2.1 Longitudinal injection

The HALS is a vacuum ultraviolet (VUV) and soft-X ray light source with an energy of 2.4 GeV and a circumference of 672 m. In the latest version of the HALS lattice design, longitudinal gradient bends and antibends are employed, and a lower natural emittance of 23.2 pm·rad is achieved. Meanwhile, the nonlinear dynamics for this version of the lattice is improved. The main parameters of the latest design of the HALS are listed in Table 1.

To keep the circulating bunches undisturbed, the pulse of the kicker field has to be shorter than the bunch spacing. The main RF frequency $f_{\text{rf}}$ was chosen as 100 MHz to reduce the technical difficulties of the kickers. The synchronous phase ${\phi }_0$ and momentum acceptance ${\delta }_{\text{acc}}$ are given by:

where $U_0$ is the energy loss per turn caused by synchrotron radiation, $V_{\text{rf}}$ is the RF voltage, h is the harmonic number, and $\alpha$ is the momentum compaction factor. The momentum compaction factor of the HALS is 3.5 × 10^-5. The bucket height is 5.15% without considering the synchrotron radiation effect, when the RF voltage is set as 350 kV. Fig. 1 shows the longitudinal phase space of the HALS. The red lines indicate the border of the stable bucket, the gray lines indicate trajectories outside the acceptance, and the cyan lines show the particle trajectories that have been tracked for 65000 turns in the longitudinal phase space with synchrotron radiation loss.

Fig. 1.Full-screen View

Longitudinal phase space of the HALS

The bunch spacing is 10 ns when a 100-MHz fundamental RF cavity is adopted. If a bunch is injected with a time offset of -5 ns with respect to the circulating bunch, its upper limit and lower limit of momentum offset are 7.94% and 7.03%, respectively. Particles with higher or lower momentum offset would be lost. Particles in the stable bucket would merge to the circulating bunches because of radiation damping in the longitudinal phase space. This process takes four to five times of the longitudinal damping time; the higher energy the particle has, the shorter the time it requires.

2.2 Double-frequency RF system

In a storage ring, an HHC is necessary to reduce the Touschek scattering and, thus, improve the beam lifetime. In addition, the longitudinal acceptance can be enlarged with an HHC. Both a second harmonic cavity (2HC) and a third harmonic cavity (3HC) are taken into consideration. For a passive HHC, the induced voltage is ^[11]

where the longitudinal coordinate $\tau$ is the time distance from the synchronous particle, A is a factor, ${\phi }_{\text{h}}$ is the tuning angle of the HHC, n is the harmonic cavity number, and ${\omega }_{\text{rf}}=2\pi f_{\text{rf}}$ is the resonant frequency of the main RF. The total voltage is then ^[12]

When an HHC is included, the synchronous phase of the main RF changes, and it is expressed by ${\phi }_{\text{s}}$. The optimal bunch lengthening condition of the passive HHC is given by ^[13-15]

The potential of a passive HHC is

Two types of double-frequency RF system were investigated: a main RF with a 2HC and main RF with a 3HC. Detailed parameters of these two RF systems are listed in Table 2. The harmonic number is 224; then, the bunch spacing of two stored bunches is 10 ns, or 3 m. Fig. 2 shows waveforms of cavity voltages and potential function with and without an HHC when n is 2 or 3. The green solid ellipse in the figure indicates the stored bunch. As can be seen in the lower picture, the potential function is obviously flattened when an HHC is attached.

Fig. 2.Full-screen View

a Waveforms of RF voltage. b Waveforms of potential function

As for the parameters of Table 1, Elegant ^[16] is used for particle tracking. The longitudinal acceptances of the HALS with and without an HHC are depicted in Fig. 3. When radiation damping in longitudinal phase space is taken into account, the bucket height of an RF system with a 3HC is approximately 5.8%, the same as that of an RF system without an HHC. The bucket height of an RF system with a 2HC reduces to 4%. The energy offset is approximately 8% when the bunch is injected at -5 ns. Although the MA at a long straight section may achieve 8%, one must consider the off-momentum DA for a better injection efficiency, because the injection bunches have an energy offset in a longitudinal injection scheme. Fig. 4 shows that the off-momentum DA is small when a bunch is injected with a momentum offset of 8%, and the DA is larger when the momentum offset is less than 6%. In addition, the smaller the time offset is, the larger the energy and phase acceptance are. The longitudinal acceptance of a 3HC system is larger, but it does not perform better in injection simulations than the 2HC system. Although the 3HC system has a higher bucket height, its off-momentum DA is smaller than that of the 2HC system. After repeated simulations, (dt, δ) = (3.6 ns, 3.2%) was chosen as the injection point for the 2HC case. The corresponding energy and phase acceptance are approximately 0.85% and 0.14 rad (190 ps), respectively — sufficient to tolerate jitters from the injection beam.

