Matter and Radiation at Extremes, Volume. 8, Issue 3, 034401(2023)

Diagnosis of ultrafast ultraintense laser pulse characteristics by machine-learning-assisted electron spin

Zhi-Wei Lu1... Xin-Di Hou1, Feng Wan1, Yousef I. Salamin2, Chong Lv3, Bo Zhang4, Fei Wang5, Zhong-Feng Xu1 and Jian-Xing Li1 |Show fewer author(s)
Author Affiliations
  • 1Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an 710049, China
  • 2Department of Physics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
  • 3Department of Nuclear Physics, China Institute of Atomic Energy, P.O. Box 275(7), Beijing 102413, China
  • 4Key Laboratory of Plasma Physics, Research Center of Laser Fusion, China Academy of Engineering Physics, Mianshan Rd. 64#, Mianyang, Sichuan 621900, China
  • 5School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
  • show less

    The rapid development of ultrafast ultraintense laser technology continues to create opportunities for studying strong-field physics under extreme conditions. However, accurate determination of the spatial and temporal characteristics of a laser pulse is still a great challenge, especially when laser powers higher than hundreds of terawatts are involved. In this paper, by utilizing the radiative spin-flip effect, we find that the spin depolarization of an electron beam can be employed to diagnose characteristics of ultrafast ultraintense lasers with peak intensities around 1020–1022 W/cm2. With three shots, our machine-learning-assisted model can predict, simultaneously, the pulse duration, peak intensity, and focal radius of a focused Gaussian ultrafast ultraintense laser (in principle, the profile can be arbitrary) with relative errors of 0.1%–10%. The underlying physics and an alternative diagnosis method (without the assistance of machine learning) are revealed by the asymptotic approximation of the final spin degree of polarization. Our proposed scheme exhibits robustness and detection accuracy with respect to fluctuations in the electron beam parameters. Accurate measurements of ultrafast ultraintense laser parameters will lead to much higher precision in, for example, laser nuclear physics investigations and laboratory astrophysics studies. Robust machine learning techniques may also find applications in more general strong-field physics scenarios.

    I. INTRODUCTION

    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 S̄i, mean energy ɛi, and beam radius we;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 S̄i to a final S̄f.38,54,55 The difference (i.e., degree of depolarization) δS̄S̄iS̄f will be the key factor for determining the laser pulse parameters. Finally, the initial and final electron beam parameters are encoded as a dataset Ij, which is then used to train a neural-network ML algorithm and benchmark the training efficiency. Here, E0, ω0, ξeE0/0, 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 (εi,we,S̄i,S̄f) and the laser parameters (ξ, w0, τ) can be a formidable task, because only one output is of relevance, namely, S̄f. To determine the three unknown laser parameters (ξ, w0, τ) simultaneously, at least three sets of output values of S̄f 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.

    Left: Three different electron beams propagating along the z direction with parameters ɛi, we, and S̄i scatter off the same laser pulse and produce final spin degrees of polarization S̄f. Right: Topology of the BPNN used for parameter prediction, which takes I⃗j;j=1,2,3=[εi,we,S̄i,S̄f,ln(S̄f/S̄i)] as input data and produces (ξ, w0, τ) as output; for details, see Sec. II B.

    Figure 1.Left: Three different electron beams propagating along the z direction with parameters ɛi, we, and S̄i scatter off the same laser pulse and produce final spin degrees of polarization S̄f. Right: Topology of the BPNN used for parameter prediction, which takes Ij;j=1,2,3=[εi,we,S̄i,S̄f,ln(S̄f/S̄i)] as input data and produces (ξ, w0, τ) as output; for details, see Sec. II B.

    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.

    II. SPIN-BASED LASER-PARAMETER DIAGNOSTIC METHODS

    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.

