Analysis and Measurement of the Transfer Matrix of a 9-cell 1.3-GHz Superconducting Cavity

0.9 MB·

by  A. Halavanau

FERMILAB-PUB-17-020-APC
Analysis and Measurement of the Transfer Matrix of a
9-cell 1.3-GHz Superconducting Cavity
A. Halavanau1,2, N. Eddy2, D. Edstrom2, E. Harms2, A. Lunin2,
P. Piot1,2, A. Romanov2, J. Ruan2, N. Solyak2, V. Shiltsev2
1 Department of Physics and Northern Illinois Center for Accelerator &
Detector Development, Northern Illinois University, DeKalb, IL 60115, USA
2 Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
(Dated: January 31, 2017)
Superconductinglinacsarecapableofproducingintense,stable,high-qualityelectronbeamsthat
have found widespread applications in science and industry. The 9-cell 1.3-GHz superconducting
7 standing-waveacceleratingRFcavityoriginallydevelopedfore+/e− linear-colliderapplications[B.
1 Aunes,et al. Phys. Rev. STAccel. Beams3,092001(2000)]hasbeenbroadlyemployedinvarious
0 superconducting-linac designs. In this paper we discuss the transfer matrix of such a cavity and
2 present its measurement performed at the Fermilab Accelerator Science and Technology (FAST)
n facility. The experimental results are found to be in agreement with analytical calculations and
a numerical simulations.
J
PACSnumbers: 41.75Ht,41.85.-p,29.17.+w,29.27.Bd
7
2
I. INTRODUCTION tion theory yields the focusing strength, K¯ = (E0e)2,
] r −8(γm)2
h for the case of a “pure” standing wave resonator. The
p
The 1.3-GHz superconducting radiofrequency (SRF) equation of motion then takes form:
-
c accelerating cavities were originally developed in the
c (cid:18)γ(cid:48)(cid:19) (cid:18)γ(cid:48)(cid:19)2
contextoftheTESLAlinear-colliderproject[1]andwere x(cid:48)(cid:48)+ x(cid:48)+K¯ x=0, (1)
a includedinthebaselinedesignoftheinternationallinear γ r γ
.
s
collider (ILC) [2] and in the design of various other
c where x is the transverse coordinate, x(cid:48) dx, γ(cid:48) dγ =
ysi obpaseerdatoinngsuocrhpalcaanvnietydinacclcuedleeraetleocrtrfoanc-il[i2ti,e3s.], mPuoronj-e[c4t]s, eE0cos(φ)/m0c2 ≡ G¯rf/m0c2 is the no≡rmdazlized≡endezrgy
h and proton-beam accelerators [5] supporting fundamen- gradient, where γ is the Lorentz factor.
p tal science and compact high-power industrial electron The solution of the Eq. 1 through the cavity is of the
[ accelerators [6]. Such a cavity is a 9-cell standing-wave form xf = Rxi, where x (x,x(cid:48))T, here R is a 2 2
≡ ×
accelerating structure operating in the TM mode. matrix,andthesubscriptsiandf indicateupstreamand
1 010,π
The transverse beam dynamics associated to such a downstream particle coordinates respectively. According
v
7 cavity has been extensively explored within the last toChambers’model, theelementsofR aregivenby[14–
8 decade [7–10]. Most recently, experiments aimed at 17]:
1 characterizingthetransversebeamdynamicsinthistype
8 of SRF cavity were performed [11–13]. In this paper we R11 = cosα √2cos(φ)sinα,
−
0 γ
discuss the measurement and corresponding analysis of R = √8 i cos(φ)sinα,
1. the transverse transfer matrix of a 9-cell 1.3-GHz SRF 12 γ(cid:48)
0 cavity. In particular, we compare the results with the γ(cid:48) (cid:20)cos(φ) 1 (cid:21)
17 Chambers’ analytical model [14]. R21 = −γf √2 + √8cos(φ) sinα, (2)
v: Inbrief,theChambers’modelisderivedbyconsidering R22 = γγi[cosα+√2cos(φ)sinα],
i f
X the transverse motion of the particle in a standing wave
ar ERF0 ifiseltdheEpz(eza,kt)fi=eldE,0n(cid:80)knisanthceosw(anvkez)nsuimn(bωetr+aφss)o,cwiahteerde wLohreernetzαf≡act√o8r.co1Ts(hφ)eldneγγtfeir,mγfin≡antγia+ssoγc(cid:48)ziactoesdφtoistthhee2fina2l
with n-th harmonic of amplitude an, φ is an arbitrary block of the matrix is R2×2 = γi/γf. The latter eq×ua-
phase shift, and z is the longitudinal coordinate along tion also holds for the|ve|rtical degree of freedom (y,y(cid:48))
the cavity axis. owing to the assumed cylindrical symmetry. Under such
The ponderomotive-focusing force is obtained un- an assumption the equations for the vertical degree of
der the paraxial approximation as F = e(E freedom are obtained via the substitutions x y, 1 3
r r
vBφ) ≈ er∂∂Ezz where v (cid:39) c is the partic−le veloc−- and 2↔4. ↔ ↔
ity along the axial direction. Using the identity (1 The assumed axially-symmetric electromagnetic field
±
n)cos(x)sin[y(1 ny)]+(1 n)sin(x)cos[y(1 n)] = invoked while deriving Eq. 2 is often violated, e.g.,
± ± ±
(1 n)sin[(1 n)y x],Ref.[17]showsthattheforceav- due to asymmetries introduced by the input-power (or
± ± ±
eraged over one RF-period in the first order of perturba- forward-power) and high-order-mode (HOM) couplers.
