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Interplay of multiphoton absorption, Raman scattering, and third-order dispersion in soliton fiber lasers

Gurkirpal Singh Parmar, Boris A. Malomed, and Soumendu Jana

Open Access Open Access

Abstract

We theoretically investigate the generation of dissipative solitons (DSs) and interactions between them in a fiber laser with higher-order dispersion and nonlinearity, multiphoton absorption, and gain dispersion or spectral filtering. A random component of the group-velocity dispersion (GVD) is taken into account too. The DSs are stabilized by the dynamical balance of the dispersion terms by the cubic–quintic nonlinearity, along with the balancing of the losses by a linear gain. Novel findings are presented for effects of the third-order GVD and intra-pulse stimulated Raman scattering on the formation and interactions of DSs in the system. A possibility of all-optical control of interactions between DSs by means of phase and temporal shifts between them is elaborated. The stability of the DSs against relatively large random-noise perturbations is explored too.

© 2021 Optical Society of America

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References

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  72. G. S. Parmar, S. Jana, and B. A. Malomed, “Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion,” J. Opt. Soc. Am. B 34, 850–860 (2017).
    [Crossref]

2020 (1)

Y. Zeng, W. Fan, and X. Wang, “The combined effects of intra-cavity spectral filtering on the fiber mode-locked laser,” Opt. Commun. 474, 126152 (2020).
[Crossref]

2019 (1)

D. Y. Song, X. Shi, C. Wu, D. Tang, and H. Zhang, “Recent progress of study on optical solitons in fiber lasers,” Appl. Phys. Rev. 6, 021313 (2019).
[Crossref]

2018 (3)

2017 (4)

2016 (2)

S. Jana, Shivani, G. S. Parmar, B. Kaur, Q. Zhou, A. Biswas, and M. Belić, “Evolution of bell-shaped dissipative optical solitons from super-Gaussian pulse in parabolic law medium with bandwidth limited amplification,” Optoelectron. Adv. Mater. Rapid Commun. 10, 143–150 (2016).

C. Cartes and O. Descalzi, “Periodic exploding dissipative solitons,” Phys. Rev. A 93, 031801 (2016).
[Crossref]

2015 (5)

G. S. Parmar and S. Jana, “Bistable dissipative soliton in cubic-quintic nonlinear medium with multiphoton absorption and gain dispersion,” J. Electromagn. Waves Appl. 29, 1410–1429 (2015).
[Crossref]

A. Chong, L. G. Wright, and F. W. Wise, “Ultrafast fiber lasers based on self-similar pulse evolution: a review of current progress,” Rep. Prog. Phys. 78, 113901 (2015).
[Crossref]

M. F. Saleh, A. Armaroli, A. Marini, and F. Biancalana, “Strong Raman-induced noninstantaneous soliton interactions in gas-filled photonic crystal fibers,” Opt. Lett. 40, 4058–4061 (2015).
[Crossref]

Z. Luo, Y. Li, M. Zhong, Y. Huang, X. Wan, J. Peng, and J. Weng, “Nonlinear optical absorption of few-layer molybdenum diselenide (MoSe2) for passively mode-locked soliton fiber laser,” Photon. Res. 3, A79–A86 (2015).
[Crossref]

E. M. Gromov and B. A. Malomed, “Solitons in a forced nonlinear Schrödinger equation with the pseudo-Raman effect,” Phys. Rev. E 92, 062926 (2015).
[Crossref]

2014 (5)

V. L. Kalashnikov and E. Sorokin, “Dissipative Raman solitons,” Opt. Express 22, 30118–30126 (2014).
[Crossref]

X. Gai, D. Choi, and B. Luther-Davies, “Negligible nonlinear absorption in hydrogenated amorphous silicon at 1.55 µm for ultra-fast nonlinear signal processing,” Opt. Express 22, 9948–9958 (2014).
[Crossref]

B. Fu, Y. Hua, X. Xiao, H. Zhu, Z. Sun, and C. Yang, “Broadband graphene saturable absorber for pulsed fiber lasers at 1, 1.5, and 2 µm,” IEEE J. Sel. Top. Quantum Electron. 20, 411–415 (2014).
[Crossref]

S. C. Latas and M. F. S. Ferreira, “Impact of higher-order effects on pulsating and chaotic solitons in dissipative systems,” Eur. Phys. J. Spec. Top. 223, 79–89 (2014).
[Crossref]

Y. Chen, M. Wu, P. Tang, S. Chen, J. Du, G. Jiang, Y. Li, C. Zhao, H. Zhang, and S. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode-locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11, 055101 (2014).
[Crossref]

2013 (2)

2012 (4)

L. Duan, X. Liu, D. Mao, L. Wang, and G. Wang, “Experimental observation of dissipative soliton resonance in an anomalous-dispersion fiber laser,” Opt. Express 20, 265–270 (2012).
[Crossref]

M. I. Carvalho and M. Facao, “Evolution of cubic–quintic complex Ginzburg–Landau erupting solitons under the effect of third-order dispersion and intrapulse Raman scattering,” Phys. Lett. A 376, 950–956 (2012).
[Crossref]

H. Huang, L. Yang, and J. Liu, “Qualitative assessment of laser-induced breakdown spectra generated with a femtosecond fiber laser,” Appl. Opt. 51, 8669–8676 (2012).
[Crossref]

