1. Introduction

Supercontinuum (SC) generation in optical fibers and waveguides has attracted considerable research interest in the last few years [1–8]. Applications of SC generation in optical fibers can now be found in fields such as optical frequency metrology [9, 10], optical coherence tomography [11, 12], and multiple optical carrier generation for wavelength-division multiplexing [13, 14]. This renewed research interest in SC light sources has been made possible thanks to the recent development of special fibers with high nonlinearities, namely photonic crystal fibers [1,3] and tapered fibers [2]. SC is typically obtained in those fibers by pumping them with high peak-power input pulses with durations ranging from several nanoseconds to several tens of femtoseconds [1, 3, 15]. It results from the combined effects of self- and cross-phase modulation (SPM and XPM), four-wave mixing (FWM), and stimulated Raman scattering (SRS), that together lead to multisoliton dynamics and dispersive wave generation [16–19].

It is important to realize however that pulsed laser sources are in principle not necessary to generate a broadband SC. As a matter of fact, SC can also be obtained in the continuous-wave (cw) regime as has been recently demonstrated experimentally [4–8]. Here, SRS and FWM are the main causes of spectral broadening, while SPM a priori plays a limited role [3, 20]. Of course, since cw sources do not benefit from the high peak-to-average power ratio of pulsed lasers, longer interaction lengths are required. The characteristic fiber lengths used in cw SC generation are in the km range even when highly nonlinear fibers are used [7], while in the pulsed case a few meters is generally sufficient [5]. Also, for a given average input power, short pulses produce much wider SC spectra than cw pumps. However, few pulsed lasers are able to generate the multi-watt average output powers that are typical of modern cw lasers such as cascaded Raman fiber lasers. Accordingly, for similar spectral widths, cw SC have a higher average power and exhibit a higher power spectral density than pulsed SC. Mean power densities of typical pulsed-pump SC usually range from –20 to –10 dBm/nm [1, 3, 15, 21]. In comparison, we have demonstrated in a previous work a cw SC source with a peak power spectral density in excess of 8 dBm/nm, which represents an improvement of nearly two orders of magnitude [6].

Given the high power density of cw SC and the simplicity of the pump scheme, SC generation in the cw regime certainly deserves more attention. The aim of our work is to understand this process better and to implement a numerical model able to reproduce fully the experimental results [6]. As we will see, this problem is closely related to the problem of femtosecond Raman soliton formation from picosecond pump pulses that has been widely studied at the end of the 80s [22–25]. Our numerical simulations consolidate these early results and extend previous cw numerical models [5, 25, 26] by including the partially coherent nature of the pump wave. In particular, our results confirm that a strong pump incoherence is actually the key to large cw SC bandwidths as has been suggested by recent experimental results [7]. The incoherence of the cw source plays the role of the pump laser noise that, in the pulsed case, is known to seed the initial spectral broadening [27]. Moreover, the random character of the fluctuations present in the incoherent pump beam naturally explains the smoothness of the cw SC spectra. Note that, in Ref. [5], the noise on the pump source was already shown as a cause of cw spectral broadening but the nanosecond simulations failed to give a complete understanding of cw SC generation. In Ref. [26], the propagation of a partially coherent cw laser beam in a fiber with anomalous dispersion at the pump wavelength was analyzed and adequately modeled, but the delayed Raman nonlinear response of the fiber was neglected. Therefore, to the best of our knowledge, our model which takes into account dispersive, Kerr and Raman effects as well as the pump incoherence is the first complete numerical study of cw SC generation. In this way, we are also able to make quantitative comparisons with experimental results. The paper is organized as follows: in Section 2, we briefly present our previous experimental results [6] and we describe the main features of SC generation in the cw regime. The details of our numerical model is then described in Section 3 where we also present a first set of numerical simulations. Finally, in Section 4, we discuss the ensemble averaging procedure that we have used to match the numerical and experimental results and that reveals the essential role of the pump incoherence in the cw SC generation process. Our conclusions are drawn in Section 5.

