Modelling self-similar parabolic pulses in optical fibres with a neural network

We expand our previous analysis of nonlinear pulse shaping in optical fibres using machine learning [Opt. Laser Technol., 131 (2020) 106439] to the case of pulse propagation in the presence of gain/loss, with a special focus on the generation of self-similar parabolic pulses. We use a supervised feedforward neural network paradigm to solve the direct and inverse problems relating to the pulse shaping, bypassing the need for direct numerical solution of the governing propagation model.


I. Introduction
Machine learning is transforming the scientific landscape, with the use of advanced algorithmic tools in data analysis yielding new insights into many areas of fundamental and applied science [1]. Photonics is no exception [2][3][4], and machine-learning methods have been applied in a variety of ways to optimise and analyse the output of optical fibre systems. For example, from a control and feedback perspective, various machine-learning strategies have been deployed to design and optimise mode-locked fibre lasers [5][6][7][8][9], optimise optical supercontinuum sources [10], analyse the complex nonlinear dynamics occurring during the buildup of supercontinuum [11][12][13], and control pulse shaping [14].
In parallel with these developments, pulse shaping based on nonlinear propagation effects in optical fibres has developed into a remarkable tool to tailor the spectral and temporal content of light signals [15,16], leading to the generation of a large variety of optical waveforms such as ultra-short compressed pulses [17], parabolic- [18], triangular- [19,20] and rectangular- [21] profiled pulses. Very different features of nonlinear pulse evolution can be observed depending on the sign of the fibre group-velocity dispersion (GVD), which results in specific changes of the pulse temporal shape, spectrum and phase profile. Specifically, the temporal and spectral features of pulses propagating in fibres with anomalous GVD are typically governed by soliton dynamics.
Conversely, because the nonlinear dynamics of pulses propagating in fibres with normal GVD are generally sensitive to the initial pulse conditions, it is possible to nonlinearly shape conventional laser pulses into various specialised waveforms through control of the initial pulse temporal intensity and/or phase profile [15]. The presence of gain or losses can further modify the nonlinear propagation dynamics. For instance, propagation in a normally dispersive fibre with distributed gain leads to the attraction of the temporal and spectral properties of any initial pulse towards a parabolic intensity profile which is then maintained with further propagation. Parabolic selfsimilar pulses in fibre amplifiers are, along with solitons in passive fibres, the most well-known classes of nonlinear attractors for pulse propagation in optical fibre [22,23]. The unique properties of parabolic similaritons have stimulated numerous applications ranging from high-power ultrashort pulse generation to optical nonlinear processing of telecommunication signals. Yet, due to the typically large number of degrees of freedom involved, optimising nonlinear pulse shaping for application purposes may require extensive numerical simulations based on the integration of the nonlinear Schrödinger equation (NLSE) or its extensions. This is computationally demanding and potentially creates a severe bottleneck in using numerical techniques to design and optimise experiments in real time.
In our previous work [24], we presented a solution to this problem using a supervised machine-learning model based on a feedforward neural network (NN) to solve both the direct and inverse problems relating to pulse shaping in a passive, lossless fibre, bypassing the need for numerical solution of the governing propagation model. Specifically, we showed how the network accurately predicted the temporal and spectral intensity profiles of the pulses that form upon nonlinear propagation in fibres with both normal and anomalous dispersion. Further, we demonstrated the ability of the NN to determine the nonlinear propagation properties from the pulses observed at the fibre output, and to classify the output pulses according to the initial pulse shape. In this paper, we extend the use of our model-free method and show that a feedforward NN can excellently predict the behaviour of nonlinear pulse shaping in the presence of distributed gain or loss. The network is able to infer the key characteristics of parabolic pulses with remarkably high accuracy, and to successfully handle variations of the pulse parameters over more than two orders of magnitude.

