We consider the self-similar amplification of two optical pulses of different wavelengths in order to investigate the effects of a collision between two similaritons. We theoretically demonstrate that similaritons are stable against collisions in a Raman amplifier: similaritons evolve separately in the amplifier without modification of the scaling of their temporal width and chirp and by conserving their velocities, only interact during their overlap and regain their parabolic form after collision. We show both theoretically and experimentally that the collision of two similaritons induces a sinusoidal modulation inside the overlap region, whose frequency decreases during the interaction. Theoretical and experimental studies of the pulse spectrum evidence that similaritons interact with each other through cross phase modulation.
©2005 Optical Society of America
Linearly chirped parabolic pulses generated in amplifiers with normal group-velocity dispersion, also called similaritons, have given rise to considerable interest during the last years [1–17]. Similaritons propagate self-similarly subject to exponential scaling of their amplitude, temporal duration and spectral width [2–4,11] and resist to the negative influence of optical wave-breaking . Experimental studies on similariton generation have been performed in fiber amplifiers where the gain is given either by the addition of dopants in the fiber core, such as Erbium [1,2] and Ytterbium [19,20], or by Raman scattering [7,8,12,16]. Similaritons are of wide-ranging practical significance for the design of high-power amplifiers [20,21], efficient temporal compressors [2,20,21] and similariton lasers . Recent experimental studies have also used similaritons to develop new methods for optical pulse synthesis , 10-GHz telecom multi-wavelength sources  and optical regeneration of telecom signals . To date, most of the studies have been devoted to the dynamic of a single similariton pulse, besides a recent work which has considered the interaction of two similaritons of the same wavelength during their propagation in a Raman amplifier .
In the present study we consider the collision between two similaritons of different wavelengths that intersect, due to their different group velocities, as they propagate through a Raman amplifier. Numerical integration of the nonlinear SchrÖdinger equation (NLSE) with a constant gain will allow us to determine the various phenomena arising during the collision. In particular we will demonstrate that similaritons preserve their characteristics and velocities and are stable after a collision in an amplifier. We will then highlight both theoretically and experimentally the appearance of a sinusoidal oscillation, inside the similariton overlap, whose frequency varies during the collision. Theoretical and experimental analysis of the similariton spectra will reveal the role played by cross phase modulation during the collision.
2. Numerical simulations
Let us assume that the pulse with a carrier frequency Ωo+Ω/2 is centred at T = -ΔTo/2, whereas the second pulse with a carrier frequency Ωo -Ω/2 is centred at T =ΔTo/2. During propagation through the Raman amplifier, each pulse transforms itself into a similariton, exhibiting an exponential increase of its temporal width TP [2–4]. On the other hand, the two pulses propagate at different group velocities as a result of their different carrier frequencies. In particular, the similariton with the smallest frequency propagates faster than the high-frequency similariton, leading to a time delay TG between the two similaritons that increases linearly with the propagation distance: TG (z) = β2 Ω z, where β2 is the second-order dispersion coefficient. The temporal separation ΔT between the two pulses can be then expressed by: ΔT(z) = ΔT 0 - TG(z). The time separation between the opposite edges of the two similaritons is given by:
If ΔTS is negative, the similaritons overlap within a temporal region of width |ΔTS|/2. Figure 1 shows the evolution of the parameter ΔTS for three values of the frequency separation Ω, for pulses with an initial energy of 2.9 pJ. The parameters of the Raman amplifier considered in this paper are a second-order dispersion β2 = 4.6 × 10-3 ps2 m-1 at 1550 nm and a Kerr nonlinear coefficient γ= 2.0 × 10-3 W-1 m-1. The linear gain coefficient g is 0.869 × 10-3 m-1, which leads, for an amplifier length of 5.3 km, to a total integrated gain of 20 dB. Initial time separation ΔTo was selected so that collision occurs, in each case, at the same distance zC fixed to 3000 m. As can be seen in Fig. 1, in the case of high frequency separation (Ω = 1.25 THz, solid line), the group velocity mismatch is sufficiently large to induce a complete separation of the similaritons after the collision. On the other hand, for small frequency separation (Ω = 0.75 THz, dashed red line), the temporal broadening TP (circles) increases more rapidly than the time delay TG so that the two similaritons will not separate each other after the collision. It is also possible to observe intermediate situations as for Ω = 1 THz (dotted blue line), where similaritons separate after the collision but temporal broadening becoming prevalent, overlapping of the pulses takes place again. Let us note that, whatever the frequency separation Ω is; beyond a certain distance of propagation (which depends on the specific value of Ω), the exponential increase of TP will prevail over the linear evolution of TG and the pulses will unavoidably overlap.
