Recent rapid advances in ultrafast ultraintense laser technology1,2 have opened up broad prospects for vital investigations in laser–plasma physics,3–5 laser nuclear physics,6,7 laboratory astrophysics,8,9 and particle physics.10,11 In particular, laser systems with peak intensities in the hundreds of terawatt to multi-petawatt ranges have achieved laboratory intensities of the order of 10²⁰–10²² W/cm², recently even reaching ∼10²³ W/cm² with a pulse duration of tens of femtoseconds.12 These achievements are paving the way for explorations of strong-field quantum electrodynamics (SF-QED), among other significant applications. Meanwhile, the unprecedented laser intensities not only cause large fluctuations in the laser output (∼1%–20% in peak power12), but also make accurate determination of the laser parameters increasingly difficult. These parameters play key roles throughout the laser-driven physical processes. For instance, in detection of the quantum radiation reaction effects, energy loss of the scattered electron beam serves as the SF-QED signal and is strongly correlated with the laser intensity and pulse duration.13,14 In the fast ignition of inertial confinement fusion, specific and precise pulse duration and intensity (∼10²⁰ W/cm²) of the ignition laser are required for improving the energy conversion from laser to fuel and suppressing uncertainties in the laser–plasma interactions.6,15 In laser–plasma acceleration, the peak intensity and pulse duration affect the electron and proton acceleration efficiency and stability.16–18 Uncertainties in the focal spot, pulse duration, and intensity of the laser pulse can lead to significant deviations from the parameters present in experiments. Thus, accurate determination of the spatiotemporal properties of ultrafast ultraintense laser pulses is a fundamental concern for today’s laser–matter interaction experiments.

Currently, laser spatiotemporal characteristics are diagnosed via separate measurements: for example, focal spot size via the high-resolution optical imaging technique12 and temporal pulse duration via the frequency-resolved optical grating (FROG) technique.19 Both of these techniques can reach an extremely high resolution, and their accuracy is limited only by the detector, experimental noise, or post-processing.19 However, for high-power lasers, the pulse energy has to reduced by several orders of magnitude to minimize damage to the optical instruments, with the results then being extrapolated to the case of full laser power.20–22 Owing to nonlinear effects in the amplifying and focusing systems, the characteristics of a space–time coupled laser pulse obtained with these methods may deviate significantly from the exact values.23–25 More reliable parameter diagnosis may be achieved via laser–matter interactions, making it possible to directly extract spatial and temporal information on ultrafast ultraintense (I₀ ≳ 10²⁰ W/cm²) laser pulses. Three mainstream diagnostic approaches are currently in use. The first of these is atomic tunneling ionization, in which the nonlinear dependence of the multiple-tunneling-ionization rate on the field strength can be used to diagnose the laser peak intensity alone, with an accuracy of ≲30%–50%. However, the barrier suppression effect destroys the accuracy, and the atom species needs to be carefully chosen to match the laser intensity requirements.21,26,27 Second, with vacuum acceleration of charged particles, the laser peak intensity, focal spot size, and pulse duration can be retrieved from particle spectral analysis. Here, though, the prepulse and plasma effects and the low statistics substantially influence the final spectra, and therefore one still needs more elaborate considerations.20,28–31 Third, SF-QED effects can be used, for example, to predict the laser intensity and pulse duration separately via analysis of the spectra of electrons,32,33 photons,22,34–36 and positrons,37 with a detection accuracy of ≳10% for laser intensities within the range of 10²⁰–10²³ W/cm². Clearly, these methods either require separate diagnoses or can only measure low-precision laser parameter values (the maximum inaccuracy can reach ≃50%). Thus, there remains an urgent need for new detection methods that can achieve high accuracy and simultaneously diagnose the laser intensity, pulse duration, and focal information.

Recent studies have indicated that the spin polarization of the electrons is sensitive to the field strength and profile of the intense laser pulse and thus can be manipulated by a laser pulse via the radiative spin-flip effect.38–40 These findings have motivated us to explore the possibilities of decoding the pulse information from the spin polarization of the laser-scattered electron beam.

