Knock Out Slow Extraction Using Betatron Sidebands at High Harmonics

Knock Out Slow Extraction UsingBetatron Sidebands at High HarmonicsPhilipp Niedermayer* and Rahul SinghGSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, GermanyEike Feldmeier, Christian Schömers, and Marcel HunHeidelberg Ion-Beam Therapy Center (HIT), Heidelberg, Germany(Drafted 10 November 2025)Radio frequency knock out resonant slow extraction is a standard method for extracting stored particle beams from synchrotrons by transverse excitation. Excitation signals comprising many betatron sidebands have shown to reduce intensity fluctuations of the extracted beam spill and are used at several facilities. In this contribution, the effect of individual sidebands at different harmonics on the spill quality is systematically studied using efficient noise excitation signals fully covering the respective sidebands. Particle tracking simulations show a clear correlation between the chosen excitation frequency and the spill quality. This relation is attributed to the spectrum of particle motion shortly before their extraction, and an empiric relation between the spectrum, the excitation frequency, and the resulting spill quality is given. Experiments show an additional reduction of pile-up when using higher excitation frequencies. These insights have implications for an optimal design of excitation waveforms and hardware for Knock Out extraction systems.DOI: 10.48550/arXiv.2503.11285I.INTRODUCTIONRadio frequency Knock Out (rf-KO) resonant slow extraction is used to extract stored particle beams from synchrotron rings [1,2]. It utilizes a working point near a sextupole-driven third-integer resonance to create a transverse, betatron amplitude dependent instability [3]. The beam is driven into the instability in a controlled manner by increasing the particle amplitude through transverse radio frequency (rf) excitation. The excitation system consists of a signal generator, amplifiers and a stripline kicker [4], inside which the electromagnetic rf field deflects traversing particles on each turn. Over several hundred turns, these transverse kicks lead to a net change of the particle amplitudes, bringing them closer to the separatrix. On crossing the separatrix, the motion of particles becomes unbound and septa are used to deflect these particles into an extraction beam line. A spill of extracted particles is then delivered to experiments or used for medical therapy.A particular challenge in resonant slow extraction is posed by fluctuations of the spill intensity on various timescales. As large fluctuations reduce the efficiency of beam usage, this is * Contact author: p.niedermay[email protected]Licensed under the terms of the Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution to the authors and document title.referred to as a low spill quality [5,6]. Numerous efforts have been made in the past seven decades to address this particular challenge[7–16]. More recently, the simultaneous excitation of multiple betatron sidebands at different revolution harmonics has been investigated. A study at the Wakasa Wan Energy Research Center (WERC) used simulations and experiments to show that excitation signals covering the first 𝑛 sidebands progressively improve the spill quality as more sidebands are added [17]. These excitation signals consisted of a series of noise bands around each betatron sideband and are referred to as many-band signals in the following. In a study at the Heidelberg Ion Beam Therapy Center (HIT), excitation signals comprising three frequencies at two different sidebands were found to significantly improve the spill quality compared to the previously used single-band signal covering the 1st betatron sideband only [18]. This excitation scheme used narrowband random binary phase-shift keying (RBPSK) signals with frequencies near 1/3 and 4/3 of the revolution frequency.In this contribution, the excitation of individual sidebands is systematically investigated using simulations and experiments. The effect of each sideband on the spill quality is characterized and compared to the combined excitation of two or more sidebands at low and high harmonics. Finally, the implications of these findings for the design of rf-KO excitation systems are discussed.page 1 / 12 A.Betatron sidebandsThe betatron tune 𝑄x=𝑄x,ref+𝜉x𝛿 describes the number of betatron oscillations a particle performs each turn. It is given by the working point 𝑄x,ref and the individual particle’s momentum deviation 𝛿=Δ𝑝/𝑝ref via the chromaticity 𝜉x=d𝑄x/d𝛿. In the following, the lowercase 𝑞x=𝑄x−⌊𝑄x⌋<1 is used to