Extracting chromatic properties of electron beams from spectral analysis of turn-by-turn beam position data
EE STIMATING CHROMATIC PROPERTIES OF ELECTRON BEAMSFROM FREQUENCY ANALYSIS OF TURN - BY - TURN BEAMPOSITION DATA
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Panagiotis Zisopoulos
CERNGeneva, CH-1211Switzerland [email protected]
Yannis Papaphilippou
CERNGeneva, CH-1211SwitzerlandSeptember 21, 2020 A BSTRACT
A method to estimate linear chromaticity, RMS energy spread, and chromatic beta-beating, directlyfrom turn-by-turn beam position data in a circular electron accelerator, is presented. This technique isbased on frequency analysis of a transversely excited beam, in the presence of finite chromaticity.Due to the turn-by-turn chromatic modulation of the beam’s envelope, betatron sidebands appeararound the main frequency of the Fourier spectra. By determining the amplitude of both sidebands,chromatic properties of the beam can be estimated. In this paper, analytical derivations justifying theproposed method are given, along with results from tracking simulations. To this end, results frompractical applications of this technique at the KARA electron ring are demonstrated. K eywords chromaticity, RMS energy spread, Fourier Analysis, NAFF, Decoherence, Beam Dynamics The measurement of chromaticity [Verdier(1993)] is an indispensable process in a circular accelerator, due to the impactit has on beam lifetime and quality. At the modern high-intensity proton and electron rings, the thresholds of severalbeam instabilities are controlled through the value of chromaticity.A plethora of methods for measuring chromaticity, each with different beam parameters as observables, have beendeveloped so far. Some of the existing techniques are mentioned below, however a more detailed review of thecorresponding approaches can be found in [Steinhagen(2009), Minty and Zimmermann(2003)]. The basic relationshipthat defines chromaticity in terms of beam dynamics parameters is presented in Sec. 6.1.1 of the Appendix.One of the most simple and widely used techniques of chromaticity is the
RF-sweep , where the operating frequency of theradio-frequency (RF) system is changed, in order to induce a change in the energy of the beam. At the same time, betatrontune measurements [Serio(1989), Bartolini et al. (1996)Bartolini, Giovannozzi, Scandale, Bazzani, and Todesco] areperformed, usually by analysing turn-by-turn (TbT) data from beam position monitors (BPMs) and employing Fourieralgorithms. From there, chromaticity can be determined from the correlation of the measured tunes with the varyingenergy of the beam. This method usually requires dedicated experimental time, thus making it impossible to use duringnormal operation of the accelerator.The estimation of chromaticity is also possible by measuring the incoherent power spectrum of the beam with Schottkymonitors [Boussard(1989)]. This method is particularly useful for high-intensity, high energy proton machines like the
Large Hadron Collider (LHC), however in the case of bunched beams, strong coherent beam modes can pollute theSchottky spectra and thus reduce the efficiency of the method [Betz et al. (2017)Betz, Jones, Lefevre, and Wendt]. a r X i v : . [ phy s i c s . acc - ph ] S e p PREPRINT - S
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21, 2020A very interesting technique to measure the relative chromaticity ξ = Q (cid:48) /Q , where Q is the betatron tune, isby observing the phase shift of the head and the tail of a bunch [Cocq et al. (1998)Cocq, Jones, and Schmickler,Fartoukh and Jones(2002)] during one synchrotron period. This phase-shift is correlated with the value of chro-maticity and it can be measured with instruments that can resolve intra-beam movements, like the Head-TailMonitor [Catalan-Lasheras et al. (2003)Catalan-Lasheras, Fartoukh, and Jones]. This method has been found to beaffected by the quality of the TbT signal [Ranjbar et al. (2017)Ranjbar, Marusic, and Minty], where kinematic decoher-ence [Meller et al. (1987)Meller, Chao, Peterson, Peggs, and Furman, Lee(1991)] arising from the frequency distribu-tion over the particles in the bunch, can severely diminish the signal-to-noise ratio. As a result, only a few hundreds ofturns can be left for meaningful frequency analysis, whereas the synchrotron period of a proton machine can be muchlarger. Note that this technique cannot be easily applied in a high-energy electron ring, where the bunch lengths are inthe range of ps, rendering the phase differences between the head and the tail of the bunches hard to resolve.Another class of methods that can be employed are the ones that extract information from the chromatic decoherence ofthe beam [Siemann(1989)]. Due to this mechanism, the envelope of the beam is modulated in amplitude with respectto the number of revolutions. By determining the maximum and the minimum of the coherent TbT signal i.e. themodulation depth of the betatron oscillations, information on chromaticity and the RMS energy spread can be acquired.In fact, measuring and controlling the RMS energy spread is of paramount importance as well for the modernhigh-intensity electron accelerators. For example, the modern light sources are designed as to achieve very lowemittances, in the regime that intra-beam scattering becomes important [Kubo et al. (2005)Kubo, Mtingwa, and Wolski,Bane et al. (2002)Bane, Hayano, Kubo, Naito, Okugi, and Urakawa, Antoniou(2012)]. At an electron ring, the RMSenergy spread is typically measured by evaluating information from the synchrotron light of the electron bunches withspecific instrumentation e.g. a streak camera. However, the same goal can be achieved by analysing TbT BPM data inthe time domain [Hsu(1990), Bassi et al. (2015)Bassi, Blednykh, Choi, and Smaluk], by performing multi-parametricfits, based on the existing analytical models of the beam centroid motion in the presence of chromaticity. Unfortunately,the unavoidable noise in the TbT data and small values of the chromaticity can drive large measurement errors whenthese methods are employed [Manukyan et al. (2011)Manukyan, Sargsyan, Amatuni, and Tsakanov].In this paper, by acknowledging the necessity of an operationally efficient technique, an alternative methodis proposed for estimating the chromaticity or the RMS energy spread through the Fourier spectra ofa transversely excited beam. The proposed technique is based on the analytical relationships derivedin [Rumolo et al. (2004)Rumolo, Schmidt, and Tomas], which govern the Fourier spectra of a longitudinally excitedbeam, in the presence of chromaticity. The method takes advantage of the fact that chromatic decoherence modulates theenvelope of the beam, with the modulation period to be exactly the synchrotron period. As a result, chromatic sidebandsappear around the main betatron tune in the transverse spectra of the excited beam, with an amplitude proportional tochromaticity. With the proposed method, the estimation of chromaticity is performed through the measurement of boththe chromatic sidebands of the beam, and by using the following simple equation Q (cid:48) z = ± Q s σ δ (cid:114) A + A − A , (1)where Q (cid:48) z , with z = x, y , is the horizontal or vertical chromaticity respectively, Q s is the synchrotron tune, σ δ is the RMSenergy spread, A and A − are the amplitudes of the first order chromatic sidebands that appear around the horizontal orvertical betatron frequency which has an an amplitude of A . Precise measurements of chromaticity with the aforemen-tioned method are possible only if the Fourier amplitudes A ± are known with high certainty. Fortunately, the existenceof powerful numerical tools for the Fourier decomposition of the beam’s motion allow for this. The Numerical Analysisof Fundamental Frequencies (NAFF) [Laskar et al. (1992)Laskar, Froeschle, and Celletti, Laskar(1993), Laskar(2003),Papaphilippou(2014)] algorithm has been used with success in the field of accelerator physics, for precise optics anddynamical stability measurements through frequency and amplitude analysis. The first results from the applicationof this method can be found in [Zisopoulos et al. (2014)Zisopoulos, Antoniou, Papaphilippou, Streun, and Ziemann],where TbT RMS energy spread measurements are performed at the Swiss Light Source (SLS).It is important to mention that a method which follows a similar approach i.e. it is based on the frequency responseof the chromatic motion of the beam, has been developed in [Nakamura(1999)] and employed for measurementsin some applications [Nakamura et al. (1999)Nakamura, Takano, Masaki, Soutome, Kumagai, Ohshima, and Tsumaki,Kiselev et al. (2007)Kiselev, Muchnoi, Meshkov, Smaluk, Zhilich, and Zhuravlev]. A limitation of this method is thatit assumes equal amplitudes for both chromatic sidebands, and it uses only one sideband for measurements. However,this would be true only in the absence of non-linearities which can cause beating of the optics in the lattice.In order to by-pass this limitation, the method uses both amplitudes of the chromatic sidebands asit is testified in Eq. (1). The same methodology has been also applied to the Diamond LightSource [Rehm et al. (2010)Rehm, Abbott, Morgan, Rowland, and Uzun], where a piecewise fit of the ratio of the chro-matic sidebands, for various chromaticities, has been implemented and used as a reference, in order to estimate2
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21, 2020chromaticity during operation. However, in the present work, the estimation of chromaticity is model independent, bysolving analytically the equations of decoherence, and by using the amplitudes of the sidebands as an observable. Anadvantage of this method requires the same experimental procedure as the typical betatron tune measurements in a ring.The present paper is organised as follows: In Section 2 the theoretical basis of this method is presented. In Sec-tion 3, the method is employed in tracking simulations for estimating chromaticity and chromatic beta-beating fromTbT BPM data, by using the accelerator model of the KARA light source and NAFF. In Section 4, the proposedmethod is employed in experimental measurements at the KARA light source, including an application to characterisethe optics response during the commissioning of the Compact Linear Collider (CLIC) Superconducting Wigglerprototype [Bernhard et al. (2016)Bernhard et al. ]. Analytical formulas of the Fourier spectra of a Gaussian beam, which is longitudinally excited in the presence ofchromaticity, have been derived in [Rumolo et al. (2004)Rumolo, Schmidt, and Tomas]. These expressions take intoaccount non-linearities, which arise from the distortion of the beam’s motion due to finite dispersion at the sextupoles.From these relationships, the Fourier amplitudes A q of the synchrotron sidebands of order q are given from A q = e − s | ˜ α | (cid:12)(cid:12)(cid:12)(cid:12) I q ( s − isk ) + ∆ β z iβ z σ δ ( k + is ) (2) (cid:2) I q − ( s − isk ) − I q +1 ( s − isk ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) , where s = Q (cid:48) z σ δ /Q s , with Q (cid:48) z the chromaticity for the z = x, y planes, σ δ is the RMS energy spread of the bunch, Q s the synchrotron tune defined from the RF cavities, ˜ α is the initial transverse amplitude of the kicked beam, I q ( x ) is the q -order modified Bessel function of the first kind for argument x , k is the longitudinal kick in units of theRMS bunch length σ t of the beam, and ∆ β z β z is the chromatic beta beating. Analytical derivations of the chromaticbeta-beating for circular accelerators can be found in [Fartoukh(1999)], which includes the effect of linear and non-linear magnetic fields. The appearance of the I q ( s − isk ) term is a signature of amplitude modulation, wherethe argument s − isk represents the modulation index [Oppenheim et al. (1997)Oppenheim, Nawab, and Willsky].The same modulation occurs in the case of a pure transverse excitation instead of a longitudinal one. A numericalexample is shown in Fig. 1, where TbT pseudo-data are produced, based on the kinematic decoherence relationshipsin [Rumolo et al. (2004)Rumolo, Schmidt, and Tomas]. The envelope (red curve) of the oscillation (blue curve) exhibitsa modulation period of N s = 100 turns i.e. the inverse of the synchrotron tune Q s = 0 . . Every N s / turns theenvelope passes a trough, thus exhibiting a minimum projection of the synchrotron oscillations in the transverse betatronmotion. The carrier frequency is the betatron frequency, which is excited with a transverse impulse at N = 0 turns.Figure 1: Example of betatron oscillations recorded at a fictitious BPM with respect to the number of turns N in thepresence of chromaticity. The betatron oscillations are shown in blue and the upper envelope of the signal in red. Theamplitude modulation due to chromaticity has a period of N = 100 turns. The synchrotron tune is Q s = 0 . . In thisexample, the transverse motion of the centroid is linear i.e. no amplitude detuning is present.Concerning the application which is presented in this paper, the longitudinal excitation is dropped in favour of atransverse excitation. The motivation behind this choice is that the goal of this analysis is to develop a methodfor chromaticity or RMS energy spread measurements, based on the same procedure of transverse betatron tunemeasurements. In order to do so, the parameter of the longitudinal excitation k , is set to k = 0 and Eq. (2) now becomes3 PREPRINT - S
