September -
SLAC-PUB-5062 1989 (4
c,
LONG-RANGE ACCELERATING
WAKE POTENTIALS STRUCTURES*
IN DISK-LOADED
D. U. L. YU
DULY Consultants Ranch0 Palos Verdes, California 90732
P. B. WILSON
Stanford Stanford Linear Accelerator Center University, Stanford, California 94309
..
To achieve a reasonable efficiency for the conversion of RF power to beam power, and consequently, to attain a higher luminosity, most designs for future high energy linear colliders require the acceleration of ten or so bunches per RF pulse. In such bunch trains, however, the long-range longitudinal wake potential produces a cumulative energy deviation from the first to the last bunch, while the transverse wake potential leads to cumulative growth in the transverse emittance. We have investigated the dependence of these long-range potentials on the disk aperture radius for typical 2w/3-mode disk-loaded structures. We find that, excluding the lowest frequency mode, the rms long-range longitudinal and transverse wake potentials scale approximately as (U/X)-~ and (U/X)-~, respectively, where a is the aperture radius and X is the wavelength. Furthermore, by using a structure with two different disk apertures, it is possible to produce beats in the long-range transverse wake potential. By placing the following bunch at one of the relatively broad nulls in this wake, it is possible to substantially reduce the transverse bunch-to-bunch coupling and hence, the emittance growth in a multibunch train.
Abstract
INTRODUCTION
Wake potentials . a computer synchronous representative sum method can be computed in two ways: equations directly in the time domain by code which solves Maxwell s modes computed frequency results. on a mesh, or by a sum over code. In the next section, time, the modal for several are compared to show that
by a frequency Since it involves
domain
and time domain calculations
they give equivalent
less computation
is then used to investigate
the long-range wake potentials
values of u/X, where a is the disk aperture radius and X is the free space wavelength. Presented at the XIV International Conference on High Energy Accelerators, Tsukubu, Japan, August 22-26, 1989.
* Work SBIR supported by the Department contract DE-AC9347ER80529. of Energy, contract DE-AC03-76SF00515 and
D. U. L. YU and P. B. WILSON
-
It is found that both the longitudinal c 4. higher modes only (the lowest frequency a simple scaling as a function An important dominance apertures, If an accelerating occur at slightly of u/x.
and transverse
wake potentials
due to the out) follow
mode having been subtracted
feature of the dipole wake potential structure is composed of alternating frequencies.
for large u/x is the strong sections having different disk to
of the lowest frequency mode compared to the sum of all higher modes. then the lowest frequency different mode in the two types of structure will also
These frequencies will then beat together
- form a wake with periodic
nulls. The nulls will be relatively structure TIME
clean if the amplitude
of the higher mode wake is small compared to the wake due to the lowest frequency mode. An example of such a beat-wave COMPARISON _ In a cylindrically -multipole OF FREQUENCY symmetric structure, is given in the final section. DOMAIN CALCULATIONS can be expressed as a angle d, according to cos modes, m = 1 a Gaussian behind
AND
the wake potentials symmetric
expansion
with terms that vary with azimuthal The longitudinal structure,
rnd, where m = 0 gives the azimuthally gives the dipole modes, etc. of such a cylindrically wz (s) = C
n
(longitudinal)
wake potential
bunch of-rms bunch length cr2 and unit charge, moving in the beam aperture region symmetric
2hh
is given by3 cos (wo&) . (14
exp ( -L&732C2)
Here, s is the distance behind the center of the bunch, ~0~ is the angular frequency of the nth synchronous . ,ken = E,2,/4 1~0, mode and Icon is the loss factor for a point bunch, given by , length in the nth mode and E,, field. (lb) is the is
where uo,, is the stored energy per unit synchronous given by3
gd(s) = z$ c n Icln (wlnulc>
component
of the axial electric
The dipole wake potential
exp (-wfn4/2c2)
sin (wins/c)
,
where 7 is the offset of the driving 0 follow) and
charge (ra is set equal to a in the examples to
kin =
[&a(~ = +I2
4Uln
electric field component evaluated
Here, EZn(r = u) is the s8ynchronous longitudinal at the disk hole radius.
Values of the wan and ICon for the longitudinal s s the program KN7C.*
modes can be obtained using (dipole) modes
Values of the ~1, and k1, for the deflecting s s 2
LONG-RANGE
-
WAKE
POTENTIALS..
.
can be obtained f, L. structures structure aperture
from the program periodic
TRANSVRS.5
Both codes model a disk-loaded irises and cylinders. The the radius of the iris No allowance using
by an infinite
sequence of alternating
geometry is entirely specified by only four parameters: a, the radius of the cylinder
between the irises b, the length of one period
p, and the cylinder geometry.
length g (period length minus the iris thickness). details of the wake potentials Alternatively,
is made for rounding this simplified
at the edges of the iris or for other details of the structure calculated the meaningful features of z 1 for
Thus, the high-frequency
model are not meaningful.
- the wake can be obtained by summing a limited the lowest 25-30 modes). the highest frequency is abruptly .-potentials terminated. A bunch length
number of modes (we take typically provided by the exponential
is used such that w,c,/fic
mode in the sum. The damping
factor then helps prevent the nonphysical
ringing that can occur when such a series and dipole wake and a,/x =
The plots at the top of Figs. 1 and 2 show the longitudinal behind a driving calculated (u/X bunch with 0*/X 0.050 (dipole), X = 10.5cm). directly The qualitative
= 0.025 (longitudinal),
from a sum over modes for the avera.ge cell in the SLAC = 0.111, p/X = l/3, g/X = 0.278, b/X = 0.391, and computed of Figs. 1 and 2 are the wake potentials computed
disk loaded structure
At the bottom
by the time domain code TBCI
for the same bunch length and structure. in these two independent TBCI dipole wake
features of the wake potentials
ways in general agree quite well, although there are some differences in detail for the dipole wake. For example, there is less detail in the long-range potential than in the modal sum dipole potential. The reason for these differences
is not well understood. Since the modal sum method uses less computer time and : probably represents the long-range wake potentials more accurately, we use this method in the following . SCALING OF WAKE section to compute the scaling of the potentials POTENTIALS WITH BEAM APERTURE as given by Eqs. (la) geometry discussed in vs. a/X.