denote the fractional part of the tune. At a fixed location in the synchrotron, the betatron motion modulates the transverse particle position𝑥(𝑠,𝑛)=√𝜖x𝛽x(𝑠)cos(2𝜋𝑄x𝑛+𝜇x(𝑠)+Φx)where 𝑛∈ℕ0 is the integer turn number and 𝑠<𝐶 the location along the orbit with circumference 𝐶. The modulation amplitude is given by 𝜖x and the beta-function 𝛽x(𝑠). The phase is given by Φx, 𝑄x and the phase advance function 𝜇x(𝑠) with 𝜇x(𝐶)=2𝜋𝑄x. Due to this turn-by-turn modulation, the spectrum of the transverse beam motion is dominated by the well known betatron sidebands at𝑓±=(ℎ±𝑘𝑞x)𝑓rev(1)where ℎ∈ℕ0 is the nearby harmonic of the revolution frequency 𝑓rev and 𝑘∈ℕ accounts for the appearance of second and higher order betatron sidebands. These higher order sidebands are the consequence of an additional frequency modulation caused by the nonlinear motion in the vicinity of the sextupole-driven third-integer resonance [19]. In the following, the discussion is limited to the dominant first-order sidebands for which 𝑘=1.B.Spill qualityTo assess the quality of a spill, the intensity of the extracted beam is recorded as the particle count 𝑁 per time interval Δ𝑡count. Fluctuations of the intensity are then evaluated over a time period Δ𝑡eval using the standard deviation 𝜎N=√⟨(𝑁−⟨𝑁⟩)2⟩ and mean 𝜇N=⟨𝑁⟩. From these, the coeffi-cient of variation𝑐v and the equivalent spill duty factor𝐹 are calculated as [6]𝑐v=𝜎N𝜇N=√⟨𝑁2⟩⟨𝑁⟩2−1𝐹=11+𝑐2v=⟨𝑁⟩2⟨𝑁2⟩where a smaller 𝑐v and larger 𝐹 correspond to a better spill quality. For the ideal case of a Poisson process with 𝜎N=√𝜇N, the coefficient of variation is 𝑐v=⟨𝑁⟩−1/2. In the following, Δ𝑡eval=1000Δ𝑡count=100 ms is used.In addition, the delay between the extraction of consecutive particles is of interest to quantify the amount of pile-up. Considering a detector system which requires a separation time of at least Δ𝑡pile-up to distinguish and resolve single events, pile-up is defined by the fraction of inter-particle-delays smaller than Δ𝑡pile-up. In the following, Δ𝑡pile-up=20 ns is used, motivated by the pulse width observed in the experimental data.C.Simulation frameworkParticle tracking simulations of the rf-KO extraction process are performed with Xsuite [20], using the lattice of the HIT synchrotron [21] with the parameters listed in tableI. Thereby, an unbunched (coasting) beam comprising 106 particles evenly distributed along the circumference of the synchrotron is tracked.The individual tune of particles deviates from the working point 𝑄x,ref due to chromatic and amplitude detuning [22], and is limited by the nearby third-integer resonance 𝑄res=5/3 where particles are extracted. To cover this range, an excitation signal with a rectangular spectral density with full bandwidth Δ𝑄ex=0.02 is used. A band-filtered noise signal is chosen because of its homogeneous spectral density. This avoids the large influence of the central frequency on the spill quality as observed for narrowband excitation signals [23], which would dominate and hinder the investigation of the behaviour at different sidebands. The amplitude of the excitation signal is controlled with a feedback system to achieve a constant spill rate of 𝑁=5⋅105particles/s as given by the number of ions and the extraction time of 2 s used in the simulations. To model disturbances from the electrical grid, ripples on the quadrupole magnets comprising the dominant harmonics of the mains frequency at 100 Hz, 250 Hz and 300 Hz as well as noise up to 10 kHz with a relative magnitude of 10−5 are added [24].TABLE I. Beam, optics and excitation parametersIonCarbon C126+Energy𝐸kin/𝑚=251 MeV/nucleonMomentum𝑝ref=8.74 GeV/cRigidity𝐵𝜌=4.86 T m𝐸𝜌=897 MVLorentz factor𝛾=1.27𝛽=0.616Rev. frequency𝑓rev=2.842 719 MHzNorm. emittance𝜀xn=1.5 mm mradMomentum spread𝛿1𝜎=1⋅10−3Betatron tune𝑄x,ref=1.6790𝑄y,ref=1.7550Chromaticity𝜉x=−1.59𝜉y=−1.99Norm. sextupole𝑆=27.9m−1/2𝜇S=−2.9°Transition𝛾tr=1.82𝛼𝑝=0.30Slip factor𝜂=𝛾−2−𝛾−2tr=0.32Excitation signalBand-filtered noiseCentral frequency𝑄ex=𝑓ex/𝑓rev=ℎ±0.327BandwidthΔ𝑄ex=Δ𝑓ex/𝑓rev=0.02page 2 / 12 II.SPILL QUALITY FOR SINGLE-BAND EXCITATIONFigure1 shows the spill quality as a function of the excited sideband obtained from the simulation of the rf-KO extraction process. A significant improvement is observed when exciting sidebands of higher revolution harmonics. This trend is in agreement with the findings obtained for many-band excitation at WERC [17], however, the improvement is already observed for single-band excitation at the higher frequencies. The excitation power required at these frequencies is only marginally larger compared to lower frequencies (about 3 dB, see Fig.1, bottom). At