Beam Loading Compensation of Traveling Wave Linac to a Multi-bunch Pulse with Gaps

In the electron driven ILC (International Linear Collider) positron source, the beam is generated and accelerated in a multi-bunch format with mini-trains. The macro-pulse contains 2 to 4 mini-trains with several gaps, because the pulse format is a copy of a part of the bunch storage pattern in DR (Damping Ring). This pulse format causes a variation of the accelerator field in the pulse due to the transient beam loading and an intensity fluctuation of captured positron. The beam loading is compensated by amplitude modulation on the input RF in the positron booster composed from L-band and S-band traveling wave RF cavity. In this article, we derive the exact solution for the compensation with the gaps. In addition, we evaluate the effect of the time constant (delay) of the input RF modulation due to klystron Q-value.


Introduction
ILC is an e+e-linear collider with center of mass energy 250 GeV -1000 GeV [1].It employs super-conducting accelerator as the main accelerator.The beam is accelerated in a macro pulse with 1300 bunches by 5 Hz repetition.The bunch charge is 3.2 nC resulting the average beam current 21 µA.This is a technical challenge, because the amount of positron per second is 40 times larger than that in SLC [2], which was the first linear collider.
The configuration of the positron source is schematically shown in Fig. 1.The positron generated by electron beam as the electro-magnetic shower is captured and boosted up to 5 GeV by two linacs.In the E-Driven ILC positron source, the drive beam energy is 3.0 GeV and the target is 16 mm thick W-Re alloy rotating with 5 m/s tangential speed.FC (Flux Concentrator) generates a strong magnetic field along the beam axis to compensate the transverse momentum.36 1.3 m L-band Standing Wave (SW) cavities with 0.5 Tesla solenoid field are placed for positron capture.This section is called as the capture linac.At the downstream, a chicane is placed to removes electrons.The positron booster is composed from 2.0 m L-and and 2.0 m S-band Traveling Wave (TW) cavities.ECS (Energy Compression Section) is composed from 3.0 m L-band TW cavities with chicane.
A first simulation was performed by T. Omori [3] only for the capture linac.A simulation with the tracking down to DR (Damping Ring) was made by Y. Seimiya [4], but no beam loading effect was accounted.A new simulation accounting the beam-loading effect was done by Kuriki and Nagoshi [5] [6].For those simulations, the peak energy deposition density on the target is kept less than 35 J/g [7], which is considered to be a practical limit of the safety operation.To obtain uniform intensity positrons over the pulse, the transient variation of the acceleration field by the beam loading has to be compensated so that positrons are accelerated uniformly.Compensation for the transient beam loading by Amplitude Modulation (AM) implemented by mixing of two inputs of klystron with Phase Modulation (PM) was proposed by Urakawa [8] [9].The detail study of the compensation is discussed in Ref. [10], [11].
Figure 2 shows the pulse structure of the positron generation which has a 80 ns gap in the middle.Although the compensation of the beam loading to obtain a uniform intensity bunch train has been studied, the treatment for the gap was studied only conceptually.In this article, we studied the compensation including the gap for the positron booster composed from TW cavities.

Perfect Beam Loading Compensation with trapezoidal pulse AM
The analytical solution of the beam loading compensation with the trapezoidal AM had been derived in Ref. [10], but only for a uniform pulse.In this section, we derive the analytical solution for the mini-train with gaps.
The accelerating voltage of TW accelerator at t is determined by integral over the preceding time window of t f which is the filling time defined as t f = L 0 1/v g dz, where L is accelerator length and v g is the group velocity.That causes the transient beam loading at the beam pulse head.It can be compensated by AM initially proposed by Satoh [12] with a matrix formalism.Here, we derive the analytic formula.The accelerating field by a TW accelerator in Laplace transformation where κ = ω/Q, τ is attenuation constant, s is variable of Laplace transformation, E(s) is input RF in field, I(s) is the beam current.To compensate the variation of E a (t) in the pulse, we consider AM on E(t) as where E 0 is the initial field, E 1 is DC component AM, and E 2 is a linear component AM, u(t) is Heaviside step function.In this case, RF input is started at t = 0, and the beam acceleration is started at t = t f .Please note that the dimension of E 2 is V/(m.sec).Acceleration field E a (t) can be obtained by the inverse Laplace transformation as [14] Here we set E 1 and E 2 as [14] resulting E a (t) to be a constant as If the mini-train length and the gap length is larger than t f , the solution with the gap is very simple.Since the cavity is in a steady state at the time the beam current changes, the solution is completely iterative.
If the mini-train length and the gap length are less than t f , the solution is a superposition of multiple Heaviside step functions that alternate their signs.This is the case of the ILC E-Driven positron source.Right after the end of the first mini-train, i.e. in the gap, the input RF for the compensation is where t BP is the duration of the mini-train.Another pulse corresponding to −I 0 beam current starting at t = t f + t BP is added.The corresponding terms will be appeared in E a (t) as When the second mini-train is started, we add another term corresponding I 0 beam current starting at t = t f +t BP +t G , where t G is the gap duration.In this way, we can derive the solution for the beam loading compensation with any number of mini-trains.

