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Transverse wakefield calculations
Nuclear Instruments and Methods 212 (1983) 23-35 North-Holland Publishing Company
23
TRANSVERSE WAKEFIELD CALCULATIONS G. A H A R O N I A N , R. M E L L E R and R.H. S I E M A N N Laboratory of Nuclear Studies, Cornell University, Ithaca, N Y 14853, USA
Received 5 July 1982 and in revised form 7 December 1982
A computer program, DBCI, is described. This program is used to calculate transverse wakefields for relativistic beams passing through cylindrically symmetric cavities. Some examples of results are given, and comparison is made with measurements.
1. Introduction Theoretical studies of single beam stability in accelerators require knowledge of the electromagnetic fields generated by the beam. The "wakefields" needed for stability studies are the synchronous components of these electromagnetic fields. For an infinite, periodic, disk-loaded waveguide both longitudinal and transverse wakefields have been calculated [1,2]. For single cavities it is possible to calculate longitudinal wakefields using either S U P E R F I S H [3] or BCI [4]. S U P E R F I S H calculates the resonant frequencies and shunt impedances of cavity modes; the longitudinal wakefield is constructed from a linear combination of the contributions of the individual modes. BCI calculates the longitudinal wakefield by a numerical integration of Maxwell's equations in the time domain. Neither of these programs can calculate transverse wakefields because they can only calculate electromagnetic fields with no azimuthal dependence. To study transverse beam stability it is necessary to calculate the electromagnetic fields excited by the passage of a charge displaced from the symmetry axis of a cavity, and these fields have azimuthal dependence. In this paper we describe an extension of BCI, which we call DBCI; this program solves the problem of the fields generated by a charge with an infinitesimal displacement from the symmetry axis. DBCI is limited to cavities with cylindrical symmetry as are S U P E R F I S H and BCI. A similar program TBCI has recently been described by Weiland [5]. The work on DBCI and TBCI was concurrent, and both programs use essentially the same technique for solving the transverse wakefield problem. DBCI is a key element in a study of beam stability in large electron-positron storage rings, and as such it is important to describe in detail the techniques used and the comparison of results with measurement.
2. DBCI description DBCI is an extension of BCI; the input and geometrical conventions are the same. The significant differences are: 1) all 6 field components ( H r, H~,, I4:, Er, E , , E : ) are used in the difference equations, 2) Mesh near the cavity axis is redefined, 3) An on-axis dipole is used as the source; this is discussed in detail below. The difference equations come from the integral forms of Maxwell's equations applied to small regions of space. Fig. 1 defines the mesh and the locations of field components. The mesh consists of J-2 equal size steps in the z direction. The step sizes in r are not constrained to be equal; the default for step sizes is ~Ar] = A r 2 = . . . = A r i.
In the time dimension, the alternating-time scheme of BCI is used [4,6]. 0167-5087/83/0000-0000/$03.00 © 1983 North-Holland
24
G. A h a r o n i a n et al.
/
T r a n s v e r s e w a k e f i e l d calculations
For the off-axis mesh (k>~J) three of the field components (H~, E r, E;) are defined at the same locations as in BCI. The other three components are zero in BCI; the points where they are defined in fig. 1 are natural extensions of BCI [6,7]. The on-axis mesh (k < J -- l ) is defined differently than in BCI. This allows the off-axis mesh to be easily extended to the axis without encountering an ambiguity about the definitions of H r and E,~. The electromagnetic fields are written as a Fourier series in the azimuthal angle. The Fourier expansion coefficients are r, z, and t (time) dependent, and the leading order radial dependence is determined by the azimuthal dependence. The d o m i n a n t transverse wake for small displacements of the beam from the symmetry axis will come from fields with cos do (or sin do) azimuthal dependence. We focus our attention on these fields and define field amplitudes (denoted by a - ) with the azimuthal and leading radial dependences removed:
Er(r, z, do, t) = F-r(r, z, t) cos do,
E~,(r, z, do, t)
E:(r, z, do, t) = i?~(r, z, t ) r cos do,
Hr(r, z, do, t) = [lr(r, Z, t) sin do,
H~(r,
Hz( r, z, do, t) = [-I:(r, z, t)r sin do.
z , do,
t)
t)
= [-Iq,(r, z,
c o s dO
=
E~(r, z, t) sin do
(l)
The dipole source is oriented with positive charge at do = 0. Since the sourse is an on-axis dipole, the difference equations off-axis are homogeneous equations. Fig.
o) k+J-I
k+J Hzk --
~EE~Hk +Jk zk
~
k
k÷J*l (~ H~bk
J+J*l
I
("
E~k
ri
T Ai2
..........................
j=l . . . . . J - I
k = (i-I)J',...,(i-I)J+(J-I)
.
.
.
.
.
.
.
.
............
.
.
.
.
re
rA
ri-i k-J+l
J
zj+r
b)
2
-- E z k
(~-%~ L
4,= 4,-,W,
I
I
k+l
Hrk
)
for
.
k+i
'k-d
j-,
r=r,=o
J+l zj +, /~Z
.
I
Hzk-J
......... ~ . H @d......... 2 rz ,~ ' zj "~Hrj ................. ~ r J zj
.
ri+l
~
J+J
AXIS
7
k+l
I T
--
ri+.l
I
k-J-I
-E
~
I
Ari
THrk )Ezk
k
T
~
k+J
If
F= 0 for
r= r i
)~"
UNIT VECTORS
1
zj
F= r i
¢=~,a4,
Ezk (q~ +Aq6) ' -
zj+l
Fig. 1. The spatial mesh used in D B C I shown in the r - z plane at a r b i t r a r y q). Field c o m p o n e n t s are defined at the locations shown. See the text for details a b o u t the mesh step sizes. The coordinate system used t h r o u g h o u t the p a p e r is defined by the unit vectors which are shown. Fig. 2. The d a s h e d lines indicate paths of integration used for two of the off-axis equations. (a) The p a t h for the ~'~ equation. The q u a n t i t y r B is equal to 0.5(6 + 6+ l); for r, = r2, rA = 0.667r 2, and for r, > r 2, rA = 0 . 5 ( 6 _ ~ + 6). (b) The p a t h for the H~ equation.
G. Aharonianet al. / Transversewakefieldcalculations
25
2a shows the path used for the E¢ equation. Following the indicated path from
n.d'= off e.dA,
(2)
one obtains
. . . . (rB--rA)(n•k--U;k_l)+AZ(n;k
j - - H ~ k ) = - ~Co ( r , - - r A ) A z ( E ¢ k ,+,/2 -- E,~-'/2)
(3)
The superscript is an index representing time; the first subscript is the field component, and the second subscript indicates the mesh point. Rearranging terms and substituting the expression in eq. (1)
rA/):k_j__--_rB/):k] ~ , + , / 2 = ~ g ; , / 2 + At ( /);• - H :~,k ' ' E*k ~o 5z + rB _ rA ].
(4)
Fig. 2b shows the path used for Hrk. Starting with
fiE.d,=
3 - off ;n.dA,
(5)
one obtains
Az[ F"+'/2(eP +
__
n+ gTn+ 1/2 E~k ,/2(q,_&/,)] + 2r,AO( E~X,/2 __ -,k+,
/~
/%A zr, (2A~) -~ (H/~, +1 - Hj~,).
--
(6)
Substituting from eq. (1) and taking the limit A~ --, 0
(
~.n+l/2
~,n+l/2)
/),+1 -, At ~n+1/2 --q~k+ 1 - - ~ q , k .k = H/k + ~0 E~k + -~-)
"
(7)
The other off-axis equations are derived similarly and are presented in appendix A. The equations for the mesh points nearest the axis are derived similarly. If the problem were source free, eq. (2) could be used along the path shown in fig. 3 to obtain "n+l/2
E.k
~n-l/2 27At m~k H;k)" ~n = E~k + 4eor2z ( ~" _
(8)
Note that the leading r dependence has been explicitly integrated in obtaining eq. (8). In the presence of sources eq. (8) is valid for the fields which satisfy eq. (2). Superimposed on these fields must be those from the source equation X7 x H = j . The source is to be a charge distribution of linear density ( C / m ) X(z, t) a distance D off-axis moving with velocity v~ ( = flc~):
j = 3(r-D)x(z,t)3(q~)v~. r
3(r-D)x(z, r
t)
-~( ½+ ~
cos mq~) .
