I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. CONCEPTIII. SAMPLE IMPLEMENTATIO...IV. RESULTSV. DISCUSSIONVI. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESSupercontinuum generation (SCG) using ultrashort optical pulses in optical fibers is a unique non-linear approach for (i) observing sophisticated non-linear processes in the time domain [e.g., soliton fission, dispersive wave (DW) formation] and (ii) efficiently transferring light into selected spectral domains. The key to this ultrafast nonlinear effect, in addition to third order nonlinearity, is the dispersion of the underlying fiber, which must be precisely managed to control dispersive and non-linear pulse propagation. One example is soliton fission and the associated emission of excess energy into DWs, in which the emission is essentially controlled by a dispersion-dominated phase-matching (PM) condition.11. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).SCG based on temporal solitons commonly involves the transfer of electromagnetic energy into well-defined spectral domains according to the PM condition. In many situations, the output spectra do not overlap with the spectral domain of interest. Moreover, in cases requiring broadband spectra, the spectrally separated generation of solitons and DWs leads to outputs that show strong variations in power density across the spectral domain of the generated radiation, i.e., are non-flat. Such variations appear to be disadvantageous in several cases, particularly in spectroscopic and astronomical calibration applications.2,32. R. A. Probst, T. Steinmetz, T. Wilken, G. Wong, H. Hundertmark, S. Stark, P. S. J. Russell, T. Hänsch, R. Holzwarth, and T. Udem, “Spectral flattening of supercontinua with a spatial light modulator,” Techniques and Instrumentation for Detection of Exoplanets VI (SPIE, 2013), Vol. 8864, pp. 706–713.3. A. Ravi, M. Beck, D. F. Phillips, A. Bartels, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “Visible-spanning flat supercontinuum for astronomical applications,” J. Lightwave Technol. 36, 5309–5315 (2018). https://doi.org/10.1109/jlt.2018.2872423 One approach to generate flat supercontinua relies on pumping in the normal dispersion region, which is typically associated with all-normal dispersion (ANDi) fibers.44. A. M. Heidt, “Pulse preserving flat-top supercontinuum generation in all-normal dispersion photonic crystal fibers,” J. Opt. Soc. Am. B 27, 550–559 (2010). https://doi.org/10.1364/josab.27.000550One solution to the aforementioned issues is axial dispersion control, which creates multiple PM opportunities for DWs. Successful implementations include gas filled fibers with axial pressure gradients,5,65. K. F. Mak, J. C. Travers, P. Hölzer, N. Y. Joly, and P. S. J. Russell, “Tunable vacuum-UV to visible ultrafast pulse source based on gas-filled Kagome-PCF,” Opt. Express 21, 10942–10953 (2013). https://doi.org/10.1364/oe.21.0109426. C. Brahms, F. Belli, and J. C. Travers, “Resonant dispersive wave emission in hollow capillary fibers filled with pressure gradients,” Opt. Lett. 45, 4456–4459 (2020). https://doi.org/10.1364/ol.398343 microstructured fibers with thickness controlled nano-film coatings,7,87. T. A. K. Lühder, H. Schneidewind, E. P. Schartner, H. Ebendorff-Heidepriem, and M. A. Schmidt, “Longitudinally thickness-controlled nanofilms on exposed core fibres enabling spectrally flattened supercontinuum generation,” Light: Adv. Manuf. 