Particle beam eigenemittances, phase integral, vorticity, and rotations

Particle beam eigenemittances comprise the lowest set of rms-emittances that can be imposed to a beam through symplectic optical elements. For cases of practical relevance this paper introduces an approximation providing a very simple and powerful relation between transverse eigenemittance variation and the beam phase integral. This relation enormously facilitates modeling eigenemittance tailoring scenarios. It reveals that difference of eigenemittances is given by the beam phase integral or vorticity rather than by angular momentum. Within the approximation any beam is equivalent to two objects rotating at angular velocities $\ifmmode\pm\else\textpm\fi{}\ensuremath{\omega}$. A description through circular beam modes has been done already in [A. Burov, S. Nagaitsev, and Y. Derbenev, Circular modes, beam adapters, and their applications in beam optics, Phys. Rev. E 66, 016503 (2002)]. The new relation presented here is a complementary and vivid approach to provide a physical picture of the nature of eigenemittances for cases of practical interest.