# Artificial Neural Network Trained to Predict High-Harmonic Flux

^{1}

^{2}

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## Abstract

**:**

## Featured Application

**We present a method based on artificial neural networks—as an efficient procedure—aimed to predict the outcome of high-order harmonic generation experiments with previously unexplored parameters. The main goal is to provide quick help for attosecond science laboratories in designing experiments.**

## Abstract

## 1. Introduction and Motivation

_{1}; ω

_{2}] is the spectral domain of interest, R is the radial extent of the interaction region, E

_{h}(ω,r) is the harmonic field in spectral representation.

## 2. Theoretical Models and Numerical Methods for High-Harmonic Generation and Artificial Neural Networks

#### 2.1. High-Harmonic Generation

#### 2.2. Artificial Neural Networks

_{i}) and medium length (L

_{med}). Consequently, these four process parameters are chosen to be the ANN inputs, whereas the harmonic yield is the model output. Based on the 3D non-adiabatic model, a total number of 453 input-output data sets were obtained and, subsequently, were divided in: (i) training data which represent 77.9% of the total amount of input-output data sets, (ii) validation data (15.7%) and (iii) testing data (6.4%). The testing data subset is obtained by uniformly extracting every sixteenth sample from the initial data. From the remaining data, every sixth sample was extracted to obtain the validation subset, whereas the remaining data represent the training data subset. The higher percentage of the training and validation subsets with respect to the testing subset is explained by the fact that the process is extremely nonlinear and, for good training results, more data are needed for the training and validation steps.

_{si}is the simulated output value, X

_{ANNi}is the model predicted output value, $\overline{{X}_{S}}$ and $\overline{{X}_{ANN}}$ are the arithmetic means of the experimental and respectively, of the predicted output values.

## 3. Results and Discussion

^{2}intensity) of the beam at the focusing element is 30 mm, total length of the interaction region is 20 cm, gas medium is argon. The scanned parameters with the values chosen for the full 3D calculations are the following: input laser pulse energy E (0.77; 3; 6; 9; 11.6) mJ; argon pressure p (0.1; 0.38; 1.41; 5.31; 20) mbar; cell entrance position relative to the nominal focus z

_{i}(−50; −25; −20; −10; 0; 10; 25; 50) cm, medium length L

_{med}(4; 8; 12; 16; 20) cm. Negative values for z

_{i}stand for gas cell beginning before the focus position. Although the parameters are very sparsely spaced, scanning along all these parameters and performing complete 3D simulations, while also to individually analyze and interpret the results means an enormous data pool and working time and effort, which would be very inefficient. In Figure 3 we represent in a very schematic manner the general configuration (Figure 3a) and the cases for which the complete 3D simulations have been performed, checked, validated and analyzed (Figure 3b).

_{i}= −25 cm; L

_{med}= 16 cm. In this study we also test this result, and present other parameter combinations which guarantee high yield comparable to the one found with the multi-parameter scan. We also propose experimentally feasible configurations for existing attosecond science laboratories.

#### 3.1. Testing the ANN against the Full 3D Simulation Results

#### 3.2. Prediction Potential of the ANN

_{med}as parameter, a quantity which is proportional to the optical density. In the absorption limited HHG regime the harmonic yield (i.e., the total photon number) should grow with the square of the optical density in a configuration with fixed phase-matching conditions [64]. In order to have better insight into the physics of how the harmonic yield evolves with the optical density of the medium, we show the results as a function of the log

_{10}(p[mbar] × L

_{med}[cm]).

_{med}= 20 cm; (b) p = 5.31 mbar and L

_{med}= 20 cm. Comparing the data in (a) and (b), we expect that the ratio of the yields for the same pulse energy should be close to the ratio of the square of the pressures (around 200) which is clearly not the case. This is because the cell position is a critical factor in every HHG process, influencing the yield via propagation of the fields, phase-matching effects and absorption of the harmonics.

_{i}= −25 cm, L

_{med}= 16 cm. The maximum harmonic yield in Figure 5b (p = 5.31 mbar) is two orders of magnitude higher than the maximum yield in Figure 5a (p = 0.38 mbar). The physical reason for this behavior is that the conditions for the formation of a self-guided beam propagation are fulfilled. A stable “working intensity” is maintained in a large interaction volume (both axially and radially) which creates good phase-matching conditions for a range of harmonics. The conditions that have to be fulfilled in order to have plasma-core induced self-guiding, as well as its effect on the phase-matching of HHG was also studied [67], but goes beyond the scope of this paper.

