Analytic expression for the ideal one-dimensional mirror potential yielding perfect energy focusing in TOF mass spectrometry

The resolving power of a Wiley-McLaren time-of-flight mass spectrometer can be severely limited by the longitudinal velocity spread of ions originating from the same position along the flight path. This effect is normally minimized by using an electrostatic mirror to reflect the incident ions from their original direction back toward the spectrometer detector, in which case higher velocity ions have a greater delay time in the mirror due to their further penetration into the mirror potential, and spatial focusing occurs at the plane of the detector. In this paper, an analytic expression is obtained for the ideal one-dimensional mirror potential that yields perfect spatial focusing along the flight path. The mathematical techniques used are those commonly employed in the solution of calculus of variations problems.