II. Governing propagation model and neural network Situation under study, model and analytical results
Optical pulse propagation in an optical fibre with distributed gain/loss can be well described by a modified NLSE including the effects of linear GVD, nonlinear self-phase modulation (SPM) and linear gain/loss [25]. Using the dimensionless variables 0 / uP   , ξ = z/LD and τ = t/T0, this equation is written as Here, ( , ) zt  is the complex electric field envelope in a comoving system of coordinates, T0 and P0 are a characteristic temporal value and the peak power of the initial pulse, respectively, LD = T0 2 /|2|, LNL = 1/(P0) and / D NL N L L  are the dispersion length, the nonlinear length and the 'soliton-order' number (or power parameter), respectively,  =  LD, and β2, γ and  are the respective GVD, Kerr nonlinearity and gain/loss coefficients of the fibre, where  >0 ( <0) describes distributed gain (loss). We consider here propagation at normal dispersion (characterized 4 by β2 >0), and we neglect higher-order linear and nonlinear effects as the leading-order behavior is well approximated by Eq. (1) for picosecond pulses. Moreover, neglecting higher-order gain effects is well-suited to describe experiments that use broadband rare-earth fibre amplifiers. We note that the dimensionality reduction entailed by Eq. It is useful to review here the main features of the pulse evolution in a normally dispersive fibre as described by Eq. (1) in the nonlinearity-dominant regime of propagation, i.e. when ≫ 1. In the absence of gain, during the initial stage of propagation, the combined action of SPM and normal GVD makes a standard laser pulse (such as a Gaussian or hyperbolic secant waveform) acquire a frequency chirp with a linear variation over the pulse center, which results in the broadening of the pulse and reshaping into a convex-up parabola. However, the parabolic shape formed is not maintained with propagation, but the pulse evolves toward a near-trapezoidal form with a linear frequency variation over most of the pulse and ultimately, when the shifted light overruns the pulse tails, the wave breaks and develops oscillations on its edges [26]. After the distance at which wave breaking occurs, and with the accumulation of a parabolic spectral phase induced by dispersion, the temporal and spectral content of the pulse become increasingly close to each other and the temporal intensity profile does not evolve anymore. This long-term far-field evolution corresponds to the formation of a spectronic nonlinear structure in the fibre [27].
The situation is different for an intense pulse with an initial parabolic profile, which keeps its shape and acquires a linear chirp upon propagation in the fibre, thereby avoiding the degrading effect of wave breaking [28]. Moreover, in contrast to passive propagation where the parabolic waveform that develops after initial propagation represents a transient state of the pulse evolution over a distance depending on the initial conditions [29], the presence of gain in the fibre results in any arbitrary pulse asymptotically reshaping into a parabolic intensity profile, and this pulse form 5 is maintained with further propagation. The dynamic evolution of the intensity of any pulse in the limit   can be in fact accurately described by [22]     22 1/ where the dynamic equation for the characteristic width P has the explicit solution

Numerical simulation and neural networks employed
The data from numerical simulations of the NLSE (1) is used to train a NN and validate its predictions. Equation (1) is solved with a standard split-step Fourier propagation algorithm [25] using a uniform grid of 2 14 points on a time window of length 280 T0. The latter is chosen wide enough to accommodate the large temporal broadening experienced by the pulses at normal GVD.
Our simulations do not include any source of noise such as, e.g., quantum noise. Propagation in the fibre is studied up to a normalized distance of  = 8, for input powers N up to 4.5 and  ranging from -0.36 to 0.8 (i.e. covering both loss and gain). Our main interest here is in temporal and spectral intensity profiles (I = and S, respectively) that can be directly recorded in experiments, rather than in the complex envelope of the electric field. As we decide to normalize both I and S by their peak value, information regarding the pulse energy is lost. The simulation pulse profiles are then anamorphically sampled along the temporal and spectral dimensions so as to capture both the details of the short pulses observed at relatively short distances in the fibre and the much longer pulses that are formed at larger distances. Moreover, exploiting the symmetry of the problem, we restrict our sampling to the positive times and frequencies only. Hence, we consider A = 90 points on the interval 0 to 150 T0 to represent the temporal intensity profiles of the pulses, and B = 60 points on the interval 0 to 5/T0 for the spectral intensities. Note that the total number of data points A+B is significantly larger than that used in our previous study [24]. The sampled data with initial conditions taken randomly are used to train a NN and validate its predictions.
We employ a feedforward NN structure relying on the Bayesian regularization back propagation algorithm and including three hidden layers with eighteen, fourteen and ten neurons, 6 respectively, as shown in Fig. 1. This NN is programmed in Matlab using the neural network toolbox. The NLSE numerical simulations are performed on a computer cluster (one and half days is approximately required to produce 25×10 3 data samples when 12 processor cores are used). The training and use of the NN are achieved on a standard personal computer (Intel Xeon processor, 6 processor cores, 3.6 GHz, 32 GB RAM).

