Simulation and Analysis of TE Wave Propagation as a Probe for Electron Clouds in Particle Accelerators
Kiran G. Sonnad, Kenneth Hammond, Robert Schwartz, Seth Veitzer
SSimulation and Analysis of TE Wave Propagation for Measurement of Electron CloudDensities in Particle Accelerators
Kiran G. Sonnad a , Kenneth Hammond b , Robert Schwartz a , Seth A. Veitzer c a CLASSE, Cornell University, Ithaca NY, USA b Department of Physics, Harvard University, Cambridge MA, USA c Tech-X Corporation, Boulder CO, USA
Abstract
The use of transverse electric (TE) waves has proved to be a powerful, noninvasive method for estimating the densities of electronclouds formed in particle accelerators. Results from the plasma simulation program VSim have served as a useful guide forexperimental studies related to this method, which have been performed at various accelerator facilities. This paper provides resultsof the simulation and modeling work done in conjunction with experimental e ff orts carried out at the Cornell electron storage ring“Test Accelerator” (CesrTA). This paper begins with a discussion of the phase shift induced by electron clouds in the transmissionof RF waves, followed by the e ff ect of reflections along the beam pipe, simulation of the resonant standing wave frequency shiftsand finally the e ff ects of external magnetic fields, namely dipoles and wigglers. A derivation of the dispersion relationship of wavepropagation for arbitrary geometries in field free regions with a cold, uniform cloud density is also provided. Keywords:
1. Introduction
Electron clouds formed in circular particle accelerators withpositively charged beams are known to degrade the quality ofthe beam. They are a concern for future accelerator facilitiessuch as the International Linear Collider (ILC) damping rings,the SuperKEKB, and also upgrade of existing facilities suchas the Large Hadron Collider (LHC) and the Fermilab MainInjector (MI). Their study is also important for the optimumperformance of spallation neutron sources.The detection of electron clouds has been a topic of studyever since their e ff ects were first observed. The methods usedfor such a detection have included retarding field analyzers,clearing electrodes, shielded pickup detectors, TE waves andstudy of the response of the beam in the presence of an electroncloud. The TE wave method involves transmitting microwavesthrough a section of the beam pipe, and then studying the ef-fect of the cloud on the microwave properties. The microwavescan be introduced either as traveling or standing waves withina section of the beam pipe.The method of using microwaves as a probe for investigat-ing the presence of electron clouds was first proposed by F.Caspers [1, 2], in which experiments were conducted at theSPS at CERN based on this method. The measurement tech-nique involves measuring the height of modulation side-bandso ff the carrier frequency of the microwave. The electron cloud,constituting a plasma modifies the dispersion relationship of themicrowave. The periodic production and clearing of the cloud,based on the bunch train passage frequency, leads to modulationof the phase advance of carrier wave as it travels. This modula-tion gives rise to side bands in the spectrum of the propagated wave. The side bands are spaced from the carrier frequency by avalue equal to integer multiples of the train passage frequency.The details of the distribution of all the side band heights de-pends upon the nature of the build up and decay of the cloud.The study of this paper is restricted to the e ff ect of the wavedispersion from a static cloud under various conditions. A con-firmation of the electron cloud induced modulation was shownin the PEP II Low Energy Ring (LER) [3]. In this experiment,the wave was transmitted across a solenoidal section of the ringand the cloud density was controlled by adjusting the strengthof the solenoidal magnetic field. The estimations of the electroncloud density in this experiment, done based on the formulationgiven in [4], were reasonable when compared to earlier build upsimulations.As shown in this paper, reflections within the beam pipe canalter the signal and thus misrepresent the cloud density in the re-gion being sampled. Instead of transmitting the wave at a pointand receiving it from another called traveling wave RF diag-nostics, one could also trap the wave within a section calledresonant wave RF diagnostics. This has the advantage that onecan sample a known section of the beam pipe. This methodwould not be a ff ected by waves being reflected from other seg-ments of the pipe coming back into the section of interest andthus compromising the precision of the measurement. Anotheradvantage of trapping, is that there is an enhancement of the sig-nal as long as the point of measurement is close to a peak of thestanding wave. Additionally, at resonance, as demonstrated inRef [5], there is improved matching of the signal transfer fromthe electrodes into the waveguide, which is the beam pipe. Asdiscussed in Ref [6], the modulation of the cloud density wouldresult in a modulation of the resonant frequency enabling one to Preprint submitted to Nuclear Instruments and Methods in Physics Research A November 9, 2018 a r X i v : . [ phy s i c s . acc - ph ] J a n elate the frequency modulation signal to an actual cloud den-sity. The draw back of this technique is the need for havingreflectors at both ends of the desired section.The electron cloud induced phase shift is known to undergoan enhancement in the presence of an external magnetic fieldunder certain conditions, due to a modification in the disper-sion relationship. This occurs when the wave magnetic fieldhas a component perpendicular to the external magnetic field,and the frequency of the wave is close to the electron cyclotronfrequency, corresponding to the value of the external magneticfield. This e ff ect was demonstrated through simulations [7, 8],and was later confirmed through experiments done at PEP IIacross a chicane [9]. Further measurements done at the samechicane, now installed in CesrTA also confirm this e ff ect. Whileit is good to have an enhanced signal, the drawback of such ameasurement is that there is no available formulation that re-lates the enhanced side-band amplitude with the expected elec-tron cloud density. In addition, as discussed in this paper, thephase shift progressively reduces as the electron cyclotron fre-quency