In recent decades, machine learning (ML) techniques have come to be widely used in particle physics41 and astrophysics,42 and they are now having an increasing impact on the study of multiscale, highly nonlinear physical processes such as those arising in condensed matter physics and quantum materials science.43–45 ML-assisted methods are more specialized than humans in comprehending multimodal data (acoustic, visual, and numerical) and optimizing nonlinear extreme physical systems46 and thus can save much time and human effort when integrated into working practices.47,48 In particular, data-driven methods are reshaping our exploration of extreme physical systems, such as in the interaction of ultrafast ultraintense lasers with materials.49 The experimental realization of such extreme conditions in millimeter-sized plasmas can provide laboratory models of astrophysical scenarios.50 The large quantities of data from such experiments or simulations need to be systematically managed. For instance, around 150 GB of data can be generated in each shot of the National Ignition Facility (NIF), and over 70 GB per minute in the Linac-Coherent-Light-Source (LCLS).51 Handling data this size is becoming beyond the capabilities of conventional methods, with the consequence that the underlying physics of the phenomena under study may become obscured. By contrast, ML-assisted methods can be data-driven and run in parallel on large-scale central processing unit (CPU) or graphics processing unit (GPU) platforms to extract internal correlations between desired physical quantities.

In this paper, we propose an ML-assisted method to directly diagnose the spatiotemporal characteristics (peak intensity, focal spot size, and pulse duration) of a linearly polarized (LP) laser pulse, based on spin analysis of nonlinear Compton-scattered electron beams. The interaction scenario and framework of the ML algorithm are illustrated in Fig. 1. First, a high-energy transversely spin-polarized (TSP) electron beam is generated via laser-wakefield acceleration with polarization degree ${\bar{S}}_i$, mean energy ɛ_i, and beam radius w_e;52,53 subsequently, this TSP electron beam is scattered by an ultrafast ultraintense LP laser (with peak intensity ξ, focal radius w₀ and pulse duration τ) via nonlinear Compton scattering (NCS). Owing to the radiative spin-flip effect, the degree of polarization changes from an initial ${\bar{S}}_i$ to a final ${\bar{S}}_f$.38,54,55 The difference (i.e., degree of depolarization) $\delta \bar{S}\equiv {\bar{S}}_i-{\bar{S}}_f$ will be the key factor for determining the laser pulse parameters. Finally, the initial and final electron beam parameters are encoded as a dataset ${\vec{I}}_j$, which is then used to train a neural-network ML algorithm and benchmark the training efficiency. Here, E₀, ω₀, ξ ≡ eE₀/mω₀, w₀, and τ are the electric field strength, frequency, normalized intensity, focal radius, and pulse duration, respectively, of the LP laser pulse, and −e and m are the charge and mass, respectively, of the electron. Relativistic units with c = ℏ = 1 will be used throughout. In addition, a one-to-one mapping between the beam parameters $({\epsilon }_i,w_e,{\bar{S}}_i,{\bar{S}}_f)$ and the laser parameters (ξ, w₀, τ) can be a formidable task, because only one output is of relevance, namely, ${\bar{S}}_f$. To determine the three unknown laser parameters (ξ, w₀, τ) simultaneously, at least three sets of output values of ${\bar{S}}_f$ are required. Therefore, three independent beams with different parameter combinations are employed here. These complex multidimensional relationships can be effectively handled by the neural network topology shown in Fig. 1. Note that this method can induce a spin depolarization of ≃30% for 1 GeV electrons, and ≃40% for 2 GeV ones (laser parameters ξ ≃ 80 and τ = 14T₀). Currently available spin polarimetries for electrons are based on Mott scattering,56 Møller scattering,57 linear Compton scattering,58 or more efficient NCS.59 Some recent studies have indicated that the detection precision of NCS-based polarimetry can reach about 0.3%,59 which qualifies the spin-based method as a new type of high-accuracy diagnostic scheme for ultrafast ultraintense laser pulses.

