Open Access
Add to BibSonomy
Abstract
We report on the multidimensional characterization of femtosecond pulse nonlinear dynamics in a tellurite glass graded-index multimode fiber. We observed novel multimode dynamics of a quasi-periodic pulse breathing which manifests as a recurrent spectral and temporal compression and elongation enabled by an input power change. This effect can be assigned to the power dependent modification of the distribution of excited modes, which in turn modifies the efficiency of involved nonlinear effects. Our results provide indirect evidence of periodic nonlinear mode coupling occurring in graded-index multimode fibers thanks to the modal four-wave-mixing phase-matched via Kerr-induced dynamic index grating.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Going from one-dimensional nonlinear dynamics in single mode fibers (SMFs) to three-dimensional (3-D) propagation in multimode fibers (MMFs) a plethora of new interesting nonlinear phenomena could be discovered, as justified by the numerous research articles published over the last years [1–13]. Indeed, recent experimental and theoretical studies have revealed that interactions among a multitude of guided modes can lead to complex spatiotemporal dynamics of beam propagation beyond those offered by SMFs. Most of these works have used kW-level sub-nanosecond or picosecond pulses at the 1 µm central wavelength to pump standard commercially available silica graded-index (GRIN) MMFs. In this regime, the effect of dispersion on the pulse propagation can be neglected facilitating the discovery of new multidimensional dynamics. Among others, spectral reshaping, including geometric parametric instability (GPI), multimode Four-Wave-Mixing (FWM), or multimode supercontinuum generation (SCG) has been demonstrated [1,7,12,13]. In the spatial domain, an intriguing effect has been observed called the Kerr beam self-cleaning (KBSC), where an initially speckled beam becomes quasi single-mode with a weak multimode background when the pump power grows larger [2,3]. KBSC has been interpreted as a two-step physical process, where in the first step, the nonlinear mode coupling occurs, which becomes non-reciprocal in the second step as a result of the further appearance of Self-Phase-Modulation (SPM). It has also been shown that KBSC is accompanied by a significant recurrent temporal reshaping of the pulse envelope, measured at the beam center, which is composed of a series of temporal broadenings, followed by the formation of a dip at the pulse center and then up to fourfold pulse shortening. Such temporal reshaping has been explained in the framework of the nonlinear mode coupling between the fundamental mode and the higher-order-modes (HOMs) [2,14]. However, differently from the picosecond or sub-nanosecond regimes, limited studies have been conducted so far with 1 µm femtosecond pulses, which enable a significant contribution of chromatic dispersion to the pulse propagation dynamics. Here, KBSC and the corresponding progressive pulse compression have been reported for the pump power increase [11].
Only recently, the multimode nonlinear beam propagation has also been investigated in lead-bismuth-gallate GRIN MMFs. KBSC and two-octave-spanning supercontinuum generation involving GPI, SPM, and Stimulated Raman Scattering (SRS) has been demonstrated when pumping with femtosecond pulses [15]. Note, however, that the spatiotemporal dynamics in other than silica MMFs remains comparatively unexplored.
In this work, we fill this gap by further studying complex multimode interactions occurring in non-silica MMFs and in the femtosecond pulse regime. We pump an in-house developed GRIN MMF made of highly nonlinear tellurite glasses with a 300 fs pump centered at a wavelength of 1030 nm. We provide a deeper insight into a multidimensional analysis of femtosecond nonlinear dynamics in GRIN MMFs beyond what has been reported so far. We carried out a detailed experimental characterization of the output pulse in the spatial, temporal, and spectral domains. Unlike the previous studies, we exploit the second harmonic generation based cross-correlation frequency resolved optical gating (SHG-XFROG) to perform a spectro-temporal characterization of the multimode output beam. Results of this characterization allow us to demonstrate indirect evidence of nonlinear mode coupling manifesting through the new spectro-temporal quasi periodic pulse reshaping.