2, 262–273 (2021). https://doi.org/10.37188/lam.2021.0218. T. A. K. Lühder, M. Chemnitz, H. Schneidewind, E. P. Schartner, H. Ebendorff‐Heidepriem, and M. A. Schmidt, “Tailored multi-color dispersive wave formation in quasi-phase-matched exposed core fibers,” Adv. Sci. 9, 2103864 (2022). https://doi.org/10.1002/advs.202103864 and fibers with adjusted geometry achieved during drawing9–119. A. Bendahmane, F. Braud, M. Conforti, B. Barviau, A. Mussot, and A. Kudlinski, “Dynamics of cascaded resonant radiations in a dispersion-varying optical fiber,” Optica 1, 243–249 (2014). https://doi.org/10.1364/optica.1.00024310. M. Billet, F. Braud, A. Bendahmane, M. Conforti, A. Mussot, and A. Kudlinski, “Emission of multiple dispersive waves from a single Raman-shifting soliton in an axially-varying optical fiber,” Opt. Express 22, 25673–25678 (2014). https://doi.org/10.1364/oe.22.02567311. A. Bendahmane, A. Mussot, M. Conforti, and A. Kudlinski, “Observation of the stepwise blue shift of a dispersive wave preceding its trapping by a soliton,” Opt. Express 23, 16595–16601 (2015). https://doi.org/10.1364/oe.23.016595 or via post-processing.3,123. A. Ravi, M. Beck, D. F. Phillips, A. Bartels, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “Visible-spanning flat supercontinuum for astronomical applications,” J. Lightwave Technol. 36, 5309–5315 (2018). https://doi.org/10.1109/jlt.2018.287242312. H. H. Chen, Z. L. Chen, X. F. Zhou, and J. Hou, “Ultraviolet-extended flat supercontinuum generation in cascaded photonic crystal fiber tapers,” Laser Phys. Lett. 10, 085401 (2013). https://doi.org/10.1088/1612-2011/10/8/085401 Thus, soliton fission and DW formation in fibers with axially controlled dispersion enables substantially more complex frequency conversion scenarios, which helps in filling the spectral gaps.Efficient SCG has been demonstrated in liquid-core fibers (LCFs)1313. J. Stone, “Optical transmission loss in liquid-core hollow fibers,” IEEE J. Quantum Electron. 8, 386–388 (1972). https://doi.org/10.1109/jqe.1972.1076966 consisting of silica fiber-type capillaries filled with inorganic liquids such as carbon disulfide (CS2).14–1814. D. Churin, T. N. Nguyen, K. Kieu, R. A. Norwood, and N. Peyghambarian, “Mid-IR supercontinuum generation in an integrated liquid-core optical fiber filled with CS2,” Opt. Mater. Express 3, 1358–1364 (2013). https://doi.org/10.1364/ome.3.00135815. M. Chemnitz, M. Gebhardt, C. Gaida, F. Stutzki, J. Kobelke, J. Limpert, A. Tünnermann, and M. A. Schmidt, “Hybrid soliton dynamics in liquid-core fibres,” Nat. Commun. 8, 42 (2017). https://doi.org/10.1038/s41467-017-00033-516. C. Wang, G. Feng, W. Li, and S. Zhou, “Highly coherent supercontinuum generation in CS2-infiltrated single-core optical fiber,” J. Opt. 21, 105501 (2019). https://doi.org/10.1088/2040-8986/ab3bb517. M. Vieweg, T. Gissibl, S. Pricking, B. T. Kuhlmey, D. C. Wu, B. J. Eggleton, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18, 25232–25240 (2010). https://doi.org/10.1364/oe.18.02523218. K. Kieu, L. Schneebeli, R. A. Norwood, and N. Peyghambarian, “Integrated liquid-core optical fibers for ultra-efficient nonlinear liquid photonics,” Opt. Express 20, 8148–8154 (2012). https://doi.org/10.1364/oe.20.008148 This platform has unique properties, such as a non-instantaneous, non-linear response, and has led to the observation of new states of light (e.g., hybrid solitary states1515. M. Chemnitz, M. Gebhardt, C. Gaida, F. Stutzki, J. Kobelke, J. Limpert, A. Tünnermann, and M. A. Schmidt, “Hybrid soliton dynamics in liquid-core fibres,” Nat. Commun. 8, 42 (2017). https://doi.org/10.1038/s41467-017-00033-5), while offering wide mid-IR transmission windows,1919. S. Junaid, W. Huang, R. Scheibinger, K. Schaarschmidt, H. Schneidewind, P. Paradis, M. Bernier, R. Vallée, S.