1. Introduction

Though nonlinear propagation of ultrashort laser pulses in dispersive single-mode optical fibers has steadily been investigated over the last three decades [1], studies on continuous-wave partially coherent (PC) light have been scarce. In recent months, however, several authors have reported significant spectral broadening of high-power continuous-wave (CW) lasers propagating in the zero-dispersion wavelength (ZDW) region of optical fibers [2, 3, 4, 5, 6]. As initially demonstrated theoretically by Cavalcanti et al. [7], this broadening mechanism occurs because the PC optical beam suffers from modulation instability (MI) in the anomalous dispersion regime while it remains stable under normal dispersion. As a matter of fact, MI is responsible for broadband noise amplification through phase-matched four-wave mixing, enhancing initial phase and amplitude fluctuations of the partially coherent beam, and thus resulting in a strong spectral broadening. This nonlinear effect is obviously detrimental in long-haul optical communication systems [8], but it has been recently used advantageously to generate supercontinuum light [4, 6], and to flatten and increase the bandwidth of Raman fiber amplifiers [2, 3]. Despite these advances, it is worth noting that the underlying mechanism responsible for spectral broadening of PC wave has never been the object of a detailed study. The aim of our work is to improve our understanding of this spectral broadening mechanism and the associated ultra-fast temporal phenomena. To this end, we present a numerical study of the nonlinear propagation of continuous-wave PC laser beam in single-mode fibers using a phase-diffusion model [9]. With this model, which accounts for the partial coherence of the incident beam, i.e., the linewidth of the laser, we verify that this broadening effect indeed relies on modulation instability in the low anomalous dispersion regime. However, we will point out that the strong spectral broadening reported in the experiment is not entirely encompassed in the MI bandwidth. Instead, theory reveals that it is associated with the fission of higher-order solitons into frequency-shifted fundamental solitons and dispersive waves, as it has been demonstrated recently in the framework of supercontinuum generation [10]. The most striking difference, here, is that these higher-order solitons and blue-shifted radiation emerge initially from continuous-wave light instead of femtosecond pulses. Experimental spectra and autocorrelation traces obtained from a CW Raman fiber laser propagating in a dispersion-shifted fiber agree well with these interpretations.

2. Theoretical model

Let us consider a linearly-polarized PC continuous-wave laser beam propagating in a single-mode optical fiber. By assuming a non birefringent fiber and including higher-order dispersion terms, the field amplitude A(z,t) satisfies the following scalar nonlinear Schrödinger equation (NLSE) [1],

where t is the time expressed in a frame moving at the group velocity of the PC wave, β _m (m=2..4) are the m^th dispersion coefficients. γ is the nonlinear coefficient, $\gamma =\frac{2\pi n_2}{\lambda A_{\mathrm{eff}}}$, with n₂ the nonlinear refractive index and A_eff the effective mode area. To model the PC field, we use the so-called phase-diffusion model [9, 11]. The complex field, which is assumed free from amplitude fluctuations, can be expressed as,

where the phase φ(t) exhibits a simple Brownian motion and obeys a Gaussian probability distribution [9]. This model implies that the power spectrum of the wave has a Lorentzian shape which is a general assumption for all-known CW lasers with finite linewidth [7, 11]. We also define the mutual coherence function as

where angle brackets denote the ensemble average. The spectral full width at half maximum Δf can be expressed as Δf=(πt_c)^-1, where t_c is the coherence time of the laser field. Using the PC field expression Eq. (2), we perform a numerical integration of the NLSE Eq. (1) with the parameters of a high-power CW Raman fiber laser (RFL) as the input PC field. The results of our simulations are presented in Fig. 1 that shows the intensity profile and power spectrum of the PC wave, respectively, for three propagation distances. A complete data series of the field dynamics (from z=100 m to z=3100 m) can be watched as a movie by clicking on the link from the Fig. 1(a–b). For comparison, the initial continuous-wave field at z=0 and its spectrum are plotted in red line. The time coherence and its spectral width are 6 ps and 50 GHz, respectively. At z=300 m, the green line shows that the continuous-wave background becomes modulationally unstable and exhibits a high-frequency amplitude modulation. Note also on Fig. 1(b) that a small pedestal appears in its spectrum. Here, we would like to emphasize that MI appears very quickly in the fiber in comparison to the usual case of coherent-MI with a monochromatic wave [1]. This is because the initial ramdom phase fluctuations (or the incoherence spectrum) of the PC wave act as a seed to initiate MI [7, 12]. Moreover, we can observe a broadband noise amplification around the central frequency instead of the two distinct symmetric MI side-bands generally observed in the coherent MI process. The MI spectrum is indeed modified by the incoherence of the field depending on the time coherence [7, 12]. At z=3100 m, the field intensity depicted in blue on Fig. 1(a) shows that the PC wave evolves towards a train of ultrashort pulses randomly spaced in time, and with peak powers more than 6 times above the input power. The power spectrum exhibits a strong asymmetric spectral broadening with, in particular, a down-shifted anti-Stokes peak. To get better insight, we plotted on Fig. 2 the spectral width output/input ratio of the PC wave as a function of the input power. Numerical results (circles and crosses) are then compared to the usual analytical formula of the full MI bandwidth (solid lines), calculated from the well-known dispersion relation [1],

