Abstract
The generation of a twooctave supercontinuum from the visible to midinfrared (700–2800 nm) in a nonsilica gradedindex multimode fiber is reported. The fiber design is based on a nanostructured core comprised of two types of drawn leadbismuthgallate glass rods with different refractive indices. This yields an effective parabolic index profile and ten times increased nonlinearity when compared to silica fibers. Using femtosecond pulse pumping at wavelengths in both normal and anomalous dispersion regimes, a detailed study is carried out into the supercontinuum generating mechanisms and instabilities seeded by periodic selfimaging. Significantly, suitable injection conditions in the high power regime are found to result in the output beam profile showing clear signatures of beam selfcleaning from nonlinear mode mixing. Experimental observations are interpreted using spatiotemporal 3+1D numerical simulations of the generalized nonlinear Schrödinger equation, and simulated spectra are in excellent agreement with experiment over the full twooctave spectral bandwidth. Experimental comparison with the generation of supercontinuum in a silica gradedindex multimode fiber shows that the enhanced nonlinear refractive index of the leadbismuthgallate fiber yields a spectrum with a significantly larger bandwidth. These results demonstrate a new pathway towards the generation of bright, ultrabroadband light sources in the midinfrared.
Introduction
Understanding the physics of complex nonlinear optical systems has been the focus of intense research in recent years and, in this context, the generation of broadband supercontinuum (SC) in gradedindex multimode fibers (GRIN MMFs) has attracted particular attention^{1,2,3,4,5,6,7,8}. In addition to multimode fibers providing additional degrees of freedom to optimize SC for specific applications, the spatiotemporal propagation in such fibers reveals a rich landscape of nonlinear dynamics, with close links to universal phenomena such as wave turbulence^{9,10,11,12}.
In contrast to singlemode fibers where the spatial intensity distribution remains constant with propagation, the parabolic index profile of GRIN fibers leads to a periodic selfimaging phenomenon enabling spatiotemporal coupling and mode mixing associated with complex and unique nonlinear dynamics. Spatiotemporal effects that have been observed include among others the generation of multimode solitons^{1,2,3,13,14}, the development of geometric parametric instabilities (GPI)^{4,5,6,7}, and the formation of GRIN lenses^{15,16}. Spatiotemporal dynamics in GRIN silica fibers have been exploited to manipulate the transverse beam profile^{4,17} and, under particular injection conditions, the output beam was observed to exhibit a quasi singlemode profile as the result of nonlinear selfcleaning dynamics^{18,19,20,21}. Numerical studies have shown that such selfcleaning is a particular feature of pulse propagation in GRIN MMFs^{22,23,24} associated with strong nonlinear coupling leading to preferential energy transfer to the loworder modes^{19,25}. Beam selfcleaning has been reported under various pumping configurations using nanosecond^{25}, picosecond^{19,26} and femtosecond^{18} pulses, both in the normal^{5,18,27,28} and anomalous dispersion regime^{29}. In addition to being of fundamental importance through its links to universal nonlinear physics, dynamical selfcleaning is also of great practical interest in the development of highpower SC sources.
To date, however, all studies and demonstrations of SC generation in gradedindex fibers have been restricted to silica fibers, and bandwidths limited only to the visible and nearinfrared spectral regions^{18,26,27,29,30}. Yet because of the ability of GRIN MMFs to provide power scaling with a nearGaussian spatial intensity distribution, there is major interest in extending the GRIN MMF platform into the midinfrared regime where high spatial beam quality and high power^{31} are required in applications including, e.g., molecular fingerprinting^{32}, microscopy^{33,34}, medical diagnostics^{35,36}, gas monitoring^{37,38}, spectroscopy^{39,40}, optical coherence tomography^{41} and LIDAR^{42}.
