Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

THE EFFECTS OF RF ASYMMETRIES ON

PHOTOINJECTOR BEAM QUALITY *

J. B. Rosenzweig, S. Anderson, X. Ding, and D. Yu+

UCLA Department of Physics and Astronomy,
405 Hilgard Ave., Los Angeles, CA 90095

Abstract the particle experiencing Lorentz forces in the rf cavity

environment moves parallel to the axis at constant
A general multipole-based formalism to study the effects velocity. The approximation of constant velocity is as
of RF asymmetries on the production of ultra-high badly violated as can be close to the photocathode, but in
brightness beam is presented, which employs both this region there are few transverse forces. Thus the
analytical and computational techniques. These field
application of the theorem is still reasonable in this case.
asymmetries can cause the degradation of beam emittance
Our version of the Panofsky-Wenzel theorem gives
due to time dependent and nonlinear focusing effects. Two
the integrated transverse and longitudinal momentum
cases of interest are examined: the dipole asymmetry
“kicks” inr terms of the components of the rf vector
produced by a coupling slot in a standard high gradient rf
potential A
gun, and the higher multipole content introduced by the
support/cooling rods in a PWT structure. Practical r zf r
q  f ∂A⊥ 
z
r
implications of our results, as well as comparison to cold ∆p⊥ =  ∫ dz + ∫ ∇ ⊥ Az dz , and (1)
test and beam-based experimental tests, are discussed. c  z i ∂ζ zi 

q  f ∂Az 
z
1 RF FORCE-DERIVED EMITTANCE ∆pz = ∫ dz , with ζ = z − ct. (2)
In a high gradient rf photoinjector, the necessity of using c  z i ∂ζ 
violent longitudinal acceleration also implies the existence
of large transverse forces. These forces are time-dependent,
The first term in Eq. 1 vanishes because it is a perfect
and may be nonlinear or non-axisymmetric as well. All of
differential, and the transverse components of the vector
these attributes can give rise to transverse emittance
potential vanish at the cathode, and outside of the cavity.
growth. Time dependent monopole[1] and dipole[2] fields
Thus we have the relation
can cause correlations between the beam’s transverse and
longitudinal phase spaces, which, while not contributing
∂ ( ∆p⊥ ) r
to the so-called slice emittance (the transverse emittance of = ∇ ⊥ ( ∆pz ). (3)
a narrow longitudinal slice of the beam), can increase the ∂ζ
total projected transverse emittance. We shall discuss the
relationship between these emittance contributions and the We consider a multipole standing wave field with a
Panofsky-Wenzel theorem, as well as observations sinusoidal dependence of the phase on distance away from
verifying the conclusions we reach from this analysis. the power coupler,
Nonlinear fields have typically been considered in the
( )
∞
context of the axisymmetric, non-synchronous spatial Ez = E0 sin ωt − κ y y + θ 0 cos(kz ) ∑ an r n cos(nφ ) (4)
harmonics of the rf field. In this paper, we examine the n=0
contribution to synchronous rf multipole fields to the
emittance, and analytically estimate the amplitude of these The asymmetry term inside of the sine function is the
multipoles for rf structure types of interest. We compare phase asymmetry due to power flow (finite Q effect), and
the analytical estimates with experimental evidence and the series expansion is the multipole content of the mode
computer simulations. fields. The vector potential associated with Eq. 4 is

( )
2 PANOFSKY-WENZEL THEOREM E0 ∞
Az = cos ωt − κ y y + θ 0 cos(kz ) ∑ an r n cos(nφ ). (5)
As cavities are designed first and foremost to accelerate, it k n=0

is of interest to relate the longitudinal acceleration which

is imparted to a given particle. This is accomplished by 3 POWER FLOW EFFECTS
an updated version of the Panofsky-Wenzel theorem[3], in To isolate the transient power flow component of the
which we take into account the fact that the electrons acceleration, we use only the lowest multipole component
which are accelerated from rest starting from a point (a field, to arrrive at longitudinal momentum gain in a gun
photocathode) where the field is not zero. The Panofsky- of length Lg of approximately
Wenzel theorem explicitly assumes in its derivation that

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 Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

( )
eE0 Lg
∆pz = cos kζ − κ y y pz ( y)c ≅ eE0 Lg (1 + a1 y / k ). (10)
2c

eE0 Lg  ( ∆φ )2 ( ) 
2
κ yy If one measures this asymmetry, then one can determine
≅ 1− − , ∆φ = k∆ζ . (6) a1 , and the expected emittance growth due to dipole kicks
2c  2 2 
  can be estimated. This was done on a 1.5 cell gun
obtained from BNL at UCLA[4], with a view of the
The transient power flow wave number can be estimated momentum spectrometer shown in Fig. 1. Here the
as κ y ≅ k / Q , where Q is the unloaded quality factor, particles at larger y (actually at large x in the gun, as the
which is of order 10 4 . Thus for reasonable beam size focusing solenoid provides a nearly 90 degree rotation) are
parameters, the effect of the power flow can be neglected. seen to have a smaller momentum in the spectrometer.