In Eq. (4) we neglect β ₄ because the central wavelength is a few nanometers away from the ZDW. For the ratio measured at -10 dB below the maximum (in red), the results show a rather good agreement, except in the very low-power regime below the MI threshold and in the high-power regime for which the ratio increases. This increase in the spectral width is most important when measured at -20 dB (in blue) because it encompasses the wide anti-Stokes peak. This comparison between numerical simulations and a simple analytical prediction clearly demonstrates that the spectral broadening cannot be accounted for by simply considering the MI effect. Higher-order nonlinear and dispersive effects occur for long propagation distances. Indeed, the presence of a blue-shifted frequency component in the power spectrum shown in Fig. 1(b) is the signature of the generation of dispersive waves. These waves are radiations emitted by fission of ultrashort optical pulses formed from a propagation distance z≈800 m (see movie). We have found numerically that some of pulses have a mean FWHM of 1.5 ps and a mean peak power of 1 W. With these parameters, the mean equivalent soliton number [1] is about N≈3, meaning that these pulses can be considered as third-order solitons. As previously demonstrated in the framework of soliton propagation [13, 14] and for supercontinuum generation [10], these pulses are not stable under the presence of high-order dispersion. In particular, when third-order dispersion acts, they undergo fission into stable red-shifted fundamental solitons and blue-shifted dispersive waves. These so-called non solitonic radiations (NSR) are emitted at particular frequencies for which a precise phase-matching condition is fulfilled with fundamental solitons. The radiation frequency detuning depends both on the dispersion coefficients and on the CW power. It can be easily written as [14, 15]

With the parameters given in the caption of Fig. 1 and for P=0, the radiation frequency is up-converted at δω/(2π)=2.28 THz, i.e., at 1536 nm, which is in excellent agreement with the numerical simulation of Fig. 1(b). We have also calculated the mean soliton number at z=3100 m for the pulses shown in Fig. 1(b). From the assessment of a mean pulse duration of 500 fs and a mean pulse peak power of 3 W, we obtain N≈1, in accordance with the prediction of fundamental soliton pulses [13, 14, 15]. Note however on the movie that both higher-order and fundamental solitons coexists during propagation in a recurrent manner.

 

Fig. 1. (a) Temporal intensity and (b) power spectrum of a PC laser beam in a single-mode optical fiber at three propagation distances (z=0, red, z=300m, green, and z=3100 m, blue). The entire sequence can be viewed as a movie (avi, 1865 kb). PC wave’s parameters are λ=1555 nm, P=600 mW, Δf=50 GHz. Fiber’s parameters are β ₂=-5.5.10^-28s²m^-1, β ₃=1.15.10^-40s³m^-1, β ₄=-2.85.10^-55s⁴m^-1, λ₀=1549 nm, γ=2W^-1km^-1, α=4.6.10^-5m^-1. [Media 1]

