Ionization Cooling in all Phase Space Planes with Various Absorber   Shapes, Including Parallel-Faced Absorbers
  Ionization cooling in a straight beamline reduces the transverse emittance of a beam, and has little effect on the longitudinal emittance (generally, in fact, it increases the longitudinal emittance). Once the beamline bends, the introduction of dispersion creates a coupling between the transverse and longitudinal planes. If this coupling is handled properly, one can achieve cooling in all three phase space planes. This is usually done by placing a wedge-shaped absorber in a region where there is dispersion. I will demonstrate using an eigenvalue analysis that there are other configurations of dispersion and absorber shape that will achieve ionization cooling in all phase space planes. In particular, I will show that a one can even achieve cooling in all phase planes with a parallel-faced absorber in a dispersion-free region. I will use perturbation theory to approximate the change in the cooling rates due to longitudinal-transverse coupling. I will then describe how the cooling of longitudinal oscillations can be understood via the projection of the "longitudinal" eigenmodes into the transverse plane. 
J. Scott Berg