## Periodic Electrostatic Focusing of a Hollow Electron Beam |

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### Contents

Derivation of basic focusing properties | 8 |

Trajectories of electrons at positions inside | 13 |

the beam 41 | 14 |

22 other sections not shown

### Common terms and phrases

approximate backward wave oscillator beam boundaries beam deflection beam electrode beam interior beam position beam shape beam tester beam thickness beam trajectories beam velocity bifilar helix carbon screen cathode cd cd charge fields circuit CoshpY CospZ current density curve cylinder voltage electric fields electrostatic focusing equilibrium position expression focused beam focusing electrode focusing fields focusing forces focusing mechanism focusing method focusing potential focusing rings focusing strength focusing structure focusing system Harris flow high-voltage ring hollow-beam ibrium inner and outer inner boundary Laplace's Equation magnet microwave tubes obtained oscillations outer boundary overfocusing overshoot periodic electrostatic perturbation plotted on Fig Poisson's Equation poor entrance conditions radial fields radial velocity restoring forces ring spacing shown in Fig SinhB 1-Y Sinhp space-charge field space-charge forces space-charge potential stable flow stiffness trajectory equation trigonometric identities value of pT velocity variation yi yi