1. Introduction

Linearly chirped optical pulses with a parabolic intensity profile, called similaritons, are asymptotically generated in amplifiers with normal group-velocity dispersion, independently of their initial shape or noise, and propagate self-similarly subject to exponential scaling of their amplitude, temporal duration and spectral width [1–3]. Similaritons are of wide-ranging practical significance since their intrinsic resistance to wave breaking allows the scaling of amplifiers to higher-power regimes [4,5]. Moreover, their linear chirp facilitates efficient temporal compression [1,4,5]. Combination of a parabolic pulse amplifier with an optical feedback has resulted in the first similariton laser [6]. Recent experimental studies have also exploited the remarkable properties of similaritons to propose new methods for optical pulse synthesis [7], 10-GHz telecom multi-wavelength sources [8] and optical regeneration of telecom signals [9].

To date, most of the theoretical [1,2,10] and experimental [1,11] studies have been devoted to the dynamic of a single similariton pulse. However, as similariton-based techniques have been recently demonstrated with high-repetition rate sources [8,9], it has become crucial to better understand the various phenomena which can be involved when similaritons overlap and interact during their propagation in the amplifier. The only numerical work, which shortly deals with self-similar amplification of a pair of pulses, was carried out by Peacock et al. [12]. They concluded that the quality of the pulses after recompression was not too severely affected by the interaction process in the case of a non-adiabatic amplification based on a rare-earth doped fiber. In this Letter, we investigate the interaction properties of two parabolic pulses which overlap during their Raman amplification. In particular, we demonstrate that the pulses evolve independently to generate self-similar parabolic profiles, only interacting in their overlap region. At the beginning of their overlap the interaction between the two linearly chirped parabolic pulses is almost linear and leads to a frequency beating of the intensity profile. Beyond the first stage of overlap, the beating, subject to the combined effects of non-linearity, normal dispersion and adiabatic Raman gain, evolves into a train of dark solitons. Numerical integrations of the nonlinear Schrödinger equation (NLSE) with gain are compared with experimental results.

2. Numerical simulations

The evolution of the complex slowly varying envelope E of an initial pulse during its propagation in a normal-dispersion Raman fiber can be modeled by the NLSE with a constant longitudinal and spectral gain g :

where β₂ and γ are the second-order dispersion and Kerr nonlinear coefficients, respectively.

The similariton, which corresponds to the self-similar asymptotic solutions of Eq. (1), is characterized by a parabolic intensity profile and a positive linear chirp C_p = g/6πβ₂ .^1,2 Let us now consider the amplification of a pair of identical pulses separated by a time-delay ΔT = 55.5 ps. Each initial pulse has a chirp-free Gaussian intensity profile with a duration of 5 ps, a carrier wavelength of 1550 nm and an initial energy of 2.9 pJ. The Raman amplifier parameters at 1550 nm are γ = 2.0 × 10^-3 W^-1 m^-1 and β₂ = 4.6 × 10^-3 ps² m^-1. The linear gain coefficient g is 0.972 × 10^-3 m^-1, which leads, for an amplifier length of 5.3 km, to a total integrated gain of 22.3 dB. Figure 1 shows the evolution of the pair of pulses as a function of the propagation distance in the amplifier, as calculated from numerical integration of Eq. (1). It is remarkable to note from Fig. 1 that there is no noticeable difference between the intensity profile of a single pulse (circles) and the intensity profile of a pulse in the presence of its neighbor (solid line). Therefore, both before and after the pulses overlap, each pulse evolves separately with a parabolic profile, and only interact in the overlap region. Owing to their compact nature, parabolic pulses are clearly unaffected by their neighbor outside the overlap region. In consequence, their mean temporal positions remain constant along the amplifier. This is in contrast to the case of two interacting solitons, where both the position and shape of the pulses are modified by the interaction.

