Baboi, N.

Eddy, N.

Frisch, J.

Hensler, O.

May, J.

McCormick, D.

Molloy, S.

Petrosyan, L.

Piccoli, L.

Rechenmacher, R.

Smith, T.

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Measurement of the beam’s trajectory using the Higher Order Modes itgenerates in a superconducting accelerating cavity.∗S. Molloy† , J. Frisch, J. May, D. McCormick, T. Smith, SLAC, Menlo Park, CaliforniaN. Baboi, O. Hensler, L. Petrosyan, DESY, HamburgN. Eddy, L. Piccoli, R. Rechenmacher, M. Ross, M. Wendt, Fermilab, Batavia, IllinoisAbstractIt is well known that an electron beam excites HigherOrder Modes (HOMs) as it passes through an acceleratingcavity [1]. The properties of the excited signal depend notonly on the cavity geometry, but on the charge and trajec-tory of the beam. It is, therefore, possible to use these sig-nals as a monitor of the beam’s position. Electronics wereinstalled on all forty cavities present in the FLASH [2] linacin DESY. These electronics filter out a mode known to havea strong dependence on the beam’s position, and mix thisdown to a frequency suitable for digitisation. An analysistechnique based on Singular Value Decomposition (SVD)was developed to calculate the beam’s trajectory from theoutput of the electronics. The entire system has been inte-grated into the FLASH control system.INTRODUCTIONAn electron beam moving through a superconducting ac-celerating cavity will produce an electromagnetic ‘wake’.Short range fields produced by the head of the bunch mayact on the tail, thereby degrading the quality of the beam,and longer range fields may act on subsequent bunches ina way that reduces their quality. In the worst case, thesefields may be resonant with the inter-bunch spacing, caus-ing beam blow up (BBU). This paper address the possibil-ity of using these long range fields for diagnostic purposes.The wake fields can be expanded as a multipole series,and, due to the cyclindrical symmetry of the acceleratingcavities, each term in this series may be classified accord-ing to its azimuthal symmetry as being monopole, dipole,quadrupole, or higher order in nature. Each of these termsis known as a higher order mode (HOM).Since HOMs are potentially damaging to the qualityof the beam, their effect is reduced by careful design ofthe cavity geometry, and by inclusion of HOM couplerports. There are designed as wideband devices to extractthe power of the HOMs, but with a tunable bandstop filterto minimise the coupling to the accelerating mode.Since the amplitude and phase of the HOMs are deter-mined by the trajectory of the beam, high resolution beamposition information may be calculated from an analysisof these modes. This paper describes an experiment at∗Work supported by US Department of Energy Contract DE-AC02-76SF00515, and by the European Community FP6 ”Structuring the Eu-ropean Research Area” programme (CARE, contract number RII3-CT-2003-506395)† smolloy@slac.stanford.eduthe FLASH facility, DESY, designed to measure the 4-dimensional transverse position of the beam (x, x′, y, y′)using dedicated electronics to measure a particular dipolemode. The electronics and the algorithm used to extractthe position information are described, as well at the exper-imental technique used to calibrate the signal. The resolu-tion of the HOM beam position monitor (HOM BPM) ismeasured and compared with the theoretical limit.HIGHER ORDER MODESThe coupling of the beam to the different modes is gov-erned by the the loss factor, k(m),where the integer, m, in-dicates the order of the mode, with m = 0 indicating amonopole mode, m = 1 dipole, etc. The loss factor is de-fined as follows,km (r) ≡∣∣∣V(m)2L∣∣∣4U (m)(1)The amplitude of the longitudinal field of a dipole mode,V(1)L, goes in proportion to r, therefore equation (1) impliesthat the energy coupled into the mode goes as r2. Thus theamplitude of the measured mode will be proportional to r,as well as the bunch charge.In the case where the beam enters the cavity above centreof a mode, and exits by the same amount below, the signalexcited at the end of the cavity will tend to cancel out thatgenerated at the beginning (due to the fact that modes withstrong coupling tend to have a phase velocity very close tothe velocity of the beam). There will, however, be someresidual left from this ‘tilt-signal’ due to the angled trajec-tory entering and leaving the individual cells at differentdisplacements from the mode centre. This signal may beapproximated as the derivative of the offset signal, and will,therefore, have a pi2 phase difference from the main offsetsignal.Dipole modes have two orthogonal polarisations whomay or may not be degenerate in frequency. The alignmentof the polarisation axis is not necessarily coincident withthe laboratory x and y axis, and will be strongly affectedby the position of the HOM couplers, as well as cavity im-perfections.It is, therefore possible to define four defining parame-ters of an individual HOM signal – the amplitude and phaseof each of the two polarisations – that correlate to the fourdegrees of freedom of the beam’s trajectory – x, x′, y, andy′.FERMILAB-CONF-07-388-AD-CDHOM ELECTRONICSThe FLASH cavities have been simulated, and it hasbeen calculated that there is a dipole mode at a frequency of∼1.7 GHz that couples strongly to the beam [3]. Electron-ics were designed to strongly bandpass filter around thisfrequency, downmix to a lower frequency, and digitise theoutput. Figure 1 shows a simple block diagram of the elec-tronics. For more details refer to [4]. Figure 2 shows anexample of a digitised pulse.Figure 1: Block diagram of the HOM mixing electronics.Figure 2: Example of the output of the HOM electronicsfor a single bunch beam.The FLASH linac consisted of five accelerating mod-ules, known as ACC1, ACC2, etc., each of which consistedof eight accelerating cavities. As shown in figure 