In Sec. II, a brief description of the Monte Carlo (MC) simulation method of spin-resolved NCS is given, together with the simulation parameters. This is followed by an introduction to our laser-parameter retrieval technique based on ML algorithms (see Fig. 1) and the associated asymptotic formulas. Numerical results and a brief discussion are given in Sec. III. Our conclusions are presented in Sec. IV.

As an illustrative example, diagnosis of a tightly focused laser with a double-Gaussian (spatial and temporal) distribution is considered. In principle, the envelope of the laser can be arbitrary, but should be predetermined via experimental methods, for instance, from a low-power splitting beam. Once the envelope form is known, the following methods can be used to retrieve the laser pulse parameters from the spin diagnosis of the scattered electrons.

Our analysis of the radiative spin-flip effect is based on the MC simulation method proposed in Refs. 38 and 60, in which the spin-resolved probability of NCS in the laser-beam scattering is considered in the local constant field approximation (LCFA).38,61 After emission of a photon, the electron spin state collapses into one of its basis states defined with respect to an instantaneous spin quantization axis (SQA) chosen along the magnetic field in the rest frame of the electron. In Fig. 1, the laser is linearly polarized along the x direction, and so its magnetic field component is B_y. The SQA tends to be antiparallel to the magnetic field in the rest frame of the electron. Depolarization amounts to the electron spin acquiring a certain spin polarization in the y direction, which is canceled from the net polarization by the periodic magnetic field, i.e., ${\bar{S}}_{f,y}\approx 0$. Therefore, we focus our analysis in what follows on the electron polarization in the x direction. In NCS, the invariant parameter characterizing the quantum effects is61,62$\chi \equiv e\sqrt{-{(F_{\mu \nu }{p}^{\nu })}^2}/m^3$, where F_μν and p denote the electromagnetic field tensor and the four-momentum of the electron, respectively. In a colliding geometry, χ ≈ 2ξγ_eω₀/m, where γ_e denotes the electron’s Lorentz factor. To excite the radiative spin-flip process, χ should be in the range of 0.01–1, over which nonlinear Breit–Wheeler pair production can be suppressed.

The LP laser parameter set for the training data includes wavelength λ₀ = 0.8 μm, focal radius w₀ = [2, 3, 4, 5]λ₀, peak intensity ξ = [10, 15, 20, 30, 40, 45, 60, 80], and pulse duration τ = [2, 6, 10, 14]T₀, with T₀ denoting the laser period. The probe electron beam has a polar angle θ_e = π, azimuthal angle ϕ_e = 0, and angular divergence σ_θ = 0.3 mrad. The initial kinetic energies are ɛ_i = [0.5, 1, 1.5, 2] GeV, with relative energy spread σ_ɛ/ɛ_i = 0.05, and the initial average degree of spin polarization along the x direction ${\bar{S}}_{i,x}=[0.6,0.8,1.0]$ (here, χ_max ≲ 1, i.e., the pair-production effect on the final electron distribution is negligible for the present parameters). The beam radius w_e = [1, 2, 3, 4]λ₀, the beam length L_e = 5λ₀, and the total number of electrons is 5 × 10⁵ with transversely Gaussian and longitudinally uniform distributions, attainable by current laser wakefield accelerators.3

Decoding the spatiotemporal characteristics of an ultrafast ultraintense laser from information carried by a scattered electron beam is an inverse transformation that requires multidimensional input and output. In principle, the equation of motion can be embedded in the neural network, which can predict one laser parameter with high accuracy. However, predicting three laser parameters simultaneously will be inefficient owing to slow real-time integration of the equation of motion. Here, therefore, a data-driven standard backpropagation neural network (BPNN) based on the PyTorch framework is used to train and predict the scattering laser parameters.63 The input data are composed of the energy, beam radius, the initial and final average spins, and the logarithm of the ratio of final spin to initial spin of the electron beam, in the vector $\vec{I}\equiv [{\epsilon }_i,w_e,{\bar{S}}_i,{\bar{S}}_f,\ln ({\bar{S}}_f/{\bar{S}}_i)]$; see Fig. 1. About 1000 sets of input data are obtained via the MC simulation and rearranged/recombined to about 3 × 10⁴ sets for training. Then the input data $({\vec{I}}_1,{\vec{I}}_2,{\vec{I}}_3)$ are normalized via the StandScaler function. After random permutation, the input information is preprocessed by the second-order polynomial feature function PolynomialFeatures to construct implicit connections between them.

