Diagnostics with quadrupolar pick-ups

The spectrum of a quadrupolar pick-up gives access to coherent second-order modes of a beam in a non-invasive way. The signature of bunched beams not only features the first-order and second-order dipolar modes, but also most notably (1.) bands of even envelope modes, (2.) odd (skew) envelope modes and (3.) coherent dispersion modes. The odd (skew) envelope modes can provide information on linear coupling and mismatch thereof. The tunes of even envelope modes and the coherent dispersion modes shift significantly with space charge and can thus provide a measurement thereof. The paper presents measurements of the various modes at the CERN Proton Synchrotron, demonstrates how the spectral amplitudes depend on the mismatch conditions, and suggests a new, alternative way to quantify space charge at high brightness conditions via the coherent dispersion mode.