January, 2013

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge MA 02139 USA This work was supported by the U.S. Department of Energy, Grant No. DE-FG02-91ER40648. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.

PSFC/JA-12-58

Calculation of Wakefields in a 17 GHz Beam-Driven Photonic Bandgap Accelerator Structure

Hu., M.,Munroe, B.J., Cook, A.M., Shapiro, M.A., Temkin, R. J.

CALCULATION OF WAKEFIELDS IN A 17 GHZ BEAM-DRIVEN PHOTONIC

BANDGAP ACCELERATOR STRUCTURE

Min Hu∗

Terahertz Research Center, School of Physical Electronics,

University of Electronic Science and Technology of China, Chengdu 610054, China and

Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Brian J. Munroe, Michael A. Shapiro, and Richard J. TemkinPlasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

(Dated: January 25, 2013)

We present the theoretical analysis and computer simulation of the wakefields in a 17 GHz Pho-tonic Band Gap (PBG) structure for accelerator applications. Using the commercial code CSTParticle Studio, the fundamental accelerating mode and dipole modes are excited by passing an18 MeV electron beam through a seven-cell traveling wave PBG structure. The characteristics ofthe longitudinal and transverse wakefields, wake potential spectrum, dipole mode distribution andtheir quality factors are calculated and analyzed theoretically. Unlike in conventional disk-loadedwaveguide (DLW) structures, three dipole modes (TM11-like, TM12-like and TM13-like) are excitedin the PBG structure with comparable initial amplitudes. These modes are separated by less than4 GHz in frequency and are damped quickly due to low radiative Q factors. Simulations verifythat a PBG structure provides wakefield damping relative to a DLW structure. Simulations weredone with both single-bunch excitation to determine the frequency spectrum of the wakefields andmulti-bunch excitation to compare to wakefield measurements taken at MIT using a 17 GHz bunchtrain. These simulation results will guide the design of next-generation high gradient acceleratorPBG structures.

PACS numbers: 29.17.+w

I. INTRODUCTION

Future high-energy accelerators will require high-gradient electromagnetic fields to accelerate the electronbeam to obtain high luminosity. Besides the accelerat-ing mode, high order modes (HOMs) are excited in theaccelerator cavity which will affect the electron beamand decrease the accelerating efficiency and beam sta-bility. The electromagnetic fields generated in the accel-erator structure by passing charged particles are calledwakefields. Wakefield damping is an important consid-eration for the design of future high-gradient accelera-tor structures. Many innovative methods have been pro-posed to provide sufficient wakefield damping for high en-ergy accelerators. These methods include: HOM damp-ing couplers [1], choke-mode structures [2, 3], damped,detuned structures (DDS) [4, 5], CLIC structure withwaveguide HOM damping [6], square [7] and triangularlattice metallic PBG structures [8, 9] and dielectric PBGstructures [10–13].

The goal of PBG accelerator structures is to providea structure with very high wakefield damping across allHOM frequencies. To validate this design goal, and todesign the next set of experimental PBG structures, thewakefields in a PBG structure need to be calculated andcompared to the wakefields in other types of accelerator

∗Electronic address: hu$\_$m@uestc.edu.cn

structures. This will allow the development of a theoryby which to predict the wakefields in new PBG designs,and provide a quantitative evaluation of the damping inthe PBG structure.

The general theoretical work for the analytical cal-culation of wakefields has been developed in [14–17].The wake potential is the summation over all the struc-ture modes excited by the impulse. The longitudinaland transverse wake potentials in cylindrically symmetricstructures can be written as [14, 17]:

Wz(r′, r, s) =

∞∑

m=0,n=1

2

(

r′

d

)m( r

d

)m

cos(mθ)

kzmn(d) cos(ωmns

c

)

exp

(

−ωmns

2Qmnc

)

W⊥(r′, r, s) =

∞∑

m,n=1

2m

(

r′

d

)m( r

d

)m−1

(1)

(r cos(mθ)− θ sin(mθ))k⊥mn(d)

sin(ωmns

c

)

exp

(

−ωmns

2Qmnc

)

where s is the distance behind the driving bunch, r′ isthe displacement of the driving bunch, r is the test beamtransverse position, d is the radius of the iris aperture,kzmn is the loss factor, k⊥mn is the kick factor, ωmn is thefrequency, c is the speed of light and Qmn is the qualityfactor of the TMmn mode. In Eq. 1:

2

kzmn(d) =|Vmn(d)|

2

4Umn(2)

k⊥mn(d) =kzmn(d)

ωmnd/c(3)

where Vmn is the voltage along one period and Umn

is the stored energy in the TMmn mode. From Eq. 1we can see that the initial transverse wake potential de-pends on the kick factor k⊥mn, and damping is inducedby the damping factor exp (−ωmns/2Qmnc). Low initialtransverse wake potential can be reached by decreasingthe kick factors of the modes, but effective wake poten-tial damping can be achieved primarily by decreasing thequality factor Qmn.The longitudinal and transverse wake potentials are

described in the time domain. We define correspondinglongitudinal and transverse impedances for the frequencydomain description [16]:

Zz(ω) =1

c

∫

∞

−∞

Wz(s)e−jωs/cds (4)

Z⊥(ω) =1

jc

∫

∞

−∞

W⊥(s)e−jωs/cds

The above formulas can be used to analytically cal-culate the wake potentials in right-cylinder structuresin which the electromagnetic fields can be expressed ina closed form, such as pillbox resonators [18] and disk-loaded waveguide structures [19, 20] with excitation by apoint charge [21] or charged rings [22].The frequency selectivity of photonic band gap (PBG)

structures makes them appealing for wakefield dampingin accelerator applications. By introducing a defect inthe center of a triangular lattice of metallic rods, a PBGcavity can be formed which confines a particular operat-ing mode and lets other modes radiate through the lat-tice. This radiative damping decreases the Q-factor of thedipole modes, making the wake damp faster. This kindof PBG structure has been extensively researched theo-retically and experimentally [23–25]. A 17 GHz six-celltraveling wave PBG accelerator structure has previouslybeen designed, simulated and tested at the MIT PSFC[8, 9].Previous theoretical work suggested that HOMs are

not supported in PBG structures [8]. Later theoreticaland experimental research indicated that HOMs do existin PBG structures, but with low Q and high loss [9].Previous experimental work has shown that PBG

structures composed entirely of round rods can operateat gradients above 80 MV/m [26]. This performance canbe improved by giving the rods closest to the defect re-gion elliptical cross sections, while keeping the other rodsrounds. A PBG structure with this rod configuration hasbeen shown to operate at a gradient above 125 MV/m[27]. This work represents the first step in the analy-sis of the wakefield damping in these high-gradient PBG

FIG. 1: Schematic of a seven cell PBG structure. The struc-ture is formed by three rows of a triangular lattice of metalliccylindrical rods. The central rod is removed to form a defectregion to confine the fundamental TM01 mode. The dimen-sions are given in Table I.

structures, which in turn will guide the design of futurehigh-gradient PBG structures for testing with and with-out beam.

In this paper, we report the results of PIC simulationsof the wake potential in a seven-cell 17 GHz PBG struc-ture based on the structure tested at MIT. Both six andseven cell structures were simulated, with the seven-cellsimulation showing more initial promise. While the re-sults presented here are for the seven cell structure, allthe wakefield results have been reproduced using a sixcell model. In both cases the model is simplified fromthe structure tested at MIT; the CST models all haveuniform cells while the MIT experimental structure hascoupling cells and waveguides forming the first and sixthcells.

The results from CST Particle Studio show that thereare three dipole-like modes: TM11-like, TM12-like, andTM13-like, which are excited by an off-axis electronbeam. In this paper, the notation of TMmn means thatthe mode has m azimuthal variations and n radial vari-ations in the entire PBG cross-section, not just in thedefect area. The total wake potential and the charac-teristics of each dipole mode are analyzed. An equiv-alent disk-loaded waveguide structure is also simulatedand the wakefields are compared to the PBG structure,showing that the PBG structure damps the transversewakefield effectively. The structure of this paper is: thewake potential and spectrum calculation is in Section II,the analysis of the dipole modes is shown in Section III,Section IV demonstrates the comparison results betweenPBG and DLW structures, multi-bunch simulation andbeam loading calculation are described in Section V, andfinally the paper is concluded in Section VI.

