Progress in the linear beam dynamics study of ionization cooling channel

We review recent progresses in the study of linear beam dynamics of the ionization cooling in solenoidal focusing channels. The topics included are: understanding of the solenoidal focusing in the rotating frame; beam parameterization in terms of the envelope functions, emittance, and angular momentum; the coupled equation for emittance and angular momentum in periodic focusing channels; general solution and comparison with simulation; analytic approach for calculating the stability criteria and envelope function in periodic focusing channels.