Analytical calculation of field in electrostatic undulators with circular cross-section conductors

It is well known that it is technically difficult to fabricate small period (< 8-5 mm) PM or electromagnetic undulators (both superconducting and pulsed), and that undulator field B-u and deflection parameter K = 0.934B(u)I(u) decrease drastically with diminishing undulator period I-u. On the other hand, there are no major problems in producing electrostatic undulators with millimeter and submillimeter periods (of course, it is necessary to decrease undulator gap proportionally). Electrostatic undulators could be used in low-electron-energy FELs, where acceptable gain per pass is easily obtainable. The principal problem in this case is to obtain rather large electric field in the median plane of an undulator (note that 1 kG is equivalent to about 300 kV/cm if relative electron velocity v/c similar or equal to 1). Since electric field on electrodes is larger than in the median plane, one needs to calculate both fields and optimize undulator geometry, minimizing as far as possible the ratio of field values. Analytical methods were used in this paper (conformal mapping, expansion in Fourier series) for field calculation and analysis. They allow to quickly review and compare various geometries and to obtain some dependences of field value on various parameters, in contrast to numerical calculations. Numerical calculations were performed to verify analytical methods and evaluate their applicability. Electrostatic and magnetic undulators are compared, disclosing their merits and drawbacks.