3, thebeam’s position was controlled by two horizontal and twovertical steerers upstream of the module under investiga-tion, and the trajectory of each pulse was interpolated fromup- and down-stream BPMs.Figure 3: Cartoon of experimental setup showing two x andtwo y steerers upstream of a module, and the BPMs usedto interpolate the position at each cavity.CALIBRATION AND POSITIONMEASUREMENT ALGORITHMA possible analysis algorithm might involve calculatingthe matrix, M , that converts the amplitude, Ai, and phase,φi, of each polarisation, i (where i = 1, 2), to the 4D po-sition. This requires knowledge of the different centre fre-quencies of each of the polarisations for each of the cavi-ties. With the added complication that many of the modeswill be almost exactly degenerate, this makes it quite dif-ficult to find these frequencies for all forty cavities. Analternative analysis method based on singular value decom-position (SVD) was developed.SVD was used to calculate the eigenvectors of the HOMoutput along with their eigenvalues. The SVD input wasa n × j matrix, where n is the number of machine pulsesmeasured (≥100), and j is the number of discrete samplesrecorded by the digitiser for each pulse (∼1500). Since onepolarisation has zero coupling to one HOM port, the infor-mation from both couplers should be concatenated into onedata string, as shown in figure 4. Therefore, j will be twicethe length of a single digitiser output. Figure 5 shows theeight strongest basis vectors in the data set, along with theirassociated eigenvectors, σ.0 500 1000 1500 2000 2500 3000−2000−1000010002000Digitisation pointAmplitude (bits)Signal from upstreamHOM couplerSignal from downstreamHOM couplerFigure 4: HOM signals from both couplers concatenated inorder to build the SVD input matrix.0 1000 2000 3000 40000.010.0150.020.025σ = 3680321.25750 1000 2000 3000 4000−0.1−0.0500.050.1σ = 252053.01880 1000 2000 3000 4000−0.2−0.100.10.2σ = 198358.3460 1000 2000 3000 4000−0.1−0.0500.050.1σ = 17608.37550 1000 2000 3000 4000−0.2−0.100.10.2σ = 13892.03920 1000 2000 3000 4000−0.04−0.0200.020.04σ = 3267.13380 1000 2000 3000 4000−0.1−0.0500.050.1σ = 1479.48360 1000 2000 3000 4000−0.1−0.0500.050.1σ = 950.3947Figure 5: Orthonormal basis set calculated from SVD.The amplitude of each of the eigenvectors may be cal-culated for each of the machine pulses by finding the dotproduct of the SVD vector with the digitiser data. The ar-ray of amplitudes is then regressed against the 4D positionin order to find the conversion matrix.RESOLUTIONTo calculate the resolution, a data-set of ∼250 pulseswas recorded, and the position of the beam was calculatedfrom the HOMs, using the SVD modes and conversion ma-trix. These measurements were compared with those in-terpolated from the BPMs, and the spread of the differ-ences was found to be consistent with the resolution of theBPMs (∼10 µm), implying that the HOMs provide a mea-surement consistent with the BPM measurement, but of ahigher resolution.In order to determine the resolution of the HOM mea-surement, the position measured at a cavity close to thecentre of the module (for example cavity #5) was comparedto the position interpolated from its neighbouring cavities(i.e. cavities #4 and #6). The spread of the differences be-tween these is shown in figures (6), and (7) for x, x′, y, andy′ respectively.−0.04 −0.02 0 0.02 0.0402468101214Residual (mm)Figure 6: Resolution of the x measurement for cavity #5 inACC5.−0.015 −0.01 −0.005 0 0.005 0.01 0.01505101520Residual (mm)Figure 7: Resolution of the x measurement for cavity #5 inACC5.The resolutions, defined as the standard deviation of thespread of each of these plots, corrected by a geometric fac-tor of√2, were calculated to be,• x: 11 µm• x′: 175 µrad• y: 5 µm• y′: 140 µradGiven the coupling, k(m), of this mode, and the thermalnoise present in the electronics, it is possible to calculatethe theoretical minimum resolution of this device. Oncethe cable losses and the noise figure of the electronics, theminimum position resolution is ∼130 nm, and the mini-mum angular resolution is ∼1 µrad.The cause of the large discrepancy between the theo-retical and achieved resolutions is due to the noise of thetoroids that perform the charge measurement, and ampli-tude and phase jitter of the mixer LO.CONCLUSIONSIt has been demonstrated that strong dipole HOMs maybe used to determine the postion of the beam with a reso-lution of ∼5 µm, and that there is potential to significantlyimprove this resolution. The electronics for these deviceshave been installed onto five of the accelerating modules atFLASH, providing forty additional position measurementsin the linac, and the algorithm has been integrated into thecontrol system.A demonstration of good multi-bunch performance iscurrently in progress, with encouraging results, and the au-thors fully expect the HOM BPM system to be suitable forinclusion in the ILC.ACKNOWLEDGEMENTSWe appreciate the support of the FLASH crew duringour measurements. In spite of technical difficulties withthe facility, we managed with their help, to finish our ex-perimental program in optimum conditions.REFERENCES[1] W. K. H. Panofsky and M. Bander, Rev. Sci. Instrum. 39, 206(1968).[2] V. Ayvazyan et al., Eur. Phys. J. D 37, 297 (2006).[3] R. Wanzenberg, “Monopole, Dipole and Quadrupole Pass-bands of the TESLA 9-cell Cavity”, 2001, TESLA 2001-33,DESY[4] S. Molloy, “High precision superconducting cavity diagnos-tics with higher order mode measurements”, 2006, PRST-AB,9, 112802 (2006)

Measurement of the Beam's Trajectory using the Higher Order Modes It Generates in a Superconducting Accelerating Cavity

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Measurement of the Beam's Trajectory using the Higher Order Modes It Generates in a Superconducting Accelerating Cavity

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