the commonly used lower excitation frequencies near the first few harmonics, a systematic difference between the excitation of upper and lower sidebands is observed. In the case presented here, the spill quality is improved by exciting a sideband at ℎ−𝑞x instead of a nearby sideband at ℎ+𝑞x. With about a factor two, this effect is largest for the two sidebands around ℎ=2. However, as shown in section II B, the role of the sidebands, the magnitude and location of the maximum value of 𝑐v and the limit of 𝑐v towards high excitation frequencies are specific to the properties of the beam and lattice.FIG. 1: Spill quality (top) and excitation power (bottom) as function of excitation frequency. Simulation of single-band signals.A.Imprint of excitation frequency onto spillIn resonant slow extraction, the spill quality is dominated by intensity fluctuations of typically up to a few 10 kHz. In addition, for rf-KO, the excitation frequency and its harmonics are imprinted onto the spill to some extent. This is an intrinsic property of the rf-KO method and the consequence of coherent beam oscillations at these frequencies induced by the excitation. The resulting spill structures of typically a few MHz are not relevant for facilities like HIT dedicated to hadron cancer therapy. However, for high intensity particle physics experiments such as those at GSI and the Facility for Antiproton and Ion Research (FAIR), the imprint of these frequencies is undesired as it can facilitate the extraction of particles in short succession, leading to pile-up and inefficient beam usage. This is especially problematic when the dominant low frequency fluctuations cause a momentary increase of the extraction rate, temporarily increasing the probability for pile-up.The simulations show that the imprint of the excitation frequency can be mitigated by exciting an appropriate sideband at a higher revolution harmonic. Figure2 shows how both the fluctuations at low frequencies and the imprint at the excitation frequency are suppressed in the same manner as the spill quality improves. For high intensity particle physics experiments, this means that the probability for pile-up is reduced in two ways: First, the suppression of low frequency fluctuations avoids the temporary increase of the extraction rate that facilitates pile-up statistically. Second, the mitigation of the high frequency imprint additionally reduces the probability for particles being extracted in short succession. As a result, pile-up is reduced as a function of the excitation frequency in analogy to the spill quality, as can be seen by comparing Figs.1 and 3. Note that even when the high frequency imprint of the excitation onto the spill is not mitigated, a reduction of pile-up as function of the absolute excitation frequency is expected when the excitation frequency exceeds the average extraction rate. This is discussed in section V B, but it can not explain the differences in pile-up observed for excitation of upper and lower sidebands at low frequencies as observed in Fig.3. This is an effect of the reduced frequency imprint.FIG. 2: Spill spectra at the baseband (left) and at the respective excitation band (right). The color correlates with the coefficient of variation according to Fig.1, i.e. lighter shades correspond to higher excitation frequencies. Vertical lines mark the tune and resonance in the respective band; the gray shading marks the extent of the excitation band. Simulation result.page 3 / 12 FIG. 3: Amount of pile-up as function of excitation frequency for the case of exciting a single betatron sideband. Pile-up is expressed in terms of the fraction of particles which are separated by less than Δ𝑡pile-up=20 ns. Simulation result.B.Influence of beam and optics parametersTo further study the role of upper and lower sidebands, it is useful to generalize the description of the excitation signal to negative frequencies. For a real-valued excitation signal, frequencies below zero are the mirror image of the positive spectrum and result in the same physical signal being applied to the beam. By changing the sign of the excitation frequency for the sidebands at ℎ−𝑞x, the treatment of upper and lower sidebands can be unified. With the excitation frequency defined as 𝑄ex≈ℎ0+𝑞x where ℎ0∈ℤ can be negative, the improvement of the spill quality becomes symmetric with respect to the maximum , saturating towards large |ℎ0| and following a double exponential curve. However, this functional dependency is subject to a number of beam and machine parameters, which is discussed in the following.The relative momentum spread 