exceeds the carrier wave frequency and at very highexternal magnetic fields the signal may not be observable atall. On can suppress the e ff ect of the external magnetic field byaligning the wave electric field parallel to the external magneticfield, however it will be shown in this paper that this cannot befully eliminated unless the waveguide is rectangular.The program VSim[10], previously Vorpal, was usedthroughout to perform the simulations. The simulations usedelectromagnetic particle-in-cell (PIC) algorithms, consisting ofpropagating waves through a conducting beam pipe. The endof the pipe had perfectly-matched layers (PMLs) [7] meant toabsorb any transmissions, and thus simulate a long, continuousbeam pipe. Electrons were uniformly distributed and set to ini-tially have zero velocity (a cold plasma). Waves were excitedin the simulations with the help of a vertically pointing currentdensity near one end of the beam pipe, which covered the fullcross-sectional area and was two cells thick in the longitudi-nal dimension. In later simulations the PMLs and RF currentsource were replaced with port boundary conditions that simul-taneously absorb RF energy at a single frequency at the ends ofthe simulation while also launching RF energy into the simula-tion domain to simulate traveling waves.The waves were propagated along the channel using the Yee[5] algorithm for solving the electromagnetic field equations.The computational parameters used did not change much be-tween the various simulations performed in this paper. All thesimulations were three dimensional using a Cartesian grid. Thegrid sizes were around 2 − − × − s . The macro-particles-per-cellused was typically 10, and they were loaded uniformly in posi-tion space, with zero initial velocity, often referred to as a “coldstart”. The duration of the simulation was about 140 RF cycles.Overall, the modeling e ff ort related to measuring electronclouds using TE waves has served as a useful guide towardbetter understanding of the physical phenomenon and properinterpretation of the measured data. This paper provides a com-prehensive account of simulations performed for various tech-niques currently under study. A derivation of the wave disper- sion relationship for propagation through a beam pipe with acold, uniform electron distribution in a field free region is givenin the Appendix. Figure 1: Snapshot of a Vorpal simulation showing propagation of a TE wavethrough the CesrTA beam pipe
2. Electron Cloud induced Phase shift from TransmissionThrough Field-Free regions and the e ff ect of reflections The first experiments using microwaves to assess the clouddensities involved simply transmitting the wave using a beamposition monitor (BPM) and receiving the transmitted wavefrom another BPM downstream to the traveling wave [1, 3].As shown in Ref [4], the electron cloud induced phase shift perunit length of transmission in the absence of external magneticfields and a uniform electron distribution, can be related to theelectron density as follows, ∆ φ = ω p c ( ω − ω co ) / (1)where ω co is the angular cuto ff frequency for a waveguide invacuum, ω p = (cid:112) n e e /(cid:15) m e is the angular plasma frequency,with n e the electron number density, e the charge of the electron, c the speed of light, m e the electron mass and (cid:15) the free spacepermitivity. In Ref [4], this formula was validated through sim-ulations for a square cross section beam pipe. The derivation ofthe phase shift given by Eq 1 used the dispersion relationshipgiven in Ref [11], k = ω c − ω p c − ω co c (2)which was proved to be valid for a circular waveguides free ofexternal magnetic fields. The results shown in this paper vali-dates the formula for a Cesr beam pipe geometry. All these re-sults collectively indicate that the formula is valid for any typeof geometry. In the Appedix, we provide an explicit proof thatthis is indeed the case. The Cesr beam pipe geometry may berepresented in the form of two circular arcs (radius 0.075m)connected with flat side planes. It is about 0.090m from side toside and 0.050m between the apices of the arcs. The cuto ff fre-quency of lowest mode for this geometry is known to be around2 .
888 GHz from past experiments and calculations related tothe beam pipe. Figure 1 provides a snapshot of the simulation,showing the propagation of a wave through the Cesr beam pipe.
Figure 2: Variation of (a) phase shift with cloud density for a Cesr beam pipegeometry, and (b)the e ff ect of reflections on the phase shift Calculation of electron induced phase shift was performedthrough separate simulations of the wave transmission througha vacuum beam pipe and through a beam pipe with electronsrespectively. At a certain axial distance L from the location atwhich the wave was launched, the variation of the voltage be-tween the midpoints of the top and bottom boundaries of thebeam pipe cross section were recorded as a function of time.After normalizing the amplitudes of the two waves to unity,their di ff erence gives a sinusoidal wave with an amplitude equalto the phase shift between the waves. Suppose that the angularfrequency of the wave is ω and phase shift is δ . The phase shiftfor nominal cloud densities is small enough that sin( δ ) ≈ δ .Hence we have cos( ω t ) − cos( ω t + δ ) ≈ δ sin( ω t + δ/ ff frequency. While this is desirable because it ampli-fies the modulation side-bands relative to the carrier signal, one will encounter reduced transmission as the carrier frequency ap-proaches the cuto ff . Figure 3: A schematic of the simulations with protrusions serving as partialreflectors, showing the Cesr and a circular cross section beam pipes. The lattergeometry is used in Sec. 3
Due to the presence of several mechanical and electroniccomponents all along the vacuum chamber, it is not very likelythat one can perform phase shift experiments that are entirelyfree of internal reflections. As discussed in Ref [14], in an ex-periment, these internal reflections would potentially a ff ect thevalue of the phase shift. A wave reflected from a device thatlies beyond the segment being measured, would be received bythe detector when it comes back after reflection. At the sametime, waves could get reflected back and forth within the seg-ment before eventually being received at the detector. In bothcases, the reflected wave would have sampled a length di ff erentfrom that meant to be sampled and would thereby contaminatethe signal, because the phase shift is proportional to the lengthof transmission. In order to understand this e ff ect, the simula-tions were altered to include two protruding conductors, whichwould reflect some of the transmitted wave. The protrusionswere slabs in the transverse plane, extending from the bottomto 1cm above the apex of the lower arc (see Fig. 3). They werespaced 0.4 meters apart, including the thickness of the protru-sions, which was 1mm. The frequencies used for this study2.41 GHz and 3.87 GHz, the same as those shown in Fig 2(a),correspond to the resonant harmonics ( n = n =
