Particle beam eigen-emittances 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 eigen-emittance variation and the beam phase integral. This relation enormously facilitates modeling eigen-emittance tailoring scenarios. It reveals that difference of eigen-emittances 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 of same strength and different sign. 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 eigen-emittances for cases of practical interest.
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