𝛿1𝜎 of the beam determines how rapidly the spill quality improves as sidebands further away from the maximum are excited Fig.4, top). In the limit of a vanishing momentum spread, no difference between the excitation of different sidebands is observed. A larger momentum spread is beneficial and reduces spill fluctuations not only at lower excitation frequencies but especially when exciting sidebands at higher frequencies. Also, the difference in spill quality between upper and lower sidebands is magnified as a function of the momentum spread. The required excitation power is independent of the momentum spread, as long as the excitation covers the momentum dependent tune spread of the beam (here 𝛿1𝜎≲2⋅10−3).The chromaticity 𝜉x of the lattice influences the location of the maximum value of 𝑐v (Fig.4, middle). An inversion of the sign of the chromaticity causes the roles of upper and lower sidebands to interchange, while for zero chromaticity they behave identically and the spill quality improves purely as a function of excitation frequency or harmonic. In analogy to the momentum spread, a large chromatic tune spread exceeding the excitation band leads to an increased power requirement for|𝜉|≳2.The slip factor 𝜂=𝛾−2−𝛾−2tr depends both on the beam energy 𝐸=𝛾𝐸0 and the lattice’s gamma transition 𝛾tr. It influences the sharpness of the spill quality improvement by acting as a scaling factor on 𝑄ex (Fig.4, bottom). For a larger absolute value of the slip factor (smaller beam energy) the spill quality improvement has a stronger correlation with increasing excitation frequency. For small excitation frequencies at the first sideband, however, the spill quality is largely independent of the slip factor. While the excitation power naturally scales with the beam energy, its dependence on the excitation frequency is independent of the slip factor.FIG. 4: Spill quality as function of excited sideband and momentum spread (top), chromaticity (middle) and slip factor (bottom). In each plot, the case of 𝛿1𝜎=0.001, 𝜉=−1.5 and 𝜂=0.32 (blue stars) corresponds to Fig.1. Each data point corresponds to a distinct simulation where one of these parameters is varied systematically. Thereby, the slip factors correspond to different beam energies between 𝐸kin=150 MeV/u for 𝜂=0.44 and 500 MeV/u for 𝜂=0.12. For better visualization, sidebands at ℎ−𝑞x are shown as mirrored to the left half, while sidebands at ℎ+𝑞x are shown on the right half. The solid lines are the result of a multivariate fit using Eqs.(5) and (6).page 4 / 12 III.SCHOTTKY SPECTRA OF STRONG NONLINEAR BEAM MOTIONThe functional dependence of the spill quality on the excitation frequency as well as the momentum spread, tune, chromaticity and slip factor discussed in section II B suggests, that it is related to the spectral properties of the respective sidebands as observed in Schottky spectra of a corresponding non-excited beam. For the case of linear particle motion, transverse Schottky spectra are well understood [25]. Particles perform harmonic betatron oscillations and the sideband frequency follows Eq.(1) for 𝑘=1. Thereby each particle contributes a momentum-dependent frequency of𝑓±𝑓rev=[ℎ±𝑞x±𝜉x𝛿](1+𝜂𝛿)(2)with the parameters from tableI and the plus and minus signs for the upper and lower sidebands respectively. Considering a beam with a 1-sigma momentum spread of 𝛿1𝜎 and neglecting small 𝛿2 terms, the 1-sigma bandwidth of each sideband is given byΔ𝑓±𝑓rev≈[(ℎ±𝑞x)𝜂±𝜉x]𝛿1𝜎(3)A.Nonlinear and hollow beam spectraTo adapt the linear description of Schottky spectra to the nonlinear case, an empirical amplitude detuning term is added to Eq.(2). This term takes the amplitude dependent average detuning 𝒱︀a towards the third-integer resonance into account:𝑓±𝑓rev≈[ℎ±𝑞x±𝜉x𝛿±𝒱︀a(1−𝜉x𝛿Δres)](1+𝜂𝛿)(4)where Δres=𝑄res−𝑄x,ref is the nominal tune distance to the resonance and the factor in round brackets corrects for the fact that this tune distance is reduced for an off-momentum particle.To estimate the sideband width of a beam for the nonlinear case, in addition to the momentum distribution also the distribution in 𝒱︀a has to be taken into account. While this generally depends on the amplitude distribution of particles in transverse phase space, for simplicity an average value of ⟨𝒱︀a⟩ and a 1-sigma spread of Δ𝒱︀a is assumed in the following to estimate the resulting frequency distribution. In addition to the average amplitude detuning, the nonlinear dynamic also causes a modulation of the betatron frequency [19,26]. To account for the resulting additional frequency spread, an second detuning term Δ𝒱︀p is added. The 1-sigma bandwidth of each sideband can thus be estimated empirically (see appendix) asΔ𝑓±𝑓rev≈Δ𝒱︀p+√[(ℎ±𝑞x±⟨𝒱︀a⟩)𝜂±𝜉x(1−⟨𝒱︀a⟩Δres)]2𝛿21𝜎+Δ𝒱︀2a+𝑐cor(𝜂−𝜉xΔres)2𝛿21𝜎Δ𝒱︀2a(5)where ⟨𝒱︀a⟩, Δ𝒱︀a,p and 𝑐cor serve as fit parameters. The bilinear correction factor 𝑐cor is introduced to correct for the approximation of describing 𝛿 and 𝒱︀a by independent random distributions with mean and 1-sigma spread, as in reality the chromatic detuning affects the tune distance to the third-integer resonance. Fig.5 shows the sideband widths as obtained from simulated Schottky spectraFIG. 