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

2011 (1)

M. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[Crossref]

2010 (8)

B. Nagaraju, R. K. Varshney, G. P. Agrawal, and B. P. Pal, “Parabolic pulse generation in a dispersion-decreasing solid-core photonic bandgap Bragg fiber,” Opt. Commun. 283, 2525–2528 (2010).
[Crossref]

D. V. Skryabin and A. V. Gorbach, “Colloquium: looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[Crossref]

S. C. V. Latas and M. F. S. Ferreira, “Soliton explosion control by higher-order effects,” Opt. Lett. 35, 1771–1773 (2010).
[Crossref]

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96, 111112 (2010).
[Crossref]

L. Zhao, D. Tang, X. Wu, and H. Zhang, “Dissipative soliton generation in Yb-fiber laser with an invisible intracavity bandpass filter,” Opt. Lett. 35, 2756–2758 (2010).
[Crossref]

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives,” J. Opt. Soc. Am. B 27, B63–B92 (2010).
[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, 4428–4433 (2010).
[Crossref]

M. Facão, M. I. Carvalho, S. C. Latas, and M. F. Ferreira, “Control of complex Ginzburg–Landau equation eruptions using intrapulse Raman scattering and corresponding travelling solutions,” Phys. Lett. A 374, 4844–4847 (2010).
[Crossref]

2009 (6)

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).
[Crossref]

H. Zhang, D. Y. Tang, L. M. Zhao, X. Wu, and H. Y. Tam, “Dissipative vector solitons in a dispersion-managed cavity fiber laser with net positive cavity dispersion,” Opt. Express 17, 455–460 (2009).
[Crossref]

D. Pal, S. G. Ali, and B. Talukdar, “Evolution of optical pulses in the presence of third-order dispersion,” Pramana 72, 939–950 (2009).
[Crossref]

B. G. Bale and S. Boscolo, “Impact of third-order fibre dispersion on the evolution of parabolic optical pulses,” J. Opt. 12, 015202 (2009).
[Crossref]

S. Zhang, G. Zhao, A. Luo, and Z. Zhang, “Third-Order dispersion role on parabolic pulse propagation in dispersion-decreasing fiber with normal group-velocity dispersion,” Appl. Phys. B 94, 227–232 (2009).
[Crossref]

S. Roy, S. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).
[Crossref]

2008 (4)

L. M. Zhao, D. Y. Tang, H. Zhang, X. Wu, and N. Xiang, “Soliton trapping in fiber lasers,” Opt. Express 16, 9528–9533 (2008).
[Crossref]

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, 12618–12623 (2008).
[Crossref]

W. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

2007 (1)

S. Roy and S. Bhadra, “Study of pulse evolution and optical bistability under the influence of cubic-quintic nonlinearity and third order dispersion,” J. Nonlinear Opt. Phys. Mater. 16, 119–135 (2007).
[Crossref]

2006 (3)

2005 (2)

L. Song, X. Shi, W. Xue, Z. Li, and G. Zhou, “Analysis on femtosecond pulses generated by passively mode-locked lasers with higher-order effects,” Opt. Commun. 246, 495–503 (2005).
[Crossref]

S. C. V. Latas and M. F. S. Ferreira, “Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain,” Opt. Commun. 251, 415–422 (2005).
[Crossref]

2004 (2)

X. Shi, L. Li, R. Hao, Z. Li, and G. Zhou, “Stability analysis and interaction of chirped femtosecond soliton-like laser pulses,” Opt. Commun. 241, 185–194 (2004).
[Crossref]

F. K. Abdullaev, D. V. Navotny, and B. B. Baizakov, “Optical pulse propagation in fibers with random dispersion,” Physica D 192, 83–94 (2004).
[Crossref]

2003 (1)

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[Crossref]

2002 (2)

2001 (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, 056602 (2001).
[Crossref]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. USA 98, 14208–14211 (2001).
[Crossref]

1999 (1)

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of the third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[Crossref]

1998 (1)

1997 (1)

M. J. F. Digonne, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. 3, 44–64 (1997).
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1994 (3)

1991 (1)

B. A. Malomed, N. Sasa, and J. Satsuma, “Evolution of a damped soliton in a higher-order nonlinear Schrödinger equation,” Chaos Solitons Fractals 1, 383–388 (1991).
[Crossref]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
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Abdullaev, F. K.

F. K. Abdullaev, D. V. Navotny, and B. B. Baizakov, “Optical pulse propagation in fibers with random dispersion,” Physica D 192, 83–94 (2004).
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Agrawal, G. P.

P. Balla, S. Buch, and G. P. Agrawal, “Effect of Raman scattering on soliton interactions in optical fibers,” J. Opt. Soc. Am. B 34, 1247–1254 (2017).
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B. Nagaraju, R. K. Varshney, G. P. Agrawal, and B. P. Pal, “Parabolic pulse generation in a dispersion-decreasing solid-core photonic bandgap Bragg fiber,” Opt. Commun. 283, 2525–2528 (2010).
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S. Roy, S. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).
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J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
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G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2020).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Y. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Akhmediev, N.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6, 84–92 (2012).
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N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
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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, 056602 (2001).
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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, 515–523 (1998).
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N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations,” in Dissipative Solitons, N. Akhmediev and A. Ankiewicz, eds. (Springer, 2005), pp. 1–17.