The evolution of the complex slowly varying envelope ψ of a pulse during its propagation in a normal-dispersion Raman amplifier can be modeled by the NLSE with a constant longitudinal and spectral gain g:
Let us recall that the similariton, which corresponds to the self-similar asymptotic solution of Eq. (2), is characterized by a parabolic intensity profile and a positive linear chirp CP = g/3/β2 [2–4]. To study the propagation of a pair of pulses in the Raman amplifier, Eq. (2) was used with the following initial condition: ψ(z=0,t) = ψ + (z=0,t) + ψ - (z=0,t), where ψ + (z=0,t) and Ω - (z=0,t) are the electric fields of the initial pulses with frequencies Ω0 + Ω/2 and Ω 0 - Ω/2, respectively. Figure 2 shows the evolution of the normalized intensity (a,c) and spectral (b,d) profiles (contour plot) for two values of the frequency separation Ω, as calculated from numerical integration of Eq. (2). The initial pulses have a Gaussian shape with a full width at half-maximum (FWHM) of 6 ps. Figure 2(a) obtained for Ω = 1.25 THz shows that the collision is followed by a temporal separation of the two similaritons, whereas Fig. 2(c), corresponding to Ω = 0.75 THz clearly reveals that temporal separation does not occur after the collision.
We can notice that in both cases, a modulation appears during collision (see also Figs. 3(c,d,e)). Indeed, as similaritons are linearly chirped, the frequency difference between the different parts of the pulses induces a beating in the resultant signal. In the case represented by Fig. 2(c), we have checked that the modulation evolves into a train of black solitons, similarly to that reported in the case of the interaction of two similaritons with the same wavelength . As can be seen in Fig. 2(b) for Ω= 1.25 THz, the spectra of both similaritons undergo a significant modification during the collision, whereas they find again a self-similar evolution marked by a spectral broadening after the collision. Nevertheless, in the case of Ω= 0.75 THz (Fig. 2(d)), we notice that the interaction is more complex. Indeed, because of their broadening, the spectra of both similaritons will superimpose themselves, leading to a spectral beating in the overlap region. In the following we will focus only on the case of two initial pulses whose spectral separation Ω is sufficiently large so that the spectra of the similaritons will not overlap after the collision.
Figure 3 shows the intensity profiles of the pair of pulses at different distances of propagation in the amplifier and obtained from numerical integration of Eq. (2). Let us remark first that the initial pulses (Fig. 3(a)) acquire a parabolic intensity profile from the onset of the collision (Fig. 3(b) for z = 2110 m). At the beginning of the collision, a sinusoidal modulation appears in the overlap region (Fig. 3(c) for z = 2360 m). Then the two pulses completely overlap at the collision distance zc = 3000 m (Fig. 3(d)). At the end of the collision (Fig. 3(e) for z = 4290 m) the overlap region is still characterized by a sinusoidal modulation but with a frequency smaller than that observed just at the beginning of the collision (Fig. 3(c)). Finally, at z = 5184 m (Fig. 3(f)) the pulses are temporally separated, each pulse exhibiting again a parabolic intensity profile. It is remarkable to note from Figs. 3(b,d,f) that there is no noticeable difference between the intensity profile of a single similariton (circles) and that of a similariton in the presence of its neighbor (solid line). This feature still holds valid when considering the similariton chirp, as shown by Figs. 3(g,h). Therefore, we show that, owing to their compact nature, both similaritons are clearly unaffected by their neighbor outside the collision. In consequence, their group velocities remain unchanged along the amplifier.
In order to more easily understand the role played by the various physical phenomena occurring during the collision, we were interested in the field at the output of the amplifier of length L = 5300 m and we varied the position zc of the collision within the amplifier by adjusting the initial time separation ΔTo. We define the quantity ΔTL the time separation between the two similaritons at the output of the amplifier:
ΔTL vanishes for a collision just arising at the fiber output (the two pulses are then completely superimposed in zc = L) . Negative and positive values of ΔTL will allow us to describe the pulse characteristics preceding and following the collision, respectively. The time separation at the amplifier output can be adjusted to a specific value (positive or negative) just by controlling the initial time separation ΔTo. Such an approach also has the advantage of simplifying the comparison between numerical predictions and experimental results (it is easier to modify the time separation between the two input pulses than to change the amplifier length).
