We show that the infrared edge of supercontinua generated in solid core photonic bandgap fibers is characterized by a very different temporal behavior than the one obtained in standard fibers. In particular, pulse-to-pulse spectral power fluctuations are significantly reduced near the bandgap edge, and the statistical distribution is quasi-gaussian. The spectral dynamics of this process and statistical properties are investigated experimentally and confirmed by numerical simulations. The reduction of power fluctuations originates from the cancellation of the soliton self-frequency shift near the bandgap edge.
© 2010 Optical Society of America
Recent research in the field of supercontinuum (SC) generation in optical fibers has pointed out the key role of noise in the temporal stability of such sources [1, 2]. In the long-pulse regime, the spectral broadening originates from the modulation instability (MI) process which is seeded by noise and the long-wavelength SC edge is characterized by statistically-rare optical rogue waves , originating from soliton dynamics [3–6]. Consequently, the SC is characterized by a low coherence, and pulse-to-pulse fluctuations of the spectral power at the long-wavelength SC edge are very high [3,7]. This is a major issue for a number of applications requiring dynamics measurements. A straightforward way to control these instabilities is to seed the SC with a coherent signal rather than leaving it self-starting from noise [4, 8–10]. Although this has been demonstrated experimentally , the setup requires a seed laser whose properties (in terms of power and wavelength) must be very precisely adjusted for an efficient MI control. This makes this solution inadequate for a large number of applications and hardly compatible with commercialization aspects.
In this paper, we demonstrate that spectral power fluctuations usually characterizing the long-wavelength edge of SC sources can be reduced by the use of an optimized solid-core photonic bandgap (PBG) fiber. Although it has already been demonstrated that such fibers allow a spectral shaping of SC sources [11, 12], we show here that they also allow a control of their temporal properties. We demonstrate a significant reduction of statistically rare rogue events at the long-wavelength SC edge which results in a greatly improved pulse-to-pulse stability. This phenomenon is interpreted in terms of soliton self-frequency shift (SSFS) suppression thanks to the specific guiding properties of the PBG fiber. These explanations are supported by numerical simulations. The remaining of the paper is organized as follows. In section 2, we provide the experimental evidence for the reduction of pulse-to-pulse fluctuations at the SC long-wavelength edge in solid-core PBG fibers as compared to standard fibers. The first part of section 3 is devoted to experimental results about the dynamics of this phenomenon in solid-core PBG fibers, while the second part focuses on their confirmation by numerical simulations. In section 4, we discuss our results and provide a physical interpretation based on numerical spectrograms, before summarizing our study.
2. Experimental evidence
The solid-core PBG fiber used in the experiments is based on the same design than those previously employed for efficient SC generation . It is characterized by a double-periodicity cladding structure incorporating germanium-doped inclusions and air holes, depicted respectively in light grey and black regions in the inset of Fig. 1(c). This ensures light to be guided by a PBG mechanism while keeping relatively low confinement losses in the first-order PBG of interest here [11,13]. Figure 1(a) shows the computed group-velocity dispersion (GVD) and nonlinear (NL) coefficient curves (deduced from the computed effective area) calculated with a finite-elements method. As previously reported , the GVD slightly increases from its zero value (located around 920 nm) to about 1580 nm where the increase suddenly becomes much more important because of the vicinity of the PBG edge. In this region (delimited by the vertical dotted line), the GVD slope β3 reaches extreme values of about 10−37 s3/m (i.e. about three orders of magnitude larger than in standard fibers), and the NL coefficient drastically drops down to less than 1 W−1.km−1. These strong spectral dependencies of guiding properties are typical of the PBG guidance in fibers [11, 14].