    A. Spin-resolved NCS and interaction scenario

    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 By. 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., S̄f,y0. 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 is61,62χe(Fμνpν)2/m3, where Fμν and p denote the electromagnetic field tensor and the four-momentum of the electron, respectively. In a colliding geometry, χ ≈ 2ξγeω0/m, where γe 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 θe = π, azimuthal angle ϕe = 0, and angular divergence σθ = 0.3 mrad. The initial kinetic energies are ɛi = [0.5, 1, 1.5, 2] GeV, with relative energy spread σɛ/ɛi = 0.05, and the initial average degree of spin polarization along the x direction S̄i,x=[0.6,0.8,1.0] (here, χmax ≲ 1, i.e., the pair-production effect on the final electron distribution is negligible for the present parameters). The beam radius we = [1, 2, 3, 4]λ0, the beam length Le = 5λ0, and the total number of electrons is 5 × 105 with transversely Gaussian and longitudinally uniform distributions, attainable by current laser wakefield accelerators.3

    B. Neural network assisted diagnosis

    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 I[εi,we,S̄i,S̄f,ln(S̄f/S̄i)]; 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 (I1,I2,I3) 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.

    Training loss (mean squared errors for all training samples) evolutions of ξ, w0, τ, and total loss (tot.) vs training time. Learning ratios of ξ, w0 and τ are 1:1:1 in (a) and 1:1:2 in (b).

    Figure 2.Training loss (mean squared errors for all training samples) evolutions of ξ, w0, τ, and total loss (tot.) vs training time. Learning ratios of ξ, w0 and τ are 1:1:1 in (a) and 1:1:2 in (b).

    C. Analytical asymptotic models

    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 bydSdη=dSdηT+dSdηR,dSdηT=eγekpiS×g21γeγe+1βBβ+g21+1γeBg2γeγe+1β×E,dSdηR=Pψ1(χ)S+ψ2(χ)(Sβ)β+ψ3(χ)nB.Here, E and B are the laser electric and magnetic fields, respectively; pi, k, η, and g are the electron momentum 4-vector, the laser wavevector, the laser phase, and the electron gyromagnetic ratio, respectively; andP=αfm23π(kpi),ψ1(χ)=0uduK2/3(u),ψ2(χ)=0uduudxK1/3(x)ψ1(χ),ψ3(χ)=0uduK1/3(u),u=2u3χ,u=u2(1+u)3,u=εγε0εγ,where ɛ0 and ɛγ are the electron energy before radiation and the emitted photon energy, respectively, Kn is the nth-order modified Bessel function of the second kind, and αf = 1/137 is the fine structure constant. The SQA is chosen along the magnetic field nB=β×â, with β = v/c the scaled electron velocity and â=a/|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 δST 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 βzβx(βy), 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 fromdSxdηCψ1(χ)γeSx,dSydηCψ1(χ)γeSy,dSzdηC(ψ1(χ)+ψ2(χ))γeSz,where C=αf23πω0m.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 Sx,f ≈ 0.8201, Sy,f ≈ 0.8211, and Sz,f ≈ 0.8741, for S̄i,x=1, S̄i,y=1, and S̄i,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 whichdγedη0,χ(η)2ω0mγeξsin2η,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 S̄f,x, using the laser-beam parameters, will be given bylnS̄f,x(τ)S̄i,x(0)M1γeξ2τ,where the factorM1=32αfω0mf14.81×109and τ is the pulse duration in units of the laser period T0.

    To be precise, the radiated photon energy (radiation loss ε̄γ) should be taken into account for 0.1 ≲ χ ≲ 1. Here, we use the quantum-corrected LL equation to include the radiation loss68 viadPdt=FL+Frad,Frad=Cχ2G(χ)β/|β|2,where FLq(E + v ×B) is the Lorentz force, Frad is the radiation reaction force, C=2αf2m/3re (with re the classical electron radius), and G(χ)[1+4.8(1+χ)ln(1+1.7χ)+2.44χ2]2/3 is the quantum correction function.69 For 0.1 ≲ χ ≲ 1, assuming χ(η) ≃ (2ω0/m)γeξ sin2η and making the approximation χ2G(χ)f2χ2 (with a fitting factor of f2 ≈ 0.077), the radiation loss (averaged over all electrons, i.e., ignoring the stochastic effect) is given byε̄γ=0ηdηFraddtdηM2τγe2ξ2,where M2 = (παfω0/m)f2 ≈ 5.36 × 10−9. Then, replacing γe in Eq. (3) by γeε̄γ, an analytical asymptotic estimate of the final spin S̄f,x is given bylnS̄f,x(τ)S̄i,x(0)M1γeξ2τ(1M2γeξ2τ).

