## Soliton fission management by dispersion oscillating fiber

Optics Express, Vol. 15, Issue 25, pp. 16302-16307 (2007)

http://dx.doi.org/10.1364/OE.15.016302

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### Abstract

We report the experimental observation of the fission of picosecond solitons in a fiber with sine-wave variation of the core diameter along the longitudinal direction of propagation. The experimental pulse dynamics is reproduced by numerical simulations. The fission of high-intensity solitons caused by both the variation of the fiber dispersion and stimulated Raman scattering is demonstrated. The number of output pulses and their frequencies can be managed by periodical modulation of the fiber dispersion even under the strong effect of the Raman scattering.

© 2007 Optical Society of America

## 1. Introduction

2. K. Tai, A. Hasegawa, and N. Bekki“Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. **13**, 392–394 (1988). [CrossRef] [PubMed]

3. P. K. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. **11**, 464–466 (1986). [CrossRef] [PubMed]

4. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Benion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. , **32**, 54–55 (1996). [CrossRef]

5. R. Driben and B. A. Malomed, “Split-step solitons in long fiber links”, Opt. Commun. **185**, 439–456 (2000). [CrossRef]

6. T. Inoue, H. Tobioka, and S. Namiki, “Stationary rescaled pulse in alternately concatenated fibers with O(1)-accumulated nonlinear perturbations” Phys. Rev. E **72**, 025601(R) (2005). [CrossRef]

7. M. Böhm and F. Mitschke, “Soliton propagation in a dispersion map with deviation from periodicity” Appl. Phys. B **81**, 983–987 (2005). [CrossRef]

*sech*-shaped fundamental solitons. A stepwise change of dispersion, a localized loss element or filter will generate pairs of fundamental solitons with shifted central wavelengths [8

8. K. Lee and J. Buck, “Wavelength conversion through higher-order soliton splitting initiated by localized channelperturbations”, J. Opt. Soc. Am. B **20**, 514–519 (2003). [CrossRef]

9. S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons”, Phys. Rev. Lett. **84**, 1902–1905 (2000). [CrossRef] [PubMed]

10. B.A. Malomed, D.F. Parker, and N.F. Smyth “Resonant shape oscillations and decay of a soliton in a periodicallyinhomogeneous nonlinear optical fiber,” Phys. Rev. E **48**, 1418–1425 (1993). [CrossRef]

11. R.G. Bauer and L.A. Melnikov “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. **115**, 190–195 (1995). [CrossRef]

12. A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. **66**, 161–164 (1991). [CrossRef] [PubMed]

13. H. Sakaguchi and B.A. Malomed “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E **70**, 066613 (2004). [CrossRef]

14. B.A. Malomed and N.F. Smyth “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E **50**, 1535–1542 (1994). [CrossRef]

12. A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. **66**, 161–164 (1991). [CrossRef] [PubMed]

*z*

_{0}[15], the soliton splits into pulses propagating with different group velocities [11

11. R.G. Bauer and L.A. Melnikov “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. **115**, 190–195 (1995). [CrossRef]

12. A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. **66**, 161–164 (1991). [CrossRef] [PubMed]

16. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva “Nonautonomous solitons in external potentials” Phys. Rev. Lett. **98**, 074102 (2007). [CrossRef] [PubMed]

## 2. Soliton fission in dispersion oscillating fiber

*A*(

*z*,

*t*) is the complex pulse envelope,

*α*is the loss coefficient, and

*ν*

_{0}is the carrier frequency of the pulse. Functions

*β*

_{2}(

*z*) and

*β*

_{3}(

*z*) describe dispersion varying along the fiber length. Nonlinear media polarization includes the Kerr effect and delayed Raman scattering

*P*(

_{NL}*z*,

*t*)=

*γ*(

_{K}*z*)|

*A*|

^{2}

*A*+

*γ*(

_{R}*z*)

