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From spectral broadening to recompression: dynamics of incoherent optical waves propagating in the fiber
PhotoniX volume 2, Article number: 15 (2021)
Abstract
Interplay between dispersion and nonlinearity in optical fibers is a fundamental research topic of nonlinear fiber optics. Here we numerically and experimentally investigate an incoherent continuouswave (CW) optical field propagating in the fiber with normal dispersion, and introduce a distinctive spectral evolution that differs from the previous reports with coherent modelocked fiber lasers and partially coherent Raman fiber lasers [Nat. Photonics 9, 608 (2015).]. We further reveal that the underlying physical mechanism is attributed to a novel interplay between groupvelocity dispersion (GVD), selfphase modulation (SPM) and inverse fourwave mixing (IFWM), in which SPM and GVD are responsible for the first spectral broadening, while the following spectral recompression is due to the GVDassisted IFWM, and the eventual stationary spectrum is owing to the dominant contribution of GVD effect. We believe this work can not only expand the light propagation in the fiber to a more general case and help advance the physical understanding of light propagation with different statistical properties, but also benefit the applications in sensing, telecommunications and fiber lasers.
Introduction
Dispersion and nonlinearity play important roles in fiberbased devices and systems. For example, dispersioninduced pulse broadening limits the capacity of fiberoptic communication systems [1]. Nonlinear Kerr effects such as selfphase modulation (SPM) and fourwave mixing (FWM) may contribute to the spectral broadening of optical waves [2], which is undesirable for some applications, e.g., narrowlinewidth fiber lasers [3], but benefits the novel light sources such as supercontinuum generation [4]. In general, the optical waves propagating in the fiber are affected by dispersion and nonlinear effects simultaneously, and the interplay between dispersion and different nonlinear effects leads to a multitude of interesting phenomena in nonlinear fiber optics, such as the conservative optical solitons which are enabled by the exact balance between dispersion and Kerr nonlinearity [5], and supercontinuum generation whose spectral evolution is affected not only by a variety of nonlinear effects but also by the dispersive properties of the fiber [6]. Several decades of investigations on nonlinear fiber optics have undoubtedly, not only promoted the growth of fundamental nonlinear physics [7, 8], but also benefited the engineering of fiberbased devices and systems [9, 10].
The propagation of optical waves in the fiber has become a simple but effective way for the investigations of interplay between dispersion and nonlinear effects [11,12,13,14]. The most extensively studied case is the propagation of ultrashort pulses. The interplay between dispersion, nonlinearity and other effects (such as mode coupling and dissipative effect) has resulted in a variety of optical solitons [15,16,17,18] and extremeamplitude rogue waves [19, 20]. Focusing on the spectral evolutions, the interplay between groupvelocity dispersion (GVD) and SPM effects leads to a monotonic but gradually saturated spectral broadening in the normal dispersion regime [2, 21], while soliton fission, dispersive wave, Raman solitons, and nonlinear interactions among them are responsible for the supercontinuum generation in the anomalous dispersion regime [22]. The partially coherent continuouswave (CW) laser beams propagating in the fiber also attracted considerable attention in recent years [23,24,25,26,27,28,29,30,31]. A series of distinctive spectral evolutions such as symmetric spectral broadening [23, 24], asymmetric spectral broadening [25], damped oscillations of the spectral tails [26], and supercontinuum generation are reported [27]. And besides the nonlinear Schrödinger equation (NLSE) [28], kinds of theories including the fully stochastic model [29], the phenomenological model [30], and the wave kinetics approach are established to describe the light propagation [31]. In 2015, S. K. Turitsyn et al. investigated the partially coherent Raman fiber lasers propagating in the fiber, and demonstrated a new nonlinear effect called inverse fourwave mixing (IFWM) [32], which may occur in the normal dispersion regime and cause the spectral compression of incident light. It’s an exciting progress but remains an open question as how to expand the light propagation in the fiber to a more general case. As it was known that, the statistical properties of incident waves play an important role in the spectral evolutions [33]. In fact, the propagation of coherent waves (e.g., modelocked lasers) has been studied extensively for decades, and the propagation of partially coherent waves such as Raman lasers has also been investigated in recent years. However, the incoherent waves (such as thermal light sources and freerunning lasers) propagating in the fiber and the underlying dynamics, which are both scientifically important and of great interest, have been relatively unexplored.