    (a) Relative errors R=Rξ,Rτ,Rw between predicted and theoretical values of (ξ, τ, w0) for the facilities in Table I, where I1⃗, I2⃗, and I3⃗ are respectively (ɛi = 1 GeV, we = λ0, S̄i,x=1) (ɛi = 1 GeV, we = 3λ0, S̄i,x=1), and (ɛi = 1.5 GeV, we = λ0, S̄i,x=1). (b) Distribution of total relative error R1=Rξ2+Rw2 in the ξ–w0 plane, where τ = 10T0 and (I1⃗, I2⃗, I3⃗) are the same as in (a). (c) Distribution of total relative error R2=Rξ2+Rτ2 in the ξ–τ plane, where w0 = 5λ0, and I1⃗, I2⃗, and I3⃗ are respectively (ɛi = 500 MeV, we = λ0, S̄i,x=1) (ɛi = 500 MeV, we = 4λ0, S̄i,x=0.8), and (ɛi = 2 GeV, we = λ0, S̄i,x=0.6).

    Figure 3.(a) Relative errors R=Rξ,Rτ,Rw between predicted and theoretical values of (ξ, τ, w0) for the facilities in Table I, where I1, I2, and I3 are respectively (ɛi = 1 GeV, we = λ0, S̄i,x=1) (ɛi = 1 GeV, we = 3λ0, S̄i,x=1), and (ɛi = 1.5 GeV, we = λ0, S̄i,x=1). (b) Distribution of total relative error R1=Rξ2+Rw2 in the ξw0 plane, where τ = 10T0 and (I1, I2, I3) are the same as in (a). (c) Distribution of total relative error R2=Rξ2+Rτ2 in the ξτ plane, where w0 = 5λ0, and I1, I2, and I3 are respectively (ɛi = 500 MeV, we = λ0, S̄i,x=1) (ɛi = 500 MeV, we = 4λ0, S̄i,x=0.8), and (ɛi = 2 GeV, we = λ0, S̄i,x=0.6).

    III. RESULTS AND DISCUSSION

    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 R1(2)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 (zR=πw02/λ). Thus, the beam-averaged spin-flip effect is relatively more sensitive to variations in the electron beam parameters, and the relative error R1 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 Rw15%. 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 χγeξ), 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), 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.

    • Table 1. Operational parameters of some international ultrafast ultraintense laser facilities: total energy EL, central wavelength λ, peak intensity I0, pulse duration τ, and focal radius w0.

      Table 1. Operational parameters of some international ultrafast ultraintense laser facilities: total energy EL, central wavelength λ, peak intensity I0, pulse duration τ, and focal radius w0.

      ProjectEL (J)λ (μm)I0 (W/cm2); ξτ (fs); T0w0(λ)
      ELI-NP70200.825.6 × 1021; 52.4318.75; 6.863.63
      J-KAREN7128.40.83.8 × 1021; 42.1432.9; 12.334.75
      GIST7244.50.811022; 69.2130; 11.13.79
      SILEX-II73300.85 × 1020; 15.2830; 11.246.16
      APOLLON74100.8152 × 1021; 31.1424; 8.832.92

    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 δS̄xMC,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, δS̄x increases as ξ and τ both increase, and a specific spin change δS̄x 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 F(,) that maps the laser pulse parameters (ξ, τ) to the degree of depolarization of the electron beam F(ξ,τ)δS̄x. Quite remarkably, the corresponding relative error Rs in the parameter ranges of ξ ∈ (10, 60) or τ ∈ (2, 6)T0 is Rs1%; 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 R21%) for high laser intensity and long pulse duration in Fig. 3(c).

