Electron Trigonometry-A New Tool for Electron-Optical Design

Electron-trigonometry is a new method to predict image formation in an electron-optical system, which may consist of one, or more, electron lenses of finite length, including electrostatic, magnetic or mixed lenses. Electron-trigonometry by-passes trajectory tracing and the determination of cardinal points. Instead, it provides immediate information about size and position of an electron image. To analyze an electrostatic system by "electron-trigonometry," a given axial potential distribution is approximated by segments, which are either voltage-linear, or parabolic. These segments are joined together with continuity of function and slope. Linear sections are replaced by equivalent drift tubes whose lengths are scaled appropriately. Parabolic sections are shown to observe certain basic triangle rules about the sum of three phase angles of flight, which are specifically defined for object, image, and lens volume. Magnification and position of electron images are immediately known from these same phase angles. This includes both real and virtual images, and lenses of either polarity. A system including magnetic lenses is similarly treated. Such fields are subdivided into segments where the magnetic field is replaced by its average value while the axis potential is either linear or parabolic. As a test case, electron-trigonometry is applied to the well-known system of a two-cylinder, bi-potential lens. This is a double lens system whose axis potential distribution is not simple. The results from electron-trigonometry are in excellent agreement with recorded data of performance. In another application, electron-trigonometry is used to compute the angular magnification of a cathode ray tube with post-deflection acceleration.