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21, 2020 A q = e − s | ˜ α | (cid:12)(cid:12)(cid:12)(cid:12) I q ( s ) + s ∆ β z β z σ δ (cid:2) I q − ( s ) − I q +1 ( s ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) , (3)where now the Fourier amplitudes of the beam, A q , are determined from the transverse impulse given to the beam. As amatter of fact, one of the assumptions of the current study is that the transverse kick given to the beam is adequatelysmall, such as not to induce rapid decrease of the beam’s envelope due to decoherence, as a result of tune-shift withamplitude.The previous expression can be written in a simpler form by using the following relation-ships [Abramowitz and Stegun(1965)]: I q − ( x ) − I q +1 ( x ) = 2 qx I q ( x ) (4) I q ( x ) = I − q ( x ) (5) I q ( x ) ≈ q + 1) (cid:18) x (cid:19) q , < | x | (cid:28) (cid:112) q + 1 . (6)Indeed, by using Eq. (4) in Eq. (3) and computing the absolute value of the result, the expression A q = | ˜ a | e − s I q ( s ) (cid:18) q Q s Q (cid:48) z ∆ β z β z (cid:19) , (7)is obtained.It is evident from Eq. (7) that chromatic beta-beating becomes significant for higher-order modes i.e. it is linear with q .In addition, it is clear that by using only one synchrotron sideband for the estimation of the chromatic properties inducesan uncertainty due to the beta-beating. However, by forming the sum of both symmetric sidebands A q + A − q withthe help of Eq. (5), and normalising it with the main betatron amplitude A for q = 0 , results in the elimination of theperturbation term ∆ β β . Moreover, by normalising the difference ±| A q − A − q | by A q + A − q , an analytical relationshipfor the estimation of the chromatic beta-beating ∆ ββ is obtained. The aforementioned operations result in the followingequations: A q + A − q A = 2 I q ( s ) I ( s ) (8) (cid:12)(cid:12)(cid:12)(cid:12) A q − A − q A q + A − q (cid:12)(cid:12)(cid:12)(cid:12) = ± q Q s Q (cid:48) z ∆ β z β z . (9)The expression in Eq. (8) can be numerically solved in order to estimate the s = Q (cid:48) z σ δ Q s parameter. The synchrotrontune Q s is usually known with fair accuracy from the properties of the RF system, but it can also be inferred fromthe frequency offset of the chromatic sidebands, with respect to the main frequency line. Then, by knowing the RMSmomentum spread σ δ , one can estimate the chromaticity Q (cid:48) z of the machine or vice-versa .On the other hand, simple analytical relationships that determine the chromaticity or the RMS energy spread can bedeveloped by combining the approximation in Eq. (6) with Eq. (8).Assuming that s (cid:28) √ q + 1 , for the q order chromatic sideband under consideration, the analytical form of Eq. (8)becomes A q + A − q A = 2Γ( q + 1) (cid:18) s (cid:19) q = 2 − q Γ( q + 1) s q . (10)4 PREPRINT - S
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21, 2020Solving Eq. (10) with respect to s , and further with respect to chromaticity Q (cid:48) z , yields Q (cid:48) z = ± Q s σ δ q (cid:115) q − Γ( q + 1) (cid:20) A q + A − q A (cid:21) (11)The previous expression allows for the estimation of chromaticity Q (cid:48) z or the RMS energy spread σ δ based on theamplitudes of the chromatic sidebands A q and A − q , for q ≥ . Since in a real Fourier spectrum of TbT data, assumingthat the second order chromaticity Q (cid:48)(cid:48) z is not large, the beam is longitudinally matched to the RF bucket, and thatcollective effects are not important, the first order chromatic sidebands A and A − are most easily resolved. Thus, bysetting q = 1 in Eq. (11) and in Eq. (9), the following expressions are obtained: Q (cid:48) z = ± Q s σ δ (cid:114) A + A − A (12) ∆ β z β z = ± Q (cid:48) z Q s (cid:12)(cid:12)(cid:12)(cid:12) A − A − A + A − (cid:12)(cid:12)(cid:12)(cid:12) (13)These relationships constitute of a method, which is independent from the calibration of the BPMs due to normalizationto the main amplitude A , for estimating the chromaticity or the RMS energy spread of the beam. As a by-productof the method, a relationship which determines the chromatic beta-beating at the position of the BPMs is recovered.Note that the previous expressions do not differentiate between positive and negative solutions, since they depend onthe Fourier amplitudes of the beams. Thus, the proposed method cannot recover the sign of the chromaticity Q (cid:48) z andof the chromatic beta beating ∆ β z β z . Analytical expressions for the measurement errors of chromaticity and chromaticbeta-beating through Eq. (12) and Eq. (13), can be found in Sec. 6.1.2 of the Appendix. Tracking simulations with the model of the KARA electron accelerator are undertaken using
MADX-PTC [Schmidt et al. (2002)Schmidt, Forest, and McIntosh], in order to numerically investigate the proposed methodand to identify the dependence of the proposed method on the initial excitation of the beam. The fundamental parametersof the simulations can be found in Table. 1. During the simulations, the particles are tracked around the KARA latticefor different cases of initial excitation, and for N = 1200 turns. The lattice of the KARA accelerator consists of DoubleBend Achromat (DBA) cells and exhibits a four-fold symmetry. The number of BPMs is M = 35 and the opticsof at the position of the BPMs are shown in Fig. 2, where the maximum horizontal beta function (top) is at around β x = 19 m at the position of the arcs of the ring, the maximum vertical beta function is approximately β y = 30 m andthe horizontal dispersion (bottom) exhibits an average value of around (cid:104) D x (cid:105) = 0 . m. The simulations employ anerror-free lattice, with no radiation effects taken into consideration, which result in a vanishing vertical dispersion D y around the accelerator.Table 1: Parameters of the tracking simulations with the KARA model. All the position dependent parameters aremeasured at the injection point of the ring. Parameter Value
Energy 2.50 [GeV]Circumference 110.40 [m]Tune Q x , Q y , Q s π ]Chromaticity Q (cid:48) x , Q (cid:48) y π ]RMS beam size σ x , σ y at injection 0.94, 0.080 [mm]Beta function β x , β y at injection 18.89, 1.67 [m]RMS bunch length σ z σ δ − ]Initial excitation at injection − [ σ x ], − [ σ y ]Dimensionality 6-DDistribution GaussianNumber of particles N p The TbT evolution of the centroid of the beam, is inferred from the arithmetic mean of all the particles for each turn N .The beam is transversely excited in a range of initial amplitudes, which are referred in this paper in units of the RMS5 PREPRINT - S
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21, 2020Figure 2: The optics for the accelerator model of the KARA light source, with respect to the position of the M = 35 BPMs of the KARA light source. Top and middle plots show the horizontal β x and vertical β y functions, and thebottom plot depicts the horizontal dispersion D x .transverse beam size σ z , where z = x, y for the horizontal and vertical planes respectively. In Fig. 3a, the horizontalTbT are shown for initial excitations of 1 σ x to 4 σ x , where for that particular position in the ring, the initial amplituderanges from . mm to mm. Kinematic decoherence is responsible for the simultaneous chromatic oscillations andthe TbT damping of the beam’s envelope due to non-linear tune-shift. The TbT simulated data for the vertical plane, arepresented in Fig. 3b, after initial excitations of 20 to 50 σ y and initial amplitudes similar to the horizontal case, Fig. 3a.The trend of the centroid’s evolution in the vertical plane exhibits a smaller non-linear detuning and larger chromaticoscillations, compared to the horizontal plane, due to the higher vertical chromaticity. (a) Horizontal TbT data for excitations of σ x (top left), σ x (top right), σ x (bottom left) and σ x (bottom right).(b) Vertical TbT data for excitations of σ y (top left), σ y (top right), σ y (bottom left) and σ y (bottomright). Figure 3: The simulated TbT tracking data of the beam’s centroid for the KARA accelerator model, with respect to thenumber of turns N . The initial amplitude of the beam, in terms of the transverse beam sizes σ x , σ y , is indicated in thelegend. 6 PREPRINT - S
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21, 2020 (a) Horizontal normalized envelopes of the centroid for thecase of 1 σ x (blue line), 2 σ x (orange line), 3 σ x (greenline), and 4 σ x (red line).(b) Vertical normalized envelopes of the centroid for thecase of 20 σ y (blue line), 30 σ y (orange line), 40 σ y (greenline), and 50 σ y (red line). Figure 4: The upper envelopes for the horizontal (a) and vertical (b) oscillations of the centroid of the beam for theKARA model, with respect to the number of turns N . The envelopes are normalised to the oscillation value at the firstturn N = 1 . The initial excitation for each simulation is shown in the legend.As it has been discussed already, chromatic decoherence results in the periodic modulation of the beam’s envelopewith a period τ s equal to the inverse of the synchrotron tune τ s = Q s . For the present simulations, the TbT evolutionof the horizontal envelopes, normalised to their maximum value at N = 1 turns, is shown in Fig. 4a for a range ofinitial excitations. The synchrotron period is about τ s = 63 turns and during the first synchrotron period, the centroidof the beam oscillates with the same phase, regardless of the initial excitation. However, during and after the secondsynchrotron period, a distinct amplitude-dependent damping of the synchrotron oscillations appears. This behaviourcan be attributed to the non-linear detuning of the centroid, which becomes dominant after around N = 300 turns i.e.almost 5 synchrotron periods τ s . In addition, the phases of the oscillations for all cases appear to become different afterthe first two synchrotron periods τ s . This indicates that the synchrotron oscillations exhibit a frequency shift, whichdepends to the initial excitation amplitude. For the σ x case and for that particular BPM, the value of the envelope is at50 % of the initial amplitude at around N = 300 turns already, indicating a relatively small decoherence time. Thesame value is at around 70 % for the excitation of σ x .The vertical envelopes are shown in Fig. 4b. The amplitude modulation is visible for every N = τ s turns, with themodulation occurring around the high-frequency component i.e. the betatron oscillations. Note that the chromaticoscillations are visible also in the vertical plane, even if there is no vertical dispersion. Due to the small verticalemittance, the TbT evolution of the vertical envelopes exhibit more coherent behaviour with respect to the horizontalplane, and the de-phasing of the oscillations due to decoherence begins at around five synchrotron periods. As a result,estimating the vertical chromaticity with the proposed method, is expected to be less dependent on the initial excitation.In addition, the damping of the chromatic oscillations appears to be very slow, which results in a wider range of numberof turns N for the survival of the synchrotron oscillations.7 PREPRINT - S
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21, 2020An immediate conclusion of the analysis the TbT evolution of the envelopes, is that the applicability of Eq. (12) andEq. (13) is expected to depend on the number of turns N , and the initial transverse excitation of the beam. For thisreason, the number of turns N should be restricted to as small values as possible. At the same time, that particularnumber of turns N should allow for enough resolution for the estimation of the Fourier spectra. The Fourier spectra of the centroid of the beam are determined with
Py-NAFF [Asvesta et al. (2017)Asvesta, Karastathis, and Zisopoulos], the adaptation of the
NAFF [Laskar(2003)]algorithm in
Python programming language, which offers an increased precision in the estimation of the spectralcomponents of a quasi-periodic signal. The appearance of the chromatic sidebands around the main betatron frequencyline is shown in Fig. 5a for the horizontal plane, and in Fig. 5b for the vertical plane. The amplitudes are normalised tothe maximum of the main Fourier amplitude A for both cases of initial excitation, and the measurements are performedwith N = 300 turns or almost five synchrotron periods τ s .Concerning the horizontal case, a slight increase of the horizontal tune Q x is observed for the σ x case, with respect tothe σ x case, due to tune-shift with amplitude. Note that the tune-shift appears as a decrease of the horizontal betatrontune Q x in the Fourier spectra, since the fractional part of the tune Q x is larger than 0.5.For the vertical case, the chromatic sidebands are found to be almost 14 times greater in amplitude, than the horizontalsidebands, due to the difference in the magnitude of the two transverse chromaticities. Furthermore, there is a negligibleshift of the frequencies for the σ y case. The frequency spectra of all the simulated cases can be similarly produced,and chromaticity can be inferred from Eq. (12). (a) Horizontal Fourier spectra, where the σ x case is shownin blue and the σ x case in orange.(b) Vertical Fourier spectra, the σ y case is shown in blueand the σ y case in orange. Figure 5: The frequency spectra of the beam’s centroid for the horizontal (a) and vertical (b) planes, normalised tothe amplitude of the main peak A . The chromatic sidebands A and A − are observed at a distance equal to thesynchrotron tune Q s , marked with dashed lines (green for A − and black for A ). The initial excitation amplitude ofthe beam, in terms of the transverse beam sizes σ x and σ y , is indicated in the legend.8 PREPRINT - S