 

Fig. 1. Numerical calculations of the intensity profiles of a single pulse (circles) and a pair of pulses (solid lines) at different distances of propagation.

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Figure 2(a) clearly shows that, at a distance of 4.17 km, an oscillation appears inside the overlap. Indeed, as the parabolic pulses are linearly chirped, the frequency difference between the overlapping falling and raising edges of the pulses, induces a beating in the resultant signal, as seen more precisely in Fig. 2(a) (solid line). The circles in Fig. 2(a) represent the linear superposition of two identical pulses with an exact parabolic shape whose amplitude, width and chirp, after 4.17 km of propagation, were deduced from numerical simulation of the propagation of a single pulse. Figure 2(a) shows that, at the center of the overlap region, the oscillation resulting from the linear addition of the two parabolic pulses is in good agreement with numerical simulations of Eq. (1) (solid line). However, it is noteworthy that the oscillations predicted by numerical simulations are temporally more extended than those obtained from linear superposition of the two parabolic pulses. These additional oscillations, which originate in the wings of the similariton pulses [2–10], implicitly appear from numerical integrations of Eq. (1), but are neglected by assuming that the pulses have an exact parabolic shape. In Fig. 2(a), the numerical results (solid line) are also compared with a sinusoidal fit (dotted red line) with a frequency of 617 GHz. It is straightforward to verify that the linear addition of two exact parabolic pulses with a chirp parameter C_p leads to a modulated oscillation with a frequency f_s = C_p ΔT = 622 GHz, in good agreement with the value obtained from the sinusoidal fit.

When the overlapping pulses further propagate through the amplifier subject to the combined effects of non-linearity, normal dispersion and adiabatic Raman gain, the oscillation reshapes into a train of dark solitons [13]. The normalized intensity profile of a dark soliton is given by [14] I(T) = A² (1-B²sech²T) /B² , where A² is the intensity depth of the hole, B² is a blackness factor related to the contrast and T is a dimensionless retarded time. Dark solitons with B² = 1 have been called black, whereas other dark solitons with -1 < B < 1 are called gray. The blackness parameter |B| can be calculated from the values of the power and phase jump at the dip center. For example, for the dark soliton designed as (1) in Fig. 2(b), the blackness parameter is |B| = 0.997 ≅ 1 and the phase jump is 0.97 π ≅ π, so that this hole is very close to a black soliton with a tanh² shape. Indeed, Fig. 2(b) confirms that after 5.3 km of propagation, the intensity and phase profiles of the dip (1) (solid line) are in very good agreement with the corresponding profiles of a black soliton (circles) with a temporal width T_d = 385 fs, a pulse depth P_d = 15.6 W and a phase jump of π. Taking into account that the fundamental black soliton depth is P_o = β₂${\mathit{/}\mathit{(}\gamma T}_d^{\mathit{2}}$) = 15.9 W ≅ P_d , we may conclude that the characteristics of dip (1) are very close to those of a fundamental black soliton.

 

Fig. 2. Numerical calculations of the intensity profiles of a a pair of pulses. (a) Central part of the pulse overlap at z = 4.17 km: simulations (solid line), linear superposition (circles) and sinusoidal fit (red dotted line). (b) Intensity and phase profiles at z = 5.3 km: numerical simulations (solid lines) and black soliton fit (circles).