(9)
m~[
The m = 1 component drives the fields of interest; therefore, the fields contributing to the transverse wake are sourced by a dipole current distribution Jl
3 ( r -- D ) -
v~ cos ~X(z, t)--.
(10)
Multiplying the fields from the source equation by x ( = r cos @) and denoting these fields by a subscript I to indicate that we are interested for the moment only in the solution to the inhomogeneous equation curl(xH,
) = e( .~ " H, ) + x curl H, = f. ( .~ . H, ) + Xjl .
(11)
26
G. Aharonian et al. / Transverse wakefield calculations
PATH
~k
H
jr.---
J ~\\ Hrk(~+ A
zrz r= --~--
=-
OF
INTEGRATION
?
---'7 ///
\ Ezk /
\\
Hrk ( ~ - A ~ )
/I \x
/
\
i1 \l
Fig. 3. The path of integration and field components for the on-axis L'. equation. Fig. 4. The contribution of the dipole source is calculated by integrating the left-hand side of eq. (12) around the path of integration shown. The fight-hand-side of eq. (12) is integrated over the area enclosed by the indicated path.
Integrating this equation over the area in fig. 4, and applying Stokes' theorem to the left hand side
~xU,.dt=Sf(;.
H,)•.dA + f fxS, .dA.
(12)
Hl•(r, z, ep, t) = ( D~t(z, t)v/Z~rr 2) cos q).
(13)
Performing the integrals
External to the source the curl of a dipole field must be zero, therefore
n , r ( r , Z, dp, t ) = -- ( D)k(z, t)vl2~rr 2) sin q~
(14)
The source fields depend only on the product X(z, t)D. External to the source they are independent of D when X(z, t)D is held fixed. We work in the limit D ~ 0 while the product X(z, t)D is constant. Replacing the source-free fields in eq. (8) with the total fields minus the source fields gives
~n+l/2_~n E;k
--Ejk
I/2
At 1 ( - , +~
e0 r22 H'~k - H;k -,
45DX~v) 8qrr22
(15)
where X~ is X(z, t) evaluated at mesh point k and time step n. Other equations for the on-axis mesh are derived by using appropriate paths of integration. These equations are presented in appendix A. The boundary conditions at metal surfaces are that the tangential electric fields and the perpendicular magnetic fields are equal to zero. At the beginning and end of the mesh open boundary conditions are used; these are discussed in detail in appendix B. The wakefields are the synchronous components of the appropriate fields; the transverse and longitudinal wakes respectively are defined as
wl (,o)= ~ f d,f dz(E + ~oV× n),~[,-(,o + z/v)],
(16)
and 1
w, (,o) cos+ =-O-D-;/ d,f dzE,~[,- (,o + z/v)]
(17)
In these equations t o specifies a coordinate in the bunch, and Q is the total charge of the bunch. The limits
G. Aharonian et al. / Transverse wakefield calculations
27
of the z integration are from some arbitrary coordinate before the cavity, z B, to ~v. It is necessary to extend the integrals to m because some of the fields radiated at the cavity interact with particles in the bunch well beyond the end of the cavity, and even well beyond the end of the DBCI mesh. Eqs. (16) and (17) are broken up into integrals over the length of the DBCI mesh (z B to zn) and integrals beyond the DBCI mesh. The integrals over the mesh are performed directly with the fields from DBCI. At small r the fields in eq. (16) are independent of r, and the projection (onto any direction transverse to the symmetry axis) of the sum of fields in the integrand is independent of q,. Therefore, we evaluate eq. (16) at q~ = 0 and r = 0.5 (r 2 + r 3) to get
(18) as the contribution to the transverse wake from the integral over the mesh. Similarly, small r and varies with q~ as cos q~. We evaluate eq. (17) at r = r2 to get
W,,(to) =-~D f dt f[R dzEzS[t - (to + Z/V)],
E z is linear in r at
(19)
for the contribution to the longitudinal wake from the integral from zB to zR. The procedure for performing the integrals beyond the mesh is described in detail in appendix C. Fields in the beam pipe downstream of the cavity are described as an expansion in waveguide modes; the expansion coefficients are calculated from the DBCI fields. The contribution to the wakefields from each waveguide mode can be determined by performing the integral from zR to ¢¢ analytically [8]. All results presented in this paper include the contributions of the fields downstream of the cavity. In BCI the ratio of the rms of the beam (measured in meters) to the mesh step size Z~z influences convergence of the program. For o/az less than or of order ten; it has been found that the wake for an infinite conductivity pipe which should be equal to zero is not. This is a problem associated with the finite size of the mesh and open boundary conditions. The procedure for resolving the problem is to make a BCI run with the cavity, make a BCI run with the beam pipe alone, and obtain the wakefield by subtracting the results of the two runs [4]. DBCI has the same problem with o/az < 10, and the same procedure is used in this case. All results are obtained by performing the DBCI calculations for the cavity and subtracting the wakefields obtained from a DBCI calculation for the beam pipe alone. Note that the beam pipe equations discussed in appendix C also require a run with beam pipe only to obtain the fields needed in the expansion.
Note added in proof" Recently Weiland suggested that both the downstream fields and the beam pipe subtraction can be avoided for speed-of-light particles by calculating the wakefield at the beam pipe radius instead of on the axis [9]. For these particles this offers an alternative approach to calculating the complete wakefield which is simpler since it is necessary to integrate only over the cavity opening; the fields at the pipe radius are zero everywhere except in this region. Beam particles near the axis experience the same wakefield as those at the beam pipe radius, but the fields interact with the particles at different distances downstream of the cavity. 3. Results
The first results reported are for a relatively long Gaussian (o = 72.8 ps) bunch passing through a CESR 500 M H z cavity cell. The cavity geometry is shown in fig. 5, and the resultant transverse wakefield in fig. 6a. The transverse impedance, which is equal to the Fourier transform of the wakefield divided by the Fourier transform of the charge distribution is shown in fig. 6b. The cavity mode frequencies and shunt impedances can be determined from the transverse impedance by equating the area under the resonance peak to the integral of high Q resonance over its bandwidth. Table i compares DBCI results with
28
G. Aharonian et al. / Transverse wakefield calculations
- - - 8EAM SHAPE --W KE
I
+1o A 0
>_~
~9 -I0 ,
I
,
I
1.6
,
3.2
I
=
4.8
6.4
TIME ( n s )
+10
LL
oc~
,0445
J~
i
~-r= .003175
i
Z~=.0400
,, I
Z n -2,29
AXIS 3156 ALL DIMENSIONS IN METERS
0
O ~
>
I
-5 136osez
0
I
J
500 FREQUENCY
I
I000
i
1500
(MHZ)
Fig. 5. Geometry of the CESR 500 M H z cavity. The cross-hatched regions are outside the DBCI mesh. The coordinates z B and z R which are discussed in the text are indicated. Fig. 6) (a) The transverse wakefield for a Gaussian bunch with o = 72,8 ps passing through the CESR cavity. (b) The real part of the deflectirLg impedance derived from the Fourier transform of the wakefield and the Fourier transform of the bunch shape.
measurements [10]. The level of agreement is typical of that obtained by a similar comparison of BCI result with measurement. In fig. 7 we present the results for a significantly shorter bunch (Gaussian with o = 20 ps) passing through
the same cavity. The contributions
to the wakes from the DBCI
mesh [eqs. (18) and (19)] and the
downstream fields (appendix C) are separated and shown in this figure. The contributions Of the downstream pipe are important and must be included.
Table 1 Comparison of deflecting mode shunt impedances and frequencies. Mode
TMHo TMH~
Measurements [10]
DBCI results
Z"T2/Q
Freq. (MHz)
Z"T2/Q (~/m 3)
Freq. (MHz)
($2/m3)
877. 1140.
3.80 × 104 1.08 x 105
875; 1140.