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The synchrotron tune Q s , is usually well known from the parameters of the RF system, however it can be also estimatedfrom the distance between the main betatron frequency line A and the first order chromatic sidebands A and A − .Although for the present study the value of the synchrotron tune Q s is obtained from MAD-X , the synchrotron tune Q s is also estimated from the response of the transverse Fourier spectra of the beam. Such an analysis can provide usefulinsights on the behaviour of the TbT data. The estimation of the average synchrotron tune Q s , where the average isperformed on the M = 35 BPMs of the KARA model, is graphically presented in Fig. 6 against the value from
MAD-X (solid black line) for the horizontal (top) and vertical (bottom) planes, and for various initial excitations.Figure 6: The synchrotron tune Q s measured from the Fourier spectra for each case of initial excitation, and averagedover all the BPMs of the KARA model. The standard deviation σ Q s of the tunes for all cases is at the order of σ Q s = 10 − . The tunes are shown with respect to the number of turns N for the horizontal (top) and vertical (bottom)planes. The legend indicates the initial excitation amplitude, while the black solid line shows the theoretical synchrotrontune Q s , which is obtained in MAD-X .For the horizontal plane, it is evident that amplitude dependent effects can impact the survival of the chromatic sidebands.For example, while for the σ x case the synchrotron tune can be recovered for up to N = 800 turns, the 4 σ x case ex-hibits synchrotron sidebands up to N = 500 turns. In addition, a longitudinal tune-shift with amplitude is evident duringthe first six synchrotron periods i.e. up to N = 300 turns, which is highlighted at N = 180 turns, and converges to a con-stant value after the characteristic decoherence time [Meller et al. (1987)Meller, Chao, Peterson, Peggs, and Furman]of each initial excitation. The nature of this effect stems from the injection of the beam in the RF bucket. A largertransverse excitation results in a beam which covers more space in the longitudinal phase space, resulting in largerlongitudinal emittance and synchrotron tune Q s .The synchrotron tune inferred from the vertical TbT data exhibits no amplitude-dependent effects, and the extractedvalue remains constant with respect to the number of turns, and close to the expected synchrotron tune.The TbT dependency of the synchrotron tune Q s on the initial excitation can introduce systematic errors in the estimationof chromaticity with Eq. (12), which is more pronounced for stronger initial excitations. On the other hand, larger initialexcitations might be favorable for resolving the chromatic sidebands. In order to correct for this source of systematicerror in the synchrotron tune measurement, analytical models can be employed which describe the relationship of thesynchrotron tune and amplitude. However, in order to avoid this error, the value of the theoretical synchrotron tune, orthe value estimated with the parameters of the RF system, could be used for chromaticity measurements. From Eq. (12), the inferred chromaticity is given as the ratio of the sum of the chromatic sidebands A ± , to theamplitude of the main frequency line A . Due to the effect of decoherence, the amplitudes of the spectral lines decreasewith respect to the number of turns N , with a decay rate that is similar for every chromatic sideband. As a consequence,systematic errors inhibit the estimation of any parameters from the spectral lines of the beam, with the magnitude of theerror depending on the optics and the TbT evolution of beam dynamics.It is worth mentioning that combined decoherence models, which describe the motion of the centroid of the beamin the presence of finite chromaticity and tune-shift with amplitude, do exist and could be "fed-back" to the data, inorder to correct for the aforementioned turn-by-turn error. This would require a proper tuning of the model and a goodknowledge of the optics, as to fit the experimental data as better as possible. Another alternative would be to estimatethe frequency response of the combined decoherence, and correct the amplitude of the the spectral lines of the centroidof the beam. 9 PREPRINT - S
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21, 2020For the sake of estimating the effect of amplitude-dependent decoherence, an analysis of the spectral lines of the beam’scentroid for various initial excitations is employed in these simulations. First the chromatic ratios R x = A x + A − x A x (14) R y = A y + A − y A y (15)are defined, where A x , A − x , A y , A − y are the first order chromatic sidebands of the main betatron amplitudes A x , A y of the horizontal and vertical planes respectively.The dependence of √ R x on the number of turns N and on the initial excitation amplitude, can be visualized in Fig. 7a,while the same dependence for the vertical chromatic ratio (cid:112) R y are shown in Fig. 7b. The values represent the averageacross all the BPMs of the KARA model, and the error-bars represent one standard deviation from the average. Bothfigures contain the theoretical chromatic ratio values that are expected from the model, by taking into account thestatistical error in the estimation of the RMS energy spread, arising from the usage of different particle distributions foreach initial excitation. This spread, which is taken as the standard deviation, SD ( σ δ ) , of the RMS energy spread σ δ ofall the simulated distributions, is at the range of − for both transverse planes, and it results in an uncertainty σ √ R z of the chromatic ratios, for z= x,y, which by simple error propagation is estimated to be: σ √ R z = Q z Q s SD ( σ δ ) (16)where the SD ( σ δ ) is the standard deviation of the RMS energy spread σ δ .The horizontal ratio √ R x exhibits a significant TbT spread between the different excitations of the beam, which cannotbe explained by the uncertainty in the estimation of the RMS energy spread σ δ . More specifically, while the cases of , , and σ x excitations produce the same √ R x ratio at N = 180 turns, the σ x case exhibits an offset of around . from that value. At the same time, the σ x case exhibits the slowest damping due to decoherence. This suggests that thefor the current simulations, the σ x excitation is not enough for precise chromaticity estimations. The curves of σ x and σ x initial excitation, exhibit similar trends, albeit a small separation is visible after N = 360 turns, or after aboutsix synchrotron periods. The σ x excitation undergoes a steep decrease in the very first turns, due to non-linearities,and then quickly decreases in magnitude. This behaviour points to the fact that the available number of turns for precisechromaticity measurements, is limited for this case. Note that for the σ x excitation, the systematic errors are almost10 times larger than the excitations of smaller magnitude.Concerning the vertical plane, the (cid:112) R y curves can be visualized in Fig. 7b, and suggest an insignificant effect ofdecoherence on the chromatic oscillations of the beam’s centroid. This is expected from the small vertical beam size σ y .All the simulated cases demonstrate a uniform damping of the vertical chromatic ratio (cid:112) R y , with a total drop of . from N = 180 to N = 480 turns. The distribution of the curves falls inside the uncertainty interval of the theoreticalvalue, however the case of σ y exhibits a damping rate which is larger the rate of the σ y . Although this effect iscounterintuitive, it can be explained by statistical fluctuations.Overall, the measurements of the vertical chromatic ratio (cid:112) R y fall inside the expected range, and since the impact ofnon-linear dynamics is not significant for the vertical plane, accurate measurements of the vertical chromaticity arepossible. The findings of the previous section lead to the conclusion that the chromaticity estimations, via the proposed method,depend on the number of turns N of the TbT BPM signal. An immediate conclusion is that one should use theleast number of turns N for the estimation of the chromatic properties of the beam, in order to avoid the effects ofdecoherence. At the same time, the number of turns N must allow for the observation of chromatic sidebands withenough precision i.e. the BPM signal should contain several synchrotron periods. Depending on the exact value ofthe synchrotron tune Q s and the characteristic decoherence time, this is typically achieved at four to six synchrotronperiods τ s = Q s , where τ s = 63 turns. In this section, the goal is to investigate the output of the proposed methodwith respect to different initial excitations. 10 PREPRINT - S
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21, 2020 (a) The evolution of the average horizontal chromatic ratio (cid:104)√ R x (cid:105) for excitations of 1 σ x in blue, 2 σ x in orange, 3 σ x in green, and 4 σ x in red. The theoretical value is shownin black, while the uncertainty around this value shown inlight green.(b) The evolution of the average vertical chromatic ratio (cid:104) (cid:112) R y (cid:105) for excitations of 20 σ y in blue, 30 σ y in orange,40 σ y in green, and 5 0 σ y in red. The theoretical value isshown in black, while the uncertainty around this value dueto statistical errors, is shown in light green. Figure 7: The TbT evolution of the chromatic ratios √ R x (a) and (cid:112) R y (b) for the horizontal and vertical planesrespectively, averaged across all the BPMs of the KARA model, for different initial excitations. The error-bars representone standard deviation from the average values. The theoretical values are shown in black, and the uncertainty due tostatistical errors in the estimation the RMS energy spread, σ δ , is shown in light green. For the current simulations, the chromaticity is estimated by direct application of Eq. 12 and by using a fixed rangeof number of turns N , for the frequency analysis of the TbT data. The value of the synchrotron tune Q s used in theanalysis, corresponds to the reference value i.e. the value which is estimated by the simulation program MAD-X .The number of turns N correspond to a range of three to six synchrotron oscillations τ s . By using the chromaticities Q (cid:48) x calculated from MAD-X as reference values, the absolute normalized error | ∆ Q (cid:48) /Q (cid:48) | x = | Q (cid:48) x − Q (cid:48) x | /Q (cid:48) x ofthe estimated horizontal chromaticity Q (cid:48) x is presented in Fig. 8a, with respect to the initial excitation amplitude x ,in units of the RMS horizontal width σ x . The absolute error of the vertical chromaticity | ∆ Q (cid:48) /Q (cid:48) | y with respect tothe vertical excitation in units of RMS vertical width σ y are shown in Fig. 8b. For both planes, the measurements aregiven as a percentage of the reference value, and they refer to the average values from all the BPMs, with the error-barsrepresenting one standard deviation from the average value.As it has been already discussed, the σ x case exhibits the least accurate chromaticity estimations, with a normalizederror of just below
10 % . Integrating the analysis over a longer number of turns seems to improve the error, but notadequately. However, if the initial excitation amplitude x increases, the value of the chromaticities converge aroundthe expected value with a normalized error | ∆ Q (cid:48) /Q (cid:48) | x = | Q (cid:48) x − Q (cid:48) x | /Q (cid:48) x of below for the σ x excitation anda number of turns N of six synchrotron periods. As a matter of fact, further increase of the excitation amplitudeand appropriate choice of the number of turns N leads to even smaller errors, which correspond to below of thereference value of the horizontal chromaticity Q (cid:48) x . Such a case is presented for the σ x excitation, and for a number of11 PREPRINT - S
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21, 2020 (a) Horizontal chromaticity error with respect to the initialhorizontal excitation of the beam. The error-bars indicateone standard deviation from the average value of all theBPMs.(b) Vertical chromaticity error with respect to the initialvertical excitation of the beam. The error-bars indicate onestandard deviation from the average value of all the BPMs.
