An analytic study of TTF of standing wave RF gap based on Bessel-Fourier expansion

Transit time factor (TTF) is important in design and simulation of standing wave RF gaps. The TTF is usually constructed on the basis of a square wave model, and it is always expanded as a function of reduced velocity and structure factors. In order to express the particle׳s motion more authentically, the TTF is studied based on the Bessel-Fourier (B-F) expansion which is realized in BEAMPATH code. By expanding square wave electric field into harmonic electric fields, the voltage component and the TTF component of each order are obtained from corresponding harmonic electric field. The ratios of each order of voltage components and integral voltage form the weights working as coefficients of TTF components. Consequently, the effective resultant TTF depends on not only the particle׳s velocity and the structure factors, but also the RF phase the particle experiences. Simple expressions are derived after simplifying the complicated TTF equation in this paper.