Response of quadrant detectors to structured beams via convolution integrals

We propose a new expression for the response of a quadrant detector using convolution integrals. This expression, exploiting the properties of the convolution, is easier to evaluate by hand. Computationally, it is also practicable to use since a large number of computer programs can evaluate convolutions right away. We use the new expression to obtain an analytical form of the response of a quadrant detector for a Gaussian beam and for 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 the 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 by calculating the sensitivity of the quadrant detector to the Bessel beam.