*QA*(

*z*,

*t*), where

*γ*(

_{K}*z*) and

*γ*(

_{R}*z*) are nonlinear coefficients. The Raman delayed response

*Q*(

*z*,

*t*) is approximated by damping oscillations [17

17. V. G. Bespalov, S. A. Kozlov, Yu. A. Shpolyanskiy, and I. A. Walmsley “Simplified field wave equations for the nonlinear propagation of extremely short light pulses” Phys. Rev. A **66**, 013811 (2002). [CrossRef]

*∂*

^{2}

*Q*/

*∂t*

^{2})+2

*T*

^{-1}

_{2}(

*∂Q*/

*∂ t*)+Ω

^{2}

*Q*(

*z*,

*t*)=Ω

^{2}|

*A*(

*z*,

*t*)|

^{2}, where

*T*

_{2}=32fs, Ω(2

*π*)

^{-1}=13.1THz. The equation (1) was solved using standard split-step Fourier algorithm [15]. Simulations were carried out with hyperbolic secant input pulses having intensity full-width at half maximum duration

*T*

_{FWHM}=2.05ps.

*d*(

*z*)=

*d*

_{0}(1+

*d*sin(2

_{m}*πz*/

*z*)), where

_{m}*d*

_{0}=133

*µ*m,

*z*=0.16km is the modulation period,

_{m}*d*

_{m}=0.03 is the modulation depth. The linear loss at 1550 nm is 0.69 dB/km that corresponds to

*α*=0.159km

^{-1}in eq.(1). Dispersion measured for three different fibers drawn from the same preform is shown in Fig. 1(b). Using measurements shown in Fig. 1 the coefficients of eq.(1) were defined as follows:

*β*

_{2}〉=-12.76ps

^{2}km

^{-1}, 〈β

_{3}〉=0.0761ps

^{3}km

^{-1}, 〈

*γ*〉=8.2W

_{K}^{-1}km

^{-1}, 〈

*γ*〉=1.8W

_{R}^{-1}km

^{-1},

*β*

_{2}(

*m*)=0.02,

*β*

_{3}(

*m*)=0.095,

*γ*=0.028,

_{m}*φ*is the modulation phase. Fourthorder dispersion coefficient

_{m}*β*

_{4}is around -1.4×10

^{-4}ps

^{4}km

^{-1}at

*λ*=1550nm. The magnitude of

*β*

_{4}is not sufficiently high to play any significant role in considered regimes.

*T*

_{FWHM}=2.05ps the soliton period [15] is

*z*

_{0}=0.16

*π*|〈

*β*

_{2}〉|

^{-1}

*T*

_{2}

_{FWHM}=0.166km. When

*z*is near

_{m}*z*

_{0}, the soliton splitting is possible at a short propagation distance (

*z*<0.8km). The soliton with

*N*=1.72 can be split into two pulses (Fig. 3). At the certain range of

*z*the soliton splitting is not clearly discernible. That is why the curves in Fig. 3(a) have discontinuities. The change between stable and unstable propagation regimes of two bound solitons with variation of modulation parameters was predicted in [12

_{m}**66**, 161–164 (1991). [CrossRef] [PubMed]

*ϕ*=0 and

_{m}*ϕ*=

_{m}*π*. For the last the soliton splitting have higher efficiency. That is agree with simulations (Fig. 3(a)). Due to the Raman scattering the maximum separation between pulses Δ

*T*is achieved at

*z*that not coincide with

_{m}*z*

_{0}=0.166km (Fig. 3(a)). However, with

*z*≃

_{m}*z*

_{0}only one modulation period of DOF is sufficient for the soliton splitting (Fig. 3(b)). In Fig. 3(b) the normalized intensities of output pulses are 0.9 and 1.0. Such an asymmetry is caused mainly by the effect of the Raman scattering. For regime shown in Fig. 3 the self-steepening operator (

*∂P*/

_{NL}*∂t*) in (1) can be neglected. While for solitons of higher order the self-steepening leads to considerable changes in amplitudes and group velocities of the pulses.

