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Transmissive volume holograms are considered as mode-selective optical elements for the de-multiplexing and detecting of optical vortex modes according to the topological charge or mode number. Diffraction of vortex modes by a fundamental mode hologram is modeled using a physical optics model that treats the volume hologram as an angle-dependent transfer function. Diffracted irradiance profiles and diffraction efficiencies are calculated numerically as a function of the incident mode number. The results of the model are compared with experimental results obtained with volume holograms of fundamental and higher-order vortex modes. When considered as a function of detuning between the incident and recorded mode numbers, the measured diffraction efficiencies are found to be invariant with respect to the recorded mode number, provided that the order difference remains unchanged, and in close agreement with the predictions of the model. Measurements are made with a 1.3mm thick permanent photo-thermo-refractive glass hologram and a 9mm thick re-writable photorefractive lithium niobate hologram. A liquid-crystal spatial light modulator generates the vortex modes used to record and read the holograms. The results indicate that a simple volume hologram can discriminate between vortex modes; however, adjacent mode discrimination with low crosstalk would require a very thick hologram. Furthermore, broadening of the vortex angular spectrum, due to diffraction at a finite aperture, can adversely affect diffraction efficiencies. [ABSTRACT FROM AUTHOR]