Current modulations, current spikes, and current horns, are observed in a range of accelerator physics applications including strong bunch compression in Free Electron Lasers and linear colliders, trains of microbunching for terahertz radiation, microbunching instability and many others. This paper considers the fundamental mechanism that drives intense current modulations in dispersive regions, beyond the common explanation of nonlinear and higher-order effects. Under certain conditions,

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... oring electron trajectories merge to form caustics, and often result in characteristic current spikes. Caustic lines and surfaces are regions of maximum electron density, and are witnessed in accelerator physics as folds in phase space of accelerated bunches. We identify the caustic phenomenon resulting in cusplike current profiles and derive an expression which describes the conditions needed for particle-bunch caustic formation in dispersive regions. The caustic expression not only reveals the conditions necessary for caustics to form but also where in longitudinal space the caustics will form. Particle-tracking simulations are used to verify these findings. We discuss the broader implications of this work including how to utilize the caustic expression for manipulation of the longitudinal phase space to achieve a desired current profile shape.