-E. Stanca, G. Zieger, and M. A. Schmidt, “Attenuation coefficients of selected organic and inorganic solvents in the mid-infrared spectral domain,” Opt. Mater. Express 12, 1754–1763 (2022). https://doi.org/10.1364/ome.455405 or substantial thermo-optical tuning capabilities.2020. M. Chemnitz, R. Scheibinger, C. Gaida, M. Gebhardt, F. Stutzki, S. Pumpe, J. Kobelke, A. Tünnermann, J. Limpert, and M. A. Schmidt, “Thermodynamic control of soliton dynamics in liquid-core fibers,” Optica 5, 695–703 (2018). https://doi.org/10.1364/optica.5.000695 A new approach uses higher-order modes (HOMs) in LCFs, featuring two zero-dispersion wavelengths (ZDWs) embracing an interval of anomalous dispersion (AD).2121. R. Scheibinger, N. M. Lüpken, M. Chemnitz, K. Schaarschmidt, J. Kobelke, C. Fallnich, and M. A. Schmidt, “Higher-order mode supercontinuum generation in dispersion-engineered liquid-core fibers,” Sci. Rep. 11, 5270 (2021). https://doi.org/10.1038/s41598-021-84397-1 Such a dispersion landscape is well known for its ability to spectrally trap solitons and to generate intense DWs at both sides of the pump.2222. R. K. W. Lau, M. R. E. Lamont, A. G. Griffith, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Octave-spanning mid-infrared supercontinuum generation in silicon nanowaveguides,” Opt. Lett. 39, 4518–4521 (2014). https://doi.org/10.1364/ol.39.004518 Of note, non-linear frequency conversion involving HOMs has increasingly attracted attention in recent years because it provides access to novel physical effects (e.g., intermodal DW generation2323. N. M. Lüpken, M. Timmerkamp, R. Scheibinger, K. Schaarschmidt, M. A. Schmidt, K. J. Boller, and C. Fallnich, “Numerical and experimental demonstration of intermodal dispersive wave generation,” Laser Photonics Rev. 15, 2100125 (2021). https://doi.org/10.1002/lpor.202100125 and intermodal third-harmonic generation2424. C. K. Ha, K. H. Nam, and M. S. Kang, “Efficient intermodal third-harmonic generation in adiabatic silica nanofiber,” in 2020 Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR) (IEEE, 2020), pp. 1–2.) and application-relevant beam shapes (e.g., doughnut-shaped beams in stimulated emission depletion microscopy25,2625. C. Tressler, M. Stolle, and C. Fradin, “Fluorescence correlation spectroscopy with a doughnut-shaped excitation profile as a characterization tool in STED microscopy,” Opt. Express 22, 31154–31166 (2014). https://doi.org/10.1364/oe.22.03115426. G. Vicidomini, P. Bianchini, and A. Diaspro, “STED super-resolved microscopy,” Nat. Methods 15, 173–182 (2018). https://doi.org/10.1038/nmeth.4593 and particle trapping2727. M. A. Khosravi, A. Aqhili, S. Vasini, M. H. Khosravi, S. Darbari, and F. Hajizadeh, “Gold cauldrons as efficient candidates for plasmonic tweezers,” Sci. Rep. 10, 19356 (2020). https://doi.org/10.1038/s41598-020-76409-3). The combination of axial dispersion control and soliton-based SCG in LCFs, therefore, represents a promising approach, which can principally provide selected non-linear light states and tailored spectra.

In this work, we introduce longitudinal dispersion control in LCFs through axial modulation of the core diameter as a new degree of freedom of the LCF platform in the context of dispersion management and SCG. Dispersion modulation markedly influences the generated spectra, owing to the strong dependence of the HOM-dispersion on the core diameter. Together, the experimental details and comparisons with simulations herein demonstrate the unique potential of the LCF platform.

