We propose a simple and general analytical model describing the operation of a velocity-map-imaging spectrometer. We show that such a spectrometer, possibly equipped with a magnifying lens, can be efficiently modeled by combining analytical expressions for the axial potential distributions along with a transfer matrix method. The model leads transparently to the prediction of the instruments operating conditions as well as to its resolution. A photoelectron velocity-map-imaging spectrometer with a magnifying lens, built and operated along the lines suggested by the model has been successfully employed for recording images at threshold photoionization of atomic lithium. The models reliability is demonstrated by the fairly good agreement between experimental results and calculations. Finally, the limitations of the analytical method along with possible generalizations, extensions, and potential applications are also discussed. The model may serve as a guide for users interested in building and operating such spectrometers as well as a tutorial tool.
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