MSU, East Lansing, Michigan, USA
Reed (2006) presented a 1D mean-field model of initially cold pancake-beam expansion demonstrating that the evolution of the entire spatial distribution can be solved for all time where the 1D assumption holds. This model is relevant to ultra-fast electron microscopy as it describes the evolution of the distribution within the photoelectron gun, and this model is similar to Anderson’s sheet beam density time dependence (Anderson 1987) except that Reed’s theory applies to freely expanding beams instead of beams within a focussing channel. Our recent work (Zerbe 2018) generalized Reed’s analysis to cylindrical and spherical geometries demonstrating the presence of a shock that is seen in the Coulomb explosion literature under these geometries and further discussed the absence of a shock in the 1D model. This work is relevant as it offers a mechanistic explanation of the ring-like density shock that arises in non-equilibrium pancake-beams within the photoelectron gun; moreover, this shock is coincident with a region of high-temperature electrons providing an explanation for why experimentally aperturing the electron bunch results in a greater than 10-fold improvement in beam emittance(Williams 2017), possibly even resulting in bunch emittance below the intrinsic emittance of the cathode. However, this theory has been developed for cold-bunches, i.e. bunches of electrons with 0 initial momentum. Here, we briefly review this new theory and extend the cylindrical- and spherical- symmetric distribution to ensembles that have non-zero initial momentum distributions that are symmetric but otherwise unrestricted demonstrating how initial velocity distributions couple to the shocks seen in the less general formulation. Further, we derive and demonstrate how the laminar condition may be propagated through beam foci.