II. CONCEPT

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. CONCEPT <<III. SAMPLE IMPLEMENTATIO...IV. RESULTSV. DISCUSSIONVI. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESDW generation relies on the fission of a higher-order soliton into its fundamental counterparts, thus releasing excess energy to DWs. This fission results from perturbation of the propagating higher-order soliton through several effects, such as third-order dispersion, self-steepening, and intrapulse Raman scattering. The frequency of the DW is determined by a PM condition given by1,281. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).28. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006). https://doi.org/10.1103/revmodphys.78.1135n(λs)+(λ−λs)dndλ(λs)+γPsλ4π=n(λ).(1)Here, λ and λs are the wavelengths of the DW and soliton, respectively, Ps is the peak power of the assumed first soliton (with input peak power P0 and soliton number N, Ps = P0(2N − 1)2/N2),2828. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006). https://doi.org/10.1103/revmodphys.78.1135 and γ = 2πn2/(λsAeff) is the non-linear coefficient (Aeff is the effective mode area, and n2 is the non-linear refractive index of CS22929. M. Reichert, H. Hu, M. R. Ferdinandus, M. Seidel, P. Zhao, T. R. Ensley, D. Peceli, J. M. Reed, D. A. Fishman, S. Webster et al., “Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide,” Optica 1, 436–445 (2014). https://doi.org/10.1364/optica.1.000436). The condition Eq. (1) determines the spectral location of the phase-matched DW(λ) if the soliton wavelength (λs) is given or vice versa.The basic idea of our fiber design involves continuous perturbation of the fundamental soliton generated after the initial high-order soliton fission, by changing the dispersion properties along the pulse propagation to trigger multiple processes, releasing excess energy (i.e., DW formation) at different wavelengths. This change in dispersion properties along the fiber can be achieved by longitudinally decreasing the core diameter of the CS2-LCF. Of note, the fundamental core mode (HE11-mode) shows hardly any change of the dispersion properties for the core diameters considered3030. R. Scheibinger, J. Hofmann, K. Schaarschmidt, M. Chemnitz, and M. A. Schmidt, “Temperature-sensitive dual dispersive wave generation of higher-order modes in liquid-core fibers,” Laser Photonics Rev. (in press) (2022). https://doi.org/10.1002/lpor.202100598 (details can be found in the supplementary material Sec. 5).A sketch of a TE01-mode propagating in a longitudinally modulated circular step-index LCF is shown in Fig. 1(a). Of note, the LCFs considered here consist of CS2 as the core and silica as the cladding material. The core size pattern leads to modulation of dispersion along the LCF and, thus, yields multiple PM opportunities for soliton-based DW generation, as suggested by Eq. (1). As shown in Fig. 1(b), multiple DWs (e.g., including several pairs of DW1 and DW2) are generated successively and approach the soliton, thus effectively filling the gaps between the soliton and the first-generated pair of DW1 and DW2. To reveal the effects of core-diameter tuning on the dispersion of the TE01-mode in detail, we calculated the group velocity dispersion semi-analytically by using the dispersion equation of a step-index fiber.3131. M. Chemnitz and M. A. Schmidt, “Single mode criterion-a benchmark figure to optimize the performance of nonlinear fibers,” Opt. Express 24, 16191–16205 (2016). https://doi.org/10.1364/oe.24.016191 The material dispersions of CS2 and silica were taken from Refs. 1515. M. Chemnitz, M. Gebhardt, C. Gaida, F. Stutzki, J. Kobelke, J. Limpert, A. Tünnermann, and M. A. Schmidt, “Hybrid soliton dynamics in liquid-core fibres,” Nat. Commun. 8, 42 (2017). https://doi.org/10.1038/s41467-017-00033-5 and 3232. J. W. Fleming, “Dispersion in GeO2–SiO2 glasses,” Appl. Opt. 23, 4486–4493 (1984). https://doi.org/10.1364/ao.23.004486. Figure 1(c) shows the evolution of the two ZDWs (dark green curves) for the TE01-mode with decreasing core diameters. In contrast to its short wavelength counterpart (ZDW1), the long-wavelength ZDW (ZDW2) is highly sensitive to changes in the core diameter, showing a spectral shift of ∼0.67 µm (ΔλZDW) for a change in core diameter of only 0.46 µm (Δd), thus yielding a tuning slope of ΔλZDW/Δd = 1.4 µm/μm. This substantial dependence suggests unique opportunities for generating multiple DWs at distinct wavelengths in core-diameter modified LCFs. Concordantly, the corresponding DW phase-mismatch [magenta and sky–blue curves in Fig. 1(c)] calculated by Eq. (1) indicate highly effective tuning of the phase-matched DWs through changing of the core diameter, particularly for the long-wavelength DW (DW2): for a fixed soliton wavelength (λs = 1.8 µm, extracted from experimental results) and soliton peak power (Ps = 14 kW), a core diameter decrease of 0.4 µm (from 3.9 to 3.5 µm) causes DW2 to blue-shift by more than 1.1 µm, whereas the short-wavelength DW (DW1) is only slightly red-shifted by 0.1 µm.Of note, although changing the core diameter affects the effective mode area and, consequently, the non-linear parameter, the overall dependence is fairly small, as shown in Fig. S1. Nonetheless, the small change in the non-linear parameter is included in the simulation. In addition, the modal cut-off of the TE01-mode decreases from 3.7 to 2.96 µm, as depicted in Fig. S1 in the supplementary material, when the core diameter decreases from 3.9 to 3.3 µm. Herein, the main non-linear frequency conversion covers the spectral range between 1 and 2.5 µm, which is far from the cut-off wavelength, and, thus, the spectral broadening is not influenced in our experiment.

