Direct fabrication of complex diffractive optical elements (DOEs) on photosensitive thin films is of critical importance for the development of advanced optical instruments. In this paper, we design and investigate DOEs capable of generating optical vortices. Analog and digital approaches for one-step polarization holographic recording of vortex DOEs on new carbazole-based azopolymer thin films are described. First configuration involves analog polarization holographic recording using a vortex phase retarder and has as a result the DOE producing a diffraction pattern with phase singularities aligned in a single line. Similar diffraction picture is achieved by the single-beam digital holographic recording setup with an integrated spatial light modulator. In the third system, the implemented double-beam digital polarization holographic recording setup yields simultaneously a spatial multiplexed vortex pattern. Diffraction efficiency evolution of these three types of DOEs are monitored and compared. The phase-shifting digital holographic microscope with an electrically controlled liquid crystal variable retarder is applied to investigate the phase and surface topography of the inscribed diffractive optical elements. The comparison between the digital and analog micro-patterning techniques contributes new evidence to limited data on the influence of the analog and digital generation of the spiral wavefront on the performance of vortex DOEs.