Here, we fill this gap and report the generation of a twooctave SC expanding from 700 nm to 2800 nm in a nonsilica gradedindex multimode fiber. The fiber is designed using two types of leadbismuthgallate (PBG) glass rods with different refractive indices drawn to yield a nanostructured core^{43,44,45,46}. The result is a multimode fiber with an effective parabolic refractive index profile, enhanced nonlinear refractive index, and transmission window up to 2800 nm. Injecting femtosecond pulses into the fiber, we observe the generation of a twooctave SC from 700 to 2800 nm. We conduct a systematic investigation of the SC generating mechanism as a function of pump wavelength, with selfphase modulation (SPM) and GPI seeding the SC generation process in the normal dispersion regime, while in the anomalous regime soliton dynamics and parametric dispersive waves excitation are found to dominate. The relative intensity noise (RIN) has also been characterized in several wavelength bands across the SC spectrum and, under particular controlled injection conditions, we see clear signatures of selfcleaning dynamics with a near singlemode spatial intensity distribution at the fiber output. In order to confirm and interpret our experiments, we perform spatiotemporal 3+1D numerical simulations of the generalized nonlinear Schrödinger equation. Remarkably, the simulations reproduce the spatial intensity distributions measured at the fiber output and confirm the selfcleaning dynamics, with spectra in excellent agreement over the full SC bandwidth. Finally, we perform an experimental comparison with the SC generated in a GRIN silica fiber showing that the spectrum generated in the PBG fiber extends further into the midinfrared as the result of an enhanced nonlinear refractive index. These results not only open up novel perspectives for the study of nonlinear spatiotemporal instabilities in nonsilica gradedindex fiber platforms but also provide an avenue for power scaling of SC sources in the midinfrared.
Results
Fiber design characteristics
The fiber preform was designed to have a parabolic index profile using two types of inhouse developed PBG glasses (see Methods for details on the fabrication process). Figure 1 shows the characteristics of the fabricated GRIN PBG fiber with R = 40 μm core radius. The relative index difference Δ = (n_{co}−n_{cl})/n_{co}=0.0101 corresponds to a numerical aperture NA = 0.26 with n_{co} and n_{cl} the refractive index at the core center and in the cladding, respectively. Both glasses have a transmission window extending from 400 nm to about 2800 nm limited by OH absorption (Fig. 1a). The measured attenuation of the fabricated fiber (see Methods) is found to be similar to that of the bulk glass, showing that it is primarily the intrinsic attenuation of the glass that limits the fiber transmission. The attenuation is significantly larger than that in silica fiber, yet because the PBG nonlinear refractive index is ten times larger than that of silica^{46} only a short length is needed to observe nonlinear dynamics and massive spectral broadening. The measured refractive index values are around 1.9 (Fig. 1b, see also Methods). The simulated refractive index distribution of the designed fiber is shown in Fig. 1c and exhibits a parabolic variation from the cladding to the core center.
Figure 1d shows the simulated propagation constant Δβ of the first 30 modes of the fiber relative to that of the fundamental mode. In contrast to stepindex fibers, one can see how the parabolic refractive index profile yields discrete clusters of modes with equally spaced propagation constant and with a cluster population that grows with Δβ. This particular feature results in periodic multimodal interference and selfimaging, where the spatial field focuses and defocuses periodically with a period \({z}_{p}=\pi R/\sqrt{2{{\Delta }}}\)^{7}.
In the experiments reported below, we operate in the regime where the injected peak power is below the critical power for catastrophic selffocusing such that the selfimaging condition is essentially independent of the nonlinear Kerr contribution (see Supplementary Information). Modes within a particular cluster experience identical propagation constant resulting in minimum modal dispersion and walkoff^{47}. The propagation constants difference between modes of distinct clusters is large, leading to strong intracluster modal interaction but limited intercluster mode coupling, such that the fundamental mode is the most stable propagating mode. We also numerically simulated the group velocity dispersion (GVD) of the fundamental and selected higherorder modes vs. wavelengths as shown in Fig. 1e with the corresponding spatial amplitude of the modes illustrated in Fig. 1f. Unlike in stepindex multimode fibers^{48}, the GVD characteristics of the different modes do not differ significantly with a nearconstant zerodispersion wavelength (ZDW) at ~1980 nm for all modes.
