In this paper, we describe Fourier-based Wave Front Sensors (WFS) as linear integral operators, characterized by their Kernel. In a first part, we derive the dependency of this quantity with respect to the WFSs optical parameters: pupil geometry, filtering mask, tip/tilt modulation. In a second part we focus the study on the special case of convolutional Kernels. The assumptions required to be in such a regime are described. We then show that these convolutional kernels allow to drastically simplify the WFSs model by summarizing its behavior in a concise and comprehensive quantity called the WFSs Impulse Response. We explain in particular how it allows to compute the sensors sensitivity with respect to the spatial frequencies. Such an approach therefore provides a fast diagnostic tool to compare and optimize Fourier-based WFSs. In a third part, we develop the impact of the residual phases on the sensors impulse response, and show that the convolutional model remains valid. Finally, a section dedicated to the Pyramid WFS concludes this work, and illustrates how the slopes maps are easily handled by the convolutional model.
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