II. WAKE POTENTIAL CALCULATION

The PBG structure with three rows of metallic rods isshown in Fig. 1 and the dimensions of the PBG structure

3

are shown in Table I. All the parameters of this 17 GHzPBG structure come from the traveling-wave PBG struc-ture design [8] which has been tested at MIT [9]. Wakepotentials and wake potential impedances are calculatedper meter of structure length. The Wakefield module inCST Particle Studio is used to calculate the longitudinaland transverse wake potentials. The simulations consid-ered both the ohmic loss due to the finite conductivity ofthe copper structure and diffractive losses due to powerincident on the absorptive boundary layer surroundingthe structure. In the experimental test of the structurethe diffracted power was dissipated by diffuse reflectionsin a large external vacuum chamber; dissipation of thispower in an accelerator structure represents a future en-gineering challenge.

For single-bunch measurements, an electron beam ofzero transverse dimension and a 1 mm Gaussian lengthis injected into the beam tunnel with a 0.8 mm displace-ment in the x direction, as shown in Fig. 1. We havealso calculated the wake potentials with the same beamdisplacement in the y direction, and the results are thesame.

The longitudinal and transverse wake potential areshown as a function of distance behind the bunch inFig. 2(a) and Fig. 3(a). The longitudinal and transversewake potential impedances are obtained to analyze fre-quency domain features in Fig. 2(b) and Fig. 3(b).

The longitudinal wake potential calculated on thebeam track decays slowly, as shown in Fig. 2(a). The nor-malized frequency spectrum of the longitudinal wake po-tential shows that the fundamental mode has three peaksat 17.002 GHz, 17.092 GHz and 17.176 GHz, as shownin top right of Fig. 2(b). These peaks reveal the differentphase advances (0, π/3, 2π/3) of the fundamental modeexcited by a single bunch in the 7-cell PBG structurewhich agree with the previous theoretical analysis andthe cold test in [8, 28]. The fundamental frequency reso-lution in the spectrum is 6 MHz which is limited by thefinite simulation length. The dipole modes do not con-tribute to the longitudinal wake because the longitudinalelectric fields of dipole modes are zero in the center, sothere is no peak.

The transverse wake potential, shown in Fig. 3(a),can be seen to damp much faster than the longitudinalwake potential. The transverse wake potential damps

TABLE I: Dimensions of the PBG structure

Geometric parameters ValueRod radius, a 1.04 mmRod spacing, b 6.97 mmIris thickness, t 1.14 mmIris radius, d 2.16 mmCavity length, L 5.83 mmTM01 mode frequency 17.14 GHzAccelerating mode 2π/3 modeNumber of cells 7

FIG. 2: Calculation of longitudinal wake potential andimpedance:(a) Longitudinal wake potential on a line paral-lel to the structure centerline and displaced 0.8 mm in the xdirection, i.e. following the path of the beam; (b) Longitudi-nal wake potential impedance.

quickly because its main contributions come from thedipole modes and the dipole modes have small Q-factors.The fundamental mode present in the frequency spec-trum of the transverse wake potential is formed due to theirises and confined in the defect area. The high Q-factorof the TM01 mode keeps the transverse wake potentialat a small but stable value even 50000 mm behind theelectron beam. Three dipole modes are excited at 23.0GHz, 24.8 GHz and 27.5 GHz with comparable initialamplitudes. The field distributions of these modes areshown in Fig. 4. These modes are distinguished as TM11-like, TM12-like and TM13-like modes separately and allof these modes are not confined within the defect region,but radiate away from the center to the outer boundary.The detailed analysis of the dipole modes will be givenin Sec. III.

4

FIG. 3: Calculation of transverse wake potential andimpedance: (a) Transverse wake potential on a line paral-lel to the structure centerline and displaced 0.8 mm in the xdirection, i.e. following the path of the beam; (b) Transversewake potential impedance.