2
The input-power coupler couples the RF power to
(a)
the cavity while the HOM couplers damp the harmful
HOM coupler
trapped fields potentially excited as long trains of RF pickup
bunches are accelerated in the SRF cavities. In addition
to the introduced field asymmetry, the coupler can also beam
impact the beam via geometrical wakefields [18, 19].
HOM coupler power coupler
The measurement of the transverse matrix of a stand-
ing wave accelerating structure (a plane-wave trans-
1.0
former, or PWT) was reported in Ref. [20] and bench-
0.5
marked against an “augmented” Chambers’ model de- 0
E
tailed in [15]. This refined model accounts for the pres- / 0.0
z
ence of higher-harmonic spatial content in the axial field E �0.5 (b)
profile E (r = 0,z). The present paper extends such a 1.0
z � 0.6 0.4 0.2 0.0 0.2 0.4 0.6
measurement to the case of a 1.3-GHz SRF accelerating
� � �
z (mm)
cavity and also investigates, via numerical simulation,
the impact of the the auxiliary couplers on the transfer E0 1.0⇥10�3
/
mmaentrtisxgoefnetrhaellycaivnitdyi.catTehtehsaetshimiguhleartiospnastaianldhamremaosunrices- B]x,y 00..05 RRee((EExy))
dcaovnitoyt. pAldayditaiosniganlliyfi,cwanetnrootleetfhoarttthheecparseeseonftetdhemTeaEsSuLreA- &cx,y�10..05 (c) IImm((ccBBxy))
ments are performed in a regime where the energy gain E � 0.6 0.4 0.2 0.0 0.2 0.4 0.6
[
through the cavity is comparable to the beam injection � � �
z (mm)
energy [γ γ(cid:48)L (where L is the cavity length)]. In such
i
∼
a regime, the impact of field asymmetries are expected
to be prominent. FIG.1: (Color)SchematicsoftheTESLA-typecavityconsid-
eredinthepresentstudy(a)alongwiththeaxialEz ≡Ez(r=
0,z) (a) and transverse (b) electromagnetic fields simulated
on the cavity geometric axis r = 0. In (a) and (b) the field
II. NUMERICAL ANALYSIS
are dumped at a time where the electric Ex,Ey, and Ez are
realwhilethemagneticcBx andcBy fieldsareimaginary. All
To investigate the potential impact of the couplers, the fields are normalized to the maximum axial electric field
a 3D electromagnetic model of the cavity, including E0.
auxiliary couplers, was implemented in hfss [21]. The
simulated 3D electromagnetic field map was imported
as an external field in the astra particle-tracking of the mode, also introduces time-dependent transverse
program [22]. The program astra tracked particles in electromagneticfieldsthatwillimpactthebeamdynam-
the presence of external field from first principle via ics. Given the field map loaded in astra, the program
a time-integration of the Lorentz equation. Addition- introduces the time dependence while computing the ex-
ally, astra can include space-charge effects using a ternal Lorentz force experienced by a macroparticle at
quasistatic particle-in-cell approach based on solving positionrrr (x,y,z) at a given time t as
≡
Poisson’s equation in the bunch’s rest frame [22].