9, re-spectively) of a 0.4 meter “resonant cavity”. This was donein order to maximize reflections. Fig. 2(b) shows the resultingphase shifts from these calculations. The solid shapes representthe data for no reflections and are the same data that appearin Fig. 2(a). The open shapes represent the phase shifts in thepresence of reflection. These results clearly indicate that inter-nal reflections modify the expected phase shift. The nature ofthe alteration of phase shift depends upon the complexities ofthe transmission-reflection combination, and the instrumenta-tion used for the method. However, the results show that thelinear relationship between phase shift and electron density isalways preserved.3 . Standing Waves from Partial Reflectors and ElectronCloud Induced Resonant Frequency Shift
While internal reflections may interfere with phase shift mea-surements, they can also be used to trap a wave. This trappedwave can be used to measure the electron cloud density, as dis-cussed in Ref [6]. This section shows the results of numericalsimulation of such an experiment. The geometries used were(a) Cesr beam pipe also used in the previous section and (b)beam pipe with a circular cross section of radius 4.45cm. Bothof them had conducting protrusions that were 1mm thick and1cm high from the base as shown in Fig 3. The cuto ff frequencyfor the circular beam pipe can be calculated from the analyticexpression, and is 1.9755GHz for the lowest ( T E ) mode. −5 Wave Frequency (GHz) A v e C en t e r E ne r g y F l u x ( a r b ) Cesr Beampipe (a) −5 A v e . C en t e r E ne r g y F l u x ( a r b ) Wave Frequency (GHz)
Circular beampipe (b)
Figure 4: The average energy flux calculated over a range of frequencies, withthe local minima indicating resonance points for (a) the Cesr beam pipe, (b)beam pipe with circular cross section
A trapped mode results when standing waves are induced be-tween the reflectors. In order to test the e ff ectiveness of induc-ing such a standing wave using partial reflectors, simulationswere done with an empty wave guide over a large range of fre-quencies. It should be noted that, since there is only a par-tial reflection of the wave taking place, there is always a nettransmission of energy across the segment between the protru-sions. The wave energy that escapes the partial reflectors getsabsorbed into the PML regions. Identification of a resonancewas done as follows. At each frequency and each time step,the wave energy flux was computed by integrating the Poyntingvector across a plane located at the mid point between the twoprotrusions. This plane was oriented transverse to the axis andcovered the entire cross section. For each of these frequencies,the mean of the energy flux was calculated over the period ofthe simulation. It is expected that as the frequency approachesthat of a standing wave, this averaged flux would reach a localminimum. This is because of increased ”back and forth” trans- mission which does not contribute to the average flux due tocancellation.Figure 4 shows the average energy flux calculated for a va-riety of frequencies spanning over several resonance points for(a) the Cesr beam pipe (b) the circular cross section beam pipe.The local minima seen on these plots correspond to a standingwave mode. The length of the section, including the width ofthe protrusions was 0.4m for the Cesr beam pipe. The length ofthe circular beam pipe, including the width of the protrusionswas 0.88m Resonance Number Squared F r equen cy S qua r ed ( G H z ) Cesr Beampipe resonance pointsstraight line fit (a)
Resonance Number Squared F r equen cy S qua r ed ( G H z ) Circular beam piperesonance pointsstraight line fit (b)
Figure 5: Resonance points obtained from the data in Fig 4 showing the linearrelationship between f and n according to Eq 3 A standing wave occurs when the wavelength λ and thelength L of the segment are related such that for any inte-ger n , L = n λ/
2. The dipersion relationship of a waveguidewith wave frequency f and cuto ff frequency f co , is given by c /λ = f − f co . Expressing λ in terms of L , for the standingwave, this then gives f = c L n + f co (3)This indicates the linear relationship between f and n , andalso the relationship of L with the slope, and f co with the in-tercept of the straight line. Equation (3) was used to con-firm that the local minima in Fig 4 correspond to resonancepoints. Figure 5 shows the value of f plotted as a func-tion of the corresponding value of n for (a) the Cesr beampipe and (b) beam pipe with a circular cross section. Per-forming a straight line fit on these points for case (a), yieldedthe relationship f ( GHz ) = . n + . L = . m and f co = . GHz . For case (b), a similar oper-ation gives f ( GHz ) = . n + . L = . m and f co = . GHz . These values are close enough to theexpected ones and ascertain the accuracy of such a method indetermining standing waves between partial reflectors.4he presence of an electron cloud would result in a shift inthe standing wave frequency. Experimentally, it is possible tomeasure this in the form of frequency modulation side-bandsassociated with the periodic passage of a train of bunches cre-ating electron clouds. Using, Eq (2) we can show that the con-dition for standing waves given by Eq (3) is modified by anelectron cloud as follows, f e = c L n + f co + f p (4)where the wave frequency is now denoted by f e and f p is theplasma frequency. Subtracting Eq 3 from Eq 4, and in the limitof small frequency shifts, we get, f e − f ≈ ∆ f f = f p , where ∆ f = f e − f . On inserting the expression for f p , this then givesa simple expression relating the shift in resonant frequency as afunction of electron density. ∆ f = n e e (cid:15) m e π f = n e r e c π f (5)in which r e is the classical electron radius. This shows that thefrequency shift is proportional to the