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3. Experimental results

We have recorded both the power spectrum and the intensity autocorrelation traces at the output of two single-mode optical fibers having different ZDWs. As a PC optical source, we used a specifically-designed CW P₂O₅-doped Raman fiber laser (OSYRIS SA) emitting at a wavelength of 1555 nm and with power ranging from 600mWto 2.5W. RFLs are known to generate low relative intensity noise (RIN) and are therefore appropriate with our model of Section 2. The laser linewidth of 50 GHz is plotted in red on Fig. 3(a) and theoretically modelled in Fig. 3(b). Note that for a better fit one could also introduce a filtering function in the theoretical spectrum to match exactly that of our actual fiber laser. We observed that the laser linewidth slightly increases with the pump power from 40 GHz till 80 GHz because of the nonlinear interaction between the laser modes [16]. We assume here that this line broadening has a little impact on the spectral broadening mechanism. The pump power was then coupled in the silica fiber through a fused 99/1 fiber coupler. The coupler was used to check that no backward stimulated Raman and Brillouin scattering occurs in the fibers. An optical attenuator and an optical spectrum analyser were placed directly at the output of the fiber to record the spectrally-broadened laser at different power levels. We analyzed under the same conditions the intensity autocorrelation traces by means of an autocorrelator. The fiber under test was a 3100m-long dispersion-shifted fiber (DSF) with a ZDWat λ ₀=1549 nm and a dispersion slope D_s=0.07 ps.nm^-2.km^-1. Note that the laser wavelength has been particulary chosen close to the ZDW in the low anomalous dispersion regime. For comparison with the normal dispersion case, we also performed the experiment in a 5 km-long standard single-mode fiber (SMF) having a ZDW at 1300 nm far from the pump wavelength (in green in Fig. 3(b)).

 

Fig. 2. Ouput/Input spectral widths ratio of a PC wave after 3100 m of propagation in a single-mode optical fiber. Solid lines: analytical prediction Eq. (4), Crosses and circles: numerical results.

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Fig. 3. (a) Experimental and (b) simulated spectra for increasing pump power. blue : output, from bottom to top (a) P=0.8, 1, 1.4, and 1.8 W and (b) P=0.4, 0.5, 0.7, 0.9 W. Green : normal dispersion P=1.4 W. Red: input at (a) 0.8 W and (b) 0.4 W.

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The output spectra are illustrated in Fig. 3(a) for increasing input power and compared to both the input one (in red) and to simulated spectra (Fig. 3(b)). Because the Raman laser was unpolarized, the pump power used in our scalar numerical simulations (linearly polarized) has been divided by 2. A more complete model should also take into account the polarization mode dispersion of the fibers. But in view of the good agreement between experimental and theoretical spectra shown in Fig. 3, we can conclude that polarization has only an impact on the pump power requirement. For a better comparison, one can also introduce a filtering function in the theoretical spectrum to match exactly that of our actual fiber laser. We can see on Fig. 3(a) the strong asymmetric spectral broadening of the laser, as expected by theory. The anti-Stokes frequency peak is located at 1536 nm, exactly the same predicted by our simple analytical prediction Eq.(5). Moreover, the measured frequency detuning δω increases linearly with the pump power in both experimental and theoretical spectra [14].

 

Fig. 4. (a) Experimental and (b) theoretical intensity autocorrelation functions for same increasing power levels as in Fig. 3. Red line: Input. Blue lines: Output.

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Figures. 4(a) and (b) show the experimental and theoretical intensity autocorrelation traces, respectively, for increasing pump power. As one can see, the input autocorrelation trace (red) is flat while at the output (blue) it is characterized by a peak on a background, meaning a substantial coherence degradation of the laser during propagation. The autocorrelation peak FWHM is δt=1.5 ps and the background decreases with power, in good agreement with the theoretical autocorrelation shown in Fig. 4(b) (δt=1 ps). We can notice that a small damped modulation appears on the theoretical traces, which is characteristic of a pseudo-periodic pulse train at a repetition rate of the order of 1 THz. This modulation is not present on the autocorrelation measurement and we believe that this discrepancy comes probably from the fact that the RFL was unpolarized.

4. Conclusion

In this work, we have investigated the nonlinear propagation of a continuous-wave partially-coherent laser beam propagating in the neighborhood of the zero-dispersion-wavelength of single-mode optical fibers by using a phase-diffusion model. We have shown that modulation instability is responsible for spectral broadening and for the break-up of the continuous background. Additionally, we have identified the asymmetric spectral broadening mechanism as resulting from the fission of higher-order solitons into fundamental solitons and blue-shifted dispersive radiations. Experimental results obtained with a Raman fiber laser and a dispersion-shifted fiber have shown a very good agreement with numerical and analytical predictions. Finally, this study provides a good guide to help in the design of Raman fibers amplifiers and CW supercontinuum generation.