Abstract

Due to the weak birefringence of single mode fibers, solitons generated in fiber lasers are indeed vector pulses and exhibit periodic parameter change including polarization evolution even when there is a polarizer inside the cavity. Period doubling eigenstates of solitons generated in a fiber laser mode-locked by the nonlinear polarization rotation, i.e., period doubling of polarization components of the soliton, are numerically explored in detail. We found that, apart from the synchronous evolution between the two polarization components, there exists asynchronous development depending on the detailed operation conditions. In addition, period doubling of one polarization component together with period-one of another polarization component can be achieved. When the period tripling window is obtained, much complexed dynamics on the two polarization components could be observed.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

Z. Luming, S. Chaojie, W. Yufei, and L. Lei, “Research progress of period doubling bifurcation in ultrafast fiber lasers (invited),” Infrared and Laser Eng. 47(8), 803002 (2018).
[Crossref]

2016 (3)

2009 (3)

2008 (2)

L. M. Zhao, D. Y. 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. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

2007 (1)

L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

2004 (3)

2001 (1)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63(5), 645–651 (2001).
[Crossref]

2000 (1)

1998 (1)

1994 (1)

K. Tamura, C. R. Doerr, H. A. Haus, and E. P. Ippen, “Soliton fiber ring laser stabilization and tuning with a broad intracavity filter,” IEEE Photonics Technol. Lett. 6(6), 697–699 (1994).
[Crossref]

1990 (1)

1988 (1)

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), 691–738 (2004).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63(5), 645–651 (2001).
[Crossref]

Akhmediev, N. N.

Almanee, M.

Ankiewicz, A.

Armas-Rivera, I.

Bao, Q. L.

Beltran-Perez, G.

Bergman, K.

Bolton, S. R.

G. Sucha, S. R. Bolton, S. Weiss, and D. S. Chemla, “Nonlinear dynamics of additive pulse mode-locked lasers: period doubling and chaos,” Quantum Electronics & Laser Science Conference, JTuC6, 1993.

Chaojie, S.

Z. Luming, S. Chaojie, W. Yufei, and L. Lei, “Research progress of period doubling bifurcation in ultrafast fiber lasers (invited),” Infrared and Laser Eng. 47(8), 803002 (2018).
[Crossref]

Chemla, D. S.

G. Sucha, S. R. Bolton, S. Weiss, and D. S. Chemla, “Nonlinear dynamics of additive pulse mode-locked lasers: period doubling and chaos,” Quantum Electronics & Laser Science Conference, JTuC6, 1993.

Cheng, T. H.

L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

Collings, B. C.

Cui, Y.

Y. Cui, Y. Zhang, Y. Xiang, Y. Song, and X. Liu, “XPM-forced frequency-oscillating soliton in mode-locked fiber laser,” https://arxiv.org/abs/1910.09225 .

Cundiff, S. T.

Doerr, C. R.

K. Tamura, C. R. Doerr, H. A. Haus, and E. P. Ippen, “Soliton fiber ring laser stabilization and tuning with a broad intracavity filter,” IEEE Photonics Technol. Lett. 6(6), 697–699 (1994).
[Crossref]

Doran, N. J.

Fermann, M. E.

Fu, S.

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), 691–738 (2004).
[Crossref]

Grelu, P.

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), 691–738 (2004).
[Crossref]

Haberl, F.

Haus, H. A.

K. Tamura, C. R. Doerr, H. A. Haus, and E. P. Ippen, “Soliton fiber ring laser stabilization and tuning with a broad intracavity filter,” IEEE Photonics Technol. Lett. 6(6), 697–699 (1994).
[Crossref]

Haus, J. W.

Hochreiter, H.

Hofer, M.

Ibarra-Escamilla, B.

Ippen, E. P.

K. Tamura, C. R. Doerr, H. A. Haus, and E. P. Ippen, “Soliton fiber ring laser stabilization and tuning with a broad intracavity filter,” IEEE Photonics Technol. Lett. 6(6), 697–699 (1994).
[Crossref]

Jablonski, Mark

Knox, W. H.

