  There has been considerable interest recently in the generation of azimuthal phase functions associated with photon orbital angular momentum (OAM) for high-dimensional quantum key distribution (QKD). The generation of secure quantum keys requires not only this pure phase basis, but also additional bases comprised of orthonormal superposition states formed from the pure states. These bases are also known as mutually unbiased bases (MUBs) and include quantum states whose wave functions are modulated in both phase and amplitude. While modulo 2\pi optical path control with high-resolution spatial light modulators (SLMs) is well suited to creating the azimuthal phases associated with the pure states, it does not introduce the amplitude modulation associated with the MUB superposition states. Using computer-generated holography (CGH) with the Leith-Upatnieks approach to hologram recording however, both phase and amplitude modulation can be achieved. This paper presents a description of the OAM states of a 3-dimensional MUB system and analyzes the construction of these states via CGH with a phase modulating SLM. The effects of phase holography artifacts on quantum-state generation are quantified and a prescription for avoiding these artifacts by preconditioning the hologram function is presented. Practical effects associated with spatially isolating the first-order diffracted field are also quantified and a demonstration utilizing a liquid crystal SLM is presented. 
Holographic generation of complex fields with spatial light modulators:   application to quantum key distribution