Ali, S. G.

D. Pal, S. G. Ali, and B. Talukdar, “Evolution of optical pulses in the presence of third-order dispersion,” Pramana 72, 939–950 (2009).
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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, 515–523 (1998).
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N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations,” in Dissipative Solitons, N. Akhmediev and A. Ankiewicz, eds. (Springer, 2005), pp. 1–17.

Armaroli, A.

Baizakov, B. B.

F. K. Abdullaev, D. V. Navotny, and B. B. Baizakov, “Optical pulse propagation in fibers with random dispersion,” Physica D 192, 83–94 (2004).
[Crossref]

Bale, B. G.

B. G. Bale and S. Boscolo, “Impact of third-order fibre dispersion on the evolution of parabolic optical pulses,” J. Opt. 12, 015202 (2009).
[Crossref]

Balla, P.

Bao, Q.

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96, 111112 (2010).
[Crossref]

Belic, M.

S. Jana, Shivani, G. S. Parmar, B. Kaur, Q. Zhou, A. Biswas, and M. Belić, “Evolution of bell-shaped dissipative optical solitons from super-Gaussian pulse in parabolic law medium with bandwidth limited amplification,” Optoelectron. Adv. Mater. Rapid Commun. 10, 143–150 (2016).

Bhadra, S.

S. Roy, S. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).
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S. Roy and S. Bhadra, “Study of pulse evolution and optical bistability under the influence of cubic-quintic nonlinearity and third order dispersion,” J. Nonlinear Opt. Phys. Mater. 16, 119–135 (2007).
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Biancalana, F.

Biswas, A.

S. Jana, Shivani, G. S. Parmar, B. Kaur, Q. Zhou, A. Biswas, and M. Belić, “Evolution of bell-shaped dissipative optical solitons from super-Gaussian pulse in parabolic law medium with bandwidth limited amplification,” Optoelectron. Adv. Mater. Rapid Commun. 10, 143–150 (2016).

Boscolo, S.

B. G. Bale and S. Boscolo, “Impact of third-order fibre dispersion on the evolution of parabolic optical pulses,” J. Opt. 12, 015202 (2009).
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Buch, S.

Cartes, C.

C. Cartes and O. Descalzi, “Periodic exploding dissipative solitons,” Phys. Rev. A 93, 031801 (2016).
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Carvalho, M. I.

M. I. Carvalho and M. Facao, “Evolution of cubic–quintic complex Ginzburg–Landau erupting solitons under the effect of third-order dispersion and intrapulse Raman scattering,” Phys. Lett. A 376, 950–956 (2012).
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M. Facão, M. I. Carvalho, S. C. Latas, and M. F. Ferreira, “Control of complex Ginzburg–Landau equation eruptions using intrapulse Raman scattering and corresponding travelling solutions,” Phys. Lett. A 374, 4844–4847 (2010).
[Crossref]

Chen, S.

Y. Chen, M. Wu, P. Tang, S. Chen, J. Du, G. Jiang, Y. Li, C. Zhao, H. Zhang, and S. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode-locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11, 055101 (2014).
[Crossref]

Chen, Y.

Y. Chen, M. Wu, P. Tang, S. Chen, J. Du, G. Jiang, Y. Li, C. Zhao, H. Zhang, and S. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode-locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11, 055101 (2014).
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Chen, Z.

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).
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Chertkov, M.

M. Chertkov, I. Gabitov, P. M. Lushnikov, J. Moeser, and Z. Toroczkai, “Pinning method of pulse confinement in optical fiber with random dispersion,” J. Opt. Soc. Am. B 19, 2538–2550 (2002).
[Crossref]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. USA 98, 14208–14211 (2001).
[Crossref]

Choi, D.

Chong, A.

A. Chong, L. G. Wright, and F. W. Wise, “Ultrafast fiber lasers based on self-similar pulse evolution: a review of current progress,” Rep. Prog. Phys. 78, 113901 (2015).
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W. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
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Clarkson, W. A.

Descalzi, O.

C. Cartes and O. Descalzi, “Periodic exploding dissipative solitons,” Phys. Rev. A 93, 031801 (2016).
[Crossref]

Digonne, M. J. F.

M. J. F. Digonne, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. 3, 44–64 (1997).
[Crossref]

Du, J.

Y. Chen, M. Wu, P. Tang, S. Chen, J. Du, G. Jiang, Y. Li, C. Zhao, H. Zhang, and S. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode-locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11, 055101 (2014).
[Crossref]

Duan, L.

Facao, M.

M. I. Carvalho and M. Facao, “Evolution of cubic–quintic complex Ginzburg–Landau erupting solitons under the effect of third-order dispersion and intrapulse Raman scattering,” Phys. Lett. A 376, 950–956 (2012).
[Crossref]

Facão, M.

S. C. Latas, M. F. S. Ferreira, and M. Facão, “Ultrashort high-amplitude dissipative solitons in the presence of higher-order effects,” J. Opt. Soc. Am. B 34, 1033–1040 (2017).
[Crossref]

M. Facão, M. I. Carvalho, S. C. Latas, and M. F. Ferreira, “Control of complex Ginzburg–Landau equation eruptions using intrapulse Raman scattering and corresponding travelling solutions,” Phys. Lett. A 374, 4844–4847 (2010).
[Crossref]

Fan, W.

