We theoretically study broadband supercontinuum generation in photonic crystal fibers exhibiting two zero dispersion wavelengths and under continuous-wave pumping. We show that when the pump wavelength is located in between the zero-dispersion wavelengths, a wide and uniform spectral broadening is achieved through modulation instability, generation of both blue-shifted and red-shifted dispersive waves and subsequently through soliton self-frequency shift. This supercontinuum is therefore bounded by these two dispersive waves which allow the control of its bandwidth by a suitable tuning of the fiber dispersion. As a relevant example, we predict that broadband (1050–1600 nm) continuous-wave light can be generated in short lengths of microstructured fibers pumped by use of a 10-W Ytterbium fiber laser.
©2007 Optical Society of America
Research on supercontinuum (SC) generation in optical fibers still attracts much attention both from fundamental and applied viewpoints (See Review ). So far, wide SCs (up to two octave spanning) has been reported and soon considered as a promising tool for metrology, biomedical and telecommunication applications, due to their high spectral brightness compared with traditional broadband light sources (incandescent or fluorescent sources). Primary results were obtained with pulsed pump [1,2,3,4,5,6] launched in Photonic Cristal Fibers (PCF), in tapered fibers  or in classical telecommunication fibers . Except for Ref. , the pump wavelength is close to the Zero Dispersion Wavelength (ZDW) of the fiber or located between them [5,6] for fiber exhibiting two ZDWs. It has been demonstrated that SCs generated in fibers with a single ZDW result from a complex interplay of nonlinear effects which relative contribution depends both on the pump temporal duration and on the dispersion profile of the fiber. In PCFs owing two ZDWs, most of the studies have been performed in the femtosecond regime [5,6]. In this case, two Dispersive Waves (DWs) can be generated because of the two ZDWs of the fiber and it has been demonstrated that the SC is bounded by these DWs. It is worth noting that extra-blue shifted waves can be generated through Cross-Phase Modulation (XPM) between the DWs and the soliton [5,6].
Overall principles for SC generation in the continuous wave (CW) regime is similar to that of the pulsed regime, except that the initial CW field is first converted into a train of ultra-short pulses which duration, peak power and repetition rate are not constant [9,10,11,12]. Since different pulses propagate in the fiber, the availability to generate new spectral components and the location of these new frequencies depends on each pulse characteristics. As a consequence, SCs obtained in the CW regime are usually smoother and more stable than those obtained in the pulse regime. From a practical point of view, recent advances in high power CW fiber lasers allows for the achievement of compact and robust all-fiber SC light sources with higher Power Spectral Density (PSD) compared with pulsed SCs [9,10,13,14,15,16]. This of course extends their field of application . In first experiments, Raman fiber lasers emitting around 1500 nm were used, i.e. in the vicinity of the ZDW of dispersion-shifted fiber or highly nonlinear fibers [10,13,14]. It has been demonstrated that it is necessary to pump around the ZDW of the fibers and that longer wavelengths (compared with the pump one) are mainly due to the Soliton Self Frequency Shift (SSFS) process while blue ones originate from DWs [9,10]. SC extension up to 1000 nm have been reported [10,14] with strong PSDs (10 mW/nm, i.e. 10 dBm/nm). More recently, to further increase the PSD of CW-SCs, Avdokin et al.  and Travers et al.  have taken advantage of more powerful fiber lasers, Ytterbium (Yb) fiber lasers, and of the high nonlinearity of PCFs. One of the major drawbacks of this configuration is the high value of the water absorption peak at 1380 nm presents in such PCFs. It prevents the extension of these SCs or dramatically reduces there PSD. Very recently, Travers et al.  design a low loss water PCF (73.2 dB/km) leading to a SC ranging from 1100 nm to 1500 nm with a PSD of about 30 mW/nm (i.e. 15 dBm/nm). However, since the SC is mainly based on Stimulated Raman Scattering (SRS), a quite long fiber length is required (100 m) and even if the water absorption peak has tremendously been reduced, 50% of the injected pump power was lost inside the PCF.
