Coupler Design For The LCLS Injector L0-1 Structure
Zenghai Li, Lynn D. Bentson, David H. Dowell,
Cecile Limborg, John Schmerge, Liling Xiao
Abstract
The impact of multipole fields in the single-feed SLAC S-band structure is studied for the
LCLS injector linac. Both the dipole and quadrupole fields were determined to significantly
increase the beam emittance. A racetrack dual-feed coupler design will be used to replace the
single-feed coupler in the L0-1 section to minimize the multipole field effect on beam emittance.
The new coupler eliminates the dipole fields and minimizes the quadrupole fields to about 10
times smaller than that in the single-feed coupler. The parameters for mechanical drawing are
presented. The sensitivity of the matching on geometry errors is provided.
Introduction
The photo injector for the LCLS is required to produce a 1-nC, 10-ps bunch with a
normalized rms transverse emittance of 1.0-µm. A photo-induced electron beam is
produced in the 1.6-cell S-band RF gun by illuminating the cathode with a temporally
shaped laser pulse. The electrons are accelerated to 6-MeV in the RF gun. Two 3-m Sband accelerator structures, operating at 18-MV/m gradient, are required to drive the
beam out of the space charge dominated regime. The SLAC 3-m S-band structure will be
used for this booster acceleration. In the SLAC S-band structure, the RF power feed is
through a single coupling hole (single-feed). Time dependent multipole fields in the
coupler induce transverse kicks along the bunch, causing head-tail beam emittance
degradation. Although the multipole field in the coupler is not significant for the SLC
beams, it is significant in the LCLS injector.
In the SLAC structures the measured asymmetries take the form of a linear amplitude
and phase variation across the coupler cell in the direction of power flow. The
longitudinal electric field can be expressed as shown in Eq. 1 [1] where E/E is
approximately 0.1 and  is approximately 1.5° with 2a = 0.7517 inches. However, no
values for the non-linear terms of the amplitude or phase were reported. The values of
the linear field and phase terms vary by about 30-40% between the entrance and exit
coupler cells due to changes in coupler cell geometry.
x
 E x  j  t kz  2 a 
Ez  Ez 0 1 
1
e
 E z 0 2a 
The transverse variation of longitudinal electric field creates a transverse magnetic field
which deflects the beam. A linear electric field variation leads to a magnetic dipole and a
quadratic dependence creates a quadrupole field. The deflection due to the dipole is
significant and was recognized before the SLAC linac was fabricated. In order to reduce
the effect the coupler cells were offset to compensate the linear amplitude term with the
natural Bessel function dependence of the cavity. This effectively reduced the amplitude
term by a factor of 100 or more but did nothing to compensate the phase term. The
resulting beam deflection for both the dipole and quadrupole term is shown in Fig. 2.
The deflections are for a compensated SLAC structure (E/E = .001) and include the
effect of a standing wave in the upstream half of the entrance coupler cell. The deflection
for the quadrupole assumes a 1 mm beam size and a 10 ps long bunch is shown in red on
the figure injected at -5.5°. An electron on crest is defined as 0 degrees phase. Both the
dipole and quadrupole terms are computed from Eq. 1. Since the quadratic field and
phase dependence for the SLAC coupler were not reported the quadrupole term computed
from the linear variations most likely underestimates the real quadrupole term. The
dipole term is almost entirely due to the phase term in Eq. 1. In the compensated SLAC
structure the dipole term from the amplitude asymmetry is approximately one order of
magnitude less than the dipole term from the phase asymmetry.
0.02
quadrupole
dipole
cpx /m0
0.01
0.00
-0.01
-0.02
-0.03
-0.04
-100
-50
0
50
100
Linac Phase (˚)
Figure 1 The change in transverse momentum versus linac phase is plotted for both the dipole term
and a quadrupole term assuming a 1 mm beam size. Also shown is a 10 ps long bunch for each case
showing the different kicks received by the head and tail for a -5.50 lunch phase.
Due to the time dependent nature of the dipole and quadrupole fields, the head and
tail of the bunch can be deflected by different amounts leading to an increase in the
projected emittance. The estimated increase in the normalized projected emittance for
both the dipole and quadrupole term is shown in Eq. 2 where px is the difference in
momentum between the head and tail.
 n  final  
2
n initial

 11  px 


4  mc 
2
2
Since the change in transverse momentum for a relativistic beam does not depend on
energy, the normalized emittance increase is independent of energy. However, the
increase in projected emittance depends on the beam size and thus is typically most
important in the injector where the emittance is small and the beam size large.
