Y. Feng, “Generation of ultra-stable period-multiplied pulsing in a figure-of-9 fiber cavity,” in OSA/FIO, JW4A.42 (2019)

L. Zhao, D. Tang, T. Cheng, H. Tam, and C. Lu, “Period-doubling of dispersion-managed soliton in Erbium-doped fiber lasers at around zero dispersion,” in OSA/FIO FTuR4 (2007).

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Figures (9)

Fig. 1.
Fig. 1. Scheme of the figure-of-9 period-multiplied mode-locked fiber laser, WDM: wavelength division multiplexer; LD: laser diode; CFBG: chirped fiber Bragg grating.
Fig. 2.
Fig. 2. Output pulse trains and RF spectra (insert) of (a) fundamental mode-locking (b) period-quadrupling operation of port #I (c) period-quadrupling operation of port #II, (d) simultaneous pulse train measurements of port #I (red) and #II (black).
Fig. 3.
Fig. 3. Output pulse trains of period-tripling operation at (a) port #I and (b) port #II.
Fig. 4.
Fig. 4. TS-DFT recording of (a) fundamental mode-locked operation (b) period-quadrupling mode-locked operation at port #I. The insert is the local enlargement of the measured spectra.
Fig. 5.
Fig. 5. A typical configuration of NALM
Fig. 6.
Fig. 6. (left) NALM's transmission function of port #II, (right) corresponding four groups autocorrelation traces.
Fig. 7.
Fig. 7. (left) NALM's transmission function of port #I, (right) corresponding four groups autocorrelation traces.
Fig. 8.
Fig. 8. Bifurcation route map from fundamental pulse operation to period-32 operation as a function of the CFBG's reflectivity.
Fig. 9.
Fig. 9. Power stability test of the period-quadrupling operation over 13 hours.

Equations (4)

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| E OUT _ 01 | 2 = | E IN | 2 × 2 g α ( 1 α ) { 1 + cos [ ( 1 ( g + 1 ) α ) | E IN | 2 × 2 π n 2 L / λ  +  θ ] }
| E OUT _ 02 | 2 = | E IN | 2 × g { 1 2 α ( 1 α ) { 1 + cos [ ( 1 ( g + 1 ) α ) | E IN | 2 × 2 π n 2 L / λ  +  θ ] } }
x n + 1 = R × f ( x n )
δ n = lim n R n + 1 R n R n + 2 R n + 1 = 4.6692

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