4.6 × 104 7.6 × 104
G. Aharonian et al. / Transverse wakefield calculations
29
For both BCI and DBCI wakefields and for short bunches, the contribution of the downstream beam pipe is important. For extremely short bunches, and therefore for short time structure in longer bunches, most of the interaction takes place downstream of the cavity. Careful design of the structure downstream of a cavity may allow significant modification of the short time wake.
.... BEAMSHAPE --16. - - - -.--
hi
(3)
DBCl MESH TM MODES TE MODES SUM
'~ ,/.,
/
/, /
//
','8
0 I
I 50.
I tO0.
I
i tSO.
I
200.
TIME (PS)
/
t3sosez
'
b)
bJe~ ' 0 ..d
......
•
//
___
" - - " ~ . :"-k ,
~ '.. t
-
~"
,7'
z_
I~"
\
.p~
4=_ ___3 ~ _ ~ ' _ t ~ : ' ~ _ _ _ / "/- - Z _ - _. .-. _. . -. _
J
t \\ / i >
-8.
~,\
. . . . TM MODES - - SUM
x
_ 16.0
i
i 50.
t/
\ ~,~/'
I
I iO0.
I
I 150.
A
200
TIME ( p s )
Fig. 7. T h e t r a n s v e r s e (a) a n d l o n g i t u d i n a l (b) w a k e s for a G a u s s i a n b u n c h w i t h o = 20 ps p a s s i n g t h r o u g h the C E S R cavity. T h e c o n t r i b u t i o n o f eqs. (18) a n d (19) are i n d i c a t e d b y the d a s h e d lines labeled " D B C I M e s h , " a n d the c o n t r i b u t i o n s of T E a n d T M m o d e s d o w n s t r e a m o f the c a v i t y are s h o w n . T h e " S U M " curves a r e the total wakefields.
4. Conclusions
Using DBCI it is possible to calculate the deflecting wakefields necessary for beam stability studies. These studies are presently being carred out for existing and planned e + - e - storage rings. T. Weiland's program BCI served as the basis for this work; we gratefully acknowledge his fundamental contribution. We have had useful and stimulating conversations with H. Henke, R. Sundelin, and M. Tigner. This work was supported in part by the National Science Foundation.
G. Aharonian et al. / Transversewakefieldcalculations
30
Appendix A
Difference equations The equations below are used for the on-axis mesh points (k < J - 1). For the field components the superscript indicates the time index; the first subscript indicates the field component, and the second subscript indicates the mesh site (see fig. 1). The coordinate r 2 is also defined in fig. 1.
f
~n+,
H;k ~n+l Ork
At / 3 / ~ - , + 1 / 2
,/2
- -"
~Ezk At
--H;k + __
~ n
~n+l
2
EJk / +
~¢,k + 1
~ g ; , / 2 = eg;'/~ + ,at ;I;~
)+ --~k
j'
'
2~.+,/2] ~k +J -- ~--.k J.
/~.+,/2
" ~z
_
S-~
SZ
""+ -- "" At[8'" 1 ( [~;ffl/23 HJk ' -- HJk + ~o t ~6 ) r2--
1/2__ ~-n+l/2 ] ~rk+, /
/2r~+
,,11 ""~I°
I~A
,
H~,k_~ - H~,k ) ~" "°
~ . k + , / 2 = ~ f Z , / 2 + , A t % r 21(/7/, k _/~¢,,q,
45DA'~.v 8~rr2 ) .
The equations below are for the off-axis mesh points (k > J ) . The superscripts and subscripts have the same meaning as for the on-axis equations, and the coordinates rg are defined in fig. 1. The quantity N F U L is defined in fig. 8.
• "~k~n+]= n,~k~, + Attzo(
N= 1 forNFUL=0;N=2forNFUL=2 14.+1 "ark
. + At
= H~'k
-~0
( ~n+l/2__ ~'n+l/2 ~k+l
~--Z-
~k
At-
3
k
Erk
~,,-I/2
= E,~k
At + -~0
NFUL = I
NFUL: 2
NFUL =5
NFUL=4
NFUL=5
/~zt~+1/2 "
) '
(~,~,;,/,_~ ~o+,/2 ' t + 1 ~, ; b k + J
Ar
~ n
H~k I H~k +
NFUL:O
N.
. . . . . 4; N = 0 for N F U L = 5.
D;+, =OJk+--..t I.to r,Zl + r,r,+l + r,2 -,,+1/2
~-z~rk+ 1
I~zk+J ~'"+l/2Ar,-- riE"~+l/2 4 Elk
-~z
~n+l/2
-- E~k
) '
~'' '
Fig. 8. The'boundary conditions for a mesh cell are described by NFUL. In this figure the cross-hatched area is metal.
G. Aharonian et al. / Transverse wakefield calculations
-
-°+,/2
+a~
e,k
= eg;'/2
~n+l/2 E~ k
= E~ k 1/2 +
rA-
2r2 3
--ri rB =
ri +
~;"~ __ Ork_l -"
~o
-.az
At
3 d + r,r,, + d
31
"" "" ) FAHjk....~J~-F_BHjk +
r.-
r.
~n
'
~tl
)
rBH;~_-__rAH,;~_~ rB-
-
,
for i = 2,
Ari 1 2 A
for i > 2,
ri/2.
Appendix B Open boundary conditions in DBCI In this appendix we discuss the open boundary conditions used in DBCI. These reduce to boundary conditions similar to those used in BCI when it is assumed that the phase velocity of all waves propagating down the beam pipe is equal to the velocity of the charge bunch. When considering the beam pipe on either side of the cavity, two classes of fields must be considered: fields radiated from the cavity and fields due to the source (the homogeneous solution and the particular solution). The radiated fields can be represented as a linear superposition of waveguide modes. In the upstream pipe these modes travel in the - z direction, and they travel in the + z direction in the downstream pipe. The general form for these radiated fields in the downstream pipe is
H~,(r,~,z,t)=Y',
am(r,d~,w)ei~kmz-~t)do~,
m --oo
zj_ I
zj
+1 V )
UPSTREAM
-I
DOWNSTREAM
zj
BEAM PIPE
zj+ I
BEAM PIPE
Fig. 9. The beam pipe upstream and downstream of the DBCI mesh. The cross-hatched areas are outside the mesh.
(20)
G. Aharonian et aL / Transverse wakefield calculations
32
where the q, c o m p o n e n t of H is chosen as an illustration, and the sum is a sum over waveguide modes. Consider three equally spaced planes (see fig. 9) at z = zj ~, z = za, and z = za+ ~. Let the plane at zj be within the D B C I mesh, The b e a m traveling at speed v takes time At to travel from zj to z j+l. Then expanding the exponentials to first order H~,(r,q~,zj, t ) - H v , ( r , ~ , z j + l , t + A t ) = ~ _ , rtt
f
oO
d o a a , , , ( r , e ~ , o o ) [ i ( o a - k m v ) A t ].
(21)
o(3
In BCI the right-hand side of eq. (21) is set equal to zero to get the open b o u n d a r y conditions. This is equivalent to assuming the phase velocity is equal to the beam velocity for all values of frequency. It is possible to improve upon this approximation by writing an equation similar to eq. (21) at zj ~ and zj. Both of these planes are within the mesh. To the same order of expansion of the exponential as used in eq. (21). H , ( r, dO, z j + , , t + 2At) = 2H4,( r , ep, z j , t + At) -- H e ( r , ~),Zj ], t ).
(22)
For an ultrarelativistic bunch the fields due to the dipole source satisfy the same equation. For an infinitely long pipe, in the rest frame of the dipole the fields can be derived from a potential. This potential has negligible z dependence in the dipole rest frame since the bunch length is y times the length in the laboratory, and the potential must satisfy 1 _O (rOq) 1 1 02~=0 r Or~ Or ] + r ~ ~Oq)
(23)
external to the charge. Separating variables and solving for the potential with the appropriate symmetry ~=q~0
-r- -
cos q~,
(24)
where q)0 is an overall constant and R is the pipe radius. This satisfies the b o u n d a r y condition g' = 0 at r = R and eq. (23). The electric fields in the dipole rest frame can be obtained from eq. (24), and the magnetic fields in the pipe rest frame follow from Lorentz transformation of these electric fields. The magnetic fields are H~,(r, q~,z, t)
D)t(z, t)v( 1 1 ) 2¢r r~- + ~-S c o s ~ , (25)
H r ( r ' g ~ ' z ' t ) - D ) t ( z ' t ) 1v ( 2 ~ r
R 2
r 12)
sin qS"
The constant q)0 was chosen such that these agree with eqs. (13) and (14) for
r 2 << R 2.