Beam Loading Compensation with square pulse AM
A perfect compensation is possible for the beam loading with the trapezoidal AM as derived [10], but the acceleration field is suppressed, because it is E a = E 0 = E 0 + E 1 − E 2 t f although the peak field E 0 + E 1 is limited by the input RF power.A way to recover E a giving up the perfect compensation is square pulse AM.In this case, a small field variation appears, but the average E a is recovered.
Here we consider the two-train case.The beam current is given as where t B i is time to start the i-th train.The field by the input RF E(t) is defined as where E i is the amplitude of the input RF in i-th region, t RF i is time changing the input RF amplitude.
We require E a (t B 0 ) = E a (t B 0 +t BP ), i.e.E a for the first and last bunches in the first mini-train has to be the same.We got two solutions depending on t B 0 .The first solution is for t B 0 = t RF 1 , i.e. the first AM is synchronized with the beam acceleration start.We call it solution 1.
The second solution is for t B 0 = t RF 1 +t f −2t BP −t G , the end of beam acceleration is synchronized with the point in time when the filling time has elapsed since the first AM modulation, t RF 1 + t f .We call it solution 2.
E 2 is the input RF field during the gap.That is determined when we require E a (t E 3 is the RF field in the second mini-train.The solution is expressed as The positron booster is composed from L-band [15] and S-band [16] Traveling Wave (TW) cavities.The parameters are summarized in Tab. 1. Cavity voltage evolution is calculated for S-band TW cavity.By assuming 4.8 nC bunch charge, the beam loading current is 0.78 A. Input RF power is assumed to be 36 MW. Figure 3 shows the cavity voltage evolution with the beam loading.This is solution 1.The solid line, dotted line, and dashed line show the cavity voltage, RF voltage, and the beam loading voltage, respectively.The orange color area corresponds to the beam pulse.The input RF amplitude is modulated according to Eq. (10,13,15).The average cavity voltage over the pulse is 26.0 MV and the energy spread is 0.34 MV (peak-to-peak).Figure 4 shows the cavity voltage evolution in solution 2 with the same manner as those in Fig. 3.The average cavity voltage over the pulse is 27.9 MV and the energy spread is 0.34 MV (peak-to-peak).According to these results, solution 2 gives a higher average voltage with an

Effect of klystron bandwidth
In the real operation, two klystron outputs are combined.AM is made by anti-sign phase modulations for each klystron input.The cavity for the intensity modulation in the klystron has a finite bandwidth determined by Q value of the cavity.It causes a time delay of AM on the input RF.Here we consider the effect of the klystron bandwidth.Instead of a square pulse, input RF profile is given as where τ is the time constant of klystron.The cavity field evolution can be obtained with Laplace transformation, but the formula is too complex to define the optimal state analytically.Instead of discussing with the analytical formula, we present a numerical example.In the calculation, we assume τ = 0.05µs.Figure 5 shows the cavity voltage evolution evaluated with the effect of klystron bandwidth.This is a numerical example based on solution 2 of the square pulse AM.The figure is plotted with the same manner as those in Fig. 3.The average cavity voltage over the pulse is 27.9 MV and the energy spread is 0.83 MV (peak-to-peak).The average cavity voltage is comparable to that with the square pulse AM case (Fig. 4), but the energy spread is increased as expected.Please note that this is a numerical example and it is not fully optimized.Comparing to those in solution 2 of the square pulse AM, t RF 1 is delayed.Based on the t RF 1 , t B 0 is 0.058 �s delayed, t RF 2 is 0.026 �s earlier, and t RF 3 is 0.026 �s earlier.Each RF amplitude is modified as ; E 0 is 1.045 times larger, E 2 is 0.58 times lower, E 3 is 1.094 times larger.

summary
The beam loading can be perfectly compensated with the trapezoidal pulse AM including gaps, but the average field is suppressed.With the square pulse AM, the compensation is not perfect, but the voltage is recovered, especially with solution 2. We derive the analytical formula of the cavity voltage including the effect of klystron bandwidth and demonstrated the compensation is possible with a larger energy spread.The solution is not fully optimized and this is a next issue.

Acknowledgment
This work is partly supported by Grant-in-Aid for Scientific Research (B) and US-Japan Science and Technology Cooperation Program in High Energy Physics.

Figure 1 .
Figure 1.Configuration of E-Driven ILC positron source is schematically shown.

Figure 3 .
Figure 3. Voltage evolution with the beam loading (solution 1).The solid line, dotted line, and dashed line show the cavity voltage, RF voltage, and beam loading voltage, respectively .The orange area is the beam part.

Figure 4 .
Figure 4. Voltage evolution with the beam loading (solution 2) in the same manner as Fig. 3.

Figure 5 .
Figure 5. Voltage evolution with the beam loading evaluated with the effect of klystron bandwidth.This is a numerical example based on the solution 2 with the square pulse AM.The figure is drawn in the same manner as Fig. 3.

Table 1 .
Parameters of L-band and S-band TW Cavities in the Booster.