1. Introduction

Ultrashort intense laser pulses interacting with atoms and molecules can produce high-order harmonic generation (HHG) [1,2]. The harmonic spectrum shows a plateau structure, i.e. the intensity of the harmonics remains almost constant as the frequency of the light emitted increases. HHG is therefore an important tabletop type of coherent light source in the XUV and soft X-ray bands [3,4]. Due to the broad-spectrum plateau characteristic of harmonics, HHG has been applied to generate ultrashort pulses on the attosecond time scale for real-time detection of electron motion [5,6].

The mechanism of HHG can be explained by a semi-classical three-step model [7]: the electrons are first ionized by tunneling through a potential barrier formed by the atomic potential and the laser electric field, then they have the opportunity to return to the nuclear region to recombine with the parent ion under the action of the driving laser, producing HHG. Using this model, the cut-off position of the HHG plateau can be accurately predicted. However, it fails to present information on the intensity and spectral distribution of the HHG.

Since the HHG process is the coherent radiation resulting from the nonlinear response of all atoms and molecules in a gas target driven by the laser field. To achieve an accurate simulation of the HHG behavior, we need to numerically solve the time-dependent Schrödinger equation (TDSE) for the multi-electron atomic/molecular system in an intense laser field and the propagation equations of the driving laser field and the resulting harmonic field in the medium [8–11]. The demand of computational resources for numerically solving the TDSE which describes the multi-electron systems in an intense laser field increases exponentially with the computational dimension and the number of particles [12], and thus in many calculations concerning HHG, for example, the strong field approximation (SFA) scheme [13,14] or the time-dependent density functional theory (TDDFT) [15,16] is adopted. The TDDFT can efficiently calculate the HHG of the multi-electron system, whereas an accurate exchange-correlation generalized function suitable for strong-field calculations is lacking [17]. The SFA [13] or quantitative rescattering theory (QRS) [18] can give a clear quantum picture of the HHG mechanism, however, the effect on the excited states of the system cannot be accurately described. The influence of excited states on harmonics is a recent topic of interest due to its ability to reflect the internal structure of atomic and molecular systems [19–21].

With the development of the machine learning, researchers began to introduce the machine learning scheme into the study of strong field ionization and high-order harmonic radiation. Shevtov et al. used the convolutional neural network to calculate the nuclear spacing of hydrogen molecular ions [22]. Wang et al. reconstructed the induced dipole moment of molecules under the action of the light field by using high-order harmonic information and the machine learning method according to the photoelectron emission spectrum [23]. He et al. applied the machine learning scheme to high-order harmonic signal analysis to realize direct imaging of the molecular rotation [24]. Ofer et al. analyzed the high-order harmonic spectrum of chiral molecules and proposed a deep learning scheme to reconstruct the components of chiral molecules [25]. For the harmonic simulation process, Mihailescu proposed to study the plasma harmonic emission process by combining particles in cell simulation and the machine learning [26]. Gherman et al. directly skipped the time-dependent process simulation, studied the high-order harmonic emission considering the propagation effect by using the artificial neural network scheme, and analyzed the influence of laser focusing position and pressure on a certain harmonic [27]. Lytova et al. studied the high-order harmonic emission process of a two-dimensional system by using a deep learning scheme, and applied it to the classification of the molecular symmetry [28]. Through the above analysis, it is found that these studies can not solve the dilemma of high-order harmonic simulation of complex systems, the demand for computational resources increases exponentially as the number of electrons raises, making it impossible to obtain accurate results for harmonic calculations. To accurately study the HHG process theoretically, the three-dimensional TDSE needs to be solved [29,30]. Once the exact wave function of the system is obtained, the induced dipole moment of an atom in an intense laser field can be calculated, and the HHG spectrum can then be produced by the Fourier transform of the dipole moment. The induced dipole moment varies with time and contains the information about the harmonic emission with different frequencies at different times, and it can therefore be regarded as a time series [31] and HHG can be calculated using a time series prediction scheme, without the demand of computational resources to solve the corresponding multi-electron high-dimensional TDSE, for harmonic simulations, it is the process of high-frequency light generation in atomic and molecular systems under the action of a strong laser pulse. For a higher frequency light, which is also a higher frequency wave from the point of view of time, and thus periodic behavior is a characteristic of such systems, it is more common.

