Fig. 1. (a) Comparison of normalized differential radiation probability of NCS (orange/blue solid curves for χ = 0.1/0.3) with the approximation (black/red dotted curves for χ = 0.1/0.3). (b) Comparison of normalized differential radiation probability of QSR (orange/blue solid curves for χ = 0.1/0.3) with the approximation (black/red dotted curves for χ = 0.1/0.3).

Fig. 2. (a) Comparison of P_i distributions given by simulations (symbols) and their analytic approximations (curves) of p₀ = 1.5 GeV electrons propagating 1–4 μm (r = 1.15, 2.3, 3.45, and 4.6) perpendicularly in a λ₀ = 800 nm, circularly polarized laser with I = 1 × 10²¹ W/cm². (b) Functions f_i.

Fig. 3. Comparison of the analytic formulas (curves) and simulations (symbols) for the electron spectrum after multiple NCS. Electrons with p₀ = 0.5 GeV (a), 1.0 GeV (b), 2.0 GeV (c), and 5.0 GeV (d) propagate perpendicularly through the waist of a λ = 800 nm circularly polarized laser that lasts for τ = 30 fs. The peak laser intensity is 5 × 10²⁰ W/cm² and the laser waist is 3.6 μm. χ₀ = a₀p₀k₀/m² is 0.05, 0.1, 0.2 and 0.5, respectively, in (a)–(d).

Fig. 4. Peak splitting process. (a), (b), and (c) Spectra of a 1 GeV electron beam after it has propagated respectively D = 2, 3, and 4 μm perpendicularly in a λ = 800 nm circularly polarized laser with I = 4 × 10²¹ W/cm², corresponding respectively to r = 4.6, 6.9, and 9.2. (d), (e), and (f) Spectra for respectively all electrons, i ≤ 3 electrons, and i ≥ 4 electrons in the whole peak splitting process. The curves in (d) show the peaks given by classical theory (white dot-dashed), analytic quantum theory (blue dashed), and simulation (gray solid). (g) Normalized spectra of emitted γ rays after the electron beam has propagated different distances.

Fig. 5. Sketches of (a) conventional and (b) new diagrams for quantum radiation reaction stages. The purple lines are peaks of the electron spectrum, r is the average number of emissions, and η is the energy normalized by the primary energy.