FFF(rrr,t) = q[EEE(rrr)sinΨ(t)+vvv BBB(rrr)cosΨ(t)], (3)
×
The electromagnetic field map EEE(x,y,z),BBB(x,y,z)
from hfss was generated over a {rectangular computa}- where Ψ(t) ωt+φ (with ω 2πf and f = 1.3-GHz
≡ ≡
tional domain with x,y [ 10,+10] mm from the cav- is the frequency) and q and v are respectively the
ity axis and for z [ 69∈7.5−,+697.5] mm with respect to macroparticlechargeandvelocity. Inthelatterequation
the cavity center∈alo−ng the cavity length; see Fig. 1(a). the time origin is arbitrarily selected to ensure φ = 0
The mesh sizes in the corresponding directions were re- corresponds to on-crest acceleration.
spectively taken to be δx = δy = 0.5 mm and δz = 1
mm. The electromagnetic simulations assume a loaded In order to deconvolve the impact of the auxiliary
quality factor Q 3 106 as needed for the nominal couplers from the dominant ponderomotive focusing of
ILC operation. Su(cid:39)ch a×loaded Q corresponds to the in- the cavity, numerical simulations based on a cylindrical-
ner conductor of the input-coupler having a 6-mm pen- symmetric model were also performed. For these calcu-
etration depth [23]. Figures 1(b) and (c) respectively lations the axial electric field Ez(r = 0,z) displayed in
present the axial and transverse fields simulated along Fig. 1(b) is imported in astra where the correspond-
the cavity axis and normalized to the peak axial field ing transverse electromagnetic fields at given positions
E0 max[Ez(r = 0,z)]. As can be seen in Fig. 1(c) (r,θ,z) are computed assuming a pure TM010 mode and
the≡impact of the coupler, aside from shifting the center under the paraxial approximation as E = r∂Ez(r=0,z)
r −2 ∂z
3
getic distribution of macroparticles arranged on the ver-
(a) 5 (b) tices of a 2 2 transverse grid in the (x,y) plane with
) ×(cid:80) (cid:80)
5 mm 0 distribution i jδ(x−i∆x)δ(y−j∆y) where δ(x) is
Dirac’s function and taking ∆x = ∆y = 0.3 µm. The
(
m) y0 5 macroparticles, with vanishing incoming transverse mo-
m 0 − menta and located within the same axial position, are
(
0 5 0 5 tracked through the cavity field and their final trans-
y −
x (mm)
0 verse momenta recorded downstream of the cavity. Fig-
5
− ure 2(a) displays the change in transverse momentum
δP imparted by the auxiliary couplers normalized to
5 (c) ⊥
5 0 5 m) the change in longitudinal momentum δP . This is com-
− x0 (mm) (m 0 putedasthedifferencebetweenastrasim(cid:107)ulationsusing
0.08 0.12 0.16 0.20 0.24 0.28 0.32y0 5 the cylindrical-symmetric field [Fig. 2(b)] from the ones
− based on the 3D field map [Fig. 2(c)]. Figure 2(a) indi-
δr/ 103
× 5 0 5 cates a strong dipole-like field and also hints to the pres-
⊥ −
x0 (mm) ence of higher-moment components. To further quantify
theimpactoftheauxiliarycouplers,wewritethechange
FIG.2: (Color)Transversemomentummagnitude(falsecolor in transverse momentum as an electron passes through
contours) and directions (arrows) simulated downstream of the cavity δPPP (δp ,δp )T as an affine function of the
⊥ x y
the cavity as a function of initial positions. Plot (a) displays input transverse≡coordinates rrr (x ,y )T (here the
⊥,0 0 0
the momentum-kick contribution from the auxiliary couplers superscript T represents the transp≡ose operator)
only, i.e. δr(cid:48)⊥ ≡ δP1(cid:107)|δPPP⊥−kprrr⊥,0| (see Eq. 4 and 5) where
δP(cid:107) istheincreaseinlongitudinalmomentum,whileplots(b) δPPP⊥ =ddd+Mrrr⊥,0, (4)
and(c)showrespectivelythetransversemomentumsimulated
usingthecylindrical-symmetric(b)andthe3-Dfieldmap(c)
where ddd (d ,d ) is a constant vector accounting for
x y
models for the cavity. Plot (a) is obtained as the difference ≡