A time series prediction scheme is a method for predicting the probability of an event occurring at a future moment through past observations. Numerous theoretical methods have been developed to forecast future data with the help of theories depending on the data relationships [32–35,40–42]. Traditional time series forecasting methods are mainly linear model-based schemes that assume a linear relationship between past observations and the values of the variables predicted, however, nonlinear relationships cannot be well predicted by these models [43]. With the rise of neural networks, it has become a new paradigm for time series forecasting. From the original artificial neural network(ANN) to the later Long-Short Term Memory(LSTM), Gated Recurrent Unit(GRU), Transformer [36], Deep State Space Model [37], DeepAR [38], and Informer [39], all give different solutions to the time series classification and regression problems [40–42,44–49]. In this paper, the dipole moments of atoms or molecules induced by the laser field are predicted using the Adaptive k nearest Neighbour (AKN) method with the aid of neural networks [50,51]. In addition, we have also applied some new models such as LSTM for the prediction of time-dependent dipole moments, all with good results. More importantly, the time complexity of this method does not change with the increase of the number of electrons in the system, so it is expected to provide a new opportunity for fast and accurate simulation of HHG in complex atoms and molecules.

2. Theoretical method

In order to calculate the HHG from an atom or a molecule, we numerically solve the TDSE which describes the electrons in the atom or the molecule irradiated by a laser field (Unless otherwise stated, atomic units are used in this paper).

In detail, the AKN method use historical dipole moment data $X_{r}=\left (x_{r}, x_{r-1}, \ldots, x_{r-m+1}\right )$ operated by sliding time windows process to choose k sets of past time sequence $X_{r_{v}}, \mathrm {v}=1,2 \ldots, \mathrm {k}$ that are closest to the current time window based on Adaptive metrics. The length of each time windows is m, and the average value of linear transformation $\lambda x_{r+1}+\mu$ of the corresponding $r+1$ step value of each past time windows $x_{r+1}$ , is used as the predicted value of the current time sequence [52]. The main Adaptive Neural Network(ADNN) method differs from the AKN method in that the main ADNN method requires pre-training a neural network $g$ using historical data and select k sets of past time sequence $X_{r_{v}}(v=1,2 \ldots, k)$ that are closest to the current time sequence. Then k sets of past time sequence $X_{r_{v}}$ which are closest to the current time sequence are fed into the neural network $g$ and obtain output result $b_{v}, v=1,2, \ldots, k$. The average value of linear transformation of the k neural network outputs $\lambda b_{v}+\mu$ is taken as the prediction result of the main network of ADNN. In this paper, the length of the forward prediction is set as 1, the size of the sliding window m is set as 300, and the number of nearest neighbours k is set as 1.

3. Results and discussions

 

Fig. 1. HHG spectrum of a hydrogen atom irradiated by a laser pulse whose central wavelength is 800 nm and peak electric field strength is 0.042. The solid red line is the result obtained by numerically solving the TDSE, and the dotted blue line line is the result obtained by using AKN. The subplot shows the harmonic spectrum calculated by using the dipole moment of the predicted time.