2. Experiment

In order to understand better the results of the numerical simulations that are presented later in this paper, we briefly describe here the experiment that our simulations tend to reproduce. This cw SC generation experiment was recently performed by some of us [6] and has led to the generation of a 200 nm wide SC, covering the S, C and L transmission bands defined by the International Telecommunication Union (ITU).

The pump laser used in our experiment is a cw cascaded Raman fiber laser (RFL) with a depolarized single-mode output tuned at a wavelength of 1455.3 nm [6]. The laser output power can be as high as 2.1 W and, thanks to a feedback control loop, is stabilized to within ±10 mW. The spectral linewidth of the RFL grows as the power increases. For the maximum 2.1 W output power, the full-width-at-half-maximum (FWHM) of the output laser spectrum is Δλ = 1.1 nm (or Δv = 150 GHz), with a RIN < -110 dBc/Hz (measured in the range 0-1 GHz). CW supercontinuum generation is observed by launching the light generated by the RFL in a 7-km-long spool of nonzero dispersion-shifted fiber (NZDSF). The nonlinear coefficient of this fiber, measured with a Sagnac interferometer [28], is found to be γ = 2.7 W^-1 km^-1. Direct probing with a tunable laser has also provided us with an independent measurement of the Raman gain coefficient, g_R = 0.7 W^-1 km^-1. The dispersion coefficients at the pump wavelength have been obtained by the phase shift method, β ₂ = -0.11 ps²/km, β ₃ = 0.06 ps³/km, β ₄ = -1.55 × 10^-4 ps⁴/km, and β ₅ = 1.33 × 10^-6 ps⁵/km. The uniformity of the chromatic dispersion along the fiber length has also been evaluated using a method developed by some of us [29] and is found to be better than ±0.13 ps²/km at the wavelength of 1555 nm (spatial resolution of ~ 500 m). The fiber absorption is nearly uniform from 1450 to 1620 nm and is about α = 0.2 dB/km. It then increases up to 0.4 dB/km at 1700 nm. The pump wavelength was chosen to lie in a spectral region of small anomalous dispersion (β ₂ < 0) in order to favor phase-matched FWM processes. Also, the use of a long fiber with low losses ensures strong FWM-mediated power transfer between the pump and all the generated spectral components as well as a low SRS threshold.

The generated cw SC spectrum has been measured at room temperature for different input power levels by means of an optical spectrum analyzer (OSA) with an 0.05 nm resolution. The results are presented in Fig. 1. Note that all the pump power levels we quote in this paper represent the power that is effectively launched into the NZDSF, taking into account an 80% coupling efficiency. For low pump power levels (P _p ≃ 0.24 W), the output spectrum exhibits a clear signature of the influence of FWM through the generation of two nearly symmetric modu-lational instability (MI) sidebands around the pump frequency. The asymmetry in the intensity of those sidebands is caused by SRS amplification of the longer wavelengths and attenuation of the shorter ones. For higher pump powers, the effective FWM and Raman gains increase, which leads to a continuous broadening of the generated spectrum. Note that this broadening mainly occurs on the long wavelength side of the pump wave but that short wavelength components are also significantly generated. Beyond a pump power P _p ≃ 1.2 W, all the pump energy has been transfered to the SC and nearly complete pump depletion is achieved (see the inset of Fig. 1). At this point, our SC has a 20-dB bandwidth of over 200 nm and a peak power spectral density > 8 dBm/nm. We must point out that, apart from the initial MI sidebands, our SC spectrum exhibits a very smooth spectral structure at all pump power levels and that it broadens very regularly with increasing pump powers. Such a smooth evolution is observed in all reported cw continua [4–7]. It is in contrast with what is observed in most pico- or femtosecond experiments that are usually characterized by SC spectra with complex spectral structures [1,2,15,21,27] (although pulsed SC may exhibit good flatness at high pumping powers). One important aim of the numerical model we present in the next Section will be to account for this particular aspect.