2. Tellurite GRIN MMF and its properties
We developed an MMF with a parabolic index profile thanks to combining two types of tellurite glasses. These two tellurite glasses, labeled TWPN/I/6 and TWZN-22, were fabricated in-house such that the refractive index of TWZN-22 was lower than that of TWPN/I/6. The difference in the refractive indices of the two glasses is essential to obtain an effective parabolic index profile in the fiber core. These glasses also have a high nonlinear refractive index of 5.1 × 10−19 m2/W at 1064 nm. The GRIN MMF was fabricated using the stack and draw technique, where a combination of 10267 rods of the two glasses was stacked according to a designed structure arrangement and drawn at a fiber drawing tower. The developed GRIN MMF has a core diameter of 60.4 µm as illustrated by an SEM image shown in Fig. 1(left). Note that minor structural imperfections observable in the cladding area of the fiber sample in the image, i.e., chips at the outer circumference, do not impact the propagation dynamics in general, as long as the core area is intact. The starting diameter of each of the individual rods was about 0.5 mm, which was next thinned down during the drawing procedure to below 200 nm, satisfying the λ/5 sub-wavelength dimension condition.
Fig. 1. (left) Scanning electron microscope images of the in-house developed GRIN tellurite fiber. (right) Measured dispersion profile of the fabricated tellurite GRIN fiber.
The chromatic dispersion of the fiber was measured using a balanced Mach-Zehnder interferometer pumped with a compact supercontinuum source (Leukos) spanning over the near-infrared spectral range. As can be inferred from the measured data in Fig. 1 (right), the chromatic dispersion remains steeply normal in the wavelength range covered by the light source used in the experiments, and it corresponds to the material dispersion of the used tellurite glasses [16,17].
3. Experimental setup
Figure 2 shows a schematic depiction of the experimental setup to study the nonlinear dynamics in femtosecond-pumped GRIN MMF. We pumped the tellurite GRIN MMF with a 1030 nm laser delivering 300 fs pulses at a repetition rate of 100 kHz (Jasper 20, Fluence). A half-wave plate (HWP) was used to control the polarization of the input pulses in order to obtain a phase-matched signal at the XFROG. 10% of the input beam was reflected to a delay line of the XFROG for being used as a gate pulse against the output pulse from the fiber. The rest of the pulse energy was focused with a 19.7 µm in diameter spot on the input face of the 23.5 cm long tellurite GRIN MMF by using a plano-convex lens with a focal length of f = 50 mm. Such injection conditions allowed us to limit the number of excited modes owing to an estimated large number of 2167 modes that can be guided in our fiber. We measured the average input power after the HWP and before the input lens. The output beam from the GRIN fiber was analyzed for its spectral, spatial, and temporal properties. The power at the output was measured to be 20% of that at the input, considering the fiber loss and the coupling efficiency. The attenuation of the fiber was roughly equal to 6 dB/m; therefore, with the 23.5 cm fiber length, the input/output power ratio would correspond to around 28% coupling efficiency to the GRIN MMF. We note that owing to the abundance of the pump power from the driving laser, we did not optimize the coupling optics for maximum coupling efficiency. To study the spatial properties of the output near-field beam, we used a CMOS beam profiler (M2MS-BC207VIS, Thorlabs) and neutral density filters (NDFs) to avoid camera saturation. The spectral properties of the output light were recorded via an optical spectrum analyzer (OSA, HP). The temporal properties were studied instead by using an in-house built cross-correlation frequency-resolved optical gating (XFROG) system, where a gate pulse with the delay line was used to measure against the stretched pulse coming from the GRIN MMF. Both beams were focused on a 0.05 mm thick β-barium borate crystal (BBO) crystal cut for sum frequency generation, being next recorded using a high resolution spectrometer (Ocean HDX, 200-1100 nm sensitivity, 0.7 nm resolution).
Fig. 2. Experimental setup to study pulse characteristics on propagating in a GRIN MMFs with femtosecond pumping. L1,L2,L3: Lenses; BS: beam splitter.
