## 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

The development of highly nonlinear microstructured fibers (MFs) has made possible the generation of ultra broad-band supercontinuum radiation in the infrared and visible by using ultrashort pulses and fiber lengths in the tens of centimeters range [1–4]. The reported results indicate that the phenomenology of supercontinuum generation is due to the interplay of many nonlinear effects such as self phase modulation, self steepening (SS), Raman self frequency shift (RSFS) and four wave mixing (FWM). The scenario is very rich and not completely understood.

Great attention has been recently devoted to the situation in which the wavelength *λ*_{p}
of the ultrashort pulses is in the anomalous dispersion region, close to the zero dispersion wavelength *λ*_{ZD}
. In such a case the spectral distribution of supercontinuum is constituted by a broad-band infrared component and a visible component. Typically, a wavelength gap of some hundreds of nanometers can be observed between the infrared and the visible components. It has been demonstrated both numerically and experimentally that the generation and the evolution of the infrared supercontinuum is mainly ruled by the fact that the input pulse behaves as a higher-order soliton that undergoes a fission process during propagation in the nonlinear fiber [5–11]. On the other hand the origin of the visible component is still subject of scientific debate. Some authors [10] have suggested that the generation of visible light can be explained by invoking a FWM interaction in which the pump is identified as a spectral component close to *λ*_{ZD}
. Herrmann *et al*. [5,6] have proposed a different explanation, based on the hypothesis that, after fission of the Nth-order soliton, each of the N fundamental solitons emits a dispersive wave (DW), called by them nonsolitonic radiation, that propagates in the fiber in a linear regime. As it is well known, a resonance condition involving high-order dispersion terms can be found at which the soliton and the dispersive wave have the same phase. [12] At that wavelength a coupling takes place and the dispersive radiation is coherently enhanced during the propagation. Due to the peculiar dispersion behavior of the holey fiber, the resonance condition is satisfied in the visible wavelength range. The experiment described in Ref. [9], in which 190-fs 800-nm pulses were launched in a highly nonlinear MF having *λ*_{ZD}
=710 nm, shows a particularly striking result: the output spectrum (see Fig. 1) presents a visible component consisting of a narrow-band peak at 430 nm that has an intensity more than one decade larger than the infrared supercontinuum and collects up to 25% of the total input power. The generated blue light is in the fundamental fiber mode, at variance with other situations in which the visible light, generated by a third harmonic process, presents the spatial pattern of a higher-order mode and is produced with a much lower efficiency [13,14].

In order to clarify the origin and time-dependence of blue radiation, we have undertaken a numerical study of the nonlinear propagation in a MF, and we have performed new measurements by using the same set-up and MF described in Ref. [9]. Our results confirm that the visible radiation can be ascribed to the resonant coupling between solitons and dispersive wave. A very important point stressed in this work is the following: since the dispersive-wave generation is a linear process, the dispersive radiation can be effectively observed only if the wavelength determined by the resonance condition belongs to the soliton spectrum, so that the generation of visible dispersive radiation can occur only in presence of a strong temporal compression of the input pulse. Our simulation can reproduce in great detail the dynamics of the input pulse that leads to soliton fission and to visible light generation. In particular, we demonstrate that the peak in the visible can be mainly ascribed to the dispersive radiation emitted by the first and most intense fundamental soliton generated by soliton fission. We find indeed a strong temporal compression of the pulse such that the short-wavelength tail of the spectrum deeply extends even into the normal dispersion region. We calculate the resonance condition both numerically and analytically by considering the real dispersion curve of the fiber and we find an excellent agreement with the experiments. In addition, the simulation describes correctly the observation that the visible peak is generated in the first few centimeters of fiber and that its spectral position is strongly dependent on the shape of the dispersion curve.

Together with the simulation, we present also new experimental data concerning the temporal auto-correlation of the blue component of the output spectrum. We find pulse durations of a few picoseconds, in good agreement with simulation. We also find that, by increasing the input intensity, the main blue pulse is accompanied by satellite pulses having similar wavelength and duration, but lower intensity. This latter finding is consistent with the simulation showing that the input pulse can undergo a sequence of compressions.

## 2. Numerical model

Before describing the nonlinear pulse-propagation, it is useful to consider the condition that determines the resonance frequency *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]:

where *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}
:

where *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], by taking *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]

Using the series expansion around the point *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}
[6,11], nevertheless the inclusion of the higher order terms in Eq. (3) allows a reliable description of the fiber dipersion behaviour over a wide wavelength range and leads to a more accurate result (the correction can be of the order of tens of nanometers). Figure 2 reports the value of the phase matching wavelength as a function of the pump wavelength with and without the contribution of the nonlinear dephasing (we considered a peak power *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.