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Figure 3 represents the evolution of the normalized intensity profile of the dark soliton train with further amplification distance. The time-shifts of the dips during propagation, which are clearly observed if Fig. 3, indicate that dark solitons propagate at different relative velocities β with respect to the group velocity at the initial wavelength. It has been previously shown that β has the same sign as B and is proportional to A√ 1-B ² /B [14]. On the other hand B > 0 (B < 0) when the phase increases (decreases) across the pulse [14]. So, according to the phase profile shown in Fig. 2(b), at t < 0 (B < 0) the dark soliton velocity decreases, whereas at t > 0 (B > 0) the dark soliton velocity increases, in agreement with the numerical results of Fig. 3. Since |β| is a decreasing function of |B|, the highest relative velocities |β| are expected at the edges of the overlap region. Moreover, since |β| is an increasing function of A, the relative intensities |β| increase with the amplification distance, so that the temporal positions of the dark solitons do not vary linearly through the amplifier, as can be seen in Fig. 3. Indeed, the repetition rate of the soliton train rapidly decreases from 380 to 135 GHz when the propagation distance only increases from 5.3 to 7.3 km. It is noteworthy to remark from Fig. 3 that the temporal width of the dark solitons decreases with propagation distance, e.g. from 380 fs at 5.3 km to 200 fs at 7.3 km. As a matter of fact, the solitons adiabatically adapt their temporal width in order to maintain their area constant during the amplification process [15].

 

Fig. 3. Evolution of the normalized intensity profile of the dark soliton train from 5.3 to 7.3 km.

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3. Experimental results

The generation of the dark soliton train via interaction of two parabolic pulses was studied by direct measurements of the autocorrelation and spectrum of light at the amplifier output. The experimental setup is described in detail elsewhere [11]. The pulses emitted from a picosecond laser were split in two and recombined in a Mach-Zehnder interferometer with a delay between the pulses set at ΔT = 55.5 ps. Pairs of pulses with an initial energy of 2.9 pJ were injected into a 5.3-km Raman fiber pumped in a backward configuration [16].

Due to signal to noise limitations, we were not able to take advantage of the FROG technique [17] for a complete intensity and phase characterization of the pair of parabolic pulses. On the other hand the temporal width of the dark pulses after 5.3 km (below 500 fs) is well below the temporal width of the initial pulses so that a cross-correlation measurement is very difficult to implement. In those conditions, we have simply characterized the amplified pulse pair by recording the autocorrelation trace and the spectrum of the output pulses.

Figure 4(a) shows the autocorrelation trace recorded at the center of the overlap region (circles). The experimental trace, which exhibits 410-GHz oscillations, is in good agreement with numerical integrations of a generalized NLSE that rigorously includes the Raman amplification process through an appropriate integral term, as well as other effects such as higher-order dispersion and self-steepening [11]. We can also note that the modulation contrast of the autocorrelation signal is consistent with numerical prediction and is very different from that obtained with a pure sinusoidal modulation, 0.33.

The experimental output spectrum, displayed in Fig. 4(b), exhibits several sidebands separated by 407 GHz, consistent with the repetition rate measured on the autocorrelation signal. Let us remark that the small asymmetry of this spectrum, which is also present in the experimental output spectrum of a single similariton (dashed red line), is mainly due to the asymmetry of the initial pulse spectrum (not shown here), slightly accentuated by the effects of Raman intra-pulse scattering. The experimental spectrum is in good agreement with the corresponding numerical spectrum, shown in Fig. 4(c), obtained from integration of the generalized NLSE.

 

Fig. 4. (a) Autocorrelations of the pair of pulses at z = 5.3 km: experimental results (circles) numerical calculations (solid line). (b) Output experimental spectra for a single pulse (dashed red curve) and a pair of pulses (solid line). (c) Corresponding theoretical spectrum for the pair of pulses..

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The very good agreement between theoretical and experimental autocorellation and spectrum signals confirms that after 5.3 km of propagation in the optical amplifier the interaction of two optical similariton generate a train of dark solitons.

4. Conclusion

In conclusion, we have analyzed the dynamic of the self-similar amplification of a pair of time-delayed identical pulses. During the amplification process, due to the exponential scaling of their temporal duration, each pulse broadens, so that, after a given distance of propagation, the two pulses overlap and interact. Owing to their compact nature, parabolic pulses are clearly unaffected by their neighbor outside the overlap region. Inside the overlap region, the superposition of the linearly-chirped pulses creates an oscillation which adiabatically evolves into a high-repetition rate dark soliton train.