Experiments and results
A schematic illustration of our experimental setup is shown in Fig. 2 (see also Methods for additional details). A tunable optical parametric amplifier (OPA) producing 350 fs pulses at a repetition rate of 500 kHz is used as the pump source. The PBG fiber is 20 cm long. Two different optical spectrum analyzers are used to measure the SC spectrum in different wavelength ranges and a monochromator was used to filter out selected wavelength bands and characterize the pulsetopulse fluctuations (see Methods for details). The spatial intensity at the fiber output was characterized in the farfield using a beam profiling camera. In all experiments below, we use a very short length of fiber (20 cm) and therefore polarization effects are not expected to play a significant role in the dynamics. Here the injected field is always linearly polarized and a previous study in GRIN silica fibers has shown that, in the nonlinear regime, the degree of linear polarization is generally increased^{49}.
We first characterized the output spectrum and spatial intensity distribution at the fiber output for input pulses at 1700 nm in the normal dispersion regime of the fiber. These results are shown in Fig. 3 for three different injection conditions corresponding to increasing tilt between the input beam and the fiber input facet. In all cases, the SC spans from 800 nm to 2800 nm (−40 dB bandwidth) with apparent discrete spectral components on the shortwavelength side. These arise from the nonlinear refractive index grating induced by the Kerr effect and periodic selfimaging along propagation^{5,50,51} that exponentially amplifies small perturbations in frequency bands determined by a specific phasematching condition (see Methods). In contrast to conventional modulation instability in singlemode fibers which is a pure temporal effect restricted to the anomalous dispersion regime, geometric parametric instability (GPI) is a spatiotemporal phenomenon that can occur regardless of the dispersion sign^{7}. The bandwidth and amplitude of the GPI sidebands decrease with the order, in agreement with the theory^{50}. In principle, GPI leads to the generation of multiple sideband pairs which are widely separated and symmetrically located around the pump. Here, however, because the longwavelength components fall outside the transmission window of the fiber, we only observe the shortwavelength sidebands except for the first order where we see both spectral components. Note that this was also the case in previous studies in GRIN silica fibers^{5,27}.
In order to confirm the origin of the discrete spectral components, we computed the theoretical location of the GPI sidebands for our fiber, and they are shown as dotted lines in Fig. 3 with the sideband order as indicated. For completeness, the experimental GPI highfrequency components and corresponding analytical values up to the seventh order are listed in Table 1 where we see excellent agreement for all generated GPI orders.
Similar to conventional modulation instability, an important feature of the geometric parametric instability observed here is that it is seeded by noise outside the pump spectral bandwidth. This is expected to lead to significant shottoshot fluctuations yielding a relatively smooth SC spectrum without fine structure when measured with an integrating optical spectrum analyzer^{52}. Intensity fluctuations were characterized in selected wavelength bands indicated by the circles in Fig. 3a (see Methods). One can see that the fluctuations are minimal near the pump at 1700 nm and increase significantly away from the pump residue confirming the influence of noise amplification in the SC development.
