Abstract
We consider the mathematical background of the wavefront sensor type that is widely used in adaptive optics systems for astronomy, microscopy, and ophthalmology. The theoretical analysis of the pyramid sensor forward operators presented in this paper is aimed at a subsequent development of fast and stable algorithms for wavefront reconstruction from data of this sensor type. In our analysis we allow the sensor to be utilized in both the modulated and non-modulated fashion. We derive detailed mathematical models for the pyramid sensor and the physically simpler roof wavefront sensor as well as their various approximations. Additionally, we calculate adjoint operators which build preliminaries for the application of several iterative mathematical approaches for solving inverse problems such as gradient based algorithms, Landweber iteration or Kaczmarz methods.
Highlights
Ground-based telescope facilities suffer from degraded image quality caused by atmospheric turbulence
Advanced Adaptive Optics (AO) systems [35, 64] are incorporated in innovative telescope systems to mechanically correct in real-time for the distortions with deformable mirrors
In Adaptive Optics one is interested in the reconstruction of the unknown incoming wavefront Φ from available data in order to calculate the optimal shape of the deformable mirror
Summary
Ground-based telescope facilities suffer from degraded image quality caused by atmospheric turbulence. In Adaptive Optics one is interested in the reconstruction of the unknown incoming wavefront Φ from available data in order to calculate the optimal shape of the deformable mirror. The main goal of this paper is to provide an extensive mathematical analysis of the PWFS operators in order to develop suitable wavefront reconstruction methods. The simplifications of the non-linear pyramid operator allow for acceptable wavefront reconstruction quality. Wavefront reconstruction from pyramid sensor data consists in solving two non-linear integral equations.
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