In this paper we numerically demonstrate that most of the CW power of an Yb fiber laser can be converted into wide and nearly flat SC. We promote nonlinear effects allowing the achievement of SC in shorter fibers and consequently we circumvent the water absorption peak drawback. Contrary to previous studies with CW pumps, the dispersion of the PCF is tailored such that a blue-shifted DW as well as a red-shifted one are generated simultaneously. As it was previously reported in the pulsed regime , it can be achieved in PCF with two ZDWs located around the pump wavelength. It is worth noting that besides the speeding-up of the SC formation, both DWs bound the SC allowing a control of the SC extension, in other words, of the PSD of the SC. This numerical study constitutes a step toward an experimental demonstration with high-power Yb fiber lasers since realistic parameters are used.
The paper is organized as follows: In section 2 we define the geometrical parameters of the air-hole cladding structure, the dispersion and nonlinear properties of PCFs used in this study. We present the computational method as well as the numerical modeling of a CW Yb laser, in section 3. In section 4 we present our results before concluding in section 5.
2. Photonic crystal fibers with two zeros dispersion wavelengths
The pitch varies from 1.60 (PCF1) to 1.70 (PCF3) with a constant ratio d/Λ of 0.39, assuring that all PCFs are endlessly single mode (<0.43 ). The first ZDW (λ01), varies from 975 nm to 1014 nm and the second one (λ02) from 1114 nm to 1314 nm. The effective area (Aeff) have been evaluated from Ref.  which corresponds to a nonlinear coefficient ranging from 5 W-1.km-1 at 1800 nm to about 50 W-1.km-1 at 700 nm (calculated from γ=n2ω/[c Aeff])  with the nonlinear refractive silica index n2=3.10-20 m2/W, c the speed of light in vacuum and ω is the carrier angular frequency of the field). From a practical point of view, Aeff value is large enough, about 7.5 µm2 (γ=23 W-1.km-1) at the pump wavelength (1070 nm), to reach an efficient coupling of the pump source inside the PCFs.
3. Numerical model
We modeled the propagation of the CW field using the generalized nonlinear Schrödinger equation (NLSE) :
Where E(z,τ) is the electric field envelope in a retarded time frame τ=t-β1z moving at the group velocity 1/β1 of the pump. ω0 is the carrier angular frequency of the CW input field and γ(ω0) is the nonlinear coefficient at the pump wavelength (23 W-1.km-1 @ 1070 nm). The dispersion parameters βm are estimated from a polynomial fit of order 12 to reach a good interpolation of the fiber dispersion profile (values are listed in table 2).
Dispersion effects are described by the first term on the right hand side of Eq. (1) while nonlinear optical effects such as Self-Phase Modulation, SRS, self steepening and shock formation correspond to the second one. R(τ)=(1-fR)δ(τ)+fRxhR(τ) is the nonlinear response of silica (fR=0.18) and we used experimentally measured Raman response of silica fibers for hR(τ) [1,21]. Note that, the convolution integral in Eq. (1) between the field intensity and the delayed Raman response is calculated as a simple product in the frequency domain and the time derivative term of the self steepening effect is numerically calculated in the time domain. Equation 1 is solved numerically using the adaptive step size method outlined by Sinkin et al.  since it reduces the total number of Fourier transforms and thus increases computation speed. We used 215 points leading to a temporal resolution of 3.8 fs, a time window of 130 ps and a spectral resolution of 7 GHz. The local goal error used in the adaptive step size method was set to δG=10-5. It corresponds to a good trade off between accuracy and computation speed. In all the simulations presented in this work, the relative change of the photon number is less than 1.3 % assuring that no radiations are emitted outside the numerical spectral window.
Up to now, the modeling of a CW fiber laser represents an important challenge since no model has raised the agreement of the community [10,11,12,23,24]. A few years ago, Cavalcanti et al.  modeled a partially coherent beam by adding small intensity and phase fluctuations to a perfect CW in order to investigate the impact of the partial coherence of the pump on the MI process. More recently, we proposed to model this continuous field exhibiting a broad spectrum with the phase diffusion model [12,25]. We obtained quite good agreement with experimental results. Vanholsbeeck et al.  and Barviau et al.  considered that all the longitudinal modes of a fiber laser can not be specially phase matched in order to generate a perfectly continuous wave in the time domain. They modeled a CW fiber laser using a measured power spectral density  or assuming a Gaussian shape  with a randomly spectral phase.
