Real-time Adaptive Optics with pyramid wavefront sensors: A theoretical analysis of the pyramid sensor model

TL;DR

This paper provides a detailed theoretical analysis of pyramid wavefront sensors used in adaptive optics, deriving mathematical models and adjoint operators to facilitate the development of fast, stable wavefront reconstruction algorithms.

Contribution

It introduces comprehensive mathematical models for pyramid and roof wavefront sensors, including their approximations and adjoint operators, supporting advanced iterative reconstruction methods.

Findings

Derived detailed mathematical models for pyramid and roof wavefront sensors

Calculated adjoint operators for use in iterative reconstruction algorithms

Supported development of fast, stable wavefront reconstruction techniques

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.