^{14}W/cm

^{2}. Figure 6c shows the spatial (r,z) evolution of the harmonic order H25. In accordance with the power spectrum, it is confirmed that H25 builds up constructively and attains maximum yield towards the end of the interaction region, predominantly off-axis. In correlation with the evolution of the fundamental pulse (Figure 6b) the spatial region of maximum H25 intensity is the same where the off-axis refocalization of the fundamental beam happens.

_{i}= 10 cm. In this parameter combination there are few 3D simulation results available, being a good opportunity to test the accuracy of the ANN against the basic physics of the HHG process. Figure 7c shows that the ANN works correctly in this respect, predicting quadratic growth of the yield in the low density regime. Moreover, the end of the region for the absorption limited HHG is also reproduced in a similar manner by the ANN as by the complete 3D simulations. It is not possible to show all results, but the main conclusion is that the ANN is capable to predict regions in the multi-dimensional parameter space where the absorption limited HHG can take place. Furthermore, ANN also predicts for which parameter combinations there is a transition from the initially absorption limited HHG to a saturation, then to a drop in obtainable harmonic yield. This dynamics is represented for example in Figure 7b,c at pressure values 0.38 mbar (green symbols and lines) and 1.41 mbar (blue symbols and lines).

## 4. Conclusions

_{i}= [−50; 50] cm, medium length L

_{med}= [0; 20] cm, while we kept unchanged the pulse duration, wavelength, focal length, beam size, gas type.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Sketch of the geometrical arrangement: laser pulse propagates from left to right; interaction cell filled with gas at p pressure, begins at z

_{i}position relative to focus; L

_{med}is the length of the interaction medium. (

**b**) Schematic representation of the parameter space where the full 3D calculations have been performed. For each pulse energy full 3D calculations were performed for several cell positions relative to focus (x axis) and gas pressures (y axis). Red bullets represent the particular cases solved by full 3D simulations and used by ANN for training, validation and testing. The forth scanned parameter is the medium length which is included in every simulation with values 4, 8, 12, 16, 20 cm.

**Figure 4.**Plot of ANN calculated log

_{10}(harmonic yield) against the values obtained with the complete 3D model for the following data subsets: (

**a**) training, (

**b**) validation, (

**c**) testing and (

**d**) overall data.

**Figure 5.**Harmonic yield as function of the gas cell position relative to the nominal focus. Dots are data obtained from the complete 3D simulations, “+” symbols are the results of the ANN predictions. Panel (

**a**) synthesizes the results obtained for different laser pulse energies at the fixed parameter values p = 0.38 mbar and L

_{med}= 20 cm, i.e., at the cell end. Panel (

**b**) shows the results obtained with p = 5.31 mbar and L

_{med}= 20 cm. Colors of the symbols dots and “+” are the same for the same energy. At the intermediate energies, 6 and 9 mJ, we have performed merely a few but computationally demanding 3D simulations and only with p = 5.31 mbar.

**Figure 6.**Results of the complete 3D simulation performed with the following values of the parameters: E = 9 mJ; p = 5.31 mbar; z

_{i}= −25 cm. (

**a**) Radially integrated power spectrum calculated at every 4 cm of medium length. (

**b**) (r,z) map of the driving pulse’s intensity spatial evolution. (

**c**) (r,z) map of how the H25 builds up during propagation. Other parameters are specified in the main text at the beginning of Section 3.

**Figure 7.**Harmonic yield as function of the optical density of the medium. We represent the logarithm of both quantities, the units are arbitrary. Slope = 2 indicates the quadratic growth of the harmonic yield with the optical density.

|Mean Relative Error| [%] | |Maximum Relative Error| [%] | |
---|---|---|

Training Data Subset | 0.33 | 1.80 |

Validation Data Subset | 1.04 | 4.48 |

Testing Data Subset | 1.41 | 3.94 |

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**MDPI and ACS Style**

Gherman, A.M.M.; Kovács, K.; Cristea, M.V.; Toșa, V. Artificial Neural Network Trained to Predict High-Harmonic Flux. *Appl. Sci.* **2018**, *8*, 2106.
https://doi.org/10.3390/app8112106

**AMA Style**

Gherman AMM, Kovács K, Cristea MV, Toșa V. Artificial Neural Network Trained to Predict High-Harmonic Flux. *Applied Sciences*. 2018; 8(11):2106.
https://doi.org/10.3390/app8112106

**Chicago/Turabian Style**

Gherman, Ana Maria Mihaela, Katalin Kovács, Mircea Vasile Cristea, and Valer Toșa. 2018. "Artificial Neural Network Trained to Predict High-Harmonic Flux" *Applied Sciences* 8, no. 11: 2106.
https://doi.org/10.3390/app8112106