electron cloud density.An e ff ort is underway at CesrTA to use this method to mea-sure the density of the electron cloud within the beam pipe sec-tion where the reflections are occurring [15]. Thus, it becamenecessary to test this phenomenon with simulations. Simulationof the frequency shift of standing waves was done for a Cesrbeam pipe cross section as well as a beam pipe with a circularcross section. All the parameters were the same as before ex-cept that the length of the section between the partial reflectorsfor the Cesr beam pipe was modified from 0.4m to 0.9 m, whichwas somewhat close to one the setups under study at CesrTA.The length of the circular cross section was retained at 0.88m.Since the frequency shift induced by electrons is very small, it isrequired that the resonant frequency be determined accurately.To do this, a parabolic fit was made to the averaged energy fluxin the vicinity of the minimum point, using the available pointsobtained from simulation. The expression for the parabola maybe obtained from a Taylor expansion of the function around theminimum. This gives the mean energy flux as a function offrequency f near the n th minimum point f n . Thus, we have E ( f ) = E (cid:48)(cid:48) ( f n ) f − E (cid:48)(cid:48) ( f n ) f n f + [ E ( f n ) + E (cid:48)(cid:48) ( f n ) f n ] (6)where E ( f ) is the averaged energy flux, E (cid:48)(cid:48) ( f n ) is the secondderivative of E evaluated at f n . The first derivative of E at theminimum point vanishes. Using the computed coe ffi cients ofthe parabola one can solve for f n .Figure 6 shows the frequency shift induced by electronclouds for the n = m − , which is rather high, but helpsvalidate Eq (5) with simulations more accurately. For these pa-rameters, the n = m − and 2 × m − . The expectedfrequency shift for the 10 m − case is 2.013MHz. The simu-lated frequency shift for this was 2.1MHz, and for an electron density of 2 × m − , it was 4.2MHz. Thus we were ableto establish that reasonably accurate values of frequency shiftfor such an experiment may be determined from simulations.It is interesting to note that the error obtained in the resonantfrequency itself was around 2 MHz , however the shift inducedby electron clouds always had reasonable agreement with Eq 5.The agreement with theory provides confidence in estimatingelectron densities based on Eq 5 from measurements. Typicalelectron cloud densities are of the order of 10 m − leading tofrequency shifts about a factor of 100 smaller than those sim-ulated here. Simulating frequency shifts this small would inprinciple be possible by scaling all numerical parameters ap-propriately, but this would have been a far more intensive com-putational process. However, it is well known that in practice,such small frequency modulations can be easily measured withstandard spectrum analyzers. Wave Frequency (GHz) A v e C en t e r E ne r g y F l u x ( a r b ) Cesr beampipe
No e−10 /m e−2.05MHz (a) −4 A v e C en t e r E ne r g y F l u x ( a r b ) Wave Frequency (GHz)
Circular beampipe no e−1e14/m e−2e14/m e− (b) Figure 6: Simulation results showing a shift in the n =
4. Transmission and Phase Shift Through Dipole and Wig-gler Field Regions
This section discusses the e ff ect of external dipole and wig-gler magnetic fields on the phase shift measurements. Simu-lations done in the past, have revealed that the phase shift isgreatly amplified in the presence of an external magnetic fieldif the electron cyclotron frequency lies in the vicinity of thecarrier frequency [7, 8]. Following these results, the cyclotronresonance was soon confirmed at an experiment performed atthe SLAC chicane [9]. The SLAC chicane has since been trans-ferred to CesrTA, where these studies continue to be made [16].In the presence of an external magnetic field and electronclouds, the medium is no longer isotropic and the polarization5f the transmitted microwave plays an important role in the out-come of the measurement. When the wave electric field is ori-ented perpendicular to the external magnetic field, the mode isreferred to as an Extraordinary wave or simply X-wave. In thissituation, if the external dipole field corresponds to an electroncyclotron frequency close to the carrier wave frequency, we seean enhanced phase shift. The phenomenon is well understoodin the case of open boundaries. It is usually referred to as up-per hybrid resonance. The dispersion relationship for the openboundary case is given as follows (see for example ref [17]). c k ω = − ω p ( ω − ω p ) ω ( ω − ω h ) (7)The quantity ω h is the upper hybrid frequency which is givenby ω h = ω p + ω c , ω c = eB / m e being the electron cyclotronfrequency for the given magnetic field B . When ω h → ω , it isclear that k → ∞ . It can also be seen that as ω h → ∞ , i.e.,for very high magnetic fields, the relationship between k and ω approaches that of propagation through vacuum. In the case ofelectron clouds in beam pipes, the plasma frequency is of theorder of a few 10 MHz while the carrier frequency is around2 GHz. In this regime, it is reasonable to state that resonanceoccurs when ω c → ω . Since the phase advance is the productof the wave vector k and the length of propagation, we see thatthe electron cloud induced phase shift will theoretically go toinfinity. Equation (7) is not valid for waveguides, which havefinite boundaries. Nevertheless, simulations show that the samequalitative features are exhibited also for propagation throughwaveguides. ph a s e s h i f t m r a d / m magnetic field (10 -2 T) x wave circular cross section ph a s e s h i f t ( l og s ca l e ) m r a d / m magnetic field (10 -2 T) Figure 7: Phase shift vs. magnetic field for dominant X wave propagation fordi ff erent cloud densities. The peak corresponds to a cyclotron resonance. Figure [7] shows the enhanced phase shift for three valuesof cloud densities when the cyclotron frequency approaches thecarrier frequency. The beam pipe cross section was circularwith a radius of 4.45 cm, which leads to a cuto ff at 1.9755GHz at the fundamental T E mode. These parameters match withthe beam pipe geometry of the PEP II / CesrTA chicane sec-tion. The wave frequency used in the simulation was 2.17 GHz.The magnetic field corresponding to this cyclotron frequencyis 0.077576T. The wave was excited with the help of a sinu-soidally varying electric field pointing perpendicular to the ex-ternal magnetic field. ph a s e s h i f t ( m r a d / m ) magnetic field (10 -2 T) Figure 8: Wave launched with a dominant O wave component showing weakresonance