Y. Zeng, W. Fan, and X. Wang, “The combined effects of intra-cavity spectral filtering on the fiber mode-locked laser,” Opt. Commun. 474, 126152 (2020).
[Crossref]

Feng, X.

Ferreira, M. F.

M. Facão, M. I. Carvalho, S. C. Latas, and M. F. Ferreira, “Control of complex Ginzburg–Landau equation eruptions using intrapulse Raman scattering and corresponding travelling solutions,” Phys. Lett. A 374, 4844–4847 (2010).
[Crossref]

Ferreira, M. F. S.

S. C. Latas, M. F. S. Ferreira, and M. Facão, “Ultrashort high-amplitude dissipative solitons in the presence of higher-order effects,” J. Opt. Soc. Am. B 34, 1033–1040 (2017).
[Crossref]

S. C. Latas and M. F. S. Ferreira, “Impact of higher-order effects on pulsating and chaotic solitons in dissipative systems,” Eur. Phys. J. Spec. Top. 223, 79–89 (2014).
[Crossref]

S. C. V. Latas and M. F. S. Ferreira, “Soliton explosion control by higher-order effects,” Opt. Lett. 35, 1771–1773 (2010).
[Crossref]

S. C. V. Latas and M. F. S. Ferreira, “Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain,” Opt. Commun. 251, 415–422 (2005).
[Crossref]

Frantzeskakis, D. J.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of the third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[Crossref]

Fu, B.

B. Fu, Y. Hua, X. Xiao, H. Zhu, Z. Sun, and C. Yang, “Broadband graphene saturable absorber for pulsed fiber lasers at 1, 1.5, and 2 µm,” IEEE J. Sel. Top. Quantum Electron. 20, 411–415 (2014).
[Crossref]

Gabitov, I.

M. Chertkov, I. Gabitov, P. M. Lushnikov, J. Moeser, and Z. Toroczkai, “Pinning method of pulse confinement in optical fiber with random dispersion,” J. Opt. Soc. Am. B 19, 2538–2550 (2002).
[Crossref]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. USA 98, 14208–14211 (2001).
[Crossref]

Gai, X.

Gorbach, A. V.

D. V. Skryabin and A. V. Gorbach, “Colloquium: looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[Crossref]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[Crossref]

Grelu, P.

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

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

Gromov, E. M.

E. M. Gromov and B. A. Malomed, “Solitons in a forced nonlinear Schrödinger equation with the pseudo-Raman effect,” Phys. Rev. E 92, 062926 (2015).
[Crossref]

Guo, X.

Haelterman, M.

Hao, R.

X. Shi, L. Li, R. Hao, Z. Li, and G. Zhou, “Stability analysis and interaction of chirped femtosecond soliton-like laser pulses,” Opt. Commun. 241, 185–194 (2004).
[Crossref]

Hizanidis, K.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of the third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[Crossref]

Hua, Y.

B. Fu, Y. Hua, X. Xiao, H. Zhu, Z. Sun, and C. Yang, “Broadband graphene saturable absorber for pulsed fiber lasers at 1, 1.5, and 2 µm,” IEEE J. Sel. Top. Quantum Electron. 20, 411–415 (2014).
[Crossref]

Huang, H.

Huang, Y.

Jana, S.

G. S. Parmar, R. Pradhan, B. A. Malomed, and S. Jana, “Dispersion-managed soliton fiber laser with random dispersion, multiphoton absorption and gain dispersion,” J. Opt. 20, 105501 (2018).
[Crossref]

G. S. Parmar, S. Jana, and B. A. Malomed, “Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion,” J. Opt. Soc. Am. B 34, 850–860 (2017).
[Crossref]

G. S. Parmar, S. Jana, and B. A. Malomed, “Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion,” J. Opt. Soc. Am. B 34, 850–860 (2017).
[Crossref]

S. Jana, Shivani, G. S. Parmar, B. Kaur, Q. Zhou, A. Biswas, and M. Belić, “Evolution of bell-shaped dissipative optical solitons from super-Gaussian pulse in parabolic law medium with bandwidth limited amplification,” Optoelectron. Adv. Mater. Rapid Commun. 10, 143–150 (2016).

G. S. Parmar and S. Jana, “Bistable dissipative soliton in cubic-quintic nonlinear medium with multiphoton absorption and gain dispersion,” J. Electromagn. Waves Appl. 29, 1410–1429 (2015).
[Crossref]

Jiang, G.

Y. Chen, M. Wu, P. Tang, S. Chen, J. Du, G. Jiang, Y. Li, C. Zhao, H. Zhang, and S. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode-locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11, 055101 (2014).
[Crossref]

Kalashnikov, V. L.

Kang, Y.

Kaur, B.

S. Jana, Shivani, G. S. Parmar, B. Kaur, Q. Zhou, A. Biswas, and M. Belić, “Evolution of bell-shaped dissipative optical solitons from super-Gaussian pulse in parabolic law medium with bandwidth limited amplification,” Optoelectron. Adv. Mater. Rapid Commun. 10, 143–150 (2016).

Kivshar, Y.

Y. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Knize, R. J.

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96, 111112 (2010).
[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, 4428–4433 (2010).
[Crossref]

Kodama, Y.