For the source fields
H~,(r, (/),Zj+ l, t -}- 2At) "4-H o ( r , ~, za ,, t) - 2Hv,(r, ep, z j , t + A t ) Dv(1 + 1 ) - 2~r 75 - ~ c o s q ~ [ X ( z j + , , t + 2 A t ) + X ( z a
,,t)-2X(zj,
t+At)],
(26)
because za ~, za, and z j+ ~ were chosen to be separated by vAt. Hence the source fields satisfy eq. (22). This equation together with an identical one for H r are used for open b o u n d a r y conditions at the downstream end. A similar analysis for waves traveling from the cavity towards the upstream end of the mesh leads to an equation for the radiated fields at z, ~ which is now outside the mesh. H , ( r, +, za_ ,, t + 2At) = 2 H + ( r , +, z , , t + At) -- H+( r, ~a, zj+ ., t ).
(27)
G. Aharonian et al. / Transverse wakefieM calculations
33
Replacing the radiated fields by the total fields minus the source fields
H , ( r , rk, zj_,, t + 2At) = 2Hee(r , (p, z,, t + At) - H~,(r, dO, za+ t, t)
ov(,
+~-~ - ~ +
X(Zs_l,t+2at)-2X(zj,
t + z ~ t ) + X ( z j + t , t ) ] c o s e O.
(28)
We use this equation as the open boundary condition at the upstream edge of the pipe. The Hr equation is the same as the one above except the factor ( l / r 2 + l / R 2) is replaced by ( 1 / R 2 - 1/r 2) and cos q~ by sin ~.
Appendix C
Beam pipe equations The transverse and longitudinal wakefields in a smooth beam pipe of infinite conductivity are zero an ultrarelativistic beam. The wakefields from a cavity are produced by fields diffracted from discontinuities in the beam pipe which form the cavity. In the rest frame of the cavity and for ultrarelativistic beam these fields cannot interact with the beam instantaneously; time is required for
for the an the
Table 2 Beam pipe equations. The expansion coefficients a . ( w ) e ik-~R are given by the Fourier transforms of the quantities in column A. The wakefield is given by eq. (36) using these coefficients and the factors X. given below, k. are given in column B. Column A
Column B
Column C
Longitudinal wakes - f o r BCI calculations:
2
rR
[~.r'~
R 2 [ J , ( X - ) ] 2 )O r drJo~ ~ - }Ez(r, z R, t)
k. =
R2
Jo(X.) = 0 - f o r DBCI calculations:
2
1rRr2
R2[j2(Xn)]2JO
[ X.r ~ _ drJt~'~R-)Ez(r'zR't)
kn=
R2
Jl(X.) = 0
m
}k n
2R
Transverse wakes - f o r TM modes:
2 ItRr2 [ ~nr ~ ~ RZ[J2(X.)12aO drJ'[~} Ez(r'z"'t)
~/~ 2
k. =
~
~kn2
R2 ik"~R 2X. (1 - ~.flc )
JI(X.) = 0 - f o r TE modes: 2X2 R2[ j , ( X . ) ] 2 (X2
1)
f?r2
[ ~.r ~ . drJl['--R-) H z ( r ' z R ' t )
kn =
7
J;(X.) = o i For k. real F. = [~r3(k. - w/tic]+ k. - w/tic )×[Factor in column C] For k. imaginary F.
k. = iK.
1
K . +ion/tic ×[Factor in column C]
R2
ik"~°cR 2X. (T~.~,o B)
G. Aharonian et al. / Transversewakefieldcalculations
34
fields to "catch-up" with the beam particles. For extremely short bunches this occurs in the beam pipe well beyond the cavity and typically well beyond the end of the DBCI (or BCI) mesh. This appendix describes a procedure used to correct the wakefields to include the beam pipe contribution. The discussion below concerns the TM mode contribution to the transverse wake. Table 2 presents other results. The field downstream of the cavity after subtracting the source field obtained from a beam pipe only run can be written as linear combination of waveguide modes:
E.(r, z, ~, t) = cos epZJ , ~
a,(w) e i(k'-'-'°'' dw.
n
(29)
o0
The sum is over waveguide modes; the X,'s are the zeros of the Bessel function J~; R is the beam pipe radius; the k n are the wave numbers for the modes, and to insure that E_. is real and propagating away from the cavity a,(-w)
= a*(c0),
k * ( - c0) = - k,,(w).
(30)
At reference plane z R within the DBCI mesh the field after subtracting the source fields is given by
E~(r, z., q~, t) = F.z(r, z R, t ) r cos ,;b.
(31)
Substituting this into eq. (29), multiplying by
f_
r Jl(Xmr/R), and integrating from 0 to R
2 Jo a,(w) e '(k . . . . ') dw - RZ(Jz()tn)) 2
[rRr2drJ1/Xnr\" ~-)Ez(r,
zR, t).
(32)
Or, an(W ) e ik-~" is the Fourier transform of the right-hand-side of eq. (32); the a,,(w)'s are considered known. For a given waveguide mode the fields contributing to the transverse wake are [11]
(E+t~°vxH)±-
X~
1-
v l E ~ ( r , q,, ,o) e "k . . . . '.
(33)
Taking the gradient and then the limit as r ---, 0
(E+t%v×H).
-
ik, 2X,R ( 1 - ~ cwfl ) e,(koz_,~,)( ~ c o s O - 4 s i n 4 , ) .
(34)
To calculate the contribution to the wakefield from the beam pipe this field must be integrated from z R to oc at a fixed bunch coordinate specified by t n = t o + zR/flc, the time the test particle arrives at z R Wl(tn)=
f~ d z f ~ dt(JE+ldol) XHlra[l--(tR-t-(Z--ZR)/t~c)].
(35)
Substituting eq. (34) into this equation and performing the z and t integrals
W±(t.)=~--~-~_.f ~ a . ( . ) e i k " : ~ e n
i ' " ~ + : " / ' " ' F , , ( c o ) dco,
~o
where
ro( 0)=
5X7[ l-
for k , real and
1
ikR(
F n ( £ ° ) - K n+i~o/flc 2X n 1 -
) ,
'
(36)
G. Aharonian et al. / Transverse wakefieM calculations
35
for k, = i K n imaginary. The wakefield given by eq. (36) must be added to the wakefield calculated by DBCI. Similar expressions are needed for the TM mode contributions to the longitudinal wake, and the TE contribution to the transverse wake. For reference these are given in table 2 along with the longitudinal wakefield which should be used when using BCI.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
E. Keil, NIM 100 (1972) 419. K. Bane and B. Zotter, 1lth Int. Conf. on High-energy accelerators, ed., W.S. Newman. K. Halbach and R.F. Holsinger, Part. Accel. 7 (1976) 213. T. Weiland, 1lth Int. Conf. on High-energy accelerators, ed., W.S. Newman; T. Weiland, ISR-TH/80-07 (unpublished). T. Weiland, DESY 82-015 (March, 1982). K.S. Yee, IEEE Trans. Antennas Propagation AP-14 (1966) 302. BCI is a simplification of the general problem discussed in T. Weiland AEf.) 31 (1977) 116. Earlier work on this was discussed in R. Siemann, Cornell Note CBN 81-32 (Nov. 1981, unpublished). T. Weiland, Letter to BCI users (August 1982, unpublished). R. Sundelin, M. Billing, R. Kaplan, J. Kirchgessner, R. Meller, D. Morse, L. Phillips, D. Rice, J. Seeman and E. von Borstel, IEEE Trans. Nucl. Sci. NS-28 (1981) 2844. [11] J.D. Jackson, Classical electrodynamics (Wiley, New York, 1963).