Lederer, M. J.

Lei, L.

Z. Luming, S. Chaojie, W. Yufei, and L. Lei, “Research progress of period doubling bifurcation in ultrafast fiber lasers (invited),” Infrared and Laser Eng. 47(8), 803002 (2018).
[Crossref]

Li, L.

Z. Wu, D. Liu, S. Fu, L. Li, M. Tang, and L. Zhao, “Scalar-vector soliton fiber laser mode-locked by nonlinear polarization rotation,” Opt. Express 24(16), 18764–18771 (2016).
[Crossref]

Q. Zhang, L. Li, D. Tang, and L. Zhao, “Period-timing Bifurcations in a Dispersion-managed Fiber Laser with Zero Group Velocity Dispersion,” IEEE Photonics J. 8(6), 1504608 (2016).
[Crossref]

Lin, F.

Liu, D.

Liu, X.

Y. Cui, Y. Zhang, Y. Xiang, Y. Song, and X. Liu, “XPM-forced frequency-oscillating soliton in mode-locked fiber laser,” https://arxiv.org/abs/1910.09225 .

Loh, K. P.

Lu, C.

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “Gain dispersion for dissipative soliton generation in all-normal-dispersion fiber lasers,” Appl. Opt. 48(27), 5131–5137 (2009).
[Crossref]

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “High fundamental repetition rate fiber lasers operated in strong normal dispersion regime,” IEEE Photonics Technol. Lett. 21(11), 724–726 (2009).
[Crossref]

L. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

Luming, Z.

Z. Luming, S. Chaojie, W. Yufei, and L. Lei, “Research progress of period doubling bifurcation in ultrafast fiber lasers (invited),” Infrared and Laser Eng. 47(8), 803002 (2018).
[Crossref]

Luther-Davies, B.

Set, Sze Y.

Song, Y.

Y. Cui, Y. Zhang, Y. Xiang, Y. Song, and X. Liu, “XPM-forced frequency-oscillating soliton in mode-locked fiber laser,” https://arxiv.org/abs/1910.09225 .

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), 691–738 (2004).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63(5), 645–651 (2001).
[Crossref]

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]

Sucha, G.

G. Sucha, S. R. Bolton, S. Weiss, and D. S. Chemla, “Nonlinear dynamics of additive pulse mode-locked lasers: period doubling and chaos,” Quantum Electronics & Laser Science Conference, JTuC6, 1993.

Tam, H. Y.

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “High fundamental repetition rate fiber lasers operated in strong normal dispersion regime,” IEEE Photonics Technol. Lett. 21(11), 724–726 (2009).
[Crossref]

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “Gain dispersion for dissipative soliton generation in all-normal-dispersion fiber lasers,” Appl. Opt. 48(27), 5131–5137 (2009).
[Crossref]

L. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

Tamura, K.

K. Tamura, C. R. Doerr, H. A. Haus, and E. P. Ippen, “Soliton fiber ring laser stabilization and tuning with a broad intracavity filter,” IEEE Photonics Technol. Lett. 6(6), 697–699 (1994).
[Crossref]

Tanaka, Yuichi

Tang, D.

Q. Zhang, L. Li, D. Tang, and L. Zhao, “Period-timing Bifurcations in a Dispersion-managed Fiber Laser with Zero Group Velocity Dispersion,” IEEE Photonics J. 8(6), 1504608 (2016).
[Crossref]

Tang, D. Y.

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

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “Gain dispersion for dissipative soliton generation in all-normal-dispersion fiber lasers,” Appl. Opt. 48(27), 5131–5137 (2009).
[Crossref]

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “High fundamental repetition rate fiber lasers operated in strong normal dispersion regime,” IEEE Photonics Technol. Lett. 21(11), 724–726 (2009).
[Crossref]

L. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

L. M. Zhao, D. Y. 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. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

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

Tang, M.