5: Top: simulated transverse Schottky spectrum of a hollow beam with betatron sidebands. Insets show zooms for the two highlighted sidebands with Gaussian fits. The small additional peaks visible in the insets are the sidebands of second order. Bottom: 1-sigma widths of the sidebands as obtained from the Gaussian fits for a full and hollow beam scenario. The dotted gray line shows the width expected from Eq.(3) and the dashed and solid lines are fits using Eq.(5) and the parameters from tableI.page 5 / 12 for a full and a hollow beam in comparison to the predictions from the linear and empirically adopted nonlinear theory. For the full beam, the fit using Eq.(5) and the parameters from tableI yields an amplitude detuning ⟨𝒱︀a⟩ and Δ𝒱︀a with values smaller than their uncertainty, indicating that the amplitude detuning is negligible as the spectrum is dominated by the densely populated beam core with small amplitudes. The phase detuning Δ𝒱︀p=(1.93±0.18)⋅10−3 is not negligible and can explain the increase of the sidebands frequency spread with respect to the linear theory in reasonable agreement considering the simplistic empiric approximation of the complex nonlinear dynamics involved. However, as the spectrum is dominated by the densely populated beam core it can not describe the spill quality improvement in Fig.1.The study of a hollow beam distribution as depicted in Fig.6 is motivated by the fact, that the extraction process is governed by particles near the separatrix performing strongly nonlinear motion. The Schottky spectrum obtained from a simulation of such a hollow beam (without rf-KO excitation) is also shown in Fig.5. Simulations of different hollow beam distributions comprising the outer 10 %, 25 % or 44 % of particles near the separatrix (in terms of the Hamiltonian) confirm the robustness and convergence of the depicted widths. In this case, the width of the sidebands shows the same behavior as the spill quality obtained by exciting the respective sideband (Fig.1): The smallest width corresponding to the largest spill fluctuations near ℎ=2 and as the width increases towards higher frequencies also the spill fluctuations reduce, resulting in a better spill quality.B.Correlation with spill qualityThe correlation between the spill quality and the width of the sidebands in the spectra of a hollow beam also remains valid if the parameters discussed in section II B are changed. FIG. 6: Hollow beam distribution in horizontal phase space (corrected for closed orbit and dispersion) occupying the outer 10 % in terms of the Hamiltonian (equipotentials marked by thin gray lines). Red lines mark the separatrix of the third-integer resonance for momentum offsets of 𝛿=0 (solid) and 𝛿=±10−3 (dashed).Fig.7 shows this correlation for various optics settings. The functional dependency found empirically based on the data is:𝑐v=𝑐v,∞+𝑎Δ𝑓±/𝑓rev−𝑏(6)where 𝑐v is the resulting spill quality by means of the coefficient of variation, Δ𝑓±/𝑓rev is the 1-sigma width of the sideband as obtained from hollow beam simulations without excitation, and 𝑎, 𝑏 and 𝑐v,∞ are fit parameters. The fit shown in Fig.7 yields 𝑎=(4.4±0.4)⋅10−4, 𝑏=(7.0±0.4)⋅10−4 and the spill quality limit 𝑐v,∞=0.24±0.02.By combining Eq.(5) and (6) and performing a meta-fit, the detuning parameters ⟨𝒱︀a⟩, Δ𝒱︀a,p and 𝑐cor representative for the spectral composition of the particles governing the extraction process can be found. The resulting dependency is shown by the solid lines in Fig.4 and as theory fit for the hollow beam case in Fig.5. For the meta-fit, all data points from the considered parameter space are used, except for the largest momentum spread and chromaticities as for these the beam spectrum becomes wider than the excitation bandwidth (dashed lines in Fig.4). The meta-fit yields ⟨𝒱︀a⟩=(−7.80±0.05)⋅10−3, Δ𝒱︀a=(4.82±0.14)⋅10−4, Δ𝒱︀p=(7.55±0.13)⋅10−4 and 𝑐cor=(2.2±0.3)⋅10−6. These values are in good agreement with the expected magnitude considering the distance to the resonance of Δres=−0.0123. The average detuning of |⟨𝒱︀a⟩/Δres|=63 % as obtained