Emittance growth of kicked and mismatched beams due to amplitude-dependent tune shift

We derive evolution equations for the first and second moments of an initially mismatched, coupled, and displaced arbitrary Gaussian phase-space distribution under the influence of decoherence due to amplitude-dependent tune shift. Moreover, we find expressions for the asymptotic values of the beam matrix and the emittance and use them to evaluate error tolerances for injection.

Figure 1

Amplitude of the beam centroid (in units of $\sqrt{{ϵ}_0}$) versus the turn number $n$, parametrized as $\theta =2\kappa {ϵ}_{0}n$ with parameters ${ϵ}_0=1$ and $\kappa =0.1$ and for two values $a_0=1$ and $a_0=2$ of the initial displacement.

Figure 2

The centroid $X_1$ (top), the beam matrix elements ${\sigma }_{11}$ and ${\sigma }_{12}$ (middle), and the emittance (bottom) as a function of the turn number $n$ for a matched beam that is injected with initial offset $X_1=2$. The parameters used are $\mu /2\pi =0.028$ and $\kappa {ϵ}_0=0.001$. The vertical axes are normalized to appropriate powers of ${ϵ}_0$.

Figure 3

The parameters ${\widehat{X}}_1$, ${\widehat{\sigma }}_{11}$, and ${\widehat{\sigma }}_{12}$ and emittance as a function of the turn number $n$ for a beam that is injected on axis but with a beta function ${\beta }_0$ that is twice the matched value $\beta$. All other parameters are equal to those used in Fig. 2. The vertical axes are normalized to appropriate powers of ${ϵ}_0$.

Figure 4

The same parameters that are shown in Fig. 3 but with an additional steering error at injection $X_2=1$. The slow decrease of ${\widehat{X}}_1$ motivates the extended range of turns.

Figure 5

The horizontal and vertical beam positions ${\widehat{X}}_1$ and ${\widehat{X}}_3$ (top), beam matrix elements (middle), and emittance (bottom) as a function of the number of turns for a beam with initial emittance ratio ${ϵ}_a/{ϵ}_b=10$, initial beta mismatch, and displacement. The beam is rotated by $\eta =30°$. The initial mismatch decoheres, and the emittance reaches its asymptotic value, given by Eq. (59).