# Response of QD to structured beams via convolution integrals

## TL;DR

This paper introduces a convolution integral-based expression for the response of quadrant detectors to structured beams, simplifying calculations and enabling analytical and computational analysis of detector sensitivity to various beam types.

## Contribution

A novel convolution integral formulation for quadrant detector response that simplifies evaluation and allows analytical solutions for different beam profiles.

## Key findings

- Derived analytical response for Gaussian beams
- Compared new response with previous approximations and simulations
- Demonstrated computational efficiency for Bessel beam sensitivity

## Abstract

We propose a new expression for the response of a quadrant detector using convolution integrals. This expression is easier to evaluate by hand, exploiting the properties of the convolution. Computationally, it is also practicable to use since a large number of computer programs can right away evaluate convolutions. We use the new expression to obtain an analytical form of the response of a quadrant detector to a Gaussian beam and to Hermite-Gaussian beams in general. We compare this analytic expression for the response for the Gaussian beam with the approximations from previous studies and with a response obtained through simulations. From the response, we also obtained an analytical form for the sensitivity of the quadrant detector to a Gaussian beam. Lastly, we demonstrate the computational ease of using our new expression for the response calculating the sensitivity of the quadrant detector to the Bessel beam.