Figure 8: Normalized absolute error of the horizontal (a) and vertical (b) chromaticity measurements for the simulationswith the KARA accelerator model. Each chromaticity measurement is the average from all BPMs, and it is shown as apercentage of the reference value against the initial excitation amplitudes of the simulation. The error-bars measure onestandard deviation from the average value. The different colours refer to the number of turns N that were used for themeasurements, which are multiples of the synchrotron period τ s = 63 turns.turns equal to N = 4 τ s . Note that for that particular case, larger number of turns increase the error rapidly due to theeffect of decoherence. Concerning the vertical chromaticity Q (cid:48) y , the estimations are very close to the value provided bythe model, exhibiting values of the normalized error of below . The overall high accuracy is a result of the largervertical chromaticity, with respect to the horizontal plane, and the low impact of amplitude dependent effects as it istestified in the evolution of the chromatic sidebands, which is presented in Fig. 7b. The particular trend of the verticalnormalized error curves can be attributed to statistical fluctuations which change the RMS energy spread σ δ of thebeam’s distribution.For both transverse planes, the spread of the chromaticity estimations across the BPMs of the KARA model is observedto be below the order of . , with respect to the average value. This observation confirms one of the assumptions ofthe method that the chromaticity measurement should be independent of the location in the ring.An immediate conclusion from the results of these simulations is that accurate chromaticity estimations, by using theFourier sidebands on the main betatron lines, are possible, if the RMS energy spread of the beam and the synchrotrontune are known. However, care has to be taken for the choice of initial excitation of the beam, as it is found to play arole in the estimations of chromaticity. Concerning the current simulations, the dependence is more pronounced for thehorizontal plane, where a minimum of the error in the chromaticity estimations can be achieved by a proper choice ofthe number of turns N , and the initial amplitude of the beam.12 PREPRINT - S
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The chromatic beta-beating is defined as the perturbation of the beta function β , with respect to the momentum offset δ .Expanding the beta function up to first order gives β ( δ ) = β + ∂β∂δ δ , (17)where β is the unperturbed beta function. The chromatic beta-beating ∆ ββ is defined as ∆ ββ = 1 β ∂β∂δ , (18)and it can be calculated by integrating along the whole circular accelerator with the relationship [Fartoukh(1999)] ∆ ββ ( s ) = ±
12 sin(2 πQ ) (cid:73) β ( s (cid:48) )[ ∓ K ( s (cid:48) ) (19) ± K ( s (cid:48) ) D x ( s (cid:48) )] cos(2 | φ ( s ) − φ ( s (cid:48) ) |− πQ ) ds (cid:48) , where s is the path length, Q is the betatron tune, K ( s ) and K ( s ) are the quadrupolar and sextupolar integratedmagnetic strengths, D x ( s ) is the horizontal dispersion and φ ( s ) is the phase advance of the betatron oscillations.The experimental measurements of chromatic beta-beating are similar to the measurements of chromatic-ity [Calaga et al. (2010)Calaga, Aiba, Tomas, and Vanbavinckhove]: The RF system is ramped up and down to inducedeviations of the energy of the particles from the reference value, and the response of the beta function is then measuredby other experimental methods. A fit of the beta function measurements to the known RF energy deviations estimatesthe chromatic beta-beating of the lattice.The proposed method in this paper, simply utilizes an initial transverse excitation of the beam, in order to producecoherent betatron oscillations. With knowledge of the chromaticity Q (cid:48) z and the synchrotron tune Q (cid:48) s , the chromaticbeta-beating can be inferred from Eq. (13). Note that, if chromaticity is not known, then the knowledge of the RMSenergy spread σ δ can be used instead. One arrives to this conclusion by substituting Eq. (12) in Eq. (13).Concerning the current simulations, Eq. (13) is used to estimate the chromatic beta-beating, by measuring the chromaticsidebands with PyNAFF , and by using the model values of the synchrotron tune Q s and the chromaticity Q (cid:48) z , where z = x, y . By utilizing the conclusions of Sections 3.2.2, a number of turns N equal to 5 synchrotron periods is used forthe analysis. It is found that, for both planes, there is no significant change in the estimated chromatic beta-beatingwith respect to the initial excitation, however for very large excitations (above 5 σ x for the horizontal plane, andabove 70 σ y for the vertical plane) accurate chromatic beta-beating measurements were not possible due to the impactof non-linearities. The weak dependence on the initial excitation can be explained by the fact that in the chromaticbeta-beating estimations, only the chromatic sidebands A ± are used, while for the chromaticity measurements, themain amplitude A of the beam is needed, which depends strongly on the initial excitation.The estimated horizontal chromatic beta-beating averaged across σ x , σ x , σ x and σ x initial excitations is shownin Fig. 9a, while the vertical chromatic beta-beating, averaged across σ y , σ y , σ y and σ y initial excitations,is shown in Fig. 9b. The light green band around the measurements defines one standard deviation from the average.The response of the model beta functions are obtained by using the PTC module of
MAD-X , and it is superimposed onthe results.It is important to note that the proposed method does not differentiate between negative and positive solutions i.e. thesign of the chromatic beta-beating is ambiguous. However, for the current simulations, the phase of the beta-beatingwave of the model can be introduced in the estimations of the proposed method, by first estimating absolute chromaticbeta-beating with Eq. (13), and then multiplying them with the sign of the chromatic beta-beating of the model. Thisresults in two chromatic beta-beating waves, one wave from the response of the model and one from the estimations ofthe proposed method, that can be easily understood and compared for the current simulations, since they demonstratesimilar phases. In the absence of reference measurements from the model, this operation would be of course impossible,and only absolute values of the chromatic beta-beating could be used.The results from the horizontal plane exhibit a good agreement of with the expected chromatic beta-beating. Themaximum chromatic beta-beating from the simulations is around ∆ β x β x = 10 for the current simulated KARA optics.The good agreement in the results for the horizontal plane is also explained by the presence of horizontal dispersion D x ,13 PREPRINT - S
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21, 2020 (a) Horizontal chromatic beta-beating for the 35 BPMs ofKARA. The model values are superimposed on the results.(b) Vertical chromatic beta-beating for the 35 BPMs ofKARA. The model values are superimposed on the results.