4. Spectro-spatio-temporal (SST) analysis
We started our investigation by measuring the spectral dynamics of multimode beam propagation. To couple the light to OSA, we used a standard silica MMF. During the measurements we increased the input average power of the pump up to 32 mW. This corresponds to 1.06 MW of pump peak power, which is below the critical power of self-focusing, which in our experiment was estimated to be ${P_{crit}} = 1.86\frac{{{\lambda ^2}}}{{4\pi {n_0}{n_2}}}$=1.8 MW [18]. Note that because of large normal dispersion of our fiber (see Fig. 1(right)), the initial 1030 nm femtosecond pulse stretches to the picosecond regime, resulting in further lowering of the pump peak power. The spectral evolution at the output of tellurite MMFs as a function of average input power is shown in Fig. 3(a). Whereas, Fig. 3(b) displays the examples of the corresponding output spectra at selected pump powers. As we can see, after initial spectral broadening induced by SPM [19], when the pump power grows larger, the spectrum becomes asymmetric with more spectral components at the redshifted spectral side. At the lowest detectable average power of 6 mW, the spectrum spans from 970 nm up to 1116 nm. As more power is pumped into the fiber, the redshifted part continues to broaden up to 1210 nm when the power reaches 29 mW. Above 29 mW, no significant spectral broadening is further observed.
Fig. 3. (a) Pulse spectra measured at the output of the tellurite GRIN MMF for the entire range of the input pulse average powers. Spectra are shown in linear scale. (b) The pulse spectra measured at the fiber output for selected input average powers showing the spectral-domain quasi-periodic reshaping with the monotonically increasing pump power.
It has been previously reported that the tellurite glass TWPN/I/6, which is used to form a parabolic index profile of the fiber core, has a Raman contribution of fR = 0.2 and gives the first order Raman gain peak at 20.4 THz shift from the pump wavelength [16]. In our case, this means that the first SRS sideband should appear at 1107 nm, which agrees well with our experiments. Therefore, SRS is likely to be involved in the generation of the redshifted part of the broadened spectrum. Interestingly, in Fig. 3(b), we can also observe the presence of a recurrent spectral reshaping manifesting as an increase and a decrease of the frequency conversion efficiency responsible for the generation of the redshifted part of the spectrum. To verify the presence of this new reshaping, we calculate the first three statistical moments of the measured spectra (mean wavelength, variance, and skewness) and then the Fourier transform of their dependence on input power. In all the cases, we found a component with a period of around 4.3 mW (except the case for the 4th moment: Kurtosis). Similar reshaping behavior has already been demonstrated in silica GRIN MMFs, where spectral broadening via SRS was decreased when modal distribution at the fiber input was in favor of fundamental or low-order modes (LOMs). In contrast, SRS was diminished when HOMs were dominant [20]. Shaping the SRS-induced spectra by changing modal distribution was also demonstrated in Ref. [21]. In the case of our tellurite GRIN MMF, this reshaping effect attains a sort of saturation for the high powers, i.e., the spectral broadening ceases following a proportional increasing trend with the input power. This “saturation” effect requires clarification which should be addressed in the future. Importantly, we observe the recurrent spectral reshaping only by increasing the pump power while keeping fixed the initial modal excitation conditions. Thus, a tentative hypothesis explaining this novel breathing dynamics is based on the nonlinear mode coupling similar to those already described in silica GRIN MMFs that provides a first step responsible for KBSC [2,22]. Here, we also use a GRIN MMF. However, owing to its large number of modes and the limited pump power it is reasonable to think that in our case, only the first step towards KBSC can be achieved. Indeed, we have not observed the typical signature of KBSC that is a formation of the bell-shaped spatial pattern in the center of the fiber core surrounded by the multimode background. It is known that in MMFs with a parabolic index profile, the multimode beam experiences a periodic oscillation along its propagating in the fiber due to the equal spacing of the modal wave numbers. In the nonlinear regime, this effect of spatial self-imaging is translated by the Kerr effect in a periodic longitudinal modulation of the refractive index. This nonlinear index grating allows, in turn to phase-matched modal FWM and thus to obtain nonlinear mode coupling. The experimental evidence of such an energy exchange among the guided modes, particularly between the fundamental and HOMs has recently been demonstrated using a novel time-resolved mapping technique [23].
Let us remind that the physical conditions of our experiment are distinct from those of previous femtosecond pulse-driven multimode propagation in GRIN silica fibers [11] in a way that the number of modes excited in our fiber is significantly larger despite the fact that the nonlinear response of tellurite glass is much stronger than silica. This can be seen in Fig. 4(a) showing typical mode field images at the output of GRIN tellurite fiber for different incident pump powers. Note that the excitation condition was optimized in terms of translation, tilt, and focus to obtain the most tightly confined mode structure guided by the fiber during the following XFROG measurements. As mentioned above, no occurrence of Kerr self-cleaning was observed; instead, we noted that after the initial growth in the diameter up to 52 µm (at FWHMI) at 10 mW, the beam was propagating without any significant change in the diameter in the range from 10 mW to 32 mW of input average power.