23
TRANSVERSE WAKEFIELD CALCULATIONS G. A H A R O N I A N , R. M E L L E R and R.H. S I E M A N N Laboratory of Nuclear Studies, Cornell University, Ithaca, N Y 14853, USA
Received 5 July 1982 and in revised form 7 December 1982
A computer program, DBCI, is described. This program is used to calculate transverse wakefields for relativistic beams passing through cylindrically symmetric cavities. Some examples of results are given, and comparison is made with measurements.
1. Introduction Theoretical studies of single beam stability in accelerators require knowledge of the electromagnetic fields generated by the beam. The "wakefields" needed for stability studies are the synchronous components of these electromagnetic fields. For an infinite, periodic, disk-loaded waveguide both longitudinal and transverse wakefields have been calculated [1,2]. For single cavities it is possible to calculate longitudinal wakefields using either S U P E R F I S H [3] or BCI [4]. S U P E R F I S H calculates the resonant frequencies and shunt impedances of cavity modes; the longitudinal wakefield is constructed from a linear combination of the contributions of the individual modes. BCI calculates the longitudinal wakefield by a numerical integration of Maxwell's equations in the time domain. Neither of these programs can calculate transverse wakefields because they can only calculate electromagnetic fields with no azimuthal dependence. To study transverse beam stability it is necessary to calculate the electromagnetic fields excited by the passage of a charge displaced from the symmetry axis of a cavity, and these fields have azimuthal dependence. In this paper we describe an extension of BCI, which we call DBCI; this program solves the problem of the fields generated by a charge with an infinitesimal displacement from the symmetry axis. DBCI is limited to cavities with cylindrical symmetry as are S U P E R F I S H and BCI. A similar program TBCI has recently been described by Weiland [5]. The work on DBCI and TBCI was concurrent, and both programs use essentially the same technique for solving the transverse wakefield problem. DBCI is a key element in a study of beam stability in large electron-positron storage rings, and as such it is important to describe in detail the techniques used and the comparison of results with measurement.
2. DBCI description DBCI is an extension of BCI; the input and geometrical conventions are the same. The significant differences are: 1) all 6 field components ( H r, H~,, I4:, Er, E , , E : ) are used in the difference equations, 2) Mesh near the cavity axis is redefined, 3) An on-axis dipole is used as the source; this is discussed in detail below. The difference equations come from the integral forms of Maxwell's equations applied to small regions of space. Fig. 1 defines the mesh and the locations of field components. The mesh consists of J-2 equal size steps in the z direction. The step sizes in r are not constrained to be equal; the default for step sizes is ~Ar] = A r 2 = . . . = A r i.
In the time dimension, the alternating-time scheme of BCI is used [4,6]. 0167-5087/83/0000-0000/$03.00 © 1983 North-Holland
24
G. A h a r o n i a n et al.
/
T r a n s v e r s e w a k e f i e l d calculations
For the off-axis mesh (k>~J) three of the field components (H~, E r, E;) are defined at the same locations as in BCI. The other three components are zero in BCI; the points where they are defined in fig. 1 are natural extensions of BCI [6,7]. The on-axis mesh (k < J -- l ) is defined differently than in BCI. This allows the off-axis mesh to be easily extended to the axis without encountering an ambiguity about the definitions of H r and E,~. The electromagnetic fields are written as a Fourier series in the azimuthal angle. The Fourier expansion coefficients are r, z, and t (time) dependent, and the leading order radial dependence is determined by the azimuthal dependence. The d o m i n a n t transverse wake for small displacements of the beam from the symmetry axis will come from fields with cos do (or sin do) azimuthal dependence. We focus our attention on these fields and define field amplitudes (denoted by a - ) with the azimuthal and leading radial dependences removed:
Er(r, z, do, t) = F-r(r, z, t) cos do,
E~,(r, z, do, t)
E:(r, z, do, t) = i?~(r, z, t ) r cos do,
Hr(r, z, do, t) = [lr(r, Z, t) sin do,
H~(r,
Hz( r, z, do, t) = [-I:(r, z, t)r sin do.
z , do,
t)
t)
= [-Iq,(r, z,
c o s dO
=
E~(r, z, t) sin do
(l)
The dipole source is oriented with positive charge at do = 0. Since the sourse is an on-axis dipole, the difference equations off-axis are homogeneous equations. Fig.
o) k+J-I
k+J Hzk --
~EE~Hk +Jk zk
~
k
k÷J*l (~ H~bk
J+J*l
I
("
E~k
ri
T Ai2
..........................
j=l . . . . . J - I
k = (i-I)J',...,(i-I)J+(J-I)
.
.
.
.
.
.
.
.
............
.
.
.
.
re
rA
ri-i k-J+l
J
zj+r
b)
2
-- E z k
(~-%~ L
4,= 4,-,W,
I
I
k+l
Hrk
)
for
.
k+i
'k-d
j-,
r=r,=o
J+l zj +, /~Z
.
I
Hzk-J
......... ~ . H @d......... 2 rz ,~ ' zj "~Hrj ................. ~ r J zj
.
ri+l
~
J+J
AXIS
7
k+l
I T
--
ri+.l
I
k-J-I
-E
~
I
Ari
THrk )Ezk
k
T
~
k+J
If
F= 0 for
r= r i
)~"
UNIT VECTORS
1
zj
F= r i
¢=~,a4,
Ezk (q~ +Aq6) ' -
zj+l
Fig. 1. The spatial mesh used in D B C I shown in the r - z plane at a r b i t r a r y q). Field c o m p o n e n t s are defined at the locations shown. See the text for details a b o u t the mesh step sizes. The coordinate system used t h r o u g h o u t the p a p e r is defined by the unit vectors which are shown. Fig. 2. The d a s h e d lines indicate paths of integration used for two of the off-axis equations. (a) The p a t h for the ~'~ equation. The q u a n t i t y r B is equal to 0.5(6 + 6+ l); for r, = r2, rA = 0.667r 2, and for r, > r 2, rA = 0 . 5 ( 6 _ ~ + 6). (b) The p a t h for the H~ equation.
G. Aharonianet al. / Transversewakefieldcalculations
25
2a shows the path used for the E¢ equation. Following the indicated path from
n.d'= off e.dA,
(2)
one obtains
. . . . (rB--rA)(n•k--U;k_l)+AZ(n;k
j - - H ~ k ) = - ~Co ( r , - - r A ) A z ( E ¢ k ,+,/2 -- E,~-'/2)
(3)
The superscript is an index representing time; the first subscript is the field component, and the second subscript indicates the mesh point. Rearranging terms and substituting the expression in eq. (1)
rA/):k_j__--_rB/):k] ~ , + , / 2 = ~ g ; , / 2 + At ( /);• - H :~,k ' ' E*k ~o 5z + rB _ rA ].
(4)
Fig. 2b shows the path used for Hrk. Starting with
fiE.d,=
3 - off ;n.dA,
(5)
one obtains
Az[ F"+'/2(eP +
__
n+ gTn+ 1/2 E~k ,/2(q,_&/,)] + 2r,AO( E~X,/2 __ -,k+,
/~
/%A zr, (2A~) -~ (H/~, +1 - Hj~,).
--
(6)
Substituting from eq. (1) and taking the limit A~ --, 0
(
~.n+l/2
~,n+l/2)
/),+1 -, At ~n+1/2 --q~k+ 1 - - ~ q , k .k = H/k + ~0 E~k + -~-)
"
(7)
The other off-axis equations are derived similarly and are presented in appendix A. The equations for the mesh points nearest the axis are derived similarly. If the problem were source free, eq. (2) could be used along the path shown in fig. 3 to obtain "n+l/2
E.k
~n-l/2 27At m~k H;k)" ~n = E~k + 4eor2z ( ~" _
(8)
Note that the leading r dependence has been explicitly integrated in obtaining eq. (8). In the presence of sources eq. (8) is valid for the fields which satisfy eq. (2). Superimposed on these fields must be those from the source equation X7 x H = j . The source is to be a charge distribution of linear density ( C / m ) X(z, t) a distance D off-axis moving with velocity v~ ( = flc~):
j = 3(r-D)x(z,t)3(q~)v~. r
3(r-D)x(z, r
t)
-~( ½+ ~
cos mq~) .