III. DIPOLE MODE ANALYSIS

Fig. 3(a) shows that the transverse wake potential inthe PBG structure is quickly damped. The transversewake is dominated by three dipole modes, with similarinitial amplitudes and with frequencies within 4 GHz ofeach other. This indicates that the previous approachof considering only the lowest-frequency dipole mode isinsufficient for predicting the transverse wake potential.To isolate the contribution to the wake potential from

each dipole mode independently, numerical filters wereused to separate the different modes within the wake po-tential. The filtering was done in Matlab using the digi-tal signal processing toolbox. For each frequency a band-pass filter was applied in frequency space. An inverse fastFourier transform was then applied to the filtered spectrato obtain the time-domain signal showing the dampingfor each dipole mode.Fig. 4 shows the separate frequency components of the

transverse wake potential versus time (t = s/c) and the

field distribution at each dipole mode frequency. The fun-damental mode (Fig. 4(a)) is confined in the defect regionin a PBG cavity with a high Q-factor, providing a small,slow-decaying transverse wake potential. The transverseelectric field component of the TM01 mode is caused bythe irises. The beating visible in the fundamental modeis caused by the different fundamental modes shown inFig. 2(b).The lowest-frequency TM11-like dipole mode

(Fig. 4(b)) decays more slowly than the two otherdipole modes. The TM12-like dipole mode (Fig. 4(c))reaches a maximum amplitude a short time after thebunch passes and decays faster than the TM11-likemode. The TM13-like mode (Fig. 4(d)) has the highestfrequency and is excited with less than half of theintensity of the previous dipole modes and decays fasterthan either of the lower-frequency dipole modes. Therelative intensity of these dipole modes agrees well withthe wake potential spectrum shown in Fig. 3(b).From Eq. 1, the decay of the wake potential is related

to the damping factor exp (−ωmns/2Qmnc). Because thefrequencies of these dipole modes are close, the qualityfactor of each mode is the dominant contributor to thedamping of each mode.To calculate the quality factor of each mode, eigen-

mode simulations are run with the same simulation en-vironment as in CST Particle Studio, i.e. a lossy copperstructure with an absorbing boundary layer located out-side of the PBG lattice. Because CST Microwave Studiodoes not support the use of absorptive boundary layersin eigenmode simulations, the Ansys code HFSS, whichdoes support absorptive boundary layers in eigenmodesimulations, is used instead. All three dipole modes arefound to have a phase advance per cell of approximatelyπ in the CST simulation, so the phase advance per cellis set to 2π/3 for the fundamental mode and π for thedipole modes in the HFSS simulations.The results of the eigenmode calculations in HFSS are

shown in Table II. The quality factors in Table II con-tain both the ohmic losses and diffractive losses in thestructure. The frequencies of the modes in the HFSSsimulations agree well with the frequencies observed inthe CST simulations.The HFSS results provide good qualitative agreement

with the damping times of the respective dipole modescalculated in the CST simulation. The 1/e distance ofthe wake potential, (2Qmnc)/ωmn from Eq. 1, is calcu-lated for each mode from the Q-factor and ω obtained

TABLE II: Comparison of 1/e distances calculated fromHFSS and CST simulation results.

Mode Freq. (GHz) Q value HFSS 1/e dist. CST 1/e dist.TM01-like 17.1 3955 2.2× 104mm 3.0× 104mmTM11-like 23.2 71 290mm 480mmTM12-like 24.7 57 220mm 210mmTM13-like 27.0 45 170mm 140mm

5

FIG. 4: Transverse wake potential as a function of time for each mode in the PBG structure: (a) TM01-like mode at 17.1 GHz;(b) TM11-like mode at 23.0 GHz; (c) TM12-like mode at 24.8 GHz;(d) TM13-like mode at 27.5 GHz.

from HFSS. The 1/e distance is also calculated from thedecay distance of the wakes observed in CST. These twovalues of 1/e distance are shown in columns 3 and 4 of Ta-ble II. The two different simulation techniques give rea-sonable agreement between the decay distances for eachmode. This validates the use of simple single-cell eigen-mode models to predict approximate decay distances formodes excited by the beam.

Unlike in conventional disk-loaded waveguide struc-tures, the different dipole modes in the PBG structure areexcited with small differences in frequency. In order todetermine the relationship between the number of dipolemodes excited and the number of rows of rods in thePBG structure, two-row and four-row PBG structures

were modeled in addition to the three-row structure. Allthe other parameters of each structure were unchangedfrom the three-row PBG structure, with the only changebeing the number of rows of rods in the structures.