the dipole kick along each axis, and M is a 2 2 corre-
betweenplots(c)and(b). Thesesimulationswereperformed ×
lation matrix. The latter equation can be rewritten to
for10-MeVelectronswith E0 =30 MV/m(correspondingto
G¯ (cid:39)15 MeV/m) and φ=0◦. decompose the final momentum in terms of the strength
characterizing the various focusing components [26]
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
and Bφ = i2ωc2rEz(r =0,z) [24]. δδppx = ddx +kp xy0 +kq xy0
y y 0 0
−
(cid:18) (cid:19) (cid:18) (cid:19)
y y
MeV)15 (a) eV/c) 0 (b) +ksk x00 +ks −x00 , (5)
(10 k
gain 5 kick(−1 wrehspereectkivp,eqly≡a(cMco1u1n±tMfo2r2t)h/e2,aaxniadllkys-ks,ysm≡m(eMtr1i2c±poMn2d1e)r/o2-
energy 0-90.0 0.0 90.0 dipole−2-90.0 ddxy 0.0 90.0 mcuostinivge,eqffueactdsr.upIotles,hsokuelwd-qbueadprouipnoteledaonudtstohleantotidhaelcfoo--
efficients introduced in the latter equation are implicit
m) φ(deg) %) 1 φ(deg) functions of the cavity field and operating phase. Fur-
c/0.1 kq (c) g( (d) thermore, the linear approximation resulting in Eq. 4 re-
eV/ kkssk usin 0 kq/kp quires validation. In order to find the focusing strength
ng(k0.0 efoc−1 kkssk//kkpp wFieg.pe2r(fco)rmanedddsiimreucltalytiocnosmspimutielarthteoothffeseotndddepanredsecnotrerdelain-
cusi ativ 2 tionmatrixM necessarytodevisethefocusingstrengths
fo -90.0 0.0 90.0 rel− -90.0 0.0 90.0 in Eq. 5. Such an analysis was implemented to provide
φ(deg) φ(deg) the steering and focusing strength as a function of the
injection phase φ as summarized in Fig. 3. Our analy-
FIG.3: (Color)Energygain(a),dipolekicks(b),absolute(c) sis confirms the presence of higher-moment components
andrelative(d)focusingstrengthsasfunctionofphase(φ=0 such as quadrupole and skew-quadrupole terms as inves-
corresponds to on-crest acceleration). The relative focusing tigatedinRef.[27]. Italsoindicatesthestrengthofthese
strengthisnormalizedtotheponderomotivefocusingstrength quadrupolar components is very small compared to the
kpinEq.5. Thesimulationconditionsareaninjectionenergy cylindrical-symmetric ponderomotive focusing, specifi-
of 10 MeV and E0 =30 MV/m. cally ksk kq (10−2 kp). Finally, we observe that
thesoleno∼idalc∼onOtributio×nk (10−4 k )isinsignif-
s p
∼O ×
In order to quantitatively investigate the transverse icant. Therelativelyweakfocusingstrengtharisingfrom
beam dynamics in the cavity, we consider a monoener- the presence of the auxiliary couplers confirm that the
4
transfermatrixwillbeessentiallydominatedbythepon- dipole correctors upstream of the cavity, CAV2, were
deromotivefocusing. Thereforeweexpectthecouplersto then employed as free variables to minimize χ using a
have negligible impact on the transfer-matrix measure- conjugate-gradient algorithm while continuously scan-
ment reported in the next Section. It should however be ningthephaseofCAV2intheintervalφ [ 30◦,+30◦].
∈ −
noted that the time dependence of these effects can lead
to significant emittance increase especially when acceler-
ating low-emittance beams [7, 9, 10].
101
5
⇥
III. EXPERIMENTAL SETUP & METHOD T CAV1 CAV2 T
4
V) GUN HV101 HV103 B104/106
Theexperimentwasperformedintheelectroninjector Me
of the IOTA/FAST facility [28, 29]. The experimental y(3
g
setup is diagrammed in Fig. 4(a). In brief, an electron er
n
beam photoemitted from a high-quantum-efficiency e2
c
semiconductor photocathode is rapidly accelerated to eti (a)
n
5 MeV in a L-band 1+ 1 radiofrequency (RF) gun. ki1
∼ 2
The beam energy is subsequently boosted using two Ekinetic
1.3-GHz SRF accelerating cavities [labeled as CAV1 0
0.0 0.2 0.4 0.6 0.8
and CAV2 in Fig. 4(a)] up to maximum of 52 MeV.