Although Frosz et al.  considered that a randomly spectral phase has no physical justification, which is strictly speaking exact, we believe that a special phase matching between the overall longitudinal modes leading to a perfect CW in the temporal domain seems difficult to achieve in fiber lasers. Besides, we believe that the Vanholsbeeck-Barviau (VB) approach is the more realistic model since it leads to the best agreement between experimental and numerical results. Furthermore, Vanholsbeeck et al. provide an experimental autocorrelation trace obtained with two different Raman fiber lasers demonstrating that these lasers are not continuous and emit light with ultra-fast intensity fluctuations of nearly 100 % contrast. This experimental evidence is well reproduced with the VB model while it is not the case with the phase diffusion one. We therefore use a VB model type in our simulations in order to be as close as possible of experimental conditions. The power spectrum shape of the pump is assumed to be Gaussian with a full width at half maximum of 400 GHz [Fig. 2(a)]. Temporally, this random phase relation between the longitudinal modes results in spikes of about 2.5 ps (1/400 GHz) whose mean power is 10 W [Fig. 2(b)]. We would like to emphasize that these parameters correspond to those of commercially available Yb fiber lasers.
4. Results and discussions
4.1 SC formation in PCF2
Mechanisms involved in the SC formation are the same for the three fibers, and for example we focus on the SC formation in PCF2. The output spectra obtained in the other fibers will be compared in the next subsection. The evolution of the spectrum in PCF2 as a function of the fiber length is plotted on Figs. 3(a) to 3(e). At L=5.57m, we observe side lobes around the pump on Fig. 3(a) which are the spectral signature of the MI process . It results in a modulation in the time domain at the top of the most powerful spikes of the pump. By further propagating inside the fiber, as the dispersion is anomalous, the modulation evolves in a train of ultra-short soliton-like pulses of about 20 fs. It has ever been demonstrated that these soliton-like pulses are not stable under the presence of higher-order dispersion and nonlinear effects [26, 27] and as a result, they undergo fission into stable fundamental soliton-like pulses and DWs. The pump sheds away energy to these so-called non-solitonic radiations if the following phase matching relation is fulfilled [6,26,27]:
Where β(ωP) and β(ωDW) represent the propagation constants at the angular frequency of the pump ωP and the dispersive wave ωDW respectively. Furthermore, it is important to point out that a spectral overlap must exist between the pump and these DWs. In optical fibers owing a single ZDW, DWs are blue shifted [26,27] while in optical fibers with two ZDWs, a DW is still blue shifted below the first ZDW (λ01) (DW1) and another is red shifted above the second ZDW (λ02) (DW2) [5,6].
We observe on Fig. 3(b) that two DWs are generated at 861 nm and 1500 nm respectively. From Eq. 2, at 1070 nm with PP=1 kW (which is the peak power of the most powerful soliton-like pulses at about 6.33 m) we found that two DWs must be generated at 862 nm and 1500 nm. This is in excellent agreement with our theoretical predictions.
