The design and testing of computer-generated holograms for complex light illumination are introduced and described. The high-order beams carrying angular momentum (OAM) are generated by means of novel spiral phase plates (SPP) for the control of the radial index. Besides the common helical profile, these phase plates exhibit radial π-discontinuities located at the zeros of the associated Laguerre polynomial. SPP samples were fabricated by electron beam lithography over glass substrates coated with a polymethylmethacrylate (PMMA) resist layer. A numerical code based on iterative Fourier transform algorithm has been developed for the computation of phase-only diffractive optical element for illumination under OAM beams. Numerical analysis and preliminary experimental results confirm the applicability of these computer-generated holograms as high-security optical elements for anti-counterfeiting applications.