From Fig. 1 the difference between the head and tail momentum due to the dipole
term is approximately .002. Thus for an initial emittance of 1 micron and a 1 mm beam
size the final emittance is 1.4 microns. As discussed above the emittance increase due to
the quadrupole term is likely to be underestimated due to the lack of knowledge of the
quadratic field asymmetries. The emittance increase due to the quadrupole term can
theoretically be compensated using solenoidal emittance compensation. However the
emittance increase due to the dipole deflection can not. The solenoid can not compensate
for a temporally correlated deflection which lead to offsets in phase space but can
compensate for misaligned phase space ellipses. Thus it appears that the overall
deflection must be reduced by about an order of magnitude to reduce the emittance
growth to acceptable levels.
More rigorous studies on the tolerances on dipole and quadrupole time dependent
kicks from the power flow in the entrance cell of L01, where the beam is large thus the
head-tail effect is more important, have been performed using PARMELA. To simplify
the simulations, the head-tail kicks were introduced at the entrance of the cell as a single
kick. The dipole time dependent kick was represented by a linear angle offset from headto-tail. The quadrupole kick was introduced as a linear quadrupole strength varying from
head-to-tail. For both cases, we looked at the growth of the 80%-emittance (over 100
slices). The tolerance on the kick is set by limiting the emittance growth to 2%. For the
dipole kick, the angle offset should not be larger than 120 micro-radiant from head-to-tail
of a 10ps bunch as illustrated in Fig. 1. For the quadrupole kick, the quadrupole moment
should not exceed 0.075 rad/m from head-to-tail of a 10ps bunch. Those two values set
the tolerance to 60 micro-radiant for the dipole and 0.0375 rad/m for the quadrupole as
we would like to leave the possibility open to run 20ps long pulses.
Figure 2 Tolerances on dipole and quadrupole head-tail effects in the S-band accelerator. Simulated
using PARMELA. Left: dipole field; Right: quadrupole field.
From Fig.1 the head-tail angle due to the dipole field in the input coupler of L0-1 is
about 200 micro-radiant (6-MeV), which is a factor of 4 larger than the tolerance. Thus
the dipole deflection in the coupler must be reduced by at least a factor of 4 in order to
preserve the beam emittance. The quadrupole effect, in Fig. 1, is likely not estimated
correctly due to lack of quadratic field information and need to be calculated numerically.
Since the dipole term is mainly due to the phase asymmetry, the only feasible solution to
reduce the dipole head-tail effect is to replace the single-feed coupler on the existing
structure with a dual-feed coupler. The design of such a coupler is the main purpose of
the work. The dual-feed scheme totally eliminates the dipole fields however do little to
the quadrupole fields. A racetrack cell profile will be needed in the dual-feed design to
minimize the quadrupole fields.
In this note, we will first present a full RF analysis of the dipole and quadrupole fields
in the existing SLAC structures to understand fully the issues of the multipole fields in
the single-feed coupler. We will then present the design and analysis for a new dual-feed
racetrack coupler for the injector S-band structures. This dual-feed racetrack coupler will
be used to replace the single-feed coupler in the L0-1 section. Dimensions of the dualfeed coupler will be provided for the mechanical design. Tolerance on design parameters
will be analyzed.
3-D RF Simulation Method
Full 3-D RF simulation will be used to study the matching and multipole fields of the
existing and new coupler designs. This section describes the code and method used in this
work for the RF simulation and analysis.
1 Parallel code S3P
The finite element code S3P is used to simulate the coupler matching and to obtain
the traveling wave fields to be used in the beam simulation. S3P is a parallel S-parameter
simulation code developed in ACD, SLAC. It is implemented on multi-processor
computer platforms (e.g., IBM-SP and Linux clusters) to take the advantage of massive
memory and CPU power of these high performance computing machines, which allows
to simulate large and complex geometries accurately and quickly. This code has been
used in the design of X-band acceleration structure and RF components. It has been
benchmarked with measurements of X-band coldtest prototypes.
2 Multipole analysis
The impact of the coupler fields on the beam dynamics is studied by analyzing the
particle momentum change after traversing the coupler fields. The equation of motion
d ( )
e

( E  c  B)
3
dt
m0 c
is integrated to obtain the momentum change (), which is directly related to the beam
emittance. The momentum change is RF phase and initial position dependent. A number
of particles of different initial positions will be traced and () calculated. Then a spatial
harmonic decomposition is performed to obtain the multipole moments of the impact.