The transverse wake potential impedances defined by

TABLE III: Dipole modes frequency of different rows PBG

Dipole Mode 2-row PBG 3-row PBG 4-row PBGTM11-like 23.6 GHz 23.0 GHz 22.7 GHzTM12-like 26.6 GHz 24.8 GHz 23.9 GHzTM13-like / 27.5 GHz 25.7 GHzTM14-like / / 27.8 GHz

6

FIG. 5: Transverse wake potential impedances: (a) Trans-verse wake potential impedance in a two-row PBG structure,two dipole mode peaks are at 23.6 GHz and 26.6 GHz; (b)Transverse wake potential impedance in a three-row PBGstructure, three dipole mode peaks are at 23.0 GHz, 24.8 GHzand 27.5 GHz; (c) Transverse wake potential impedance in afour-row PBG structure, four dipole mode peaks are at 22.7GHz, 23.9 GHz, 25.7 GHz and 27.8 GHz

Eq. 4 of these structures are shown in Fig. 5, and dipolemodes frequencies are listed in Table III. In the two-row PBG structure, however, there are only two dipolemodes; the TM11-like mode at 23.6 GHz and the TM12-like mode at 26.6 GHz. The impedance of the TM11-likemode is higher than that in the three-row PBG, but theimpedance of the TM12-like mode is much smaller. Inthe four-row PBG structure there are four dipole modesexcited; the TM11-like mode at 22.7 GHz, the TM12-likemode at 23.9 GHz, the TM13-like mode at 25.7 GHz, andthe TM14-like mode at 27.8 GHz. The highest-impedancedipole mode in the four-row PBG is the TM12-like mode,as in the three-row PBG. The impedance of the TM14-like mode is smaller by a factor of 4 than that of thehighest-impedance dipole mode. From the above threesituations, we can find that as the number of rows in-creases, more dipole modes are excited, but the frequen-cies of the dipole modes decrease. Unlike the DLW, thedipole modes of the PBG structure are spread out overa narrow frequency range. All modes listed in Table IIIare confirmed by the field distribution graphs in CST.Although the number of dipole modes are different andthe dipole modes are at different frequency, the initialtransverse wake potentials in the 2-row, 3-row and 4-rowPBG structures are almost the same.

IV. COMPARISON OF WAKE POTENTIAL IN

PBG AND DLW

The previous results confirm that the PBG structureprovides wake potential damping, although three dipolemodes are excited. To compare the strength of the wakepotential damping in the PBG structure we compare thewake potential between the PBG structure and an equiv-alent 7-cell traveling wave disk-loaded waveguide struc-ture with the same iris geometry, as shown in Table IV.We used the CST PS Wakefield module to simulate theequivalent disk-loaded waveguide structure.The transverse wake potentials for the disk-loaded

waveguide structure and the PBG are shown in Fig. 6.From the theory, the quality factor of the PBG struc-ture includes two parts: Qohm and Qdiff . Qohm dependson the ohmic loss of copper, and Qdiff accounts for thediffractive loss in the structure. The Qdiff in the disk-loaded waveguide structure is very large because of theclosed cavities, so the total Q in the disk-loaded waveg-

TABLE IV: Dimensions of the DLW structure

Geometric parameters ValueCavity radius, b 6.91 mmIris thickness, t 1.14 mmIris radius, d 2.16 mmCavity length, L 5.83 mmAccelerating mode 2π/3 modeNumber of cells 7

7

FIG. 6: Comparison of transverse wake potential betweenDLW and PBG structures. The blue line is DLW wake po-tential, and the red is PBG wake potential: (a) The envelopeof transverse wake potential in DLW and PBG along 50000mm; (b) The first 100 mm transverse wake potential in bothstructures.

uide structure depends only on the ohmic Q. The diffrac-tive Q of the fundamental mode in the PBG is finite butmuch larger than Qohm because this mode frequency liesin the band gap of the structure.