We show in Fig. 3 the optical spectrum measured whith different initial wavelengths at moderate input power. The figure shows a rather good agreement between the positions of the visible peaks and phase matching wavelength calculated by Eq. (3).

In order to monitor the birth of the dispersive wave during the nonlinear propagation process, we have numerically solved the generalized nonlinear Schrödinger equation (GNLS):

Equation (4) is derived under the slowly varying field envelope approximation (SVEA) [15]. In Eq. (4)
*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:

$$T=t-\frac{z}{{u}_{g}\left({\omega}_{o}\right)}$$

where *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}
=${A}_{o}^{\mathit{2}}$
.

The second term at the left-hand side of Eq. (4) represents the linear propagation loss whereas the third term takes into account the dispersive behaviour of the fiber. The right-hand side of Eq. (4) describes the nonlinear effects involved in the propagation of the optical pulse. The temporal derivative represents the self-steepening effect. The convolution integral describes the Raman response of the fiber. The function *R*(*T*) describes both the instantaneous and the delayed material response [16]:

The constant *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]. Numerical solutions of Eq. (4) were obtained by using the symmetrized Split-Step Fourier method [15] through which it is possible to solve the dispersive part of the equation completely in the frequency domain while the nonlinear part is solved partially in the time domain and partially in the frequency domain. In particular, considering only the nonlinear terms, we have

where we indicated with “∗” the convolution operator. Considering the SS effect as a perturbation, i.e., neglecting the time derivative [16], Eq. (7) can be treated as a homogeneous differential equation whose solution is given by

Introducing a new function *V*(*z,T*), defined on the interval *z*_{0}
<*z*<*z*_{0}
+*d*_{z}
, given by:

it is possible to rewrite Eq. (7) as

where *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

where “ℑ” indicates the Fourier transform operator. Finally from Eq. (11) it is possible to solve the nonlinear step using fourth-order Runge-Kutta integration method, obtaining *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] contain some misprints that have propagated in the subsequent literature.

In our simulations we used 2^{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] that appear in the spectrum as pronounced peaks at fixed frequencies. By appropriately reducing *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.

As the spectral window adopted in our simulation was limited by the memory of the computer, we could only simulate the propagation of soliton pulses with *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

By using the numerical model described in the previous section we investigated the nonlinear propagation of optical pulses with wavelength inside the anomalous dispersion region. The dispersion curve we utilized was that of the microstructured silica fiber used in Ref. [9]. We assumed a nonlinear coefficient *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], we assume that the monomodality condition is guaranteed for every analysed spectral component, so that we can confidently use a numerical model that intrinsically does not take into account the transverse modal distribution of each propagating field.

By solving the complete GNLS (1) it results that the higher order solitons are unstable degenerate solutions [6]. Higher order solitons, which obey the relation

where *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.

Our results clearly show the important role of Raman scattering in the generation of a broad infrared supercontinuum, in agreement with the trend discussed in Ref. [8], so that we limit ourselves to a short comment. The temporal and spectral dynamics of the soliton decay is strictly connected to the phenomenon of the soliton self-frequency shift (SSFS) [16]. The partial overlap of the pulse spectrum with that of the odd Raman response function causes the enhancing of the red-shifted spectral component and a frequency dependent absorption for the blue-shifted components. By studying such phenomenon as a function of the propagation coordinate *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.

Since bound-state solitons are not stable in presence of perturbations, the original pulse splits into an intense fundamental soliton and a *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] so that, using Eq. (12) with *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.

Considering now the generation of visible radiation, it is important to remark that the fundamental condition that allows a resonant power transfer from the soliton to the dispersive wave is given by the superposition of the soliton spectrum with the spectral region where are located the solutions of Eq. (3). In addition, the amplitude of the dispersive peak should coincide with the amplitude reached by the soliton spectrum in the dispersive wave region. These conditions give to the first contraction that characterizes the soliton periodical dynamics a primary importance in the coupling process: as said above, the strong temporal contraction corresponds to the maximum expansion of the soliton spectrum and makes the coupling process between soliton and dispersive wave more efficient. Since the temporal contraction represents the first step of the higher-order soliton dynamics, such a consideration explains also the experimental observation that blue light radiation is generated mainly in the first few propagation centimetres inside the holey fiber. The picture shown in Fig. 5 indicates indeed that the light scattered out of the fiber cladding becomes blue after a few centimeters. By changing the pump wavelength we found a shift of the visible line spectral position consistent with the results reported in Figs. 2 and 3. We found both experimentally and numerically that the amplitude of the visible peak decreases by increasing the pump wavelength. This fact can be easily explained by considering that for longer wavelengths the spectral overlap between the soliton spectrum and the phase matching wavelength becomes less efficient, leading to a lower spectral density in correspondence to *λ*_{D}
.

