Adaptive optics (AO) is a technique to improve the resolution of ground-based telescopes by correcting, in real-time, optical aberrations due to atmospheric turbulence and the telescope itself. With the rise of Giant Segmented Mirror Telescopes (GSMT), AO is needed more than ever to reach the full potential of these future observatories. One of the main performance drivers of an AO system is the wavefront sensing operation, consisting of measuring the shape of the above mentioned optical aberrations. Aims. The non-modulated pyramid wavefront sensor (nPWFS) is a wavefront sensor with high sensitivity, allowing the limits of AO systems to be pushed. The high sensitivity comes at the expense of its dynamic range, which makes it a highly non-linear sensor. We propose here a novel way to invert nPWFS signals by using the principle of reciprocity of light propagation and the Gerchberg-Saxton (GS) algorithm. We test the performance of this reconstructor in two steps: the technique is first implemented in simulations, where some of its basic properties are studied. Then, the GS reconstructor is tested on the Santa Cruz Extreme Adaptive optics Laboratory (SEAL) testbed located at the University of California Santa Cruz. This new way to invert the nPWFS measurements allows us to drastically increase the dynamic range of the reconstruction for the nPWFS, pushing the dynamics close to a modulated PWFS. The reconstructor is an iterative algorithm requiring heavy computational burden, which could be an issue for real-time purposes in its current implementation. However, this new reconstructor could still be helpful in the case of many wavefront control operations. This reconstruction technique has also been successfully tested on the Santa Cruz Extreme AO Laboratory (SEAL) bench where it is now used as the standard way to invert nPWFS signal.