Fig. 3.Full-screen View

Longitudinal acceptance of the HALS

Fig. 4.Full-screen View

Off-momentum DA of the HALS

3.1 Error analysis

In a realistic machine, various errors may occur. This study classifies errors from the entire injection process into three types: beam mismatch error, beam trajectory error, and misalignment error ^[17,18]. The beam mismatch error comes from errors of the injector and transport line, leading to mismatched injected bunches with unexpected beam energy, emittance, energy spread, and bunch length. The injector of the HALS is a full energy linac. The design requirements for injected bunches are a normalized emittance of 2 mm·mrad, a bunch length of approximately 10 ps (FWHM), and an energy spread less than 0.1%. To simulate beam mismatch error, the injected bunch is assumed to have an emittance of 2.0 nm·rad, an energy spread of 0.1%, and 20 ps rms bunch length.

Beam trajectory error resulted from errors of the injection system, including trajectory error at the injection point, pulse-to-pulse jitter, and bending angle errors of the septum and strip-line kickers. The beam trajectory error was equivalent to deviations of (x, x’, y, y’) of the injected bunches, a four-dimensional error. First, for the convenience of simulation, the impact of errors in each single dimension on the capture efficiency of injected bunches was analyzed. The variation trends of capture efficiency under respective deviation of x, x’, y, and y’ are shown in Fig. 5. Fig. 5 shows that the HALS has less tolerance to the deviations of y and x’. Table 3 summarizes the ranges of respective deviations when the capture efficiency is more than 80%. These four types of deviation have no correlations with each other. Their correlations are shown in the next section.

Fig. 5.Full-screen View

a Capture efficiency under deviations of x and y. b Capture efficiency under deviations of x’ and y’

3.2 Error analysis and simulation study

To investigate how the longitudinal injection scheme performs on the HALS, both beam mismatch error and beam trajectory error were then introduced to simulate a realistic machine. The parameters of beam mismatch error were presented in Section 3.1. The beam trajectory error was equivalent to a four-dimensional variable $(\Delta x,\Delta y,\Delta x\text{'},\Delta y\text{'})$. The injected beam was set as $\Delta x$ = -0.2 mm, $\Delta y$ = -0.1 mm, $\Delta x\text{'}$ = -0.06 mrad, and $\Delta y\text{'}$ = -0.06 mrad. A bunch of 1000 particles was tracked for 60,000 turns. The simulation results are shown in Fig. 6. Figure 6a is the distribution of particles in the longitudinal plane, and Figs. 6b and 6c show the particles in transverse phase space. After 60,000 turns, the injected bunch was successfully damped, and no particle was lost in the simulation. This means longitudinal injection performs well on the HALS when beam mismatch error and beam trajectory error are introduced.

Fig. 6.Full-screen View

Particle distributions in the longitudinal and transverse planes

To get closer to the realistic machine condition, errors of misalignments were then introduced. One hundred random errors were assigned to the quadrupole and sextupole magnets: quadrupole and sextupole roll errors of 0.2 mrad, quadrupole and sextupole fractional magnet strength of $5\times {10}^{-4}$, and transverse sextupole misalignments of 20 μm. In this simulation, the beam mismatch error was the same as Section 3.1, and the beam trajectory error was at a lower level: $\Delta x$ = -0.05 mm, $\Delta y$ = -0.05 mm, $\Delta x^{\text{'}}$ = -0.05 mrad, and $\Delta y^{\text{'}}$ = -0.05 mrad. One thousand particles with a Gaussian distribution were tracked for 3600 turns for 100 random cases. As a result of the simulation, 100 groups of data were generated. Fig. 7 shows a bar chart of the capture efficiency distribution over 100 random errors. The abscissa indicates ranges of capture efficiency, and the ordinate indicates the number of occurrences corresponding to capture efficiency. The performance was poor with misalignments; only 27 of 100 random seeds gave an efficiency of more than 80%. The small DA of the HALS was thought to be the problem, so the DA was tracked for another 100 random cases with misalignments, and 100 groups of DA data were generated. The blue line in Fig. 8 is the DA without misalignments, and the red point is the DA of 100 seeds with misalignments. The minimum DA with misalignments was almost 0. The lattice needs to be further optimized for a larger DA to adopt longitudinal injection scheme for the HALS.

Fig. 7.Full-screen View

Distribution of capture efficiency over misalignments

Fig. 8.Full-screen View

DA of the HALS with and without misalignments