Realizing Laminar-Like Flow In Charged Bunches With Density Evolution Equations

In the ultra-fast electron microscopy community, electron bunches with much smaller longitudinal widths than transverse widths are emitted from the cathode surface. The community has believed that these bunches evolve to a uniform ellipsoid, but recent simulations by our group and others suggest that if the bunch has an initially transverse Gaussian profile, a ring-like density "shock" emerges at the median of the bunch during evolution. To explain these results, we generalized Reed's 1D fluid model of charged bunch expansion to cylindrical and spherical geometries demonstrating such a shock emerges analytically under these symmetric geometries. Mathematically, the shock in these models occurs when particles more toward the middle "catch-up" to outer particles, and eventually the trajectory of the more central particle crosses-over the outer particle's trajectory. This cross-over marks the transition from the laminar to nonlaminar regime. However, this theory has been developed for cold-bunches, i.e. bunches of electrons with zero initial momentum. Here, we briefly review this new theory and extend it to the cylindrically- and spherically-symmetric cases that have nonzero initial momentum. This formulation elucidates how charge-dominated bunches may be manipulated to maintain laminar conditions even through focussing of the bunch.