Simple analytical model of gain saturation in microchannel plate devices

We derive, discuss, and test against experimental data an analytical model of the gain saturation in microchannel plate (MCP) devices. By introducing a simple recharging circuit for each dynode, we extend the well‐known, unsaturated gain model of Eberhardt to a microchannel operating in condition of gain saturation and show that the amplification of a current pulse and the voltage drop along the channel can be described by a pair of coupled differential equations. Solutions of these equations are given in various conditions, including an approximate solution, valid in the case of weak saturation and a general solution in implicit form. The behavior of a microchannel operating in current mode is studied by finding the transient and steady‐state solutions obtained with an input step current wave form. Exact solutions are given for the charge gain of pulses with a short duration, compared to the dynode recharging time, and for the gain recovery of a microchannel after the amplification of a short pulse. The single channel saturation model is then extended to multistage MCP assemblies by taking into account the statistical distribution of the photoelectrons at the input and the spread of the multiplied electron cloud in the interplate gaps. The expressions found in this way are used for the best fit of experimental data from a Z‐stack MCP photomultiplier operated in single and double pulse mode. Satisfactory agreement between the model and experimental data is obtained in the case of single pulse measurements, finding a reduced chi squared χ²=4.67. Less satisfactory agreement is found for double pulse data, giving χ²=7.46 and a clear indication that the model may be significantly improved by taking into account the charge redistribution among the dynodes during the recharging process, neglected in the present formulation.