SC generation experiments were performed using a microchip laser at 1064 nm delivering 600 ps pulses with a peak power of a few kilowatts, at a repetition rate of 7 kHz. Figure 1(b) shows the spectrum measured for a launched peak power of 3 kW and a fiber length of 7 m. In these conditions, the infrared spectral broadening is limited to about 1580 nm by the PBG edge as demonstrated in Ref. . The peak centered around 1555 nm corresponds to an accumulation of solitons for which the SSFS effect has been cancelled [11, 15]. Our aim here is to quantify pulse-to-pulse fluctuations of the SC spectral power as a function of wavelength in the solid-core PBG fiber, and to compare it with a SC generated in a standard photonic crystal fiber (PCF). We thus fabricated an air/silica PCF whose linear properties were designed to produce a SC with approximately the same spectral extent. Its computed GVD and NL coefficient are plotted in Fig. 1(d). In order to generate a spectral broadening mainly on the long-wavelength side of the pump, the zero-GVD wavelength is located at 950 nm, i.e. relatively far from the pump wavelength to minimize the dispersive wave generation on the short-wavelength edge of the SC. The NL coefficient is 25 W−1.km−1 at 1064 nm, which is much higher than the one of the solid-core PBG fiber. The pump peak power launched in the standard PCF was adjusted so that the spectral extent of the SC (calculated for a 20 dB power drop from the spectral power at 1200 nm) was similar to the one produced in the solid-core PBG fiber. The pump peak power was thus 1.5 kW in the standard PCF (against 3 kW in the PBG fiber). Although SC extensions are very similar in both fibers, their spectrum shape at long wavelengths is very different.
Pulse-to-pulse power fluctuations of the SC generated in both fibers were then studied experimentally. To do this, the SC output was collimated and spectrally filtered with 10 nm bandpass filters. Pulse-to-pulse fluctuations in these 10 nm-wide spectral regions were measured using the method firstly proposed in Ref. . The energy of spectrally filtered pulses is proportional to their peak power, so that rogue events and their associated characteristics can be captured through a simplified measurement of shot-to-shot pulse energy . This was done in our experiments using a photodiode and an analog oscilloscope, with the same photodiode maximum signal amplitude and trigger level. Note that in our experiments, this setup provides a measurement of the average energy of each pulse, but it does not allow to quantify noise-sensitive instabilities related to the actual temporal coherence of the SC, as explained in Ref.  for instance. The percentage of shot-to-shot variations, σ was evaluated using σ = 100× (Vmax – Vmin)/(Vmax + Vmin) where Vmax and Vmin are respectively the maximum and minimum photodiode signal amplitude measured at least for 10 of the 10,000 recorded pulses.
Full squares in Fig. 1(b) and (e) correspond to pulse-to-pulse variations measured every 50 nm over the whole SC spectrum in the solid-core PBG fiber and in the standard PCF, respectively. For both fibers, it slightly increases from 10 % at 1100 nm to about 25 % at 1400 nm. For higher wavelengths, the temporal behavior becomes drastically different : pulse-to-pulse fluctuations significantly increase in the standard PCF to reach σ values of about 90 % at 1550 nm, while in the solid-core PBG fiber, σ values remain lower than 40 % until 1500 nm, and significantly drop down to 10 % at 1550 nm. Note that increasing the launched pump peak power in the standard PCF would result in a larger spectral broadening towards the infrared, and would reduce shot-to-shot fluctuations at the fixed wavelength of 1550 nm. We arbitrarily chose to compare both fibers for a fixed spectral width, which required a lower pump peak power in the standard PCF due to its higher NL coefficient, so that the NL phase shift γPL was comparable in both fibers. Besides, the statistical power distributions of the pulses at 1550 nm have very different features, as shown in Figs. 1(c) and (f). These figures show histograms obtained over 10,000 pulses at 1550 nm in the solid-core PBG fiber and in the standard PCF, respectively. In the standard PCF, the statistical distribution is highly asymmetric and presents a long-tail profile, which means that the most powerful events have a higher probability of apparition than in a gaussian distribution. In fact, this corresponds to the spectral region of instabilities identified as so-called optical rogue waves . On the contrary, the statistical distribution measured at 1550 nm in the solid-core PBG fiber presents a quasi-gaussian shape, which is the signature of good SC pulse-to-pulse stability.
These experiments show that, besides offering the possibility of controlling the SC spectral extent [11, 12], solid-core PBG fibers allow SC fluctuations to be drastically reduced at the long-wavelength edge of the spectrum, as compared to standard PCFs.