III. Model-free modelling of nonlinear pulse shaping Prediction of the output pulse properties
The direct-problem NN learns the NLSE model from 25000 numerical simulations of the propagation of a transform-limited Gaussian pulse with randomly chosen combinations of , N and After training, the network is tested on 10 5 new simulations from random initial conditions.  where the expected profiles are interpolated to the same time or frequency points used for sampling the network output. The maps of misfit parameter values in the space of input parameters ( ,N,) are given in Fig. 3(a) and confirm that the nonlinear pulse shaping occurring in the fibre is very closely reproduced by our NN. The distributions of the misfit parameter values [ Fig. 3(b)] show that, remarkably, more than 92% of the error realizations are well confined to values below 0.02.
These results are significantly better than those reported in [24], where the anomalous dispersion regime of the fibre was more complex to deal with. Distributions of the temporal (subplot b1) and spectral (subplot b2) misfit parameter values. 10 To further assess the prediction ability of the direct-problem NN, we also compare the temporal and spectral widths (FWHM) of the output pulse as measured from NLSE simulations, which rely on a very fine time and frequency discretization (2 14 points), with those extracted by the NN from the reduced set of sampling points (only 150 unequally spaced points). The results are summarized in Fig. 4(a), and highlight that even though the pulse temporal duration varies by more than two orders of magnitude across the space of input parameters, the network is able to retrieve this characteristic with a remarkable level of confidence. The spectral width is also predicted accurately. Further to this, we also compute the excess kurtosis [32] (defined as propagation lengths longer than 1 (inset plots in Fig. 4(b)) with the full set of data, we can see that the most important discrepancy occurs during the first stage of the nonlinear propagation. We explain this as partly due to a stronger impact of the limited number of sampling points fed to the network on the computation accuracy for the shorter pulses. Finally, we note that the use of a NN that we make in this work is different from and more powerful than that presented in [14], where the NN was trained using the temporal and spectral pulse durations and kurtosis parameter directly.

Parabolic pulse properties
We now assess the ability of the network to identify the region of the parameter space (, N, ) that enable the formation of a parabolic waveform in the fibre. To this end, the trained NN is interrogated with 10 6 new simulations covering the whole input parameter space and asked to isolate the region that leads to the desired output pulse form by using the excess kurtosis of the calculated intensity profiles as a measure of shape. An excess kurtosis between -0.9 and -0.8 is typical of a parabolic intensity distribution. We note that the NN performs this task in only a few seconds. We can see in Fig. 5(a) that a parabolic pulse is found in two different regions of the parameter space. The region featuring short propagation lengths and high soliton-number values corresponds to the transient state of the nonlinear dynamic pulse evolution toward wave breaking in a passive fibre, and to the early stage of evolution toward the asymptotic solution in a gain fibre.
This reshaping stage has been discussed in [26,33] and used in several experiments to produce parabolic profiles [34,35]. The region featuring large propagation lengths and  underpins the asymptotic parabolic regime in a fibre amplifier. The results relating to the spectral shape of the pulse shown in Fig. 5(b) highlight the different nature of the dynamics enabled by the two regions.
Whereas the second region also supports a parabolic pulse form in the spectral domain, this is not the case for the first region. Note that the formation of parabolic spectronic pulses in a passive fibre [27,36,37] with a Gaussian pulse initial condition falls beyond the boundary of the parameter space being studied.   Fig. 6(b), and confirms that longer propagation length and higher gain enables pulse properties that increasingly approach the asymptotic limits. This difference, however, may still affect inverse design problems. Owing to the three-dimensional representation of the shaping problem being studied, we can deploy a graphical method [14] and, based on the NN predictions, isolate the region of input parameters that supports a given large pulse duration, e.g., As it is seen in Fig. 6(c), the enabling parameters are not limited to a single combination but are spread over a large surface area. The quite significant deviation of this region from that obtained from Eq. (3) indicates that the asymptotic analysis may not be an efficient tool for solving inverse problems. Raman amplifiers [38]. The initial pulse profiles shown in Fig. 7(a) show a non-negligible difference in terms of pulse duration and peak power.