∼ distance(m) 101
In the present experiment the average accelerating ⇥
gradient of the accelerating cavities was respectively set
tsoimG¯uClaAteVd1 (cid:39)bu1n5chMterVan/smvearsnedsG¯izCeAsVa2nd(cid:39)l1e4ngMtheVa/lomn.gTthhee mm)3.2 ��xy 1.04mm)
(
InOomTAin/aFlAbSuTncphhcohtaoringjeec(tQor=ap2p5e0arpCin)Faingd. 4s(ebtt,cin)gfsorusthede msize(2.4 �z 0.96msize
in the experiment. The corresponding peak current, a ea
e b
Iˆ(cid:39)30 A, is small enough to ensure wakefield effects are msb1.6 ms
vionafslCiugeAn.iVfi2cainst.KT(cid:39)he3s4imMuelVatecdonksiinsetetinctewnietrhgythdeowmnesatsrueraemd transverser0.8 (b) 00..8808ongitudinalr
l
The available electron-beam diagnostics include
0.0
cerium-doped yttrium aluminum garnet (Ce:YAG) scin- 0.0 0.2 0.4 0.6 0.8
tillating crystals for transverse beam size measurement distance(m) 101
⇥
upstream of CAV1 and downstream of CAV2 and beam
position monitors (BPMs) which were the main diag-
nostics used duing our experiment. Each BPM con-
FIG. 4: (Color) Experimental setup under consideration and
sists of four electromagnetic pickup “button” antennae
associated energy gain (a) and transverse and longitudinal
located 90◦ apart at the same axial position and at a ra- bunch size (b) simulated with Astra . In the diagram dis-
dial position 35-mm from the beamline axis. The beam playedin(a),thelabels“CAV”,“T”,“HV”,and“B”respec-
position u = (x,y) is inferred from the beam-induced tively correspond to the SRF cavities, the integrated-current
voltage on the antenna using a 7-th order polynomial monitors (ICM), the magnetic steerers, and beam-position
u = (cid:80) a F(Φ ) where Φ (j = 1,2,3,4) are the in- monitors.
i u,i j j
ducedvoltagesoneachofthefourBPMantennaandthe
coefficients au,i are inferred from a lab-bench calibration In order to measure the transfer matrix, we used a
procedure using a wire-measurement technique; see Ref. standard difference-orbit-measurement technique where
[30]. At the time of our measurements, the BPM sys- beam-trajectoryperturbationsareappliedwithmagnetic
tem was still being commissioned and the resolution was steererslocatedupstreamofCAV2andresultingchanges
limited to 80 µm in both dimensions [31]. are recorded downstream of the cavity with a pair of
(cid:39)
A priori to performing the transfer-matrix measure- BPMs. In our experiment, the perturbations were ap-
ment, the beam was centered through both cavities plied using two sets of horizontal and vertical magnetic
CAV1 and CAV2 using a beam-based alignment pro- steerers (HV101 and HV103) with locations displayed in
cedure. The beam positions [Bi = (xi,yi), where Fig. 4(a). Orbit perturbations were randomly generated
i = 104,106] downstream of the CAV2 were recorded to populate a large range of initial conditions in the 4D
for tw(cid:112)o phase points (φ1,2 = ±30◦) and the function trace space Xi ≡ (xi,x(cid:48)i,yi,yi(cid:48)). Only the perturbations
χ = (x x )2+(y y )2 quantifying the relative for which the beam was fully-transmitted were retained
1 2 1 2
− −
beam displacement was evaluated. The settings of the [the charge transmission is inferred from two integrated-
5
current monitors (ICM) shown in Fig. 4(a)]. For each For each set of perturbation the beam positions along
measured cavity phase point, 20 different sets of pertur- the beamline were recorded over 4 shots to account for
bations (associated to a set of upstream dipole-magnet possible shot-to-shot variations arising from beam jitter
settings)wereimpressed. Thebeamwasthenpropagated or instrumental error. The corresponding set of 80 or-
through CAV2 up to a pair of downstream electromag- bits were subsequently used in in the analysis algorithm
neticbutton-styleBPMs. Themeasurementofbeampo- described in the previous Section.