III. SAMPLE IMPLEMENTATION AND OPTICAL SETUP

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. CONCEPTIII. SAMPLE IMPLEMENTATIO... <<IV. RESULTSV. DISCUSSIONVI. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESSample design: In the present work, four types of LCFs are implemented to reveal the concept of core-diameter tuning. Beyond a reference sample with constant diameter (c-LCF), LCFs with single (s-LCF), multiple (m-LCF), and linear (l-LCF) collapsed sections are implemented [the corresponding evolutions of the core diameter of the four fiber-type capillaries along the longitudinal (z) direction are shown in Figs. 2(a) and 2(b)]. All core-diameter modified LCFs have an initial 2 cm long, uniform section at the beginning.The s-LCF (initial diameter dc = 3.9 µm) has only one collapsed section (dc = 3.6 µm) over 2 cm in length [magenta curve in Fig. 2(b)]. The m-LCF (initial diameter dc = 3.9 µm) has multiple collapsed sections (dc = 3.8, 3.7, 3.65, 3.6, and 3.55 µm) of different lengths over a distance of >1 cm each [cyan curve in Fig. 2(b)]. The core diameter of the l-LCF decreases linearly from 3.8 to 3.3 µm over a distance of 9 cm and becomes constant again for the final 3 cm long section (dc = 3.8 µm). Here, we deduce the local core diameter by scaling the original core diameter to the reduced outer diameter [Eq. (S1), supplementary material]; this deduction is valid on the basis of the assumption that the central hole does not collapse during tapering.Sample implementation: The implementation of CS2-LCFs with varying core diameters relies on controlled, partial collapse of the hole of the fiber-type silica capillary (fabricated in-house) before liquid filling. This collapse was performed in a post-processing step: s-LCF and m-LCF were generated by an optical fiber glass processor (Vytran, GPX-3200), with precise control of the feed and pull velocities, as well as the filament power. The l-LCF was realized using a CO2 tapering machine, which relies on a power stabilized, 70 W, CO2 laser, a double axicon for beam shaping, and a metallic, freeform mirror focusing the laser beam on the circumference of the fiber to heat it uniformly.3333. J. Nold, M. Plötner, S. Böhme, B. Sattler, O. deVries, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication of longitudinally arbitrary shaped fiber tapers,” Components and Packaging for Laser Systems IV (SPIE, 2018), Vol. 10513, pp. 287–292. The cladding diameter evolution of the fabricated taper was measured in the case of the I-LCF by clamping both fiber ends in holders mounted on a translation stage. Within the corresponding travel range, two crossed optical micrometers (Keyence LS-9000) are positioned, allowing the measurement of the fiber diameter along the tapered length using a LabView interface. During tapering, the pressure inside the hole of capillaries was kept at atmospheric pressure. After tapering, both ends of the post-processed capillaries were inserted into optofluidic mounts including fluidic access ports and transparent windows, thus allowing for both launching and collection of light from the core mode, as well as liquid filling. Filling with CS2 (≥99.9%, Sigma-Aldrich) was based on the capillary effect (described by Washburn’s law3434. E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17, 273 (1921). https://doi.org/10.1103/physrev.17.273) and required ∼5 min over a length of ∼15 cm for the capillary diameters used (3.5 µm dcµm). Notably, the variation of the core diameter did not lead to any measurable changes in filling behavior, and the use of optofluidic mounts ensured long-term stable operation.Ultrafast setup: The optical setup used for the non-linear experiments is shown in Fig. 2(c). Femtosecond pulses [central wavelength λp = 1570 nm; pulse duration (FWHM) τ = 36 fs] delivered by an ultrafast fiber laser (Toptica FemtoFiber pro IRS-II, repetition rate ν = 80 MHz) were sent through a combination of a half-wave plate and polarizer to adjust the transmitted energy. Conversion to radial polarization was achieved with an s-waveplate (Altechna; details in Ref. 2121. R. Scheibinger, N. M. Lüpken, M. Chemnitz, K. Schaarschmidt, J. Kobelke, C. Fallnich, and M. A. Schmidt, “Higher-order mode supercontinuum generation in dispersion-engineered liquid-core fibers,” Sci. Rep. 11, 5270 (2021). https://doi.org/10.1038/s41598-021-84397-1), followed by an in-coupling lens (Thorlabs C230TMD-C, f = 4.5 mm, NA = 0.6) to excite the TE01-mode in the LCFs. The corresponding coupling efficiency was determined to be ∼18%, corrected for reflection losses at the various interfaces and for the attenuation of the LCFs. The generated supercontinuum (SC) was collimated by an out-coupling lens (Thorlabs C036TME-D, f = 4 mm, NA = 0.56) and guided to the diagnostic tools [optical spectrum analyzer (OSA, Yokogawa AQ6375, working wavelength range: 1–2.5 µm); IR camera (ABS InGaAs camera, IK 1513); and power meter], which could be used as desired. For the spectral measurements, the output light was guided to the OSA by an InF3 patch cable (Thorlabs, MF22L1, core diameter: 200 µm, length: 1 m). Considering the chromaticity of the out-coupling lens and the large bandwidth of the SC, we stitched the presented spectra together from multiple measurements. For example, in Fig. 4, three spectral datasets were involved, each of which was optimized through adjusting the out-coupling lens for a specific spectral feature measured with the same OSA [spectrum 1 (1 µm λµm): DW1 at λ = 1.2 µm; spectrum 2 (1.5 µm λµm): soliton at λ = 1.8 µm; spectrum 3 (1.8 µm λµm): DW2 at λ = 2.4 μm]. Of note, an infrared long-pass filter (cut-on wavelength: 1.65 µm, Edmund optics) was inserted before the patch cable during the third spectrum measurement to remove second-order grating effects. All spectra from the different coupling conditions were stitched together at a wavelength at which the spectral power densities coincide, thus yielding a final spectrum that was corrected for the short-, central-, and long-wavelength spectral power. An in-coupled energy of 0.3 nJ was reached, and no damage was observed even at the highest input power level in the experiment. A slight spectral drifting during the spectral measurements, especially occurring between 2.4 and 2.5 µm, presumably resulted from a changing ambient temperature in the laboratory.Non-linear simulations: The non-linear pulse dynamics was simulated on the basis of the model presented in Ref. 1515. M. Chemnitz, M. Gebhardt, C. Gaida, F. Stutzki, J. Kobelke, J. Limpert, A. Tünnermann, and M. A. Schmidt, “Hybrid soliton dynamics in liquid-core fibres,” Nat. Commun. 8, 42 (2017). https://doi.org/10.1038/s41467-017-00033-5 through solving the generalized non-linear Schrödinger equation,11. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007). considering the varying dispersion [Fig. 1(c)] and non-linear properties (Fig. S1). Of note, owing to the short pump pulse duration (τ = 36 fs), the CS2 responded almost instantly to the femtosecond pulses, which is reflected by a small molecular fraction of fm = 0.18 (calculated by fm = n2,mol/n2,total). As shown in our previous work,15,2115. M. Chemnitz, M. Gebhardt, C. Gaida, F. Stutzki, J. Kobelke, J. Limpert, A. Tünnermann, and M. A. Schmidt, “Hybrid soliton dynamics in liquid-core fibres,” Nat. Commun. 8, 42 (2017). https://doi.org/10.1038/s41467-017-00033-521. R. Scheibinger, N. M. Lüpken, M. Chemnitz, K. Schaarschmidt, J. Kobelke, C. Fallnich, and M. A. Schmidt, “Higher-order mode supercontinuum generation in dispersion-engineered liquid-core fibers,” Sci. Rep. 11, 5270 (2021). https://doi.org/10.1038/s41598-021-84397-1 this fraction was so small that the non-instantaneous contribution had no measurable effect on SCG and, therefore, could safely be neglected.