(9)
m~[
The m = 1 component drives the fields of interest; therefore, the fields contributing to the transverse wake are sourced by a dipole current distribution Jl
3 ( r -- D ) -
v~ cos ~X(z, t)--.
(10)
Multiplying the fields from the source equation by x ( = r cos @) and denoting these fields by a subscript I to indicate that we are interested for the moment only in the solution to the inhomogeneous equation curl(xH,
) = e( .~ " H, ) + x curl H, = f. ( .~ . H, ) + Xjl .
(11)
26
G. Aharonian et al. / Transverse wakefield calculations
PATH
~k
H
jr.---
J ~\\ Hrk(~+ A
zrz r= --~--
=-
OF
INTEGRATION
?
---'7 ///
\ Ezk /
\\
Hrk ( ~ - A ~ )
/I \x
/
\
i1 \l
Fig. 3. The path of integration and field components for the on-axis L'. equation. Fig. 4. The contribution of the dipole source is calculated by integrating the left-hand side of eq. (12) around the path of integration shown. The fight-hand-side of eq. (12) is integrated over the area enclosed by the indicated path.
Integrating this equation over the area in fig. 4, and applying Stokes' theorem to the left hand side
~xU,.dt=Sf(;.
H,)•.dA + f fxS, .dA.
(12)
Hl•(r, z, ep, t) = ( D~t(z, t)v/Z~rr 2) cos q).
(13)
Performing the integrals
External to the source the curl of a dipole field must be zero, therefore
n , r ( r , Z, dp, t ) = -- ( D)k(z, t)vl2~rr 2) sin q~
(14)
The source fields depend only on the product X(z, t)D. External to the source they are independent of D when X(z, t)D is held fixed. We work in the limit D ~ 0 while the product X(z, t)D is constant. Replacing the source-free fields in eq. (8) with the total fields minus the source fields gives
~n+l/2_~n E;k
--Ejk
I/2
At 1 ( - , +~
e0 r22 H'~k - H;k -,
45DX~v) 8qrr22
(15)
where X~ is X(z, t) evaluated at mesh point k and time step n. Other equations for the on-axis mesh are derived by using appropriate paths of integration. These equations are presented in appendix A. The boundary conditions at metal surfaces are that the tangential electric fields and the perpendicular magnetic fields are equal to zero. At the beginning and end of the mesh open boundary conditions are used; these are discussed in detail in appendix B. The wakefields are the synchronous components of the appropriate fields; the transverse and longitudinal wakes respectively are defined as
wl (,o)= ~ f d,f dz(E + ~oV× n),~[,-(,o + z/v)],
(16)
and 1
w, (,o) cos+ =-O-D-;/ d,f dzE,~[,- (,o + z/v)]
(17)
In these equations t o specifies a coordinate in the bunch, and Q is the total charge of the bunch. The limits
G. Aharonian et al. / Transverse wakefield calculations
27
of the z integration are from some arbitrary coordinate before the cavity, z B, to ~v. It is necessary to extend the integrals to m because some of the fields radiated at the cavity interact with particles in the bunch well beyond the end of the cavity, and even well beyond the end of the DBCI mesh. Eqs. (16) and (17) are broken up into integrals over the length of the DBCI mesh (z B to zn) and integrals beyond the DBCI mesh. The integrals over the mesh are performed directly with the fields from DBCI. At small r the fields in eq. (16) are independent of r, and the projection (onto any direction transverse to the symmetry axis) of the sum of fields in the integrand is independent of q,. Therefore, we evaluate eq. (16) at q~ = 0 and r = 0.5 (r 2 + r 3) to get
(18) as the contribution to the transverse wake from the integral over the mesh. Similarly, small r and varies with q~ as cos q~. We evaluate eq. (17) at r = r2 to get
W,,(to) =-~D f dt f[R dzEzS[t - (to + Z/V)],
E z is linear in r at
(19)
for the contribution to the longitudinal wake from the integral from zB to zR. The procedure for performing the integrals beyond the mesh is described in detail in appendix C. Fields in the beam pipe downstream of the cavity are described as an expansion in waveguide modes; the expansion coefficients are calculated from the DBCI fields. The contribution to the wakefields from each waveguide mode can be determined by performing the integral from zR to ¢¢ analytically [8]. All results presented in this paper include the contributions of the fields downstream of the cavity. In BCI the ratio of the rms of the beam (measured in meters) to the mesh step size Z~z influences convergence of the program. For o/az less than or of order ten; it has been found that the wake for an infinite conductivity pipe which should be equal to zero is not. This is a problem associated with the finite size of the mesh and open boundary conditions. The procedure for resolving the problem is to make a BCI run with the cavity, make a BCI run with the beam pipe alone, and obtain the wakefield by subtracting the results of the two runs [4]. DBCI has the same problem with o/az < 10, and the same procedure is used in this case. All results are obtained by performing the DBCI calculations for the cavity and subtracting the wakefields obtained from a DBCI calculation for the beam pipe alone. Note that the beam pipe equations discussed in appendix C also require a run with beam pipe only to obtain the fields needed in the expansion.
Note added in proof" Recently Weiland suggested that both the downstream fields and the beam pipe subtraction can be avoided for speed-of-light particles by calculating the wakefield at the beam pipe radius instead of on the axis [9]. For these particles this offers an alternative approach to calculating the complete wakefield which is simpler since it is necessary to integrate only over the cavity opening; the fields at the pipe radius are zero everywhere except in this region. Beam particles near the axis experience the same wakefield as those at the beam pipe radius, but the fields interact with the particles at different distances downstream of the cavity. 3. Results
The first results reported are for a relatively long Gaussian (o = 72.8 ps) bunch passing through a CESR 500 M H z cavity cell. The cavity geometry is shown in fig. 5, and the resultant transverse wakefield in fig. 6a. The transverse impedance, which is equal to the Fourier transform of the wakefield divided by the Fourier transform of the charge distribution is shown in fig. 6b. The cavity mode frequencies and shunt impedances can be determined from the transverse impedance by equating the area under the resonance peak to the integral of high Q resonance over its bandwidth. Table i compares DBCI results with
28
G. Aharonian et al. / Transverse wakefield calculations
- - - 8EAM SHAPE --W KE
I
+1o A 0
>_~
~9 -I0 ,
I
,
I
1.6
,
3.2
I
=
4.8
6.4
TIME ( n s )
+10
LL
oc~
,0445
J~
i
~-r= .003175
i
Z~=.0400
,, I
Z n -2,29
AXIS 3156 ALL DIMENSIONS IN METERS
0
O ~
>
I
-5 136osez
0
I
J
500 FREQUENCY
I
I000
i
1500
(MHZ)
Fig. 5. Geometry of the CESR 500 M H z cavity. The cross-hatched regions are outside the DBCI mesh. The coordinates z B and z R which are discussed in the text are indicated. Fig. 6) (a) The transverse wakefield for a Gaussian bunch with o = 72,8 ps passing through the CESR cavity. (b) The real part of the deflectirLg impedance derived from the Fourier transform of the wakefield and the Fourier transform of the bunch shape.
measurements [10]. The level of agreement is typical of that obtained by a similar comparison of BCI result with measurement. In fig. 7 we present the results for a significantly shorter bunch (Gaussian with o = 20 ps) passing through
the same cavity. The contributions
to the wakes from the DBCI
mesh [eqs. (18) and (19)] and the
downstream fields (appendix C) are separated and shown in this figure. The contributions Of the downstream pipe are important and must be included.
Table 1 Comparison of deflecting mode shunt impedances and frequencies. Mode
TMHo TMH~
Measurements [10]
DBCI results
Z"T2/Q
Freq. (MHz)
Z"T2/Q (~/m 3)
Freq. (MHz)
($2/m3)
877. 1140.
3.80 × 104 1.08 x 105
875; 1140.