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Using the numerical method, we have calculated the harmonic spectrum of a hydrogen atom irradiated by a laser pulse whose central wavelength is 800 nm and peak electric field strength is 0.042, as shown in Fig. 1. The solid red and blue dot lines display the results obtained by numerical solving the TDSE and by using the AKN prediction, respectively. Here, the dipole moment in the range of 0-992 from TDSE is used to calculate the harmonic spectrum, then taking advantage of the dipole moment between the time area 0-425, the dipole moment in the range of 425-992 can be predicted by the AKN method. The single-atom response of the harmonic emission spectrum is calculated using the dipole moment for the full driving laser action time. Here, in order to compare the differences between the predicted partial time-dependent dipole moments and the true values in detail, we show in subplots a comparison of the spectra calculated using the predicted partial dipole moments; detailed calculations of the harmonic emission spectra are presented in the supplementary material. As can be seen from the figure, odd harmonic generation is clearly observed in the results of both calculations, and there are some differences between the two simulations at lower non-odd harmonic intensities. The subplots give a comparison of the predicted part and the results obtained by truthfully numerically solving the TDSE at the corresponding time range. It can be observed that the harmonic spectrum can be reproduced accurately by the time-dependent dipole moment from the AKN.

On this basis, we further analyzed the time-frequency behavior of the HHG, as displayed in Fig. 2. It can be noticed from the figure that the harmonic spectrum shows a distinct periodical character, and distinct trajectories can be observed at every half optical cycle when the energy of the harmonics is above the ionization threshold. The time-frequency distribution of HHG predicted by the AKN (Fig. 2(b)) and ADNN(Fig. 2(c)) appears overall to be very close to that obtained by numerically solving the TDSE (Fig. 2(a)), whereas there are some differences at some time points. There are also some differences in the HHG behaviour for the period before the onset forecasting point (i.e., 425), because the analysis of the time-frequency distribution here is chosen to be done with wavelets, and a certain time width needs to be chosen in the calculation. As can be seen from the above analysis, both the AKN and ADNN method can give good predictions of the HHG process of an atom irradiated by an optical field based on the results of the dipole moment at part of the time.

 

Fig. 2. Time-frequency diagram of HHG calculated with time-dipole moments by (a) TDSE (b)AKN and (c)ADNN approaches.

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To further support the feasibility and reliability of the method, we deliberate the predicted results for different onset forecasting points. For a valid evaluation of the accuracy and error of the results, the evaluation metrics chosen here are mean absolute percentage error (MAPE) and normalized mean square error (NMSE). MAPE, takes the form as follows:

 

Fig. 3. Results of the statistical criterion of the AKN method for backward prediction of dipole moments at different onset forecasting points for a laser electric field with a peak amplitude of 0.042.

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Fig. 4. Distribution of the predicted values of the normalized dipole moment by (a) AKN and (b) main ADNN methods versus the true values, The peak amplitude of laser electric field strength is 0.042. The backward forecasting is conducted to 992.0. The horizontal axis is the predicted value of the dipole moment and the vertical axis is the true value of the dipole moment, and the red diagonal line represents the perfect predicted value, i.e. the predicted value is completely equal to the true value. The subplots show the error distribution of the model predictions.

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Since the statistical criterion alone does not reflect the difference between the predicted and true values, we calculated the distribution of the predicted and true values of the dipole moment of an atom irradiated, as shown in Fig. 4(a). Here, the time-dependent dipole moment of the system is first normalized for the convenience of the calculation and the normalized result is added by 0.1, because the MAPE evaluation index requires that the true value cannot be zero. It can be seen from the figure that the transformed values of the time-dependent dipole moment are mainly distributed around 0.6. If these points are located on the diagonal, it means that the predicted value is equal to the true one. The distribution of these points is more concentrated, but shifted downwards from the diagonal, as can be seen, more clearly from the statistical information in the subplots of the figure. This result reflects that the method has great accuracy, but slightly less precision. This means that there is a bias between the predicted and true values.To further improve the accuracy of the forecasting, we used the main ADNN method to forecast the time-dependent dipole moment, and the results are shown in Fig. 4(b). Different from the AKN method, after selecting the k sequences with the smallest Adaptive metrics values, the main ADNN method does not take the average of the linear transformations of the next step values of these sequences as the predicted values, but feds the k sequences into a fully connected neural network which has been trained with the historical data in advance, and the average of the linear transformations of the k outputs of the neural network is then used as the single-step predicted value. As can be seen from Figs. 4(a) and 4(b), the results predicted using the main ADNN method are mostly better than those predicted by the AKN, with the predictions scattered around the true value with a mean error around 0, definitely improving the accuracy of the predictions.