sitionwithCAV2“off”and“on”,where“off”meanszero
acceleratinggradient,(indirectly)providedtheinitialX
i
andfinalXf beampositionsanddivergencesrespectively 1.0 2.0
upstream and downstream of CAV2. 1.8
0.8
Correspondingly,giventhe4 4transfermatrixofthe 1.6 m)
AscuanvchitinytihtRaiat,lXtphieer=steuXrvbe0acitt+ioorδnsXaδ0rXiew0riielltl×aorteetsdhuelvtinainomXanifnao=lrboRirtbXcihtia+Xng0δei. R,R1133000...246 0111....8024 R,R(1234
downstream of CAV2 given by 0.6
0.0
30 20 10 0 10 20 30 30 20 10 0 10 20 30
− − − − − −
δX =RδX . (6)
0f 0i 0.4 1.0
)
1− 0.2 0.8
Thereforeanyselectedorbitcanserveasareferenceorbit (m R44
ttliounrefibanerddlytt(rhwaejheitccrthaonrissiefotshrameroaesutsinoednnctReh,oisfartsehsfueemrpeiannrcgaextiihasletarsapenptsrfooofxrimpmeeard-- R,R2143−00..20 00..46 R,22
0.4 0.2
tion). Consequently,impressingasetofN initialpertur- −
30 20 10 0 10 20 30 30 20 10 0 10 20 30
bations δX(n) where n = [1...N] results in a system of − − −φ(deg) − − −φ(deg)
0i
N equations similar to Eq. 6 which can be casted in the
matrix form
0.4 0.4
Ξf =RΞi, (7) 0.2 0.2
R31 R32
0.0 0.0
wthheerpeosΞitjion(js a=ndid,fiv)eragreenc4e ×assNociamteadtritcoesthceonNtaionribnigt R,13 0.2 0.2 R,14
− −
perturbations. This system can then be inverted via a 0.4 0.4
− −
least-squares technique to recover R. 30 20 10 0 10 20 30 30 20 10 0 10 20 30
− − − − − −
0.4 0.4
The error analysis includes statistical fluctuations
0.2 0.2
(whicharisefromvarioussourcesofjitter)anduncertain- R41 R42
ties on the beam-position measurements. The statistical 0.0 0.0
error bars were evaluated using an analogue of a boot- R,23 0.2 0.2 R,24
− −
strappingtechnique. Giventhatthetransformation(6)is
0.4 0.4
linear, anycoupleofinitialX andfinalX beampo- − −
k,i k,f 30 20 10 0 10 20 30 30 20 10 0 10 20 30
sition measurements can define the reference orbit while − − − − − −
φ(deg) φ(deg)
the other couples (X ,X ) for j N =k are taken as
j,i j,f
∈ (cid:54)
perturbedorbitsandthetransfermatrixcanbeinferred.
FIG. 5: (Color) Diagonal (top four plots) and anti-diagonal
Consequently,weretrievedthetransfermatrixRj associ- (top lower plots) blocks of the transport matrix. The
ated to a reference orbit (Xk,i,Xk,f). Such a procedure solid(blue)linesrepresentChambers’approximation,dashed
is repeated for all orbits k [1,N] and the resulting (green/red)linesareobtainedfrom3Dfieldmapsimulations
transfer matrix R is recorde∈d. A final step consists in for(x,x(cid:48))and(y,y(cid:48))planesrespectively,circularmarkersand
k
computing the average R and variance R2 R 2 /N purple lozenges correspond to experimental values for (x,x(cid:48))
overtheN realizationso(cid:104)fR(cid:105) . Finally,them(cid:104) eas−ur(cid:104)ed(cid:105)va(cid:105)lue and(y,y(cid:48))planesrespectively. Shadedarearepresentsmatrix
j
element variation due to RF calibration uncertainties (simu-
is reported as R= R 2 R2 R 2 1/2/√N. lation).
(cid:104) (cid:105)± (cid:104) −(cid:104) (cid:105) (cid:105)
The comparison of the recovered transfer matrix
IV. EXPERIMENTAL RESULTS elements with the Chambers’ model along with the
matrix inferred from particle tracking with astra
The elements of the transfer matrix were measured appear in Fig. 5. The shaded areas in Fig. 5 and subse-
for nine values of phases in the range φ [ 20◦,20◦] quent figures correspond to the simulated uncertainties
aroundthemaximum-acceleration(or“crest∈”)−phasecor- giventheCAV2cavitygradientG¯ =14 1MeV/m.
