High Precision Sc Cavity Alignment Measurements with Higher Order Modes

Experiments at the FLASH linac at DESY have demonstrated that the higher order modes (HOMs) induced in superconducting cavities can be used to provide a variety of beam and cavity diagnostics. The centres of the cavities can be determined from the beam orbit which produces minimum power in the dipole HOM modes. The phase and amplitude of the dipole modes can be used as a high resolution beam position monitor. For most superconducting accelerators, the existing HOM couplers provide the necessary signals, and the downmix and digitising electronics are straightforward, similar to those for a conventional BPM. HIGHER ORDER MODES IN SC ACCELERATOR CAVITIES In addition to the fundamental accelerating mode, superconducting (SC) cavities support a spectrum of higher order modes (HOMs)[1]. While HOMs can be a source of a variety of accelerator problems (e.g. beam breakup, heating, etc.), they can also be used as beam and cavity diagnostics. Mode Coupling to the Beam HOMs can be characterized by their azimuthal dependence as monopole, dipole, or higher multipole modes. Here we consider the response of these modes to a single electron bunch propagating near the axis of the cavity, and we assume bunches with lengths short compared with the wavelength of the HOM modes. The interaction of an electron beam with a particular mode is characterised by the loss factor, k, or by R/Q, whose values may be calculated from numerically calculated fields[2]. The integer, m, indicates the order of the mode, with m=0 indicating a monopole mode, m=1 a dipole mode, etc. The loss factor and R/Q are defined as follows, k r ≡ ∣V L ∣ 4U  (1) R  Q ≡ 1 r 2m ⋅2k  r   (2) where VL is the longitudinal potential of the mode (of order m) at a radius, r, U is the stored energy of the mode, and ω is its central frequency. The longitudinal potential induced by the beam for each mode will depend on the magnitude of the longitudinal component of the electric field, Ez. Since the longitudinal electric field due to the monopole mode has no first order variation with r, equation 1 implies that the coupling to the beam does not depend on its offset from the mode axis. The amplitude and phase of monopole modes will, therefore, be governed only by the charge of the bunch, and its time of arrival in the cavity. The amplitude of the longitudinal electric field of a dipole mode varies in proportion to r, so equation 1 implies that the energy coupled into a mode is proportional to r. The amplitude of the potential of a dipole mode will, therefore, be directly proportional to r. This argument may be easily extended to determine the r dependence of quadrupole and higher modes. Since strongly coupled dipole modes are excited by a beam traversing the cavity with an offset, they may also be excited by a beam entering the cavity with a positive offset, but leaving with a negative offset (i.e. a tilted trajectory that brings it through the centre of the cavity). A tilted bunch will also cause the same effect, and will result in a signal that is 90° out of phase with the offset signal. A derivation of this may be found in [3]. Dipole modes are therefore excited with an amplitude proportional to bunch charge, and to: • Transverse position relative to the cavity axis • Transverse angle relative to the cavity axis. This signal is excited at 90° phase relative to the position signal. • Bunch tilt, with amplitude proportional to the bunch length, and 90° phase relative to the position signal, however, for the short bunch length in FLASH this signal is not significant. Dipole modes occur in doublets with orthogonal polarisations corresponding to two orthogonal transverse degrees of freedom. The frequencies of these polarisations may be degenerate, or may be split due to asymmetries caused be the cavity couplers and fabrication imperfections. [email protected] SLAC-PUB-12349