By further propagating inside the fiber, we can see the starting point of the SSFS process on Fig. 3(c) due to the SRS effect [9,10,11] (indicated by a black arrow). We verified the solitonic nature of this broad wave by spectrally filtering it. The part of the SC filtered is represented in red in Fig. 4(a) and its inverse Fourier transform in Fig. 4(b). The filter has a supergaussian shape, centered at 1220 nm and 240 nm wide in order to include most of the spectral components of the expected soliton [red curve on Fig. 4(a)]. From the inset in Fig. 4(b), we can clearly associate the filtered part of the SC with the pulse of 1 kW peak power (PP) and 20 fs duration (TFWHM) which is closed to fundamental soliton (N=1.2 with with β2=-3.68×10-27 s2/m at 1130 nm ). Then, the soliton shifts away from 1070 nm to stop around 1200 nm, just below the second ZDW of the fiber located at 1237 nm (black arrows on Figs. 3 (c) and 3(d) and movie). This is clearly illustrated on Fig. 3(e) and on the movie where the red shift is cancelled at about 7.5 m . This cancellation of the SSFS has recently been demonstrated in PCF owing a negative dispersion slope  since the spectral recoil associated with DW generation (blue-shift in PCF owing a negative dispersion slope) is balanced by the Raman frequency shift (red-shift). During its spectral shift, the soliton still sheds away energy to both DWs provided that the phase matching (Eq. 2) is fulfilled and a spectral overlap exists. Figure 5(a) represents the evolution of this phase matching when λP increases. In this case, DW1 and DW2 are tuned toward the blue wavelengths (860 nm to 837 nm for DW2 and 1500 nm to 1350 nm for DW2). Numerical simulations shown in Figs. 3 confirm this behavior. We found that DW2 moves from 1500 nm to 1350 nm which is in good agreement with theory. A quantitative comparison is quite difficult to achieve for DW1 since as the soliton moves to the red, the spectral overlap with DW1 decreases and as a consequence, the amount of energy transfer between the soliton toward DW1. So we can not clearly distinguish a shift due to spectral broadening toward short wavelengths. Indeed, XPM process between the soliton and the DWs can also be involved in the generation of blue spectral components [5, 6], as shown by the spectral modulations on the left side of DW1 [inset in Fig. 3(c)].
It can also be viewed in Fig. 3(c) for DW2 that also exhibits these spectral modulations. We believe that the tendency of DW1 to move toward the blue side of the SC results of all these processes.
In addition, the major role of the SSFS process in the SC generation can be highlighted by performing a simulation in PCF2 without stimulated Raman effect. The average output spectrum is represented on Fig. 5(b) in blue to compare with the green one from Fig. 3(d) where the SRS effect is included.
We used averaged spectrum in order to make easier the comparison. Without SRS, the beginning of the SC formation is quite similar than with SRS effect. Indeed, MI induces break up of input pump pulses and two DWs are generated around 860 nm and 1500 nm. However, by further propagating in the fiber no SSFS effect starts and as a consequence, blue shifts of DWs do not occur and especially that one of DW2. Consequently a weak spectral broadening of the pump and generation of two DWs is achieved, but no SC is generated.
The red shift of the SSFS combined with the blue shift of DW2 allows for the achievement of a quite flat SC between the pump wavelength and the second DW2. The whole SC is bounded by the two DWs and as a consequence its extension can be controlled by an appropriate tuning of the dispersion curve of the optical fiber (i.e. of the center wavelengths of each DWs) as in the case of pulsed pump [5,6].
Finally, the onset of the SC formation in a fiber owing two ZDWs under CW pumping can be divided in four steps: (i) the pump is launched in the anomalous dispersion region of the fiber and is split into soliton-like pulses through MI, (ii) these solitons are not stable due to higher dispersion and nonlinear effects and shed away energy to DWs located on both sides of the ZDWs of the fiber. (iii) a SSFS is induced via SRS and the central frequency of each DWs is blue shifted. (iv) the SSFS effect stopped just below the second ZDW of the fiber and the SC spectrum is not modified by further propagating in the fiber.
Note that in Ref  where effects of soliton collision in a fiber with two ZDWs and a CW fiber laser of 10 W have been numerically studied, the spectra achieved are comparable to those presented on Fig. 3(e). However, they are obtained in longer fibers (64 m against 20 m in our study). We believe that as they used the phase diffusion model to reproduce the Yb fiber laser behavior, MI is induced from a perfectly continuous wave of 10 W rather than short spikes with about 60 W of peak power (Fig. 2(b)). As a consequence, the MI induced pulse-break up occurs more quickly in our case since MI gain depends on pump peak power .
4.2 Comparison of SCs in PCF 1, 2 and 3
Figure 6(a) represents the phase matching curves (Eq. 2) for the three optical fibers under test (PCF1, 2 and 3, see Fig. 1). They are plotted up to the second ZDWs of each fiber because as we explained above, the SSFS is stopped just below this ZDW. We observe that the center wavelengths of DW1s are all located around 900 nm while for DW2s their central wavelengths varies from 1300 nm (PCF1) to 1750 nm (PCF3). From dispersion curves on Fig. 1(a), we notice that the most red shifted DW2 is achieved in the fiber for which the second ZDW is also the most detuned toward longer wavelengths (PCF3), or vice versa (PCF1). As it was demonstrated in previous subsection, the SC extension is bounded by the central wavelengths of each DWs.