from the fit matches the prediction from the Kobayashi theory of 60 % or more for particles near the resonance [27].C.InterpretationThis correlation can be understood as follows: A larger sideband width in the hollow beam spectrum means that the frequency spread of the nonlinear betatron motion particles FIG. 7: Correlation between the 1-sigma width of sidebands in the Schottky spectrum of a hollow beam (compare Fig.5) and the spill quality resulting from its excitation, obtained from simulations with various optics settings (compare Fig.4). The excitation bandwidth is with Δ𝑄ex=0.02 always larger than the sideband width. The dashed fit function is described in the text.page 6 / 12 perform near the third-integer resonance separatrix is larger. A broadband excitation with frequencies ranging from the working point to the resonance can then couple more homogeneously to the beam as particles transit from the core to the separatrix. The result is a “resistive” excitation process with a larger net energy transfer to the beam. On the other hand, if the resonant frequency spread of the beam is small compared to the excitation bandwidth, the process is “reactive”, resulting in a perturbation of the particle motion prior to extraction rather than an energy transfer. This perturbation is observed as a stronger imprint of the non-resonant part of the excitation spectrum on the spill. It can also be understood as a demodulation of the excitation spectrum by the beam, causing the “overhanging” frequencies in the excitation spectrum not matched to the beam spectrum to be transformed onto the spill spectrum in baseband as can be seen in Fig.2 (left). While it seems reasonable to mitigate this perturbation also for smaller sidebands by using an excitation signal with a smaller bandwidth, such signals cannot efficiently drive the rf-KO extraction, because they cannot account for the intrinsic amplitude detuning from the beam core to the separatrix. As such, these methods rely on a higher excitation power, which in turn can amplify the perturbation and negate the desired effect, although the specific details for narrow-band excitation signals also depend on thir exact centre frequency. Only a sinusoidal excitation signal with zero bandwidth can achieve the desired effect, as it is not modulated in the first place and has no spectral bandwidth which could perturb the extraction process — even with the high excitation strength required. In fact, by speeding up the extraction and separatrix crossing for individual particles, the sinusoidal excitation can even suppress perturbations from power supply induced separatrix size fluctuations or those resulting from combinations with broadband excitation signals [23].Generally, to maximize the net energy transfer throughout the transition of particles from the core to the separatrix, a broadband excitation signal has to be used. In this case, exciting the beam at a sideband where the spectrum of particle motion is broader has the same effect of mitigating the perturbation of the extraction process and resulting imprint of the excitation, since the excitation matches the spectrum of particles near the separatrix better, which results in a better spill quality as discussed above. For this process, the width as obtained from a hollow beam distribution is a determining factor, since only the spectrum of particles near the separatrix is relevant for the extraction process.IV.COMPARISON TO DUAL ANDMANY-BAND EXCITATIONFigure8 compares the results obtained for single-band excitation as discussed above to different multi-band excitation methods: Dual-band signals as the combination of two single-band signals at different sidebands; and many-band signals as the sum of 𝑛 single-band signals at the first 𝑛 sidebands. For the many-band cases, the spill quality improves with the number of bands, which is in agreement with the previous findings at HIT [18] and WERC [17]. In view of the single-band excitation study, this improvement can be attributed to the contribution of the covered single sidebands at higher harmonics. In a simplified picture, the resulting spill quality can be understood as the average obtained from each distinct excitation sideband. This is demonstrated in Fig.9 where the FIG. 8: Spill quality (top) and excitation power (bottom) as function of excitation frequency for excitation of sidebands in different combinations. Left: Simultaneous excitation at many sidebands, i.e. the first 𝑛 upper, first 𝑛 lower, or all first 𝑛 sidebands up to the indicated maximum frequency. Center: Excitation of single sidebands (same data as Fig.1 for reference). Right: Simultaneous excitation at two sidebands, i.e. two upper, two lower, or one upper and one lower sideband respectively at the pairwise indicated frequencies. The power 𝑃∝(𝑘0𝑙)2 is shown relative to the reference deflection 𝑘0𝑙ref=1 µrad. Simulation according to the parameters listed in tableI.page 7 / 12 FIG. 9: Correlation between the average 1-sigma width of sidebands in the Schottky spectrum of a hollow beam (Fig.5) and the spill quality resulting from their combined excitation (Fig.8). The dashed line is the fit function Eq.