Figure 9: Chromatic beta-beating measurements using the proposed method, for each BPMs of the KARA model.Horizontal chromatic beta-beating is shown in (a), averaged across the results from σ x , σ x , σ x and σ x simulations.Vertical chromatic beta-beating is shown in (b), averaged across the results from the σ y , σ y , σ y and σ y simulations. The light green band across the measurements signifies one standard deviation from the average. Themodel values are shown in red for both planes.as it is testified from Eq. (19). For the same reason, the results from the vertical plane do not agree as well with themodel, since vertical dispersion D y is zero everywhere. For the vertical chromatic beta-beating, the maximum beatingis found at a value of ∆ β y β y = 15 . As it can been for the uncertainty of the measurements for both planes, there is quite agood agreement in the results from the current simulated cases. Note that, as expected from theory, the results from theproposed method return a well defined beta-beating wave, with a frequency of almost twice the corresponding betatrontunes. The proposed method for estimation of chromaticity through TbT data, is tested experimentally at the KARA lightsource. The KARA ring is equipped with M = 35 BPMs, and a total of n b = 110 bunches can be injected in the ring,with a range of nominal beam currents at I b = 90 − mA. At flat-top energy of operation E = 2 . GeV and for theKARA optics, the synchrotron tune is around Q s = 0 . and the measured bunch length for the nominal machinesettings at KARA is σ z = 1 . · − m, which can be used to estimate the RMS energy spread at σ δ = 8 . · − .The TbT data are generated by the use of the injection kicker, which can induce horizontal betatron oscillations. Thevertical excitation is transferred to the beam by virtue of betatron coupling, which will unfortunately result in TbTsignals with a much lower Signal-to-Noise Ratio (SNR). The data are gathered from the M = 35 BPMs for around N = 2000 turns. The kick signal from the magnet lasts for about N = 8 turns or about . ms, and for the currentexperiment, a range of kicker currents from A to
A is used, in order to generate initial excitations of different14
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21, 2020amplitudes. A sample of the experimental horizontal TbT data can be visualised in Fig. 10a, for different magnitudesof the kicker current, which give a maximum RMS oscillation amplitude of (cid:112) (cid:104) x (cid:105) = 0 . m for kicker current of I k = 750 A. The vertical TbT data for the same kicker configurations are shown in Fig. 10b, where the loss of signalpurity is evident from the random spikes arising from the low oscillation amplitudes. At a kicker current of I k = 750 A,the RMS vertical oscillation amplitude (cid:112) (cid:104) y (cid:105) is 10 times smaller than the horizontal one. As a result, larger systematicerrors are expected in the vertical chromaticity measurements. (a) Horizontal TbT data, after excitation of the injectionkicker in a range of currents, which is shown in the legend.(b) Vertical TbT data, generated from betatron coupling,following the horizontal excitation of the beam. Figure 10: Experimental TbT data at KARA light source. The horizontal oscillations are presented in (a) and thevertical oscillations in (b). The TbT data are generated by the KARA injection kicker and the current at the kicker foreach case is shown in the legend of the graphs.
The first chromaticity estimations from TbT data, based on the application of the proposed method for a number of turnsequal to 5 synchrotron periods, is shown in Fig. 11, for the horizontal chromaticity Q (cid:48) x (top) and vertical chromaticity Q (cid:48) y (bottom), with respect to each KARA BPM. The chromaticity estimations from the raw experimental data areshown with blue markers, while the estimations from the same TbT data, but after post-processing with Singular ValueDecomposition (SVD) [Wang(1999)] analysis in order to increase SNR, are shown in orange. Both markers correspondto the average value from 10 injections of bunches, and the error-bars represent one standard deviation from the average.In addition, the method is benchmarked against the traditional method for chromaticity measurements that is usedat KARA ring i.e. with the RF sweep. More specifically, for the generation of the coherent betatron oscillations,the current at the horizontal injection kicker is set to the nominal value ( I k = 450 A), the RF frequency is changedover a range of values, and the betatron tune measurements are performed with the bunch-by-bunch (BBB) feed-backsystem [Hertle et al. (2014)Hertle et al. ] feedback system.For the horizontal chromaticity measurements, it is evident that the proposed method can be used for on-line chromaticitymeasurements, as the estimations agree very well with the RF sweep method. Even for the raw data the agreement isimpressive, while the outliers are absent in the same measurements with the SVD filtered data. The statistical errorfrom 10 consecutive shots is at around while the agreement with the RF sweep method is below for the SVDmeasurements. 15
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21, 2020Figure 11: Horizontal (top) and vertical (bottom) chromaticity at KARA, by using the proposed method (blue andorange) and the traditional method (black line). The blue markers correspond to measurements without noise filteringof the TbT data, while orange ones, to data that have been filtered with the SVD method.Concerning the vertical chromaticity measurements, it is clear that not all BPMs could yield Fourier spectra withdetectable chromatic sidebands. However, for the SVD filtered data, the population of the successful BPMs increasesdue to the improvement of SNR, yielding a relatively good agreement with the RF sweep results. A systematic erroris also present in the measurements, probably coming from the betatron coupling mechanism, which results in theoverestimation of the vertical chromaticity for all the BPMs. In any case, the vertical chromaticity is estimated with anaccuracy of around for the SVD measurements.Similar measurements can be employed, by computing the BPM average with respect to the excitation kicker current, inorder to quantify the effect of the initial excitation in the TbT chromaticity estimation.Such a measurement is graphically presented in Fig. 12a, where in the top plot the average horizontal chromaticityfrom all BPMs is plotted with respect to the kicker current I k , for a number of turns equal to three (blue), four (orange)and five (green) synchrotron periods. The error-bars correspond to one standard deviation from the average value. Thevertical amplitude-dependent chromaticity measurements are shown at the bottom plot for the same number of turns asbefore and with the same colour code as before.Clearly, as the kicker current increases, the horizontal chromaticity estimations converge to a value which agreesvery well with the measurement provided by the RF sweep method, with five synchrotron periods yielding the mostaccurate and precise results. More specifically, for the aforementioned number of turns, the horizontal chromaticitymeasurements converge to the expected value at around I k = 650 A, while at I k = 100 A the error in accuracy isestimated at around . Note that, when the least number of turns is used, the error between the two methods increasesafter the current of the kicker is set to I k = 420 A.The impact of the SNR of TbT beam position signal is visible at the bottom plot, where the vertical chromaticityis only reproducible for an excitation kick of I k ≥ A. The choice of a large number of turns seems to improvethe measurement error, which reduces significantly at I k = 800 A. The reproducibility of the vertical chromaticitymeasurements is lower that the horizontal chromaticity measurements, due to the non-existence of a pure vertical kick.The particular dependence of the accuracy of the proposed method, for an increasing excitation amplitude, is alsoobserved in the simulations of Sec. 3, Fig. 8a and Fig. 8b. As an immediate conclusion, the excitation amplitude shouldbe large enough as to induce chromatic sidebands which can be accurately resolved. Empirical studies show that such aconfiguration would be an excitation amplitude that allows for around − synchrotron periods, before it reaches
50 % of the initial amplitude of the centroid at the BPMs. On the other hand, a limit to the maximum excitation should alsoexist due to non-linearities, but it was not observed until the maximum current of the horizontal kicker.
The KARA light source has been recently selected to commission the new prototype of the Compact Linear Collider(CLIC) [Aicheler et al. (2012)Aicheler, Burrows, Draper, Garvey, Lebrun, Peach, Phinney, Schmickler, Schulte, and Toge]Superconducting (SC) damping wiggler [Bernhard et al. (2016)Bernhard et al. ], which will be used at the DampingRings [Papaphilippou et al. (2012)Papaphilippou, Antoniou, Barnes, Calatroni, Chiggiato, Corsini, Grudiev, Koukovini, Lefevre, Martini, Modena, Mounet, Perin, Renier, Russenschuck, Rumolo, Schoerling, Schulte, Schmickler, Taborelli, Vandoni, Zimmermann, Zisopoulos, Boland, Palmer, Bragin, Levichev, Syniatkin, Zolotarev, Vobly, Korostelev, Vivoli, Belver-Aguilar, Faus-Golfe, Rinolfi, Bernhard, Pivi, Smith, Rassool, and Wootton]of CLIC in order to cool-down the electron and positron beams, i.e. reduce the initially large transverse emittances. Thebasic parameters of the CLIC SC wiggler, which define the linear dynamics, are summarised in Table 2.16
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21, 2020 (a) Horizontal Chroma VS Kick.(b) Vertical Chroma VS Kick.
Figure 12: Estimation of chromaticity at KARA by using the proposed method, with respect to the current of theexcitation kicker. The synchrotron period is τ s = Q s = 90 turns. The measurements are produced from analysis ofseveral synchrotron periods, which correspond to τ s (blue), τ s (blue) and τ s (green). The error-bars indicate onestandard deviation from the average value all the BPMs at KARA. The measurements from the traditional RF sweepmethod are superimposed with black lines.Table 2: Basic parameters of the CLIC Superconducting Damping Wiggler prototype, commissioned at the KARAlight-source Max. on-Axis Magnetic field B w [T] 2.90Period length λ w [mm] . Total length L [m] 1.85Horizontal beta function β x [m] 18.96Vertical beta function β y [m] 2.17 Since an insertion device, as powerful as the CLIC SC wiggler, can potentially influence the beamdynamics of the KARA ring, a series of measurement campaigns have been deployed for the char-acterisation of the linear and non-linear beam dynamics’ response in the presence of the wig-gler [Bernhard et al. (2013)Bernhard, Huttel, Peiffer, Bragin, Mezentsev, Syrovatin, Zolotarev, Ferracin, and Schoerling,Gethmann et al. (2017)Gethmann et al. , Papash et al. (2018)Papash et al. ].The first set of measurements during the commissioning of the CLIC SC wiggler at KARA, focuses on the evaluation ofthe transverse betatron tunes and chromaticities for various magnetic fields of the wiggler.Due to the symmetries of the magnetic field components at the wiggler, it acts as a focusing quadrupole in bothplanes [Venturini(2003)]. For the present study, the following points from theory are taken into consideration:i) A vertical focusing component is expected, due to the existence of a non-vanishing longitudinal field componentat the wiggler, coupled to the trajectory of the beam ("wiggling") along the insertion device. The focusingresults in vertical betatron tune-shift which depends on the period and the peak field of the wiggler.ii) The horizontal focusing in the wiggler is compensated by feed-downs of the sextupolar components. However,a slight horizontal defocusing might be observed, in the case that the feed-downs are larger than the linear17
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21, 2020focusing of the wiggler. A beam which enters the wiggler with a non-vanishing horizontal position, experiencesa gradient in the distribution of the magnetic field at the successive poles of the wiggler. The net effect is aslight horizontal deflection, which can be observed as a horizontal tune-shift. This defocusing depends on theperiod and the peak field of the wiggler.iii) The previous horizontal defocusing highlights the existence of sextupolar components at the wiggler magneticfield. In addition, the vertical focusing of a wiggler can have contributions from higher-order multipolesas well. As a result, the wiggler is expected to alter the chromaticity of the ring in both planes, albeit theeffect will be smaller in the vertical plane. Therefore, chromaticity measurements are also important in thecommissioning of a wiggler.The measurement campaign at KARA consisted of gathering and analysing TbT data, while ramping up the magneticfield of the CLIC SC wiggler, in steps of . T until the maximum field of . T is encountered. The analysisis performed with the
PyNAFF software [Asvesta et al. (2017)Asvesta, Karastathis, and Zisopoulos], for each of the M = 39 BPMs available at KARA, during the measurements. The stored current during the measurement procedure isaround I b = 100 mA. The horizontal excitation is delivered through the horizontal injection kicker, while the verticalexcitation is induced by virtue of betatron coupling. The linear chromaticities are set to low enough values as to assertthe observation of any sextupolar components due to the ramp-up of the wiggler. In order to assess the contribution ofthe CLIC SC wiggler on the linear beam dynamics, experimental measurements of the CLIC SC wiggler tune-shift andbeta-beating are performed, and the results can be inspected in sections 6.1.3 of the Appendix. The dependence of the RMS energy spread σ δ in the magnetic field of a wiggler can be expressed as [Wiedemann(1979)] σ δ = σ δ L w πρ (cid:0) ρ ρ w (cid:1) L w πρ (cid:0) ρ ρ w (cid:1) , (20)where σ δ is the value of the natural RMS energy spread without the wiggler, L w is the length of the wiggler, ρ isthe bending radius of the main dipoles, and ρ w is the bending radius of the wiggler magnet. The previous relationshipdepends on the wiggler magnetic field B w through the conservation of the beam rigidity ρ w = B ρ B w , (21)where B is the magnetic field of main dipoles.The RMS energy spread σ δ of the beam is experimentally measured for each step of the CLIC SC wiggler. Thesynchrotron light diagnostics at KARA allow for the measurement of the RMS bunch length σ z by using a Hamamatsustreak camera [Kehrer et al. (2015)Kehrer et al. ]. The RMS energy spread σ δ is inferred by σ δ = Q s a p R σ z , (22)where Q s is the synchrotron tune, a p is the momentum compaction factor, and R is the radius of the KARA ring.Note that the momentum compaction factor a p depends also on the magnetic field of the wiggler B w , however thecontribution for the CLIC SC wiggler is negligible. The resolution of the streak camera is ∆ σ z = 1 . ps, which definesthe error of a single RMS energy spread measurement to be ∆ σ δ = 1 . · − . For the current value of the RMS energyspread σ δ , the normalised uncertainty of the measurement is around ∆ σ δ /σ δ = 1 . .For the current experimental measurements at KARA, the ratio (cid:0) σ δ /σ δ (cid:1) is estimated for wiggler fields of B w = 0 . T, . T, . T and . T, due to the unavailability of the streak camera for the intermediate steps of B w = 0 . T, . Tand . T. In order to estimate the RMS energy spread in the missing steps, a non-linear fit of the available RMSenergy spread measurements to Eq. (20) is performed. The results are graphically presented in Fig. 13, where the ratio (cid:0) σ δ /σ δ (cid:1) is plotted with respect to the CLIC SC wiggler field B w . The experimental measurements are marked withblue, with the error-bars corresponding to the uncertainty of the streak camera measurements, while the aforementionedfit is shown in orange.From the trend of the measurements, the total increase of the initial RMS energy spread σ δ at B w = 0 . T, is around for the CLIC SC wiggler operating at B w = 2 . T. 18
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21, 2020Figure 13: RMS Energy spread measurements (blue markers) from the KARA streak camera, with respect to themagnetic field of CLIC SC wiggler. The error-bars represent one standard deviation from the average. The fit to thetheoretical model is shown in orange.