3. Dynamics of the SC stabilization in solid-core PBG fibers
3.1. Experimental investigation
Following the experimental evidence of the reduction of pulse-to-pulse fluctuations in the solid-core PBG fiber, we experimentally studied the dynamics of this phenomenon as a function of fiber length for a fixed peak pump power of 3 kW. A cutback experiment was performed to record the spectral evolution of the SC, as well as pulse-to-pulse fluctuations across the spectrum every 0.5 m. First of all, Fig. 2(a) shows the evolution of the spectral broadening along fiber length. For propagation distances longer than 3 m, the spectrum is limited by the long-wavelength PBG edge located at 1580 nm, as previously reported in Ref. . Corresponding spectra measured at 2.5 m, 3 m and 7 m are reported in Fig. 2(b). Although the spectral extent is comparable in each case, the shape of the spectrum is slightly different. Indeed, while the long-wavelength edge is relatively smooth for a fiber length of 2.5 m, it becomes much sharper after 3 m, where the cancellation of the SSFS imposes a saturation of the spectral broadening [11,15]. This phenomenon causes a soliton accumulation whose spectral signature is the peak located at 1555 nm that clearly appears in the spectrum recorded after 7 m.
Secondly, the spectral evolution of pulse-to-pulse fluctuations σ has been studied as a function of fiber length with the same method as described in the previous section. Figure 2(c) shows examples of pulse trains measured for each of the three investigated lengths after a 10 nm bandpass filter (centered at 1550 nm in this example), from which pulse-to-pulse fluctuations σ can be evaluated. The pulse amplitude has been normalized to its average value over the 10,000 measurements for this plot. This measurement of σ has also been performed every 50 nm all over the SC. Figure 2(d) shows that σ increases from about 10 % in the pump region to 40–50 % at 1500 nm whatever the fiber length. However, the observed behavior is radically different at 1550 nm. Indeed, pulse-to-pulse fluctuations at 1550 nm are as high as 80 % after 2.5 m, and they drop down to 40 % after 3 m and to 10 % after 7 m. Besides, the corresponding statistical distributions at 1550 nm have very different shapes, as attested by Fig. 2(e) which represents histograms calculated over 10,000 measurements for fiber lengths of 2.5 m, 3 m and 7 m (respectively bottom to top traces). The distribution is highly asymmetric after 2.5 m, and becomes more and more gaussian-like for longer propagation distances, i.e. once solitons reach the PBG edge (for fiber lengths larger than 3 m). These observations thus suggest that there is a correlation between the cancellation of the SSFS occurring near the PBG edge (leading to a saturation of the SC spectrum for propagation distances longer than 3 m), and the enhanced pulse-to-pulse stability and statistical distributions in the same spectral region.
3.2. Confirmation with numerical simulations
In order to confirm these results, we performed numerical simulations by integrating the generalized nonlinear Schrödinger equation (GNLSE) in the frame of the model firstly proposed by J. Laegsgaard . Our aim at this point is to try to reproduce both the experimental spectral and temporal features. To do this, all available fiber characteristics and experimental conditions have been taken into account with no free parameter, with the exception of the pulse duration that has been reduced to 50 ps (against 600 ps in experiments). Indeed, shortening the pulse duration in simulations (while staying in the long-pulse regime though) allows to significantly reduce the computation time without significantly affecting the SC dynamics. The input pump peak power was 3 kW, and quantum noise was modeled by adding one photon per mode with random phase on each spectral discretization bin of the input field . In order to study the statistical behavior of the pulse train, we performed 200 simulations with random initial noise conditions.