The modal sums for the longitudinal
and dipole wake potentials, structure
and (2a), have been carried out for the disk-loaded of u/X, the cylinder accelerating
the previous section for a/X = 0.100, 0.111, 0.150, 0.200, and 0.276. For each value radius b adjusted was to keep the accelerating mode frequency 2~/3 fixed at 2856 MHz (X0 = 10.5 cm). The group velocities mode corresponding 0.085, and 0.185, respectively. of all higher mode are plotted The wake potentials separately. for the synchronous
to these apertures are vus/c = 0.0095,0.0135,0.038, for the last four cases are shown for the lowest frequency mode and for the sum Note the rapid decrease in the amplitudes is increased, especially in the as the aperture 3
in Figs. 3 and 4. The wake potentials of the higher mode wake potentials
D. U. L. YU and P. B. WILSON
-
-0.2 0 a-e* I 20 I 1 40 s a I 60 , 1 80 I 100 L111A1
(cm)
FiGURE 1 (a) Longitudinal wake potential for the SLAC disk-loaded structure (u/X = 0.111) calculated f rom a sum of modes with a,/x = 0.025. (b) Longitudinal wake potential for the SLAC structure calculated by TBCI (three cells with beam tubes, aJ/X = 0.025).
-0.1 -0.2 0 I-80
" I 20
I I 40 s I I 60 I , 80 , 100 II61AP
(cm)
FIGURE 2 (a) Dipole wake potential for the SLAC structure calculated from a sum of modes (a,/x = 0.050). (b) D o 1e wake potential for the SLAC structure p 1 calculated by TBCI (0*/X = 0.050).
4
LONG-RANGE
-
WAKE
POTENTIALS..
.
L. 0 d
2 d 8 '3 -1.0 0.8 0.4 0 -0.4 .-0.8
.
S-88
0
20
40 s
60 (cm)
80
0
20
40 s
60 (cm)
80
100 6461A3
FIGURE 3 Longitudinal wake potential for a disk-loaded structure calculated from a modal sum; cr,/X = 0.025 and (a) a/X = 0.111, (b) u/x = 0.150, (c) u/x = 0.200, and (d) u/X = 0.278. Accelerating mode (dashed); higher modes (solid).
0.8 0.4 0
: -
-0.4 -0.8
0
S-OS
20
40 s
60 (cm)
80
0
20
40 s
60 (cm)
80
100 6461 AS
FIGURE 4 Dipole wake potential for a disk-loaded structure calculated from a modal sum; a,/X = 0.050 and (a) u/X = 0.111, (b) u/X = 0.150, (c) u/X = 0.200, and (d) u/X = 0.278. Lowest frequency mode (dashed); higher modes (solid).
5
D. U. L. YU and P. B. WILSON
-
case of dipole wake. This illustrates i I. beam aperture turbations in an accelerating
in a graphic manner the advantage of a large for a linear collider. are sensitive to small percannot be used to predict in a multibunch train with can, how-
structure
Since the details of the higher mode wake potentials in structure geometry, the present calculation at bunch locations the precise values of those potentials ever, be estimated - wake potentials, longitudinal by (u/X)-~.~,
bunch spacing nXo . The probable amplitude is also shown for comparison. Wrms(all), Numerical
of the higher mode potential
from the rms values plotted in Fig. 5. The lowest frequency mode values are listed in Table I for the total IVr,,(hm). The variation with aperture is best fitted Although modes and for the higher mode potentials,
rms higher mode wake potential
while the dipole higher mode potential
varies as (U/X)-~. .
these fits for the higher mode scaling are quite good, the lowest frequency for the longitudinal
.
1.0
and transverse cases do not follow a simple power law scaling.
0.6
0 0 .lO
P-18
0.15
0.20
0.25
allr
6446lAl
FIGURE 5 (a) Scaling of longitudinal higher mode (dashed curve) and accelerating mode (solid curve) wake potentials with beam aperture. (b) Scaling of dipole higher mode (dashed curve) and lowest frequency mode (solid curve) with beam aperture. Also given in Table I is the ratio of the rms value of the higher mode wake potential to the peak value of the potential for the lowest frequency mode, @(rz = 1). wake) and multibunch transThis ratio is useful for estimating the residual effects of higher modes on the multi-
bunch energy spread (in the case of the longitudinal 6
LONG-RANGE
WAKE
POTENTIALS..
.
TABLE
I
Variation
of rms values of wake potential (V/PC per cell) Wrms(hm) tiQ2 = 1) 0.37 0.31 0.20 0.14 0.11 1
with beam aperture. per cell) Wrms(hm) I?&2 = 1) 0.51 0.34 0.15 0.07 0.05
T I. 4
-1
0.100 0.111 0.150
Longitudinal Wrms(all) 1.134 1.052 0.806 0.582 0.351 growth
Dipole (V/PC W,,,(all) 0.681 0.697 0.635 0.470 0.254 Wrns(hm) 0.396 0.302 0.128 0.047 0.020
W,,,(hm) 0.520 0.427 0.222 0.113 0.057
- _ 0.200 0.276
verse emittance
(in the case of the dipole wake), after the effect due to the
lowest frequency mode has been taken into account.*y2
.-
EEAT-WAVE STRUCTURE . As noted previo...