Town, G.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63(5), 645–651 (2001).
[Crossref]

Wai, P. K. A.

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “High fundamental repetition rate fiber lasers operated in strong normal dispersion regime,” IEEE Photonics Technol. Lett. 21(11), 724–726 (2009).
[Crossref]

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “Gain dispersion for dissipative soliton generation in all-normal-dispersion fiber lasers,” Appl. Opt. 48(27), 5131–5137 (2009).
[Crossref]

Weiss, S.

G. Sucha, S. R. Bolton, S. Weiss, and D. S. Chemla, “Nonlinear dynamics of additive pulse mode-locked lasers: period doubling and chaos,” Quantum Electronics & Laser Science Conference, JTuC6, 1993.

Wen, S. C.

L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

Wood, David

Wu, X

L. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

Wu, X.

L. M. Zhao, D. Y. 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.

Xiang, Y.

Y. Cui, Y. Zhang, Y. Xiang, Y. Song, and X. Liu, “XPM-forced frequency-oscillating soliton in mode-locked fiber laser,” https://arxiv.org/abs/1910.09225 .

Yaguchi, Hiroshi

Yufei, W.

Z. Luming, S. Chaojie, W. Yufei, and L. Lei, “Research progress of period doubling bifurcation in ultrafast fiber lasers (invited),” Infrared and Laser Eng. 47(8), 803002 (2018).
[Crossref]

Zhang, H.

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

L. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

L. M. Zhao, D. Y. 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, Q.

Q. Zhang, L. Li, D. Tang, and L. Zhao, “Period-timing Bifurcations in a Dispersion-managed Fiber Laser with Zero Group Velocity Dispersion,” IEEE Photonics J. 8(6), 1504608 (2016).
[Crossref]

Zhang, Y.

Y. Cui, Y. Zhang, Y. Xiang, Y. Song, and X. Liu, “XPM-forced frequency-oscillating soliton in mode-locked fiber laser,” https://arxiv.org/abs/1910.09225 .

Zhao, B.

Zhao, L.

Q. Zhang, L. Li, D. Tang, and L. Zhao, “Period-timing Bifurcations in a Dispersion-managed Fiber Laser with Zero Group Velocity Dispersion,” IEEE Photonics J. 8(6), 1504608 (2016).
[Crossref]

Z. Wu, D. Liu, S. Fu, L. Li, M. Tang, and L. Zhao, “Scalar-vector soliton fiber laser mode-locked by nonlinear polarization rotation,” Opt. Express 24(16), 18764–18771 (2016).
[Crossref]

Zhao, L. M.

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

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “Gain dispersion for dissipative soliton generation in all-normal-dispersion fiber lasers,” Appl. Opt. 48(27), 5131–5137 (2009).
[Crossref]

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “High fundamental repetition rate fiber lasers operated in strong normal dispersion regime,” IEEE Photonics Technol. Lett. 21(11), 724–726 (2009).
[Crossref]

L. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

L. M. Zhao, D. Y. 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. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

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

Appl. Opt. (1)

IEEE Photonics J. (1)

Q. Zhang, L. Li, D. Tang, and L. Zhao, “Period-timing Bifurcations in a Dispersion-managed Fiber Laser with Zero Group Velocity Dispersion,” IEEE Photonics J. 8(6), 1504608 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (2)

K. Tamura, C. R. Doerr, H. A. Haus, and E. P. Ippen, “Soliton fiber ring laser stabilization and tuning with a broad intracavity filter,” IEEE Photonics Technol. Lett. 6(6), 697–699 (1994).
[Crossref]

L. M. Zhao, C. Lu, H. Y. Tam, P. K. A. Wai, and D. Y. Tang, “High fundamental repetition rate fiber lasers operated in strong normal dispersion regime,” IEEE Photonics Technol. Lett. 21(11), 724–726 (2009).
[Crossref]

Infrared and Laser Eng. (1)