*sech*

^{2}envelope. After the splitting the oscillations appear in the spectrum (Fig. 3(d)). Such a structure of output spectrum arises due to the interference between two pulses with shifted carrier frequencies.

*λ*<1560nm (Fig. 4(b)) corresponds to the pulse with the highest peak intensity (Fig. 4(a), pulse No.3). Spectra of other two pulses are overlapped in the range 1545nm<λ<1555nm. The splitting into three pulses was detected experimentally using the oscilloscope record (Fig. 4(c)).

## 3. Soliton fission management

*d*and modulation period

_{m}*z*on the soliton fission. The constants

_{m}*β*

_{2(m)},

*β*

_{3(m)}and

*γ*in eq. (2),(3) are practically linearly dependent on the modulation depth

_{m}*d*for considered fiber diameters.

_{m}*N*=1.72, the input pulse splits into two pulses (Fig. 3(b)) with different carrier frequencies. For

*d*=0.03 (Fig. 3) the difference between carrier frequencies of output pulses is Δ

_{m}*ν*=0.214THz. By varying the magnitude of the perturbation, one can vary the difference Δ

*ν*and temporal separation of output pulses accordingly. From numerical simulations the differences Δ

*ν*=0.276THz (

*d*=0.039) and Δ

_{m}*ν*=0.103THz (

*d*=0.021) were obtained for

_{m}*z*=0.160km.

_{m}*N*=2.33) the number of output pulses becomes dependent on the modulation period

*z*(Fig. 5). Without modulation (

_{m}*z*=∞) SRS leads to the soliton fission into two pulses (Fig. 5(a)). Periodical modulation of the fiber dispersion with modulation period (

_{m}*z*=0.16)km) allows to obtain three output pulses (Fig. 5(b)). DOF with reduced value of the modulation period (

_{m}*z*=0.08km) splits the initial pulse into two pulses with nearly identical amplitudes (Fig. 5(c)).

_{m}*N*=3.02) into three pulses (Fig. 6(a), red curve). The modulation of the fiber diameter with

*z*=0.16km only increase the temporal separation between these pulses (Fig. 4(a)). At the modulation period

_{m}*z*=0.08km the pulse dynamics becomes quite different. The pulse splits into several lowintensity pulses (Fig. 6(b), red curve). As a result the output spectrum does not reach the infrared region λ>1557nm (Fig. 6(b), blue curve). This example demonstrates that the DOF can be used for the management of the fission of high-order soliton even under the strong effect of SRS.

_{m}## 4. Conclusion

*N*>2) the stimulated Raman scattering becomes important for the fission of 2-picosecond solitons. Change of the modulation period and modulation depth of the DOF allows to manage the number of output pulses and their frequencies even in presence of the strong effect of the Raman scattering.

## Acknowledgments

## References and links

1. | E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. |

2. | K. Tai, A. Hasegawa, and N. Bekki“Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. |

3. | P. K. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. |

4. | N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Benion, “Enhanced power solitons in optical fibers with periodic dispersion management,” Electron. Lett. , |

5. | R. Driben and B. A. Malomed, “Split-step solitons in long fiber links”, Opt. Commun. |

6. | T. Inoue, H. Tobioka, and S. Namiki, “Stationary rescaled pulse in alternately concatenated fibers with O(1)-accumulated nonlinear perturbations” Phys. Rev. E |

7. | M. Böhm and F. Mitschke, “Soliton propagation in a dispersion map with deviation from periodicity” Appl. Phys. B |

8. | K. Lee and J. Buck, “Wavelength conversion through higher-order soliton splitting initiated by localized channelperturbations”, J. Opt. Soc. Am. B |

9. | S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons”, Phys. Rev. Lett. |

10. | B.A. Malomed, D.F. Parker, and N.F. Smyth “Resonant shape oscillations and decay of a soliton in a periodicallyinhomogeneous nonlinear optical fiber,” Phys. Rev. E |

11. | R.G. Bauer and L.A. Melnikov “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun. |