When the wave is polarized with an electric field that is paral-lel to the external magnetic field, the mode is often referred to asthe Ordinary wave, or the O-wave. In this case one would notexpect any e ff ect created by the external magnetic field. Thiswould be the case for a rectangular cross-section waveguide, orwith open boundaries. However, with a circular cross sectionas in our study, the boundary conditions would force a com-ponent perpendicular to the magnetic field in the wave electricfield even if it is launched with a purely parallel electric field.Thus, it is inevitable that a weak component of a wave with anorthogonal polarization gets excited. Additionally, the methodemployed in launching the wave in the simulations is similarto that performed in experiments, where an electric field waveis excited along a particular direction over a surface area. Thefunctions describing the wave for a cylindrical geometry areBessel functions involving the radial and azimuthal variables.Unless care is taken to excite a wave having the given func-tional form, the wave is expected to couple itself to two orthog-onal modes with varying degrees of intensity. Additionally, thedetection system, would receive the e ff ect of the two modes tovarying degrees. Disentangling this combination will involvemore analysis, guided by simulations. Due to these e ff ects, wesee a weak resonance e ff ect even in a wave excited with a purelyvertical electric field, that is aligned to the external magneticfield as indicated. Figure 8 shows the presence of such a weakresonance and this e ff ect has been observed at CesrTA as well.Figure [9] shows the variation of phase shift with electroncloud density at di ff erent settings of external magnetic fields.In these simulations, the wave was excited with an electric fieldperpendicular to the external magnetic field. These densities aretypical of what is produced in CesrTA. The plots show that thevariation of phase shift with density remains linear even whenone is close to resonance. This is expected to be true as long asthe plasma frequency is much smaller than the wave frequency,regardless of how complex the dispersion relationship of thewave is. Thus, one could easily amplify the signal with the6 ph a s e s h i f t ( m r a d / m ) electron density (1e11/m ) Figure 9: Variation of phase shift with cloud density for di ff erent magneticfields for dominant X-wave propagation. help of an external magnetic field to monitor relative changesin cloud density, if not the absolute density.Experiments have been done to study the phase shift acrossthe damping wigglers at CesrTA. These experiments corre-spond to various bunch currents and wiggler field settings. Thewiggler field setting influences the measurement in more thanone way. The wiggler field a ff ects the motion of the electrons,which influences the secondary production of the cloud. Thesynchrotron radiation flux is determined by the strength of thewiggler field, and this in turn determines the photoemission rateof the cloud. Both these e ff ects determine the density of thecloud. The electron density is not uniform across the length ofthe wiggler, as shown in Ref [18]. As already shown in thispaper, the external magnetic field, by itself, alters the phaseshift for a given cloud density. Given that the wiggler fieldis rather complex, along with a cloud density that is longitu-dinally nonuniform, simulations become particularly importantto fully interpret results from such an experiment. In this paper,we examine just the e ff ects of the nonuniform wiggler field onthe phase shift. Figure 10: The full Wiggler field in 3 dimensions
The wiggler chamber cross section in CesrTA is close to thatof a rectangle. The height is 5cm and width is 9cm. The cornersof the rectangle are chopped, so that the actual width of the baseand top is 64.6mm and the height of the side walls are 24.6mm.This which would only moderately alter the results obtainedfrom using a perfect rectangle. Thus, for the sake of simplic-ity, the simulations were done with a perfect rectangle with theabove parameters. The length of the section simulated is 80cm,which corresponds to half the length of the wiggler. This issu ffi cient to account all the variations in the wiggler magneticfield. The computed wiggler magnetic field used was based on the formulation given in [19]. Figure 10 shows the magneticfield, in three dimensions. For most of the region, the field isoriented vertically, while in the transition region between polesthere is a longitudinal component to the field. There is almostno magnetic field in the horizontal direction. Figure 11: Variation of phase shift with cloud density for di ff erent magneticfields in a rectangular chamber with a vertical wave electric field.Figure 12: Variation of phase shift with cloud density for di ff erent magneticfields in a rectangular chamber with a horizontal wave electric field. Even before performing simulations with the full wigglerfield turned on, some preliminary studies were done with a justa constant dipole field within the same geometry. Since theshape of the vacuum chamber cross section is rectangular, thethe cuto ff frequency of the wave is determined by the polar-ization of the TE wave. If the wave electric field is pointingin the vertical direction, the cuto ff frequency is 1.66GHz andif the field is horizontal, the cuto ff is 3 GHz. All simulationswere done at a frequency 10% above the respective cuto ff . Fig-ure 11 shows the phase shift associated with propagation of avertically polarized wave under di ff erent conditions. The figureshows that when the wave electric field is polarized along theexternal magnetic field, the phase shift matches with that pre-dicted by Eq (1). This result is expected to be true in the caseof a rectangular cross section, where the wave electric field ispointing parallel to the external magnetic field everywhere inthe pipe, and is thus una ff ected by the external magnetic field.As already shown, this would not be true in the case of a curva-ture in the cross-section boundary. The figure also shows thatthe phase shift is suppressed in the case of the wave electricfield pointing perpendicular to the external field, referred to asextraordinary wave. The cyclotron resonance in this case occurswhen the magnetic field is equal to 0 .