4 MULTIPOLE FIELDS

4.1 Monopole effects y

For the monopole component of the field, with
normalization a0 = 1 , the acceleration is independent of
transverse offset (e.g. y ), and the transverse emittance
growth for a beam with a uniform density distribution
propagating near the peak acceleration phase is
p
ε ny = γ
2
y 2 y ′ 2 − yy ′

2 σ y ( kσ z ) .
eE0 2
= 13
20
2
(7)
2 me c

This is the rf emittance contribution first analyzed by

Figure 1. Electron beam (green) image in focal plane of
Kim[1], and can be mitigated by keeping the beam sizes spectrometer, with smaller energy electrons at larger y.
small.
From this image, and knowledge of the bunch size
4.2 Dipole effects and length, it was deduced that the gun dipole asymmetry
contributed 3.5 mm-mrad to the normalized rms
The lowest significant order asymmetry has traditionally emittance. The low charge vertical emittance in this gun
arisen from the existence of a coupling slot on one side was measured to be 5 mm-mrad, most of which came not
(in y )of the cavity. In the first 1.5 cell BNL designed S- from the more familiar monopole effects, but from rf
band gun, the coupling was in both cells, and thus initial dipole components. This is partially due to the
condition on the transverse vector potential is Ay = 0 , overcoupling of the device ( β = 1.6 ); the coupling slots
giving a transverse momentum kick of were anomalously large, and the dipole component of the
rf field was larger than necessary.l
eE0 a1
∆py = ∂ y ∫ ∆pz dζ = Lg sin(kζ )
2c k
eE0 4.3 Higher multipole effects.
≅ a1 Lgζ (8)
2c In the next-generation rf guns[Palmer,Colby] beyond the
1.5 cell BNL style model, several design innovations were
This phase dependent dipole kick gives rise to an effective implemented, including the coupling of external power
projected emittance only through the full cell, and the use of a dummy slot
opposite to the coupling slot for dipole symmetrization.
eE0
ε n, y = a1 Lgσ yσ z . (9) These schemes worked well, and have additionally been
2 me c 2 supplemented by the use of a race-track outer wall
geometry[5] to eliminate the quadrupole components of
According to the Panofsky-Wenzel theorem, the transverse the field left after dipole symmetrization.
momentum kick is accompanied by an acceleration which In the new PWT photoinjector structure, under
is dependent on the offset of the electron in y, development by a UCLA/DULY Research collaboration,
the structure is based on disks which are not connected to

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 Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

the outer wall, but are supported by four rods (the cross- the emittance is ε n, y ≅ 2.3 × 10 −8 m-rad, which is almost
sections of two are shown in Fig. 2). The cell-to cell two orders of magnitude smaller than the expected
coupling in this device is obtained through the annular emittance due to monopole rf and space-charge effects.
region between the disks and the outer wall, and can be On the other hand, for the proposed X-band PWT
very strong, leading to excellent mode separation. The photoinjector[6] under study by a DULY/UCLA/LLNL
external coupling is through the outer wall, and is so far collaboration, the rods must expand by a factor of 50%
from the axis that it does not give rise to significant relative to the disk size in order to provide adequate
dipole components of the field. In fact, the rods, which are cooling water flow. In this case Eq. 11 gives
relatively close the axis, give rise to a dominant octupole a4 a0 ≅ 0.15 cm −4 , while the beam, for 1 nC operation
field perturbation. (as in S-band), is smaller by a factor of 3 in all
We have examined this perturbation both analytically dimensions. In this case, the expected octupole
and through field simulations. The rods effect lasts for the contribution to the normalized emittance is
entire structure, just as the dipole component, and so the ε n, y ≅ 2.1 × 10 −7 m-rad. This is now significant, as it is
normalized emittance in such a long device may be roughly 20% of the design monopole emittance. In
impacted more severely than in a short gun. All of the addition, it implies that it would be unwise to raise the
higher multipole components which have a strong effect charge Q significantly in this device, as this would result
on the beam will have a speed-of-light phase velocity, and octupole-induced emittance scaling[8] as Q 5 / 3 .
thus have a transverse field profile which obeys the
equation ∇ 2⊥ Ez = 0 with solutions as in Eq. 4. The
boundary conditions for the situation with the rods may be
approximated as the field being constant at the rod offset
radius ρ , but dropping to zero in the region of the rods
(which have radius b). Fourier analysis of this rectangular
profile in φ gives the ratio of the octupole to monopole
components of the field,

a4 2 sin( 4b ρ )
≅ 4 (11)
a0 ρ (( 4b ρ ) − π )

x
Figure 3. Calculated contours of constant Ez in S-band
PWT at mid-cell, from GdfidL simulation.
Figure 2. S-band PWT photoinjector cross-section, with 2 _____________________
*
support/cooling rods showing. Work supported by US DoE Contracts DE-FG03-92ER40693 and
DE-FG03-98ER45693.
For the S-band PWT linac, Eq. 11 gives a4 a0 ≅ 10 −3 #
Email: rosenzweig@physics.ucla.edu
+Duly Research.
cm −4 , while the GdfidL 3-D field simulations shown in
Fig. 3 give a4 a0 ≅ 1.3 × 10 −3 cm −4 , which is good
5 REFERENCES
agreement for so rough of a model.
1. K.J.Kim, Nucl. Instr. Methods A 275, 201 (1988)
The effect of the octupole component on the emittance 2. D.T.Palmer, et al., Proc. PAC’97, 2687 (IEEE,1998)
in this device can be estimated as 3. W. K. H. Panofsky and W. A. Wenzel, Rev. Sci. Inst. 27, 967
(1956).
4. J.B. Rosenzweig, et al., NIM A 341, 379 (1994).
3
ε n, y = γ f a4σ 4y σ z , (12) 5.
6.
J. Haimson, private communication.
J.B. Rosenzweig, et al., Proc. PAC’97, 1968 (IEEE, 1998)
14
7. D. Yu, et al., Proc. PAC’97, 2806 (IEEE, 1998)
8. J.B. Rosenzweig and E. Colby, Advanced Accel. Concepts, 724
where we have written it as proportional to the final (AIP Conf. Proc. 335, 1995).
energy γ f me c 2 to emphasize that the emittance is linearly
dependent on the length of the structure. For the 20 MeV
S-band PWT photoinjector at UCLA, σ z = 0.7 mm,
σ y ≅1.5 mm, and the expected octupole contribution to

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