Fig. 4. a) Near-field images of mode field profiles recorded at the output of the tellurite GRIN MMF, b) the spectrogram retrieved from the measured input laser pulse (a FROG signal), c) retrieved input pulse spectrum and time shape. Note: the side plots of the spectrum and time profile of the laser pulses are not simple projections (sums) over respective axes but instead are profiles retrieved from the FROG data using the generalized projections computer algorithm.
Next, we used the SHG-FROG and SHG-XFROG to perform spectro-temporal characterization of the spectrally broadened multimode beam. Results of input pulse characterization using FROG are shown in Fig. 4(b) and Fig. 4(c). In contrast, the XFROG data on the pulses at the fiber output, recorded as a function of the input power, are presented in Fig. 5. The pulse durations shown in boxes at each of the subplots were obtained as full width at half maximum (FWHM) of the intensity of the pulse shapes retrieved from the measured XFROG data by using the generalized projections algorithm following Refs. [24,25]. Background and noise corrections were carried out for the raw data before retrieval. The typical retrieval error of the full-field information from the XFROG data was estimated to be at the level of 2 × 10−2. Note that this error level was achieved for the set of input conditions resulting in the tightly focused mode structure, examples of which are shown in Fig. 4(a). The integration time of the spectrometer was set to 300 ms to eliminate the impact of any random intensity fluctuations on the recorded data. As mentioned above, in our experiment, the femtosecond pulse experiences a strong positive chirp up to several picoseconds during the propagation in the tellurite MMFs, which now is seen as a tilt in the spectrograms. Note that we can also see fine structures in both spectral and temporal traces in the spectrograms. As it has already been reported in SMFs, such a structuring of XFROG measurements is due to the non-ideal shape of the input femtosecond pulses [26], namely the existence of weak pre- and post-pulses, which are common in real femtosecond laser systems. Figure 4(b) displays the frequency-resolved optical gating (FROG) measurement of the input laser pulse used in our experiment. Note that this is not cross-correlation but a second-order auto-correlation map of the input pulse; thus, it is symmetric around the time axis t = 0 (corresponding to the center of the pulse). The pulse is 300 fs long and has a 10 nm spectral width at FWHMI. Indeed, as expected, the pulses feature a complex structure with an additional two low intense pre-pulses at around 250 fs and 150 fs ahead of the main pulse, which can be observed in the subplot of Fig. 4(c). These pre-pulses are related to uncompensated 3rd-order dispersion in the pump laser system. Again, note that this, in turn, is a pulse shape retrieved from the FROG data; therefore, it does not exhibit the symmetry around t = 0, which is specific for the case of second order autocorrelation. The retrieval error of this FROG measurement was estimated to be 3 × 10−3. Now, if we look closer at the spectro-temporal results obtained as a function of pump power, we can observe the same recurrent spectral behavior as discussed above. The spectra of the retrieved pulses agree well with the spectra previously measured with OSA; This was checked for every incident pump power level and is quite remarkable, considering the complexity multimode complexity of the output pulses. A comparison between the OSA spectra (orange traces) and the retrieved spectra (blue traces) is shown in Fig. 5 for two pump powers of 9 mW and 10 mW. Interestingly, Fig. 5 also shows that such a recurrent behavior in the spectral domain can be transferred to the temporal domain. The enhancement of the redshifted part of the spectral broadening corresponds to the temporal pulse elongation up to around 10-12 ps owing to the large normal dispersion of tellurite MMF. The further diminution of the redshifted part of the spectrum obtained for the increased power leads instead to the pulse shortening down to the value of 6.5 ps on average. This breathing spectro-temporal dynamics repeats in a quasi-periodic manner when the pump power continues to grow. To let the readers better appreciate this phenomenon, in Fig. 6, we present a quasi-periodic pulse reshaping as a function of input average power, as deduced from the retrieved XFROG data, in the form of a scatter plot. We also calculate the variance, skewness, and kurtosis of the temporal profile and perform Fourier transform of their dependence on input power. Again, in all these signals, as well as in the behavior of temporal FWHM we found a significant component with a period of around 4.3 mW. This observation provides another confirmation that beam propagation in MMFs is inherently multidimensional and cannot be analyzed separately.