Fig.6 shows the comparison of the transverse wakepotential between the disk-loaded waveguide structureand the PBG structure. This clearly indicates that thewake potential in the PBG structure damps much fasterthan in the disk-loaded waveguide structure. In the PBGstructure the transverse wake potential is highly dampedwithin the first 500 mm, whereas the transverse wakepotential in the disk-loaded waveguide structure is notsignificantly damped even at 50000 mm after the drivebunch. This proves that the PBG structure provides sig-nificant damping of the transverse wake potential relativeto a disk-loaded waveguide structure. Fig.6(b) shows adetail of the first 100 mm of the transverse wake poten-tial, indicating that the initial amplitude of the trans-verse wake potential in both structures is approximatelythe same, but the wake in the PBG structure shows sig-

FIG. 7: Comparison of transverse wake potential impedancesbetween PBG and DLW structures: (a) The transverse wakepotential impedance in the DLW; (b) The transverse wakepotential impedance in the PBG (same as Fig. 3(b)).

nificant damping after the first 30 mm.Fig. 7 shows the comparison of the transverse spec-

trum between the PBG and the disk-loaded waveguidestructure. Fig. 7(a) shows that in the disk-loaded waveg-uide structure the TM11 π mode (25.1 GHz) has an ex-tremely high peak, and the next higher peaks appear at38.5 GHz and 54.5 GHz. In conclusion, all of these sim-ulations demonstrate the effective damping of wakefieldsin PBG structures designed for accelerator applications.

V. MULTI-BUNCH SIMULATION AND BEAM

LOADING CALCULATION

A six-cell PBG structure with the same iris geome-try has been experimentally tested at MIT using theMIT/Haimson accelerator [9]. This experiment was con-ducted with a train of 1 ps bunches spaced at 17 GHz.This was modeled in the PIC module in CST using amulti-bunch excitation. The simulated PBG structure,

8

shown in Fig. 1, had 7 cells to be consistent with theother simulations shown here, and to reduce computa-tion time used uniform cells instead of the coupling cellsand waveguides used in [9]. The simulation used an on-axis train of 1 ps long bunches separated by 58.1 ps tosimulate the long bunch train of the experiment. Thetotal charge of one bunch is 20 pC, corresponding to 344mA current. The longitudinal E-fields are calculated on-axis.The longitudinal E-fields at the center of a cell on

the beam axis and 35 mm off the beam axis are cal-culated in CST, showing that most of the energy in thefields is in the fundamental mode, with the second andthird harmonics observed approximately 30 dB below thefundamental. The magnitude spectrum of the longitudi-nal E-field calculated outside the structure about 35 mmaway from the beam reveal that the fundamental modeis well confined to the defect area, but the second har-monic, whose frequency is in the pass-band of the lattice,propagates out of the lattice with a small decrease inamplitude. Three dipole modes can be recognized withmuch lower energy. Due to the coherence, the fields atthe beam train’s frequency (17.2 GHz) and its harmonics(34.4 GHz, 51.6 GHz, etc) are enhanced, so the energyin the dipole modes is much lower than the fundamentalmodes and the harmonics. The calculated E-field at thefundamental is about 10dB lower than that at the sec-ond harmonic, which agrees well with the experimentalresults measured outside the chamber window [9].As a check on the applicability of the simulation re-

sults to experiment, we can compare the beam loadingin the simulation to the beam loading in the theory andthe experiment [9]. From the theory [29], the longitudi-nal electric field induced by the beam after traversing adistance l should be:

Eb = irs(

1− e−Il)

(5)

where, i is the beam current, rs is the shunt impedanceper unit length and I is the voltage loss per unit length.I can be described as a function of frequency (ω), groupvelocity (Vg), and quality factor (Q) of the structure [30]:

I =ω

2VgQ(6)

From Eq. 5 and Eq. 6, we can calculate the magni-tude of the E-field induced by the bunched beam. Thetheoretical magnitude of Eb in the middle of the fifthcell (i.e. at z = 32.1mm) , is calculated to be 3.4 × 106

V/m, using Vg = 0.013c, rs = 98MΩ/m, Q = 4200 [8],and beam current i = 344 mA. In the CST simulationthe steady-state amplitude of Eb measured on-axis in themiddle of the fifth cell, when driven by a train of on-axisbunches, is observed to be 4.6 × 106 V/m. This resultagrees reasonably well with the theory.We can calculate the induced power using:

Pb =E2

b

2Irs=

E2

bVgQ

ωrs=

(

i2rs2I

)

(1− e−Il)2 (7)

Substituting the Eq. 6 into the Eq. 7, we obtain:

Pb =

(

i2rsVgQ

ω

)(

1− exp

(

−ωl

2VgQ

))2

(8)