In this article, we numerically and experimentally investigated the spectral properties of incoherent CW optical waves propagating in the fiber with normal dispersion, and introduced a distinctive spectral evolution that differs from the previous reports with coherent modelocked fiber lasers and partially coherent Raman fiber lasers. The numerical simulations revealed that the underlying physical mechanism was attributed to a novel interplay between GVD, SPM and IFWM effects. The spectral recompression is particularly found to be the result of the GVDassisted IFWM effect. And propagation experiments employing an amplified spontaneous emission (ASE) source as the input were carried out, which agreed well with the numerical simulations. We believe this work can help expand the light propagation in the fiber to a more general case, broaden the physical understanding of light propagation with different statistical properties, and benefit the applications in sensing, telecommunications and fiber lasers.
Results
Numerical simulation of spectral evolutions
To investigate the influence of initial spectral width on the spectral evolutions of an incoherent CW optical field propagating in the fiber, we first carried out the simulations with a relatively narrow input spectrum (0.1 nm fullwidth at half maximum (FWHM) linewidth) and a relatively broad input spectrum (0.5 nm FWHM linewidth), and the spectral profile is assumed to be Gaussian for the sake of simplicity (see ‘Methods’ section for details). A similar trend of spectral evolutions is confirmed in the simulations, where first spectral broadenings followed by recompressions are observed (although the broad input spectrum shows no major changes). The input spectra, the broadest spectra and the eventual spectra after a rather long propagation distance (35 km) are shown in Fig. 1A, in which the input power P_{0} is fixed at 1.2 W. The spectrum with 0.1 nm initial FWHM linewidth shows a considerable broadening over ~ 2 km propagation distance, after that a recompression of the central part takes place while the tails are nearly unchanged. The spectral evolution with 0.5 nm initial FWHM linewidth undergoes a similar but less pronounced process, and the recompression point (the propagation distance where the spectral recompression occurs) moves backwards to ~ 350 m.
The spectral evolution shows a convergence after a quite long propagation distance. Interestingly, the eventual stable spectrum is neither fully Gaussian nor fully Lorentzian, and the corresponding spectra plotted in a logarithmic scale further verified this characteristic (see Fig. 1B). Indeed, the output spectrum looks more like a sum of two Gaussians, thus it cannot be fully expressed by a single parameter [23]. Here we introduce both FWHM and rootmeansquare (RMS) to describe the spectral width, and the corresponding spectral broadening factors (BFs) defined as Δλ_{out}/Δλ_{in} are plotted in Fig. 1C, D [30, 32]. The FWHMBF shows a rapid increase in the early stage of propagation, after that the spectral recompression occurs, the FWHMBF gradually decreases and eventually tends to a stable value as the propagation distance increases. The maximum FWHMBF decreases dramatically, and the recompression also takes place at a shorter propagation distance as the initial spectral FWHM increases. For example, the maximum FWHMBF with 0.1 nm initial FWHM linewidth reaches ~ 2.34 at the propagation distance of ~ 2 km, while that with 0.5 nm initial FWHM linewidth decreases to ~ 1.05, and the recompression point also moves backwards to ~ 350 m. Further simulations find that with a broader input spectrum, the maximum FWHMBF will tend towards 1, and the recompression point will move to 0, indicating that the spectrum will keep almost unchanged as the optical field propagates down the fiber (details are provided in Supplementary Section S1). In addition, an interesting feature is observed with 0.25 and 0.5 nm initial FWHM linewidth, where the eventual FWHMBF decreases to ~ 0.95 and ~ 0.99, respectively, indicating that the eventual output spectra become narrower than the input ones (although the narrowing seems not so pronounced). The RMSBF shows a similar evolution, with the only difference that the decrease looks less evident (Fig. 1D), which implies that the spectral tails are nearly unchanged during the recompression stage. Besides, in contrast to FWHMBFs, the eventual spectral BFs calculated with RMS linewidth will not decrease to < 1.
Dynamics of spectral evolution