4.6 × 104 7.6 × 104
G. Aharonian et al. / Transverse wakefield calculations
29
For both BCI and DBCI wakefields and for short bunches, the contribution of the downstream beam pipe is important. For extremely short bunches, and therefore for short time structure in longer bunches, most of the interaction takes place downstream of the cavity. Careful design of the structure downstream of a cavity may allow significant modification of the short time wake.
.... BEAMSHAPE --16. - - - -.--
hi
(3)
DBCl MESH TM MODES TE MODES SUM
'~ ,/.,
/
/, /
//
','8
0 I
I 50.
I tO0.
I
i tSO.
I
200.
TIME (PS)
/
t3sosez
'
b)
bJe~ ' 0 ..d
......
•
//
___
" - - " ~ . :"-k ,
~ '.. t
-
~"
,7'
z_
I~"
\
.p~
4=_ ___3 ~ _ ~ ' _ t ~ : ' ~ _ _ _ / "/- - Z _ - _. .-. _. . -. _
J
t \\ / i >
-8.
~,\
. . . . TM MODES - - SUM
x
_ 16.0
i
i 50.
t/
\ ~,~/'
I
I iO0.
I
I 150.
A
200
TIME ( p s )
Fig. 7. T h e t r a n s v e r s e (a) a n d l o n g i t u d i n a l (b) w a k e s for a G a u s s i a n b u n c h w i t h o = 20 ps p a s s i n g t h r o u g h the C E S R cavity. T h e c o n t r i b u t i o n o f eqs. (18) a n d (19) are i n d i c a t e d b y the d a s h e d lines labeled " D B C I M e s h , " a n d the c o n t r i b u t i o n s of T E a n d T M m o d e s d o w n s t r e a m o f the c a v i t y are s h o w n . T h e " S U M " curves a r e the total wakefields.
4. Conclusions
Using DBCI it is possible to calculate the deflecting wakefields necessary for beam stability studies. These studies are presently being carred out for existing and planned e + - e - storage rings. T. Weiland's program BCI served as the basis for this work; we gratefully acknowledge his fundamental contribution. We have had useful and stimulating conversations with H. Henke, R. Sundelin, and M. Tigner. This work was supported in part by the National Science Foundation.
G. Aharonian et al. / Transversewakefieldcalculations
30
Appendix A
Difference equations The equations below are used for the on-axis mesh points (k < J - 1). For the field components the superscript indicates the time index; the first subscript indicates the field component, and the second subscript indicates the mesh site (see fig. 1). The coordinate r 2 is also defined in fig. 1.
f
~n+,
H;k ~n+l Ork
At / 3 / ~ - , + 1 / 2
,/2
- -"
~Ezk At
--H;k + __
~ n
~n+l
2
EJk / +
~¢,k + 1
~ g ; , / 2 = eg;'/~ + ,at ;I;~
)+ --~k
j'
'
2~.+,/2] ~k +J -- ~--.k J.
/~.+,/2
" ~z
_
S-~
SZ
""+ -- "" At[8'" 1 ( [~;ffl/23 HJk ' -- HJk + ~o t ~6 ) r2--
1/2__ ~-n+l/2 ] ~rk+, /
/2r~+
,,11 ""~I°
I~A
,
H~,k_~ - H~,k ) ~" "°
~ . k + , / 2 = ~ f Z , / 2 + , A t % r 21(/7/, k _/~¢,,q,
45DA'~.v 8~rr2 ) .
The equations below are for the off-axis mesh points (k > J ) . The superscripts and subscripts have the same meaning as for the on-axis equations, and the coordinates rg are defined in fig. 1. The quantity N F U L is defined in fig. 8.
• "~k~n+]= n,~k~, + Attzo(
N= 1 forNFUL=0;N=2forNFUL=2 14.+1 "ark
. + At
= H~'k
-~0
( ~n+l/2__ ~'n+l/2 ~k+l
~--Z-
~k
At-
3
k
Erk
~,,-I/2
= E,~k
At + -~0
NFUL = I
NFUL: 2
NFUL =5
NFUL=4
NFUL=5
/~zt~+1/2 "
) '
(~,~,;,/,_~ ~o+,/2 ' t + 1 ~, ; b k + J
Ar
~ n
H~k I H~k +
NFUL:O
N.
. . . . . 4; N = 0 for N F U L = 5.
D;+, =OJk+--..t I.to r,Zl + r,r,+l + r,2 -,,+1/2
~-z~rk+ 1
I~zk+J ~'"+l/2Ar,-- riE"~+l/2 4 Elk
-~z
~n+l/2
-- E~k
) '
~'' '
Fig. 8. The'boundary conditions for a mesh cell are described by NFUL. In this figure the cross-hatched area is metal.
G. Aharonian et al. / Transverse wakefield calculations
-
-°+,/2
+a~
e,k
= eg;'/2
~n+l/2 E~ k
= E~ k 1/2 +
rA-
2r2 3
--ri rB =
ri +
~;"~ __ Ork_l -"
~o
-.az
At
3 d + r,r,, + d
31
"" "" ) FAHjk....~J~-F_BHjk +
r.-
r.
~n
'
~tl
)
rBH;~_-__rAH,;~_~ rB-
-
,
for i = 2,
Ari 1 2 A
for i > 2,
ri/2.
Appendix B Open boundary conditions in DBCI In this appendix we discuss the open boundary conditions used in DBCI. These reduce to boundary conditions similar to those used in BCI when it is assumed that the phase velocity of all waves propagating down the beam pipe is equal to the velocity of the charge bunch. When considering the beam pipe on either side of the cavity, two classes of fields must be considered: fields radiated from the cavity and fields due to the source (the homogeneous solution and the particular solution). The radiated fields can be represented as a linear superposition of waveguide modes. In the upstream pipe these modes travel in the - z direction, and they travel in the + z direction in the downstream pipe. The general form for these radiated fields in the downstream pipe is
H~,(r,~,z,t)=Y',
am(r,d~,w)ei~kmz-~t)do~,
m --oo
zj_ I
zj
+1 V )
UPSTREAM
-I
DOWNSTREAM
zj
BEAM PIPE
zj+ I
BEAM PIPE
Fig. 9. The beam pipe upstream and downstream of the DBCI mesh. The cross-hatched areas are outside the mesh.
(20)
G. Aharonian et aL / Transverse wakefield calculations
32
where the q, c o m p o n e n t of H is chosen as an illustration, and the sum is a sum over waveguide modes. Consider three equally spaced planes (see fig. 9) at z = zj ~, z = za, and z = za+ ~. Let the plane at zj be within the D B C I mesh, The b e a m traveling at speed v takes time At to travel from zj to z j+l. Then expanding the exponentials to first order H~,(r,q~,zj, t ) - H v , ( r , ~ , z j + l , t + A t ) = ~ _ , rtt
f
oO
d o a a , , , ( r , e ~ , o o ) [ i ( o a - k m v ) A t ].
(21)
o(3
In BCI the right-hand side of eq. (21) is set equal to zero to get the open b o u n d a r y conditions. This is equivalent to assuming the phase velocity is equal to the beam velocity for all values of frequency. It is possible to improve upon this approximation by writing an equation similar to eq. (21) at zj ~ and zj. Both of these planes are within the mesh. To the same order of expansion of the exponential as used in eq. (21). H , ( r, dO, z j + , , t + 2At) = 2H4,( r , ep, z j , t + At) -- H e ( r , ~),Zj ], t ).
(22)
For an ultrarelativistic bunch the fields due to the dipole source satisfy the same equation. For an infinitely long pipe, in the rest frame of the dipole the fields can be derived from a potential. This potential has negligible z dependence in the dipole rest frame since the bunch length is y times the length in the laboratory, and the potential must satisfy 1 _O (rOq) 1 1 02~=0 r Or~ Or ] + r ~ ~Oq)
(23)
external to the charge. Separating variables and solving for the potential with the appropriate symmetry ~=q~0
-r- -
cos q~,
(24)
where q)0 is an overall constant and R is the pipe radius. This satisfies the b o u n d a r y condition g' = 0 at r = R and eq. (23). The electric fields in the dipole rest frame can be obtained from eq. (24), and the magnetic fields in the pipe rest frame follow from Lorentz transformation of these electric fields. The magnetic fields are H~,(r, q~,z, t)
D)t(z, t)v( 1 1 ) 2¢r r~- + ~-S c o s ~ , (25)
H r ( r ' g ~ ' z ' t ) - D ) t ( z ' t ) 1v ( 2 ~ r
R 2
r 12)
sin qS"
The constant q)0 was chosen such that these agree with eqs. (13) and (14) for
r 2 << R 2.
