Objective The pyramid wavefront sensor (PWFS) has been successfully applied to astronomical adaptive optics, mirror testing, and microscopy imaging due to its advantages of high energy utilization and spatial resolution. Modulation is often performed to expand the linear and dynamic ranges of the PWFS. Classical modulation methods include mechanical modulation, static modulation, and dynamic aberration modulation. Mechanical modulation involves the oscillation of the pyramid itself or a tip-tilt mirror at the entrance pupil of the system; static modulation adds a diffuser into the system; dynamic aberration modulation uses rapidly changing and undetectable aberrations as the signals to be modulated. However, the above methods all sacrifice the sensitivity of the PWFS for wider dynamic and linear ranges, which reduces the practicality of the PWFS. This paper proposes a novel non-modulation PWFS to expand the application of the PWFS in the field of phase detection. The proposed PWFS iteratively optimizes the wavefront to be measured with a phase retrieval algorithm based on a light-field propagation model of the PWFS. This PWFS based on phase retrieval has the features of high accuracy, fast convergence speed, and favorable noise immunity. Moreover, the proposed sensor covers a large dynamic range with no need for modulation. Methods A PWFS with a 4f configuration is adopted, and a phase retrieval algorithm based on a light- field propagation model of the PWFS is designed to reconstruct the wavefront to be measured. Due to the beam-splitting effect of the pyramid tip, four sub- images of the pupil are recorded by the detector of the PWFS. The whole image from the detector is used as one constraint on the phase retrieval algorithm, while the assumed uniform intensity on the pupil plane serves as the other constraint. Owing to the abundant information provided by this detector image, the phase retrieval algorithm in the proposed sensor usually converges quickly. A series of simulation experiments are performed to evaluate the performance of the proposed sensor. Firstly, three different kinds of wavefronts, including a complex randomly combined aberration, a random phase of atmospheric turbulence, and a freeform surface with a large amplitude, are selected as the wavefronts to be measured to explore the generality of the proposed sensor. Secondly, convergence comparisons with the classical phase retrieval algorithm are conducted in the form of reconstruction experiments on wavefronts with different amplitudes. Thirdly, the dynamic range of the proposed sensor is investigated in a simulated scenario, in which the wavefronts to be measured exceed the dynamic range of the traditional PWFS. This experiment is also expected to verify the non- modulation property of the proposed sensor. Fourthly, the performance of the proposed sensor under different noise conditions is evaluated by inputting simulated detector images with different signal- to-noise ratios into its phase retrieval algorithm. Last but not least, a plateau always emerges at the tip of the pyramid due to limited processing technology. The impact of the central plateau on the pyramid on the performance of the proposed sensor is examined by using pyramids with different flat tips to simulate detector images and employing a phase retrieval algorithm based on the desired pyramid shape to reconstruct the wavefronts to be measured. Results and Discussions As shown by the experimental results, the proposed sensor can quickly and accurately reconstruct different kinds of wavefronts with different amplitudes, indicating its favorable wavefront-sensing capability (Figs. 4- 6). The results of convergence comparison experiments reveal that the convergence speed decreases as the amplitude of the wavefront to be measured increases. Nevertheless, the performance of the proposed sensor is always better than that of the classical phase retrieval algorithm ( Fig. 7). The experimental results on the dynamic range show that the proposed PWFS based on phase retrieval performs still well when the amplitudes of the wavefronts to be measured are far beyond the dynamic range of the traditional PWFS. This indicates that the proposed sensor covers a large dynamic range with no need for modulation (Table 1). A slight degradation in the performance of the proposed sensor is observed when noise is present. According to the results, the root- mean- square error of the reconstruction results is smaller than (1/25). even under heavy noise, representing high noise immunity of the proposed sensor (Table 2). In addition, when the size of the flat tip of the pyramid is smaller than 20% of the diameter of the system Airy disk in front of the pyramid, the proposed sensor remains practically useful ( Fig. 11). The results of a series of numerical simulations show that the proposed PWFS based on phase retrieval has the features of high accuracy, fast convergence speed, high noise immunity, and large dynamic range with no need for modulation. Conclusions This study proposes a non- modulation PWFS based on phase retrieval. The sensor avails the beamsplitting property of the pyramid to obtain four sub-images of the pupil, and these sub- images contain the information on the wavefront to be measured. Then, the wavefront to be measured is obtained by iterative optimization with the light- field propagation model of the PWFS. The simulation results show that the proposed phase retrieval algorithm based on the PWFS model is more accurate and converges faster than the classical phase retrieval algorithm. Compared with the traditional PWFS, the proposed sensor can obtain a larger dynamic range with no need for modulation. Its performance is still robust in the presence of noise. Wavefronts can be reconstructed with high accuracy when the central plateau on the pyramid tip is relatively small. As the computational platform and the pyramid processing technology further develop, the proposed sensor is expected to serve as a practical wavefront sensor for adaptive optical systems in the fields of astronomy and biomedicine.
