Transverse structure of field-emission electron wave packets from nanometer-scale hemispheroidal cathodes and two-dimensional film edge emitters

The transverse structure of field-emission electron wave packets is governed by the cathode material and geometry through the near-apex electrostatic potential, which induces electrostatic defocusing and spatial transmission filtering. To study these effects, we develop a framework for field-emission wave-packet propagation that combines finite-element electrostatic potentials with paraxial Schrödinger propagation under an effective parabolic transverse-potential approximation, together with barrier-side spatial filtering. Applied to nanometer-scale prolate hemispheroidal emitters, the method enables efficient wave-packet calculations that reproduce the expected electron-optical defocusing and show how tunneling-induced narrowing can amplify subsequent transverse expansion. We further apply the framework to a two-dimensional film-edge model of a graphene edge emitter, where the Å-scale initial width leads to stronger transverse expansion. The resulting wave packets are then used to study coherent multi-site superposition, predicting either overall beam narrowing or fragmentation into multiple beamlets, which may contribute to the striated field-emission-microscopy patterns reported for graphene edges.