Kolobov, M.

M. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[Crossref]

Krasieva, T. B.

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).
[Crossref]

Kudlinski, A.

M. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[Crossref]

Latas, S. C.

S. C. Latas, M. F. S. Ferreira, and M. Facão, “Ultrashort high-amplitude dissipative solitons in the presence of higher-order effects,” J. Opt. Soc. Am. B 34, 1033–1040 (2017).
[Crossref]

S. C. Latas and M. F. S. Ferreira, “Impact of higher-order effects on pulsating and chaotic solitons in dissipative systems,” Eur. Phys. J. Spec. Top. 223, 79–89 (2014).
[Crossref]

M. Facão, M. I. Carvalho, S. C. Latas, and M. F. Ferreira, “Control of complex Ginzburg–Landau equation eruptions using intrapulse Raman scattering and corresponding travelling solutions,” Phys. Lett. A 374, 4844–4847 (2010).
[Crossref]

Latas, S. C. V.

S. C. V. Latas and M. F. S. Ferreira, “Soliton explosion control by higher-order effects,” Opt. Lett. 35, 1771–1773 (2010).
[Crossref]

S. C. V. Latas and M. F. S. Ferreira, “Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain,” Opt. Commun. 251, 415–422 (2005).
[Crossref]

Li, L.

X. Shi, L. Li, R. Hao, Z. Li, and G. Zhou, “Stability analysis and interaction of chirped femtosecond soliton-like laser pulses,” Opt. Commun. 241, 185–194 (2004).
[Crossref]

Z. Xu, L. Li, Z. Li, and G. Zhou, “Soliton interaction under the influence of higher-order effects,” Opt. Commun. 210, 375–384 (2002).
[Crossref]

Li, Y.

Y. Li, L. Wang, Y. Kang, X. Guo, and L. Tong, “Microfiber-enabled dissipative soliton fiber laser at 2 µm,” Opt. Lett. 43, 6105–6108 (2018).
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Z. Luo, Y. Li, M. Zhong, Y. Huang, X. Wan, J. Peng, and J. Weng, “Nonlinear optical absorption of few-layer molybdenum diselenide (MoSe2) for passively mode-locked soliton fiber laser,” Photon. Res. 3, A79–A86 (2015).
[Crossref]

Y. Chen, M. Wu, P. Tang, S. Chen, J. Du, G. Jiang, Y. Li, C. Zhao, H. Zhang, and S. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode-locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11, 055101 (2014).
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Li, Z.

L. Song, X. Shi, W. Xue, Z. Li, and G. Zhou, “Analysis on femtosecond pulses generated by passively mode-locked lasers with higher-order effects,” Opt. Commun. 246, 495–503 (2005).
[Crossref]

X. Shi, L. Li, R. Hao, Z. Li, and G. Zhou, “Stability analysis and interaction of chirped femtosecond soliton-like laser pulses,” Opt. Commun. 241, 185–194 (2004).
[Crossref]

Z. Xu, L. Li, Z. Li, and G. Zhou, “Soliton interaction under the influence of higher-order effects,” Opt. Commun. 210, 375–384 (2002).
[Crossref]

Lin, Z. B.

Liu, J.

Liu, X.

Loh, K. P.

H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96, 111112 (2010).
[Crossref]

Louvergneaux, E.

M. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[Crossref]

Luo, A.

S. Zhang, G. Zhao, A. Luo, and Z. Zhang, “Third-Order dispersion role on parabolic pulse propagation in dispersion-decreasing fiber with normal group-velocity dispersion,” Appl. Phys. B 94, 227–232 (2009).
[Crossref]

Luo, A. P.

Luo, Z.

Luo, Z. C.

Lushnikov, P. M.

Luther-Davies, B.

Malomed, B. A.

G. S. Parmar, R. Pradhan, B. A. Malomed, and S. Jana, “Dispersion-managed soliton fiber laser with random dispersion, multiphoton absorption and gain dispersion,” J. Opt. 20, 105501 (2018).
[Crossref]

H. Sakaguchi, D. V. Skryabin, and B. A. Malomed, “Stationary and oscillatory bound states of dissipative solitons created by third-order dispersion,” Opt. Lett. 43, 2688–2691 (2018).
[Crossref]

G. S. Parmar, S. Jana, and B. A. Malomed, “Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion,” J. Opt. Soc. Am. B 34, 850–860 (2017).
[Crossref]

G. S. Parmar, S. Jana, and B. A. Malomed, “Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion,” J. Opt. Soc. Am. B 34, 850–860 (2017).
[Crossref]

E. M. Gromov and B. A. Malomed, “Solitons in a forced nonlinear Schrödinger equation with the pseudo-Raman effect,” Phys. Rev. E 92, 062926 (2015).
[Crossref]

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of the third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[Crossref]

B. A. Malomed, “Optical domain walls,” Phys. Rev. E 50, 1565–1571 (1994).
[Crossref]

B. A. Malomed, N. Sasa, and J. Satsuma, “Evolution of a damped soliton in a higher-order nonlinear Schrödinger equation,” Chaos Solitons Fractals 1, 383–388 (1991).
[Crossref]

Mao, D.

Marini, A.

Midrio, M.

Moeser, J.