To reveal the underlying physical mechanism of the distinctive spectral evolution, we focus on a snapshot of the optical field and analyze its evolution in both spectral and temporal domains. Without loss of generality, the case with 0.1 nm initial FWHM linewidth is chosen as an example. At the input, the temporal intensity profile within a time range of 200 ps mainly consists of two intense ‘unchirped pulses’ (see Fig. 2A, a broader view is shown in Supplementary Fig. S2) [34]. After a propagation distance of 0.6 km, the optical spectrum broadens due to the SPM effect, which generates redshifted frequency components at the leading edges and blueshifted frequency components at the trailing edges of the pulses [2]. At the same time, the GVD effect makes the original lowfrequency and newly generated redshifted frequency components move to the leading edges, and the original highfrequency and newly generated blueshifted components move to the trailing edges, resulting in the broadened and chirped pulses (see Fig. 2B).
When the optical field further propagates to 1.2 km, owing to the combined effects of SPM and GVD, the blueshifted frequency components near the trailing edge of the preceding pulse and the redshifted frequency components near the leading edge of the following pulse overlap in the temporal domain [35], the socalled IFWM effect takes place and regenerates the centralfrequency components [32], resulting in a bridgelike area in the spectrogram and a new pulse in the temporal profile (see Fig. 2C). With more and more centralfrequency components being regenerated, the optical spectrum starts to recompress, and the newly generated pulse becomes more and more intense (see Fig. 2D). As the propagation distance further increases, the whole process described above will repeat itself. Eventually, the temporal profile evolves into a bunch of random pulses with much narrower durations. And due to the fiber attenuation, the pulse intensities decrease exponentially so that the nonlinear effects become weaker and weaker (see Fig. 2E, F). After a rather long propagation distance, the GVD effect dominates the field evolution, and the adjacent pulses exchange their energies frequently, resulting in a stationary optical spectrum and a kinetic equilibrium of the spectrogram (see Supplementary Video S1 for the whole dynamic process). Besides the evolution dynamics of a snapshot, we also investigated the evolutions of statistical properties as the incoherent optical field propagates along the fiber, the results show that the spectral components are uncorrelated at the input, noticeably correlated in the middle distance, and uncorrelated again after a rather long propagation distance (see Supplementary Section S3 for details).
The physical mechanism described above also explains the evolution differences with broader input spectra. Since it was known that, a broader initial spectral width corresponds to a narrow fluctuation duration (see ‘Methods’ section) [34], thus the blueshifted frequency components of the preceding pulse and the redshifted frequency components of the following pulse will overlap at a shorter propagation distance, the IFWM effect and spectral recompression will also occur earlier. Correspondingly, the accumulation of the SPM effect as well as the maximum spectral BF decreases with the increase of initial spectral width. The evolution dynamics and statistical properties with initial FWHM linewidth of 0.5 nm are simulated and presented in Supplementary Section S4 and Supplementary Section S5, respectively.
According to the physical mechanism described above, we have derived an approximate equation governing the recompression point L_{r} (The detailed derivation is provided in Supplementary Section S6) [2]:
where β_{1} represents the firstorder dispersion coefficient, Δω_{0} is the 1/e halfwidth of the optical spectrum, δω_{max} stands for the maximum SPMinduced frequency chirp which satisfies:
where P_{0} is the average input power, α and γ stand for the attenuation coefficient and Kerr nonlinearity coefficient, respectively. By substituting Eq. (2) into Eq. (1), the solution of recompression point L_{r} can be obtained. Then the maximum RMSBF BF_{max} is given by [36]:
Figure 3A, B show the dependence of recompression point and maximum RMSBF on the initial FWHM linewidth. As can be seen, the solutions of Eq. (1) and Eq. (3) agree well with the NLSEbased simulations. As the initial spectral FWHM increases from 0.1 to 2 nm, the recompression point moves from ~ 1750 m backwards to ~ 88 m, and the maximum RMSBF decreases dramatically from ~ 2.93 to ~ 1.01.
Experimental setup