The first order chromaticity Q (cid:48) x,y of a circular accelerator with respect to the optics, is given by Q (cid:48) x,y = 14 π (cid:73) C β x,y ( s ) (cid:2) K x,y ( s ) ∓ S ( s ) D x ( s ) (cid:3) ds , (23)where the integration is performed around the circumference of the ring C , β x,y ( s ) is the transverse beta function, K x,y ( s ) is the quadrupolar focusing gradient, S ( s ) is the sextupolar focusing gradient, and D x ( s ) is the dispersiongenerated by horizontal bending in the main dipoles. In the case of a wiggler with peak magnetic field of B w , additionalquadrupolar and sextupolar gradients are superimposed to the magnetic fields of the ring. Furthermore, the wigglercreates additional dispersion, whose average value (cid:104) D w (cid:105) scales linearly with the magnetic field of the wiggler B w as [Walker(1994)] (cid:104) D w (cid:105) = − B ρ k B w , (24)where k = πλ w , λ w is the period of the wiggler, and B ρ is the magnetic rigidity of the ring. For the KARA ring (cid:104) D w (cid:105) /B w ∝ − m/T, which gives negligible dispersion at the maximum field of B w = 2 . T, compared to theaverage dispersion from the main dipoles at the wiggler region (cid:104) D (cid:105) ∝ − m.In order to establish the presence of sextupolar components in the location of the CLIC SC wiggler, dedicatedchromaticity measurements are performed by recording TbT data for around N = 2000 turns, while ramping up theCLIC SC wiggler from T to . T in steps of . T. The analysis is performed with
PyNAFF by using two methods:i)
RF-sweep : During each step, the RF frequency is modulated in order to induce a change in the relativemomentum offset of the beam δ = ∆ p/p in the range of | δ | ≤ . . Chromaticity is estimated from thechromatic response of the betatron tunes for each value of the wiggler field B w .ii) Chromatic sidebands : The suggested method for chromaticity measurements is benchmarked against theRF-sweep method. For this method, chromaticity is inferred by inspecting the Fourier spectra of the beamand applying Eq. (12), where the RMS energy spread σ δ estimations are known from previous analysis, seeSec. 4.3.1. The measurements of the chromatic sidebands are performed for around 4 to 5 synchrotron periods.It should be mentioned that the analysis of the chromatic sidebands is performed on TbT data that are gatheredwhile the beam is on the nominal chromatic orbit i.e. for δ = 0 . For each step of CLIC SC wiggler, three setsof data, where the beam follows the nominal chromatic orbit, are available. This results in fewer statistics thanthe RF-sweep method, which uses all the available data.During the initial set-up of the experiment, transverse chromaticity is trimmed to low-enough values ( Q (cid:48) x ≈ , Q (cid:48) y ≈ )by reducing the strength of the lattice sextupoles, in order to observe the effect of the CLIC wiggler. These values arethe lower limit that ensure beam stability, since the BBB feedback is not employed during the measurements to avoiddistortion of the Fourier spectra.The results for the chromaticity measurements with respect to the field of the CLIC SC wiggler B w are shown inFig. 14a for the horizontal chromaticity Q (cid:48) x and in Fig. 14b for the vertical chromaticity Q (cid:48) y . The estimations with the19 PREPRINT - S
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21, 2020traditional RF-sweep method are shown in orange and they correspond to the average chromaticity, from all the M = 39 KARA BPMs. The standard error of the mean is below − for both planes, due to the very good Signal-to-NoiseRatio (SNR) and the excellent reproducibility of the betatron tunes for each bunch injection. The output from theanalysis of the chromatic sidebands with the suggested method are shown in blue, where each marker corresponds to theaverage value from all the BPMs, and for the three available data-sets, while the error-bars correspond to the standarderror of the mean, whereas the uncertainty of the RMS energy spread σ δ measurements has been added in quadrature. (a) Horizontal chromaticity, with respect to the magneticfield of the CLIC SC wiggler.(b) Vertical chromaticity, with respect to the magnetic fieldof the CLIC SC wiggler. Figure 14: Experimental measurements of the horizontal (a) and vertical (b) chromaticities by using the proposedmethod (blue markers) and the standard method of RF sweeping (orange markers). The values from the RF-sweepmethod correspond to the average from all the 39 BPMs at KARA. The respective measurements of the proposedmethod quote the average value from all the 39 BPMs and the three available data sets. The error-bars indicate thestandard error of the average, where the uncertainty in the RMS energy spread is added in quadrature.Concerning the horizontal chromaticity, both methods agree very well, within the margin of error, and both signify thepresence of sextupolar components in the CLIC SC wiggler, revealing a positive correlation with the magnetic field ofthe wiggler. The error-bars for the suggested method is of the order of − . The trend of the measurements suggeststwo different regions of horizontal chromaticity increase: a small increase of about from T to T, and a moresteep, almost linear increase of about from T to . T, which slightly drops at the final step of B w = 2 . T. Thedrop is reported from both methods, and it is possible that ramping the wiggler at the maximum field of B w = 2 . T, cangenerate additional non-linearities which can slightly perturb the optics. A similar behaviour, albeit of less significance,is observed in the horizontal betatron tune measurements at the top of Fig. 16, where the measurement at B w = 2 . Texhibits a slight focusing with respect to the tunes measurements at lower fields. Note that magnetic quenches forthe CLIC SC wiggler prototype have been reported in [Bernhard et al. (2016)Bernhard et al. ] for higher values of themagnetic field B w , limiting the stable operating region to ≤ B w ≤ . T.As for the vertical plane, there is also a good agreement between the two methods, by considering also the technique forgenerating the vertical TbT through coupling, which results in a weak vertical TbT signal at the BPMs. Note that ingeneral the uncertainty of the proposed method is smaller than the uncertainty in the horizontal plane, due to the largervertical chromaticity. However, for some steps of the wiggler field B w , the beam measurements were observed to beless reproducible in the vertical plane, probably due to the simultaneous ramp up and down of the RF system, duringthe continuous ramping of the wiggler. 20 PREPRINT - S
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21, 2020In any case, the vertical chromaticity measurements reveal a pattern similar to the one observed in the horizontalchromaticity measurements. A pronounced difference is that from T to . T both methods report a drop in thechromaticity, which would be expected from the simultaneous increase of the horizontal chromaticity. However,from T to . T the vertical chromaticity increases, leading to the conclusion that the distribution of the sextupolarcomponents of the CLIC SC wiggler changes with respect the magnetic field. Nevertheless, the effect is very small, forthe current optics, and it should not pose a problem for the operation of the KARA ring with the CLIC SC wiggler atmaximum field.
