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.

Figure 1

Echo amplitude relative to the dipole kick as a function of quadrupole kick predicted by Eq. (2.36). The two plots are for different initial emittances. In each plot, the black curve shows the prediction without including the emittance growth from a dipole kick, the blue and red curves include emittance growth from dipole kicks of 1 mm and 3 mm, respectively.

Figure 2

Left: relative amplitude of the first echo vs quadrupole strength for two dipole kicks. Right: relative amplitude of the second echo for the same two dipole kicks. The initial emittance is the same in both plots.

Figure 3

Time evolution of the centroid after the dipole kick at turn 200, with different strength quadrupole kicks applied at turn 1600. The echo pulse is centered around turn 3000. Both the initial emittance (corresponding to ${\sigma }_0=1\mathrm{mm}$ at the BPM) and dipole $\mathrm{kick}=3\mathrm{mm}$ were kept constant. Quadrupole kicks increase from left to right in the three plots. The relative amplitude of the echo has a maximum at $q=0.08$ (center plot) at the chosen emittance.

Figure 4

Centroid evolution with different dipole kicks, constant emittance ($\sigma =0.5\mathrm{mm}$), and constant quadrupole kick $q=0.1$. The dipole kicks increase from left to right. The decoherence time decreases with increasing dipole kick.

Figure 5

The relative amplitude of the first echo as a function of the quadrupole strength parameter $q$. Simulations (red dots) are compared with QT, the nonlinear quadrupole theory (green curve) and DQT, the nonlinear dipole-quadrupole theory (black curve). The initial emittances increase from top to bottom, at each emittance the left plot corresponds to a dipole $\text{kick}=1\mathrm{mm}$, the right plot to a dipole $\text{kick}=3\mathrm{mm}$.

Figure 6

Relative echo amplitude vs the quadrupole strength from simulations for different dipole kicks. Left: initial beam size at the BPM $\sigma =0.5\mathrm{mm}$. Right: $\sigma =1\mathrm{mm}$. In both plots the maximum echo amplitude is not significantly affected by increasing the dipole kick, but the value of $q_{\mathrm{opt}}$ changes significantly.

Figure 7

Optimum quadrupole strength as a function of the initial emittance for two initial dipole kicks: simulations (red dots) compared with fits to the form in Eq. (4.1).

Figure 8

Maximum relative echo amplitude $A_{\max }$ as a function of the beam size (left plot) and the relative dipole kick (right plot).

Figure 9

Multiple echoes at constant emittance, quadrupole kick, and increasing values of the dipole kick strength, left to right. The second echo (centered at turn 5800) and third echo (centered at turn 8600) become visible at the larger dipole strengths.

Figure 10

Left: tune shifts (without echoes) vs the emittance. The emittance was changed by varying dipole kicks. Also shown is the straight line fit which yields the tune slope parameter ${\nu }^{\prime }$. Right: spectra without echo and with echoes. The spectrum without an echo was obtained with $q=0$ while the echo spectra were obtained with the same dipole kick (1 mm), the same value of $q=0.01$, and two initial emittances corresponding to ${\sigma }_0=1.5\mathrm{mm}$ and ${\sigma }_0=2\mathrm{mm}$. The vertical dashed line shows the bare lattice tune.