We present a mathematical model for the generation of vortex-beams by using a square profile amplitude fork diffraction grating with arbitrary topological charge. The mathematical framework of aberrations in the forked-shape diffraction grating is analysed, and the resulting diffracted pattern is simulated. Three cases of desired distortions (aberrations) in the diffraction grating are considered, obtaining phase modulation from the amplitude grating. Experimental optical vortices are generated by using a transmission spatial light modulator, which is used as a dynamic diffraction grating, allowing us to aberrate it. We show the effect of aberrations in the experimental diffracted vortex-beams and compare it with the numerical simulation.