## Dispersive wave generation by solitons in microstructured optical fibers

Optics Express, Vol. 12, Issue 1, pp. 124-135 (2004)

http://dx.doi.org/10.1364/OPEX.12.000124

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### Abstract

We study the nonlinear propagation of femtosecond pulses in the anomalous dispersion region of microstructured fibers, where soliton fission mechanisms play an important role. The experiment shows that the output spectrum contains, besides the infrared supercontinuum, a narrow-band 430-nm peak, carrying about one fourth of the input energy. By combining simulation and experiments, we explore the generation mechanism of the visible peak and describe its properties. The simulation demonstrates that the blue peak is generated only when the input pulse is so strongly compressed that the short-wavelength tail of the spectrum includes the wavelength predicted for the dispersive wave. In agreement with simulation, intensity-autocorrelation measurements show that the duration of the blue pulse is in the picosecond time range, and that, by increasing the input intensity, satellite pulses of lower intensity are generated.

© 2004 Optical Society of America

## 1. Introduction

1. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. **25**, 25–27, (2000). [CrossRef]

4. J.M. Dudley, L. Provino, N. Grossard, H. Maillotte, R.S. Windeler, B.J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B **19**, 765–771, (2002). [CrossRef]

9. L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B **77**, 307–311, (2003). [CrossRef]

## 2. Numerical model

*ω*

_{d}of the dispersive radiation. The phase of the soliton at frequency

*ω*

_{p}in its moving frame should coincide with that of the dispersive radiation at frequency

*ω*

_{d}in the same frame. If we call

*β*(

*ω*) the frequency-dependent wave vector of the optical signal, such a phase matching condition can be written as [5

5. A.V. Husakou and J. Herrmann, “Supercontinuum generation of higher - order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. **87**, 203901, (2001). [CrossRef] [PubMed]

*u*

_{g}is the group velocity at frequency

*ω*

_{p}

*, γ*is the fiber nonlinear coefficient and

*P*

_{o}is the soliton peak power. The last term in Eq. (1) accounts for the soliton nonlinear dephasing. The dispersive radiation having a low spectral density is supposed to propagate in the linear regime. As usual, we express

*β*(

*ω*) by a Taylor expansion in powers of

*Ω*=

*ω*-

*ω*

_{p}:

*β*

_{k}is the

*k*-th order derivative of

*β*(

*ω*), calculated at

*ω*

_{o}. However, in dealing with supercontinuum generation, it is important that the reconstructed dispersion curve closely follows the original curve for values of

*ω*appreciably different from

*ω*

_{o}. For that reason, we considered the Taylor expansion up to the 13

^{th}order. In the case of the MF used in the experiment of Ref. [9

9. L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B **77**, 307–311, (2003). [CrossRef]

*λ*

_{o}=810 nm as the central wavelength of the input pulse, the calculated

*β*

_{n}values of the expansion are:

*β*

_{2}=-5.7887 ps

^{2}/Km,

*β*

_{3}=0.015254 ps

^{3}/Km,

*β*

_{4}=3.9595×10

^{-6}ps

^{4}/Km,

*β*

_{5}=1.2258×10

^{-8}ps

^{5}/Km,

*β*

_{6}=1.9712×10

^{-10}ps

^{6}/Km,

*β*

_{7}=-3.9784×10

^{-12}ps

^{7}/Km,

*β*

_{8}=2.5761×10

^{-14}ps

^{8}/Km,

*β*

_{9}=-9.6603×10

^{-17}ps

^{9}/Km,

*β*

_{10}=2.3525e×10

^{-19}ps

^{10}/Km,

*β*

_{11}=-3.7319×10

^{-22}ps

^{11}/Km,

*β*

_{12}=3.5563×10

^{-25}ps

^{12}/Km and

*β*

_{13}=-1.5619×10

^{-28}ps

^{13}/Km. At the power levels used in our simulation the contribution of the nonlinear term in Eq. (1) is negligible. In such a case the frequencies at which the phase condition is satisfied are the real solutions of the following polynomial equation [12

12. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A **51**, 2602–2607, (1995). [CrossRef] [PubMed]

*λ*

_{o}=810 nm and solving for

*Ω*, we find a real solution at

*λ*

_{D}=548 nm. It is well known that the main contribution to the resonance condition is due to the third order dispersion term