Simulations results are displayed in Fig. 3, with the same organization as Fig. 2 for easy comparison. Figures 3(a) and (b), which show the dynamics of the SC formation, correspond to the averaged spectrum over 200 simulations shots. The agreement with experiments is excellent, and the typical spectral features discussed above about the spectrum shape at its long-wavelength edge are accurately reproduced. Temporal properties shown in Fig. 3(c) and (d) have been respectively calculated by using a numerical gaussian filter of 10 nm (FWHM) centered at 1562 nm, which corresponds to the maximum of the spectral power peak observed near the PBG edge [plot (c)], or using a sliding 10 nm filter across the spectrum [plot (d)]. Taking the inverse Fourier transform gives the temporal profile of the SC filtered at the corresponding wavelength, whose average power has been calculated. A sample of the modeled pulse train corresponding to 40 simulation shots is displayed in Fig. 3(c), and shows good agreement with the measured one represented in Fig. 2(c). Simulated shot-to-shot fluctuations σ are represented in Fig. 3(d) as a function of wavelength, and show qualitative agreement with experiments. In particular, the inset of Fig. 3(d) shows that the σ value at 1562 nm (depicted by squares) strongly decreases from about 100 % to 20 % for fiber lengths increasing from 2.5 m to 7 m, respectively. Note that pulse-to-pulse fluctuations do not further decrease for a longer fiber of 8 m. Corresponding statistical distributions calculated from all the 200 simulations are shown in Fig. 3(e), and also show a good agreement with experiments. Note that in simulations, the σ parameter is calculated with Vmax and Vmin corresponding respectively to the maximum and minimum power of the filtered SC for 1 occurrence of the 200 simulations shots. This explains the slightly higher fluctuations obtained in simulations as compared to experiments (in which they are defined for 10 occurrences over 10,000 measurements).
All these simulation results therefore allow to reproduce the main spectral features observed in the experiments (dynamics of the SC formation, typical spectral shape at the long-wavelength edge), but also the measured statistical features of the pulse train (pulse-to-pulse fluctuations as a function of wavelength and propagation distance, shape of the distribution). At this point however, numerical simulations of Fig. 3 do not allow to discuss on the physical mechanisms responsible for the enhanced pulse-to-pulse stability near the PBG edge.
As shown above, numerical simulations using the GNLSE accurately reproduce our experiments and can consequently be used to discuss the physical mechanisms causing the enhanced stability of the SC long-wavelength edge. To this end, the spectro-temporal representation has proved to be a powerful tool in studying the dynamics of SC generation . Figure 4 shows the evolution of a simulated spectrogram for fiber lengths of 2.5 m, 3 m and 7 m (from top to bottom). Figure 4(a) shows how MI initially generates solitons that subsequently red-shifts through Raman-induced SSFS. In such a long-pulse pumping regime, the presence of statistically rare temporal events in SC experiments can find two complementary physical origins in the recent literature. Firstly, it can be due to a single high peak power soliton generated from MI for particular initial noise conditions [3, 4]. Because of its higher peak power, it experiences a more efficient SSFS than other solitons, and consequently becomes statistically rare at highest wavelengths due to the low probability for these particular noise conditions to happen . Note that, in this case, the requirement for a soliton to be statistically rare at the long-wavelength SC edge, is only to experience a slightly more efficient SSFS than the other ones . Secondly, following these early interpretations, it has been suggested that rare and brief events can arise from the collision of two or more solitons travelling with different group-velocities [5,6,19,20] because of the convective nature of the system . Note that these two explanations are complementary for explaining long-tail statistical distributions usually observed at the SC long-wavelength edge .
In the case of the present experiments however, this dynamics is strongly affected by the SSFS cancellation occurring near the PBG edge [11, 15], as can be seen from Figs. 4(b) and (c). Firstly, these figures show that, as long as solitons are frequency-locked around 1562 nm, they all have very close group velocities at this location in the spectrum, as confirmed by the movie of Fig. 4. Consequently, once they have reached the PBG edge, they cannot collide anymore, which prevents the formation of brief spikes  and leads to an enhanced pulse-to-pulse stability at the SC edge. Secondly, since all solitons whose SSFS has been cancelled near the PBG edge have very close characteristics, as can be seen from the spectrogram movie of Fig. 4, power fluctuations are reduced in this spectral region as compared to the usual case in which only a few solitons with slightly higher peak power are rare at the long-wavelength edge [3, 4].
As a consequence, the reduction of pulse-to-pulse fluctuations at the long-wavelength edge as a function of propagation distance is intimately linked to the cancellation of the SFSS recently reported [11, 15]. It is thus in fact due to the specific linear properties inherent to solid-core PBG fibers, and in particular to the strong third-order dispersion near the PBG edge .