M. Chertkov, I. Gabitov, P. M. Lushnikov, J. Moeser, and Z. Toroczkai, “Pinning method of pulse confinement in optical fiber with random dispersion,” J. Opt. Soc. Am. B 19, 2538–2550 (2002).
[Crossref]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. USA 98, 14208–14211 (2001).
[Crossref]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[Crossref]

Mussot, A.

M. Kolobov, A. Mussot, A. Kudlinski, E. Louvergneaux, and M. Taki, “Third-order dispersion drastically changes parametric gain in optical fiber systems,” Phys. Rev. A 83, 035801 (2011).
[Crossref]

Nagaraju, B.

B. Nagaraju, R. K. Varshney, G. P. Agrawal, and B. P. Pal, “Parabolic pulse generation in a dispersion-decreasing solid-core photonic bandgap Bragg fiber,” Opt. Commun. 283, 2525–2528 (2010).
[Crossref]

Navotny, D. V.

F. K. Abdullaev, D. V. Navotny, and B. B. Baizakov, “Optical pulse propagation in fibers with random dispersion,” Physica D 192, 83–94 (2004).
[Crossref]

Nilsson, J.

Ning, Q. Y.

Nistazakis, H. E.

B. A. Malomed, D. J. Frantzeskakis, H. E. Nistazakis, A. Tsigopoulos, and K. Hizanidis, “Dissipative solitons under the action of the third-order dispersion,” Phys. Rev. E 60, 3324–3331 (1999).
[Crossref]

Olivier, M.

Pal, B. P.

B. Nagaraju, R. K. Varshney, G. P. Agrawal, and B. P. Pal, “Parabolic pulse generation in a dispersion-decreasing solid-core photonic bandgap Bragg fiber,” Opt. Commun. 283, 2525–2528 (2010).
[Crossref]

Pal, D.

D. Pal, S. G. Ali, and B. Talukdar, “Evolution of optical pulses in the presence of third-order dispersion,” Pramana 72, 939–950 (2009).
[Crossref]

Pantell, R. H.

M. J. F. Digonne, R. W. Sadowski, H. J. Shaw, and R. H. Pantell, “Resonantly enhanced nonlinearity in doped fibers for low-power all-optical switching: a review,” Opt. Fiber Technol. 3, 44–64 (1997).
[Crossref]

Parmar, G. S.

G. S. Parmar, R. Pradhan, B. A. Malomed, and S. Jana, “Dispersion-managed soliton fiber laser with random dispersion, multiphoton absorption and gain dispersion,” J. Opt. 20, 105501 (2018).
[Crossref]

G. S. Parmar, S. Jana, and B. A. Malomed, “Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion,” J. Opt. Soc. Am. B 34, 850–860 (2017).
[Crossref]

G. S. Parmar, S. Jana, and B. A. Malomed, “Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion,” J. Opt. Soc. Am. B 34, 850–860 (2017).
[Crossref]

S. Jana, Shivani, G. S. Parmar, B. Kaur, Q. Zhou, A. Biswas, and M. Belić, “Evolution of bell-shaped dissipative optical solitons from super-Gaussian pulse in parabolic law medium with bandwidth limited amplification,” Optoelectron. Adv. Mater. Rapid Commun. 10, 143–150 (2016).

G. S. Parmar and S. Jana, “Bistable dissipative soliton in cubic-quintic nonlinear medium with multiphoton absorption and gain dispersion,” J. Electromagn. Waves Appl. 29, 1410–1429 (2015).
[Crossref]

Peng, J.

Piche, M.

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L. Song, X. Shi, W. Xue, Z. Li, and G. Zhou, “Analysis on femtosecond pulses generated by passively mode-locked lasers with higher-order effects,” Opt. Commun. 246, 495–503 (2005).
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X. Shi, L. Li, R. Hao, Z. Li, and G. Zhou, “Stability analysis and interaction of chirped femtosecond soliton-like laser pulses,” Opt. Commun. 241, 185–194 (2004).
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S. Jana, Shivani, G. S. Parmar, B. Kaur, Q. Zhou, A. Biswas, and M. Belić, “Evolution of bell-shaped dissipative optical solitons from super-Gaussian pulse in parabolic law medium with bandwidth limited amplification,” Optoelectron. Adv. Mater. Rapid Commun. 10, 143–150 (2016).

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B. Fu, Y. Hua, X. Xiao, H. Zhu, Z. Sun, and C. Yang, “Broadband graphene saturable absorber for pulsed fiber lasers at 1, 1.5, and 2 µm,” IEEE J. Sel. Top. Quantum Electron. 20, 411–415 (2014).
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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Y. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

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Figures (18)
Fig. 1.
Fig. 1. Scheme of the DS fiber laser. SMF1, a longer single-mode fiber; SMF2, a shorter single-mode fiber; GF, the gain fiber (the Yb- or Er-doped one); WDM, wavelength-division multiplexing; C, the collimator; $\frac{{\lambda}}{2}$ WP, the half-wave plate; $\frac{{\lambda}}{4}$ WP, the quarter-wave plate; PBS, the polarization beam splitter; SF, the spectral filter; I, the isolator.

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Fig. 2.
Fig. 2. Variation of excess gain $\Delta g$ required to generate DSs at different values of the TOD coefficient $\beta$ under the action of TPA, 3PA, and combined TPA-3PA losses as indicated in the figure. Other parameters are $K = {0.01}$ , $\upsilon = {0.01}$ , $d = {0.05}$ , $\gamma = - {0.01}$ .