 

Fig. 1. SC spectrum with different pump powers at room temperature (≃ 20 °C). All the curves have been normalized to the same peak value. Inset shows how increasing pump power leads to a stronger pump depletion.

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3. Numerical model

Our cw simulations of SC generation are based on a generalized nonlinear Schrodinger Eq. (GNLSE) for the scalar electric field envelope E(z,t) that is suitable for studying broadband light evolution [3,21,27,30,31],

Here z is the longitudinal fiber coordinate and t is the retarded time. R(t) = (1 - f_R)δ(t) + f_Rh_R(t) is the nonlinear response of silica that includes instantaneous Kerr and delayed Raman nonlinearities with f_R = 0.18. For h_R(t), we used the experimentally measured Raman response of silica fibers. The stochastic variable Γ_R(z,t) represents the spontaneous Raman noise and has frequency-domain correlations [21, 31]

 

Fig. 2. (a) Temporal intensity profile of the cw input beam used in the simulations for one particular realization of the random initial spectral phase (shown here with an average power of 1.7 W). (b) Intensity autocorrelation of the RFL output at 2.1 W.

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where n _th(Ω) = [exp(ħΩ/(kT)) - 1]^-1 is the thermal Bose distribution of the phonons and U is the Heaviside step function. All the other symbols have their usual meaning and are set to values identical to those of the experiment.

Equation (1) has been numerically integrated using a split-step Fourier algorithm [30] (the validity of the use of this integration technique in the cw regime is discussed at the end of Section 4). Particular attention has been paid to the initial condition, which in our case is supposed to represent the light emitted by a cw cascaded RFL. It turns out that modelling this cw input is significantly more challenging than a pulsed input. A somewhat naive approach would be to use a rigorously-continuous input wave as the initial condition, i.e, an electric field envelope with constant temporal amplitude and phase, and a perfectly monochromatic input spectrum. However, this approach is not able to reproduce the experimental results [5]. In practice, real cw lasers are only partially coherent and present a non vanishing spectral width. This means that these lasers exhibit random phase and/or intensity fluctuations on time scales of the order of their coherence time. Earlier works have shown that such partially coherent cw beams can undergo a significant amount of spectral broadening through, e.g, SPM [26,32,33] and this contribution has to be taken into account in our cw SC generation simulations.

To this aim, we adopt a phenomenological approach in which the cw input wave is represented as having the measured spectral density of our pump laser with an underlying random spectral phase. This approach leads to an initial temporal intensity profile that is essentially made up of a random succession of pulses of a few ps durations (about the inverse of the Δv = 150 GHz spectral width of the pump laser), as illustrated in Fig. 2(a) for one particular realization of the random spectral phase. Such strong intensity fluctuations could at first appear unrealistic for a stable cw RFL exhibiting a RIN < - 110 dBc/Hz at 1 GHz. We have however confirmed their existence through an intensity autocorrelation measurement of the RFL output [Fig. 2(b)]. As can be seen, a peak about twice as strong as the cw background is detected, a feature compatible with intensity fluctuations of nearly 100 % contrast. We note that such behavior has been observed with two RFLs from different manufacturers. Although this description might not necessarily represent the real nature of a partially coherent cw beam [26,33], strong intensity fluctuations do appear to be present in the input beam and we will show in the following that it leads to a very good agreement with experimental results. We must also point out that these fluctuations should not affect the noise properties of the Raman amplifiers used in wavelength-division-multiplexed telecommunication systems [34] because these amplifiers are mostly based on backward pumping schemes and because of typically large walk-off effects between the pump and the signals (that are separated by about 13 THz). Accordingly, fast pump fluctuations are averaged out along the amplifier length.

 

Fig. 3. (a) Numerical output spectra for various pump power levels and (b) temporal intensity output at P _p = 1.7 W for the initial condition shown in Fig. 2(a). (c) Measured intensity autocorrelation of the SC at P _p = 1.7 W.