In our BPNN, we choose eight fully connected hidden layers, with 128, 256, 512, 512, 512, 512, 256, and 128 nodes respectively. The numbers of hidden layers and nodes here ensure adequate prediction accuracy and appropriate computing resources. The activation functions use tanh and PReLU alternatively between different layers. The mean squared error MSELoss is used as the loss function, and the stochastic gradient descent SGD method is used as the optimizer. After each training iteration, the optimizer clears old gradients, and losses are backpropagated for the calculation of new gradients. Finally, the network parameters are updated according to the new gradients. The initial learning rate is set as 0.3, and the adjustment factor of the exponential learning rate ExponentialLR scheduler is set as 0.9. In our calculations, the total number of training iterations is 4 × 10⁴. To enhance the learning efficiency of the model on the laser pulse duration τ, we consider two models with learning ratios of ξ, w₀, and τ set as 1:1:1 and 1:1:2, respectively [see Figs. 2(a) and 2(b)]. Note that the training loss measures the training efficiency of the model. The training loss may increase as a result of inappropriate network structure design and will decrease with effective learning. In the final stable stage, there may be overfitting to the training data. However, the overfitting can be restrained by using a technique such as weight upper limit64 or dropout.65 For instance, the losses of ξ, w₀, and τ are reduced for the learning ratio of 1:1:2, and further increasing the ratio of τ will produce larger losses in other parameters. This BPNN model will be used in the subsequent prediction. In principle, the ML-assisted method is not limited to the current application, but can also be used for other inverse problems.

Asymptotic estimation of the depolarization effect is done below analytically from the radiative equations of motion for the dynamics [the Landau–Lifshitz (LL) equation66] and the spin [the modified Thomas–Bargmann–Michel–Telegdi (T-BMT) equation67]. A dependence of the spin dynamics on the electron energy follows, assuming weak radiation. The quantum-corrected LL equation is then used to obtain the approximate electron energy, which is then plugged into the solution for spin dynamics.

The radiative spin evolution is composed of Thomas precession (subscript T) and radiative correction (subscript R) terms67 and is governed by$$\frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\eta }={\left(\frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\eta }\right)}_T+{\left(\frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\eta }\right)}_R,$$(1a)$$\begin{array}{ll}\hfill {\left(\frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\eta }\right)}_T& =\frac{e{\gamma }_e}{\left(k\cdot p_i\right)}\mathbf{S}\times \left[-\left(\frac{g}{2}-1\right)\frac{{\gamma }_e}{{\gamma }_e+1}\left(\mathit{\beta }\cdot \mathbf{B}\right)\mathit{\beta }\right.\hfill \\ \hfill & \left.+\left(\frac{g}{2}-1+\frac{1}{{\gamma }_e}\right)\mathbf{B}-\left(\frac{g}{2}-\frac{{\gamma }_e}{{\gamma }_e+1}\right)\mathit{\beta }\times \mathbf{E}\right],\hfill \end{array}$$(1b)$${\left(\frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\eta }\right)}_R=-P\left[{\psi }_1(\chi )\mathbf{S}+{\psi }_2(\chi )(\mathbf{S}\cdot \mathit{\beta })\mathit{\beta }+{\psi }_3(\chi ){\mathbf{n}}_B\right].$$(1c)Here, E and B are the laser electric and magnetic fields, respectively; p_i, k, η, and g are the electron momentum 4-vector, the laser wavevector, the laser phase, and the electron gyromagnetic ratio, respectively; and$$P=\frac{{\alpha }_f{m}^2}{\sqrt{3}\pi (k\cdot p_i)},$$$${\psi }_1(\chi )={\int }_0^{\infty }{u}^{″}\mathrm{d}u{K}_{2/3}(u^{\prime }),$$$${\psi }_2(\chi )={\int }_0^{\infty }{u}^{″}\mathrm{d}u{\int }_{u^{\prime }}^{\infty }\mathrm{d}x{K}_{1/3}(x)-{\psi }_1(\chi ),$$$${\psi }_3(\chi )={\int }_0^{\infty }{u}^{″}\mathrm{d}u{K}_{1/3}(u^{\prime }),$$$$u^{\prime }=\frac{2u}{3\chi },u^{″}=\frac{u^2}{{(1+u)}^3},u=\frac{{\epsilon }_{\gamma }}{{\epsilon }_0-{\epsilon }_{\gamma }},$$where ɛ₀ and ɛ_γ are the electron energy before radiation and the emitted photon energy, respectively, K_n is the nth-order modified Bessel function of the second kind, and α_f = 1/137 is the fine structure constant. The SQA is chosen along the magnetic field ${\mathbf{n}}_B=\mathit{\beta }\times \widehat{\mathbf{a}}$, with β = v/c the scaled electron velocity and $\widehat{\mathbf{a}}=\mathbf{a}/|\mathbf{a}|$ the unit vector along the electron acceleration a.

