Novel perturbation method for judging the stability of the equilibrium solution in the presence of passive harmonic cavities

When using the harmonic cavity to stretch the beam in electron storage rings, we expect the beam can be lengthened effectively and also stably. Semianalytical approaches can be used to obtain the equilibrium distribution but generally failed in the prediction of its stability. In this work, a novel perturbation method is proposed to check whether the calculated equilibrium solution can survive from the $l=1$ mode perturbation. For the case of uniform beam filling, we derive a simple formula that can determine the threshold of $l=1$ mode instability under specific parameters, which is well verified with tracking simulation for the Hefei Advanced Light Facility storage ring.

Figure 1

Schematic of steady-state harmonic voltage phasors contributed by all bunches at the reference phase for bunch 1.

Figure 2

The value of $f(\theta ,800)$ (top) and the relative deviations of $f(\theta ,8)$ and $f(\theta ,80)$ to $f(\theta ,800)$ (bottom) as functions of detuning.

Figure 3

Threshold detuning obtained by tracking simulation (in red) and using Eq. (22) (in blue).

Figure 4

Bunch density profiles for the case of $R/Q=41\mathrm{\Omega }$ and $Q=5\times {10}^5$.

Figure 5

Amplification factor as a function of detuning for the case of $R/Q=175\mathrm{\Omega }$.

Figure 6

Schematic diagram of simplifying Eq. (10) to Eq. (11).

Figure 7

Schematic of an asymmetric bunch distribution with nonzero bunch form factor phase. The resulting distribution is obtained with the HC parameters of $R/Q=90\mathrm{\Omega }$, $Q=5\times {10}^5$, and $\mathrm{\Delta }f=125\mathrm{kHz}$. The corresponding form factor phase is 0.08414 and the deviation phase of bunch centroid is ${\omega }_{\mathrm{rf}}⟨\tau ⟩=-0.02834$, which is negative since the centroid phase is ahead of the synchronous phase. The form factor phase for the reference of bunch centroid is ${\phi }_F+n{\phi }_{⟨\tau ⟩}=-8.8\times {10}^{-4}\approx 0$.

Figure 8

Schematic of the perturbation phase for the four bunches.

Figure 9

Schematic of the influence of the perturbation on the voltage phasors. (a) For the first bunch, (b) for the second bunch, (c) for the third bunch, and (d) for the fourth bunch. The red arrow shows the rotation direction of the voltage phasor caused by the perturbation.