Z. Luming, S. Chaojie, W. Yufei, and L. Lei, “Research progress of period doubling bifurcation in ultrafast fiber lasers (invited),” Infrared and Laser Eng. 47(8), 803002 (2018).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

L. M. Zhao, D. Y. 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. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, C. Lu, and S. C. Wen, “Period-doubling of dispersion-managed solitons in an Erbium-doped fiber laser at around zero dispersion,” Opt. Commun. 278(2), 428–433 (2007).
[Crossref]

L. M. Zhao, D. Y. Tang, H. Zhang, X Wu, C. Lu, and H. Y. Tam, “Period-doubling of vector solitons in a ring fiber laser,” Opt. Commun. 281(22), 5614–5617 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. E (2)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63(5), 645–651 (2001).
[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), 691–738 (2004).
[Crossref]

Other (2)

Y. Cui, Y. Zhang, Y. Xiang, Y. Song, and X. Liu, “XPM-forced frequency-oscillating soliton in mode-locked fiber laser,” https://arxiv.org/abs/1910.09225 .

G. Sucha, S. R. Bolton, S. Weiss, and D. S. Chemla, “Nonlinear dynamics of additive pulse mode-locked lasers: period doubling and chaos,” Quantum Electronics & Laser Science Conference, JTuC6, 1993.

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Figures (5)

Fig. 1.
Fig. 1. Schematic of the fiber laser. WDM: wavelength division multiplexer; PC1/PC2: polarization controller; EDF: Erbium-doped fiber; OC: output coupler; I: isolator; P: polarizer.
Fig. 2.
Fig. 2. Soliton pulse evolution and the corresponding optical spectra numerically calculated under different pump strength. The linear cavity phase delay bias is set as ${{\Phi }_{PC}} = 1.35{\pi }$. (a)/(b) Period-one soliton state, $G = 700$; (c)/(d) Period-two soliton state, $G = 800$
Fig. 3.
Fig. 3. The detailed period doubling eigenstates of period doubling solitons under different linear cavity phase delay bias. The gain coefficient is fixed at $G = 820$. (a)/(b) synchronous when ${{\Phi }_{PC}} = 1.35{\pi }$; (c)/(d) the horizontal component is period one while the vertical component is period doubling when ${{\Phi }_{PC}} = 1.42{\pi }$; (e)/(f) asynchronous when ${{\Phi }_{PC}} = 1.46{\pi }$
Fig. 4.
Fig. 4. Evolution of the soliton (left), the horizontal component of the soliton (middle), and the vertical component of the soliton (right) in the cavity for two adjacent roundtrips. (a) The state of period one; (b) the state of synchronous period doubling evolution; (c) the state that the horizontal component is period one while the vertical component is period doubling; (d) the state of asynchronous period doubling evolution.
Fig. 5.
Fig. 5. Period tripling of soliton and the corresponding period tripling of individual polarization component. (a)/(b) Period tripling of soliton and the corresponding optical spectra; (c)/(d) Period tripling of soliton of individual polarization component

Tables (1)

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Table 1. Parameters used in the simulations

Equations (4)

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{ u z = i β u δ u t i k 2 2 u t 2 + i k 6 3 u t 3 + i γ ( | u | 2 + 2 3 | v | 2 ) u + i γ 3 v 2 u + g 2 u + g 2 Ω g 2 2 u t 2 v z = i β v + δ v t i k 2 2 v t 2 + i k 6 3 v t 3 + i γ ( | v | 2 + 2 3 | u | 2 ) v + i γ 3 u 2 v + g 2 v + g 2 Ω g 2 2 v t 2
g = G exp [ ( | u | 2 + | v | 2 ) d t P s a t ]
T = sin 2 θ sin 2 φ  +  cos 2 θ cos 2 φ  +  1 2 sin 2 θ sin 2 φ cos [ Φ l + Φ n l ]
Φ l = Φ P C + 2 π ( 1 Δ λ λ c ) L L b

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