ASE source which originates from the random thermal noise, is assumed to be a good approximation of incoherent CW optical field [34, 37]. With the aid of a linewidthtunable 1064 nm ASE source [38] and sections of G.652.D fibers, we experimentally studied the spectral evolutions of ASE light propagating in the normal dispersion regime of optical fiber. The experimental setup is illustrated in Fig. 4, since the minimal FWHM linewidth of the ASE source is ~ 0.4 nm and to obtain a narrower spectrum, the ASE source is spliced with a circulator, followed by a highly reflective 1064 nm fiber Bragg grating (FBG). Thus, a narrower optical spectrum (orange line) reflected by the FBG can be obtained in port3 of the circulator, while the transmitted spectrum is converted to a doublepeak profile (blue line). Sections of G.652.D fibers with a length ranging from 100 m to 35 km are used as the transmission media, and an optical spectrum analyzer (OSA) with a resolution of 0.02 nm is used to measure the output spectra.
Spectral evolution with singlepeak spectrum
We first carried out the propagation experiments of the singlepeak spectrum. Three cases with different spectral widths are investigated, two of them are realized by the reflection of FBGs (0.11 and 0.23 nm reflection bandwidth, respectively), and the last one is obtained by directly removing the FBG and adjusting the FWHM linewidth of the ASE source to 1.47 nm (recalling that the ASE source is linewidthtunable with a minimal FWHM linewidth of ~ 0.4 nm). Figure 5A–C depict the spectral evolutions in a linear scale with the initial FWHM linewidth of 0.11, 0.23 and 1.47 nm, respectively. The input spectrum with 0.11 nm FWHM linewidth shows a Gaussianlike profile, while that with 0.23 nm FWHM linewidth looks a little irregular, and the spectrum with 1.47 nm FWHM linewidth approaches a superGaussian shape. As can be seen, the optical spectrum with 0.11 nm FWHM linewidth varies significantly during the propagation, showing a dramatic spectral broadening in the near 2 km and a noticeable recompression in the latter propagation distance. A similar evolution is observed with 0.23 nm initial FWHM linewidth. However, in contrast to the two cases above, the optical spectrum with 1.47 nm initial FWHM linewidth remains stationary during the whole propagation process.
Interestingly, more details can be found in the spectral wings (see Fig. 5D–F for the optical spectra in a logarithmic scale). Indeed, the input spectrum with 0.11 and 0.23 nm FWHM linewidth is not fully Gaussian especially considering the broad tails and the irregular peaks embedded in the tails. However, the spectra become smooth, regular and symmetrical after propagation, showing a selforganization effect [39]. And the optical spectrum with 1.47 nm initial FWHM linewidth is not completely unchanged during the propagation, its wings instead become more and more intense as the propagation distance increases. Figure 5 G shows the FWHMBF as functions of the propagation distance, it should be noted that here the input power P_{0} is fixed at 1.2 W (we also numerically and experimentally investigated the impact of input power, see Supplementary Section S7 for details). Regarding the case with 0.11 nm initial FWHM linewidth, the recompression occurs at ~ 2 km with the maximum FWHMBF reaching ~ 2.14, and the eventual FWHMBF recompresses to ~ 1.27. While with 0.23 nm initial FWHM linewidth, the recompression point moves backwards to ~ 1 km with the maximum FWHMBF decreasing to ~ 1.25, and the eventual FWHMBF also drops to ~ 0.81, indicating that the output spectrum has a narrower FWHM linewidth than the input one (which has been proved to be the impact of the initial spectral profile, see Supplementary Section S8). In terms of RMSBFs (Fig. 5 H), the spectral recompression seems not so pronounced, the RMSBF with 0.11 nm initial FWHM linewidth slightly decreasescan also introduce the dispersion length from maximal 2.04 to eventual 1.93, and that with 0.23 nm initial FWHM linewidth decreases from maximal 1.76 to eventual 1.66. More interestingly, the BF with 1.47 nm initial spectral FWHM keeps unchanged at 1 in terms of both FWHM and RMS linewidth. Incidentally, the distinctive spectral evolutions enabled by the interplay between dispersion and nonlinearity in the propagation naturally raises an intriguing question—how the dispersion and nonlinearity parameters influence the spectral evolution, which has been numerically studied and presented in Supplementary Section S9.
The experimental spectral evolutions of incoherent CW optical waves propagating in the fiber with normal dispersion have well confirmed the simulation results, and their unique properties contrast sharply with those of the pulsed sources (e.g., modelocked lasers), in which no spectral recompression occurs instead of a broadening saturation [2, 21, 40], and they also differ from those of the Raman fiber lasers, in which a relatively broad initial spectrum undergoes narrowing while the propagation, and the spectral narrowing becomes more and more pronounced as the initial spectral width increases (see Fig. 6A of Ref. [32]). To summarize, here the distinctive spectral evolutions indicate not only the ‘continuous’ but also the ‘incoherent’ natures of the input optical waves.
Spectral evolution with doublepeak spectrum