06T and the field here wasset to a much higher value. In general, the wave electric field7erturbs the electrons, causing them to oscillate and thereby al-ter the wave dispersion relation. When the external magneticfield is very high, the electrons tend to get locked against anymotion transverse to the magnetic field. Since the wave electricfield is perpendicular to the external magnetic field they will en-counter electrons that tend to be ”frozen”. As a result the wavewill undergo reduced electron cloud induced phase shift at mag-netic fields much higher than that causing cyclotron resonancefor the specific carrier frequency.Figure 12 corresponds to a wave with the electric field point-ing horizontally. This shows similar features as those in Figure11. The wave frequency is 3.3GHz and cyclotron resonance oc-curs at field of 0.11 T, and so the figure shows enhanced phaseshift at a magnetic field setting close to this value. Excitingthe chamber at this higher frequency could be less e ffi cient dueto poorer matching between the various hardware components.In addition, one can expect a mixing of modes to take place athigher frequencies because of the presence of various irregular-ities in a real beam pipe as opposed to a simulated one. Nev-ertheless, studying this mode is important because the wigglermagnetic field is mostly pointing in the vertical direction. Onecould amplify the signal by setting a wiggler field so that a cy-clotron resonance is excited. This e ff ect might prove useful indetecting the presence of very low density electrons in wigglerregions. The presence of low energy electrons in wiggler andundulators is of particular interest if the device is cryogenic,in which case the electrons are believed to contribute to theheat load of the system [20]. The electron cloud in such sys-tems would be produced by electron beams primarily throughphotoemission, and thus if present, they will occur at very lowdensities, requiring greater sensitivity in the detection. Figure 13: Variation of phase shift with cloud density with the full wigglermagnetic field and a vertical wave electric field.
In the end, we look at the phase shift in the presence of thefull wiggler field. Figure 13 shows that the phase shift getssuppressed by about 20% when compared that expected in theabsence of any fields. In this case, the wave electric field ispointing in the vertical direction, which is a configuration thatshould have little e ff ect over the phase shift, since the exter-nal magnetic field is largely vertical. A longitudinal magneticfield would alter the dispersion relationship, where the wavegets split into a left and right circularly polarized components.This has been analyzed for guided wave propagation in cylin-drical geometries in Ref [11]. While the 20% reduction in phaseshift in our result can be attributed to such an e ff ect, a detailed analysis of the same is beyond the scope of this paper. Over-all, it is clear that simulations are of prime importance to accu-rately interpret the observed electron cloud induced phase shiftsacross such wiggler fields.
5. Summary
In this paper, we provide a comprehensive account of thesimulation and analysis e ff ort that has been carried out in con-junction with the experimental e ff ort of using TE waves to mea-sure electron clouds in CesrTA. These simulations helped con-firm several physical phenomena either in a quantitative or in aqualitative manner. For example, they helped validate Eq 1 thatrelates the phase shift with cloud density for geometries like theCesrTA beam pipe, which does not have a regular shape suchas rectangular or circular. The e ff ect of reflections on phaseshift measurements has always been a concern, and simulationsshow that one must be careful especially of standing waves ex-cited within the beam pipe due to partial reflectors. The feasi-bility of using standing waves to measure the cloud density isclearly demonstrated by simulations and has provided valuableguidance to the experimental e ff ort being carried out at Ces-rTA. In the process, we were able to determine a novel methodof detecting the presence of standing waves in simulations byaveraging the total Poynting vector flux across a surface overtime for varying frequencies.Simulations of phase shifts in the presence of external mag-netic fields were modeled for a variety of cases. The nature ofresults vary greatly based on the the parameters present in thesystem. For example, the possibility of exciting cyclotron res-onances is clearly shown in simulations, which would be pos-sible to produce in dipole fields present in a chicane. However,dipole fields used in bend regions of an accelerator are muchhigher and they can suppress the electron induced phase shift,depending on the polarization of the wave. In the presence ofcurved boundaries, there is always a mixture of e ff ects fromordinary and extraordinary wave propagation. Since most ac-celerator vacuum chambers have a curvature, this e ff ect is im-portant to understand. A direct comparison with the analyticexpression Eq 2 would lead to an incorrect interpretation of themeasured data.The results will always be hard to interpret when a systemis near the cyclotron resonance, when the phase shift is the-oretically infinity. Simulations and experiments would neveryield an infinity, and there may be poor agreement between thetwo in such regimes. However, an enhancement of the signalthat is still predictable can always be obtained by moving rea-sonably close to a cyclotron resonance point. Electron cloudformation in wiggler fields can be very important in machinessuch as positron damping rings. Given the complexity of such asystem, experiments of determining the cloud density using TEwaves have to be accompanied by careful simulations beforeinterpreting any results obtained from measurements.It may be noted that the relationship between the phase shiftand cloud density is always linear regardless of how complexthe system is. This would be true for very low cloud densities,which would have low plasma frequencies. Thus, this method8an be of great utility if one is interested in relative changes inelectron cloud densities for example when a machine is under-going conditioning. Simulations can then be used to obtain aproportionality constant between cloud density and phase shift.The studies in this paper were always done with an electrondensity that was cold and uniformly distributed transversely andlongitudinally. Simulations have not indicated a dependence ontemperatures associated with typical electron cloud densities.In a transverse nonuniform distribution, there will be a slightenhancement in the phase shift if more electrons are populatedin regions with high peak electric fields produced by the wave.As mentioned earlier, longitudinal variation of the cloud densitybecomes important in wiggler fields because it couples with thelongitudinal variation of the magnetic field. These additionalcomplexities could be topics for future studies.The TE wave method is an attractive technique for measuringelectron cloud densities that can replace or complement othermeasurement methods. Besides CesrTA, this method is beingstudied at other accelerator facilities [21, 22, 23]. The mea-surement technique and its required instrumentation are sim-ple, the process is noninvasive, and can be kept in operationcontinuously. Thus, it holds the promise of wide usage wher-ever it is useful to monitor the electron cloud properties con-tinuously and at all locations of the accelerator. It is evidentthat a careful study toward understanding of the physical phe-nomenon through analysis and simulation are very importanttoward proper interpretation of the measured data.
