Geometry-Enabled Radiation from Structured Paraxial Electrons

Abstract

We present a microscopic calculation of spontaneous photon emission by twisted (paraxial) electrons propagating through inhomogeneous, axisymmetric magnetic fields. We construct exact electron states that incorporate transverse mode structure and wavefront curvature by combining the Foldy-Wouthuysen transformation with a geometric framework based on Lewis-Ermakov invariants and metaplectic transformations. We show that the evolution of such structured states corresponds to an open path in the space of quadratic forms, giving rise to a geometric contribution to the emission amplitude that cannot be eliminated by gauge choice or adiabatic arguments. The inverse radius of curvature of the electron wavefront emerges as an effective geometric field that enables radiation even in regions where the external magnetic field vanishes locally. This mechanism generalizes Landau-level radiation to nonasymptotic, structured electron states and establishes a direct connection between noncyclic geometric evolution and photon emission.

Figures (2)

• Figure 1: Schematic setup: two solenoids shape an axisymmetric guiding field $B_z(z)$ and imprint a nontrivial Ermakov scaling $b(z)$ (state transverse size and wavefront curvature) on the paraxial twisted-electron mode. Photon detection occurs in the central cavity located in a locally field-free region, where radiation persists due to the inherited curvature.
• Figure 2: Normalized differential rates for the field-free region calculated for $\beta=0.548$ (electron kinetic energy $100$ keV) using Eq.\ref{['eq:Rrate']} as a function of the angle $\theta$. The left panel uses interaction length $L=30$ and the right panel uses $L=250$ (both in units of $k\rho_H^2$).