Fig. 5. XFROG spectrograms of the output pulse from the GRIN fiber showing the evolution of the pulse in spectral and time domain with increase in average input power. The green labels of the pulse durations indicate pulses for which compression was observed compared to the preceding pulse; for example, the pulse recorded for 11 mW of input power was compressed compared to the pulse recorded for 10 mW. The Y-axis of the spectrograms is scaled in the sum-frequency generation wavelengths recorded in XFROG and labeled λSFG; the retrieved pulse spectrum to the right of the spectrogram is scaled in the fundamental wavelengths of the fiber output pulses and labeled λ. Right-bottom box of each plot contains values of average input power and the FWHM duration of the output pulse retrieved from the XFROG data.
Fig. 6. Quasi-periodic temporal pulse reshaping as a function of input average powers. The shown pulse durations relate to the pulses at the fiber output and are intensity FWHM durations retrieved from the XFROG data shown in Fig. 5.
5. Conclusions
We investigated multimode nonlinear beam propagation in a GRIN MMF made of a highly nonlinear tellurite glass in a highly normally dispersed femtosecond pulse regime. A large number of more than 2000 guided modes and limited pump power hindered observation of the nonlinear Kerr beam self-cleaning in our MMF. However, we experimentally demonstrated new interesting dynamics of spectro-temporal pulse breathing observed with increasing pump power that we measured using the XFROG technique. We showed that the pulse could periodically change its spectral bandwidth and temporal duration by only changing the pump power. Our tentative explanation for this observed multimode phenomenon is built on the nonlinear mode coupling leading to a power-dependent modification of the modal distribution. The current study was strictly experimental, and the hypothesis explaining the observed quasi-periodic “breathing” of the pulses at the fiber output was formulated in reference to the measured data. It was theoretically shown that relatively large negative group-velocity dispersion could be generated by off-axis propagation in a graded-index material with Kerr nonlinearity [27]. This fact may also add to the observed pulse reshaping dynamics by modifying the modal power distribution in an analogy to the dispersion control methods based on the prism or grating pair setups. Although fascinating, this problem is out of the scope of the present work, leaving several open questions, such as, for instance, whether the periodicity of the observed pulse breathing or the maximum and minimum achievable pulse duration within a period could be further influenced by either the pulse characteristics or the fiber parameters. We believe our findings provide additional building blocks for a better understanding of the complexity of multimode beam propagation.
Funding
Fundacja na rzecz Nauki Polskiej (TEAM NET POIR.04.04.00-00-1644/18, TEAM NET POIR.04.04.00-00-16ED/18).
Disclosures
Yuriy Stepanenko and Tomasz Kardaś are shareholders of Fluence Sp. z o. o., which built the Jasper ultrafast laser used in this work. Other authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of Geometric Parametric Instability Induced by the Periodic Spatial Self-Imaging of Multimode Waves,” Phys. Rev. Lett. 116(18), 183901 (2016). [CrossRef]
2. K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017). [CrossRef]
3. L. G. Wright, Z. Liu, D. A. Nolan, M. J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016). [CrossRef]
4. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015). [CrossRef]
5. P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83(3), 033838 (2011). [CrossRef]
6. N. Berti, K. Baudin, A. Fusaro, G. Millot, A. Picozzi, and J. Garnier, “Interplay of thermalization and strong disorder: Wave turbulence theory, numerical simulations, and experiments in multimode optical fibers,” Phys. Rev. Lett. 129(6), 063901 (2022). [CrossRef]
7. R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017). [CrossRef]