To summarize, we have experimentally demonstrated that large pulse-to-pulse fluctuations usually observed at the long-wavelength SC edge can be significantly reduced by using solid-core PBG fibers. By appropriately choosing the pump power and fiber length, solitons formed by MI have enough peak power to reach the PBG edge where their SSFS is cancelled , which leads to a saturated SC spectrum . In this case, the statistical pulse amplitude distribution at the SC edge has a quasi-gaussian shape, while it has a highly asymmetric shape with long tails when the SC spectrum is unsaturated. Numerical simulations confirming these experiments showed that this reduction of pulse-to-pulse fluctuations is in fact intimately related to the unique linear properties of solid-core PBG fibers which provide an efficient SSFS cancellation near the PBG edge. As long as solitons remain far from the PBG edge, pulse-to-pulse fluctuations are comparable in a solid-core PBG fiber and in a standard PCF, as shown by Figs. 1(b) and (e), but the SSFS cancellation occurring as the solitons approach the PBG edge (in solid-core PBG fibers) forces them to travel at the same group velocity, which leads to a significant reduction of pulse amplitude fluctuations.
We acknowledge Remi Habert for experimental assistance, and Karen Delplace for assistance in fiber fabrication. This work was partly supported by the Agence Nationale de la Recherche through the IMFINI ANR-09-BLAN-0065 project, by French Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and FEDER through the “Contrat de Projets Etat Région (CPER) 2007–2013” and the “Campus Intelligence Ambiante” (CIA).
References and links
1. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]
2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]
3. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature (London) 450, 1054–1058 (2007). [CrossRef]
5. A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009). [CrossRef] [PubMed]
6. G. Genty, C.M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J.M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010). [CrossRef]
7. F. Vanholsbeeck, S. Martin-Lopez, M. González-Herráez, and S. Coen, “The role of pump incoherence in continuous-wave supercontinuum generation,” Opt. Express 13, 6615–6625 (2005). [CrossRef] [PubMed]
8. G. Genty and J. M. Dudley, “Route to coherent supercontinuum generation in the long pulse regime,” IEEE J. Quantum Electron. 45, 1331–1335 (2009). [CrossRef]
9. G. Genty, J. M Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94, 187–194 (2009). [CrossRef]
11. A. Bétourné, A. Kudlinski, G. Bouwmans, O. Vanvincq, A. Mussot, and Y. Quiquempois, “Control of supercontinuum generation and soliton self-frequency shift in solid core photonic bandgap fibers,” Opt. Lett. 34, 3083–3085 (2009). [CrossRef] [PubMed]
12. B. Kibler, T. Martynkien, M. Szpulak, C. Finot, J. Fatome, J. Wojcik, W. Urbanczyk, and S. Wabnitz, “Nonlinear femtosecond pulse propagation in an all-solid photonic bandgap fiber,” Opt. Express 17, 10393–10398 (2009). [CrossRef] [PubMed]
13. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32, 1719–1721 (2007). [CrossRef] [PubMed]
14. V. Pureur, A. Bétourné, G. Bouwmans, L. Bigot, A. Kudlinski, K. Delplace, A. Le Rouge, Y. Quiquempois, and M. Douay, “Overview on solid core photonic bandgap fibers,” Fiber and Integrated Optics 28, 27–50 (2009). [CrossRef]
15. O. Vanvincq, A. Kudlinski, A. Bétourné, Y. Quiquempois, and G. Bouwmans, “Extreme deceleration of the soliton self-frequency shift by the third-order dispersion in solid-core photonic bandgap fibers,” J. Opt. Soc. Am. B 27, 2328–2335 (2010). [CrossRef]
16. C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett. 45, 217–219 (2009). [CrossRef]
17. H. Kubota, K. R. Tamura, and M. Nakazawa, “Analyses of coherence-maintained ultrashort optical pulse trains and supercontinuum generation in the presence of soliton-amplified spontaneous-emission interaction,” J. Opt. Soc. Am. B 16, 2223–2232 (1999). [CrossRef]
19. M. Erkintalo, G. Genty, and J.M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Special Topics 185, 135–144 (2010). [CrossRef]
20. K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010). [CrossRef]
21. M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Phys. Lett. A 374, 691–695 (2010). [CrossRef]