CAV2
responding to φ=0◦. ±
6
0.6 0.5
0.5
0.4
0.4
0.3
2γ)f0.3 2γ)f
/ /
(γi (γi0.2
0.2
0.1
0.1
0.0
0.0 28 30 32 34 36 38 40 42
30 20 10 0 10 20 30
− − − φ(deg) γi
FIG.7: (Color)Measuredscalingforthe4×4transfermatrix
FIG. 6: (Color) Measured 4 × 4 transfer-matrix determi-
determinantasafunctionofinjected-beamenergy(symbols)
nant(symbols)comparedwiththeChambers’approximation
compared with the Chambers’ approximation solid line) and
(solidline)andnumericalsimulationsusingthe3Dfieldmap
numerical simulations using the 3D field map (dashed line).
(dashedline). Theshadedarearepresentstheuncertaintyon
Theshadedarearepresentstheuncertaintyonthesimulations
the simulations due to RF calibration uncertainties.
due to RF calibration uncertainties.
Overall, we note the very good agreement between the 2 2 matrix) is expected to follow an adiabatic scal-
the measurements, simulations, and theory. The slight ing γ×/γ . The experimental measurement presented in
i f
discrepancies between the Chambers’ model and the Fig. 7 confirm a scaling in (γ /γ )2 as expected for the
i f
experimental results do not appear to have any cor- determinant of the 4 4 transfer matrix.
relations and are attributed to the instrumental jitter ×
of the BPMs, RF power fluctuations, cavity alignment
uncertainties,haloinducedbynon-ideallaserconditions. V. DISCUSSION
During the measurement, we were unable to set the
phase of the CAV2 beyond the aforementioned range
In summary, we have measured the transfer matrix of
as it would require a significant reconfiguration of the
a 1.3-GHz SRF accelerating cavity at IOTA/FAST facil-
IOTA/FAST beamline. Nevertheless we note that this
ity. Themeasurementsarefoundtobeingoodagreement
range of phases is of interest to most of the project
with numerical simulations and analytical results based
currently envisioned.
on the Chambers’ model. In particular, the contribu-
tions from the auxiliary couplers are small and does not
The coupling (anti-diagonal) 2 2 blocks of the 4 4 affect the 4 4 matrix which can be approximated by a
matrix, modeled by the simulation×, appear insignifica×nt. symmetric2× 2-blockdiagonalmatrixwithinourexper-
This finding corroborates with our experimental results; imental unce×rtainties. Furthermore the electromagnetic-
see Fig. 5. The latter observation indicates that for the fielddeviationsfromapurecylindrical-symmetricTM
010
range of parameters being explored the 3D effects asso- mode do not significantly affect the single-particle beam
ciated to the presence of the couplers has insignificant dynamics.
impact on the single-particle beam dynamics as already Itshouldhoweverbestressedthatnonlinearitiesalong
discussed in Sec. II. The measured matrix elements were with the time-dependence of the introduced dipole, and
usedtoinferthedeterminant R whichisinoverallgood non-cylindrical-symmetric first order perturbations yield
| |
agreement with the simulation and Chambers’ models; to transverse-emittance dilutions [9, 25]. Investigating
see Fig. 6. such effects would require beams with ultra-low emit-
tances. AuniquecapabilityoftheIOTA/FASTphotoin-
Finally, the field amplitude in CAV1 was varied, jector is its ability to produce flat beams – i.e. beams
thereby affecting the injection energy in CAV2 and the with large transverse-emittance ratios [32, 33]. The lat-
transfer matrix element of CAV2 measured. Since the tertypeofbeamscouldproducesub-µmtransverseemit-
beam remained relativistic the change did not affect the tances along one of the transverse dimensions thereby
injection phase in CAV2. The resulting determinant (for providing an ideal probe to quantify the emittance dilu-
7
tion caused by the cavity’s auxiliary couplers. their support. This work was partially funded by the
US Department of Energy (DOE) under contract DE-
SC0011831 with Northern Illinois University. Fermilab
VI. ACKNOWLEDGMENTS is operated by the Fermi Research Alliance, LLC for the
DOE under contract DE-AC02-07CH11359.
We are grateful to D. Broemmelsiek, S. Nagaitsev,
A. Valishev and the rest of the IOTA/FAST Group for
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