Bunch-length broadening of synchrotron radiation ($\Delta t_b\sim L_b/u$)

In a fixed tangent viewing geometry, an electron bunch of longitudinal length $L_b$ passes the same radiating point over approximately $L_b/u$. Different electrons then emit from nearly the same point at different times. The incoherent pulse envelope is the convolution of the single-electron intensity profile with the bunch's passage-time distribution, assuming equal particle trajectories and a linear detector. Its width generally has scale at least $\max \{L_b/u,r/(c{\gamma }^3)\}$. A bunch short compared with the orbit need not be short compared with the intrinsic single-electron pulse.