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Fig. 3.
Fig. 3. Generation of DSs under the action of the TOD and gain dispersion for different types ofmultiphoton absorption: (a), (b) in the presence of TPA, with $K = {0.01}$ and $\upsilon = 0$ ; (c), (d) in the presence of 3PA, with $K = 0$ and $\upsilon = {0.01}$ ; (e), (f) under the action of the TPA-3PA combination, with $K = 0.01$ and $\upsilon = {0.01}$ . Other parameters are $\gamma = -0.01$ and $d = {0.05}$ , $\beta = {0.01}$ . The excess gain required for the generation of stable DS in the presence of TPA is ${\Delta}g = 5.317 \times {10^{- 6}}$ ; ${\Delta}g = 5.206 \times {10^{- 6}}$ for 3PA; and in the case of combined TPA-3PA losses ${\Delta}g = 5.402 \times {10^{- 6}}$ . The left column displays the 3D profile of the simulated evolution, whereas the corresponding contour plots of the local intensity are presented in the right column.

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Fig. 4.
Fig. 4. Generation of stable DSs under the action of TOD and gain dispersion in the presence ofrandom GVD at the 5% level, for different schemes of the multiphoton absorption: (a), (b) in the presence of TPA; (c), (d) in the presence of 3PA; (e), (f) under the combined action of TPA and 3PA. Other parameters are the same as in Fig. 3. The excess gain required for the generation of stable DS in the presence of TPA is ${\Delta} g = 6.233 \times {10^{- 6}}$ ; for 3PA it is ${\Delta} g = 6.116 \times {10^{- 6}}$ ; and in the case of TPA-3PA combination it is ${\Delta} g = 6.302 \times {10^{- 6}}$ .

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Fig. 5.
Fig. 5. Variation of the quintic-nonlinearity coefficient $| \gamma |$ , required to generate stable DSs at different values of TOD coefficient $\beta$ , under the action of 3PA alone or TPA-3PA combination, while the excess gain stays constant: in the presence of both TPA and 3PA, it is ${\Delta}g = 5.402 \times {10^{- 6}}$ , and for 3PA only, it is ${\Delta}g = 5.206 \times {10^{- 6}}$ . Other parameters are $K = 0.01$ , $\nu = 0.01$ , and $d = {0.05}$ .

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Fig. 6.
Fig. 6. Interaction of identical in-phase DSs with initial temporal separation (a)  ${T_g} = 80$ and (b)  ${T_g} = 30$ . (c) The interaction of DSs with initial phase difference ${\Delta} \varphi = \pi /80$ and ${T_g} = 80$ . Other parameters are $K = {0.01}$ , $\nu = 0.01$ , $d = {0.05}$ , $\beta = 0.01$ , and $\gamma = - 0.01$ .

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Fig. 7.
Fig. 7. Merger of identical DSs with different values of the initial phase shift ${\Delta}\varphi$ between them and a fixed initial temporal separation ${T_g} = 10$ : (a)  ${\Delta}\varphi = \pi /15$ (resulting in a blueshift of the merged pulse), (b)  ${\Delta}\varphi = \pi /3$ (no frequency shift), and (c)  ${\Delta}\varphi = 2\pi /3$ (the redshift). Other parameters are $K = {0.01}$ , $\nu$ = 0.01, $d = {0.05}$ , $\beta = 0.01$ , and $\gamma = - 0.01$ . Panels (d) and (e) display the 3D view of the interaction shown in (a) and (c), respectively. Power spectra corresponding to (a), (b), and (c) are plotted in (e), (f), and (g), respectively.

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Fig. 8.
Fig. 8. Same as in Fig. 7, but in absence of TOD, $\beta = 0$ . In this case, the 3D plots in the temporal domains are omitted.

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Fig. 9.
Fig. 9. Interaction of identical DSs with different values of the initial temporal separation ${T_g}$ between them and a fixed initial phase shift ${\Delta}\varphi = \pi /15$ ; (a)  ${T_g} = 10$ , (b)  ${T_g} = 13$ , and (c)  ${T_g} = 17$ . The corresponding power spectra are plotted in (d), (e), and (f), respectively. The parameters are $K = {0.01}$ , $\nu = 0.01$ , $d = {0.05}$ , $\beta = 0.01$ , and $\gamma = - 0.01$ .

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Fig. 10.
Fig. 10. Generation of robust DSs under the combined effect of TOD, SRS, gain dispersion, and (a), (b) TPA or (c) and (d) 3PA. The 3D view of the evolution of the DSs and their initial (blue-solid) and final (green-dotted) temporal profiles are displayed in the left and right columns, respectively. In (a) and (b), $K = {0.01}$ , $\upsilon = {0}$ , and $\Delta g = 5.692 \times {10^{- 6}}$ . In (c) and (d), $K = {0}$ , $\upsilon = {0.01}$ , and $\Delta g = 5.440 \times {10^{- 6}}$ . Other parameters are $\gamma = -0.01$ , $d = {0.05}$ , $\beta = {0.01}$ , and ${T_R} = {0.1}$ .

