# Kernel formalism applied to Fourier based wave front sensing in presence of residual phases

## TL;DR

This paper models Fourier-based Wave Front Sensors as linear integral operators with kernels, simplifying their analysis and optimization, especially in the presence of residual phases, and introduces a convolutional kernel approach for fast diagnostics.

## Contribution

It introduces a convolutional kernel framework for Fourier-based WFSs, enabling simplified modeling and sensitivity analysis, even with residual phases.

## Key findings

- Convolutional kernels allow concise WFS behavior representation.
- Residual phases do not invalidate the convolutional model.
- The approach simplifies sensitivity computation and optimization.

## Abstract

In this paper, we describe Fourier-based Wave Front Sensors (WFS) as linear integral operators, characterized by their Kernel. In a first part, we derive the dependency of this quantity with respect to the WFS's optical parameters: pupil geometry, filtering mask, tip/tilt modulation. In a second part we focus the study on the special case of convolutional Kernels. The assumptions required to be in such a regime are described. We then show that these convolutional kernels allow to drastically simplify the WFS's model by summarizing its behavior in a concise and comprehensive quantity called the WFS's Impulse Response. We explain in particular how it allows to compute the sensor's sensitivity with respect to the spatial frequencies. Such an approach therefore provides a fast diagnostic tool to compare and optimize Fourier-based WFSs. In a third part, we develop the impact of the residual phases on the sensor's impulse response, and show that the convolutional model remains valid. Finally, a section dedicated to the Pyramid WFS concludes this work, and illustrates how the slopes maps are easily handled by the convolutional model.