Multipole Fields in the Existing SLAC structure couplers
In this section we present the full wave simulation of the SLAC S-band structure
couplers. Both the input and output RF power couplers in the SLAC 3-m structure feed is
through a single coupling hole (single feed) as shown in Fig. 3. The coupler cells are
racetrack shaped. The center of the racetrack cell is offset opposite to the coupling hole to
compensate the dipole asymmetry due to the perturbation of the coupling hole. The center
offset minimizes the dipole fields but enhances the quadrupole fields because the long
side of the racetrack has to be oriented along the direction of the power coupler. In
addition, the power flow from a single feed coupling hole induces phase variation across
the coupler cell, which induces phase related dipole and quadrupole effects to the beam
(even if the geometry asymmetry is compensated perfectly).
Figure 3 SLC structure input and output coupler cell geometries. The cell has a racetrack profile.
The center of the cell is shifted to minimize the dipole fields.
1 Coupler simulation
The parameters used to build the coupler model for S3P simulation are from a set of
SLAC drawings. These drawing are appended to this note in the Appendix.
The coupler perturbation is localized in the coupler and the adjacent cell. Beyond that,
the fields are cylindrically symmetric through out the whole structure. Since we are
interested in the 3D fields due to the coupler perturbation, we only need to include the
coupler and the two adjacent cells in the simulation model. In the S3P simulations, a
two-port network is required to calculate the S-parameters and the traveling wave fields.
Two symmetric models were built for the input and output couplers respectively. Each
model is a reflected symmetry model of the coupler and two adjacent cells as shown in
Fig. 4. In the beam dynamics studies for the input coupler, only the fields in the “port-1”
side of the input coupler model are used. For the output coupler, only the fields in the
“port-2” side of the output-coupler model are used.
port-1
port-2
Figure 4 Symmetric model use for S3P simulation. A traveling wave is obtained by driving port-1
with a TE10 mode. Only one half of geometry is needed because of symmetry about the vertical plane.
2 Simulation results
The matching of the couplers was verified using S3P. The input coupler is well
matched at 2.856-Ghz with a reflection (S11) of 0.04, while the output coupler has a large
reflection of 0.24. The output coupler is re-matched to reduce the standing wave content
in the RF field. A reflection of 0.04 was obtained by adjusting the opening of the
coupling iris from 22.860-mm to 22.160-mm. It is worth to note that the SLAC structure
couplers were tuned after they were built. The initial mismatch in the output coupler
simulation is consistent with what was achieved on the actual structure.
The 3D traveling RF fields in the coupler models were obtained by driving the “port1” with a TE10 mode. In Fig. 5 are plotted the transverse dipole and quadrupole moments
of the coupler fields on the beam as functions of beam RF phase. The zero RF phase is
the “crest” phase where beam gets maximum acceleration in this three-cell model. The
crest phase in the full structure will be somewhat different because of the phase slippage
due to low beam energy and the lower averaging coupler effect in the full length structure.
The thicker line segments in Fig. 5 indicate the beams which are 10 degrees in length in
RF phase. In table 1 are shown the head-tail momentum (()) and steering angle
(()/) variations for a bunch accelerated on “crest”. Notice that in the real machine
operation, the beam is off the “crest” by about -10 degrees. The head-tail effect in the real
machine will be slightly different from the numbers shown in Table 1.
Figure 5 Head-tail effect of SLAC 3-m structure: left) input coupler; right) output coupler. The
thicker line segment represents the 10-degree bunch which is on the RF crest in these plots. In the
real injector operation, the beam is about -10 degrees off crest.
Table 1 Comparison of dipole and quadrupole head-tail effects in the SLAC 3-m structure
input/output couplers. Bunch length=10degrees, Gradient=20MV/m
Beam energy ()
Dipole: ()
Quad: ()/m
Dipole head-tail angle  (rad)
Quadrupole head-tail angle  (rad/m)
Input coupler
10
2.610-3
7.810-1
2.610-4
7.810-2
Output coupler
130
1.010-3
4.410-1
1.010-5
4.410-3
Both the dipole and quadrupole head-tail effects of the single-feed input coupler are
found too large for the LCLS injector section L0-1. It is thus necessary to use a more
symmetric coupler design for the input of L0-1 to minimize the head-tail emittance
degradation.
In the output coupler of L0-1, the head-tail effects (()) are about half of that in
the input coupler. In addition, because of the 10-times higher beam energy at the output
coupler, the beam size is damped by a factor of more than 3, assuming beta function does
not very too much in the structure. The head-tail emittance growth is about 6 times
smaller. Similarly in the accelerator section L0-2 and accelerator sections down stream,
the beam size is adiabatically damped so are the head-tail effects. It is thus adequate to
use the existing single-feed coupler for the output of L0-1 and for the structures down
stream.
