Fig. 1. Left: Three different electron beams propagating along the z direction with parameters ɛ_i, w_e, 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.

Fig. 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).

Fig. 3. (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 (ɛ_i = 1 GeV, w_e = λ₀, S̄i,x=1) (ɛ_i = 1 GeV, w_e = 3λ₀, S̄i,x=1), and (ɛ_i = 1.5 GeV, w_e = 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 (ɛ_i = 500 MeV, w_e = λ₀, S̄i,x=1) (ɛ_i = 500 MeV, w_e = 4λ₀, S̄i,x=0.8), and (ɛ_i = 2 GeV, w_e = λ₀, S̄i,x=0.6).

Fig. 4. (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 ɛ_i = 1 GeV, a beam radius w_e = 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 w_e = 1λ₀ (solid line) and w_e = 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 η_f). (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.

Fig. 5. Impact of probe electron beam parameters on detection signals. (a) Final average kinetic energies ε̄f vs initial energy spreads σ_ɛ/ɛ_i of probe electron beams (σ_θ = 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 N_θ = (Δθ_f,x − Δθ_i,x)/Δθ_f,x vs initial angular spread σ_θ (σ_ɛ/ɛ_i = 0.05) of probe electron beams, where Δθ_i,x and Δθ_f,x denote the full widths at half maximum (FWHM) of the initial and final angular spectra along the x direction, and θ_x = arctan(p_x/p_z). (c) and (d) Final transverse spin degrees of polarization of scattered electron beams S̄f,x vs σ_ɛ/ɛ_i and σ_θ, respectively. (e) and (f) Relative errors R vs σ_ɛ/ɛ_i and σ_θ, 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.