A method to measure the frequencies of individual half cells in a dumbbell cavity

Dumbbell fabrication is a midprocess for manufacturing an elliptical superconducting rf cavity. In order to understand how a welding shrinkage affects a dumbbell's frequencies and length, we need to measure the exact frequencies of each individual half cell of a dumbbell. To improve such a calculation precision and to simplify the calculation formulae, based on a two-coupled oscillator model and a cavity perturbation theory, a new formula to calculate the individual half-cell frequencies of a dumbbell or the individual cavity frequencies of a two-cavity coupling system has been developed, and its performance has been confirmed by using a dumbbell simulation. This formula can be applied to any kind of rf cavities with electric, magnetic, or electromagnetic coupling, if a coupling hole between two coupling cavities is small compared to the wavelength. Compared to other calculation formulae, this formula simplifies the calculation process of the individual resonator frequencies of a coupling system considerably, and it can also improve the calculation precision than that of a normal calculation method. Another advantage of this new method is that we do not need to consider a coupling factor between two resonators during a testing for an individual resonator frequency of an oscillator. The developed formula has been successfully used to tune the PEFP dumbbells.