IV. RESULTS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. CONCEPTIII. SAMPLE IMPLEMENTATIO...IV. RESULTS <<V. DISCUSSIONVI. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESThe experimental results, in comparison to non-linear pulse propagation simulations (both TE01-modes) of the four different LCF samples, are summarized in Fig. 3. Of note, the left y-axis of the experiment-related-energy spectral evolution [Figs. 3(a-1), 3(b-1), 3(c-1) and 3(d-1)] corresponds to the in-fiber pulse energy, i.e., includes the coupling and transmission losses, thus allowing for a direct comparison with the simulations.Overall, the main spectral features in the experimental data for the four LCFs are qualitatively reproduced in the simulations. This agreement illustrates the accuracy of the non-linear pulse propagation model and the appropriateness of unlocking the relevant physical processes on the basis of simulations. Of note, quantitative differences, such as variations in the onset energies of the DWs are observed, while all the ongoing physical effects are qualitatively reproduced. The main mechanism of the non-linear frequency conversion in the four LCFs and the difference resulting from the differences in core-diameter modulation are explained in detail below. An in-depth discussion of the ongoing physical processes is presented in Sec. , on the basis of numerical simulations of the four LCF configurations.c-LCF: For the uniform structure [c-LCF, Fig. 3(a)], the initial broadening starts from the well-known effect of self-phase-modulation and temporal pulse narrowing, owing to the pump located at the AD region. This is followed by a fission of the formed higher-order soliton at an approximate pulse energy of E = 0.05 nJ, leading to the emergence of two DWs (DW1: 1.3 µm and DW2: 3.1 µm) in ND domains. Of note, the wavelength of DW2 is beyond the measurement range, and, thus, is not visible in Figs. 3 and 4, but is shown in the simulation presented in supplementary material Sec. 3 [Figs. S2(c1)–S2(c4)].s-LCF: For the single-section LCF [s-LCF, Fig. 3(b)], the initial broadening mechanism is identical to that of the c-LCF. Owing to the fulfilled PM condition in the small-core section, another two DWs at 2.5 µm and around 1.2 µm are generated. Of note, the new two DWs are newly generated because of the newly formed higher-order soliton fission process, rather than a shifting of the first pair of DWs generated in the initial part of the LCF,77. T. A. K. Lühder, H. Schneidewind, E. P. Schartner, H. Ebendorff-Heidepriem, and M. A. Schmidt, “Longitudinally thickness-controlled nanofilms on exposed core fibres enabling spectrally flattened supercontinuum generation,” Light: Adv. Manuf. 2, 262–273 (2021). https://doi.org/10.37188/lam.2021.021 as clearly seen in Fig. 5(b).m-LCF: A similar effect is observed for the multi-section LCF [m-LCF, Fig. 3(c)], which overall provides more DW-PM opportunities, owing to multiple sections with different core diameters (a detailed analysis can be found in Sec. ). Of note, although high energy-transfer-efficiency to the spectral interval between the soliton and DW2 is demonstrated, the spectral gap between the pump and DW1 remains.l-LCF: Spectra with improved spectral flatness can be achieved in LCFs with a linearly decreasing core diameter section [l-LCF, Fig. 3(d)]. To clearly reveal the advantage of the l-LCF, we compare the experimental output spectra for the highest in-fiber pulse energy for the four fiber geometries (TE01 mode) in Fig. 4. Here, the output spectrum of the l-LCF has the best flatness, with a bandwidth (−20 dB level) of 1.22 µm (an octave), thus overall efficiently filling the spectral gaps between DW1, the soliton, and DW2. To quantify this improvement, Table I shows the percentage of spectral power integrated in two characteristic spectral intervals (gap1: 1.22 µm λµm, gap2: 2 µm λµm), which are separated by DW1, the soliton, and DW2, relative to the spectral power integration across the entire spectral domain (1–2.5 µm).

TABLE I. Percentage of spectral power integration in gap1 (1.22–1.52 µm) and gap2 (2–2.3 µm) wavelength ranges relative to the spectral power integration across the entire spectral domain (1–2.5 µm).