*β*

_{3}[6

6. J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. **88**, 173901, (2002). [CrossRef] [PubMed]

11. E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B **77**, 197–204, (2003). [CrossRef]

*P*

_{o}=5 kW and

*γ*=20 W

^{-1}Km

^{-1}). It can be easily observed that, by increasing the pump wavelength, the phase matching condition is satisfied at shorter wavelengths. Also the nonlinear phase shift leads to a slight decrease of

*λ*

_{D}that becomes more appreciable by increasing the peak power. This phenomenon was observed in our experiments at high power levels; for example in Fig. 1 the phase matching wavelength is shifted at

*λ*

_{D}=430 nm due to the nonlinear phase generated by the higher order soliton propagating inside the fiber.

*A*(

*z,T*) represents the complex envelope of the propagating field in a frame of reference traveling at the group velocity of the input pulse. We consider an input pulse having a central angular frequency

*ω*

_{0}and hyperbolic-secant shape:

*T*

_{o}is connected to the full-width half-maximum duration by the relation

*T*

_{FWHM}=1.763·

*T*

_{o}, and

*u*

_{g}is the group velocity at frequency

*ω*

_{o}. The pulse peak power

*P*

_{o}satisfies the condition

*P*

_{o}=

*R*(

*T*) describes both the instantaneous and the delayed material response [16

16. K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics **25**, 2665–2673 (1989). [CrossRef]

*f*

_{r}is chosen to be equal to 0.18 and represents the fractional contribution of the instantaneous Raman response to the nonlinear refractive index, while

*h*

_{r}(

*T*) accounts for the delayed Raman response and can be expressed with good accuracy by an analytical function reported in Ref. [16

16. K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics **25**, 2665–2673 (1989). [CrossRef]

16. K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics **25**, 2665–2673 (1989). [CrossRef]

*V*(

*z,T*), defined on the interval

*z*

_{0}<

*z*<

*z*

_{0}+

*d*

_{z}, given by:

*V*

_{o}=

*V*(

*z*=

*z*

_{o},

*T*). Since convolution integrals in time domain can be expressed with products in Fourier domain, Eq. (7) can be written as

*A*(

*z*+

*dz,T*) through inversion of Eq. (5). We have written explicitly Eqs. (6)–(10) in this work because the equations reported in Ref. [16

**25**, 2665–2673 (1989). [CrossRef]

^{13}points to discretize a temporal window 64 times larger than the input pulse duration

*T*

_{o}, in this way the spectral window covered a region as larger as 1000 THz. The other key parameter that has to be set is the fiber discretization step

*Δz*: the presence of higher-order dispersion terms tends to produce numerical instabilities [17

17. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. **18**, 1499–1501, (1993). [CrossRef] [PubMed]

*Δz*it is possible to shift the instability peaks outside the spectral window, so avoiding unwanted interferences. Typical

*Δz*values used in our simulations are around 50

*µ*m.

*N*≤6. This prevented us from reproducing those experimental results showing the largest conversion efficiency toward blue light generation.

## 3. Dispersive wave and Raman driven supercontinuum generation dynamics

9. L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B **77**, 307–311, (2003). [CrossRef]

*γ*=20 (W·Km)

^{-1}. Consistently with the experimental observation that both the infrared and the visible components of the output spectrum are in the fundamental fiber mode [9

**77**, 307–311, (2003). [CrossRef]

6. J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. **88**, 173901, (2002). [CrossRef] [PubMed]

*N*indicates the soliton order and

*P*

_{o}the pulse peak power, tend to decay from the bound-state condition to a situation in which

*N*fundamental solitons propagate separately.

8. G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express **10**, 1083–1098, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083. [CrossRef] [PubMed]

**25**, 2665–2673 (1989). [CrossRef]

*z*, it is possible to observe a continuous intrapulse power flow that depletes the higher frequencies of the spectrum in favour of the lower ones. The rate at which this transfer takes place is proportional to the spectral overlap of the pulse spectrum with the imaginary part of the Raman gain function so, fixing

*N*=1 in Eq. (12), it becomes evident that the SFS rate is larger for high-peak-power solitons that have to be correspondingly very short in time.

