Differential equation model of the tune ripple effect on beam spill ripple in RFKO slow extraction

Beam uniformity is an important factor that must be considered in slow extraction optimization, and the tune ripple caused by the power supply ripple is an important factor that causes beam uniformity to deteriorate. In this study, based on the beam excitation concept for two regions (extraction and diffusion regions), a differential equation of beam spill under the influence of a mono frequency tune ripple was established. By solving and analysing the differential equation, several conclusions were obtained and verified by simulation.


Introduction
Third order resonance slow extraction [1] is widely used in hadron therapy and space radiation environment simulation devices based on a synchrotron.The extracted beam is referred to as the "spill".The spill uniformity is extremely important in the above applications.The third order resonance extraction is sensitive to the horizontal tune because it is set close to the third order resonance tune.The tune ripple therefore has a significant impact on the temporal structure of the spill.Tune ripple mainly originates from the power supply ripple, especially the main dipole and quadrupole power supply.
In radio frequency knockout (RFKO) slow extraction [2], the stable phase space region remains constant, and a transverse RF field is used to excite particle emittance growth causing it to leave the stable region.Thus, both the variation of the area of the stable phase space region and the variation of the emittance growth rate contribute to the spill ripple.
A model explaining how the tune ripple transfers to the beam spill ripple in RFKO slow extraction has been proposed by researchers at the Heavy Ion Medical Accelerator in Chiba (HIMAC) facility [3].Only the variation of the stable phase space region's area was considered and the influence of tune ripple on emittance growth was neglected.A novel beam spill ripple model was proposed by W. B. Ye [4], which included the effect of tune ripple on emittance growth.However, neither of these models can specify how to optimize the RF signal because the excitation effect of the RF signal on the beam is described as only one variable on average.The average excitation effect has already been determined by the extracted beam intensity that is demanded.
Experiments have been conducted to optimize RFKO signals and reduce beam spill ripple.In 2002, K Noda reported that there are two tune regions in the RFKO slow extraction process: the extraction and diffusion regions [5].By applying a transverse RF field with a mono frequency that matches the tune in the extraction region, the spill ripple is reduced.The reduction is due to the decrease in beamless time.This method is called the separate function method.In 2009, K Mizushima reported that the separate function method can reduce the uncontrollable extracted beam intensity after turning off the RFKO [6].This was explained by the particle density decreasing with a mono frequency signal in the extraction region.E C Cortés García [7], W B Ye [8] and other researchers successfully reduced beam spill ripple by experimentally optimizing the RF signal with qualitative explanations.
We therefore attempted to establish a model in which the excitation effect of the RF signal on the beam is not described as only one variable on average, but as two variables impacting on the extraction and diffusion regions.The excitation effect on the diffusion region was determined by the extracted beam intensity, while the effect on the extraction region could be optimized to reduce beam spill ripple.Set the number of particles in A2 as N2, and an assumption is made as follows: Assumption A2 is a small region in which particles are uniformly distributed.With the assumption above, the number of particles extracted can be considered to be proportional to the number of particles in A2 and the extraction time.If a short period of time is ∆, an equation can be obtained as follows:

Differential equation Model
In equation ( 1), k is a coefficient and represents the ratio of the number of extracted particles per unit time to the initial number.k reflects the excitation effect of the RF signal on the extraction region.
Due to the beam supply of A1, N2 varies as follows: Based on equation ( 1) and ( 2), a differential equation for N2 can be obtained: After solving N2, I2 can be naturally obtained:

Differential equation with tune ripple influences
For the RFKO slow extraction, the tune ripple is transferred to the beam spill ripple by the variation of the area of the stable phase space region and the variation of the beam emittance growth rate [8].
For the area variation, equation ( 2) is adapted to equation ( 5): Where ρ(t) represents the ratio of the particles extracted by the area variation per unit time to the initial number.
Define the total area of the stable region as A0 and the ratio of A2 to A0 as .A0 can be calculated from equation ( 6): where q=Qx-Qres, Qx is the horizontal tune, and Qres is the resonance tune.S is the virtual normalised sextupole strength.Assuming that the tune ripple is a sine wave, the amplitude is a and the frequency is f.Therefore, q can be written in the following form: 0 ( ) sin(2 ) q t q a ft    where a is far less than q0.Therefore, ρ(t) can be derived as follows, considering a first-order approximation: cos( 2) Where p=4πfa/q0γ.In equation (8), it is assumed that dS/S is far less than dq/q because the horizontal tune is close to the resonance tune and q is very small.
For the beam emittance growth rate variation, k and I1(t) are written as follows: In equation ( 9) and equation (10), it is temporarily considered that the amplitude feedback [9] or feedforward [10] is used to control I1(t) to approximately be a constant I1 for the convenience of analysis.Δk and Ir represent the amplitudes of the spill ripples of k and I1, which are caused by tune ripple.