6. Appendix: Derivation of Dispersion Relationship
In this Appendix, we provide a derivation of Eq 2 which isnot specific to the geometry of the cross-section of the waveg-uide. We also discuss the approximations and assumptions as-sociated with the derivation of this equation. This dispersionrelationship is specific to guided waves propagating throughelectron clouds in field free regions. The starting equations forsuch a system would include the fluid and Maxwell’s equations.These are, m [ ∂ v ∂ t + ( v · ∇ ) v ] + e ( E + v × B ) = ∂ n e ∂ t + ∇ · ( n e v ) = ∇ · E = en e (cid:15) ∇ · B = ∇ × E = − ∂ B ∂ t ∇ × B = µ ( − en e v + J ext ) + µ (cid:15) ∂ E ∂ t (8)where n e is the number density of the electrons, v is the velocityof the fluid, and J ext is an external current density. The otherterms have their usual meanings. The continuity equation is notindependent from the rest of the equations as it can be obtainedfrom Maxwell’s equations.We perturb all quantities about an equilibrium, so that wehave v = v + v ( ) , n e = n + n (1) , E = E + E ( ) , B = B + B ( ) where the zeroth order quantities satisfy the steady statecondition ∂/∂ t =
0. As a result, we get m ( v · ∇ ) v + e ( E + v × B ) = ∇ · ( n v ) = ∇ · E = en (cid:15) ∇ · B = ∇ × E = ∇ × B = µ ( − en v + J ext ) (9)Such an equilibrium state requires for the particles to be con-fined in a steady state indefinitely. This is normally associatedwith neutral plasmas in which the ions may be considered im-mobile, or charged particles trapped for a long period of time inconfinement devices. In this paper, the electron cloud densityis sustained for the duration of the bunch train passage. Theabove equilibrium condition may be considered valid as long as1 /τ (cid:29) f where τ is time of confinement of the charge and f isthe frequency of the perturbing wave. Periodic changes in thestate occurring over time scales greater than the wave period-icity manifest themselves as modulations of the output signal,while changes occurring over much smaller time scales, remainunresolved by the carrier wave. Thus, the spectrum of the out-put signal would depend on the variation of the electron cloudassociated with its build up and decay.Inserting the perturbation expansion into the original fluidand Maxwell’s equation, and imposing the above equilibriumconditions and ignoring terms of second and higher order, weget, m ∂ v ( ) ∂ t + e ( E ( ) + v ( ) × B ) = ∂ n (1) ∂ t + ∇ · ( n v ( ) ) = ∇ · E ( ) = en (1) (cid:15) ∇ · B ( ) = ∇ × E ( ) = − ∂ B ( ) ∂ t ∇ × B ( ) = µ ( − en v ( ) + (cid:15) ∂ E ( ) ∂ t ) (10)These equations are linear and we can seek all perturbationsto have the following form, v (1) ( x , y , z , t ) = ˜ v ( x , y ) e i ( k − ω t ) n (1) ( x , y , z , t ) = ˜ n ( x , y ) e i ( k − ω t ) E (1) ( x , y , z , t ) = ˜ E ( x , y ) e i ( k − ω t ) B (1) ( x , y , z , t ) = ˜ B ( x , y ) e i ( k − ω t ) (11)From the above, it is clear that ∂∂ z = ik , ∂∂ t = − i ω (12)This form is valid as long as the geometry along “ z ”, the lon-gitudinal coordinate is uniform and infinite. This is not entirely9rue when there are partial reflectors. In the case of perfect re-flectors, one would obtain discrete values for k , representingstanding waves. Thus one can expect that the above form to bemore accurate when close to a resonance in the presence of par-tial reflectors. In general, the reflectors would be small enoughso that the above form of solutions can be considered a validapproximation.For the sake of convenience, we drop the accent ˜ and thearguments ( x , y ) in the functions given in Eq (11). To simplifythe analysis, we make two assumptions, (1) The fluid is coldand at rest. So v =
0, and (2) the density is uniform, whichmeans n = constant. We further assume that there is no exter-nal magnetic field, which means that B =
0. In the absenceof any static external magnetic fields, the only contribution to B would be the magnetic field produced by the beam. Sincethe beam is highly relativistic, this would be confined along thelength of the bunch. It is reasonable to disregard this when thegap between the bunches is much larger than the bunch length,in which case the wave would sample mostly a field free region.Inserting the waveform solutions (Eq 11) into the perturbedmomentum and continuity equations in Eq (10), with B = − im ω v ( ) + e E ( ) = − i ω n (1) + n ∇ · v ( ) = n (1) + n em ω ∇ · E ( ) = . (14)Combining this with the perturbed electrostatic field equationin Eq (10) gives n (1) =
0, which means, up to the first order,there is no perturbation in the charge density due to the waveelectric and magnetic fields. This gives us ∇ · E ( ) = ∇ × B ( ) = − i ωµ (cid:15) E ( ) (16)where (cid:15) = (cid:15) (1 − ω p /ω ). Similarly, it is easy to see that, ∇ × E ( ) = i ω B ( ) (17)Using Eqs 15, 16 and 17, along with ∇ · B ( ) =
0, and assum-ing that the boundary conditions are perfectly conducting, onecan follow the steps given in Ref[24]. to get γ = µ (cid:15)ω − k . (18)The constant γ must be nonnegative for oscillatory solutions,and will take on a set of discrete “eigenvalues”, correspond-ing to the di ff erent modes associated with the geometry of thecross-section of the waveguide. Combining Eq (18) with a sim-ilar relationship for a vacuum waveguide where, (cid:15) = (cid:15) , one caneasily show that, k = ω c − ω p c − ω co c (19) ω co = c γ being the angular cuto ff frequency for the vacuumwaveguide. This relationship is the same as Eq (2). Acknowledgements
The authors wish to thank John Sikora for many useful dis-cussions and for suggesting us to do the simulations with par-tial internal reflections. Thanks to Jim Crittenden for help-ing us in generating the complete the wiggler magnetic field.We also wish to thank David Rubin, Mark Palmer and PeterStoltz for their support and guidance. This work was sup-ported by the US National Science Foundation (PHY-0734867,PHY-1002467, and PHY-1068662) and the US Department ofEnergy (DE-FC02-08ER41538 and de-sc0006505; DE-FC02-07ER41499 as part of the ComPASS SCiDAC-2 project, andde-sc0008920 as part of the ComPASS SCiDAC-3 project).