For the source fields
H~,(r, (/),Zj+ l, t -}- 2At) "4-H o ( r , ~, za ,, t) - 2Hv,(r, ep, z j , t + A t ) Dv(1 + 1 ) - 2~r 75 - ~ c o s q ~ [ X ( z j + , , t + 2 A t ) + X ( z a
,,t)-2X(zj,
t+At)],
(26)
because za ~, za, and z j+ ~ were chosen to be separated by vAt. Hence the source fields satisfy eq. (22). This equation together with an identical one for H r are used for open b o u n d a r y conditions at the downstream end. A similar analysis for waves traveling from the cavity towards the upstream end of the mesh leads to an equation for the radiated fields at z, ~ which is now outside the mesh. H , ( r, +, za_ ,, t + 2At) = 2 H + ( r , +, z , , t + At) -- H+( r, ~a, zj+ ., t ).
(27)
G. Aharonian et al. / Transverse wakefieM calculations
33
Replacing the radiated fields by the total fields minus the source fields
H , ( r , rk, zj_,, t + 2At) = 2Hee(r , (p, z,, t + At) - H~,(r, dO, za+ t, t)
ov(,
+~-~ - ~ +
X(Zs_l,t+2at)-2X(zj,
t + z ~ t ) + X ( z j + t , t ) ] c o s e O.
(28)
We use this equation as the open boundary condition at the upstream edge of the pipe. The Hr equation is the same as the one above except the factor ( l / r 2 + l / R 2) is replaced by ( 1 / R 2 - 1/r 2) and cos q~ by sin ~.
Appendix C
Beam pipe equations The transverse and longitudinal wakefields in a smooth beam pipe of infinite conductivity are zero an ultrarelativistic beam. The wakefields from a cavity are produced by fields diffracted from discontinuities in the beam pipe which form the cavity. In the rest frame of the cavity and for ultrarelativistic beam these fields cannot interact with the beam instantaneously; time is required for
for the an the
Table 2 Beam pipe equations. The expansion coefficients a . ( w ) e ik-~R are given by the Fourier transforms of the quantities in column A. The wakefield is given by eq. (36) using these coefficients and the factors X. given below, k. are given in column B. Column A
Column B
Column C
Longitudinal wakes - f o r BCI calculations:
2
rR
[~.r'~
R 2 [ J , ( X - ) ] 2 )O r drJo~ ~ - }Ez(r, z R, t)
k. =
R2
Jo(X.) = 0 - f o r DBCI calculations:
2
1rRr2
R2[j2(Xn)]2JO
[ X.r ~ _ drJt~'~R-)Ez(r'zR't)
kn=
R2
Jl(X.) = 0
m
}k n
2R
Transverse wakes - f o r TM modes:
2 ItRr2 [ ~nr ~ ~ RZ[J2(X.)12aO drJ'[~} Ez(r'z"'t)
~/~ 2
k. =
~
~kn2
R2 ik"~R 2X. (1 - ~.flc )
JI(X.) = 0 - f o r TE modes: 2X2 R2[ j , ( X . ) ] 2 (X2
1)
f?r2
[ ~.r ~ . drJl['--R-) H z ( r ' z R ' t )
kn =
7
J;(X.) = o i For k. real F. = [~r3(k. - w/tic]+ k. - w/tic )×[Factor in column C] For k. imaginary F.
k. = iK.
1
K . +ion/tic ×[Factor in column C]
R2
ik"~°cR 2X. (T~.~,o B)
G. Aharonian et al. / Transversewakefieldcalculations
34
fields to "catch-up" with the beam particles. For extremely short bunches this occurs in the beam pipe well beyond the cavity and typically well beyond the end of the DBCI (or BCI) mesh. This appendix describes a procedure used to correct the wakefields to include the beam pipe contribution. The discussion below concerns the TM mode contribution to the transverse wake. Table 2 presents other results. The field downstream of the cavity after subtracting the source field obtained from a beam pipe only run can be written as linear combination of waveguide modes:
E.(r, z, ~, t) = cos epZJ , ~
a,(w) e i(k'-'-'°'' dw.
n
(29)
o0
The sum is over waveguide modes; the X,'s are the zeros of the Bessel function J~; R is the beam pipe radius; the k n are the wave numbers for the modes, and to insure that E_. is real and propagating away from the cavity a,(-w)
= a*(c0),
k * ( - c0) = - k,,(w).
(30)
At reference plane z R within the DBCI mesh the field after subtracting the source fields is given by
E~(r, z., q~, t) = F.z(r, z R, t ) r cos ,;b.
(31)
Substituting this into eq. (29), multiplying by
f_
r Jl(Xmr/R), and integrating from 0 to R
2 Jo a,(w) e '(k . . . . ') dw - RZ(Jz()tn)) 2
[rRr2drJ1/Xnr\" ~-)Ez(r,
zR, t).
(32)
Or, an(W ) e ik-~" is the Fourier transform of the right-hand-side of eq. (32); the a,,(w)'s are considered known. For a given waveguide mode the fields contributing to the transverse wake are [11]
(E+t~°vxH)±-
X~
1-
v l E ~ ( r , q,, ,o) e "k . . . . '.
(33)
Taking the gradient and then the limit as r ---, 0
(E+t%v×H).
-
ik, 2X,R ( 1 - ~ cwfl ) e,(koz_,~,)( ~ c o s O - 4 s i n 4 , ) .
(34)
To calculate the contribution to the wakefield from the beam pipe this field must be integrated from z R to oc at a fixed bunch coordinate specified by t n = t o + zR/flc, the time the test particle arrives at z R Wl(tn)=
f~ d z f ~ dt(JE+ldol) XHlra[l--(tR-t-(Z--ZR)/t~c)].
(35)
Substituting eq. (34) into this equation and performing the z and t integrals
W±(t.)=~--~-~_.f ~ a . ( . ) e i k " : ~ e n
i ' " ~ + : " / ' " ' F , , ( c o ) dco,
~o
where
ro( 0)=
5X7[ l-
for k , real and
1
ikR(
F n ( £ ° ) - K n+i~o/flc 2X n 1 -
) ,
'
(36)
G. Aharonian et al. / Transverse wakefieM calculations
35
for k, = i K n imaginary. The wakefield given by eq. (36) must be added to the wakefield calculated by DBCI. Similar expressions are needed for the TM mode contributions to the longitudinal wake, and the TE contribution to the transverse wake. For reference these are given in table 2 along with the longitudinal wakefield which should be used when using BCI.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
E. Keil, NIM 100 (1972) 419. K. Bane and B. Zotter, 1lth Int. Conf. on High-energy accelerators, ed., W.S. Newman. K. Halbach and R.F. Holsinger, Part. Accel. 7 (1976) 213. T. Weiland, 1lth Int. Conf. on High-energy accelerators, ed., W.S. Newman; T. Weiland, ISR-TH/80-07 (unpublished). T. Weiland, DESY 82-015 (March, 1982). K.S. Yee, IEEE Trans. Antennas Propagation AP-14 (1966) 302. BCI is a simplification of the general problem discussed in T. Weiland AEf.) 31 (1977) 116. Earlier work on this was discussed in R. Siemann, Cornell Note CBN 81-32 (Nov. 1981, unpublished). T. Weiland, Letter to BCI users (August 1982, unpublished). R. Sundelin, M. Billing, R. Kaplan, J. Kirchgessner, R. Meller, D. Morse, L. Phillips, D. Rice, J. Seeman and E. von Borstel, IEEE Trans. Nucl. Sci. NS-28 (1981) 2844. [11] J.D. Jackson, Classical electrodynamics (Wiley, New York, 1963).