During the experimental measurements at the KARA light source, the influence of the CLICK SC wiggler on secondorder chromaticity Q (cid:48)(cid:48) z and chromatic beta-beating ∆ β z β z is determined, by employing the proposed method, shown inEq. 13, for the latter, and by using the RF-sweep method for the former. Similar to the chromaticity measurementsin Sec. 4.3.2, the field of the wiggler is ramped up from 0 T to 2.9 T, in steps of 0.5 T. Second order chromaticity ismeasured by the response of the centroid of the beam to ramping up and down the frequency of the RF cavities, whilethe chromatic beta-beating is measured after analyzing for the first 5 synchrotron periods the TbT BPM data, for eachvalue of the magnetic field of the CLIC SC wiggler, in order to estimate the amplitude of the chromatic sidebands, A ± .The values of chromaticity Q (cid:48) z , measured with the suggested method, are used in Eq. 13.Excluding second order dispersion effects, the chromatic beta-beating ∆ β z β z and the second order chromaticity Q (cid:48)(cid:48) z arecoupled together through the relationship [Luo et al. (2010)Luo, Fischer, Robert-Demolaize, Tepikian, and Trbojevic]: Q (cid:48)(cid:48) x,y = − Q (cid:48) z + 14 π (cid:73) ds [ ∓ K + K D x ]∆ β x,y , (25)where ∆ β x,y = ∂β x,y ∂δ is the chromatic dependence of the horizontal and vertical beta functions, K , K are theintegrated magnetic strength of the quadrupoles and sextupoles respectively, and D x is the horizontal dispersion alongthe ring. Assuming that the magnetic strengths are static, one concludes that an increase of the horizontal beta-beatingtranslates in a increase or decrease of the second order chromaticity, according to the plane of reference.The measurements for the horizontal plane are presented in Fig. 15a, while the measurements for the vertical planeare shown in Fig 15b. Both figures illustrate the evolution of the chromatic beta-beating ∆ β z β z , measured with thesuggested method, on the left axis (green color), while the dependence of the second order chromaticity Q (cid:48)(cid:48) z on the fieldof the wiggler is shown on the right axis (red). For both measurements and for both planes, the points are the averagemeasurements, across 3 data sets and 39 BPMs of KARA, and the error-bars represent the standard error of the mean.The normalized error for the horizontal chromatic beta-beating is at the range of σ ∆ β y / ∆ β y ≈
10 % , while for thevertical plane, the same parameter is σ ∆ β y / ∆ β y ≈ . . The small error in the vertical plane is explained by the factthat the vertical chromaticity is on average 3 times larger than the horizontal one, which results in more resolvablechromatic sidebands. In addition, the vertical data are produced through the betatron coupling mechanism, which meansthat finite vertical dispersion is generated in the sextupoles and thus chromatic beta-beating is estimated with moreconfidence. The errors of the second chromaticity measurements are negligible in both planes, due to the very goodreproducibility of the TbT data.The trend of the curves for the horizontal plane, exhibits an increase of the chromatic beta-beating of around
100 % with respect to the nominal value at B w = 0 T, followed by an increase of the second order chromaticity for around . Note that from B w = 1 T to B w = 2 . T, chromatic beta-beating appears to slow down in growth, while the lastpoint at B w = 3 T can be explained by non-linearities, as explained in Sec. 4.3.2.The results of the vertical indicate that, vertical chromatic beta-beating is resolved with higher confidence by using theproposed method, and it is found to be also increasing with respect to the CLIC SC magnetic field B w , with a totalchange of around
50 % , which is half the increase of the horizontal chromatic beta-beating ∆ β x β x . However, the growthin the vertical chromatic beta-beating is followed by a decrease in the vertical second order chromaticity Q (cid:48)(cid:48) y of about , two times larger than the change in the horizontal second order chromaticity Q (cid:48)(cid:48) x .Note that the behaviour of second order chromaticity and chromatic beta-beating is fully expected from theory, as it isjustified from Eq. (25). An immediate conclusion is that, during experimental measurement, where the reproducibility ofthe beam is at an adequate level, the chromatic beta-beating can be estimated with the proposed method. Unfortunately,there are no reference measurements in order to compare the results, neither from experiment nor from the model. Inany case, the increase of the chromatic beta-beating is another indication of the existence of additional sextupolar fieldsfrom the CLIC SC wiggler, which however do not seem to complicate the operation of the KARA ring.21 PREPRINT - S
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21, 2020 (a) Left axis (green): Horizontal chromatic beta-beating,with respect to the magnetic field of the CLIC SC wiggler.Right axis (red): Horizontal second order chromaticity mea-surement, for the same values of the magnetic field of theCLIC SC wiggler.(b) Left axis (green): Vertical chromatic beta-beating, withrespect to the magnetic field of the CLIC SC wiggler. Rightaxis (red): Vertical second order chromaticity measurement,for the same values of the magnetic field of the CLIC SCwiggler.
Figure 15: Correlation of the average chromatic beta-beating (left axis in green), during the ramping of the CLIC SCwiggler, and the corresponding change in second order chromaticity (right axis in red) for the horizontal (a) and vertical(b) planes. The error-bars indicate the standard error from the average, sampled across the KARA BPMs and for threedata-sets.
Two simple equations are proved and proposed, which can be used to estimate linear chromaticity and chromaticbeta-beating, directly from the Fourier spectra of the TbT beam position of an electron beam, in a similar manner tothe procedure of measuring the betatron tunes i.e. with a transverse excitation. Such a possibility would add furtherflexibility to the effort for continuous on-line measurements and control of the beam optics. By using the same method,the RMS energy spread can be also estimated in a TbT manner.One of the most important assumptions of the methods is that the distribution of the electron beam is Gaussian, whichis almost always the case for electron beams in high-energy circular accelerators, and that the initial excitation of thebeam is not too strong as to generate non-linearities which can possibly affect the efficiency of the proposed methods.Tracking simulations are deployed with the KARA model, where the chromaticity is estimated with the proposedtechnique. The efficiency of the method is demonstrated, as it shown that chromaticity can indeed be recovered fromTbT data. More specifically, the results indicate that two parameters can be used to fine-tune the estimations: the initialexcitation amplitude and the number of turns that is used for the frequency analysis. Since decoherence is found to playan important role in the final result, the number of turns should be as small as to allow the generation of synchrotronsidebands in the transverse Fourier spectra. For a powerful frequency analysis tool like
NAFF , this is usually achievedin four to six synchrotron periods.Similar behaviour is found for the chromatic beta-beating measurements, as it is found that it can be fully recovered inthe simulations, via the proposed method. For these particular simulations, it is found that the presence of dispersion can22
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21, 2020increase the confidence in the measurements. This is to be expected, since dispersion at the location of the sextupoles isone of the main contributors to the generation of chromatic beta-beating.The method is also deployed in experimental measurements at the KARA light source, where horizontal excitations areapplied to the beam, and the vertical TbT data are generated from betatron coupling. The concept of using the chromaticsidebands for chromaticity estimations is demonstrated with success, and the importance of the signal-to-noise ratio inthe beam position signal is highlighted. This is also reflected from the fact that the vertical chromaticity measurements,which are not produced by direct excitation, are less precise and accurate than the horizontal ones. Nevertheless, animportant outcome is that the reduction of noise with numerical methods such as the SVD, leads to more precise results.In another series of experiments, the method is utilized for the measurement of chromaticity under the influence of theCLIC Superconducting Damping Wiggler. In addition, the impact of the CLIC wiggler on the KARA beam dynamics ischaracterised qualitatively and quantitatively. The most important outcome of these measurements, is the demonstrationof the existence of non-linear fields at the position of the wiggler. The influence of these fields on chromaticity ispresented, by using the traditional RF-sweep method and measurements with the proposed methodology. In bothtransverse planes, the suggested method is benchmarked with success against the traditional technique, and report asimultaneous increase of both transverse chromaticities. The magnitude of the increase however is not severe, and as aresult, it is concluded that the operation of the CLIC wiggler at maximum, does not have an important effect on theKARA beam dynamics.Moreover, the proposed method is employed in order to estimate chromatic beta-beating and demonstrate its relationshipto the wiggler field. At the same time, measurements of the second order chromaticity are performed by using theRF-sweep method. The increase in horizontal chromatic beta-beating is followed by an increase of second orderchromaticity. Vertical chromatic beta-beating is found also to be increasing, which leads to the decrease of verticalsecond order chromaticity. Both chromaticity and chromatic beta-beating measurements with the proposed method,suggest that additional sextupolar fields are generated during operation of the CLIC SC wiggler, which however do notpose a problem for the operation of the KARA ring.Finally, as a future work, a similar methodology could be also developed for proton machines, where the on-linemonitoring and control of chromaticity and/or RMS energy spread is of high importance as well.
The linear Q (cid:48) and non-linear Q (cid:48)(cid:48) chromaticities are defined as the betatron tune-shift of a single particle, due to achange of the particle’s energy. The energy dependent betatron tune-shift is defined from the well known formula Q ( δ ) = Q + Q (cid:48) δ + 12 Q (cid:48)(cid:48) δ + O ( δ ) , (26)where δ = ∆ pp with ∆ p the deviation from the reference momentum p , Q ( δ ) is the energy dependent betatron tune,and Q is the unperturbed betatron tune, defined by the lattice. By using the formulas: Q (cid:48) z = ± Q s σ δ (cid:114) A + A − A (27) ∆ β z β z = ± Q (cid:48) z Q s (cid:12)(cid:12)(cid:12)(cid:12) A − A − A + A − (cid:12)(cid:12)(cid:12)(cid:12) (28)where z=x, y the horizontal and vertical planes respectively, Q (cid:48) z the chromaticity, Q s the synchrotron tune, σ δ the RMSenergy spread, A the amplitude of the main betatron line, and A ± the chromatic sidebands that appear around A ,one can estimate the chromaticity Q (cid:48) z and chromatic beta-beating ∆ β z β z , as it is suggested in this paper. In the followinganalysis, new symbols are introduced for the chromatic beta-beating and the chromatic ratios as:23 PREPRINT - S
