Fig. 2: The effective correlation radius of the Cherenkov radiation.
The ratio Reff to the distance \({u}_{p}{t}^{{\prime} }\) is given for the electron traveling times \({t}^{{\prime} }=1{0}^{7}\,{t}_{{{{\rm{C}}}}}\) (solid black line), \({t}^{{\prime} }=1{0}^{10}\,{t}_{{{{\rm{C}}}}}\) (dotted red line) and \({t}^{{\prime} }=1{0}^{14}\,{t}_{{{{\rm{C}}}}}\) (dot-dashed green line). At panel a \(\beta =0.7\,(\gamma =1.4),n=1.5,\,\theta ={\theta }_{{{{\rm{Ch.cl.}}}}}= \arccos (1/{u}_{p}n)\approx 17.{8}^{\circ },\,\omega =1{0}^{-6}m,\,\sigma =1{0}^{-5}m,\,{\phi }_{R}-\phi =0\) deg, θMach ≈ 107.8°. At panel b β = 0.9999 (γ = 70.7), n = 1.33, θ = θCh.cl. ≈41. 2°, ω = 10−5m, σ = 10−4m, ϕR − ϕ = 0°, θMach ≈131. 2∘. Nearby the Mach angle θMach, space-time dependence of the Wigner function quickly vanishes within the correlation radius \(R < {R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\), which is a hallmark of the wave zone.