To facilitate theoretical analysis and extract analytical formulas, some approximations will be made with the current laser and electron beam parameters in mind, i.e., a GeV electron beam interacting with an LP laser (ξ < 100) and 0.1 ≲ χ ≲ 1. Owing to laser defocusing, the Thomas-term-induced variation δS_T is ≲10⁻⁴, and only the dominant term, i.e., the radiative correction, will be considered. Furthermore, the initial velocity of the electron beam is along the z direction, with β_z ≫ β_x(β_y), and thus the ψ₂ term is negligible for initially TSP electrons. Moreover, owing to the periodic nature of the magnetic field, the contribution of the ψ₃ term vanishes on average within one laser period. Hence, the approximate evolution of the spin components may be obtained from$$\frac{\mathrm{d}{S}_x}{\mathrm{d}\eta }\simeq \frac{C{\psi }_1(\chi )}{{\gamma }_e}{S}_x,$$(2a)$$\frac{\mathrm{d}{S}_y}{\mathrm{d}\eta }\simeq \frac{C{\psi }_1(\chi )}{{\gamma }_e}{S}_y,$$(2b)$$\frac{\mathrm{d}{S}_z}{\mathrm{d}\eta }\simeq \frac{C({\psi }_1(\chi )+{\psi }_2(\chi ))}{{\gamma }_e}{S}_z,$$(2c)where $$C=-\frac{{\alpha }_f}{2\sqrt{3}\pi }\frac{{\omega }_0}{m}.$$Because ψ₁(χ) > 0 and ψ₂(χ) < 0, depolarization in the x and y directions is faster than in the z direction. For instance, for a laser with parameters ξ = 60, τ = 8T₀, and w₀ = 5λ₀, and the electron beam of Fig. 4(a), the final average spin degrees of polarization are S_x,f ≈ 0.8201, S_y,f ≈ 0.8211, and S_z,f ≈ 0.8741, for ${\bar{S}}_{i,x}=1$, ${\bar{S}}_{i,y}=1$, and ${\bar{S}}_{i,z}=1$. Thus, in this paper, we take the electron beam to be initially polarized along the x direction for a larger detection signal.

Under the assumption of weak radiation loss, for which$$\frac{\mathrm{d}{\gamma }_e}{\mathrm{d}\eta }\simeq 0,\chi (\eta )\simeq 2\frac{{\omega }_0}{m}{\gamma }_e\xi {\sin }^2\eta ,$$one can obtain, to the leading-order approximation, ψ₁(χ) ≃ f₁χ² for 0.1 ≲ χ ≲ 1, and f₁ ≈ 0.25 is obtained by curve fitting. Integrating Eq. (2a), the asymptotic ${\bar{S}}_{f,x}$, using the laser-beam parameters, will be given by$$\ln \left[\frac{{\bar{S}}_{f,x}(\tau )}{{\bar{S}}_{i,x}(0)}\right]\simeq M_1{\gamma }_e{\xi }^2\tau ,$$(3)where the factor$$M_1=-\frac{\sqrt{3}}{2}{\alpha }_f\frac{{\omega }_0}{m}{f}_1\approx -4.81\times 10^{-9}$$and τ is the pulse duration in units of the laser period T₀.