The spatial intensity profile at the fiber output corresponding to the spectra in Fig. 3 and obtained for different initial excitation conditions shows signatures of selfcleaning dynamics similar to previous observations in silica fibers^{18,19,21,25,26,27}. Specifically, the fiber theoretically supports ~750 transverse modes at 1700 nm, however, when light is injected at normal incidence (Fig. 3a) the measured spatial intensity distribution at the fiber output displays a quasiGaussian profile close to the fundamental LP_{01} mode. When the light was injected with a small angle with respect to the fiber axis in order to reduce the fraction of energy coupled to the fundamental mode, the intensity profile at the fiber output showed two distinct sidelobes (Fig. 3b) characteristic of the LP_{11} mode. Increasing the input angle further to excite a larger fraction of modes at the fiber input (Fig. 3c) yielded an output intensity profile with a fine specklelike structure indicative of multiple modes contribution. These observations are consistent with nonlinear modemixing dynamics mediated by the nonlinear refractive index grating induced along the propagation direction^{19,21}. Although the number of modes excited by the injection condition remains essentially constant with propagation^{12}, higherorder modes transfer energy via strong nonlinear coupling towards preferential modes as the result of optical wave turbulence^{11,12}. This energy flow is similar to wave condensation observed in hydrodynamics and depends on the initial spatial overlap between the input beam and fiber modes^{9,11,12,29}. In the case of normal incidence where the input energy is distributed among the lowestorder modes and concentrated around the fiber longitudinal axis, energy preferentially flows from unstable higherorder modes to the fundamental mode^{21}, while for a small input tilt the offaxis periodic local intensity oscillation generates an offaxis refractiveindex grating overlapping with the LP_{11} mode^{26}. For a larger tilt, the fraction of energy coupled to higherorder modes at the fiber input is too large for selfcleaning dynamics to fully develop^{12}. One can also see that the output SC spectrum changes only slightly with the launching conditions. This can be seen from the spectral bandwidth as well as from the frequencies of the GPI sidebands which remain unchanged in all three cases illustrated. This can be attributed to the fact that the dispersion profile of all modes is nearly identical, leading to dynamics that are essentially independent of the number of excited modes.
In order to study the influence of the spatiotemporal dynamics on the SC spectrum and spatial intensity profile at the fiber output, we performed additional measurements, gradually increasing the input peak power for normal incidence launching conditions similar to that in Fig. 3a. The corresponding measured SC spectra and farfield intensity profiles are plotted in Fig. 4. At the lowest injected power value (for which our camera can image the spatial intensity distribution), the nonlinear dynamics are dominated by SPM with nearsymmetrical spectral broadening. The corresponding farfield spatial intensity profile is smooth with a narrow diameter, consistent with previous observations in the femtosecond regime^{18}. As the injected energy is increased, discrete shortwavelength GPI sidebands develop from noise and the SPMbroadened spectrum extends into the anomalous dispersion regime where soliton dynamics develop. Concomitantly in the farfield, we see an increase in diameter of the beam profile with an apparent contribution from loworder modes. For an input pulse energy of several 100 s of nJ, the beam profile shows a dominant contribution from the LP_{01} mode.
To corroborate our experimental observations, we performed numerical simulations using the 3+1D generalized nonlinear Schrödinger equation (see Methods for details) and the results are shown in the right subpanels of Fig. 4. In principle, one can also use a modal decomposition approach^{53} and the two models are equivalent. However, unlike the 3+1D model which intrinsically includes all propagating modes in the structure, the modal decomposition requires a large number of modes to be included in the simulations for accurate results which can increase significantly the computation time (see Supplementary Information for a comparison between the two approaches). The input energy values in the simulations are adjusted to yield energies at the fiber output similar to those measured in our experiments. We also emphasize that, for the fiber studied here, Ramaninduced dynamics are limited because of the reduced Raman contribution arising both from the lower amplitude of the Raman gain in silicate glasses and the short fiber length (see Supplementary Information). The experimentally measured farfield intensity profiles correspond to the Fourier transform of the near field and for completeness, our simulations show both near and farfield distributions. Note that the simulations assume a length of 20 cm which may slightly differ from the experimental fiber length value with a discrepancy that can easily be of the order of one selfimaging period (z_{p} = 0.9 mm, see Methods). One may therefore expect that the simulated farfield distributions may not exactly match with the experimentally recorded intensity profiles as the near field can vary significantly within one selfimagine period which will also affect the farfield distribution. However remarkably, one can see overall very good agreement between both the experimentally measured and simulated spectra and intensity profiles for all energy values. The simulated evolution of spectra with injected energy follows a similar trend as observed in our experiments, with SPMdominated broadening. For sufficient increase in injected energy, we observe the emergence of GPIinduced discrete spectral components widely separated from the pump and energy transfer to the anomalous dispersion region with the formation of solitons. At the highest energy values, crossphase modulation interaction between the SPM spectral component, GPI sidebands, and ejected solitons leads to a quasicontinuous SC spectrum. This scenario is confirmed in the spectrogram animation shown in Supplementary Movie 1 based on a simplified 1+1D model^{23} which reproduces the essential features of the nonlinear propagation (see Supplementary Information). We also see how nonlinear modemixing dynamics lead to smoother nearfield spatial intensity distribution with decreased beam size.
The role of selfcleaning dynamics at higher peak power values is further highlighted in Fig. 5 where we plot the simulated evolution of the beam effective area along with the fiber for the different spectra shown in Fig. 4. For comparison, we also include the evolution in the linear regime in the absence of spectral broadening dynamics. In all cases, the effective area oscillates periodically as the result of the selfimaging dynamics (see inset) and it is this periodic perturbation that leads to the phasematched generation of parametric sidebands^{7,50}. We also see that the spatial intensity distribution varies significantly with propagation with clear contributions from higherorder modes in the defocused regions for lower injected energy (Fig. 5a–d). As the injected energy is significantly increased, one can see how the contribution from the higherorder modes decreases dramatically as the result of nonlinear mode mixing with selfcleaning already occurring in the first few centimeters of the fiber (Fig. 5e–f).
We next investigated the influence of the pump wavelength on the SC spectrum. The results are illustrated in Fig. 6 where we plot the normalized spectra measured at the fiber output corresponding to normal incidence injection conditions that maximized the SC bandwidth. The SC bandwidth is essentially independent of the pump wavelength, with the resulting SC spectrum spanning from 700–800 nm to 2800 nm in all cases, limited on the side of the longwavelength by the fiber’s intrinsic high attenuation. Interestingly, the GPI sidebands only appear visible in the spectrum when the pump is located in the normal dispersion regime (Fig. 6a–c) but they are not present when the pump is tuned to the anomalous dispersion region beyond 2000 nm (Fig. 6d–f).
This difference can be explained in light of the different SC generating dynamics at play in the two regimes. Specifically, in the anomalous dispersion regime, the initial stage of propagation is dominated by higherorder soliton compression and fission accompanied by the generation on the shortwavelength side of multiple dispersive waves phasematched by the selfimaging dynamics (see Supplementary Information and the spectrogram animation presented in the Supplementary Movie 2). At lower power values the dispersive wave components are clearly visible in the spectrum (see Supplementary Information) but when the injected power is significantly increased, the interaction between the ejected solitons and dispersive waves leads to the broadening of the dispersive wave components and their merging. Finally, we also performed an experimental comparison with the generation of a SC in a commercial GRIN silica fiber (see Methods). The result is illustrated in Fig. 6b as a black solid line for a pump wavelength at 1700 nm. The input energy was scaled to yield an injected intensity identical to the case of the PBG fiber as the two fibers have different core sizes. One can see how the SC spectrum generated in the PBG fiber extends much further to the midinfrared wavelengths as the result of both a higher nonlinear refractive index and lower dispersion value. We also note that the spectrum extends to shorter wavelengths in the silica fiber with significantly reduced power spectral density below the pump wavelength. This can be explained by the ZDW, which is further detuned from the pump as compared to the ZDW of the PBG fiber.
Discussion
Despite significant progress in midinfrared SC generation, there are still important challenges to be overcome in order to obtain SC spectra with characteristics similar to those achieved in the visible/nearinfrared. Silicabased platforms exhibit a relatively low nonlinear refractive index which requires significant injected energy to obtain an SC spectrum that extends into the midinfrared beyond 2 μm. While this has been achieved using very short lengths of singlemode silica fibers^{54,55}, it is much less straightforward in multimode fibers due to the large core diameter that reduces even further the nonlinearity. The material aspect is therefore particularly crucial and using nonsilica glasses with different refractive indices and a nanostructured core design, we have fabricated a gradedindex fiber with an order of magnitude enhanced nonlinearity compared to silica fibers. We have then reported periodic spatiotemporal instabilities and the generation of a broadband SC spanning twooctave up to the midinfrared. We performed a detailed study of the SC generation process as a function of the pump wavelength and pump power and our results show evidence of nonlinear beam cleaning. Experimental results were compared with spatiotemporal numerical simulations, and the remarkable agreement obtained over the full twooctave bandwidth allows us to confidently interpret the dominant physical mechanisms underlying the SC broadening, even in the complex multimode regime. Some limitations to the achievable spectral bandwidth remain, associated with the strong OH attenuation at longer wavelengths. We believe that improving the purification process of the constituent glasses will allow the extension of the SC spectrum even further, and open up a new avenue for the development of highbrightness SC sources in the midinfrared.