The simulation presented in Fig. 6 shows the evolution on half soliton period of the spectrum of a fourth order soliton: it can be seen that the dispersive-wave generation process occurs not only in correspondence of the first contraction but every time the soliton spectrum overlaps the resonant dispersive wave region. To better understand the dynamics of the phenomenon is interesting to consider the simulation presented in Fig. 7, in which the temporal and spectral evolution of the input pulse are reported and compared with the temporal evolution of the dispersive wave obtained by numerically filtering the complete spectrum from 150 to 200 THz. The Fig. 7 shows that after the first contraction the input pulse is split in a high-intensity fundamental soliton and in a (*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.

By considering Fig. 7(c) it is clear that at the end of the fiber the dispersive radiation in the time domain is constituted by a sequence of pulses propagating with a slightly different group velocity. The first and most intense pulse corresponds to the radiation emitted in the first contraction, whereas the weaker pulses are generated during the subsequent soliton contractions. In Figs. 8(a), 8(b), 8(c), the dispersive-wave output intensity, obtained by simulating the propagation of a fifth order soliton, is reported as a function of time. As it can be seen, by increasing the propagation length we find new satellite peaks that are generated in the subsequent contractions. At the same time the first and most intense pulse experiences a progressive broadening due to the dispersive propagation.

We have performed measurements of the second-harmonic intensity-autocorrelation of the blue light obtained at the fiber output. In order to get a sufficient signal to noise-ratio, the measurement was carried out in a high power regime. The autocorrelator used a BBO crystal in a collinear scheme. First we considered a 33 cm long span of fiber and we coupled an average power *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

As a conclusion, we have presented numerical and experimental results describing the nonlinear propagation of femtosecond pulses in a microstructured fiber, by using input wavelengths in anomalous dispersion region where soliton fission mechanisms play an important role. The output spectrum is constituted by a broad-band infrared component and a narrow-band visible component, localized at 430 nm, which carries about one fourth of the input energy. The main aim of this work was to explore the generation mechanism and the properties of the visible peak. By numerically solving the generalized nonlinear Schrödinger equation, we were able to follow in detail the evolution of the pulse spectrum as it propagates along the fiber. We have found that the peak in the visible is generated only when the input pulse undergoes a sufficiently strong compression such that the short-wavelength tail of the spectrum overlaps the wavelength determined by the resonance condition for the dispersive wave. In agreement with simulation, our intensity-autocorrelation measurements show that the duration of the blue peak is in the few ps time range, and that, by increasing the input intensity, the input pulse can undergo a sequence of compressions giving rise to satellite blue pulses of lower intensity and similar duration.

What remains to be explained is the very efficient conversion from the infrared to the visible radiation (about 25 %) that was observed in the experiment of Ref. [9]. We don’t know at present whether an extension of the simulation at higher input power could explain the results or a modification of the model could be necessary. It should be noted that in the points of strong temporal compression the soliton pulsewidth gets very short so that the validity of the SVEA approximation could be questioned

## 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. **25**, 25–27, (2000). [CrossRef]

**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 **19**, 753–763, (2002). [CrossRef]

**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 **10**, 382–387, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-382. [CrossRef] [PubMed]

**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]

**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]

**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]

**7. **Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric waveguide,” IEEE J. Quantum Electron. **QE-23**, 510–524, (1987). [CrossRef]

**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]

**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]

**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. **15**, 233–235, (2003) [CrossRef]

**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]

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

**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 **11**, 2567–2576 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2567 [CrossRef] [PubMed]

**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. **215**, 191–197 (2003). [CrossRef]

**15. **G.P. Agrawal, *Nonlinear Fiber Optics* (Academic Press, San Diego, 1995).

**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]

**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]

**18. **J.P. Gordon, “Theory of the soliton self frequency shift,” Opt. Lett. **11**, 662–664, (1986). [CrossRef] [PubMed]

**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. **19**, 165–167, (1994). [CrossRef] [PubMed]