To be precise, the radiated photon energy (radiation loss ${\bar{\epsilon }}_{\gamma }$) should be taken into account for 0.1 ≲ χ ≲ 1. Here, we use the quantum-corrected LL equation to include the radiation loss68 via$$\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}={\mathbf{F}}_L+{\mathbf{F}}_{\text{rad}},$$(4a)$${\mathbf{F}}_{\text{rad}}=-C^{\prime }{\chi }^2\mathcal{G}(\chi )\mathit{\beta }/|\mathit{\beta }{|}^2,$$(4b)where F_L ≡ q(E + v ×B) is the Lorentz force, F_rad is the radiation reaction force, $C^{\prime }=2{\alpha }_f^{2}m/3r_e$ (with r_e the classical electron radius), and $\mathcal{G}(\chi )\simeq {[1+4.8(1+\chi )\ln (1+1.7\chi )+2.44{\chi }^2]}^{-2/3}$ is the quantum correction function.69 For 0.1 ≲ χ ≲ 1, assuming χ(η) ≃ (2ω₀/m)γ_eξ sin² η and making the approximation ${\chi }^2\mathcal{G}(\chi )\simeq f_2{\chi }^2$ (with a fitting factor of f₂ ≈ 0.077), the radiation loss (averaged over all electrons, i.e., ignoring the stochastic effect) is given by$${\bar{\epsilon }}_{\gamma }={\int }_0^{\eta }\mathrm{d}\eta {\mathbf{F}}_{\text{rad}}\frac{\mathrm{d}t}{\mathrm{d}\eta }\simeq M_2\tau {\gamma }_e^2{\xi }^2,$$where M₂ = (πα_fω₀/m)f₂ ≈ 5.36 × 10⁻⁹. Then, replacing γ_e in Eq. (3) by ${\gamma }_e-{\bar{\epsilon }}_{\gamma }$, an analytical asymptotic estimate of the final spin ${\bar{S}}_{f,x}$ is given by$$\ln \left[\frac{{\bar{S}}_{f,x}(\tau )}{{\bar{S}}_{i,x}(0)}\right]\simeq M_1{\gamma }_e{\xi }^2\tau (1-M_2{\gamma }_e{\xi }^2\tau ).$$(5)

To demonstrate the efficiency of the proposed diagnostic method, some operational parameters of petawatt-scale lasers at a number of international facilities will be used; see Table I. The corresponding depolarization processes, investigated via MC simulations, indicate that the relative errors between the predicted and input parameters are of orders 0.1%–10%; see Fig. 3(a). After consecutive training, the BPNN model grasps the pattern of the radiative spin-flip effect and is therefore capable of accurately predicting the laser characteristics, i.e., (ξ, τ, w₀), simultaneously. Owing to the limited weighting parameters and training data, the relative prediction errors for ξ, τ, and w₀ (simultaneously) are of the order of ${\mathcal{R}}_{1(2)}\lesssim 10\%$; see Figs. 3(b) and 3(c). Compared with cases with w₀ ≳ 3λ₀, the number of electrons scattered by a tightly focused laser (w₀ ≲ 3λ₀) is lower, owing to the small Rayleigh range $(z_R=\pi w_0^2/\lambda )$. Thus, the beam-averaged spin-flip effect is relatively more sensitive to variations in the electron beam parameters, and the relative error ${\mathcal{R}}_1$ is larger for w₀ ≲ 3λ₀ [see Fig. 3(b)]. For a laser radius w₀ ≳ 5λ₀, which is already beyond the current training range, a certain amount of overfitting is expected. For SILEX-II, for example, the relative error in the focal radius ${\mathcal{R}}_w\sim 15\%$. Moreover, for a specific ξ, the electron energy will damp rapidly within a certain τ, and the depolarization will saturate owing to the much smaller χ ≪ 1. From the aspect of the ML algorithm, there might occur a sharp gradient along the pulse duration. As a consequence, for long pulse duration τ, the complete prediction will be inaccurate and the growth in the relative errors will be quite rapid (owing to the limited training datasets). For instance, the relative errors in the cases of J-KAREN and SILEX-II are relatively large in Fig. 3(a), and similar trends can also be found in Fig. 3(c). In addition, as another key parameter for the NCS, once the peak intensity ξ is too small (i.e., when χ will also be too small via χ ∝ γ_eξ), the statistical error in the MC calculation will increase and the prediction will be inaccurate, too. For instance, the relative errors for SILEX-II and APOLLON are larger than that of ELI-NP; see Fig. 3(a) and the general trends in Fig. 3(c). Therefore, for short pulse duration τ and large intensity ξ, the relative errors will be lower than those in other cases, for instance, in the upper-left region of Fig. 3(c), $\mathcal{R}$ can reach the order of ≲0.1%. It can be foreseen that with a larger network size, i.e., more weighting parameters, the model will be more accurate and robust.

