Description of the non-linear behaviour of Fourier filtering wavefront sensor using matrix formalism: the specific matrix

Fourier Filtering WaveFront Sensors (FFWFS) are extremely sensitive wavefront sensors that will improve the wavefront estimation of large ground-based facilities such as European Extremely Large Telescope (ELT) or Giant Magellan Telescope (GMT). Yet they suffer from inherent non-linear behaviour that prevents their optimised use. We propose a modification of the calibration process of these sensors to implement their non linearities in the matrix formalism (linear formalism). The new approach called the specific matrix tackles the loss of sensitivity and the modal confusion of the FFWFS. It allows the measurement of the absolute phase and to close the loop using a non-modulated pyramid. We demonstrate this method by use of numerical simulations.