Presents the vacuum Doppler formula for a relativistic oscillator as an ellipsoid in momentum space and derives the spectral and angular properties of its X-ray emission in a homogeneous medium, where the medium's frequency-dependent permittivity distorts the ellipsoid. Students learn how a single Doppler framework unifies several coherent X-ray sources -- parametric X-ray radiation, coherent bremsstrahlung, undulator, transition, and Smith-Purcell radiation -- produced by charged particles traversing periodic structures.
Derives that the Doppler frequency formula for a relativistic oscillator moving in vacuum is geometrically an ellipsoid in momentum space, mapping the emission angle, Lorentz factor, and velocity ont…