The physical essence of the ML-assisted pulse information decoding method is revealed by our analytical asymptotic estimation on the basis of Eq. (5), which is in good agreement with the numerical MC results over a wide range of laser parameter values; see Figs. 4(a)–4(c). The distributions of $\delta {\bar{S}}_x^{MC,AE}$ with respect to ξ and τ are shown in Figs. 4(a) and 4(b), where superscripts “MC” and “AE” denote the results from the MC and analytical asymptotic estimation (AE) methods, respectively. As expected, $\delta {\bar{S}}_x$ increases as ξ and τ both increase, and a specific spin change $\delta {\bar{S}}_x$ determines a curve that binds ξ with τ (or a hyperplane for ξ, τ, and w₀), i.e., the NCS acts as a nonlinear function $\mathcal{F}(\cdot ,\cdot )$ that maps the laser pulse parameters (ξ, τ) to the degree of depolarization of the electron beam $\mathcal{F}(\xi ,\tau )\to \delta {\bar{S}}_x$. Quite remarkably, the corresponding relative error ${\mathcal{R}}_s$ in the parameter ranges of ξ ∈ (10, 60) or τ ∈ (2, 6)T₀ is ${\mathcal{R}}_s\simeq 1\%$; see Fig. 4(c). With the analytical AE extracted subject to the condition 0.1 < χ ≲ 1, and for ξ > 60 and τ > 6, the low-order estimation deviates from the MC result, owing to the nonlinear radiative effects. By contrast, the ML-assisted method is data-driven, i.e., the algorithms can still grasp the correlations between laser pulse parameters and depolarization of the electron beam, without artificial restrictions; see the prediction accuracy (the total relative error ${\mathcal{R}}_2\sim 1\%$) for high laser intensity and long pulse duration in Fig. 3(c).