(6) as obtained from Fig.7.average width of the respectively excited sidebands from the hollow beam Schottky spectrum is correlated with the resulting spill quality. The dependency follows the same functional relation found previously for the single-band study (Fig.7). Since the many-band excitation also always contains contributions from the lower harmonics which have a worse spill quality, the spill quality improvement is weaker compared to the single-band excitation. Accordingly, dual-band excitation at (only) the high harmonics result in a better spill quality compared to the many-band case containing also all the lower harmonics.In all cases, the combination of upper and lower sidebands results in an additional improvement of the spill quality compared to exciting either type alone (Fig.9). This additional improvement goes beyond what is predicted by Eq.(6) using the average width of the involved sidebands, hinting that it is caused by a different mechanism. This can possibly be attributed to the fact, that a detuning corresponds to a frequency change in opposite directions for upper and lower sidebands respectively. The simultaneous coupling of the excitation through these opposite bands leads to a more homogeneous excitation process FIG. 10: Spill spectra at the baseband (left) and at the respective excitation band (right) for two single band excitations and the dual band case combining the two of them. For the latter, the spectrum shown is the folded average of the two excited bands. Vertical lines mark the tune and resonance and the gray shading marks the extent of the excitation. Simulation result.which is less coherent and reduces the imprint of the excitation onto the spill beyond the average of the two single bands. This can be seen in Fig.10 at the respective excitation band. The weaker imprint also leads to reduced spill fluctuations in the baseband and thus improves the spill quality.In conclusion, and subject to the parameters discussed in section II B, dual-band excitation at upper and lower sidebands of high harmonics is the most beneficial excitation method in terms of the spill quality. While slightly more excitation power is required, this method can simplify the design of excitation systems compared to the many-band case spanning many harmonics, since a much smaller system bandwidth is required. For example, exciting the upper and lower sideband of the 20th harmonic would require a system bandwidth of only 10 % of the tunable central frequency. The reduced demand in terms of frequency characteristics could allow for a higher impedance power coupling mechanisms with less power losses [17].V.EXPERIMENTAL INVESTIGATIONA.Measurement setupThe spill quality and amount of pile-up were measured at the Heidelberg Ion Beam Therapy Center. Therefore, an ion optics with 𝑄x,ref=1.6802 and 𝜉x=−0.655 was used, and a C126+ beam was extracted at 𝐸kin=251 MeV/u (𝑓rev=2.8438 MHz). While for treatment purposes the rf-KO extraction is performed with the beam bunched by the acceler-ating rf cavity, bunching was turned off for the measurement to simplify the beam dynamics interpretation and allow for a closer comparison to the simulations. The initial beam momentum spread was measured to be 𝛿1𝜎≈7⋅10−5 using a longitudinal Schottky monitor. While the measurement conditions differ from the initial baseline simulations shown in Figs.1, 2 and 3, they lie within the large parameter range considered in Fig.4.For the measurement, rf-KO extraction was performed with the RBPSK excitation and spill intensity feedback system used for treatment, and with a band-filtered noise excitation of manually adjusted amplitude. In both cases, a central frequency and bandwidth according to tableI was used. A 500 W power amplifier with a bandwidth of 20 MHz was used to power the 50 Ω terminated stripline exciter. The exciter’s electrode plates have a physical length of 0.75 m and an electrical length of 1.1 m each (measured between the vacuum feed-throughs). They are connected in series with cables and a transformer in