References [1] F. Caspers, W. Hofle, J. M. Jimenez, J. F. Malo, J. Tuckmantel, and T.Kroyer, in Proceedings of the 31st ICFA Beam Dynamics Workshop:Electron Cloud E ff ects (ECLOUD04), Napa, California 2004 (CERN Re-port No. CERN-2005-001, 2004).[2] T. Kroyer , F. Caspers, E. Mahner, Proceedings of 2005 Particle Acceler-ator Conference, Knoxville, Tennessee 2212-2214[3] S. De Santis, J. M. Byrd, F. Caspers, A. Krasnykh, T. Kroyer, M. T. F.Pivi, and K. G. Sonnad Phys. Rev. Lett. 100, 094801 (2008)[4] Kiran Sonnad, Miguel Furman, Seth Veitzer, Peter Stoltz and John Cary,Proceedings of PAC07, Albuquerque, New Mexico, USA, pp. THPAS008[5] K G Sonnad et al Proceedings of Particle Accelerator Conference 2009,Vancouver, Canada, 2009, pp.TH5RFP044[6] J.P. Sikora et al , Proceedings of International Particle Accelerator Con-ference 2011, San Sebastin, Spain, pp.TUPC170[7] K.G. Sonnad et al http: // meetings.aps.org / link / BAPS.2007.DPP.TP8.13449th Annual Meeting of the Division of Plasma Physics[8] S. Veitzer, DOE Scientific and Technical Information,http: // / bridge Identifier number 964651[9] M. T. F. Pivi, et al , Proceedings of European Particle Accelerator Confer-ence 2008, Genoa, Italy pp. MOPP065[10] C. Nieter and J. R. Cary, J. Comp. Phys. 196, 448-472 (2004).[11] H. S. Uhm, K. T. Nguyen, R. F. Schneider an d J. R. Smith, Journal ofApplied Physics, Vol 64(3), 1988, pages 1108- 1115.[12] J. Berenger, Journal of Computational Physics 114, 185 (1994)[13] K. Yee, IEEE Transactions on Antennas and Propagation, AP-14, 302(1966)[14] S. De Santis, Phys. Rev. ST Accel. Beams 13, 071002 (2010)[15] John Sikora and Stefano DeSantis, http: // arxiv.org / abs / et al , Proceedings of 2011 Particle Accelerator Conference,New York, NY, USA, pp. MOP228[17] R J Goldstone and P H Rutherford, Introduction to Plasma Physics Insti-tute of Physics Publishing, 1995[18] C. Celata Phys. Rev. ST - Accel. Beams 14, 041003 (2011)[19] D. Sagan, J. A. Crittenden, D. Rubin and E. Forest, Proceedings of Parti-cle Accelerator Conference 2003, Portland, OR, USA pp.1023[20] S. Casalbuoni, S. Schleede, D. Saez de Jauregui, M. Hagelstein, and P. F.Tavares, Phys. Rev. ST Accel. Beams 13, 073201 (2010)[21] S. Federmann, F. Caspers, and E. Mahner, Phys. Rev. ST Accel. Beams14, 012802 (2011)[22] N. Eddy, J. Crisp, I. Kourbanis, K. Seiya, B. Zwaska, S. De SantisProceedings of Particle Accelerator Conference 2009, Vancouver, BC,Canada pp. WE4GRC02[23] J. C. Thangaraj, N. Eddy, B. Zwaska, J. Crisp, I. Kourbanis, K. SeiyaProceedings of the Electron Cloud Workshop 2010, Ithaca, New York,USA pp. DIA00[24] J D Jackson. Classical Electrodynamics, Wiley, New York, NY, 3rd ed.edition, (1999), Proceedings of 2011 Particle Accelerator Conference,New York, NY, USA, pp. MOP228[17] R J Goldstone and P H Rutherford, Introduction to Plasma Physics Insti-tute of Physics Publishing, 1995[18] C. Celata Phys. Rev. ST - Accel. Beams 14, 041003 (2011)[19] D. Sagan, J. A. Crittenden, D. Rubin and E. Forest, Proceedings of Parti-cle Accelerator Conference 2003, Portland, OR, USA pp.1023[20] S. Casalbuoni, S. Schleede, D. Saez de Jauregui, M. Hagelstein, and P. F.Tavares, Phys. Rev. ST Accel. Beams 13, 073201 (2010)[21] S. Federmann, F. Caspers, and E. Mahner, Phys. Rev. ST Accel. Beams14, 012802 (2011)[22] N. Eddy, J. Crisp, I. Kourbanis, K. Seiya, B. Zwaska, S. De SantisProceedings of Particle Accelerator Conference 2009, Vancouver, BC,Canada pp. WE4GRC02[23] J. C. Thangaraj, N. Eddy, B. Zwaska, J. Crisp, I. Kourbanis, K. SeiyaProceedings of the Electron Cloud Workshop 2010, Ithaca, New York,USA pp. DIA00[24] J D Jackson. Classical Electrodynamics, Wiley, New York, NY, 3rd ed.edition, (1999)