Controlling the transverse multipole components in rf cavity modes using the azimuthal modulation method

Recent work introduced a systematic method for designing so-called azimuthally modulated rf cavities that support transverse magnetic modes composed of user-desired multipoles, enabling precision control of the magnitude and orientation of multipolar components in rf cavity design. This paper extends this method to practical implementation by deriving the multipolar expansion of the longitudinal electric field in such rf cavities with beam pipes, as well as the momentum change of ultrarelativistic particles traversing these modes. The derived equations explicitly show the radial variation of the change in longitudinal and transverse momentum follows a polynomial rather than Bessel-function relationship. The expression for the longitudinal electric field is then compared to a field map obtained from the 3D electromagnetic simulation of an azimuthally modulated cavity designed to support a mode composed of monopole, dipole, and quadrupole components. Beam dynamics studies are presented to assess the derived expressions for the change in momentum, including the effects of relaxing the ultrarelativistic assumption. Finally, two example applications are presented: the first demonstrates the removal of unwanted transverse multipoles to create a multipole-free accelerating structure with a single-port coupler, whereas the second illustrates the synthesis of desired multipoles to create an rf cavity that transforms the transverse distribution of a beam from Gaussian to uniform.