between, which is designed to invert the signal polarity at frequencies below 2 MHz in order to create the transverse deflecting rf field between both electrodes. Given this serial connection scheme, an additional frequency dependent phase shift is introduced by the total length of the signal path between the plates. At about 17 MHz≈6𝑓rev the additional shift amounts to 180°, which including the transformer leads to a total phase shift of about 360° between the plates, effectively causing the transverse rf field to vanish. The field has another maximum at about 34 MHz≈12𝑓rev, where the total phase shift amounts to 540°.page 8 / 12 The spills were recorded with an Ionisation Chamber (IC) at 50 µs resolution for analysis of the spill quality. In addition, a BC400 plastic scintillator connected to a photomultiplier was installed downstream of the IC to record the last 18 ms of the spill with an oscilloscope at 0.32 ns resolution. The scintillator pulses were analyzed with individual particle pulse detection using the standard peak detection algorithm signal.find_peaks of the scipy library [28]. The resulting timestamps are used for the pile-up analysis.B.Measured spill quality and pile-upThe spill quality, measured at an average extraction rate of 5⋅106 particles/s with the machine settings as stated above is shown in Fig.11 as a function of the central frequency of the single-band RBPSK excitation signal. The spill quality gradually improves for higher sidebands, and there is a small yet systematic difference between excitation at lower and upper betatron sidebands. There is a fair agreement of the measured FIG. 11: Measured spill quality for single-band RBPSK excitation at different sidebands. For comparison, the simulation result as obtained for the same machine settings, but reduced spill rate of 5⋅105 1/s and ideal frequency response of the exciter is shown.spill quality with the corresponding simulations, considering that the number of particles in the simulation is reduced due to limited computational resources, which results in a shift of the values of 𝑐v. Given the low beam momentum spread, the measured effect is small compared to the initial simulations with a larger momentum spread in Fig.1. While the simulations assume an ideal frequency response of the excitation system, in the measurement it was not possible to extract the desired intensity at the 𝑄ex=6.327 band due to the frequency charac-teristic of HIT’s excitation system as explained above.For analysis of pile-up, individual particles were recorded with a scintillator under the same machine settings but at a lower extraction rate of 4⋅106 particles/s. The cumulative distribution of the measured time between consecutive particles is shown in Fig.12(left). In the figures, also a selected region of 2 µs of the raw signal with piled-up pulses is highlighted. It can be seen that the predominant periodicity of particles is defined by the excitation frequency, resulting in clustering at a lower excitation frequency. With the scintillator pulse full width at half maximum (FWMH) being about 25 ns, consecutive particles arriving within this period would pile-up. By evaluating the fraction of particles arriving within less than Δ𝑡pile-up=20 ns, the amount of pile-up is shown as a function of excitation frequency in Fig.12 (right). As the excitation frequency increases with respect to the average particle rate, the clustering is avoided and pile-up reduces. Due to the small momentum spread in the experiment, the effect of spill quality improvement and associated reduction of excitation frequency imprint as discussed in section II and observed in Fig.3 is weakened, such that the measured pile-up behaviour is dominated by the clustering frequency and no systematic difference between upper and lower sidebands is observed. This also suggests that the excitation frequency can be utilized as a free parameter to mitigate pile-up independent of the momentum spread: For large momentum spreads the mechanism detailed in sections II and III can be used to reduce pile-up by suppressing the imprint of the excitation onto the spill as shown in Fig.3. For FIG. 12: Left: Cumulative distribution of particle arrival times for RBPSK excitation at 𝑄ex=0.327 and 5.673. The insets show the raw detector signals with pulses corresponding to the detection of individual particles (marked with crosses). Right: Fraction of particles arriving within less than Δ𝑡pile-up=20 ns of one another (pile-up) as a function of the excitation frequency as well as the corresponding average extraction rate of about 4⋅106 particles/s.page 9 / 12