To show the difference between the predicted results of the AKN and ADNN methods more distinctly, we give the time-dependent dipole moments obtained by numerically solving the TDSE and by the two prediction methods, as in Fig. 5. It can be seen from the figure that the predicted values by two methods are very close to the true values and the behavior of the oscillations with the time are consistent, thus reproducing the HHG results very well. Compared with the results from the main ADNN, the predicted values from AKN are generally bigger than real values at high dipole moment amplitudes, and closer to true values at low amplitudes. At some local peaks of the amplitude, both the main ADNN and the AKN methods predict the dipole moment well. The subplot shows the absolute errors of the two methods, since the AKN method always predicts bigger values at the high dipole moment amplitude, the prediction error of the AKN method reaches about 10% at this point, thus, the errors of the AKN method(dark green line) are significantly larger in periods when the dipole moment amplitude is higher, while in the other periods the errors are basically the same as those of the main ADNN method (brown line). Furthermore , the errors of the main ADNN method are more uniform and generally smaller than those of the AKN method. In terms of average significance, the average absolute error of the ADNN main method is smaller than that of the AKN method.

 

Fig. 5. The predicted dipole moments in the time range 525-992 by the AKN (blue triangle line) and main ADNN method (red circle line), with the horizontal coordinate being the time and the vertical one being the value of the dipole moment after the normalization operation. Here, the true dipole moment from TDSE (green solid line) is also presented. The vertical dashed blue line is the cut-off point for the dipole moment forecasting.

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Fig. 6. High harmonic emission spectra of (a) argon atoms, (b) nitrogen molecules, and (c) $\mathrm {C}_{4} \mathrm {H}_{6}$ molecules, the red solid line is obtained by numerically solving TDDFT and the blue dotted line is obtained using AKN.

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With the scheme which accurately predicted the harmonic spectra of the hydrogen atom, we further investigate the generation of high-order harmonics from more complex systems. Fig. 6 presents HHG spectra of argon atoms, nitrogen, and $\mathrm {C}_{4} \mathrm {H}_{6}$ molecules calculated by the TDDFT scheme. The corresponding time-dependent dipole moments are also calculated from the AKN scheme. With the dipole moment, we calculated the harmonic spectra as shown in the dotted blue line from Fig. 6. It can be seen from the figure that the harmonic spectra calculated from the numerical simulation of different systems can be reproduced by using the AKN scheme, even including the detailed structures of the harmonic spectra.

Except for the AKN, we also used other deep learning methods to predict dipole moments of the multi-electron system, such as LSTM and Bi-LSTM. In Fig.7 we give a comparison of the time-dependent dipole moment predictions for Ar, $\mathrm {N_2}$ and $\mathrm {C_4H_6}$, respectively (a comparison of the statistical indicators for these schemes is shown in the supplementary material). It can be noticed from the figures that good predictions can be obtained using AKN, LSTM and Bi-LSTM. The AKN scheme has a larger error compared to the LSTM and Bi-LSTM schemes, but it has better prediction stability. As can be observed from the analysis in the previous discussion, the harmonic emission spectra obtained using the time-dependent dipole moments calculated with these scheme are accurate enough to be applied to most high-order harmonic emission studies. In addition, the time-dependent dipole moments of the system also has been predicted using the time series models Prophet and SARIMA, which are commonly used in the industry, as well as an attention method further optimized based on the LSTM, and the results are presented in the supplementary material. Due to the complexity of the studied objects, a particular scheme has not yet been found to be very perfectly applicable to all harmonic calculations. The applicability of the various networks and the corresponding laws of the physical processes will be further investigated in future work.