To further reveal the particularity of the incoherent optical field, we carried out the propagation experiment with a doublepeak spectrum. The unique spectral profile is obtained with a relatively broad singlepeak spectrum transmitting through the FBG, and the initial FWHM linewidth of the doublepeak spectrum reaches ~ 0.68 nm. Figure 6A shows the normalized optical spectra in a logarithmic scale, the intensity of the central valley is ~ 15 dB lower than the peaks at the input, the it increases to about − 2.5 dB due to the broadening and overlapping of the two peaks at 3 km, after that a slight spectral recompression is observed, and the intensity of the central valley decreases to about − 4.2 dB at 35 km. Figure 6B depicts the spectral evolutions in a linear scale, as can be seen, no matter how long the optical field propagates through, the spectrum only shows a slight broadening in terms of the FWHM linewidth. Besides, the evolution trend is similar to that of the singlepeak spectrum. The doublepeak spectrum broadens to its maximum with 3kmlong propagation, after that a spectral recompression is observed. The intensity of the central valley also shows a similar variation, which reaches its maximum with 3kmlong propagation, then decreases as the propagation distance further increases. In contrast to the partially coherent Raman fiber lasers, the doublepeak spectrum of an incoherent CW optical field will neither evolve to a singlepeak profile nor show a spectral narrowing as propagating in the normal dispersion regime [32], it seems more like a sum of two separate singlepeak spectra propagating independently, which further reveals the incoherent nature of the optical field. Figure 6C shows the spectral BFs as functions of the propagation distance. The FWHMBF increases to maximal 1.13 at the propagation distance of 3 km, then gradually recompresses to eventual 1.04, while the RMSBF shows a maximum value of 1.08 and a convergence of 1.05.
In addition, the random phase model of the CW incoherent optical field (see Eq. (5) of the ‘Methods’ section) enables us to use the experimental spectral profile with random phases as the input and simulated the spectral evolution of a doublepeak initial spectrum. The simulation matches nicely with the experimental result (see Supplementary Section S10 for details).
Discussion
Here we would like to first discuss the evolution differences with different initial spectral widths. Regarding the dynamics, since a broader initial spectral width corresponds to a narrow fluctuation duration, the blueshifted frequency components of the preceding pulse and the redshifted frequency components of the following pulse will overlap at a shorter propagation distance, thus the IFWM effect and spectral recompression will also occur earlier. Correspondingly, the accumulation of the SPM effect as well as the maximum spectral BF will decrease with the increase of initial spectral width. We can also introduce the dispersion length L_{D}=T_{0}^{2}/β_{2} and the nonlinear length L_{NL}=1/(γP_{0}) as references. As we know, the shorter the reference length (namely, the dispersion length and nonlinear length), the stronger the corresponding effect. With the same input power, the nonlinear length is identical for different initial spectral widths. In our case, the nonlinear length is estimated to be ~ 0.46 km at the input. However, the dispersion length varies with the initial spectral width (since the temporal fluctuation duration changes with the initial bandwidth). For example, the dispersion length is estimated to be ~ 11 km with 0.1 nm initial spectral FWHM linewidth, while it decreases to ~ 0.44 km with 0.5 nm initial spectral FWHM linewidth. Therefore, since the dispersion length is much longer than the nonlinear length (L_{D}/L_{NL}≫1) for narrow initial spectral widths, the nonlinear effect (here it refers specifically to the SPM effect) will dominate the spectral evolution at the early stage, accordingly, the effect of spectral broadening is stronger for narrow initial spectral widths.
We also would like to further discuss the differences with coherent pulsed light (e.g., modelocked fiber lasers), partially coherent CW light (e.g., Raman fiber lasers) and incoherent CW optical field propagating in the normal dispersion fiber. For pulsed light, the SPM effect will first broaden the optical spectrum and enhance the pulse broadening induced by the GVD effect, and the enhanced pulse broadening in turn weakens the SPM effect, eventually, the interplay between GVD and SPM effects leads to a monotonic but gradually saturated spectral broadening with the increase of propagation distance [2, 21]. For partially coherent CW optical waves such as Raman fiber lasers, the similarity with coherent modelocked fiber lasers is that they all have a variety of longitudinal modes, the difference is the correlation degree of the multiple