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We have calculated numerically the output spectra corresponding to the initial condition shown in Fig. 2(a) after propagation in our NZDSF for various pump power levels. The results are presented in Fig. 3(a). This figure shows that the low power simulations (up to 0.72 W) agree fairly well with the experimental results (Fig. 1) as they reproduce the MI sideband growth and the pump broadening. However, higher pump power simulations depart significantly from the experiments. In particular, the supercontinuum smoothness is not reproduced at all and we can clearly see, for the highest pump power, the generation of soliton-like features red-shifted from the pump beam. The solitonic nature of these spectral features can be readily confirmed by observing the corresponding temporal intensity output [Fig. 3(b)]. In this case, two ultra-short pulses of 220 fs and 250 fs (FWHM) can be clearly seen. Numerical Fourier filtering of the spectrum enables us to associate these two pulses with the two broad spectral features observed at 1570 nm and 1660 nm, respectively. At 1570 nm, chromatic dispersion amounts -6.7 ps²/km while it reaches -12.6 ps²/km at 1660 nm. A straightforward calculation then shows that these two pulses are close to single fundamental solitons [30]. While the presence of these fs solitons might seem surprising when considering that we are in principle dealing with a cw laser output, these are not mere numerical artifacts as can be shown by observing the experimental intensity autocorrelation trace of the SC output at 1.7 W pump power [Fig. 3(c)]. This trace exhibits a subpicosecond feature much narrower than the autocorrelation trace of the pump laser [Fig. 2(b)] and confirms that ultrashort features have developed during propagation. The observed pulses are actually Raman solitons whose generation is seeded from the modulation instability (MI)-induced break-up of the temporal fluctuations present in the initial condition. As these solitons propagate down the fiber, they undergo a continuous shift to longer wavelengths because of the Raman self-frequency shift. While shifting their wavelengths, these solitons shed away some blue-shifted radiation in the form of dispersive waves, which explains the generation of the blue-side of the continuum [7, 8, 17, 23, 26]. Such femtosecond Raman soliton generation has been previously experimentally observed about two decades ago when pumping fibers with ps pump pulses [22–24] and is one of the key process underlying SC generation in the pulsed regime [16,17,27]. Theoretical studies have shown that these solitons should also be generated in the cw regime [25, 26, 35] but our work, together with Refs. [7, 8] is the first to provide a clear experimental evidence of their presence. Quite interestingly, this picture makes clear that nonlinear phenomena associated with the propagation of short pulses also play an important role in the cw regime, and that, after solitons are generated through MI-induced break-up, cw SC generation occurs essentially in an identical way than in the pulsed regime. We must also note that the initial temporal intensity fluctuations due to the beam incoherence are an important ingredient to speed up the spectral broadening since this naturally leads to an enhanced MI gain in comparison to a pure cw input beam.

At this point we have established that the dynamics of Raman solitons significantly contribute to cw SC generation in a way akin to what happens in pulsed-pumped SC. The contribution of these solitons to the spectral broadening, although very important, is however not sufficient to fully account for the experimental results, and in particular for the spectral smoothness of the generated spectra. As we will discuss below, the incoherent nature of the pump beam is actually the key ingredient to reconcile experiments with simulations.

4. Ensemble averaging

The simulation results presented in Fig. 3 have been obtained for one particular realization of the random initial spectral phase. Given the discrete nature of our numerical scheme, it means that we have only simulated the propagation of one particular temporally-limited snapshot of a partially coherent cw beam [as shown in Fig. 3(a)]. Recent studies performed under pulsed pumping conditions have however demonstrated that the SC generation process is typically very sensitive to random input noise [21,27]. Noise should a priori play an even more important role on the SC generation process in the cw regime since a partially coherent cw beam is inherently random from start to end, whereas the pulses emitted by a mode-locked laser have a relatively well defined temporal intensity and phase profile with only a limited superimposed random noise component.