c-LCF (%)s-LCF (%)m-LCF (%)l-LCF (%)1.22–1.52 µm1.92.42.4232.00–2.30 µm00.5107.4

V. DISCUSSION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. CONCEPTIII. SAMPLE IMPLEMENTATIO...IV. RESULTSV. DISCUSSION <<VI. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESIn the following, the underlying physical mechanisms in the dispersion-managed fibers are revealed through non-linear pulse propagation simulations, a well-established approach in the SCG community, because of the good match typically observed between simulations and experiments, both herein and in previous work.2121. R. Scheibinger, N. M. Lüpken, M. Chemnitz, K. Schaarschmidt, J. Kobelke, C. Fallnich, and M. A. Schmidt, “Higher-order mode supercontinuum generation in dispersion-engineered liquid-core fibers,” Sci. Rep. 11, 5270 (2021). https://doi.org/10.1038/s41598-021-84397-1 The spatiospectral evolutions (Fig. 5; peak power Ppeak = 10 kW for all four situations) clearly show that the axial dispersion modification substantially influences soliton dynamics. Of note, all dispersion-managed fibers have a 2 cm long, constant diameter section at the input, regardless of the different initial core diameters [3.8 µm for Figs. 5(a) and 5(d) and soliton number Ns = 4.3; 3.9 µm for Figs. 5(b) and 5(c) and Ns = 3.8]. This initial constant section ensures that the same initial non-linear effect happens for all structures, which is soliton fission and accompanied DW formation.c-LCF: In a uniform fiber structure [c-LCF, Fig. 5(a)], the spectral broadening and, particularly, the energy transfer from pump λp = 1.57 µm to around λ = 3.1 µm result from pulse compression, soliton fission (soliton number Ns = 4.3), and the formation of two DWs. Specifically, after the initial fission process at z = 1.3 cm, the pulse peak power increases to 22 kW until z = 2.5 cm, and another pair of DWs is generated. Thereafter, a periodical pulse compression — soliton breathing — accompanied by the emission of weak DWs is observed, as clearly evidenced by the periodic peak power distribution along z. Of note, owing to third-order dispersion, the four fundamental solitons move at slightly different speeds, thereby leading to the spectral fringes.11. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007). The final output spectrum shows a substantial gap between the soliton and DW2, because all the DW2 are created in close spectral vicinity.s-LCF: When considering a small core section after the soliton fission [z = 1.3 cm, Fig. 5(b)], the decrease in β2 at λs leads to an increase in the soliton number, i.e., a transformation of the fundamental soliton to a higher-order soliton. Because of the closer proximity to ZDW2, another soliton fission and DW generation process is triggered, thus leading to the generation of another pair of DWs. Of note, because the solitons are very close to the ZDW2, the energy transfer from the soliton to the DW2 at λ = 2.4 µm is highly efficient.m-LCF: For the multi-section LCF [m-LCF, Fig. 5(c)], the spatiospectral evolution suggests more complex soliton dynamics, which quantitatively follows the situation of the s-LCF. The key difference is that, owing to the multiple decreases in β2, more DWs are emitted during propagation, thereby partially filling the spectral interval between the soliton and DW2.l-LCF: In the linear-section LCF [l-LCF, Fig. 5(d)], the gradually decreasing core diameter leads to a steady reduction of ZDW2, thus forcing the soliton to spectrally approach ZDW2. This imposes a series of higher-order soliton formations, accompanied by soliton fission and DW generation processes at different z-positions. At z = 4.5 cm, the soliton spectrum begins to overlap with ZDW2, thus leading to slight “pushing” of the soliton toward shorter wavelengths—a phenomenon denoted “soliton blue-shift.”9,35–379. A. Bendahmane, F. Braud, M. Conforti, B. Barviau, A. Mussot, and A. Kudlinski, “Dynamics of cascaded resonant radiations in a dispersion-varying optical fiber,” Optica 1, 243–249 (2014). https://doi.org/10.1364/optica.1.00024335. A. C. Judge, O. Bang, and C. Martijn de Sterke, “Theory of dispersive wave frequency shift via trapping by a soliton in an axially nonuniform optical fiber,” J. Opt. Soc. Am. B 27, 2195–2202 (2010). https://doi.org/10.1364/josab.27.00219536. S. P. Stark, A. Podlipensky, and P. St. J. Russell, “Soliton blueshift in tapered photonic crystal fibers,” Phys. Rev. Lett. 106, 083903 (2011). https://doi.org/10.1103/PhysRevLett.106.08390337. Z. Chen, A. J. Taylor, and A. Efimov, “Coherent mid-infrared broadband continuum generation in non-uniform ZBLAN fiber taper,” Opt. Express 17, 5852–5860 (2009). https://doi.org/10.1364/oe.17.005852 This spectral recoil is accompanied by DW emission, until almost all the soliton’s energy is converted to DWs. The resulting DW continuum is spectrally broad and pulse-wise coherent, because of multiple PM wavelengths provided by the gradually decreasing core diameter. Of note, at z = 8.