    (a) and (b) Transverse spin degrees of depolarization of the probe electron beams δS̄x≡S̄i,x−S̄f,x vs laser peak intensity ξ and pulse duration τ: (a) δS̄xMC calculated by the MC method; (b) δS̄xAE calculated by asymptotic estimation from Eq. (5). Here, a laser radius w0 = 5λ0, a probe electron beam energy ɛi = 1 GeV, a beam radius we = λ0, and an initial average spin S̄i,x=1 are used. Other parameters are the same as in Fig. 3. (c) Relative error Rs=|δS̄xMC−δS̄xAE|/δS̄xMC vs ξ and τ. (d) δS(ξ, τ) = 0.12 [white circles P1(ξ = 50, τ = 6T0) in (a)–(c)] for we = 1λ0 (solid line) and we = 4λ0 (dash-dotted line). (e) S̄x vs laser phase η ≡ ω0(t − z). The solid and dashed black lines (averaged MC evolution process) correspond to the blue circles P2(ξ = 50, τ = 8T0) and P3(ξ = 50, τ = 12T0) in (a)–(c), respectively. The blue lines and circles indicate the analytical calculations (only related to the final laser phase ηf). (f) S̄x vs laser phase η. The solid and dashed lines correspond to the red circles P4(ξ = 40, τ = 6T0) and P5(ξ = 60, τ = 6T0) in (a)–(c), respectively. Black lines are from the averaged MC evolution calculation and red circles (right axis) are from the analytical calculations.

    Figure 4.(a) and (b) Transverse spin degrees of depolarization of the probe electron beams δS̄xS̄i,xS̄f,x vs laser peak intensity ξ and pulse duration τ: (a) δS̄xMC calculated by the MC method; (b) δS̄xAE calculated by asymptotic estimation from Eq. (5). Here, a laser radius w0 = 5λ0, a probe electron beam energy ɛi = 1 GeV, a beam radius we = λ0, and an initial average spin S̄i,x=1 are used. Other parameters are the same as in Fig. 3. (c) Relative error Rs=|δS̄xMCδS̄xAE|/δS̄xMC vs ξ and τ. (d) δS(ξ, τ) = 0.12 [white circles P1(ξ = 50, τ = 6T0) in (a)–(c)] for we = 1λ0 (solid line) and we = 4λ0 (dash-dotted line). (e) S̄x vs laser phase ηω0(tz). The solid and dashed black lines (averaged MC evolution process) correspond to the blue circles P2(ξ = 50, τ = 8T0) and P3(ξ = 50, τ = 12T0) in (a)–(c), respectively. The blue lines and circles indicate the analytical calculations (only related to the final laser phase ηf). (f) S̄x vs laser phase η. The solid and dashed lines correspond to the red circles P4(ξ = 40, τ = 6T0) and P5(ξ = 60, τ = 6T0) in (a)–(c), respectively. Black lines are from the averaged MC evolution calculation and red circles (right axis) are from the analytical calculations.