longitudinal modes. As we know, the longitudinal modes of a modelocked fiber laser are phaselocked — in other words, these spectral components are coherent (or totally correlated). And the interference of these resonant modes leads to a train of ultrashort pulses. While regarding Raman fiber lasers, the turbulent FWM interaction of numerous longitudinal modes gives rise to the partial correlation of the spectral components (especially at the spectral wings), and the partial correlation results in the turbulent intensity fluctuations in the temporal domain. The spectral evolution of partially coherent Raman waves propagating in the normal dispersion fiber shows two cases: one is similar to that of incoherent optical waves propagating in the fiber — first spectral broadening followed by recompression, the other is monotonic but gradually saturated spectral narrowing [32]. In fact, the first case corresponds to a narrowband Raman fiber laser (usually operating at a low power), in which the turbulent FWM interaction of numerous longitudinal modes is weak, thus its output is similar to an incoherent optical wave; while the second case corresponds to a relatively broad initial spectrum (usually operating at a higher power), where the optical spectrum can be considered as an assembly of correlated spectral components generated by the FWM effect inside the cavity [41], thus the IFWM effect can occur at the beginning and compress the spectrum. Of course, SPM also takes place in this process but the IFWM effect dominates the evolution, and due to the power attenuation of light propagation, the interplay between IFWM and SPM effects leads to a monotonic but weakening spectral narrowing as the propagation distance increases (see Fig. 6D of Ref. [32]). Accordingly, we can treat the partially coherent Raman fiber laser as an intermediate state between the coherent modelocked fiber lasers and incoherent optical waves. When the turbulent FWM interaction of numerous longitudinal modes is weak, the output field of a Raman fiber laser comes closer to an incoherent optical wave; when the turbulent FWM interaction of numerous longitudinal modes is stronger, the output field of a Raman fiber laser can be treated as the partially coherent light; in the extreme, the longitudinal modes of a Raman fiber laser are totally phaselocked (or correlated), it becomes a coherent modelocked Raman fiber laser.
In contrast to coherent modelocked fiber lasers and partially coherent Raman fiber lasers, the optical spectrum of an incoherent CW field propagating in the normal dispersion regime evolves consistently no matter how broad the initial spectrum is, that is, broadens first afterwards recompresses. In the beginning, the SPM effect dominates the evolution and the IFWM effect cannot take place since the spectral components of an incoherent CW field are uncorrelated. Thus the initial spectral broadening is due to the interplay between SPM and GVD effects, just like the case of pulsed light. However, with more and more correlated redshifted components and blueshift components being generated through the SPM effect, and overlapping in the temporal domain through the GVD effect, the IFWM effect can occur and compress the optical spectrum. And since the nonlinear effects become extremely weak due to the power attenuation of longdistance propagation, the ultimate optical spectrum remains stable owing to the dominant contribution of the GVD effect. To better understand the evolution dynamics with different input optical waves, a detailed comparison is presented in Supplementary Section S11.
Conclusions
In conclusion, we numerically and experimentally investigated the propagation of an incoherent CW optical field in the fiber with normal dispersion, and demonstrated a distinctive spectral evolution that differs greatly from the previous reports with coherent modelocked fiber lasers and partially coherent Raman fiber lasers. The results show that, no matter how broad the initial spectrum is, the output spectrum undergoes a broadening followed by a recompression as the propagation distance increases. Besides, the broader the initial spectrum, the earlier the recompression occurs, and the smaller the maximum spectral broadening factor. The underlying physical mechanism is attributed to a novel interplay between GVD, SPM and IFWM effects, in which SPM and GVD are responsible for the first spectral broadening, while the following spectral recompression is due to the GVDassisted IFWM, and the eventual stationary spectrum is owing to the dominant contribution of GVD effect. According to the physical mechanism, an approximate equation governing the recompression point was derived, which agrees well with the NLSEbased simulations. Propagation experiments employing an ASE source as the input were carried out, which confirmed the results of numerical simulations. Specifically, the result with a doublepeak input spectrum differs greatly from that with Raman fiber lasers, indicating the critical importance of spectral correlation on the propagationinduced spectral changes. We believe this work can not only expand the light propagation in the fiber to a more general case and help advance the physical understanding of light propagation with different statistical properties, but also benefit the applications in sensing, telecommunications and fiber lasers.