Methods
Fiber fabrication
The GRIN fiber was fabricated using a stackanddraw method and nanostructured core approach similar to those used to fabricate photonic crystal fibers and described in ref. ^{56}. The gradient index profile was obtained by stacking two types of PBG glass rods (PBG81 and PBG89) with distinct refractive indices. The rods have a diameter of 0.35 mm and when stacked together they yielded a structure with a total external diameter of 4.5 cm. The hexagonal structural arrangement of the rods was optimized using a stochastic simulated annealing optimization inspired by the MaxwellGarnet model^{57}. The stacked structure was subsequently drawn in order to fuse the glass rods together to a diameter of about 5 mm. The preform structure was then placed in an external tube acting as the fiber cladding and drawn to a diameter of 125 μm. After the drawing process, the subwavelength diameter of the rods effectively yields a gradient index in the core area^{16,56}.
Fiber characterization
The attenuation of the fabricated fiber was characterized over the 1200–2200 nm wavelength range using the cutback technique, a compact SC source (Leukos SM30450), and an optical spectrum analyser (Yokogawa AQ6375B). The initial fiber length of the sample used for the attenuation measurements was 150 cm and cut down to 30 cm in five consecutive cuts. For each fiber length, the spectrum of the transmitted light was measured and averaged over two different cleaves (with minimal change in the fiber length) in order to limit the possible impact of imperfect cleaving. The attenuation shown in Fig. 1 further accounts for Fresnel reflections that were measured from flatparallel polished glass samples of thicknesses 2 and 4 mm (PBG81) and 2 and 5 mm (PBG89). The group index n_{g} was measured using whitelight interferometry and the refractive (phase) index and associated Sellmeier coefficients were then calculated by fitting the first derivative of the Sellmeier equation to the measured group refractive indices^{58}
where n(λ) is the phase refractive index at wavelength λ and B_{i}, C_{i} are the Sellmeier coefficients. The Sellmeier coefficients extracted using this procedure are shown in Table 2 below for both PBG glasses.
Experimental setup
The femtosecond laser beam was focused into the fiber using a 5 cm focal length MgF_{2} planoconvex lens resulting in a beam radius of 25 μm (1/e^{2} intensity) at the input facet. The fiber holder was placed on a threeaxis precision translation stage to control the input coupling conditions. The fiber was laid straight without any bending or other stress. A dichroic filter was used to select the signal or idler depending on the pump wavelength, and a spectral longpass filter was inserted to attenuate the OPA pump residue. The maximum throughput was about 45% (calculated as a ratio of output to input power and including coupling efficiency, Fresnel reflection losses, and attenuation along with the fiber). Two optical spectrum analyzers (AQ6315B and AQ6376) were used to measure the SC spectrum in the 350–1700 nm and 1500–3400 nm wavelength range, respectively. Spatial intensity measurements were performed in the farfield at a distance of 2.5 cm using a beam profiling camera (Pyrocam IIIHR).
GRIN silica fiber
The GRIN silica fiber used for comparison in Fig. 6 is a 20 cm gradedindex multimode fiber (Thorlabs M115) with a core diameter of 50 μm, a numerical aperture of 0.2, and zerodispersion wavelength at ~1300 nm.
Noise measurements
To characterize the SC pulsetopulse intensity fluctuations, light from the fiber output was collimated and directed to a monochromator to filter out wavelength bands with a bandwidth of 6 nm. This bandwidth was selected to yield sufficient energy to be detected as well as to minimize wavelengthaveraging. An MgF_{2} planoconvex lens with a 10 cm focal length was used to focus light from the monochromator output to a 15 MHz photodetector (PbSe; PDA10DEC) connected to a fast 1 GHz realtime oscilloscope (LeCroy WaveRunner 6100 A). Spectral filtering at the monochromator output removed the secondorder diffraction of the SC shortwavelength components. The RIN defined as the standard deviation over the mean value was calculated by integrating the voltage of 4000 consecutive pulses after subtracting the noise background.
Geometric parametric instability sidebands
The frequencies of the geometric parametric instability sidebands f_{m} of order m where m = ±1, ±2, ±3... are governed by^{7,50}
where n_{2} is the nonlinear refractive index, ω_{0} is the center angular frequency, I is the pump pulses intensity, and β_{2} is the GVD coefficient at ω_{0}. The parameter \({z}_{p}=\pi {{{{{{{\rm{R}}}}}}}}/\sqrt{2{{\Delta }}}\) is the selfimaging period with Δ = (n_{co}−n_{cl})/n_{co} and R the fiber core radius. For our fiber at 1700 nm, n_{co} = 1.885, n_{cl} = 1.866, Δ = 0.0101, R = 40 μm, z_{p} = 0.89 mm, β_{2} = 6.16 × 10^{−26} s^{2} m^{−1}, n_{2} = 1.95 × 10^{−19} m^{2} W^{−1}.
Spatiotemporal numerical simulations
We simulate the propagation in the fiber using the 3+1D generalized nonlinear Schrödinger equation which is an extension of the GrossPitaevskii equation^{13,14}
where the field envelope A is expressed in \(\sqrt{W}/m\), r^{2} = x^{2} + y^{2}, \({\nabla }_{T}^{2}={\partial }_{x}^{2}+{\partial }_{y}^{2}\) is the transverse Laplacian, α represents the fiber attenuation, γ = ω_{0}n_{2}/c is the nonlinear coefficient, \(\hat{D}={\sum }_{n\ge 2}{(i{\partial }_{t})}^{n}{\beta }_{n}/n!\) is the dispersion operator expanded in terms of Taylorseries coefficients, and τ_{s} is the shock term. Note that these dimensions and the normalization of A used here are different from those conventionally used for the GNLSE with a constant mode area. For completeness, we also investigated the Raman contribution h_{R} to the overall dynamics. It was modeled as a delayed response using the conventional form
with τ_{1} = 5.5 fs, τ_{2} = 32 fs, and f_{R} = 0.05^{43}. Its effect on the propagation dynamics was found to be negligible (see Supplementary Information) and therefore was subsequently neglected in the simulations to reduce the computation time. We also use the experimentally measured attenuation (see Fig. 1). Note that the attenuation was measured over the range 1200–2200 nm and we use the bulk glass attenuation for wavelengths outside this range.
The simulations consider pulses of 350 fs duration and a Gaussian temporal intensity profile. Shot noise was added via onephotonpermode with a random phase in the frequency domain^{52} and we also include a 0.2% intensity noise in the time domain distributed across the fulltime window. The refractive index profile is defined along the radial coordinate by n(r) = n_{co}−ar^{2} for r ≤ R and n(r) = n_{cl} for r > R, where n_{co} and n_{cl} are the refractive indices at the center of the core and in the cladding, respectively, and a = (n_{co}−n_{cl})/R^{2}. Δ is the relative refractive index difference between the core and the cladding and R is the core radius.
The Taylorseries expansion coefficients for the dispersion operator are calculated for the refractive index along the fiber longitudinal axis at 1700 nm, and they are β_{2} = 6.16 × 10^{−26} s^{2} m^{−1}, β_{3} = 3.22 × 10^{−40} s^{3} m^{−1}, β_{4} = −8.96 × 10^{−55} s^{4} m^{−1}, β_{5} = 2.69 × 10^{−69} s^{5} m^{−1}, β_{6} = −4.44 × 10^{−84} s^{6} m^{−1}, and β_{7} = 3.40 × 10^{−99} s^{7} m^{−1}. The relative refractive index difference between the core and the cladding Δ = 0.0101, and the nonlinear refractive index n_{2} = 1.95 × 10^{−19} m^{2} W^{−1}.