Differential equation Solution
Equation ( 11) is a first order nonhomogeneous nonlinear differential equation.Although it can be solved, its solution is difficult to understand and then apply for physical analysis.A solution convenient for a physical analysis was obtained in a relatively simple way by sacrificing precision.Assume that the expression of N2 is as follows: where ω=2πft.Substitute equation ( 12) into equation (11) and separate the equations for different frequencies.Considering only the ω and 2ω terms, and the constant term, three equations were obtained: ( sin( 2) cos( 2)) sin( ) From equation ( 13), we can get N20: A and ϕ1 were obtained from equation ( 14) as equation ( 17) and (18).From Eq. (15), B is not constant and is quite small than A: From equation ( 13), ( 16), (17), and (18), I2 can be obtained as equation ( 19), ( 20), (21), and (22): .... IOP Publishing doi:10.1088/1742-6596/2687/5/052035 Because the parameters are difficult to calculate directly from the definitions, only qualitative conclusions can be obtained at present.
From equation ( 19), a mono frequency tune ripple will modulate the multiple frequency beam spill ripple.
From equation ( 21) and ( 22), the relative spill ripple amplitude does not vary monotonically with k.It is usually agreed that the larger the value of k the better because the smaller the N20 is the smaller the relative spill ripple is [6].

Simulation verification
The two conclusions presented above were verified by a special case with a simulation by Syntrack [11,12].The lattice used in this simulation was obtained from the Xi'an Proton Application Facility (XiPAF) [13] synchrotron.
Tune ripple was generated by the focus quadrupole strength ripple in the simulation.The amplitude and frequency of the focus quadrupole strength ripple were 200 ppm and 100 Hz, respectively.
Define the revolution frequency as frev.The RF signal was composed of a mono frequency signal (0.674 frev) corresponding to the extraction region and a dual frequency modulation (dual FM) signal corresponding to the diffusion region.The dual FM signal's centre frequency and the bandwidth were 0.6789 frev and 0.001 frev, respectively.
k represents the ratio of the number of extracted particles per unit time to the initial number in the extraction region.Thus, the larger the ratio of the mono frequency amplitude to the dual FM amplitude is, the larger the k is with the dual FM signal's amplitude fixed.
The horizontal machine tune was set to 1.6784.A 60 MeV proton beam was extracted within 3,000,000 turns under different ratios of the mono frequency signal's amplitude to the dual FM signal's amplitude.
The extraction spill's time structures and spectrum diagrams under the influence of the 100 Hz tune ripple are shown in figure 2. The square of the relative beam spill ripple amplitudes before 600 ms are shown in figure 3.
From figure 2, it was confirmed that a mono frequency tune ripple will modulate a multiple frequency beam spill ripple.Figure 3 shows the existence of a situation where k is not the larger the better and an optimal k exists.Therefore, future studies are needed to make this model's parameters computable so that the optimal k can be theoretically determined.

Conclusion
The excitation effect of the RF signal on a beam could be described as two variables corresponding to the extraction and diffusion regions, respectively.A new differential equation model of the tune ripple effect on beam spill ripple was established and solved.Two preliminary conclusions were obtained and examined by a simulation, which validated the reliability of the model.This model has the potential to theoretically provide the optimal RF signal.

2. 1 .
Differential equation without tune ripple influences Referring to the concept of diffusion and extraction regions, the extraction beam intensity of the two regions is calculated separately.A schematic diagram of the two regions is shown in figure 1.

Figure 1 .
Figure 1.Schematic diagram of the diffusion and extractions regions.A1 refers to the diffusion region and A2 refers to the extraction region.Set the beam intensity of A1 entering A2 as I1, and the beam intensity exiting A2 as I2.I1 reflects the excitation effect of the RF signal on the diffusion region.Set the number of particles in A2 as N2, and an assumption is made as follows: Assumption A2 is a small region in which particles are uniformly distributed.With the assumption above, the number of particles extracted can be considered to be proportional to the number of particles in A2 and the extraction time.If a short period of time is ∆, an equation can be obtained as follows:

Figure 3 .
Figure 3. Square of the relative beam spill ripple amplitudes under different ratios of the mono frequency amplitude to the dual FM amplitude.