*N*-1 bound state soliton. The splitting process proceeds during propagation until

*N*fundamental solitons are generated. The peak power of the fundamental solitons is a decreasing function of the generation order as discussed in [17

17. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. **18**, 1499–1501, (1993). [CrossRef] [PubMed]

*N*=1, the earlier generated fundamental solitons undergo a stronger SFS. Such a behaviour constitutes the principal mechanism that leads to the broadening of the original pulse spectrum to supercontinuum, whereas, considering central wavelengths of the input pulse appreciably distant from the first ZDW, SPM widening contributions can be neglected. The results of the simulation shown in Fig. 4 demonstrate a significant agreement with the experimental measurements.

*λ*

_{D}.

*N*-1)th order soliton. The subsequent spectral expansions seen in Fig. 7(b) correspond to temporal compressions of the first soliton, whose trajectory is shown in Fig. 7(a). The first soliton is the one with the highest peak power and, as a consequence, also the shortest one. Since the steady state for every generated soliton is not reached instantaneously, there are periodical oscillations of the spectral width during propagation. The soliton trajectory gives information about its SFS dynamic: since the trajectory is not straight line but it is curved, this indicates that the soliton group velocity is continuously decreasing, following a behavior dictated by the dispersion characteristics of the fiber. In Fig 7(c) it is shown the temporal behaviour of the dispersive wave during propagation. As discussed above, dispersive radiation is generated only in the fiber sections in which spectral expansion occurs. In addition, it is possible to observe that the successively generated dispersive waves propagate with different group velocities (this fact is suggested by the different slope of the arrows in the Fig. 7(c) which means that at every soliton contraction the emitted dispersive wave has a slightly different frequency. The temporal behaviour of the dispersive wave presents no trapping phenomenon but simply a continuous broadening due by the fact that those waves propagate in the normal dispersion region.

*P*=90 mW. The autocorrelation trace shown in Fig. 9(a) exhibits five peaks in agreement with the results of the simulation (see Fig. 8(a)). The duration of the central peak is

*T*

_{FWHM}=3.8 ps considering a gaussian shape. We measured the intensity-autocorrelation after decreasing the input power down to 50 mW. In this situation we found only two pulses (see Fig. 9(b)): this is consistent with the consideration that at lower power the higher-order soliton experiences only two compressions. We performed a third measurement by keeping the same average power and using a 18 cm long fiber span. In this situation we found only one pulse with a shorter duration (

*T*

_{FWHM}=1.3 ps), as expected, because the reduced propagation length permits just one compression of the soliton and makes less important the effect of linear dispersion on the pulse duration. In all cases the results are in good qualitative agreement with the experiment: as the average power considered in the simulation is considerably smaller than that used in the experiment, the simulation predicts, in comparison with experiment, a slower evolution of the process along the fiber and a weaker amplitude of the dispersive wave.

## 5. Conclusions

**77**, 307–311, (2003). [CrossRef]

## References and links

1. | J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. |

2. | S. Coen, A.H. L. Chau, R. Leonhardt, J.D. Harvey, J.C. Knight, W.J. Wadsworth, and P.St.J. Russell, “Supercontinuum generation by stimulated Raman scattering and parametric four-wave-mixing in photonic crystal fibers,” J. Opt. Soc. Am. B |

3. | J.H.V. Price, W. Belardi, T.M. Monro, A. Malinowski, A. Piper, and D. J. Richardson., “Soliton transmissione and supercontinuum generation in holey fiber, using a diode pumped Ytterbium fiber source,” Opt. Express |

4. | J.M. Dudley, L. Provino, N. Grossard, H. Maillotte, R.S. Windeler, B.J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B |

5. | A.V. Husakou and J. Herrmann, “Supercontinuum generation of higher - order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. |

6. | J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. |

7. | Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric waveguide,” IEEE J. Quantum Electron. |

8. | G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express |

9. | L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B |

10. | X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. |

11. | E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B |

12. | N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A |

13. | A. Efimov, A.J. Taylor, F.G. Omenetto, J.C. Knight, W.J. Wadsworth, and P.S. Russell, “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express |

14. | L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. |

15. | G.P. Agrawal, |

16. | K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics |

17. | F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. |

18. | J.P. Gordon, “Theory of the soliton self frequency shift,” Opt. Lett. |

19. | Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. |

**OCIS Codes**

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 5, 2003

Revised Manuscript: December 22, 2003

Published: January 12, 2004

**Citation**

Ilaria Cristiani, Riccardo Tediosi, Luca Tartara, and Vittorio Degiorgio, "Dispersive wave generation by solitons in microstructured optical fibers," Opt. Express **12**, 124-135 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-124

