Calibration of photomultiplier tubes

The purpose of the present article is to demonstrate the calibration of photomultipliers with a gaussian single photoelectron response using a numerical method based on the Discrete Fourier Transform (DFT). Conventional techniques, commonly employed in the literature, use approximate models or brute force numerical calculations of the convolution integrals that lead to the charge response function of the photomultiplier, $S_R(x)$. In this publication, we explain how a truncated gaussian model for the single photoelectron amplification can lead to rigorous results if one leans on the DFT approach. The distinct feature of this procedure is that $S_R(x)$ is calculated to all orders in the Poisson mean $\mu$ that characterizes the light intensity and no approximations are needed. This scheme was applied to the calibration of the Hamamatsu R7081 photomultiplier tube and a comparison of the DFT approach with the more standard numerical integration method is also presented. Last, toy Monte Carlo data were analyzed for different values of $\mu$ to understand the precision of the DFT method.