8. K. Krupa, T. M. Kardaś, and Y. Stepanenko, “Real-Time Observation of Double-Hopf Bifurcation in an Ultrafast All-PM Fiber Laser,” Laser Photonics Rev. 16, 2100646 (2022). [CrossRef]
9. L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode nonlinear fiber optics: Massively parallel numerical solver, tutorial, and outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 5100516 (2018). [CrossRef]
10. D. Majus, G. Tamošauskas, I. Gražulevičiūtė, N. Garejev, A. Lotti, A. Couairon, D. Faccio, and A. Dubietis, “Nature of Spatiotemporal Light Bullets in Bulk Kerr Media,” Phys. Rev. Lett. 112(19), 193901 (2014). [CrossRef]
11. Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016). [CrossRef]
12. K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guenard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, and S. Wabnitz, “Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber,” Opt. Lett. 41(24), 5785 (2016). [CrossRef]
13. G. Lopez-Galmiche, Z. Sanjabi Eznaveh, M. A. Eftekhar, J. Antonio Lopez, L. G. Wright, F. Wise, D. Christodoulides, and R. Amezcua Correa, “Visible supercontinuum generation in a graded index multimode fiber pumped at 1064nm,” Opt. Lett. 41(11), 2553 (2016). [CrossRef]
14. K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A 97, 1–6 (2018). [CrossRef]
15. Z. Eslami, L. Salmela, A. Filipkowski, D. Pysz, M. Klimczak, R. Buczyński, J. M. Dudley, and G. Genty, “Two octave supercontinuum generation in a non-silica graded-index multimode fiber,” Nat. Comm. 13, 2126 (2022). [CrossRef]
16. J. Cimek, N. Liaros, S. Couris, R. Stępień, M. Klimczak, and R. Buczyński, “Experimental investigation of the nonlinear refractive index of various soft glasses dedicated for development of nonlinear photonic crystal fibers,” Opt. Mater. Express 7(10), 3471–3483 (2017). [CrossRef]
17. R. Stepień, R. Buczyński, D. Pysz, I. Kujawa, A. Filipkowski, M. Mirkowska, and R. Diduszko, “Development of thermally stable tellurite glasses designed for fabrication of microstructured optical fibers,” J. Non-Cryst. Solids 357(3), 873–883 (2011). [CrossRef]
18. P. Horak and F. Poletti, “Multimode Nonlinear Fibre Optics: Theory and Applications,” in Recent Progress in Optical Fiber ResearchM. Yasin, S. W. Harun, and H. Arof, eds. (InTech, 2012).
19. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2004).
20. Y. Leventoux, G. Granger, T. Mansuryan, M. Fabert, K. Krupa, A. Tonello, S. Wabnitz, V. Couderc, and S. Février, “0.75-6 µm supercontinuum generation using spatiotemporal nonlinear dynamics in graded index multimode fiber,” in Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (Optica Publishing Group, 2021), paper cj_3_2.
21. O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018). [CrossRef]
22. E. V. Podivilov, D. S. Kharenko, V. A. Gonta, K. Krupa, O. S. Sidelnikov, S. Turitsyn, M. P. Fedoruk, S. A. Babin, and S. Wabnitz, “Hydrodynamic 2D Turbulence and Spatial Beam Condensation in Multimode Optical Fibers,” Phys. Rev. Lett. 122(10), 103902 (2019). [CrossRef]
23. Y. Leventoux, G. Granger, K. Krupa, A. Tonello, G. Millot, M. Ferraro, F. Mangini, M. Zitelli, S. Wabnitz, S. Février, and V. Couderc, “3D time-domain beam mapping for studying nonlinear dynamics in multimode optical fibers,” Opt. Lett. 46(1), 66 (2021). [CrossRef]
24. K. W. DeLong and R. Trebino, “Improved ultrashort pulse-retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11(9), 2429–2437 (1994). [CrossRef]
25. K. W. DeLong, B. Kohler, K. Wilson, D. N. Fittinghoff, and R. Trebino, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19(24), 2152–2154 (1994). [CrossRef]
26. A. Rampur, D.-M. Spangenberg, G. Stępniewski, D. Dobrakowski, K. Tarnowski, K. Stefańska, A. Paździor, P. Mergo, T. Martynkien, T. Feurer, M. Klimczak, and A. M. Heidt, “Temporal fine structure of all-normal dispersion fiber supercontinuum pulses caused by non-ideal pump pulse shapes,” Opt. Express 28(11), 16579–16593 (2020). [CrossRef]
27. Chih-Hung Chien, Shinn-Sheng Yu, Linchieh Lai, and Juhpyng Wang, “Off-axis pulse propagation in graded-index materials with Kerr nonlearity – a variational approach,” Opt. Commun. 128(1-3), 145–157 (1996). [CrossRef]