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21, 2020 ∆ β z β z ≡ ∆ β (29) (cid:114) A + A − A ≡ R (30) (cid:12)(cid:12)(cid:12)(cid:12) A − A − A + A − (cid:12)(cid:12)(cid:12)(cid:12) ≡ R (31)for reasons of convenience. In addition the index of the chromaticity is dropped and it is thus symbolized as Q (cid:48) , sincethe expressions are valid for both planes.The measurement errors of chromaticity σ Q z and chromatic beta-beating σ ∆ β are simply given by: σ Q (cid:48) = (cid:18) ∂Q (cid:48) ∂Q s σ Q s (cid:19) + (cid:18) ∂Q (cid:48) ∂σ δ σ σ δ (cid:19) + (cid:18) ∂Q (cid:48) ∂R σ R (cid:19) (32) σ β = (cid:18) ∂ ∆ β∂Q s σ Q s (cid:19) + (cid:18) ∂ ∆ β∂Q (cid:48) σ Q (cid:48) (cid:19) + (cid:18) ∂ ∆ β∂R σ R (cid:19) (33)where σ Q s is the error in the synchrotron tune Q s measurement, σ σ δ the error in the RMS energy spread σ δ measurement, σ R the error of the chromatic ratio R , and σ R the error of the chromatic ratio R .By computing the partial derivatives that are present in Eq. (32) as: ∂Q (cid:48) ∂Q s = R σ δ = Q (cid:48) Q s (34) ∂Q (cid:48) ∂σ δ = − R Q s σ δ = − Q (cid:48) σ δ (35) ∂Q (cid:48) ∂R = Q s σ δ = Q (cid:48) R (36)and continuing with the derivatives in Eq. (33) as: ∂ ∆ β∂Q s = − Q (cid:48) R Q s = − ∆ βQ s (37) ∂ ∆ β∂Q (cid:48) = R Q s = ∆ βQ (cid:48) (38) ∂ ∆ β∂R = Q (cid:48) Q s = ∆ βR (39)results in the necessary expressions to calculate Eq. (32) and Eq. (33).However, an intermediate step is the calculation of the errors of the chromatic sidebands σ R and σ R which is given bythe expressions: σ R = (cid:18) ∂R ∂A σ A (cid:19) + (cid:18) ∂R ∂A σ A (cid:19) + (cid:18) ∂R ∂A − σ A − (cid:19) (40) σ R = (cid:18) ∂R ∂A σ A (cid:19) + (cid:18) ∂R ∂A − σ A − (cid:19) (41)where σ A , σ A and σ A − are the measurement errors of the amplitudes A , A and A − respectively. With no loss ofgenerality, the errors in measuring A and A − , which mostly come from the limitations of the BPM system and thenoise in the TbT signal, can be considered similar i.e. σ A − ≈ σ A ≡ σ A ± . By computing the partial derivatives:24 PREPRINT - S
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21, 2020 ∂R A = − R A (42) ∂R A = ∂R A − = 12 A R (43) ∂R A = 2 A − ( A + A − ) (44) ∂R A − = 2 A ( A + A − ) , (45)the errors σ R and σ R in Eq. (40) and Eq. (41) are calculated as: σ R = R A σ A + 12 A R σ A ± (46) σ R = 4 A + A − (cid:0) A + A − (cid:1) σ A ± (47) = 4 (cid:18) σ A ± A + A − (cid:19) (cid:20) − A A − ( A + A − ) (cid:21) . (48)Concretely, the normalized errors in the chromaticity σ Q (cid:48) , and chromatic beta-beating σ ∆ β measurements are found tobe: (cid:18) σ Q (cid:48) Q (cid:48) (cid:19) = (cid:18) σ Q s Q s (cid:19) + (cid:18) σ σ δ σ δ (cid:19) +14 (cid:18) σ A A (cid:19) + 12 (cid:18) σ A ± A + A − (cid:19) (49) (cid:18) σ ∆ β ∆ β (cid:19) = (cid:18) σ Q s Q s (cid:19) + (cid:18) σ Q (cid:48) Q (cid:48) (cid:19) +4 (cid:20) − A A − ( A + A − ) (cid:21)(cid:18) σ A ± A − A − (cid:19) . (50)The quadratic terms in the previous expressions testify that, in the presence of measurement errors of the Fourieramplitudes A , A and A − , the chromaticity error σ Q (cid:48) increases for a vanishing betatron amplitude A and a smallsum of the synchrotron sidebands A and A − . Since the amplitude of the synchrotron sidebands depends on thequantity s = Q (cid:48) σ δ Q s , as it is shown in Sec. 2, therefore it cannot be altered except by changing the beam dynamicsparameters, one could experimentally use a sufficiently large excitation in order to increase the A term. However, carehas to be taken as to be influenced by non-linearities as less as possible, since they might perplex the results.Moreover, it is evident that similar in amplitude synchrotron sidebands, A and A − , penalize the error σ ∆ β of thechromatic beta-beating ∆ β measurement, since this would mean that the chromatic beta-beating is itself very small.Finally, as it is also found for the chromaticity measurement error, the error σ ∆ β of the chromatic beta-beating increasesfor a small sum of amplitudes A and A − . The 3-D Hamiltonian that describes the motion of an on-momentum particle in the field of a wiggler, with sinusoidalfield variation, and expanded up to fourth order is [Gao(2003)]25
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21, 2020 H ( x, y, p x , p y , s ) = p x + p y k ρ w ( k x x + k y y )+ 112 k ρ w ( k x x + k y y + 3 k k x x y ) − sin( ks )2 kρ w ( p x ( k x x + k y y ) − k x p y xy ) , (51)where x refers to the horizontal plane, y to the vertical plane, k x + k y = k = ( πλ w ) is the wave-number of the wigglerwith λ w the wiggler period, ρ w = B ρ B w the bending radius of the wiggler with peak field B w in a ring of magneticrigidity B ρ , and s is is the parameter that describes the azimuthal motion of the beam. Note that in the case of idealwiggler with magnet poles of infinite length, k x → and the focusing of the particle is entirely on the vertical plane.Inspection of Hamiltonian H , leads to the conclusion that a particle encounters focusing forces inside the wiggler whichscale with ρ − w ∝ B w . Due to non-linearities, which arise from the geometry of the wiggler and the initial conditions ofthe beam, feed-down of the octupoles can generate sextupolar fields, which can have an effect on the beam dynamics.In the linear regime, the most important contributions come from the quadrupolar terms of Eq. (51), where afterdifferentiation and averaging over the whole length of the wiggler, the additional quadrupolar magnetic strength ∆ K z per unit cell of the wiggler is ∆ K z ≈ − (cid:18) k z kB ρ (cid:19) B w , (52)with z = x, y the horizontal and vertical plane respectively.Tune-shiftThe theoretical tune-shift ∆ Q z ( B w ) due to the presence of the wiggler field B w , which induces the extra quadrupolarcomponent in Eq. (52), is ∆ Q z ( B w ) = 14 π (cid:90) L w / − L w / β z ( s )∆ K z ds = L w (cid:104) β z (cid:105) π (cid:18) k z kB ρ (cid:19) B w , (53)where z = x, y the horizontal or vertical transverse planes. For an ideal wiggler, the focusing is purely vertical ( z = y )since k x = 0 , and due to the relationship k y = k , Eq. (53) becomes ∆ Q y ( B w ) = L w (cid:104) β y (cid:105) πB ρ B w (54)Beta-beatingThe presence of a wiggler in a circular accelerator induces perturbation to the linear optics in the form of beta-beatingdue to the wiggler generated quadrupolar error, Eq. (52). In theory, only vertical perturbations are allowed in the idealcase. The theoretical average vertical beta-beating (cid:10) ∆ ββ (cid:11) W due to the presence of the wiggler is [Walker(1994)] (cid:28) ∆ β y β y (cid:29) W = 2 π ∆ Q y sin(2 πQ y ) , (55)where Q y is the unperturbed betatron tune, and ∆ Q y is the tune shift due to the finite wiggler field, given in Eq. (54). In the following sections, the results from optics measurements at the KARA ring, under the influence of the CLIC SCwiggler are presented in order to establish quantitatively and qualitatively the impact of the wiggler on the linear beam26
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21, 2020dynamics at KARA. The agreement between experiment and theory is also examined, in order to validate the efficiencyof the CLIC SC wiggler models that are currently used.
For the KARA ring and the CLIC SC wiggler, the expected vertical betatron tune-shift ∆ Q y ( B w ) , by virtue of Eq. (55)iscalculated to be ∆ Q y ( B w ) = 2 . · − B w , (56)which is indeed small due to the small vertical beta function at the position of the CLIC SC wiggler. In or-der to experimentally measure the betatron tune-shift, tune measurements are performed with the mixed BPMmethod [Zisopoulos et al. (2019)Zisopoulos, Papaphilippou, and Laskar] for around N = 50 turns. The results arepresented in Fig. 16, for the horizontal (top) and vertical (bottom) tunes with respect to the increasing magnetic field B w of the CLIC SC wiggler.Figure 16: Betatron tune-shift in the horizontal (top) and vertical (bottom) planes at KARA, as a function of themagnetic field of the CLIC SC wiggler. The fits are performed with quadratic models. For the vertical plane, thetheoretical tune-shift is shown in black.For the evolution of the horizontal tune, a slight shift is observed, with a magnitude that is estimated with a fit of themeasured tunes (blue curve) to Eq. (53), (orange curve). Although the observed tune-shift, is not dangerous for theoperation and beam quality at KARA, it proves the existence of non-linear fields at the location of the CLIC SC wiggler.Concerning the vertical tune-shift, a similar fit is performed (black curve) in order to estimate its magnitude, which isfound similar to the theoretical expectations (blue line), quoted in Eq. (56).The tune measurements reveal a total normalised tune-shift of around ∆ Q x /Q x = 0 . for the horizontal plane, androughly ∆ Q y /Q y = 2% , for the wiggler at maximum field i.e. B w = 2 . T. For the current experimental measurements with the operation of the CLIC SC wiggler, TbT data are recordedat the M = 39 BPMs, while ramping up the CLIC SC wiggler in a series of steps, from T to . T . Theestimations of the beta-beating are performed by using information on the Fourier amplitudes of the oscilla-tions [Zisopoulos et al. (2013)Zisopoulos, Papaphilippou, Streun, and Ziemann], measured with PyNAFF , for no wig-gler field ( B w = 0 T) and maximum wiggler field ( B w = 2 . T).In order to estimate the beta-beating at the maximum field, first the models are set-up. The CLIC SC wiggler is insertedin the KARA model in
ELEGANT [Borland(2000)] tracking code, and the response of the beta function is recorded forboth values of the field. The beta-beating of the model due to the maximum field of the wiggler from the model can becalculated as ∆ ββ model = β mW − β m β m (cid:12)(cid:12)(cid:12)(cid:12) W =2 . T , (57)27 PREPRINT - S
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21, 2020 (a) Horizontal beta-beating induced by the CLIC SC wigglerat maximum field.(b) Vertical beta-beating induced by the CLIC SC wigglerat maximum field.
Figure 17: Comparison of the experimental and model beta-beating, induced by the maximum of the CLIC SC wiggler,with respect to the azimuthal position of the BPMs. The horizontal beta-beating is shown in (a), for the model (green)and experimental measurements (blue). The results for the vertical plane are shown in (b), with the model estimationshown in green, and the experimental measurements in red. The azimuthal position of the CLIC SC wiggler is markedwith a black line.where β m is the value of the model beta-function for B w = 0 T, and β mW is the value of the model beta-function for B w = 2 . T.In order to compare with the experimental estimation of the beta-beating generated purely by the wiggler, one needsfirst to disentangle the baseline beta-beating, which originates from the quadrupolar errors in the lattice. The amplitudesof the beam for both cases of B w = 0 T and B w = 2 . T are fitted with the model’s beta functions at B w = 0 T. Fromthis operation, two sets of model dependent, but experimentally measured, beta functions become available, at bothfields. Then the experimental beta-beating is found as the difference of the two sets of beta functions, in a mannersimilar to Eq. (57). The difference cancels out the term of the beta-beating due to the quadrupolar errors of the lattice.The previous considerations can be visualized in Fig. 17a for the horizontal plane, and in Fig. 17b for the vertical plane,where the estimated wiggler dependent beta-beating is plotted with respect to the azimuthal position of the BPMs. Inboth plots, the position of the CLIC SC wiggler is marked with a black line.The model horizontal beta-beating (green curve) is vanishing as expected, with the experimental measurements (bluecurve) agreeing well. A portion of the measurements above s = 60 m, exhibit an irregular pattern of about RMSvalue, which is likely coming from calibration issues of these particular BPMs.Concerning the vertical beta-beating, the agreement between the model (green curve) and the experimental measurements(red curve) is very good. Some deviations are present but they are attributed to the calibration of the BPMs. Theexperimentally measured vertical beta-beating is less than
15 % , with an average value of
12 % in agreement to thetheoretical predictions of Eq. (55). As expected, due to the small value of the vertical beta-function at the position ofthe CLIC SC wiggler constraints the vertical beta-beating in relatively low values, which are of no concern for beamstability at KARA. 28
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