The optical differentiation coronagraph relies on the optical differentiation technique implemented on a standard coronagraph. The use of a coronagraphic mask to estimate the first derivative of the incoming field shows high starlight suppression (theoretically infinite), but it is highly sensitive to pointing errors and suffers from monochromaticity drawbacks. In order to overcome these limitations, we generalize the optical differentiation concept to higher order derivatives. Here we describe a novel set of coronagraphic masks that estimate the second derivative of the incoming field. A mathematical description of the optical differentiation coronagraph is presented. These new masks also achieve a theoretical perfect suppression of the on-axis light, and furthermore, they are less sensitive to pointing errors (fourth-order sensitivity to tip/tilt error leakage). Moreover, they are pure amplitude masks, and hence they do not require a complementary phase mask, which represents an additional advantage. The use of a Gaussian roll-off helps to concentrate the diffracted starlight near the pupil borders, where it could be removed more efficiently by a Lyot stop, and transforms the coronagraph into a nearly band-limited coronagraph.
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