Quantification and analysis of the nonlinear effects in spectral broadening through solid medium of femtosecond pulses by neural network

Supercontinuum generation has been widely applied in laser spectroscopy and few-cycle pulse generation. It is composed of complex and unmeasurable nonlinear optical effects, which influence the final broadened spectrum markedly. To describe and characterize the two key processes, the Kerr effect and ionization, we employ two nonlinear integrals, including the common B integral for the Kerr effect and a P integral for ionization or plasma effect. With these integrals, the contributions of Kerr and plasma effects in the supercontinuum generation are identified and determined quantitatively. Then we utilize machine learning to construct a multilayer perceptron neural network to obtain the solution of the propagation equation for a femtosecond pulse in a solid medium. We employ supervised and unsupervised training with both experimental and simulation data to train the neural network for a better accuracy of the calculation. The trained network can take the input and broadened spectra of the pulse to compute initial laser parameters and the B and P integrals instantly so that the contributions of Kerr and plasma effects in the supercontinuum generation may be quantified in real time, unveiling the nonlinearities behind the spectral evolution. This method provides a more accurate understanding of the physics of the entangled nonlinear optics effects and a faster and more convenient tool in the investigation of nonlinear optics.

Figure 1

(a) The numerically simulated integrands of B and P integrals ($\mathrm{\Delta }{n}_{\mathrm{Kerr}}$ and $\mathrm{\Delta }{n}_{\mathrm{plasma}}$) as functions of pulse propagation distance. The input parameters are the pulse duration $\tau =40\mathrm{fs}$, beam radius $r=0.1$ mm, pulse energy $E=0.02\mathrm{mJ}$, and focus length $f=150\mathrm{mm}$, and the 2 mm sapphire plate is set 5 mm before the focal point. Before the formation of the filament, plasma almost has no effect on pulse evolution, while the Kerr effect is not strong either. When optical clamping occurs ∼1.6 mm, the strengths of both nonlinear effects increase sharply. (b) The variation of pulse intensity and energy with propagation distance. During the propagation, low and high ionization stages are experienced, and the high ionization stage is covered by the yellow shadow. The plasma density generated by the peak intensity is $\sim 1.63\times {10}^{15}\mathrm{m}{\mathrm{m}}^{–3}$.

Figure 2

Evolutions of B and P integrals as functions of (a) beam radius, (b) pulse energy, (c) medium length, and (d) pulse duration. When each single parameter changes in the simulation, other ones remain constant and the same as in Fig. 1.

Figure 3

The mapping relationship between the initial and final spectra and the laser propagation parameters.

Figure 4

Spectral data processing. The autoencoder is composed of an encoder and a decoder. The input of the encoder is a spectral data array. After passing through eight hidden layers, a feature vector with five elements is obtained. The input data of the decoder are the feature vector, and the spectral data array are restored after eight hidden layers.

Figure 5

Structure of the neural network. The original and broadened spectra pass through the same encoder and serve as the multilayer perceptron (MLP) input data. After three hidden layers, the predicted parameters as the output of MLP are the initial laser parameters (pulse width, beam size, and pulse energy) and the nonlinear B and P integrals.

Figure 6

Performance of the trained neural network in computing pulse energy, beam radius, pulse duration, and B and P integrals based on simulation dataset. The deviation between predicted value (orange circles) and simulated value (blue lines) is labeled on the right side of each plot, which reveals a high computation accuracy of all five parameters. The index of the dataset along the $x$ axis corresponds to the testing dataset chosen from the $X-Y$ database. Here, 50 datasets are selected as the testing set from the whole database randomly.

Figure 7

Computation of the target parameters in an extended range. Here, we take pulse energy as an example. The orange filled area shows the computation uncertainty. The computed values match well with target values in the training data range (0.01 to 0.03 mJ), and the computation uncertainty increases when the energy moves out of the training range.

Figure 8

The schematic diagram of the experimental setup to collect initial laser pulse parameters as well as spectra before and after the nonlinear medium. The optics to measure the spectrum before the nonlinear medium are not shown. HWP: half-wave plate, TFP: thin film polarizer, BS: beam splitter, PM: plane mirror, FL: $f=150\mathrm{mm}$ focus lens, WP: wedge pair, SP: sapphire plate, OAP: off-axis parabolic mirror, CCD: camera to measure beam size, Wizzler: device to measure pulse width. In principle, there is no special requirement to the input laser pulse in experiments if it generates spectral broadening in the nonlinear medium. However, considering the consistency of parameters with the numerical simulations and reasons discussed in the text, we set the input pulse to be close to the Fourier transform limit.

Figure 9

Computing the initial pulse parameters from the experimentally measured spectra. The computed values (orange circles) fit well with reference experimental data (blue curves), especially the results of pulse duration. The reference data are extracted from the 350 sets of experimental data randomly and are not included in the training dataset for the network to ensure reliability.

Figure 10

Computing the nonlinear phases (B and P integrals) from experimental spectra, which are collected under the laser conditions where the pulse energy is 21 μJ, beam radius is 0.1109 mm, and pulse width changes from 40 to 95 fs. B and P integrals simulated by the nonlinear envelope equation (NEE) are plotted in comparison. The computed and simulation results as functions of pulse width are consistent.

Figure 11

Spectrum broadening as the pulse energy rises. (a) The frequency width variation of the final output spectrum. (b) The nonlinear integrals and the ratio of integrated spectral densities of the high frequency (HF) and low frequency (LF) sides as functions of the input pulse energy.