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### References

- J. K. Ranka, R. S. Windeler, A. J. Stentz, �??Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,�?? Opt. Lett. 25, 25-27, (2000). [CrossRef]
- S. Coen, A.H. L. Chau, R. Leonhardt, J.D. Harvey, J.C. Knight, W.J. Wadsworth, P.St.J.Russell, �??Supercontinuum generation by stimulated Raman scattering and parametric four-wave-mixing in photonic crystal fibers,�?? J. Opt. Soc. Am. B 19, 753-763, (2002). [CrossRef]
- J.H.V. Price, W. Belardi, T.M. Monro, A. Malinowski, A. Piper, and D. J. Richardson., �??Soliton transmission and supercontinuum generation in holey fiber, using a diode pumped Ytterbium fiber source,�?? Opt. Express 10, 382-387, (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-382.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-382</a>. [CrossRef] [PubMed]
- J.M. Dudley, L. Provino, N. Grossard, H. Maillotte, R.S. Windeler, B.J. Eggleton, S. Coen, �??Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,�?? J. Opt. Soc. Am. B 19, 765-771, (2002). [CrossRef]
- A.V. Husakou and J. Herrmann, �??Supercontinuum generation of higher �?? order solitons by fission in photonic crystal fibers,�?? Phys. Rev. Lett. 87, 203901, (2001). [CrossRef] [PubMed]
- J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell and J. Korn, �??Experimental evidence for supercontinuum generation by fission of higher �?? order solitons in photonic crystal fibers,�?? Phys. Rev. Lett. 88, 173901, (2002). [CrossRef] [PubMed]
- Y. Kodama, A. Hasegawa, �??Nonlinear pulse propagation in a monomode dielectric waveguide,�?? IEEE J. Quantum Electron. QE-23, 510-524, (1987). [CrossRef]
- G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, M. Kaivola, �??Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,�?? Opt.Express 10, 1083-1098, (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083</a>. [CrossRef] [PubMed]
- L. Tartara, I. Cristiani, V. Degiorgio, �??Blue light and infrared continuum generation by soliton fission in a microstructured fiber,�?? Appl. Phys. B 77, 307-311, (2003). [CrossRef]
- X. Fang, N. Karasawa, R. Morita, R.S. Windeler, M. Yamashita, �??Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber �?? Experimental and theoretical studies beyond the slowly varying �?? envelope approximation,�?? IEEE Photon. Technol. Lett. 15, 233-235, (2003). [CrossRef]
- E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, I. T. Sorokina, �??Intra-and extra-cavityspectral broadening and continuum generation at 1.5 m using compact low-energy femtosecond Cr:YAG laser,�?? Appl. Phys. B 77, 197-204, (2003). [CrossRef]
- N. Akhmediev, M. Karlsson, �??Cherenkov radiation emitted by solitons in optical fibers,�?? Phys. Rev. A 51, 2602-2607, (1995). [CrossRef] [PubMed]
- A.Efimov, A.J. Taylor, F.G.Omenetto, J.C. Knight, W.J. Wadsworth, P.S. Russell, �??Phase-matched third harmonic generation in microstructured fibers,�?? Opt. Express 11, 2567-2576 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2567">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2567</a>. [CrossRef] [PubMed]
- L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D.Faccio, M. Romagnoli, W Belardi, �??Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,�?? Opt. Commun. 215, 191-197 (2003). [CrossRef]
- G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
- K. Blow, D. Wood, �??Theoretical description of transient stimulated Raman scattering in fibers,�?? IEEE J. of Quantum Electronics 25, 2665-2673 (1989). [CrossRef]
- F. Matera, A. Mecozzi, M. Romagnoli and M. Settembre, �??Sideband instability induced by periodic power variation in long-distance fiber links,�?? Opt. Lett. 18, 1499-1501, (1993). [CrossRef] [PubMed]
- J.P. Gordon, �??Theory of the soliton self frequency shift,�?? Opt. Lett. 11, 662-664, (1986). [CrossRef] [PubMed]
- Y. Kodama, M. Romagnoli, S. Wabnitz and M. Midrio, �??Role of third-order dispersion on soliton instabilities and interactions in optical fibers,�?? Opt. Lett. 19, 165-167, (1994). [CrossRef] [PubMed]

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