    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 δS̄1; 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 we = 4λ0 produces δS̄2, the dot-dashed line labeled “4λ0” in Fig. 4(d). According to Eq. (5), two average intensities ξ̄1 and ξ̄2 can be determined from δS̄1 and δS̄2, respectively, corresponding to different beam radii. Since w0we, the average laser intensity sensed by the sub-micrometer probe can serve approximately as the peak intensity in the focusing region. Thus, ξ̄1=51.62 is identified as the peak intensity of the laser pulse, with a relative error of 3.2%, whereas ξ̄2=42.96, corresponding to we = 4λ0, is taken as the average intensity within the probe radius, i.e., ξ̄2=ξ̄1weweexp(r2/w02)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 δS̄ 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 ηf) between Eq. (5) and the temporal evolution of the average spin. Here, S̄(ηf) 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 σɛ/ɛi varies from 1% to 30%, the average energy (ε̄f500 MeV) of the final electron beam (we = 1λ0, ɛi = 1 GeV) changes by ∼1%; see Fig. 5(a). However, the effect of energy spread on the spin polarization S̄f,x of the final state is ∼0.3%; see Fig. 5(c). According to Eq. (5) expressing the analytical AE, Sf ∼ exp(−k1γe) and δSfδγek1 exp(−k1γe), which leads to the conclusion that the spin variations due to dynamics exhibit exponential decay. Similarly, while the initial angular spread σθ changes from 0.3 to 100 mrad, the normalized variation of angular spread Nθ is ∼30%, and the effect on the spin S̄f,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 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 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 R10%. Especially for w0, the accuracy of the ML-assisted method is at least twice as good as that of the analytical prediction.

    Impact of probe electron beam parameters on detection signals. (a) Final average kinetic energies ε̄f vs initial energy spreads σɛ/ɛi of probe electron beams (σθ = 0.3 mrad). Lines marked with triangles, circles, and diamonds denote probe electrons with different beam radii and energies. The initial spin polarization S̄i,x=1, and the laser parameters are the same as in Fig. 4(d). (b) Relative changes in angular spread Nθ = (Δθf,x − Δθi,x)/Δθf,x vs initial angular spread σθ (σɛ/ɛi = 0.05) of probe electron beams, where Δθi,x and Δθf,x denote the full widths at half maximum (FWHM) of the initial and final angular spectra along the x direction, and θx = arctan(px/pz). (c) and (d) Final transverse spin degrees of polarization of scattered electron beams S̄f,x vs σɛ/ɛi and σθ, respectively. (e) and (f) Relative errors R vs σɛ/ɛi and σθ, respectively. The red and blue lines are the relative errors from analytical asymptotic estimation and BPNN, respectively. Lines marked with triangles, circles, and diamonds denote R of ξ, w0, and τ, respectively.

    Figure 5.Impact of probe electron beam parameters on detection signals. (a) Final average kinetic energies ε̄f vs initial energy spreads σɛ/ɛi of probe electron beams (σθ = 0.3 mrad). Lines marked with triangles, circles, and diamonds denote probe electrons with different beam radii and energies. The initial spin polarization S̄i,x=1, and the laser parameters are the same as in Fig. 4(d). (b) Relative changes in angular spread Nθ = (Δθf,x − Δθi,x)/Δθf,x vs initial angular spread σθ (σɛ/ɛi = 0.05) of probe electron beams, where Δθi,x and Δθf,x denote the full widths at half maximum (FWHM) of the initial and final angular spectra along the x direction, and θx = arctan(px/pz). (c) and (d) Final transverse spin degrees of polarization of scattered electron beams S̄f,x vs σɛ/ɛi and σθ, respectively. (e) and (f) Relative errors R vs σɛ/ɛi and σθ, respectively. The red and blue lines are the relative errors from analytical asymptotic estimation and BPNN, respectively. Lines marked with triangles, circles, and diamonds denote R of ξ, w0, and τ, respectively.

    IV. CONCLUSION

    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.

    ACKNOWLEDGMENTS

    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).

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    Zhi-Wei Lu, Xin-Di Hou, Feng Wan, Yousef I. Salamin, Chong Lv, Bo Zhang, Fei Wang, Zhong-Feng Xu, Jian-Xing Li. Diagnosis of ultrafast ultraintense laser pulse characteristics by machine-learning-assisted electron spin[J]. Matter and Radiation at Extremes, 2023, 8(3): 034401

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    Paper Information

    Category: Fundamental Physics At Extreme Light

    Received: Dec. 31, 2022

    Accepted: Feb. 26, 2023

    Published Online: Jun. 30, 2023

    The Author Email:

    DOI:10.1063/5.0140828

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