12. | A. Hasegawa and Y. Kodama, “Guiding center solitons,” Phys. Rev. Lett. |

13. | H. Sakaguchi and B.A. Malomed “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E |

14. | B.A. Malomed and N.F. Smyth “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E |

15. | G. Agrawal, |

16. | V. N. Serkin, A. Hasegawa, and T. L. Belyaeva “Nonautonomous solitons in external potentials” Phys. Rev. Lett. |

17. | V. G. Bespalov, S. A. Kozlov, Yu. A. Shpolyanskiy, and I. A. Walmsley “Simplified field wave equations for the nonlinear propagation of extremely short light pulses” Phys. Rev. A |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: October 11, 2007

Revised Manuscript: November 15, 2007

Manuscript Accepted: November 16, 2007

Published: November 26, 2007

**Citation**

Alexej A. Sysoliatin, Andrew K. Senatorov, Andrey I. Konyukhov, Leonid A. Melnikov, and Vladimir A. Stasyuk, "Soliton fission management by dispersion oscillating fiber," Opt. Express **15**, 16302-16307 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16302

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### References

- E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, "Decay of optical solitons," JETP Lett. 42, 87-91 (1985).
- K. Tai, A. Hasegawa, N. Bekki "Fission of optical solitons induced by stimulated Raman effect," Opt. Lett. 13, 392-394 (1988). [CrossRef] [PubMed]
- P. K. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, "Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers," Opt. Lett. 11, 464-466 (1986). [CrossRef] [PubMed]
- N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Benion, "Enhanced power solitons in optical fibers with periodic dispersion management," Electron. Lett., 32, 54-55 (1996). [CrossRef]
- R. Driben and B. A. Malomed, "Split-step solitons in long fiber links", Opt. Commun. 185, 439-456 (2000). [CrossRef]
- T. Inoue, H. Tobioka, S. Namiki, "Stationary rescaled pulse in alternately concatenated fibers with O(1)-accumulated nonlinear perturbations" Phys. Rev. E 72, 025601(R) (2005). [CrossRef]
- M. Böhm, F. Mitschke, "Soliton propagation in a dispersion map with deviation from periodicity" Appl. Phys. B 81, 983-987 (2005). [CrossRef]
- K. Lee, J. Buck, "Wavelength conversion through higher-order soliton splitting initiated by localized channel perturbations", J. Opt. Soc. Am. B 20, 514-519 (2003). [CrossRef]
- S. Sears, M. Soljacic, M. Segev, D. Krylov, K. Bergman, "Cantor set fractals from solitons", Phys. Rev. Lett. 84, 1902-1905 (2000). [CrossRef] [PubMed]
- B.A. Malomed, D.F. Parker, N.F. Smyth "Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber," Phys. Rev. E 48, 1418-1425 (1993). [CrossRef]
- R.G. Bauer, L.A. Melnikov "Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter," Opt. Commun. 115, 190-195 (1995). [CrossRef]
- A. Hasegawa, Y. Kodama, "Guiding center solitons," Phys. Rev. Lett. 66, 161-164 (1991). [CrossRef] [PubMed]
- H. Sakaguchi, B.A. Malomed "Resonant nonlinearity management for nonlinear Schr¨odinger solitons," Phys. Rev. E 70, 066613 (2004). [CrossRef]
- B.A. Malomed, N.F. Smyth "Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber," Phys. Rev. E 50, 1535-1542 (1994). [CrossRef]
- G. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1989).
- V. N. Serkin, A. Hasegawa, and T. L. Belyaeva "Nonautonomous solitons in external potentials" Phys. Rev. Lett. 98, 074102 (2007). [CrossRef] [PubMed]
- V. G. Bespalov, S. A. Kozlov, Yu. A. Shpolyanskiy, and I. A. Walmsley "Simplified field wave equations for the nonlinear propagation of extremely short light pulses" Phys. Rev. A 66, 013811 (2002). [CrossRef]

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