To study in more detail the consequences of the collision, let us first neglect the nonlinear effects which occur in the interaction process between the two similaritons. According to this assumption, each pulse will acquire a parabolic intensity profile at the amplifier output, independently of its neighbor. On the other hand, in the overlap region, the resulting intensity profile will be given by the linear superposition of the two parabolic profiles, shifted in frequency by Ω and in time by ΔT:
with AP the peak amplitude and fs the frequency of the sinusoidal modulation which appears in the overlap region:
Equation (5) shows that the modulation frequency depends on ΔT(z), thus varying with the propagation distance z, as already observed in Figs. 3(c,d,e). The dependence of fs with z is induced by the group velocity mismatch between the input pulses. In the particular case of two time-delayed input pulses with identical wavelength (Ω= 0 and ΔT = ΔTo), the linear superposition of the similaritons induces a sinusoidal modulation whose frequency fs remains constant throughout the overlap region . Figure 4 shows the variation of the modulation frequency as a function of the time separation ΔTL at the amplifier output. The results obtained under the assumption of a linear superposition of the two similaritons during the collision (Eq. (5)) (dotted red line) are compared with those explicitly taking into account the nonlinear effects via a direct numerical integration of Eq. (2) (solid line). Even if the quantitative values differ slightly, the slope of the quasi-linear decrease of the modulation frequency during the collision is relatively well determined by the linear superposition of the two similaritons.
Let us now consider the effect of the collision on the spectral profiles of the similaritons. The evolution of the power spectrum at the amplifier output is represented in Fig. 5(a) as a function of the time separation ΔTL. The results of Fig. 5(a) were obtained from numerical integration of the NLSE with a constant gain (Eq. (2)). We clearly see in Fig. 5(a) that, during the collision, the spectral profile of each similariton is modified and a twist shape appears. These modifications induced by collision are symmetrical about the average carrier frequency Ωo (chosen to be zero here). For better highlighting these modifications, we introduce the quantity ΔS, representing the difference between the power spectrum obtained in the presence of nonlinear interactions and that resulting from the linear superposition of the spectra of both similaritons having evolved independently of their neighbor. Figure 5(b) shows the evolution of ΔS at the amplifier output as a function of ΔTL. We note in Fig. 5(b) the presence of narrow spectral bands of amplified frequencies (red, yellow, white) whose mean values evolve during the collision. Broader spectral bands (blue) correspond on the contrary to negative values of ΔS. The appearance of these narrow and broad spectral bands is related to the nonlinear interaction between the similaritons, induced by cross phase modulation (CPM) (incoherent coupling) or four wave mixing (FWM) (coherent coupling) . In order to determine the relative contributions of the nonlinear effects, we have modelled the pulse propagation by a system of two incoherently coupled equations:
with δ= β2 Ω the group velocity mismatch between the two pulses. The last terms of the right-hand side of Eqs.(6), which correspond to CPM between the pulses, are the only coupling terms, since FWM coherent interaction is neglected in Eqs. (6). The resulting field ψ(z,t) during the collision is obtained by:
Evolution of the parameter ΔS obtained from numerical integration of Eqs. (6) is represented in Fig. 5(c). Comparison between Figs. 5(b) and 5(c) shows that only small discrepancies are visible between the results of the two models, which implicitly include or neglect the coherent terms, respectively. In consequence, the spectral modifications observed in Fig. 5(b) are mainly due to CPM between the pulses, as expected.
Finally, Fig. 6 gives a comparison of the intensity profiles (a) and spectra (b) of the pulses at the amplifier output, on one hand, before the collision (ΔTL = -90 ps, blue solid line) and on the other hand, after the collision (ΔTL = 100 ps, circles). As can be seen from Fig. 6(a) the intensity profile of the similaritons remains unchanged after the collision, which is not surprising since CPM is essentially the only interaction process which occurs during the collision. However, a slight modification of the spectrum can be observed in Fig. 6(b), in agreement with the results of Figs. 5(b,c) showing that ΔS does not completely vanish for ΔTL = 100 ps.
3. Experimental observations
We now present some experimental confirmation of the numerical predictions presented in the previous section.
