A method to generate the optical vortex beam with arbitrary superposition of different orders of orbital angular momentum (OAM) on a photonic chip is proposed. The distributed Fourier holographic gratings are proposed to convert the propagating wave in waveguides to a vortex beam in the free space, and the components of different OAMs can be controlled by the amplitude and phases of on-chip incident light based on the principle of Fourier transformation. As an example, we studied a typical device composed of nine Fourier holographic gratings on fan-shaped waveguides. By scalar diffraction calculation, the OAM of the optical beam from the reconfigurable vortex beam generator can be controlled on-demand from -2nd to 2nd by adjusting the phase of input light fields, which is demonstrated numerically with the fidelity of generated optical vortex beam above 0.69. The working bandwidth of the Fourier holographic grating is about 80 nm with a fidelity above 0.6. Our work provides an feasible method to manipulate the vortex beam or detect arbitrary superposition of OAMs, which can be used in integrated photonics structures for optical trapping, signal processing, and imaging.
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