At the center wavelength of 2500 nm, the dispersion coefficients are β_{2} = −1.30 × 10^{−25} s^{2} m^{−1}, β_{3} = 8.51 × 10^{−40} s^{3} m^{−1}, β_{4} = −2.18 × 10^{−54} s^{4} m^{−1}, β_{5} = 4.55 × 10^{−69} s^{5} m^{−1}, β_{6} = −5.74 × 10^{−84} s^{6} m^{−1}, and β_{7} = 3.46 × 10^{−99} s^{7} m^{−1} and the relative refractive index difference between the core and the cladding Δ = 0.0096.
The spatial intensity distribution of the beam is taken to be Gaussian with an input beam radius of 25 μm (1/e^{2} intensity radius). In order to mimic imperfections in the input beam and in the freespace coupling to the fiber, we use a similar approach as in ref. ^{19} and apply to the spatial amplitude distribution a multiplicative phase mask where a random phase shift from (0 to π) is added to each spatial grid points. During the very first few centimeters of propagation, large angular frequencies leak to the cladding and a super Gaussian filter is applied to absorb the field at the boundaries of the spatial window. The experimental attenuation of the fiber is similar to that of the bulk constituent glasses over the measured wavelength range (see Fig. 1). We include the attenuation in the simulation model as that of the average of the two glasses since it covers a larger wavelength span. The total losses resulting from leakage to the cladding and infrared absorption are in the range of 3–9 dB depending on the injected power and pump wavelength. The input energy values used in the simulations are adjusted to match the experimentally measured energy at the fiber output.
The simulation grid consists of 16,384 spectral/temporal grid points with a temporal window size of 20 ps and 64 × 64 spatial points with a window size of 160 × 160 μm. A step size of 37 μm was used in the propagation direction. The beam profiles in the farfield are calculated as the Fourier transform from the nearfield at the fiber output. The simulated spectra are averaged over 10 realizations with different noise seeds, and the spectra are convolved with a super Gaussian filter with 2 nm bandwidth corresponding to our experimental resolution.
Data availability
Data are available from the corresponding author upon reasonable request.
Code availability
The simulations code used in this manuscript is available from the corresponding author upon reasonable request.
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Acknowledgements
Z.E. acknowledges the Horizon 2020 Framework Programme (722380); L.S. acknowledges the Faculty of Engineering and Natural Sciences graduate school of Tampere University. J.D. acknowledges the French Agence Nationale de la Recherche (ANR15IDEX 0003, ANR17EURE0002, ANR20CE300004). G.G. acknowledges the Academy of Finland (333949, Flagship PREIN 320165).
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Z.E. performed all the experiments. Z.E., A.F., D.P., M.K., and R.B. designed and fabricated the fiber. L.S. and Z.E. performed the numerical simulations. Z.E., L.S., J.D., and G.G. performed the data analysis and interpretation, and all authors participated in the writing of the manuscript. GG planned the research project and provided overall supervision.
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41467_2022_29776_MOESM2_ESM.mp4
Simulated spectrogram evolution as a function of distance for a pump wavelength of 1700 nm in the normal dispersion regime of the fiber.
41467_2022_29776_MOESM3_ESM.mp4
Simulated spectrogram evolution as a function of distance for a pump wavelength of 2500 nm in the anomalous dispersion regime of the fiber.
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Eslami, Z., Salmela, L., Filipkowski, A. et al. Two octave supercontinuum generation in a nonsilica gradedindex multimode fiber. Nat Commun 13, 2126 (2022). https://doi.org/10.1038/s41467022297766
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DOI: https://doi.org/10.1038/s41467022297766
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