3.1. Generation of two time-delayed pulses with different wavelengths
The basic idea to generate two time-delayed pulses with different wavelengths is described in Fig. 7. The solution implemented consists in spectral slicing the broad spectrum of a similariton generated in an erbium-doped fiber amplifier (EDFA). The idea of spectrally slicing the broad spectrum of similaritons was already used to generate multi-wavelength pulse sources . The initial pulses, delivered by a Pritel laser at a repetition rate of 22 MHz, have a duration of 8 ps and a wavelength of 1545 nm. They are spectrally filtered to eliminate part of the initial noise thanks to a bandpass filter (bandwidth of 1 THz). An isolator at 1480 nm was inserted after the filter to prevent that part of the residual pump in the EDFA does not disturb the operation of the picosecond laser. A tunable attenuator permits to adjust the energy of the input pulses sent into the EDFA. The amplifier consists of a normally dispersive erbium-doped fiber of 15 m manufactured by OFS, identical to the fiber used by Billet et al. . The erbium fiber has the following parameters at 1545 nm: dispersion β2 = 40 × 10-3 ps2 m-1 and nonlinear coefficient. γ = 6 × 10-3 W-1 m-1. The fiber is pumped in a counter-propagating configuration with a hundred of milliwatts delivered from a 1480-nm cw Raman laser from Keopsys. At the EDFA output the initial pulses are transformed into similaritons characterized by a broad spectrum. To accentuate the spectral broadening, the similaritons are injected into a passive fiber with normal dispersion, consisting in 800 m of non-zero dispersion-shifted fiber (NZ-DSF) with a dispersion of 2.33 × 10-3 ps2 m-1 and a nonlinear coefficient of 2.0 × 10-3 W-1 m-1. Let us recall that the similariton preserves, through further propagation in a normally dispersive fiber, its parabolic shape while broadening temporally and spectrally [2,12].
The broadband pulses thus generated in the passive fiber are split in two and recombined in a Mach-Zehnder interferometer with a variable delay line. Spectral slicing is carried out via two Bragg filters (from Teraxion) placed within each arm of the interferometer and separated by the desired frequency shift (here Ω = 1.25 THz). The filters have a Gaussian transfer function with a spectral width (FWHM) of 75 GHz and a middle wavelength of 1540 nm and 1550 nm, respectively. A tunable attenuator placed inside one arm of the interferometer permits to obtain two pulses of identical energy. A polarizer located at the interferometer output imposes on the two pulses the same linear polarization. Chirp compensation of the pulses is carried out by means of 500 m of standard single mode fiber (SMF 28). The various operations having involved a significant reduction of the pulse power, a second EDFA amplifies the two pulses in order to have an energy of 2.9 pJ per pulse at the input of the Raman similariton generator. The Raman amplifier, whose parameters have been reported in section 2, consists in a 5.3-km NZ-DSF fiber pumped in a backward configuration .
The spectra of the pulses delivered by the picosecond laser source and after spectral broadening and slicing are displayed in Figs. 8(a,b,c), respectively. The spectral broadening obtained through the similariton amplification in the EDFA and subsequent propagation in the passive NZ-DSF appears then clearly : the spectral bandwidth of the initial pulses (67 GHz) is thus increased up to 1.6 THz, what corresponds to a broadening factor of about 24. The flatness of the spectral profile is also remarkable over nearly 1.5 THz. The spectrum of the pair of pulses obtained after spectral slicing (Fig. 8(c)) is symmetric. The autocorrelation of the pulse generated at 1550 nm, represented in Fig. 8(d), does not show any pedestal and indicate a pulse width of 7 ps (with the assumption of a Gaussian pulse), value slightly higher than the Fourier limit of 5.9 ps, revealing the presence of a small residual chirp. Let us note that to isolate the pulse at 1550 nm, we placed at the input of the autocorrelator a flat-top Bragg filter with a central wavelength of 1550 nm and a spectral bandwidth of 1 THz.
3.2. Experimental investigation of the effects of a collision between two similaritons
Let us now study the propagation of the two time-delayed pulses with different wavelengths in the Raman amplifier. The time separation ΔTo between the two initial pulses was experimentally adjusted thanks to the delay line placed in one arm of the Mach-Zehnder interferometer (see Fig. 7). Variation of ΔTo enables recording of the spectrum and autocorrelation function at the Raman amplifier output for a collision occurring at a variable distance zC = ΔTo /δ. The autocorrelation function corresponding to ΔTL = 0 (zc = L) is represented in Fig. 9(a). The experimental results (circles) are in excellent agreement with the calculations obtained by numerical integration of NLSE with a constant gain (Eq. (2)) (solid red line) and with a sinusoidal fit (dotted blue line). We have experimentally measured the evolution of the frequency fs of the modulation which appears on the autocorrelation function as a function of ΔTL. The experimental results, represented in Fig. 4 (circles), are in excellent agreement with the numerical predictions given by Eq. (2) (solid line).
