Nonlinear theory of transverse beam echoes

Transverse beam echoes can be excited with a single dipole kick followed by a single quadrupole kick. They have been used to measure diffusion in hadron beams and have other diagnostic capabilities. Here we develop theories of the transverse echo nonlinear in both the dipole and quadrupole kick strengths. The theories predict the maximum echo amplitudes and the optimum strength parameters. We find that the echo amplitude increases with smaller beam emittance and the asymptotic echo amplitude can exceed half the initial dipole kick amplitude. We show that multiple echoes can be observed provided the dipole kick is large enough. The spectrum of the echo pulse can be used to determine the nonlinear detuning parameter with small amplitude dipole kicks. Simulations are performed to check the theoretical predictions. In the useful ranges of dipole and quadrupole strengths, they are shown to be in reasonable agreement.