It can be deduced from Fig. 6(a) that the broadest SC will be achieved in PCF3 while the shortest will be generated in PCF1. On Fig. 6(b) we superimposed the PSD of the SC generated in each fiber and we verified that the broadest SC is generated in PCF3 (1000–1700 nm at -20 dB) while the thinner is achieved in PCF1 (900–1300 nm at -20 dB). In PCF2 we obtained an intermediate case (1000–1500 nm at -20 dB). In all these cases, the PSDs of these SCs is at least 10 dBm/nm, which is comparable or better than the best published results [10,10,14,15,16]. To generalize, with this setup SCs are quite flat and most of the launched pump power is transferred into the SC because the OH absorption peak effect is negligible in such short fibers.
4.3 Sensitivity to initial pump phase conditions
Results presented above have been obtained for a single random drawn of the initial spectral phase of the pump. As it was pointed out in Refs. [10 and 23], the initial conditions strongly influences the output characteristics of the SC. This effect can not be verified experimentally since the integration time of optical spectrum analyzers is around 1 ms (to compare with numerical time window of the order of the nanosecond, in our case). For this reason in these works an average is performed over tens of identical simulations with different random initial phase to match with experimental conditions. Hence, a fairly good agreement is achieved between numerical and experimental results. In our case, it is interesting to note that we obtained very similar output spectra by performing successive simulations (see Figs. 7, for SC generated in PCF2 for example). We believe that the cancellation of the SSFS effect is responsible for the reproducibility of our simulations, because once the balance between the spectral recoil effect and the Raman frequency shift is reached, the frequency of the soliton remains constant by further propagating in the fiber . As a consequence, the frequency shift of the DWs associated with the frequency shift of the soliton [Fig. 4 and Fig. 5(a)] is also stopped. Of course, it may remain nonlinear interactions between all the waves included in the SC, but this can not strongly modify the SC shape. In our simulations, the fiber length (20 m) is always longer enough such that the SSFS cancellation (at about 10 m) is achieved well before the fiber end. Therefore, we understood that in our case the output spectra are nearly insensitive to initial pump phase conditions [Fig. 7(e)].
However, for lengths of fibers shorter than 10 m (i.e. before the cancellation of the SSFS effect), we actually verified that the SC spectra depend on initial pump phase conditions, in agreement with Refs. [10,11]. Basically, when the SSFS is not cancelled, SSFS rises up at different fiber lengths and the output SC spectra are then different from one drawn of the initial pump phase conditions to another [10,11].
In this work, we have numerically investigated continuous-wave supercontinuum generation in photonic crystal fibers characterized by two zero-dispersion wavelengths. We have shown that bounded and quite flat SCs can be readily obtained when the CW pump is located in between the two ZDWs, as with a pulsed pump. Tailoring of the SC extension is achieved by a suitable tuning of the fiber dispersion. As an example, we have demonstrated that highpower 1050–1600 nm SC generation can be achieved by use of a conventional CW Yb fiber lasers and PCFs of tens of meters only. Consequently, the OH absorption drawback is circumvented and most of the launched pump power is converted into the SC, leading to high power spectral density (>10dB/nm). It is worth noting that realistic parameters were used in this numerical study and therefore it constitutes a step toward an experimental demonstration of such nearly flat and powerful SCs.
The authors would like to thank Dr. Nicolas Jolly and Dr. Pierre Suret for helpful discussions, the Institut de Recherche sur les Composants logiciels et materials pour l’Information et la Communication Avancée (IRCICA), as well as Pole de Recherche en MIcro-Optique (PREMIO) Belgian-French Interreg III project for their support. We also thank the reviewer for its helpful comments.
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28. Note that Fig. 3(e) represents the evolution of the SC spectrum versus the fiber length in logarithm scale. In order to obtain a clear figure, it is plotted from smoothed SC spectra by using the method described in Ref. . One example of this smoothing method is represented in green on Fig. 3(d).