Dual-feed input coupler for L0-1
1 Coupler design and optimization
In this section we present a dual-feed design for the accelerator section L0-1 to
minimize the coupler multipole fields. The dual feed coupler eliminates the dipole fields
by symmetry. A racetrack coupler cell, with the coupling hole on the long side, is needed
to minimize the quadrupole fields. One additional improvement in the new design is to
round the edges of the coupling irises. The rounding of the irises minimizes field
enhancement and RF heating. Although there were no problems encountered in the
SLAC structures due to sharp coupling holes edges, it is evident in the X-band high
power test that the sharp edges in the coupling hole have caused successive pulse heating
and breakdown damage. As a result, all the new X-band couplers are designed with no
sharp edges, and the structure high gradient performance is improved greatly. With this in
mind, it is believed beneficial to have the coupling irises rounded in the new design. A
sketch and a cutaway view of the dual-feed racetrack coupler for L0-1 are shown in Fig.
6. The radius of the rounding on the coupling iris is chosen to be 1-mm. The two (+)s
mark the centers of two racetrack arcs. The separation of these two arc centers need to be
adjusted to reduce the quadrupole field. The iris opening and the arc radius are the
parameters used for matching the coupler. The design was done iteratively between
adjusting the arc separation for quadrupole field and adjusting the iris and the arc radius
for matching. For a given arc separation, the coupler is matched and quadrupole field is
analyzed. After a number of iterations, we reached a design that has a matching reflection
of 0.02 and quadrupole field about 10 times smaller than the SLAC single-feed coupler,
see Fig. 7 and Table 2.
Figure 6 Dual-feed coupler with racetrack cell profile to minimize dipole and quadrupole fields.
Two more coupler configurations were studied for comparison. One is a dual feed
with a cylindrical cell profile (cylindrical dual feed). The other is a dual-feed with a
cylindrical cell plus two slots, the same width as the coupling iris, 90 degrees away
azimuthally from the coupling iris to compensate the quadrupole effects (cross dual feed).
In Fig. 7 and Table 2 are shown the quadrupole moments of these designs for comparison.
It is clear that the racetrack dual-feed coupler is least in quadrupole fields.
Figure 7 Comparison of quadrupole fields in different type couplers.
Table 2 Comparison of quadrupole fields in different coupler designs.
Input coupler: comparison of quad head-tail ()/m: 10 Degree bunch
()/m
Head-tail angle  (rad/m)
SLAC Single feed
0.78
7.810-3
Symmetric dual
0.63
6.310-3
Race-track dual
0.04
4.010-4
Cross Dual
0.20
2.010-3
2 Racetrack coupler dimensions
This section summarizes the dimensions of the racetrack coupler design for section
L0-1. The parameters used in S3P simulation are shown in Fig. 8. The dimensions for the
mechanical drawing are listed in table 3 with arc radius “b” is corrected for the 0.5-mm
fillet (due to finite tool radius) and the skin depth. The skin depth effect almost cancels
the perturbation of the 0.5-mm fillet in the present design. All dimensions are in
millimeters. The dimensions at 450C (operating temperature) in column 2 are for RF
simulation. The dimensions at 200C (scaled by 0.99958 from the 450C dimensions) in
column 3 are for mechanical design.
Figure 8 Coupler parameters used in S3P simulation.
Table 3 Racetrack dual-feed coupler parameters for mechanical drawing.
Parameters
Beam pipe diameter
Beam pipe cutoff hole rounding R
Racetrack arc radius b
Racetrack arc separation d
Cell iris radius a
Disk thickness t
Disk rounding radius R
Disk flat part tf
Coupling iris opening w
Coupling iris rounding r
Waveguide width Wg
Waveguide height hg
Dimension at 450C
19.0932
3.0861
35.8583+0.0004
13.0000
13.1102
5.8420
3.0861
0.7874
25.0200
1.0000
61.0810
29.1338
Dimension at 200C (0.99958)
19.0852
3.0861*
35.8437
12.9945
13.1047
5.8420*
3.0861*
0.7874*
25.0095
0.99958
61.0553
29.0216
Note: *) numbers from the old SLC drawing, not scaled
Dual Feed Tolerances
1. Dimension Sensitivity on Matching
The multipole fields are much less sensitive to the geometry errors than the matching.
Here we present the sensitivity of the matching (S11) on major cell parameters. The
sensitivity data are to provide guidelines for the tolerance specifications for machining. In
Fig. 9 are shown the change of matching (S11) when one of the cell parameters is off by 0.02-mm. The complex S11 is plotted to show both the phase and amplitude. The blue
diamond represents the present design which has a reflection of about 0.02. The matching
of the coupler depends moderately on most cell parameters. The most sensitive parameter
is the cell radius “b” because it controls the cell frequency. Thus tuners are suggested on
the coupler cell.
