This dissertation is centered on the analytical and numerical study of ellipsoidal electron bunch dynamics. We are particularly interested in the focusing process of the probing electron bunches in the ultrafast electron diffraction/microscopy system, so that we can improve the temporal and spectral resolution of the ultrafast experiments. More specifically, to understand the collective space charge effects throughout the bunch evolution, we employed several analytic models to describe the bunch dynamics. We start with an extension of the mean-field model using ordinary differential equations. Analysis of this mean-field model leads to the identification of a longitudinal critical chirp, which separates two regimes for particle trajectories for the longitudinal focusing of the bunches: (1) bounce-back, where the particles reverse their direction at the waist of the focusing process, and (2) cross-over, where the bunch experiences a singularity where the bunch width reduces to zero. We show that time can be scaled by the initial plasma frequency, and that the critical chirp becoming dimensionless and depends solely on the initial bunch aspect ratio. In order to study the emittance effect on the bunch dynamics, we introduce the self-similar analytical (SSA) model, a statistical method describing the second order moments dynamics to model the evolution of an ellipsoidal electron bunch. We also discussed its linear chirp assumption, explaining how it is the key assumption that leads to the emittance conservation according to the SSA model. We discuss the statistical nature of bunch emittance noting that the space charge effect of the uniform density profile and of the Gaussian profile are close to each other in the SSA model. The impact from a changing emittance is captured by an additional term in the modified SSA model, which is then equivalent to the Kapchinsky-Vladimirsky (K-V) envelope equation in accelerator physics. We point out that the application of the statistical methods can extend beyond the uniform ellipsoidal bunch, while the accuracy of the SSA prediction is mainly related to the discrepancy between the actual density profile and the uniform density profile.We present the Molecular Dynamics (MD) simulation results for the longitudinal focusing process of uniform spheroidal electron bunches. The comparison of the longitudinal width evolution between the MD simulations and the SSA predictions shows the impact of a varying emittance on bunch evolution. We propose two competing mechanisms for the change of emittance throughout the compression process. The disorder-induced heating (DIH) increases the emittance in both degrees of freedom while the difference in the SSA temperature generates emittance transfer between degrees of freedom. In addition, the non-uniform density profile at the focal point introduces significant emittance growth in both the longitudinal and transverse directions.
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