To assess the influence of the random nature of our cw pump beam, we have run additional sets of numerical simulations. Within each set, all the simulations have been performed under identical conditions except for the random input spectral phase. The results obtained for a pump power of P _p = 1.7 W are illustrated in Fig. 4. Here we have plotted the initial input intensities and the corresponding output spectra for five different simulations. This figure clearly reveals how different initial fluctuations evolve into Raman solitons exhibiting widely different parameters. Basically, when fluctuations of higher amplitude are present in the initial condition, MI generates and fissions solitons earlier, and these solitons can then undergo a larger self-frequency shift while propagating in the remaining part of the fiber. As can be seen, this effect is rather drastic. The wavelengths of the solitons at the fiber end differ by more than 100 nm between different simulations, and even the number of generated solitons can vary. These observations are even made clearer in the movie associated with Fig. 4 and that shows how the temporal and spectral intensities of the field evolve along the fiber for the five simulations. In particular, by observing the appearance of fully developed MI oscillations on top of the initial fluctuations, we can clearly see that MI occurs at a different position along the fiber in the different simulations.

As a matter of fact, Fig. 4(b) reveals that the randomness of the fluctuations present in the cw pump beam leads to such a broad distribution of Raman solitons that they cover the entire spectral band of the experimentally observed SC [24]. We must also point out that the soliton parameters vary on a time scale of the order of the coherence time of the pump laser (about a few picoseconds), i.e., much shorter than the integration time of an optical spectrum analyzer.

 

Fig. 4. Results of five identical simulations differing only by the random initial spectral phase and performed with a pump power P _p = 1.7 W. (a) represents the initial field intensities while (b) shows the corresponding generated spectra. A movie of the simulation can be seen by clicking on the figure. [Media 1]

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Accordingly, we have to conclude that the spectral characteristics of the randomly-generated Raman solitons are significantly averaged in the measurements and that the experimental spectra are simply the envelope of the Raman soliton distributions. This explains the smoothness of cw SC spectra and why they exhibit no visible signature of individual solitons. We must stress that this averaging is qualitatively different from what typically occurs in the pulsed regime due to the pump laser noise [21, 27, 36]. In that latter case, the fluctuations are not strong enough to completely smooth the spectrum and hide the underlying spectral broadening mechanisms. Accordingly, the signature of Raman solitons is still clearly visible. In other words, the overall spectral shape of pulsed SC in mainly determined by the single-pulse response. This contrasts with the cw regime where the spectral shape is mostly determined by the statistical distribution of the initial fluctuations.

This discussion makes clear that the experimental situation does not correspond to one particular simulation. Instead, an ensemble average over many identical simulations performed with different random initial spectral phases is needed to reach a good agreement [24, 27]. To verify this interpretation, we have performed such an ensemble average numerically over 100 identical simulations. The result of these calculations is plotted in Fig. 5 (solid curves) for 0.72 and 1.7 W pump power and is compared to the corresponding experimental spectra (dashed curves). As can be seen, the agreement is excellent. The SC smoothness is now reproduced correctly. Even some fine spectral features such as the location of the maximum and some other peaks as well as the shape of the blue shoulder of the SC are correctly simulated. As regards the decreasing slope of the spectrum on the long wavelength side of the pump, we can simply interpret it from the low probability to generate solitons that undergo a large frequency shift. Note that we have not performed any adjustments of the simulation parameters to try to fit better the experimental and numerical results. The larger bandwidth of the simulated spectra observed in Fig. 5(b) can probably be explained by the imperfect modelling of the fluctuations of our pump laser. Also, our simulations do not take into account polarization effects, the frequency-dependent effective area, as well as longitudinal variations of the fiber parameters that typically reduce the efficiency of nonlinear processes. Additionally, some of the fiber parameters might have been imperfectly evaluated.