In our simulations, it is found that the accuracy of the time series prediction scheme also decreased as the simulation time became longer. According to our research, the prediction accuracy decreases slightly as the number of prediction steps increases, possibly because as the simulation time becomes longer, patterns in the future data did not appear in the historical data, resulting in the model not being able to predict the future data well through time series prediction. This can be seen in Fig. 4, where the computational accuracy oscillates less as the number of time points to be predicted increases. It can also be seen that the time series solutions all give good predictions, although the accuracy decreases as the prediction time becomes longer. More importantly, the accuracy of the time-dependent dipole moment calculation can be further improved by combining the neural network scheme (as shown in Fig. 4, the error of the AKN method is higher when we caculated the dipole moment in the laser electric field with the peak amplitude 0.042, so the ADNN model combining time series and neural network is used to further improve the prediction accuracy, similarly, the accuracy can also be improved by utilizing methods such as LSTM, as shown in Figs. 7(d) and 7(e)), which in turn can compensate for the loss of computational accuracy due to longer simulation times. In addition, experiments are usually conducted with ultrashort pulses of a few optical periods to generate attosecond pulses, and it is not necessary to calculate very long pulse effects. In general, the original AKN methods did not show any significant disadvantage over the new schemes, and even outperformed some models, while the new methods give better predictions for some dipole moments. However, none of these models achieved good prediction results on all data. Based on these results, it is found that both linear and non-linear autoregressive model works well on the prediction of high harmonic generation due to physical mechanism of the harmonic generation. Since the harmonic process is produced by ionized electrons returning to the nucleus under the electric field of the driving laser, a linear model can be used to give a good prediction overall due to the periodic nature of the driving laser. Because of the complexity of the system, harmonics produce by ionized electrons returning to the nucleus several times, interference of ionized electrons recombined with different atoms to generate harmonics, and the harmonic emission originated from different quantum electronic trajectories, which brings many non-linear features, and a non-linear scheme can be used to further enhance the accuracy of the simulation for some specific driving laser pulses and atomic or molecular system.

 

Fig. 7. Predicted-Real value distribution of dipole moment results for multi-electron systems predicted by different models

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For the AKN scheme the harmonic computational complexity is $O(nd)$, where n is the size of the dataset, approximately $10^4$, and d is the dimension of a single sample, chosen as 600 in this paper [53,54], and the computational complexity grows linearly with the size of the dataset and the dimensionality of the samples, and the complexity of the LSTM method is $O(mnd^2)$, where m is the size of the dataset, n is the length of a sequence, and d is the number of hidden layer neurons. n is 600 and d is 32 in this paper [36]. The computational complexity of these schemes does not vary with the atomic and molecular systems.

The calculation of the higher harmonics varies with the laser parameters and the system. For a single-electron system, the complexity is $O(m^2)$, where m is the number of spatial grid points or electron orbitals for harmonic calculations, and the order of magnitude is usually chosen to be $10^3$. For laser pulses with high intensities and low frequencies, m needs to be $10^4$, thus requiring the development of momentum space-based calculation schemes [55]. For the TDDFT method, the computational complexity reaches $O(n_e^2m^3)$[56], where $n_{e}$ is the number of electrons and the computational effort increases polynomially in the third order as the system increases. For the more accurate multistate time-dependent HF scheme, the computational complexity is $O(m^8)$ [57]. For simple systems, the study of harmonics from laser pulses with weak intensities and high frequencies can be accomplished by solving the time-dependent Schrödinger equation scheme directly, while for complex systems in laser pulses with higher intensities and lower frequencies, the time series prediction scheme displays great computational advantages.

4. Conclusion

In brief, by using the time series method combined with the neural network, we investigated the high-order harmonic processes of the atom and the molecule in the intense laser pulse. The harmonic spectrum is found to be well reproduced through the AKN time series scheme. The effects of the onset forecasting point as well as the peak field amplitude of the driving laser on the harmonic emissions are discussed, and the main ADNN method is further developed, which can significantly improve the prediction accuracy. Based on the results of this research, it can help to advance the study of HHG in multi-electron atomic and molecular systems where there are very great difficulties in calculations, and also can be applied to the simulation of time-dependent responses of complex systems in the strong laser pulse, such as the study of transient absorption processes, etc [58–60].