Figure 9(b) shows the pulse spectrum measured at the Raman amplifier output when the similaritons do not undergo a collision (circles). Let us note that we systematically filter from the spectra the noise due to the spontaneous emission of the various amplifiers by numerically removing the spectrum measured in the absence of any signal. We can remark some differences between the experimental spectrum and that predicted from numerical integration of Eq. (2) (dotted red line). First the pulse shifted towards low frequencies has an energy more significant than the pulse shifted towards high frequencies. Indeed, since the two pulses are frequency separated by 1.25 THz, they undergo a different Raman gain. The asymmetry observed in the spectrum of the two similaritons can be modelled by explicitly taking into account the dispersion of the Raman gain in a generalized NLSE . The results of numerical integration of the NLSE with a frequency-dependent Raman gain (solid blue line) are in better agreement with the experimental results. However, let us note that a narrow peak still exists in the experimental spectrum which is not reproduced by these calculations. This narrow peak can be related to the residual chirp of the initial pulses. Indeed, it was shown that an initial chirp of negative slope could involve a change on the spectrum at the amplifier output . The spectrum at the amplifier output is in that case much more sensitive to the initial pulse profile  and in particular to a possible pulse asymmetry.
Since the NLSE with a frequency-dependent Raman gain more accurately accounts for the pulse spectrum, we thus used this model to predict the evolution of ΔS as a function of ΔTL. The results, shown in Fig. 10(a), can be compared with those obtained from the NLSE with a constant gain (see Fig. 5(b)) : we can see that the introduction of the dispersion of the Raman gain does not modify qualitatively the evolution of ΔS. The experimental spectra recorded for various values of ΔTL are reported in Fig. 10(b). As can be seen from Fig. 10 the experimental results are in good agreement with the numerical predictions. Let us note however a slight asymmetry between the experimental spectra of the two similaritons, which does not appear in the numerical simulations. A slight asymmetry of the initial pulse profile once again could explain this asymmetry.
Finally we have recorded the autocorrelation function of the pulse at 1550 nm before and after the collision. We can observe in Fig. 11(a) that the autocorrelation signal of the pulse which has undergone a collision (solid blue line, case corresponding to ΔTL = 100 ps) is very close to the autocorrelation of the pulse which has not undergone a collision (circles, case corresponding to ΔTL = -90 ps). As a matter of fact, the signal observed in Fig. 11(a) does not rigorously correspond to the autocorrelation of the similariton at the Raman amplifier output. Indeed, using of the Bragg filter to isolate the 1550 nm pulse, involves a slight temporal compression of the similariton, the filter introducing a chirp with a negative slope. However, the chirp introduced by the filter being independent of the initial conditions at the amplifier input, it does not modify our conclusion, namely that the intensity profile evolution of a similariton is not modified by a collision with another similariton. Figure 11(b) shows a comparison between the spectral profiles of the pulse before or after collision, represented by the circles and solid blue line, respectively. It is noteworthy from Fig. 11(b) that the spectrum is slightly modified by the collision, as already mentioned in section 2.
In this study we have analyzed the dynamic of the self-similar amplification of a pair of time-delayed pulses with different wavelengths. Due to their different group velocities the similaritons collide in the amplifier and pass completely through each other. The distance of collision was varied simply by changing the time separation between the initial pulses, chosen so that collision occurs between pulses that have already reached their parabolic regime. We have theoretically shown that the collision between two similaritons of different wavelengths could lead to various evolutions, depending on whether the similaritons separate or not after the collision. We have been mainly interested in the case where the group velocity mismatch between the two similaritons was sufficiently large, to induce a separation of the two similaritons after the collision.
We have demonstrated theoretically that similaritons preserve their characteristics after their collision : owing to their compact nature, they evolve separately in the amplifier, only interact during their overlap and regain their parabolic form after the collision. On the other hand, a collision has no effect on the scaling of the temporal width and chirp of the similaritons. Finally, the velocities of the similaritons are unchanged by the collision. These results show that similaritons are stable against collisions in an amplifier. We then underlined both theoretically and experimentally that the collision of two optical similaritons induces a sinusoidal modulation inside the overlap region, whose frequency decreases during the interaction. The theoretical study of the spectrum of the pair of similaritons highlights that the similaritons interact with each other through the nonlinear effects of cross phase modulation.
In the present study we have considered the case of a Raman amplifier but our results can be straightforwardly extended to the case of doped-fiber amplifiers.
This work has been supported by the Fond National pour la Science (FNS) under contract “ACI-Photonique PH43”, by the Centre National de la Recherche Scientifique (CNRS) under contract “Equipe-Projet EPML3”, by the Institut Universitaire de France (IUF) and by the Conseil Régional de Bourgogne.
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