Because ωl2VgQ

is much smaller than 1, we can use Tay-

lor series expansion to get the following expression:

Pb = i2ωl2rs4VgQ

(9)

Using the steady-state field amplitude from the simu-lation and Eq. 7, the electromagnetic power induced bythe beam in the simulation is calculated to be 33 kW. Bycomparison the theoretical power induced by the beamafter 32.1 mm is 18 kW.From [9] we can extrapolate the experimental value of

the power induced by a 344 mA current beam, givinga value of approximately 13 kW in a 6-cell PBG struc-ture with two ports coupling the field at the first andthe sixth cells. This coupling makes the effective lengthof the structure at the measurement location shorter byapproximately the length of one full cell, making the ef-fective length 26.7 mm instead of the 32.1 mm used inthe simulation. In Eq. 8 we can see that the power in-duced by the beam scales as the square of the length.If we scale the power predicted by the CST result wefind a beam induced power of 23 kW. This representsreasonable agreement between the power induced by thebeam in the simplified geometry used in the simulationand the power induced by the beam that was observedexperimentally.

VI. CONCLUSION

The longitudinal and transverse wake potentials ex-cited by an electron beam in a traveling-wave PBG struc-ture have been simulated using CST Particle Studio.Simulation of a disc-loaded waveguide structure with thesame iris geometry shows that the PBG structure caneffectively confine the fundamental mode and damp thedipole modes excited by the bunch. While the resultspresented here are for 7-cell structures, simulations with6-cell structures have shown excellent agreement.The wake potential impedances for the longitudinal

and transverse wakes were calculated from the time-domain wake potentials using Matlab. Numerical filterswere used to isolate each mode from the wake potentialspectrum. This frequency information was then used torecover the contribution to the wake potential from eachmode; this provides the 1/e decay distance for each mode.

9

The modes observed in CST simulations of the PBGstructure were confirmed using single-cell eigenmode sim-ulations in HFSS. The single-cell simulations were usedto confirm the frequency of the observed mode, and thento provide a Q-factor for that mode for quantitative com-parison. These simulations showed reasonable agreementbetween the decay distances of the wakes observed in thefrequency domain (HFSS) and particle-in-cell (CST) sim-ulations for the fundamental mode and the three lowest-frequency dipole-like modes.The combination of particle-in-cell (CST) and

frequency-domain (HFSS) simulations provides an effec-tive method by which to study the wakefield damping inthe latest PBG structures. The testing of these struc-tures, e.g. the elliptical-rod PBG structure tested atSLAC by MIT and described in [27], has focused onachieving the highest gradient possible in a PBG struc-ture. Achieved gradient cannot be easily predicted fromsimulation, so the use of PIC simulations for wakefielddamping allows the experimental efforts to continue tofocus on achieved gradient.The PBG lattice was found to have dipole-like modes

with peaks in field amplitude within the lattice, givingrise to three dipole modes which are excited with similaramplitudes over a 4 GHz frequency range in the three-row PBG lattice. CST simulations confirmed that thenumber of distinct dipole modes in the PBG lattice isdetermined by the number of rows of rods in the struc-ture. This means that, while going to a larger numberof rows of rows gives a larger diffractive Q-factor for the

fundamental mode, more dipole modes are supported inthese larger structures.

A comparison of the transverse wake potentials of thePBG and DLW structures shows that the PBG structuredoes not significantly alter the initial excitation of HOMs,based on the initial amplitude of the wake potential, butthe PBG does damp the wake potential much faster thanthe DLW structure. This validates the use of the HOMQ as a design metric for future PBG structures.

In addition to the single bunch simulations used toinvestigate the wake potential in the PBG structure, abunch train spaced at the fundamental frequency of thestructure was also simulated. This reflects the micro-bunched beam used the MIT experiment to study wake-fields in the traveling-wave PBG structure. The full ex-perimental geometry could not be simulated, but thebeam loading calculated in the CST simulation agreeswith the experimental beam loading.

Acknowledgments

The authors thank Zenghai Li from SLAC and EmilioNanni, Sergey Arsenyev from PSFC, MIT for helpfuldiscussions. The work is supported by Department ofEnergy High Energy Physics, under contract No.DE-FG02-91ER40648 and Fundamental Research funds forthe Central Universities under the contract No.ZYGX2010J055

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