Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems

A phase mask at the aperture stop of a hybrid digital-optical imaging system can improve its tolerance to aberrations. The choice of the introduced phase modulation is crucial in the design of such systems. Several successful phase masks have been described in the literature. These masks are typically derived by searching for optical-transfer-functions that retain restorability under aberrations such as defocus. Instead of optimizing the optical-transfer-function for some desired characteristics, we calculate the expected imaging error of the joint design directly. This was used to compare thirddegree polynomial phase masks, including the cubic phase profile and a commonly used generalization. The analysis shows how the optimal phase profile depth is always limited by noise and more importantly, numerical simulations show that only a finite range of the third-degree polynomial profiles yield optimal performance.