Modal analysis of wake fields and its application to elliptical pill‐box cavity with finite aperture

The potential of the wake‐field produced by a bunch of relativistic charged particles passing through a pill‐box cavity is expressed by using Floquet’s theorem, and an obvious requirement that the energy gain over all acceleration cavity of many pill boxes must be proportional to the number of pill boxes, based on the previous modal approach (BWW theory). It is found that the wake‐field is consisted of two classes of modes: the longitudinal modes which are independent of the aperture and the pill‐box gap, the hybrid (pill‐box) modes which are dependent of the pill‐box gap. The wake field is predominated by the fundamental longitudinal mode whose wavelength is on the order of the effective diameter of the cavity, and its magnitude is inversely proportional to the cross sectional area of the cavity for practical cavities with small apertures. Both longitudinal and transverse wake fields due to the longitudinal modes in an elliptical pill box cavity are expressed analytically in a closed series form by solving exactly the longitudinal eigenmode equation in the elliptical cylindrical coordinates in terms of Mathieu functions. It is found that both longitudinal and transverse wake fields whose amplitudes per driving charge are greater than 100 MV/m/μC can be generated in an elliptical cavity.