Non-Liouvillean achromatic beam guidance systems

Achromatic beam guidance systems are described that use not only magnetic sector fields and quadrupole lenses but contain at specific positions also energy absorber foils whose thicknesses vary across the beam. For fully ionized energetic particles such absorber foils can make an initially only focusing beam guidance system to be also achromatic. In these systems the δ_tδ_K phase space can be reduced which would be impossible for systems using only magnetic fields.