We must stress that, in addition to the ensemble averaging procedure described above, the complexity of cw SC generation simulations is also compounded by the long fiber length and by the associated huge temporal walk-off. In the case of our experiment, the maximum walk-off occurs between the zero-dispersion wavelength (1453 nm) and the red-side of the continuum (i.e., 1625 nm) and is as high as 4 ns over the 7 km-long fiber. It means that a numerical window wider than 4 ns is in principle required to keep track of all the generated spectral components. Given that our numerical scheme is based on Fourier transforms, the sampling theorem then imposes the use of a very large number of discretization points to describe the SC spectral bandwidth, typically 2¹⁹ or 2²⁰, making the simulations very computationally intensive. In practice, we have tested the robustness of our numerical integration scheme by performing ensemble of simulations with a number of points ranging from 2¹² to 2¹⁸ (corresponding to temporal windows from 50 ps to 3.2 ns). After averaging over the ensembles, the simulated spectra obtained in each case are essentially identical. The numerical results shown in this paper have actually been obtained with 2¹⁴ points, which represents a good trade-off between the number of simulations required for a good averaging and the overall computation time. It is important to note, however, that the split-step Fourier scheme is implicitly based on periodic boundary conditions. Accordingly, if one uses a time window shorter than 4 ns to describe the cw field, a cyclic wrapping of the temporal envelope of the electric field occurs. This means that a Raman soliton travelling at a group velocity different than that of the pump may periodically travel across the simulated window and interact over and over with the same portion of the field. Of course, this is not what happens in the experiment where Raman solitons forever drift away from their source point and interact with other random realizations of the fluctuations. The fact that the simulation results obtained with < 4 ns-wide numerical windows are essentially independent of the number of discretization points indicate however that a limited sample of fluctuations is statistically significant and represents correctly the overall physics.

 

Fig. 5. Comparison of experimental and simulated SC spectra obtained for a pump power of (a) P _p = 0.72 W and (b) P _p = 1.7 W. The numerical results have been averaged over an ensemble of 100 simulations differing only by the random initial spectral phase.

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5. Conclusion

In this work, we have presented a detailed numerical study of the process of SC generation in the cw regime. To the best of our knowledge, our model which takes into account dispersive, Kerr and Raman effects as well as the pump incoherence is the first complete numerical study of cw SC generation. Our work confirms previous theoretical results [25, 26, 35] and some recent experimental observations suggesting that cw SC generation arises in a way akin to pulsed SC [7]. First, MI breaks-up the cw pump beam into multiple solitons. These solitons then undergo Raman self-frequency shift, extending the SC on the long wavelength side, while simultaneously shedding some energy in the form of blue-shifted dispersive waves. While this scenario explains the SC bandwidth and the differences that are observed between the normal and anomalous dispersion regimes [8], it does not account for the spectral smoothness of the measured spectra. Our numerical simulations have demonstrated that the partially-coherent character of the cw pump beam is actually the key ingredient to interpret this important feature. The pump incoherence manifests itself by random phase and intensity fluctuations in the pump wave, and that makes the dynamics of cw SC generation much more complex than in the pulsed case. Because of these fluctuations, the MI-induced Raman solitons are generated with widely different parameters, so that, on a time scale comparable to the coherence time of the pump, the output light is made up of a random succession of solitons. The SC spectral smoothness simply results from the fact that this distribution of solitons covers the entire SC bandwidth and is naturally averaged by measuring instruments. Clearly, the pump incoherence also plays a fundamental role in seeding the MI-induced break-up of the pump beam. This lead to the conclusion that low coherence pump lasers should induce a cw SC more efficiently than highly coherent cw sources. We must stress, however, that pump incoherence is known to suppress MI whenever the frequency-bandwidth of the pump becomes of the same order than the MI frequency [33], so that SC might also be inhibited for extremely broad pump spectra. Also, the exact statistical nature of the fluctuations present in the pump laser beam may also affect the spectral broadening process [37]. This latter proposal should therefore be carefully tested against experiments.