A recently proposed method is upgraded to convert two amplitude phase modulation systems (APMSs) to pure phase elements (PPEs), for generating the stable propagation Bessel beam and the axial multifoci beam, respectively. Phase functions of the PPEs are presented analytically. Numerical simulations by the complete Rayleigh-Sommerfeld method demonstrate that the converted PPE has implemented the same optical functionalities as the corresponding APMS, in either the longitudinal or the transverse direction. Compared with the traditional APMS, the converted PPE possesses many advantages such as fabrication process simplification, system complexity reduction, production cost conservation, alignment error avoidance, and experimental precision enhancement. These inherent advantages position the PPE as an ideal choice and driving force behind further advancements in optical system technology.