Retrieval of the propagation characteristics
For the inverse problem at hand, the trained network is tasked with the retrieval of the propagation parameters , N and  from a pulse shape and spectrum generated after propagation in the fibre.
As we noted in [24], the use of a reverse propagation method [39,40]

Identification of the initial pulse shape
For this problem, we train the network on an ensemble of 36000 simulated output pulses from the fiber corresponding to a mix of Gaussian, hyperbolic secant, parabolic and super-Gaussian initial pulse shapes and randomly chosen combinations of input parameters , N and .
Then we ask the trained network to categorize 4 × 10 5 new unlabeled simulated output pulses according to the initial waveform. As we can observe from Fig. 9(a), the classification accuracy of the NN algorithm is remarkably high: there are only 859 errors, which represent 0.2% of the total number of input samples. Figure 9(b) indicates that once again the errors occur in the parameter region corresponding to the onset of the asymptotic parabolic regime in a fibre amplifier [see Fig.   5(a) and Fig. 6(b)]. Indeed, as it is also seen in Fig. 7(c), the process of attraction towards the asymptotic parabolic shape tends to conceal the impact of the initial condition, thereby making the waveform recognition more challenging for the network.

IV. Conclusion
In this paper, we have successfully applied machine learning to solve the direct and inverse problems relating to the shaping of optical pulses during nonlinear propagation in a normally dispersive fibre in the presence of distributed gain or loss, and our results expand previous studies of nonlinear pulse shaping using machine learning [24]. Using a feedforward NN trained on numerical simulations of the NLSE, we have shown that the temporal and spectral properties of the output pulses from the fibre can be predicted with high accuracy. The network is able to accommodate to and maintain high accuracy for a wide dynamic range of pulse parameters. The key properties of parabolic self-similar pulses are successfully reproduced by the NN model.
Remarkably, within the range of system parameters considered, any temporal and spectral shape generated at the fibre output from the propagation of an initial Gaussian pulse can be associated with a single set of normalized propagation length, soliton-order number and gain/loss values, (, N, ).
Our results show that a properly trained network can greatly help the characterization and inverse-engineering of fibre-based shaping systems by providing immediate and sufficiently accurate solutions. However, the presence of a global attractor in a fibre amplifier system for arbitrary initial pulse conditions can potentially introduce uncertainty in the inverse design problem or in the recognition of the input pulse characteristics. Although this remains to be studied in further detail, our results suggest that the use of NNs for pattern recognition in systems that possess asymptotic self-similar solutions may possess limitations. Note that we have also limited our discussion here to initially Fourier transform-limited pulses. Future steps may expand the parameter space of the NN operation for chirped initial pulses [14,24]. The inclusion of higherorder propagation effects such as nonlinear longitudinal variation of the gain or gain saturation will add complexity to the nonlinear shaping problem by breaking the simple scaling laws from the NLSE. However, a NN algorithm should be able to tackle this problem and greatly accelerate its solution. Other future lines of research could be the analysis of the impact of noise on the accuracy of the NN predictions, and the deployment of convolutional NNs [13].