Figure 4(d) illustrates how to determine ξ and w₀ via AE for a specific set of parameters (ξ = 50, τ = 6T₀, and w₀ = 5λ₀) marked as white circles P₁ in Figs. 4(a)–4(c). Here, the pulse duration τ is pre-acquired with other diagnostics, for instance, from the low-power mode of detection. Then, a sub-micrometer probe is collided with the laser pulse, from which one obtains $\delta {\bar{S}}_1$; see the solid line labeled “1λ₀” in Fig. 4(d), which has been obtained from Eq. (5). After that, a second probe with beam radius w_e = 4λ₀ produces $\delta {\bar{S}}_2$, the dot-dashed line labeled “4λ₀” in Fig. 4(d). According to Eq. (5), two average intensities ${\bar{\xi }}_1$ and ${\bar{\xi }}_2$ can be determined from $\delta {\bar{S}}_1$ and $\delta {\bar{S}}_2$, respectively, corresponding to different beam radii. Since w₀ ≫ w_e, the average laser intensity sensed by the sub-micrometer probe can serve approximately as the peak intensity in the focusing region. Thus, ${\bar{\xi }}_1=51.62$ is identified as the peak intensity of the laser pulse, with a relative error of 3.2%, whereas ${\bar{\xi }}_2=42.96$, corresponding to w_e = 4λ₀, is taken as the average intensity within the probe radius, i.e., ${\bar{\xi }}_2={\bar{\xi }}_1{\int }_{-w_e}^{w_e}\exp (-r^2/w_0^2)dr$. Numerical calculation gives the focal radius w₀ = 5.18λ₀, with a relative error of 3.6%. Note that in Eq. (5), once τ (or ξ) is given, the map between $\delta \bar{S}$ and the other parameter is uniquely fixed. For instance, in Fig. 4(d), once ξ is fixed [points P₂ and P₃ in Figs. 4(a)–4(c)], there will be only one intersection (the final phase η_f) between Eq. (5) and the temporal evolution of the average spin. Here, $\bar{S}({\eta }_f)$ is the final degree of polarization of the electron beam. Conversely, once τ is fixed [points P₄ and P₅ in Figs. 4(a)–4(c)], the MC final results will evolve to a unique ξ value; see Fig. 4(f).

Compared with the signals from dynamical statistics, the degree of spin polarization is more accurate and more robust with respect to fluctuations in energy and angular spread of the electron beam probe; see Figs. 5(a)–5(d). As the initial energy spread σ_ɛ/ɛ_i varies from 1% to 30%, the average energy (${\bar{\epsilon }}_f\sim 500$ MeV) of the final electron beam (w_e = 1λ₀, ɛ_i = 1 GeV) changes by ∼1%; see Fig. 5(a). However, the effect of energy spread on the spin polarization ${\bar{S}}_{f,x}$ of the final state is ∼0.3%; see Fig. 5(c). According to Eq. (5) expressing the analytical AE, S_f ∼ exp(−k₁γ_e) and δS_f ∼ δγ_ek₁ exp(−k₁γ_e), which leads to the conclusion that the spin variations due to dynamics exhibit exponential decay. Similarly, while the initial angular spread σ_θ changes from 0.3 to 100 mrad, the normalized variation of angular spread N_θ is ∼30%, and the effect on the spin ${\bar{S}}_{f,x}$ is ∼0.2%. In short, the detection accuracy of the spin signal is one to two orders of magnitude higher than that of the dynamic signal. The relative errors $\mathcal{R}$ of the analytical AE and ML-assisted spin signals are shown in Figs. 5(e) and 5(f). Owing to angular and energy spread, the relative errors $\mathcal{R}$ of the analytical AE, for ξ and w₀, are both kept within 5%, while the ML-assisted method can simultaneously predict three parameter values for ξ, w₀, and τ, with relative errors $\mathcal{R}\lesssim 10\%$. Especially for w₀, the accuracy of the ML-assisted method is at least twice as good as that of the analytical prediction.

We have proposed an ML-assisted method to diagnose the spatiotemporal properties of an ultrafast ultraintense laser pulse, namely, the pulse duration τ, peak intensity ξ, and focal spot size w₀, based on the radiative spin-flip effect of the electrons while they experience strong NCS. Our trained BPNN can accurately predict the spatiotemporal characteristics of petawatt-level laser systems with relative errors ≲10%. The proposed method is accurate and robust with respect to fluctuations in the electron beam parameters, and it is suitable for use with currently running or planned multi-petawatt-scale laser facilities. Accurate measurement of ultrafast ultraintense laser parameters may pave the way for future strong-field experiments, of importance to laser nuclear physics investigations, laboratory astrophysics studies, and other fields.

Acknowledgment. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11874295, 12022506, U2267204, 11905169, 12275209, 11875219, and 12171383), the Open Fund of the State Key Laboratory of High Field Laser Physics (Shanghai Institute of Optics and Fine Mechanics), and the Foundation of Science and Technology on Plasma Physics Laboratory (Grant No. JCKYS2021212008). The work of Y.I.S. is supported by an American University of Sharjah Faculty Research (Grant No. FRG21).