Methods
Light propagation in the fiber
The propagation of an optical field along the fiber can be modeled by the wellknown NLSE [2]:
where A(z, T) is the electric field envelope, z is the propagation distance, T represents the time in a frame of reference moving at the group velocity of the central frequency. α, β_{2} and γ stand for attenuation coefficient, secondorder dispersion coefficient and Kerr nonlinearity coefficient, respectively. The NLSE is numerically integrated using the socalled splitstep Fourier method (SSFM) [2], and the following paramenters are used in the simulation: α = 0.84 dB/km, β_{2} = 25 ps^{2}/km and γ = 1.8 (W×km)^{−1}, which are the same with the fiber parameters employed in the experiment. In addition, the G.652.D fiber we used features an 8.2 μm core diameter with the numerical aperture (NA) of 0.14, supporting not only the fundamental LP_{01} mode but also the LP_{11} mode at 1064 nm. However, we have carried out the modal decomposition to verify the singlemode propagation [42], thus proving the validity of using singlemode NLSE in the simulations (see Supplementary Section S12). What’s more, the simulated spectrum via single realization is quite noisy due to the initial random phase [23], thus an ensemble average over 2400 realizations and a last filtering of the optical spectrum are conducted. Considering that each realization is independent, the simulations are performed on a parallel computing platform with 4 notes, and each note consists of 24 cores (2.65 GHz clock speed) and 64 GB RAM.
Modeling the CW input field
Before simulating the optical field propagating in the fiber, modeling the CW input field is of critical importance. However, the previously reported models such as the phasediffusion model and CW with one photon per mode approach [25, 28], cannot properly simulate the spectral property and temporal behavior at the same time. The most promising model so far is to simulate the actual laser oscillation using the set of coupled NLSEs and rate equations (if the active gain is involved) [43, 44]. However, for precisely simulating the spectral shape and the phase relationship between different spectral components, not only the experimental mirror profiles but also the exact values of fiber parameters are required [45, 46], which makes this approach quite complex and challenging. Fortunately, the case would be much simpler if assuming the incoherent optical field is a random complex field, then the input field can be expressed as a sum of Fourier components [47]:
where the Fourier modes \({\hat {X}_m}=\left {{{\hat {X}}_m}} \right\exp \left( {i{\varphi _m}} \right)\)are complex variables with random phases \({\varphi _m}\) uniformly distributed between 0 and 2π [48, 49]. Accordingly, the power spectrum \(A\left( {z=0,t} \right)\) of the random field reads:
And for the sake of simplicity, the initial optical power spectrum is assumed to be Gaussian with the FWHM linewidth of Ω_{L}:
Supplementary Fig. S13A shows the simulated temporal intensities A(z = 0, t)^{2} for four spectral FWHM linewidths as indicated. It’s clear that the temporal intensity profile of the socalled CW optical field is not constant, in contrast, it’s comprised of a random succession of fluctuations with time scales of picoseconds, and the fluctuation duration decreases as the spectral width increases [37]. The intensity autocorrelation functions (ACFs) illustrated in Supplementary Fig. S13B show the average durations of the temporal fluctuations, indicating a decrease from ~ 16.2 to ~ 3.3 ps with the spectral FWHM linewidth increasing from 0.1 to 0.5 nm. Obviously, an incoherent CW optical field with such intense temporal fluctuations propagating in the fiber will undergo a variety of nonlinear effects, e.g., the SPM effect, which will in turn reshape the temporal intensities as well as the spectral properties of the optical field.
Experimental setup of the ASE source
The ASE source is counterpumped by a 976 nm laser diode through a (2 + 1)×1 signalpump combiner, the pump light is launched into 8mlong 10/130 µm Ybdoped fiber with a cladding absorption of ~ 4 dB/m. To eliminate the end feedback induced parasitic lasing, two broadband isolators are utilized to improve the isolation ratio. The broadband ASE source is spliced with a bandwidthtunable optical filter, through which a linewidth tuning range of 0.4–15 nm can be obtained.
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 CW:

continuouswave
 GVD:

groupvelocity dispersion
 SPM:

selfphase modulation
 IFWM:

inverse fourwave mixing
 FWM:

fourwave mixing
 NLSE:

nonlinear Schrödinger equation
 ASE:

amplified spontaneous emission
 FWHM:

fullwidth at half maximum
 RMS:

rootmeansquare
 BF:

broadening factor
 FBG:

fiber Bragg grating
 OSA:

optical spectrum analyzer
 SSFM:

splitstep Fourier method
 NA:

numerical aperture
 ACF:

autocorrelation function
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Acknowledgements
The authors thank Jiaxin Song and Dr. Zhiyuan Dou for help with the spectral measurement, Yi An and Dr. Liangjin Huang for assistance with the modal decomposition, and Dr. Jian Wu for providing the computing resources. We also acknowledge Prof. Zhichao Luo and Prof. Wencheng Xu from South China Normal University for fruitful discussions.
Funding
National Natural Science Foundation of China (NSFC) (61905284, 62035015, 62061136013).
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All the authors contributed to the interpretation of the results and manuscript writing. J.Y., X.M., and Y.Z. performed the experiments; H.Z., J.X., and P.Z. conceived the idea; J.Y. conducted the numerical simulation; T.Y. and J.L. provided the technical support in experiment and data analysis; J.X. and P.Z supervised the project. The author(s) read and approved the final manuscript.
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Ye, J., Ma, X., Zhang, Y. et al. From spectral broadening to recompression: dynamics of incoherent optical waves propagating in the fiber. PhotoniX 2, 15 (2021). https://doi.org/10.1186/s4307402100037x
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DOI: https://doi.org/10.1186/s4307402100037x
Keywords
 Nonlinear fiber optics
 incoherent light
 propagation dynamics
 spectral recompression