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Fig. 11.
Fig. 11. Generation of robust DSs under the combined effect of TOD, SRS, gain dispersion, and (a) TPA or (b) 3PA, but in the presence of the random GVD term, with the mean value equal to 5% of the constant-GVD coefficient. In (a)  $K = {0.01}$ , $\upsilon = {0}$ , and $\Delta g = 6.842 \times {10^{- 6}}$ ; in (b)  $K = 0$ , $\upsilon = {0.01}$ , and $\Delta g = 6.657 \times {10^{- 6}}$ . Other parameters are $\gamma = -0.01$ , $d = {0.05}$ , $\beta = {0.01}$ , and ${T_R} = {0.1}$ .

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Fig. 12.
Fig. 12. Generation of the DSs under the combined action of SRS, TPA, 3PA, and the gain dispersion: (a) in the absence of TOD and (b) in the presence of TOD. In (a)  $\beta = {0}$ and $\Delta g =$ $5.751 \times {10^{- 6}}$ ; in (b)  $\beta = {0.01}$ and $\Delta g = 5.860 \times {10^{- 6}}$ . Other parameters are $\gamma = - {0.01}$ , $d = {0.05}$ , $K = {0.01}$ , $\upsilon = {0.01}$ , and ${T_R} = {0.1}$ .

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Fig. 13.
Fig. 13. Generation of robust DSs under the same conditions as in Fig. 12, but with the addition of random GVD at the 5% level (a) in the absence of TOD: $\beta = {0}$ , with $\Delta {g} = 6.898 \times {10^{- 6}}$ ; (b) in the presence of TOD, $\beta = {0.01}$ , with $\Delta g = 6.972 \times {10^{- 6}}$ . Other parameters are same as in Fig. 11.

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Fig. 14.
Fig. 14. Temporal shift of the DS versus the SRS strength, under the action of the multiphoton absorption, gain dispersion, and CQ nonlinearity. The solid and dashed lines present the dependence in the absence and presence of TOD, respectively. The parameters are $\gamma = -0.01$ , $d = 0.05$ , $K = 0.01$ , $\upsilon = 0.01$ , and $\beta = 0.01$ (for the dashed line only). The results are displayed for propagation distance $z = 50$ .

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Fig. 15.
Fig. 15. Interaction of two in-phase pulses with different initial temporal separations ${T_{g}}$ . (a)  ${T_{g}} = 0$ , (b)  ${T_{g}} = 20$ , and (c)  ${T_{g}} = 35$ . Other parameters are $K=0.01$ , $\upsilon = 0.01$ , $\gamma = 0.01$ , $d = 0.05$ , $\beta = 0.01$ , and ${T_R} = 0.1$ . The simulations demonstrate that the SRS-mediated interaction transfers energy from the leading pulse to the trailing one, along with the SRS-induced redshift of both pulses.

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Fig. 16.
Fig. 16. Interaction of two in-phase DSs for a fixed initial temporal separation ${T_g} = {10}$ in the presence of different types of multiphoton absorption: (a) TPA only, (b) 3PA only, (c) the combination of TPA and 3PA. In (a)  $K = 0.01$ , $\upsilon = 0$ ; in (b)  $K = 0$ , $\upsilon = 0.01$ ; in (c)  $K = 0.01$ , $\upsilon = 0.01$ . Other parameters are $\gamma = - 0.01$ , $d = 0.05$ , $\beta = 0.01$ , and ${T_R} = 0.1$ .

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Fig. 17.
Fig. 17. Phase-difference-controlled interaction under the same conditions as in Fig. 7 (i.e.,  ${T_g} = 10$ , $K = 0.01$ , $\nu = 0.01$ , $d = 0.05$ , $\beta = 0.01$ , and $\gamma = - 0.01$ ) but with inclusion of SRS. The initial phase difference between the two DSs is (a)  $\pi /30$ , (b)  $\pi /10$ , (c)  $\pi /3$ , and (d)  $2\pi /3$ . In this case, switching cannot be achieved by varying the phase difference.

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Fig. 18.
Fig. 18. Persistent evolution of the DSs with the addition of initial random noise at the amplitude level (a) 2%, (b) 4%, and (c) 6%. The parameters are $\gamma = -0.01$ , $d = {0.05}$ , $K = {0.01}$ , $\beta = {0.01}$ , $\upsilon = 0.01$ , and ${T_R} = {0.1}$ .

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Equations (5)

Equations on this page are rendered with MathJax. Learn more.

i ∂ E ∂ z + D ( z ) 2 ∂ 2 E ∂ t 2 − i β 6 ∂ 3 E ∂ t 3 + | E | 2 E + γ | E | 4 E = i 2 ( g 0 − α ) E + i d 2 ∂ 2 E ∂ t 2 − i K | E | 2 E − i ν | E | 4 E + T R E ∂ ∂ t ( | E | 2 ) ,
E ( z , t ) = A ( z ) s e c h ( t W ( z ) ) exp ⁡ ( i ϕ ( z ) ) ,
∂ E ∂ z = L ^ E + N ^ E ,
L ^ = ( i D ( z ) 2 ∂ 2 ∂ t 2 + β 6 ∂ 3 ∂ t 3 + 1 2 ( g o − α ) + d 2 ∂ 2 ∂ t 2 ) E ,
N ^ = i ( | E | 2 + γ | E | 4 + i K | E | 2 + i ν | E | 4 − T R ∂ | E | 2 ∂ t ) E .

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