0.05
design
cell radius off -0.02mm
arc center off -0.02mm
S11_imag
iris width off -0.02mm
wg location off -0.02mm
0
-0.05
-0.05
0
0.05
S11_real
Figure 9 Sensitivity of matching on geometry errors.
2. Dimension Tolerance on Field Asymmetry
Machine/construction errors and RF feed imbalance of the dual-feed coupler will
result in field asymmetry in the coupler cell. We consider the effects of four errors in the
construction of dual feed RF coupler. The four errors are:
1. Coupler position error (couplers not located 180 degrees apart)
2. Coupler iris size error (couplers with different coupling coefficients)
3. RF feed amplitude error (each coupler fed with different RF amplitudes)
4. RF feed phase error (each coupler fed with different RF phases)
The criteria used for acceptable tolerance were that the amplitude term in Eq. 4 must be
reduced by a factor of 100 just as in the compensated SLAC structures and the phase
term must be reduced by a factor of 10. The field from N couplers is shown in Eq. 4
where V0 is the amplitude necessary to produce an accelerating gradient E z0 and D0 is the
nominal dimension of the coupler iris. Alternatively it can be thought of as the nominal
coupling coefficient. It is assumed that the coupling varies linearly with small coupler
iris dimension variations. Vi, Di and i are the complex amplitude, coupler iris dimension
and coupler iris angular position of the ith coupler respectively.

D

y 
i
Ez 0 N Vi Di  E 
x
y   j 0  D0  sini 2 a cosi 2 a 
4
Ez 
 cos i

1 
 sin i
 e
N i 1 V0 D0  Ez 0 
2a
2a  
For a perfect dual feed coupler with no errors the dipole term is completely cancelled as
expected but a quadrupole term is excited since Eq. 4 assumes a cylindrically symmetric
model (no racetrack geometry). The field perturbation from a perfect dual feed coupler
due to an error was considered for each of the four case listed above. Table 4 lists the
error and the corresponding tolerance as well as the criteria setting the tolerance. The
criteria is either amplitude, meaning the amplitude term in Eq. 1 will be reduced by a
x
factor of 100 with the associated error, or phase, meaning the phase term in Eq. 1 will be
reduced by a factor of 10 with the associated error. In some cases the errors affected
both the amplitude and phase terms but only the tightest tolerance is listed. The fact that
a dual coupler will have a smaller coupler iris than the single feed coupler is not
accounted for in the analysis. Thus the actual tolerances will be somewhat loser since
the field asymmetries should be correspondingly decreased. However, rather than
estimate this effect it seemed appropriate to error on the conservative side and ignore the
change.
Table 4 The tolerance for the four coupler errors is listed. The tolerance is determined by
either reducing the amplitude to 1% of its nominal value or the phase to 10% of its nominal
value as listed in the last column.
Error
Coupler Position
Coupler Iris Size
RF Feed Amplitude
RF Feed Phase
Tolerance
 < 1°
D/D0 < .02
V/V0 < .02
 < 3°
Defining Criteria
Phase
Amplitude
Amplitude
Phase
All errors were considered independently. Thus multiple simultaneous errors will
produce larger amplitude and or phase asymmetry than desired. However, the coupler
position tolerance is quite loose and will likely be better than 0.1°. Thus the only error
affecting the phase is the RF feed error. This can in principle be set to 1° by
measurement and waveguide phase tuning so phase errors do not appear to be
problematic. The coupler iris size is also relatively loose since the nominal dimension is
more than 0.1 inches leading to a tolerance of at least 2 mils. The RF amplitude tolerance
will be challenging but in principle can be reached with network analyzer measurement
and subsequently tuning the RF power splitter.
Summary
The multipole fields in the single-feed SLAC S-band structure are analyzed and found
to be too large in the L0-1 accelerator section. A racetrack dual-feed coupler is design to
replace the single-feed input in the L0-1 structure. The new coupler eliminates the dipole
fields and minimizes the quadrupole fields to about an order magnitude smaller than that
in the single-feed coupler. The parameters for mechanical drawings for the new coupler
are provided. The sensitivity of the matching on geometry errors is analyzed.
References
[1] The Stanford Two-Mile Accelerator, Edited by R.B. Neal, 1968, W.A. Benjamin Inc,
pg 146.
Appendix
SLAC 3-m structure mechanical drawings: