Longitudinal Shaping of Relativistic Bunches of Electrons Generated by An RF Photoinjector
Longitudinal Shaping of Relativistic Bunches of Electrons Generated by An RF Photoinjector
Longitudinal Shaping of Relativistic Bunches of Electrons Generated by An RF Photoinjector
Los Angeles
by
2007
c Copyright by
Robert Joel England
2007
The dissertation of Robert Joel England is approved.
Steven Cowley
Chandrashekhar Joshi
Claudio Pellegrini
ii
To my parents . . .
for their continuing love and support.
iii
Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
iv
2 First and Second Order Beam Optics . . . . . . . . . . . . . . . . 49
v
3.1.2 Transformation Equation for the Longitudinal Coordinate 87
vi
4.4.2 Traveling vs. Standing Wave Cavities . . . . . . . . . . . . 135
vii
5.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 178
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
viii
List of Figures
1.10 Plots of (a) the wakefield for a Gaussian 1D bunch and (b) the
accelerating and retarding fields and corresponding transformer
ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Trace space ellipses for (a) three trajectories with different invari-
ants and (b) three trajectories with the same invariant but different
momenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
ix
3.1 Cartoon drawings of doglegs at (a) ATF-VISA, (b) UCLA-Neptune,
and (c) ORION-SLAC. Each drawing is scaled to fit the figure. . . 85
3.5 Allowed Twiss parameters for three different values of (f1 , f2 ) cor-
responding to points on the curve in Fig 3.4: (a) f1 = 0.267 m, f2
= -0.42 m; (b) f1 = 0.269 m, f2 = -0.63 m; (c) f1 = 0.267 m, f2
= -0.82 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.8 Plots of the z trace space and current profile from PARMELA
simulations showing (a) the beam at the entrance of the dogleg
compressor, and the same beam at the end (b) without sextupole
correction and (c) with sextupole correction. . . . . . . . . . . . . 111
x
4.5 Plot of K vs. τ for standing wave (dashed) and traveling wave
structures (solid). . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.6 Time resolution vs. power for n pillbox cells in series. . . . . . . . 139
4.9 Geometry of the input coupler design for the HFSS simulation. . . 146
4.10 Plots of axial (a) electric field, (b) magnetic field, and (c) field
gradient as functions of position along the cavity axis. . . . . . . . 148
4.11 Plots (a) electric and magnetic field magnitudes in the transverse
center plane of a cell, (b) the rod geometry at a junction between
cells, and (c) the hole geometry. . . . . . . . . . . . . . . . . . . . 150
4.12 Plots of resonant pi-mode frequency for a pair of (a) holes and (b)
rods of radius 2 mm, as a function of radial position. The curves
are polynomial fits, with the dashed curves representing the desired
polarization mode, and the solid curves the undesired mode. . . . 151
4.14 Plots of simulated coupling beta for two different coupling iris radii
vs the half-width of the coupler. . . . . . . . . . . . . . . . . . . . 153
4.15 Block diagram of the RF system layout for the UCLA Neptune
photoinjector, linac, and deflecting cavity. . . . . . . . . . . . . . 155
4.16 CAD drawing of the Neptune deflector with 1/4 section cutaway. . 157
xi
4.17 Plots of reflectance at the input coupler showing (a) all five modes
of the passband, (b) a closeup of the π mode, and (c) the reflectance
plot in the complex plane (i.e. Smith Chart) for the π mode. . . . 159
4.18 Aluminum bead pull results showing (a) frequency shift and (b)
the square root of the frequency shift vs. position along the cavity
axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1 Schematic of the Neptune beamline and the S-Bahn dogleg com-
pressor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.3 Longitudinal phase space plots and density profiles obtained from
ELEGANT results corresponding to the sextupole settings of the
(a) first, (b) third, (c) fourth, and (d) sixth data points in Fig. 5.2
respectively, illustrating the progression of the phase space com-
pression and decompression. . . . . . . . . . . . . . . . . . . . . . 176
5.4 Block diagram of the experimental setup for the deflecting cavity
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.5 Schematic of the combined beam dump and YAG profile monitor
built as a final diagnostic device. . . . . . . . . . . . . . . . . . . 179
xii
5.6 Images of the deflected beam at two different phases and measured
on-screen deflection distance y plotted against the RF phase. A
sinusoidal function is fitted to the data (solid curve). . . . . . . . 181
5.7 False color plots of uncompressed electron beam with (a) deflecting
cavity turned off, (b) deflecting cavity turned on, as well as (c) the
current profile reconstruction of the image in part (b). . . . . . . . 183
5.9 False color streak images with (a) deflecting cavity turned off, (b)
deflecting cavity turned on, as well as (c) the current profile re-
construction of the image in part (b) showing structure in the tail
region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.12 Plots of the simulated asymmetric bunch used as input for the
ELEGANT results of Fig. 5.13 showing (a) longitudinal trace
space and (b) current profile. . . . . . . . . . . . . . . . . . . . . . 192
xiii
5.13 Simulated deflecting cavity streaks and current profiles (using EL-
EGANT with 10,000 macroparticles) of an (initially) chirped Gaus-
sian electron beam for four different sextupole field values, with a
sextupole field ratio α = −1: (a) κ = 0, (b) κ = 1094 m−3 , (c)
κ = 1641 m−3 , (d) κ = 2188 m−3 . . . . . . . . . . . . . . . . . . . 193
xiv
List of Tables
4.4 Cavity and coupling iris dimensions for the field-balanced driven
modal simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2 Simulated Parameters Corresponding to Fig. 6.5 (a), (b), and (c) 204
xv
Acknowledgments
I am grateful to the many people who have supported, advised, assisted, and
encouraged me in a multitude of ways throughout my graduate career. I will
humbly attempt to acknowledge them here.
I wish to thank Gil Travish for his help in numerous aspects of my research
and for his continual willingness to lend a hand in the lab, proofread a paper,
or offer his advice. Many thanks to Chan Joshi for fostering an atmosphere of
harmony and cooperation between our two research groups, and for his excellent
management of the Neptune laboratory, which has kept it at the forefront of
research in lasers, plasmas, and beams. I also wish to thank Claudio Pellegrini
for his interest in my work and for his numerous contributions to our research
group and to the physics department.
xvi
For their assistance with aspects of the deflecting cavity design and construc-
tion, I wish to thank David Alesini for his advice on HFSS, Brendan O’Shea for
his diligent work on the CAD drawings, Joris Fourrier for his meticulous cali-
brations of the various RF components, and to Harry Lockart for his excellent
management of the physics machine shop.
Finally, I wish to thank my family who have always supported me in all things,
big and small.
xvii
Vita
Publications
xviii
R. J. England, J. B. Rosenzweig, and N. Barov, ”Plasma electron fluid motion
and wave breaking near a density transition,” Phys. Rev. E, 66, 016501 (2002).
xix
der, ”High energy gain of trapped electrons in a tapered, diffraction-dominated
inverse-free-electron-laser,” Phys. Rev. Lett., 94, 154801 (2005).
xx
Abstract of the Dissertation
xxi
tions, a dogleg beamline was designed and built at the UCLA Neptune accelerator
laboratory. A standing wave deflecting cavity was designed and built as a tem-
poral diagnostic for measuring the current profiles of the electron bunches after
passing through the dogleg. Second-order horizontal dispersion measurements
and coherent transition radiation bunch length measurements of the electron
beam after passing through the dogleg show good agreement with the predic-
tions of theory and simulation.
xxii
CHAPTER 1
Introduction
These various forces collectively limit the achievable peak beam current and
longitudinal bunch shape. In experiments that require either a higher peak beam
1
current or a different shape of current profile than what the current photoin-
jector design produces, one or more longitudinal bunch shaping techniques may
be employed. In the present chapter, we discuss several areas of current active
research for which relativistic electron bunches of high current and brightness are
important (inverse Compton scattering, the free electron laser, and the plasma
wakefield accelerator), and review various techniques for manipulating the longi-
tudinal distribution of the beam (velocity bunching, the magnetic chicane, and
the dogleg compressor). All of these three techniques have been employed at the
UCLA Neptune laboratory in recent years.
The 6D phase space particle density of a bunched electron beam at some time t
may be represented by a distribution function f (r, p, t), where r and p are vectors
in position and momentum space respectively. The charge density ρ of the bunch
may be obtained by integrating this function over momenta and multiplying by
the total charge Q:
Z
ρ(r, t) = Q f (r, p, t)d3 p (1.1)
2
given by the first moments:
Z
hr(t)i = r f (r, p, t) d3 r d3 p (1.2)
Z
hp(t)i = p f (r, p, t) d3 r d3 p (1.3)
The averaged values hri and hpi preferably coincide with the design trajectory
and the design momentum of the beam, meaning the intended trajectory and
momentum of the beam. An arbitrary position or momentum may be expressed
relative to that of the beam centroid via
r = hri + δr (1.4)
p = hpi + δp (1.5)
where δr = (x, y, z) and δp = (δpx , δpy , δpz ) are the deviations in position and
momentum relative to the centroid, and we have suppressed the explicit time
dependence. The distribution function may now be parameterized in terms of
the new coordinates, f (r, p) → F (δr, δp), and second moments may be taken:
Z
2
hµ i = µ2 F (δr, δp) d3 δr d3 δp. (1.6)
Here µ stands for any one of the coordinates (x, y, z, δpx , δpy , δpz ) relative to the
centroid. From these second moments we can define the root mean square (RMS)
quantities
p
σµ = hµ2 i (1.7)
1 p 2 2
x,N = hδpx ihx i − hδpx xi2 , (1.8)
mc
1 q 2 2
y,N = hδpy ihy i − hδpy yi2 , (1.9)
mc
3
1 p 2 2
z,N = hδpz ihz i − hδpz zi2 . (1.10)
mc
Note that the normalized emittances have the dimensions of area in each of the
three phase planes. Under the influence of a time-independent Hamiltonian and
in the absence of coupling between the phase planes they are constants of the
motion by virtue of Liouville’s Theorem. Their relationships to the unnormal-
ized (or RMS) emittances and to the Courant-Snyder invariants are discussed in
Chapter 3. Normalized transverse emittance is a general figure of merit for elec-
tron bunches, as the area occupied by the beam distribution in each transverse
phase plane limits how tightly the beam can be focused in that dimension. The
transverse emittances are related to the transverse beam brightness, defined by
2I
B= , (1.11)
x,N y,N
where I = Qv0 /σz is the beam current with v0 being the bunch velocity. Bright-
ness, a measure of phase space density, is a general figure of merit for electron
beams. However, the expression given in Eq. (1.11) poses difficulties in certain
circumstances. For example, it contains no information about the longitudinal
distribution of the beam. Consequently, various alternative definitions have been
formulated for different applications, producing some ambiguity about the mean-
ing of the word brightness. These alternative definitions may, however, introduce
additional difficulties in some cases. Averaging over the full 4D distribution in
the combined x and y phase space, for example, produces an infinite result for a
Kapchinskij-Vladmirskij distribution [1]. The expression presented in Eq. (1.11)
is the one most commonly employed.
4
1.2 Longitudinal Photoinjector Dynamics
5
Figure 1.1: Cartoon drawing of 1.5 cell photoinjector.
The electrons are not relativistic at the moment they are emitted from the
cathode, as they have not yet experienced any acceleration. The phase of the
electric field seen by a given electron changes during the acceleration process,
making it necessary to carefully phase the RF in the structure relative to the
arrival time of the photocathode laser pulse. At a longitudinal distance s from
6
the cathode, the phase of the electric field seen by an electron is given by
Z s
γ(s̃)
φ = φ0 + k (p − 1) ds̃, (1.13)
0 γ(s̃)2 − 1
where φ0 is the injection phase of the electron and γ is its instantaneous energy
(normalized by mc2 ) at position s. The function γ(s) must satisfy the differential
equation for energy conservation which takes the form
dγ
= αk[sin φ + sin(φ + 2ks)], (1.14)
ds
where α = eE0 /2mc2 k is the so-called acceleration factor, which may also be
regarded as a normalized vector potential, or as the fraction of rest energy an
electron may receive in one radian of the wave. Equations (1.13) and (1.14)
constitute a pair of coupled differential equations for γ and φ as functions of lon-
gitudinal position s. Kim derives the following approximate analytical solutions
[2]:
1 p
φ= [ γ̃ 2 − 1 − (γ̃ − 1)] + φ0 , (1.15)
2α sin φ0
1
γ = γ0 + α[ks sin φ + (cos φ − cos(φ + 2ks))], (1.16)
2
where γ̃ = γ0 + 2α sin(φ0 )ks is the lowest order approximation for γ and γ0 is
the initial energy of the electron. Since the governing equations are controlled
by two unitless parameters (the acceleration factor α and the injection phase φ0 )
this treatment has fairly universal applicability. When considered as a function
of s the expression for the phase of the electron asymptotically approaches the
1
value φ∞ = φ0 + 2α sin φ0
. Requiring that this asymptote correspond with the crest
of the RF (and the maximum accelerating field) at φ∞ = π/2 produces a simple
transcendental equation for the optimal injection phase:
7
For the photoinjector at the UCLA Neptune Laboratory, for example, the value
of the acceleration factor is α = 1.5, giving an optimal injection phase of 69.1◦ ,
or approximately 21◦ behind the crest of the RF. It should be noted that Kim’s
theory breaks down for small values of φ0 or α. The conditions for validity of the
theory may be stated as: α ≥ 0.9 and φ0 ≥ π/4 [3]. A slightly modified approach
valid at smaller values of injection phase has been proposed by Gao [4].
In practice, the injection phase is chosen to lie close to φopt as given by Eqn.
(1.17), as this choice has been found to minimize the RF contribution to the
transverse emittance [2]. With this choice of injection phase the emittances take
the following forms for the case of a Gaussian distribution:
αk 3 σx2 σz2 √
rf
x,N = √ ; rf
z,N = 3(γf − 1)k 2 σz3 (1.18)
2
The resulting longitudinal phase space of the beam upon exiting the photoinjector
has a chirp in energy, with higher energy particles at the head of the beam and
8
Figure 1.2: Plots of longitudinal phase space.
lower energy particles at the tail, as seen in part (a) of Fig. 1.2. The chirp is not
linear but rather possesses a curvature imparted by the sinusoidal variation of
the electromagnetic accelerating force. This chirp may be reduced, enhanced, or
reversed in sign by appropriate phasing of the electron beam upon injection into
whatever standing-wave or travelling-wave structure is used for post-acceleration
of the particles following the initial acceleration in the photoinjector. There is
also a distribution in initial energy, called the intrinsic (or uncorrelated) energy
spread, due to the temperature of the cathode and the spectral width of the laser
pulse. The intrinsic energy spread corresponds with the vertical thickness of the
curves plotted in Fig. 1.2.
9
charge forces tends to counteract this compression.
10
where δr is the position of the particle relative to the bunch centroid. The
momentum kick imparted to a test particle in the bunch may then be written
Z Z
1 1
δpsc = Fsc dt = F(γ, δr) ds (1.21)
c γ2β
Due to the 1/γ 2 scaling, the most significant part of the momentum kick is
delivered close to the cathode where the bunch is nonrelativistic. Consequently,
the function F may be evaluated at γ = 1, and the equation of motion Eq. (1.14)
may be evaluated at s = 0 and then used to replace the integration over ds by
one over dγ. This produces the result
γf
mc F(1)
Z
dγ
δpsc = (1.22)
eE0 sin φ0 1 γ2β
For large values of γf the integral evaluates to π/2. Defining the normalized
field E(x, y, z) = (4π0 /eλ0 ) F(1, δr), where λ0 is the line charge density at the
bunch center, the space charge contributions to the emittances as obtained via
Eqs. (1.8)-(1.10) are given by
πI
sc
i,N = µi (ζ) ; i = x, z (1.23)
4αk sin φ0 IA
where ζ = σx /σz is the aspect ratio of the beam, I is the beam current, and
IA = 4π0 mc3 /e is the Alvén current. The quantities µi are dimensionless form
factors derived from the second moments of the distribution:
p
µx (ζ) = hEx2 ihx2 i − hEx xi2 (1.24)
p
µz (ζ) = hEz2 ihz 2 i − hEz zi2 (1.25)
1 1.1
µx (ζ) ≈ ; µz (ζ) ≈ . (1.26)
3ζ + 5 1 + 4.5ζ + 2.9ζ 2
11
Due to the variation in space charge force along the bunch the resulting con-
tribution to the intrinsic energy spread for a Gaussian bunch as a function of
longitudinal position is given by:
π mc2 I
∆σγsc (z) = G(z/σz ) (1.27)
2 eE0 σz sin φ0 IA
Since the space-charge and RF contributions to the emittances are not truly
decoupled, it is not immediately clear how to combine them in order to obtain
the total emittance. It can be shown, however, that the total emittance lies
between the direct sum and the root-square sum of the two contributions [2]:
q
rf 2 th rf sc
(th 2 sc 2
i,N ) + (i,N ) + (i,N ) < i,N < i,N + i,N + i,N (1.28)
Note that although it is generally small compared with the RF and space-charge
contributions, we have included for completeness the contribution from thermal
emittance at the cathode, which may be written
r
th kT
i,N = σi . (1.29)
mc2
Comparison of Eqs. (1.23) and (1.18) indicates that for constant bunch charge
the space charge contributions to the emittances scale linearly with peak beam
current and therefore inversely with the laser pulse length (if the aspect ratio is
12
scaled accordingly), while the RF contributions scale quadratically or cubically
with pulse length for x and z respectively. Consequently, there exists an optimal
value for the laser pulse length at which the emittance in one phase plane or the
other is minimized (i.e. balanced between the two competing effects). In this
regard, the transverse emittance is usually of greater concern, and in practice
the optimal pulse length is determined experimentally or by use of photoinjector
modeling codes so as to minimize transverse emittance. For a good example of
this, see Ref. [8]. Generally the optimal pulse length is on the order of 10 de-
grees of RF phase. For an S-Band gun operating at 2.856 GHz, this corresponds
to 10 ps, which is the approximate (full-width) laser pulse length on the cath-
ode at the UCLA Neptune Laboratory. The transverse space charge emittance
can be reduced by using a solenoidal lens to remove linear correlations between
the transverse and longitudinal phase planes. This process, called space charge
compensation is discussed in the following section.
13
of the overall emittance growth in the photoinjector [9]. Consequently, proper
employment of this technique can result in a significant improvement in overall
transverse emittance.
1 1 1
R(z, s) = R0 + λ(z)(D + s)2 − [R0 s + D2 λ(z)s] (1.30)
2 f 2
1 1
R0 (z, s) = λ(D + s) − (R0 + λ(z)D2 ) (1.31)
f 2
The quantities R and R0 represent the edges of a distribution in the x − x0 phase
space, with each longitudinal slice of the beam corresponding to a line segment in
this phase space. The area of the distribution is nonzero when the slopes of these
lines are different for different longitudinal positions within the bunch. That is,
if the quantity R0 /R is a function of z. Consequently, the distribution is reduced
to a single line in phase space (and therefore the linear space charge emittance
14
is zero) if R0 /R is made to be independent of z. This is found to occur at the
position s = s0 satisfying
1 2(D + s0 )
= (1.32)
f s20
Substituting Eq. (1.32) back into (1.30), one finds that R = R0 = 0 when s0
p
has the critical value sc = D2 + 2R0 /λ. For s0 ≤ sc the analysis fails because
the beam edges cross paths and so the assumption that the space charge force is
constant cannot hold. For s0 > sc a beam waist occurs. The conclusion, then, is
that a focusing lens can be used to eliminate the linear space charge emittance,
and one expects this to occur immediately following a beam waist (i.e. focus).
While this treatment is very elementary, it helps illuminate the basic mech-
anism at work. Serafini and Rosenzweig have examined space charge emittance
compensation in a more rigorous theoretical framework using the transverse RMS
envelope equation, with an extended hard-edged solenoid field as the focusing el-
ement [10]. The envelope equation is a differential equation which governs the
evolution of the RMS transverse bunch size σx in the paraxial limit under the
influence of space-charge (κs ), acceleration (γ 0 ), emittance pressure, and external
focusing (Kx2 ):
σx00 σx0 γ0 2x,N
κs
+ + Kx2 − − =0 (1.33)
σx σx β 2γ σx2 β 3 γ 3 σx4 β 2 γ 2
15
conditions σx = σ0 < σeq and σx0 = 0 at the cathode (s = 0), the resultant
solution for σx is
√
σx (z, s) = σeq (z) + [σ0 − σeq (z)] cos( 2Kx s) (1.34)
1 δIrms √
N,x (s) ≈ γβKx σ0 σeq (z0 ) sin( 2Kx s) (1.35)
2 Ip
where δIrms is the averaged RMS deviation from the peak current value Ip which
occurs at point z0 . From Eqs. (1.34) and (1.35) we see that the minimum of σx
√
occurs at 2Kx z = 2π which is also the location of the second null in x,N . This
behavior persists in a qualitative sense even when the acceleration of the beam
is properly taken into account, although the oscillation is no longer sinusoidal in
nature.
16
beam is well-collimated during post-acceleration, which effectively ”locks in” the
minimized emittance value by virtue of the 1/γ 2 dilution of the space-charge
force. The solenoid is generally positioned such that the magnetostatic field
generated by it is zero at the cathode, in order to avoid a longitudinal magnetic
field contribution to the transverse emittance. To this end, a secondary (bucking)
coil is sometimes placed behind the gun in such a way as to zero the magnetic
field in the cathode plane.
For beams of high charge and/or current density, the force exerted on the beam
by its image charge in the conducting plane of the cathode becomes comparable
to the RF accelerating force. This results in a lengthening of the bunch due to
decceleration of the tail particles and imposes an upper limit on the extractable
charge [11]. A simple model which treats the beam as a flat disk of charge
with a Gaussian transverse profile yields a maximum surface charge density of
Σmax = 0 E0 sin φ0 and an extracted charge Q of [12]
Σmax
Σ0
[1 + ln( ΣΣmax
0
)] , Σ0 ≥ Σmax
Q = Q0 × { (1.36)
1 , Σ0 < Σmax
where Q0 and Σ0 = Q0 /(2πσx2 ) are the charge and surface charge density which
would be extracted in the absence of the image charge effect. The saturation
begins when Σ0 = Σmax at which point the extracted charge is
17
duration, is derived by Travier using the relativistic Child-Langmuir formalism
and treating the beam as a voltage gap of width d [3]. This approach yields a
saturation charge of
r 3/2
2IA σx2 σt
4π eE0 sin φ0
Qsat = (1.38)
9 d mc2
where σt is the RMS duration of the laser pulse on the cathode. This result
reduces to that of Eq. (1.37) in the limit of short pulses, except for a multiplicative
constant that is on the order of unity. For typical pulse durations (on the order
of 1 to 10 ps) the correction for nonzero duration is small and Eq. (1.37) may be
used instead. However, in order to avoid bunch lengthening, Traviers recommends
that the bunch charge not exceed one-fifth the saturation value:
1 2
Qmax = Qsat = πσx2 0 E0 sin φ0 (1.39)
5 5
18
The final beam current is therefore given by
I0 I0 (t0 )
I(t) = ≈ Σ0 /Σmax
(1.41)
∂t/∂t0 1+ I (t )
Q0 2kαc sin φ0 0 0
We may conclude from the preceding analyses that the laser pulse duration and
injection phase are somewhat predetermined by an interest in obtaining optimal
beam quality. The optimal temporal shape of the laser pulse has also been
investigated and has been found to be a square pulse. The end result is that
the longitudinal phase space of a beam generated by an RF photoinjector tends
to invariably end up resembling that of Fig. 1.2 (a).
One may increase the peak current by scaling the total charge, but in doing so
the other photoinjector parameters must also be scaled appropriately in order to
prevent blowup of the transverse emittance. In the end, although the peak current
increases under such charge scaling, the transverse brightness is diminished by
the resulting dilution of the transverse emittance.
19
Rosenzweig and Colby have derived the general charge scaling laws for an
emittance compensated photoinjector [11]. This scaling relies upon keeping the
space charge term in the envelope equation κs /(σx2 β 3 γ 3 ) constant. Noting that
the beam perveance is given by κs = 2I/I0 and that the beam current goes as I ∝
Q/σz , we see that this can be accomplished by scaling the bunch dimensions as the
cube root of charge: σx,z ∝ Q1/3 . It follows from Eqns. (1.29), (1.18) and (1.23)
that the thermal, RF, and space charge emittance contributions scale as th
x,N ∝
Q1/3 , rf
x,N ∝ Q
4/3
, and sc
x,N ∝ Q
2/3
respectively. If the various contributions are
rf
represented by th sc
0 , 0 , and 0 for an emittance compensated photoinjector with
charge Q0 and current I0 , then taking the lower limit in Eq. (1.28) gives the
following scaling for the transverse emittance:
q
x,N = (th 2 2/3 + (sc )2 q 4/3 + (rf )2 q 8/3
0 ) q 0 0 (1.43)
2I0
B= (1.44)
(th 2 sc 2 2/3
0 ) + (0 ) q + (rf 2 4/3
0 ) q
IA (1 + 53 ζ)2
B = 16(2π)9/2 αkz (1.45)
σz ζ 2
20
employ one of various techniques to manipulate the longitudinal distribution of
particles within the bunch. Several such techniques as well as applications which
may benefit from them are discussed in the following two sections.
21
Figure 1.3: Cartoon of a simple chicane compressor.
A single dipole magnet would bend the electron beam’s path away from its
original trajectory, which is often undesirable. Assuming that one wishes to
return the electron beam to a path colinear with its original one, then the simplest
configuration for a magnetic compressor of this type is a set of three consecutive
22
dipole magnets where the first and third magnet are identical, but the field in
the middle magnet is reversed, and is either twice as strong or twice the length of
the end magnets. This configuration, called a magnetic chicane, is a commonly
employed method of bunch compression in linear accelerator systems. A cartoon
drawing of a chicane is shown in Fig. 1.3. The trajectory of a particle with the
central momentum of the bunch p0 is shown in black and the trajectory of an off-
energy particle with momentum p0 +δp is shown in red. Due to the different bend
radii, the higher-energy particle will traverse a shorter path through the device
and will therefore move forward in longitudinal position z within the bunch. For
a bunch that is initially chirped such that the higher energy particles lie in the
tail of the beam (see the subset plots of momentum error vs. z in Fig. 1.3), the
tail particles will move forward and the particles at the head will move backward
within the bunch. The result is that the bunch is longitudinally compressed at the
exit of the device. The required chirp may be imparted to the beam by choosing
the phase of the RF in one of the post-accelerating structures downstream of the
photoinjector gun such that the beam is injected forward of the crest of the peak
accelerating field. The chirp may alternatively be created in the photoinjector
itself, but as discussed in Section 1.2.1, operating the photoinjector away from
the optimal phase φopt degrades the emittance of the beam.
δp
ξ = ∆s αc (1.46)
p0
23
position within the bunch and δ = δp/p0 . The quantity ∂z/∂δ is an element of
the 6 × 6 transport matrix of partial derivatives of final coordinates with respect
to initial ones. This matrix formalism will be presented in Chapter 2, where its
use in calculating quantities such as ηx and ηz for compound systems of magnets
will be explained. The momentum compaction may be written as an integral
over the curvature k(s) of the central trajectory and the horizontal dispersion
function ηx (s) as
Z sf
1
αc = k(s) ηx (s) ds (1.47)
∆s s0
24
Figure 1.4: Plots of the curvature and dispersion function for a chicane.
the description we have presented illuminates the basic mechanism for compress-
ing the electron beam, various other physical effects may be important, some of
which can have a negative impact on beam quality. These include space-charge
effects which can cause degradation or bifurcation of the horizontal phase space
distribution [17], and emittance increase due to coherent synchrotron radiation
[18]. The minimum achievable pulse length obtainable is ultimately limited by
the intrinsic energy spread of the beam and by the degree of RF curvature im-
posed upon the longitudinal phase space. Theoretical limits are in the tens of
femtoseconds (RMS), if the beam is sufficiently short to start with, has sufficiently
high energy, and the energy-time correlation is quite linear [19]. But to obtain
the required (picosecond-level) pre-compression bunch length, an additional prior
stage or stages of compression may be needed [20].
25
Figure 1.5: Illustration of ballistic bunching compression.
26
the beam is related to the normalized beam energy γ by [21]
γ3
L= (1.48)
dδ/dz
where δ = δp/p0 is the fractional momentum deviation from the centroid mo-
mentum p0 and z is longitudinal position within the bunch. The derivative in the
denominator is intended to represent the slope of the chirp imparted to the beam
distribution by the accelerating structure in the trace space of δ and z. This
chirp is related to that prior to the accelerating structure used for the velocity
bunching by the relation
!
dδ dδ ∆pmax sin φ
= +k ∆pmax
(1.49)
dz dz init p0 1+ p0
cos φ
where k is the RF wavenumber, ∆pmax is the maximum longitudinal momentum
kick the structure imparts, and φ is the RF phase relative to the crest (i.e. the
phase corresponding to maximum accelerating force).
Here φ is the phase of the particle relative to the crest of the wave, βr = vr /c, and
α is the dimensionless acceleration factor defined in Section 1.2.1. Level sets of
the Hamiltonian in the phase space of γ and φ are plotted in Fig. 1.6, for values
p
of α = 0.2 and γr = 1 − βr2 = 12. The separatrix is shown in bold and the
27
Figure 1.6: Illustration of phase space rotation velocity buching, show-
ing (a) the full separatrix, in bold, and (b) a closeup of the bottom
portion.
value of γr is marked by a horizontal dashed line. Part (a) of the figure shows the
entire separatrix, while (b) is a closeup of the bottom portion of the phase space
plot. Orbits lying within the separatrix are closed and represent the trajectories
of particles trapped in an accelerating phase of the wave. An unchirped beam
(i.e. with zero energy-time correlation and a small energy spread) injected into
the wave with a moderately relativistic energy (point A) will slip back in phase
as it is accelerated along the closed orbit in phase space and will arrive at point
B where the beam energy is equal to the equivalent γr of the wave. As seen in
the figure, the phase space contours become nearly parallel and vertical, so the
movement of the electrons along these contours produces an inherent compression
in phase, with the maximum compression occurring at point B.
If the initial energy spread were zero then under this model, the final bunch
length would differ from zero only because of nonlinearities in the phase space
28
transformation imposed upon the bunch as it travels from point A to point B
along the contours. Consequently, a reasonable picture of the compression re-
quires inclusion of the intrinsic energy spread δγ0 . If δφ0 is the deviation from
zero of the injection phase of a particle in the vecinity of point A and φr +δφr is its
phase upon extraction in the vecinity of point B, then equating the Hamiltonian
at the two locations gives
1 p
α cos(φr + δφr ) = − (γ0 + δγ0 ) + βr (γ0 + δγ0 )2 − 1 + α cos δφ0 (1.51)
γr
Expanding the cosines in power series to lowest order produces the result [15]
δφr 1 h 2 i1/2
= δφ40 + δγ0 /αγ02 (1.52)
δφ0 2δφ0 | sin φr |
from which we see that the fractional compression is ultimately limited by a
combination of the phase spread and the initial energy spread δγ0 /γ0 . In the
limit where the wave phase velocity approaches the speed of light, the top of the
separatrix in Fig. 1.6 moves upward and the contours asymptote toward vertical
lines extending to infinite γ. In this case, a full quarter rotation in phase space
can still be achieved by simply injecting the beam slightly forward in phase.
29
Figure 1.7: Drawing of a dogleg.
30
Figure 1.8: Example dispersion function for a dogleg.
Pulses of x-rays with durations on the order of a few hundred femtoseconds are of
interest in a variety of areas of research, including the probing of sub-molecular
chemical and biological processes as well as the development of polarized proton
sources for high energy physics applications. Scattering of an intense laser beam
31
off of a high-brightness bunch of electrons is a promising technique for the gen-
eration of ultrashort pulses of short-wavelength and narrow spectrum radiation
[22], and has consequently been a subject of various recent experimental efforts
[23, 24, 25]. The scattering of photons off of moving electrons is referred to as
inverse Compton scattering (ICS). Scattered photons are upshifted in energy by
a fractional amount
hν 0 1 − β cos α
= 0
(1.54)
hν 1 − β cos α + (hν/γmc2 )(1 − cos ψ)
where α and α0 are the angles between the incoming and outgoing photons and
the incident electron, ψ is the angle between the incoming and outgoing photons,
ν and ν 0 are the incident and outgoing photon frequencies, h is Planck’s constant,
and β = v/c, γ = U/mc2 are the normalized electron velocity and energy [26].
The scattering geometry is shown in Fig. 1.9 (a). If the photon energy is compa-
rable with the relativistic electron energy, then the upshift becomes increasingly
small, tending to values less than unity as the photon energy exceeds that of the
electron. Therefore, assuming that the incident photon energy is small compared
with the relativistic electron energy (hν << γmc2 ), the distribution given by Eq.
(1.54) becomes peaked in the vecinity of ψ ≈ α with a maximal value of
0
hν 2
= (1.55)
hν max (1/2γ ) + 2(hν/γmc2 )
2
32
Figure 1.9: Collision geometry for inverse Compton scattering.
electron beam thus produces a pulse of upshifted radiation directed along the
path of the electron beam, as illustrated in Fig. 1.9 (b). The interaction angle
α between the beam and the laser is generally chosen to be either 90 degrees
(right-angle collision) or 0 degrees (head-on collision). The limit (hν << γmc2 )
when observed in the rest frame of the electron beam corresponds to the nonrel-
ativistic limit of ordinary Compton scattering. The total cross-section for ICS in
the low-energy photon limit is therefore equal to the (Lorentz invariant) classical
Thomson cross-section: σ = σT = (8π/3)(e2 /mc2 ). This type of scattering is
therefore sometimes called Thomson scattering. The total photon count of the
scattered pulse is then given by Nph = L σT , where L is the luminosity of the
collision, which is proportional to the number of electrons interacting with the
laser field.
33
electron bunch must be focused to a very tight waist at the interaction point (in
the case of a 90-degree scattering geometry). In either scenario, it is desirable to
maximize the charge density in the interaction region, requiring electron beams
of high current and short bunch length, as well as strong transverse focusing.
However, the large energy spreads required for magnetic compression techniques
used to obtain high current beams may result in chromatic aberrations in the
focusing system. In recent ICS experiments conducted at the PLEIADES x-ray
source facility at Lawrence Livermore National Laboratory, very high-gradient
(500 T/m) permanent magnet quadrupole magnets were used to overcome these
chromatic aberrations and obtain spot sizes on the order of tens of microns [27].
The free electron laser (FEL) is a technique for generating intense coherent radi-
ation over a wide range of frequencies, including those which are too high for the
use of traditional optics. The seminal theory was provided by Madey [28]. In a
free electron laser an electron beam is injected into a periodic magnetic field (an
undulator or wiggler). The transverse oscillation of the electrons in the magnetic
field produces coherent radiation at the fundamental resonant wavelength
λu
λ0 = 2
(1 + K 2 ) (1.57)
2γ
where λu is the spatial period of the undulator field, γ is the relativistic Lorentz
factor of the electron beam, and K is a dimensionless parameter measuring the
field strength of the undulator. For a helical undulator, K = eBu 2π/mcλu where
Bu is the peak magnetic field of the undulator. For a planar undulator, Bu is
√
replaced by Bu / 2 in this expression. Note the similarity between K and the
normalized electric field parameter α used in the photoinjector theory of Section
1.2.1.
34
Due to poor electron beam quality, early FELs used a seed pulse and mirrors
to create an optical cavity with multiple passes of the radiation through the un-
dulator. The advent of the modern photoinjector and the resultant improvement
in electron beam brightness has allowed for sufficient improvement in the FEL
amplification process that saturation can be reached in a single pass. This has
opened the possibility of lasing in the very ultraviolet (VUV) and x-ray regimes
where optical mirrors are transparent. However, since traditional lasers do not
exist at such frequencies, the FEL in this scenario must either be operated at a
higher harmonic of the seed laser pulse (high gain harmonic generation or HGHG)
[29] or must start up from noise (self amplified spontaneous emission or SASE)
[30, 31, 32].
In the 1-D theory of single-pass FEL operation, the normalized field amplitude
A of the produced radiation as a function of distance along the undulator z is a
superposition of exponential solutions of the form
X
A= Aj eikj z (1.58)
j
where kj = 4πρΛj /λu is the growth rate of the j’th mode, and Λj are solutions of
the 1D dispersion relation, which in the absence of space charge takes the form
[33]
(Λ + δ)2 Λ + 1 = 0 (1.59)
Equation (1.59) produces one real and two complex solutions. Among the
latter, the solution which has a negative imaginary part produces a continuous
35
exponential growth of the radiation field, giving rise to the so-called “high-gain
regime.” Calling this solution Λg , we see that it will begin to dominate Eq. (1.58)
for sufficiently large z, particularly if z is greater than the so-called gain length
given by Lg = λu /(4πρRe[iΛg ]). It can be shown that the gain is maximum in
the case of zero detuning [i.e. the wavelength of highest gain is the one given by
√
Eq. (1.57)], in which case Λg = (1/2) − i( 3/2), producing the solution [32]
λu
Lg = √ (1.61)
2 3πρ
The gain length is increased above this 1D value by the effects of diffraction,
energy spread (σγ ), and slippage (S = λNu ) if the following conditions are not
met: x,y ≤ λ/4π, σγ /γ < ρ, S < σz , and ZR > Lg , where Nu is the num-
ber of undulator periods and ZR is the coherent FEL radiation Rayleigh range
[31]. Physically, the high-gain regime of the FEL corresponds with a collective
instability produced by the exchange of energy between the electron beam and
the radiation field, producing a ponderomotive microbunching of the electrons
[34]. The process saturates when the beam is maximally bunched, typically after
about 10 gain lengths.
The ρ parameter provides a figure of merit for the FEL, as maximizing its
value reduces the gain length and increases the amplification. Noting that the
1/2
plasma frequency scales with electron density ne as Ωp ∝ ne , we have that
1/3
ρ ∝ ne . So the inverse gain length, and hence the output power, scales directly
with the cube root of charge density, making high current and brightness critical
to obtaining multiple order of magnitude gain in an undulator of reasonable (a
few meters) length. Consequently, photoinjector beams are typically used in
conjunction with chicane compression prior to injection into the undulator.
36
spatial microbunching with a chicane in order to enhance the amplification of
the seed pulse in the primary undulator. In this context, it has recently been
proposed that the use of a beam with a triangularly ramped current profile may
help to counteract inhomogeneity in the microbunching due to slippage [35]. This
proposal requires further study, but is of particular interest here because the
primary experimental work of this dissertation revolves around the generation
of ramped electron bunches for application to the plasma wake-field accelerator.
The latter application is discussed in the following sections.
Due to their capacity to support large electric fields, plasmas have been con-
sidered in recent years as a means for acceleration of charged particles capable
of producing field gradients larger than those achievable with traditional radio-
frequency linear accelerating cavities by several orders of magnitude. Longitu-
dinal field gradients in excess of 1 GeV/m can be obtained by the excitation
of large-amplitude relativistic waves in a plasma. Various acceleration schemes
have been proposed which rely upon driving such plasma waves, using either a
short intense laser beam (laser wakefield accelerator, LWFA) or a short relativis-
tic electron beam (plasma wakefield accelerator, PWFA) [36, 37, 38, 39, 40]. The
37
issues of beam quality associated with the drive beam in the PWFA scenario will
be found to be relevant to the experimental goals of this dissertation, namely
the generation and measurement of ultrashort electron beams with a triangularly
ramped current profile.
38
The wavebreaking electric field EW B is then given by this equation as βm → βb .
√
We can deduce the limiting forms EW B = βb E0 and EW B = 2E0 (1 − βb2 )1/4 for
the small amplitude (Em /E0 << 1) and large amplitude (Em /E0 >> 1) cases
respectively. In terms of drive beam parameters the maximum achievable field
occurs for a delta-function beam with charge density ρb = σb δ(φ) where σb =
R
Q/ dA is the charge divided by the transverse area of the beam, in which case,
the field amplitude is Em = 4πσb . Since the driver is a delta function, the solution
for Ez with this amplitude represents the Green’s function. In the low-amplitude
case, the Green’s function is a truncated sinusoid: G(φ) = 4π cos(φ)H(φ), where
H is the Heaviside step function. The linear wakefield for an arbitrary 1D drive
beam distribution is therefore given by
Z ∞ Z ∞
0 0 0
Ez (φ) = ρb (φ )G(φ − φ )dφ = 4π ρb (φ0 )cos(φ − φ0 )dφ0 (1.66)
−∞ φ
The shape and amplitude of the wakefield are thus intimately related to the
longitudinal shape of the drive beam. For example, for a Gaussian beam with
RMS length σφ = kp σz , the wakefield obtained from Eq. (1.66) is
2
Ez (φ) = πσb e−σφ /2 e−iφ erf(w) + eiφ erf(w∗ ) − 2 cos(φ)erf(1)
(1.67)
√
where w ≡ (φ−iσφ2 )/ 2σφ . This function is plotted for an RMS value of σφ = 1/2
in Fig. 1.10 (a). For values of σφ < π, the peak amplitude of the accelerating
wakefield behind the bunch occurs very close to φ = −π, in which case it assumes
the form
2
e−σφ /2 π − iσφ2 π + iσφ2
E+ ≈ 4πσb 2erf(1) + erf( √ ) + erf( √ ) (1.68)
4 2σφ 2σφ
39
Figure 1.10: Plots of (a) the wakefield for a Gaussian 1D bunch and
(b) the accelerating and retarding fields and corresponding transformer
ratio.
However, as seen in Fig. 1.10 (a), the peak field E− inside of the bunch is
deccelerating and will tend to deplete the drive bunch energy γb after a dis-
tance of approximately ∆z ≈ γb mc2 /qE− . Over the same distance, the in-
crease in energy of a test particle in the accelerating field behind the bunch
is ∆γ ≈ qE+ ∆z/mc2 = (E+ /E− )γb . So a better figure of merit for the wake-
field accelerator is the maximum fractional energy gain which the wakefield can
produce as measured by the ratio R = E+ /E− which is termed the transformer
ratio. The peak fields E+ and E− and their ratio R are plotted in Fig. 1.10 (b) as
functions of bunch length σφ = kp σz . We see that the maximum value is R = 2,
√
which occurs at σφ = 2. In fact, it has been shown that for any symmetrical
bunch, 2 is the maximal value of the linear 1D transformer ratio [43, 44].
It should be noted that the 1D theory is valid only in the case of a pancake
beam whose transverse size is much greater than its longitudinal size σx,y >> σz .
In fact, the primary operating regime of the PWFA, the so-called blowout regime,
is one in which this assumption breaks down. In the blowout regime, the beam
density is sufficiently larger than the ambient plasma density (nb > n0 ) that the
plasma electrons are blown radially outward, forming a rarified ion channel behind
40
the drive bunch, which provides a linear focusing force of strength K = 2πre n0 /γb .
Furthermore, the accelerating wakefield (Ez ) is radially uniform across the width
of the bunch [45]. These qualities make this regime highly desirable from the
perspectives of drive beam stability and uniformity of acceleration.
The mechanics of the blowout process itself is inherently nonlinear and mul-
tidimensional and involves wavebreaking. A nonrelativistic radial fluid approx-
imation may be used to describe certain aspects of this regime [45, 46]. This
treatment gives an approximate radial equation of motion for the radius r of a
2D fluid element (i.e. a ring of plasma) of initial radius r0 as a function of wave
phase φ:
Z r 0
d2 r r 1 r02
0 nb (r , φ) 0
=− + + r dr (1.69)
dφ2 2 r 2 0 n0
from which the blowout behavior of the ion channel can be calculated (at least
up to the point of wavebreaking). This analysis gives a maximum blowout radius
√ R∞
of rmax = 2.58 Λ, where Λ = kp2 0 r nb (r, 0)/n0 dr is the normalized charge
per unit length of the drive bunch. For a bi-Gaussian bunch, Λ = (n̂b /n0 )kp2 σr2 ,
where n̂b = nb (0, 0). For sufficiently large beam density (n̂b /n0 > 10) almost all
electrons are expelled from the region inside this radius. This depletion of the
electrons in the ion column results in a saturation of the longitudinal wake-field
amplitude, which has the following approximate dependence:
Emax 1
≈ 1.3Λ ln p (1.70)
E0 Λ/10
In the linear regime (n̂b /n0 << 10) this expression is replaced by
Emax 1
≈ 1.3Λ ln √ (1.71)
E0 Λ
41
Values obtained thereby tend to be slightly lower than what is predicted by a
linear 1D calculation, such as that shown in Fig. 1.10 for the Gaussian beam.
Nevertheless, as the dependence of the transformer ratio on the current distribu-
tion is qualitatively similar in these cases, the linear theory provides a useful rule
of thumb for both the nonlinear 1D and blowout regimes. As mentioned above,
for asymmetric beams the transformer ratio can in principle exceed the usual
limit of R = 2. Consequently, in the following section we explore the question of
what type of current distribution in the drive beam produces the highest possible
value for the transformer ratio.
The theoretical and experimental investigations which will comprise the primary
focus of this dissertation initially arose from an interest in discovering methods
for tailoring the current profile of a photoinjector beam to make it optimal for
use as a plasma wakefield accelerator drive beam. As discussed in Section 1.4.3,
the transformer ratio (equal to the maximum energy gain of a witness particle
as a fraction of the drive beam energy) is a figure of merit for the PWFA. The
amplitude and transformer ratio of the wakefield generated in the PWFA was
found to be intimately connected, via Eq. (1.66), to the charge density ρb = enb
of particles in the drive beam. The linear 1D wakefield of a Gaussian beam was
calculated directly and shown in Fig. 1.10 to have a maximum transformer ratio
√
R = 2 for a bunch length of σz = 2/kp .
42
Figure 1.11: PWFA with ramped drive beam and witness bunch.
if the current profile of the drive bunch is asymmetric. In addition, the use
of a linearly growing train of bunches has recently been reported at Argonne
National Laboratory to produce an enhancement of the transformer ratio [47].
For the case of a single drive bunch, an investigation, using linear 1D theory,
of the bunch shape that produces the maximum possible transformer ratio was
conducted initially by Bane, Chen, and Wilson [48]. The optimal current dis-
tribution was found to be the one which produces a retarding field that is the
same for all particles within the drive bunch and vanishes outside of it. Since
a retarding field that arises instantaneously from zero at the head of the bunch
would be unphysical, a form is proposed with a more gradual exponential rise at
the head:
(1 − e−αφ ) , 0 < φ < Φ
Eret (φ) ∝ (1.72)
0 , otherwise
where L is the length of the bunch and Φ = kp L. Equation (1.66) is then inverted
using the Laplace transform L to solve for the current distribution:
Z +i∞
1 L[Eret (φ)] sφ
ρb (φ) = e ds (1.73)
2πi −i∞ L[G(φ)]
43
yielding the result
kp σb 2
ρb (φ) = [(α + 1)eαφ − (αφ − 1)] ; 0<φ<Φ (1.74)
AΦ
where
e−αΦ − 1 Φ2
−αΦ
AΦ = Φ − −α e − −1 (1.75)
α 2
is the normalization constant. Equation (1.74) describes a triangular ramp with
an exponentially increasing component at the head. The maximum transformer
ratio occurs in the limit α → ∞ in which case the wakefield assumes the form
4πσb Φ sin(φ + Φ) − cos(φ + Φ) ; φ < −Φ
Ez (φ) = 2
(1.76)
1 + Φ /2 −1 ; −Φ < φ < 0
Dividing the maximum amplitudes of the top and bottom lines gives a transformer
√
ratio of R = 1 + Φ2 ≈ Φ = kp L. Consequently, R > 2 if the drive bunch is
longer than 2 plasma skin-depths (L > 2kp−1 ). However, the ideal bunch shape
given by Eq. (1.74) is very nonphysical due to the sharp spike at the head. A
simple triangular ramp, ρb (φ) = −(kp σb φ)/(Φ2 /2), gives the following wake-field:
4πσb cos(φ) − cos(φ + Φ) − Φ sin(φ + Φ) ; φ < −Φ
Ez (φ) = 2 (1.77)
Φ /2 cos(φ) − 1 ; −Φ < φ < 0
The extrema of the top line can be expressed analytically in terms of inverse
trig functions, but it is not particularly illuminating to do so. Its maximum value
is approximately Φ and that of the lower line is exactly 2, giving a transformer
ratio of R ≈ Φ/2, which is half that of the ideal case. An alternative distribution
considered in Ref. [48] is that of a triangular ramp with a rectangular pulse for
the first quarter of a period. This distribution produces a transformer ratio which
p
is almost identical to the ideal one: R = 1 + (1 − π/2 + Φ)2 ≈ Φ. The three
distributions and their respective transformer ratios are displayed in Fig. 1.12.
44
Figure 1.12: Several bunch distributions and their transformer ratios.
These results apply strictly only in the 1D linear regime. However, the use
of a truncated Gaussian (which closely resembles a triangular ramp) was investi-
gated in the blowout regime by Rosenzweig [45], who found, for a bunch length of
approximately kp L = 4π a transformer ratio of 5.6 in the simulation, as compared
with the value R = 7.85 predicted by linear theory. And Lotov has recently de-
termined the optimal profile shapes for both the drive and witness bunches in the
blowout regime through 2D particle-in-cell simulations using LCODE [49]. The
optimal drive bunch current profile obtained in these cases turns out to be close
to that predicted by the 1D theory. Consequently, the highest transformer ratio
in the various regimes (linear 1D, nonlinear 1D, and blowout) is produced by a
drive beam with a ramped current profile. Physically, we can interpret this as
an induction problem. The adiabatic rise in current minimizes the electric field
inside the beam and maximizes the potential energy stored in the plasma, which
is released when the bunch current drops to zero, producing a large-amplitude
plasma oscillation. The predicted features of this oscillation (sinusoidal, nonlin-
ear, wavebreaking) depend upon the physical model used, but as the released
energy is of similar magnitude in all cases, the various models give similar results
for the overall transformer ratio.
45
Figure 1.13: Idealized current profile and 2D simulation results.
46
phase back-of-crest in the accelerating section. A linear transformation of this sort
is analogous to the compression mechanism of the dogleg, which was discussed in
Section 1.3.3, with the association ηz ≈ −αc ∆s where αc was the (positive-valued)
momentum compaction given by Eq. (1.53) and ∆s is the total path-length
traversed through the device. The quantity ηz , which is termed the longitudinal
dispersion, will be defined more rigorously in Chapter 2.
The primary experimental goals of this dissertation are: (1) the design and con-
struction of a dogleg compressor for the UCLA Neptune Laboratory suitable for
producing ramped electron bunches, (2) the development and implementation
of temporal diagnostics with which to probe the longitudinal structure of these
electron bunches on sub-ps time scales, and (3) the experimental generation and
temporal measurement of ramped bunches as a proof-of-principle verification of
47
the mechanism proposed in the previous section. The diagnostics employed are:
(1) coherent transition radiation interferometry and (2) a transverse RF deflect-
ing cavity. As the high frequency deflecting cavity is a new technology developed
expressly for this experiment, Chapter 4 will be devoted primarily to the details
of its design and construction. The experimental results will be presented in
Chapter 5. Future efforts using the Neptune dogleg compressor are discussed in
Chapter 6.
48
CHAPTER 2
49
2.1 Background and Notation
dp q dU
= v × B(r) ; =0 (2.1)
dt c dt
The basis vectors of the coordinate system attached to the design particle are
then defined by the Serret-Frenet equations [50]
dr0 / ds k
ẑ ≡ ; x̂ ≡ − ; ŷ ≡ ẑ × x̂ (2.3)
|dr0 / ds| k
where k is the curvature vector. Its definition in terms of the reference trajectory
and its relationship to the radius of curvature of the design trajectory R(s) are
given by the following relations:
dẑ 1 p0 c
k≡ ; R(s) ≡ = (2.4)
ds k qB0 (s)
50
Figure 2.1: The curvilinear coordinate system.
Here q and p0 are the charge and momentum of the design particle and B0
is the y-component of the magnetic field along the design trajectory. We will
permit R (and k ) to assume both positive and negative values, according to the
relative signs of q and B0 in Eq. (2.4). Doing so prevents the coordinate axes, as
they are defined in Eq. (2.3), from flipping about the x-z plane in the event that
k changes sign (as would happen if the direction of the vertical component of the
magnetic field were reversed). The vector position r of a given particle relative
to a fixed coordinate system may now be written:
where δr represents the test particle’s deviation from the position of the design
particle, which in the curvilinear coordinates defined by Equation(2.4) may be
written
δr(s) = x x̂ + y ŷ + z ẑ (2.6)
Here (x, y, z) denote the deviations along the directions of the unit vectors of
the test particle position from that of the design particle. These coordinates (and
51
the unit vectors themselves) are functions of the arclength parameter s, which
represents the length traversed by the design particle along the design trajectory
(as such it is effectively a temporal parameter). Defining the momenta
˙
p ≡ γmṙ ; p0 ≡ γmṙ0 ; δp ≡ γmδr (2.7)
we see that Eq. (2.5) implies that the momentum deviation of the test particle
from that of the design particle is given by
δp = p − p0 . (2.8)
However, in the coordinates of Eq. (2.4) the design momentum has only a z-
component (p0 = p0 ẑ). Hence, Eq. (2.8) reads
The next step is to obtain from Eq. (2.1) the general equations of motion in
the curvilinear coordinate system defined in Section 2.1.1. From Eqs. (2.3-2.4)
we have (assuming that the design trajectory lies in a plane normal to the y-
direction),
ṙ0 = ṡẑ
Here superscript dots denote time derivatives and ṡ = v0 is the velocity of the
design particle. Using these identities, we find that
r = r0 (s) + x x̂ + y ŷ + z ẑ;
52
where
Combining (2.13) with the first of Eqs. (2.1), which may be cast into the form
q
v̇x = (Bz vy − By vz );
γmc
q
v̇y = (Bx vz − Bz vx ); (2.13)
γmc
q
v̇z = (By vx − Bx vy );
γmc
q
x00 − zk 0 − 2z 0 k − x k 2 − k = {Bz y 0 − By (z 0 + x k + 1)} ;
γmcv0
q
y 00 = {Bx (z 0 + x k + 1) − Bz (x0 − zk)} ; (2.14)
γmcv0
q
z 00 + xk 0 + 2x0 k − zk 2 = {By (x0 − zk) − Bx y 0 } .
γmcv0
53
we can write
δpz γ
= (z 0 + xk + 1) − 1
δ= (2.15)
p0 γ0
The presence of momentum error introduces a difference in total integrated path-
length (as opposed to z, the difference in instantaneous longitudinal position)
between the test particle and the design particle, due to different radii of curva-
ture in bends. To illuminate this mathematically, define the function S(p) to be
the path-length traversed by a particle of momentum p during the time that the
design particle travels from s0 to s. The vector element dS may then be written
Inserting for the components of v from Eq. (2.12), we find that the integrated
path-length difference during the time t = s/v0 that the design particle travels a
distance s is
Z s p
S(p) − S(p0 ) = [ (x0 − zk)2 + y 02 + (z 0 + kx + 1)2 − 1]ds (2.17)
0
where we have used the fact that dt = ds/v0 . This difference in distance trav-
eled is due entirely to the disparity in velocity of the two particles, so we can
additionally write
δv
S(p) − S(p0 ) =
s. (2.18)
v0
If we set z = z 0 = 0 in Eq. (2.17), the resulting integral measures the path-length
along the test particle trajectory up to the point where it intersects the x − y
plane of the curvilinear coordinate system at time t = s/v0 . Let us call this
integral Ŝ(p) and write
Z sp
Ŝ(p) ≡ S(p)|z,z0 →0 = x02 + y 02 + (kx + 1)2 ds (2.19)
0
54
The difference between S(p) − S(p0 ) and ζ(p) is approximately equal to the
longitudinal displacement ∆z of the test particle. To see this, note that the delay
in arrival time at point s between the test particle and the design particle is given
by:
Ŝ(p) Ŝ(p0 )
∆t = − . (2.21)
v v0
Expanding the right-hand side in powers of δv/v0 we obtain
2
δv δv
v0 ∆t = ζ − (s + ζ) + (s + ζ) + ... (2.22)
v0 v0
Noting that the time delay times v0 is the negative of the longitudinal displace-
◦
ment ∆z ≡ z − z and invoking Eq. (2.18), we have, to first order in combined
powers of ζ and δv, that
δv
∆z ≈ − ζ(p). (2.23)
v0
The variable ζ(p) represents the difference in path-length between the off-momentum
particle and the design particle due to the transverse displacement x of the parti-
cle, which is also the distance by which its radius of curvature varies locally from
that of the design particle. The variable ∆z represents the change in longitudinal
displacement of the particle relative to the position of the design particle. Here,
we give a name to the quantity ζ because some references (see Ref. [51]) refer
to ζ as the longitudinal coordinate instead of z (Carey refers to this variable as
l). This is justified by noting that in the ultrarelativistic limit, in which case the
velocities of both particles are approximately equal to c and so δv/v0 → 0. In
this case, we have that ζ = −∆z as β → 1.
55
2.1.4 Vector and Tensor (Transport) Notation
Note that in Eq. (2.27) there is an implied summation on repeated indices. The
transformation from the initial to final state is executed, therefore, by a sequence
56
of tensors Q, R, T , U , . . . of increasing rank which operate upon the initial state
◦
x. These tensors may be defined in component form as follows:
Qi = xi |x→0
◦
∂xi
Rij = ◦
∂ xj ◦ x→0
1 ∂ 2 xi
Tijl = (2.28)
2 ∂ x◦ j ∂ x◦ k ◦
x→0
1 ∂ 3 xi
Uijkl =
6 ∂ x◦ j ∂ x◦ k ∂ x◦ l ◦
x→0
Note that the zeroth-order vector Qi is typically set to zero, as the origin
of the curvilinear system generally lies on the design orbit by definition. The
components of the first and higher-order tensors are obtained by expanding the
equations of motion (2.14) of an arbitrary particle traveling near the design or-
bit. The 6-vector notation and the corresponding transport matrix formalism
was originally developed for use in the particle tracking code TRANSPORT [52].
It has more recently been implemented in the code ELEGANT [53], which will be
used extensively in this dissertation. The transport matrix notation has become
widely used in the accelerator community as a means of calculating first (and
sometimes higher) order analytical expressions for the effects of various configu-
rations of magnets and drifts on the beam dynamics. In order to calculate the
elements of the lowest order transformation Rij , it is first necessary to linearize
the general equations of motion (2.14) by expanding the static magnetic field B in
powers of the transverse coordinates (and then eliminating the nonlinear contri-
butions). The resulting expansion of the magnetic field into so-called multipoles
is the topic of the following section.
57
2.2 Multipole Expansion of the Magnetic Field
In order to obtain explicit analytical forms for the elements of the 6 × 6 transport
matrix R discussed in the previous section, we must linearize the general single-
particle equations of motion (2.14). This requires expanding the magnetic field
components in powers of the transverse coordinates. The contribution to the
magnetic field from products of transverse coordinates of order n is termed the
n’th magnetic multipole.
1
γ≈p (2.29)
1 − vz2 /c2
γ0 vz p0
≈ = (z 0 + xk + 1)(1 − δ + δ 2 − δ 3 + . . .) (2.30)
γ v0 p
δv δ 3 β02 2 1 2
= 2− δ + (β0 − 6β04 + 5β06 )δ 3 + . . . (2.31)
v0 γ0 2 γ02 2
The utility of these expansions will become apparent when we linearize the single
particle equations of motion in Section 2.3.1.
58
2.2.2 The General Field Expansion
The divergencelessness of the magnetic field then requires that ψ satisfy the
Laplace equation
∇2 ψ = 0 (2.33)
ψ = ψ(x, y, s) (2.34)
With respect to these coordinates, the gradient and the Laplacian take the forms
∂ ∂ 1 ∂
∇= x̂ + ŷ + ẑ ; (2.35)
∂x ∂y 1 + kx ∂s
2 2
∂2
2 1 ∂ 1 ∂
∇ = + 2+ ; (2.36)
1 + kx ∂x ∂y 1 + kx ∂s
Expanding ψ in powers of x and y gives us
∞
X x` y m
ψ(x, y, s) = Am,` (s) (2.37)
m, ` = 0
`!m!
Substitution of Eq. (2.37) into the Laplace equation results in the following
recursion relation among the coefficients [52]
59
These recurrence relations effectively eliminate the dependence of the field coef-
ficients on the mode number m. Applying the gradient operator to ψ we obtain
the magnetic field components
∞
X x`−1 y m
Bx = Am,` (s) ;
m, ` = 0
(` − 1)!m!
∞
X x` y m−1
By = Am,` (s) ; (2.39)
m, ` = 0
`!(m − 1)!
∞
X A0m,` (s) x` y m
Bz = ;
m, ` = 0
1 + k(s)x `!m!
The requirement that the reference plane of the design trajectory be normal to
the direction of y forces ψ to be an odd function of y: ψ(x, y, s) = −ψ(x, −y, s).
If this condition were not satisfied, the design particle could not be constrained
to lie in the plane y = 0, because B would have a nonzero component tangential
to this plane. In the interest of this symmetry consideration, we therefore require
that the coefficients for even values of m vanish
The eliminated terms in the series corresponding to even m are called “skew”
terms. Their failure to vanish would violate the symmetry rule imposed above on
the magnetic potential, and would permit coupling of the x and y phase planes
and deviation of the design trajectory from the y = 0 plane. Using (2.38) and
(2.40), all coefficients may be expressed solely in terms of those with m = 1:
...
60
We can classify the terms in the field expansion of B by the combined powers of
the transverse coordinates which we shall call n. We see from (2.39) that for the
x and y components, n = ` + m − 1, while for the z-component, n = ` + m. We
can then write
∞
X ∞
X ∞
X
Bx = Bx,n ; By = By,n ; Bz = Bz,n (2.42)
n=0 n=0 n=0
where we define
X x`−1 y m
Bx,n = Am,` (s) ; Q = {m, ` : n = (` − 1) + m} (2.43)
Q
(` − 1)!m!
X x` y m−1
By,n = Am,` (s) ; Q = {m, ` : n = ` + (m − 1)} (2.44)
Q
`!(m − 1)!
X A0m,` (s) x` y m
Bz,n = ; Q = {m, ` : n = ` + m} (2.45)
Q
1 + k(s)x `!m!
is called the nth magnetic multipole field. Using (2.41) to eliminate the coeffi-
cients Am,` with m 6= 1, the kth contribution to the field can be written solely in
terms of the coefficients A1,` , which we rename
`
∂ ∂ψ
a` (s) ≡ A1,` = (2.47)
∂x ∂y
x=y=0
The first few multipole contributions to the magnetic field are then given by
B0 = a0 ŷ (dipole)
B1 = a1 yx̂ + a1 xŷ + a00 yẑ (quadrupole)
B2 = a2 xyx̂ + 1
a
2 2
(x2 − y 2 ) ŷ + (a01 − ka00 ) xy ẑ (sextupole)
(2.48)
The “multipoles” derived here are to be distinguished from the usual multipole
fields discussed in electrodynamics textbooks, which are obtained by regarding
61
all current sources as being near the origin and representing the field at a point
far away by expanding in powers of 1/r, where r is the radial distance from the
origin. In the present analysis, the field expansion is performed in the smallness
of the coordinates, and therefore represents the limit where the current sources
producing the field are located away from the origin. Evaluating the vertical (y)
component of the total field (B = B1 + B2 + B3 + . . .) at the midplane (y = 0)
gives
1 1
By (y = 0) = a0 + a1 x + a2 x2 + a3 x3 + ... (2.49)
2 6
By (y = 0) = B0 1 − nkx + βk 2 x2 + γk 3 x3 + ...
(2.50)
or
By (y = 0) = B0 R k + K1 x + K2 x2 + K3 x3 + ...
(2.51)
The first form of the expansion is convenient because the coefficients (n, β, γ, ...)
are dimensionless, having been normalized to the local radius of curvature R =
1/k. However, when considering field configurations that do not possess a dipole
component, the curvature radius becomes infinite. In deriving the equations
of motion using the general expansion of B, it is then desirable to rewrite the
coefficients in the form of (2.51) before taking the limit k → 0.
62
2.3 Beam Optics
If we assume that the deviation of the particle from the design trajectory is small,
then we can expand the equations of motion in the smallness of the coordinates
x, y, and z. If we keep only the terms linear in these coordinates (and their
derivatives), we obtain a lowest-order representation of the motion. An expansion
of this sort was performed upon the magnetic field B in Section 2.2.3. Retaining
only the linear (dipole+quadrupole) terms in this expansion we have
The local dipole field B0 is related to the curvature k by the inverse of the
formula for the cyclotron radius applied at the local position of the design particle
qB0 qB00
k= , k0 = . (2.53)
γ0 mcv0 γ0 mcv0
Now the equations of motion (2.14) read
γ0 0 0
x00 − 2z 0 k − x k 2 − k = {k yy − k(1 − nkx) (z 0 + x k + 1)} , (2.54)
γ
γ0 2
y 00 = − nk y (z 0 + x k + 1) + k 0 y (x0 − zk) ,
(2.55)
γ
γ0
z 00 + xk 0 + 2x0 k − zk 2 = k {(1 − nkx) (x0 − zk) + nkyy 0 } . (2.56)
γ
Using expansion (2.30) for γ/γ0 in powers of (z 0 + kx) and δ, keeping only terms
linear in the coordinates, we obtain
y 00 + k 2 ny = 0, (2.57)
z 00 + kx0 + xk 0 = 0.
63
2.3.2 Linear Transverse Solutions
where
Kx2 (s) = k(s)2 [1 − n(s)] , Ky2 (s) = k(s)2 n(s) . (2.59)
◦ ◦0 ◦ ◦0
x̃(s) = Cx (s)x + Sx (s)x , ỹ(s) = Cy (s)y + Sy (s)y (2.60)
◦ ◦ ◦ ◦
where x = x (s0 ), x0 = x0 (s0 ), y = y(s0 ), and y 0 = y 0 (s0 ). The C and S functions are
called “cosine-like” and “sine-like.” In the absence of any field imperfections, the
linear motion in y is completely described by the betatron solution. The complete
solution for motion in x is the sum of the homogeneous solution in (2.60) and
a particular solution ηx (s)δ, where ηx is the solution to the first of (2.58) with
δ = 1:
ηx00 + Kx2 (s)ηx = k. (dispersion function) (2.61)
The function ηx (s) is called the horizontal dispersion function. It represents the
trajectory followed by an off-momentum particle with unit momentum error that
64
is initially coincident with the design particle. Equations (2.62) do not include a
vertical dispersion function ηy (s) due to the assumption that the design trajectory
lies in a plane normal to the y direction. Vertical dispersion may be introduced
however in the case of systems with bending in the vertical plane.
dz + (1 + xk)ds = vz dt . (2.63)
If we now insert for x the linear solution from Eq. (2.62) and expand δvz /v0 to
lowest order in δ according to Eq. (2.31), we obtain the expression
◦ ◦ ◦0
z = z + Cz x + Sz x + ηz δ (2.65)
The factor αc which appears in the longitudinal dispersion term is called the
“momentum compaction”:
Z s
1
αc (s) = k(s)ηx (s)ds . (2.69)
s − s0 s0
65
The function ηz , called the longitudinal dispersion, represents the propor-
tionality between the longitudinal momentum deviation δp and the change in the
longitudinal position of the test particle over the interval from s0 to s. The expres-
sion for αc given in Eq. (2.69) is a first-order approximation to the path-length
compression in bends defined in Eq. (2.20):
δpz
ζ(p) ' (s − s0 ) αc . (2.70)
p0
Comparing this with the terms in a Taylor expansion of ζ(p) in powers of the
momentum deviation,
∂ζ 1 ∂2ζ 2
ζ(p) = δpz + δp + ... (2.71)
∂pz 2 ∂p2z z
we see that (2.70) is merely the first term in the Taylor series. We can therefore
make (to lowest order in the momentum deviation) the following association:
∂ζ p0
αc ' . (2.72)
∂pz s − s0
1
γt ≡ √ . (transition energy) (2.73)
αc
The concept of transition energy relates two competing effects: the longer
(shorter) total path-length versus the larger (smaller) velocity of a test particle
with a positive (negative) momentum dispersion. When the design energy γ0
is above the transition energy, particles with a positive momentum deviation
66
(i.e. with a momentum greater than the design momentum) take longer to pass
through the system and lag behind the design particle. When the design energy is
below transition, these particles pass through the system more quickly and move
ahead of the design particle. These statements may be summarized as follows:
δ>0 δ<0
− − −− − − −−
γ0 < γt ∆z ≤ 0 ∆z > 0 (below transition) (2.74)
γ0 > γt ∆z > 0 ∆z ≤ 0 (above transition)
γ0 = γt ∆z = 0 ∆z = 0 (at transition)
◦
Here ∆z = z - z, and the less than and greater than signs are approximate because
we are neglecting the contributions from Cz and Sz .
To linear order in the coordinates of the initial state, the tensor expansion of
◦
Section 2.1.4 reads x = Q + R x. The lowest order tensor Q is a 6-element vector
representing any offset of the beam centroid. If the centroid is assumed to adhere
properly to the design trajectory, then Q may be set to zero. In this case, to
lowest nonvanishing order, the transformation is a simple matrix multiplication:
.
◦
x=Rx (2.75)
R = RN . . . R3 R2 R1 (2.76)
67
The matrix R is the Jacobian of a transformation under linear forces. The lin-
earized system it represents has a time-independent Hamiltonian so the phase
space area is preserved and its Jacobian has unit determinant. Hence, we can
write
det(R) = 1. (2.77)
The components of the matrix R for a particular section of the beamline are
obtained by linearizing the equations of motion of an arbitrary particle traveling
near the design orbit. This linearization, performed in Sections 2.3.2 and 2.3.3,
gives an R of the form
Cx (s) Sx (s) 0 0 0 ηx (s)
0
Cx (s) Sx0 (s) 0 0 0 ηx0 (s)
0 0 Cy (s) Sy (s) 0 0
R(s) = (2.78)
Cy0 (s) Sy0 (s) 0
0 0 0
Cz (s) Sz (s) 0 0 1 ηz (s)
0 0 0 0 0 1
In the case of the quadrupole magnet, due to the fact that the transverse
equations of motion differ by a minus sign in front of the linear force term,
focusing only occurs in one transverse direction or the other. If n > 0 then
the magnet is vertically focusing and horizontally defocusing. If n < 0 then
the magnet is horizontally focusing and vertically defocusing. The quadrupole
68
Table 2.1: Matrix Elements for Common Beamline Components
Element Name Drift Quad Dipole Thin Lens
√
R11 Cx (s) 1 cosh[k ns] cos(ks) 1
1 √ 1
R12 Sx (s) s √
k n
sinh[k ns] k
sin(ks) 0
√ √
R21 Cx0 (s) 0 k n sinh[k ns] −k sin(ks) −1/f
√
R22 Sx0 (s) 1 cosh[k ns] cos(ks) 1
√
R33 Cy (s) 1 cos[k ns] 1 1
1 √
R34 Sy (s) s √
k n
sin[k ns] s 0
√ √
R43 Cy0 (s) 0 −k n sin[k ns] 0 1/f
√
R44 Sy0 (s) 1 cos[k ns] 1 1
1−cos(ks)
R16 ηx (s) 0 0 k
0
R26 ηx0 (s) 0 0 sin(ks) 0
R51 Cz (s) 0 0 − sin(ks) 0
cos(ks)−1
R55 Sz (s) 0 0 k
0
s s sin(ks)
R56 ηz (s) γ02 γ02
{ γs2 − s + k
} 1
0
69
√
matrix elements in Table 2.1 are correct for both cases. If n < 0, then n is pure
imaginary and the hyperbolic and sinusoidal functions effectively trade places by
virtue of the identities cosh(iz) = cos(z) and sinh(iz) = i sin(z). There is also a
corresponding change in sign in the focusing terms Cx and Cy .
In Section 1.1 we defined a beam distribution function F (δr, δp) over the coor-
dinates (x, y, z, δpx , δpy , δpz ) relative to the beam centroid, and used this distru-
bution to obtain second moments and to define the normalized emittances. If
we express the beam distribution instead in the trace space coordinates of the
6-vector x defined in Eq. (2.25), we can then use it to obtain a 6 × 6 matrix
of second moments whose evolution (to linear order) is given by way of a sim-
ple matrix multiplication with the transport matrix R. Let this new distribution
function be denoted f (x, s). If the beam is properly centered about its design
position and momentum then the first moments vanish:
Z
hxj i = xi f (x, s)d6 x = 0. (2.79)
70
We can define a 6 × 6 matrix Σ of second moments of the distribution:
Z
Σij (s) = xj xk f (x, s)d6 x = hxj xk i. (2.80)
◦ ◦ ◦
f (x, s) = f (x, s0 ), where x = Rx or x = R−1 x. (2.81)
Due to the fact that the transport matrix has unit determinant [Eq. (2.77)] we
can write
◦ ◦
d6 x = det(R) d6 x = d6 x. (2.82)
we see that Eq. (2.83) is the component form of the following matrix relationship
between the sigma matrices at the two points on the beamline:
◦
Σ = R Σ RT . (2.85)
71
variances in each of the 6 trace space coordinates. Taking their square roots then
gives us the RMS values for each of these quantities. We typically denote these
as follows:
p q
σxj ≡ Σjj = x2j , (2.86)
where the index j takes the values {1,2,3,4,5,6} corresponding to the trace space
coodinates (x, x0 , y, y 0 , z, δ). Because of their application to the examination of
the transverse dynamics, it is also customary to define the following transverse
submatrices:
Σ11 Σ12 Σ33 Σ34
Σx ≡ ; Σy ≡ (2.87)
Σ21 Σ22 Σ43 Σ44
and the corresponding 2 × 2 transport matrices
R11 R12 R33 R34
Mx ≡ ; My ≡ . (2.88)
R21 R22 R43 R44
The transverse RMS emittances are then defined by
p p
x ≡ det Σx ; y ≡ det Σy (2.89)
72
presence of dispersion and/or nonlinear effects, however, will tend to produce
emittance growth. The contribution to the emittance from the linear dispersion
can be seen by transporting the matrix of second moments according to Eq. (2.85)
and then calculating the emittances x,0 and x,f before and after respectively.
Because the y (vertical) elements of the matrices do not contribute to the final
result for x, we can write the matrix transport in an abbreviated 3 × 3 form
◦ T
= Mx d Σx 0 Mx d
Σx = (2.91)
0 1 0 σδ2 0 1
where d is the dispersion vector
ηx
d= (2.92)
ηx0
√
and σδ = Σ66 is the rms momentum spread. Evaluating Eq. (2.91) gives
2
= Σx d σδ ◦
Σx = , where Σx ≡ Mx Σx MxT + σδ2 dd T . (2.93)
d T σδ2 σδ2
The final rms emittance is then obtained by taking the determinant of the upper
left 2 × 2 portion of the matrix. This gives
1/2
p ◦
T 2 T
x,f = det Σx = det Mx Σx Mx + σδ dd (2.94)
In the case where the dispersion is zero (dd = 0) the above reduces, by utilizing
the identity det(AB) = det(A) det(B) and the fact that the transport matrix Mx
has unit determinant, to the expression for the initial emittance:
1/2
◦
x,f = det Σx = x,0 ; (dd → 0) (2.95)
It should be noted that the equality is valid only to linear order, and to the extent
that space charge effects on the emittance growth can be neglected.
73
2.3.6 Betatron Motion
The betatron motion was the name given earlier to the homogeneous parts (x̃,
ỹ) of the solutions to the linear transverse equations of motion, which satisfy
differential equations of the type:
where µ stands for either x̃ or ỹ. For a periodic function Kµ (s), this equation
is of Hill’s type. However, even if the system under investigation is not periodic
but is of finite length, the periodic Hill’s solution is still useful as a means of
reparametrizing the equations of motion. In terms of 2 × 2 transport matrices
the betatron solutions satisfy the transport relation
◦
χ µ = Mµχ µ , (2.97)
Substitution of Eq. (2.98) into (2.96) produces a pair of differential equations for
wµ and ψµ ,
e2µ
wµ00 + Kµ2 wµ = 3
(2.99)
wµ
eµ
ψµ0 = 2 (2.100)
wµ
where eµ is a constant of integration called the Courant-Snyder invariant. Using
the amplitude function wµ , we can define the following parameters of the test
74
particle:
wµ2 1 1 + αµ2
βµ ≡ ; αµ ≡ − βµ0 ; γµ ≡ . (2.101)
eµ 2 βµ
χ Tµ Wµ−1χ µ , = 1 (2.104)
◦
Wµ = Mµ W µ MµT . (2.105)
Equation (2.103) follows directly from the third of Eqs. (2.101). Equation (2.104)
may be verified by combining (2.98) with (2.101) and employing various trigono-
metric identities. Equation (2.105) then follows from (2.104) and (2.97), with use
of the identity (ABAT )−1 = (AT )−1 B −1 A−1 .
75
Figure 2.2: Trace space ellipses for (a) three trajectories with differ-
ent invariants and (b) three trajectories with the same invariant but
different momenta.
ellipses. All of these ellipses have the same Twiss parameters but vary according
to the areas which they enclose in the trace space plane of µ and µ0 . This is illus-
trated in Fig. 2.2(a) which shows three ellipses in the (x, x0 ) plane with the same
Twiss parameters (αx , βx , γx ) but with different values of the Courant-Snyder in-
variant ex . Consequently γµ , αµ , βµ are functions of s which are determined by
the curvature function Kµ (s) of the lattice and by their initial values. It should
also be noted that since γµ , αµ , βµ are all functions of s, the orientation of the
trace space ellipse upon which a particle’s betatron motion is constrained to lie
changes as the particle travels through the lattice. However, the area πeµ of this
ellipse does not change.
Recall that the horizontal dispersion function ηx (s) describes the trajectory
of an off-energy particle having unit momentum error and initially coincident
with the design particle. A particle that is initially displaced from the design
trajectory and having the arbitrary momentum error δ will then execute betatron
oscillations about this trajectory.
76
Consequently, the trace space ellipses in the plane of x and x0 (as opposed to x̃
and x̃0 are those of Eq. (2.106) but offset from the origin by amounts ηx δ and ηx0 δ
along the horizontal and vertical axes respectively. In this representation then
the set of all possible trace space trajectories consists of the set of concentric
ellipses described by (2.106) plus the set of all possible translations of them
along a line in the phase plane which extends from the origin through the point
(x, x0 )=(ηx ,ηx0 ). Figure 2.2(b) illustrates three trace space ellipses in the (x, x0 )
plane corresponding to the same value of ex but with different values of δ. Since
there is generally some spread in momentum σδ the phase ellipses are essentially
“smeared out” along this line in a way consistent with the momentum distribution
of the particles in the beam.
77
W-matrix and Courant-Snyder invariant satisfies
◦ ◦ ◦ ◦
W µ = Σµ and eµ = µ , (2.107)
◦
where Σµ is the actual RMS matrix of second moments of the beam. We call the
◦
corresponding ellipse the RMS ellipse. After propagating both matrices (W µ and
◦
Σµ ) via the usual linear transport relations (2.91) and (2.105), they are found at
a later position s along the beamline to be related by way of
In the limit of zero dispersion the foregoing results are much simplified. In
a dispersionless beamline, the RMS emittance (to linear order and neglecting
space-charge forces) is preserved and the envelope equations have the same form
for both transverse directions. To see the truth of these statements, we need
merely to set d = 0 in Eq. (2.108). We then have that the matrix of second
moments and the matrix of Courant-Snyder parameters are equal at all points.
This statement may be written
78
The RMS ellipse is a useful concept because it transforms under linear fo-
cusing forces according to the same matrix transport relation as the matrix of
second moments. The Courant-Snyder parameters of the RMS ellipse therefore
provide information about the size of the beam (βµ ), the angular divergence or
convergence (γµ ), the position-to-angle correlation (αµ ), and the phase space area
(eµ ). It follows that under these conditions the RMS beam size σµ is equal to
the betatron amplitude function wµ for both transverse directions. Consequently,
Eq. (2.99) takes the form
σµ00 2
2µ
+ Kµ − 4 = 0 , (dd → 0). (2.112)
σµ σµ
The RMS emittance appears on the right-hand side now, since µ =eµ for both x
and y. Note that this is identical to the RMS envelope equation, Eq. (1.33), in
the absence of the space charge and acceleration terms (κs = γ 0 = 0).
As a first approximation for the preliminary design of a linear beamline, the first
order matrix approach discussed in Section 2.3.4 is sufficient. A more accurate
description of the single-particle beam dynamics can be obtained by inclusion of
higher-order terms in the matrix expansion in Eq. (2.28). In circular machines,
where minor discrepancies in the theory can accumulate over many passes through
the machine to create significant errors, it may be necessary to include terms up
to 5th order. For linear devices, the inclusion of the 2nd order 6 × 6 × 6 tensor
elements Tijk is often sufficient, although under some circumstances (such as a
large energy spread), 3rd order terms may be required to explain observed beam
dynamics.
79
but keeping terms to second order in products of the coordinates. This produces
a set of coupled nonlinear differential equations. To second order, the transverse
equations of motion read
1
x00 + (1 − n)k 2 x − kδ = (2n − 1 − β)k 3 x2 + k 0 xx0 + kx02
2
1 1
+ (2 − n)k 2 xδ + (k 00 − nk 3 + 2βk 3 )y 2 + k 0 yy 0 − ky 02 − kδ 2 , (2.113)
2 2
y 00 + nk 2 y = 2(β − n)k 3 xy + k 0 xy 0 − k 0 x0 y + kx0 y 0 + nk 2 yδ.
◦ ◦ ◦
xi = Rij xj + T̃ijk xj xk ; i = 1, 2, 3, 4 (2.114)
into Eqs. (2.113) and then equating coefficients of like products of the initial
◦ ◦
coordinates (xj xk ). This produces a set of differential equations of the form
00
T̃ijk (s) + Kµ2 (s)T̃ijk (s) = fijk (s) ; i = 1, 2, 3, 4. (2.115)
where µ stands for either x or y and the form Kx2 = (n − 1)k 2 or Ky2 = nk 2 that
appears depends upon which of Eqs. (2.113) the particular element originated
from, and the so-called driving terms fijk (s) are functions of n, β, k, Cx,y , Sx,y ,
and their derivatives. The second order matrix elements are then obtained by
individually solving each of Eq. (2.115) by integrating the driving term over the
Green’s function Gµ for each term:
Z s
T̃ijk = Gµ (s, τ )fijk (τ )dτ ; i = 1, 2, 3, 4. (2.116)
0
80
The Green’s function is given by Gµ (s, τ ) = Sµ (t)Cµ (τ ) − Sµ (τ )Cµ (s). It
should further be noted that to maintain continuity in the second order across
field boundaries, which may change abruptly, it is appropriate to employ a change
from derivatives d/ds with respect to path-length along the reference trajectory
to derivatives d/dz with respect to the longitudinal cartesian coordinate z. That
is, we implement the transformations
dx dx dx/ds
x0 = → = , (2.117)
ds dz 1 + kx
dy dy dy/ds
y0 = → = . (2.118)
ds dz 1 + kx
This change of variable effectively adds an additional term Fijk (s), which is a
function of the first-order matrix elements, to the solution for T̃ijk in Eq. (2.116),
so that the final matrix elements Tijk are given by
Z s
Tijk = Gµ (s, τ )fijk (τ )dτ + Fijk (s) ; i = 1, 2, 3, 4. (2.119)
0
Explicit general forms for the matrix elements are derived at length in Refs.
[52, 56]. The transverse second order matrix elements extracted from these results
for a drift, a quadrupole, a dipole, and a sextupole are shown in Table 2.2. It
should be noted that in this table, we have used the abbreviations K1 = −k 2 n
and K2 = k 3 β for the quadrupole and sextupole field strengths respectively. This
is consistent with the notation of Eqs. (2.50) and (2.51).
81
Table 2.2: Second-order (relativistic) matrix elements for common optical beam-
line components.
82
second order in ζ and δ this yields
" 2 #
◦ δ 1 3 β0
z = z − ζ + 2 (ζ + s) − s 4 + δ2. (2.121)
γ0 γ0 2 γ0
◦
For an ultrarelativistic beam, only the first two terms contribute (z = z − ζ).
The remaining terms containing powers of 1/γ0 contribute nonrelativistic cor-
rections to the second order matrix elements involving the momentum error δ,
namely the nonvanishing T5jk where j and/or k is equal to 6. These correction
terms are derived in [51]. The relativistic terms are then obtained from the ex-
◦
pression z = z−ζ, which is the same expression used in the linear case except that
the integral for ζ from Eq. (2.20) is expanded to second order in the transverse
coordinates:
Z s
◦ 1 02 02
z≈z+ kx + (x + y ) ds. (2.122)
0 2
◦
The matrix elements are then obtained by substituting the form z = R5j xj +
◦ ◦
T5jk xj xk into the left-hand side and the form of Eq. (2.114) for the transverse
coordinates on the right-hand side and then equating coefficients of like pow-
◦
ers of the initial coordinates xi . General forms for the nonvanishing relativistic
matrix elements T5jk are given in Refs. [52, 57]. The explicit forms for a drift,
quadrupole, dipole, and sextupole obtained therefrom are displayed in Table 2.2.
83
CHAPTER 3
The technique for generating ramped electron bunches proposed in Section 1.5.2
relies upon the use of a dogleg as a bunch compressor. We will derive in the
present chapter the general beam optics theory of this device. We then apply
this theory specifically to the dogleg design implemented at the UCLA Neptune
Laboratory, including discussion of the optimal Twiss parameters for matching
the electron beam into the device and conditions for killing the horizontal dis-
persion and its derivative. Additionally, we describe in detail the bunch-shaping
mechanism in the language of first- and second-order beam optics. The resulting
analytical predictions are supported with simulations of the Neptune beamline.
Examples of three dogleg beamlines found at different facilities are shown in Fig.
3.1: (a) the Accelerator Test Facility (ATF) at Brookhaven National Laboratory,
(b) the UCLA Neptune Laboratory, and (c) the Stanford Linear Accelerator
Laboratory (SLAC) ORION test beamline. The drawings are cartoons drawn to
different scales, but the actual physical lengths are shown on the figure. Wedges,
blue lenses, and red rectangles represent dipoles, quadrupoles, and sextupoles,
84
Figure 3.1: Cartoon drawings of doglegs at (a) ATF-VISA, (b)
UCLA-Neptune, and (c) ORION-SLAC. Each drawing is scaled to fit
the figure.
85
respectively. In the drawing an approximate representation of the horizontal
dispersion function ηx is superimposed. We observe that the arrangement of
magnets has a mirror symmetry in the beamlines of Fig. 3.1(b) and (c), but
not (a). Although it is not strictly necessary to implement this sort of optical
symmetry, doing so simplifies both the analytical description and experimental
operation of the device as a compressor. Optimal operation of the dogleg as
a bunch compressor, under an optically symmetric geometry, relies upon the
horizontal dispersion function ηx passing through zero at the midpoint between
the bends. This ensures that the final dispersion function and its derivative
with respect to the path length parameter s are both zero at the exit of the
device. In order to further utilize the symmetry of the beamline in controlling
the beam size and eliminating net emittance growth, we additionally require that
a waist be formed at the midpoint. This ensures that the beta functions have
mirror symmetry and therefore return to their original values at the exit of the
beamline. In summary, then, the contraints we will impose upon the system are
that it
For the optical analysis we will represent the trace space coordinates of a test
particle in the beam distribution using the transport 6-vector notation x =
(x, x0 , y, y 0 , z, δ) described in Section 2.1.4. Let R(s0 , s) denote the transport
matrix between the points s0 and s, with s = 0 denoting the entrance of the
first bend magnet of the dogleg, and let Σ(s) denote the 6 × 6 matrix of second
moments of the beam distribution at the point s. Since we are concerned with
the values of the various system parameters primarily at three points (the en-
86
trance s = 0, the midpoint s = ŝ, and the exit s = ∆s), we adopt the following
simplified notation:
◦
R̂ ≡ R(0, ŝ), R ≡ R(0, ∆s), Σ ≡ Σ(0), Σ̂ ≡ Σ(ŝ), Σ ≡ Σ(∆s). (3.2)
R̂16 = 0 (3.3)
form contraints upon the system which effectively reduce the number of free
parameters. Equations (3.3) and (3.4) produce a set of simultaneous algebraic
equations for the values of the focal lengths fi (or equivalently, the field strengths
◦ ◦
Ki ) of the quadrupoles and the initial Twiss parameters α and β, which are
connected to the midpoint sigma-matrix elements by the linear transport relation
◦
Σ̂ = R̂T ΣR̂. (3.5)
For a given initial beam, Eqs. (3.3) and (3.4) provide the values of the quadrupole
strengths which optimize the lattice, to linear oder.
For considerations of beam shaping, we are concerned with the longitudinal (or
i = 5) component of the general transport relation Eq. (2.27) from the entrance
87
s = 0 to the exit of the final bend at s = ∆s which is given by
◦ ◦ ◦0 ◦ ◦0
z = z + R51 x + R52 x + R56 δ + T561 xδ + T562 δ x + . . . (3.6)
◦
z ≈ z + R56 δ + T566 δ 2 + U5666 δ 3 + . . . (3.7)
The first order coefficient R56 = (∂z/∂δ)δ→0 represents the longitudinal dis-
persion function ηx . The remaining elements, T566 , U5666 , . . ., are higher-order
momentum error contributions to the longitudinal dispersion. We may consider
Eq. (3.7) to apply to a beam of small transverse emittances and large energy
spread. Of course, it is conceivable to have a beam of very small emittance, but
which is either large in its transverse dimensions and very well collimated, or
which is very small in transverse size but with large angles. In either of these
cases, the assumption of small emittance is insufficient, so we additionally stipu-
late that the beam size is well controlled and does not undergo a sharp focus. It is
also presumed that the beam is sufficiently relativistic that space charge may be
neglected. The point at which higher-order terms in Eq. (3.7) may be truncated
depends upon the energy spread of the beam. In practice it is rarely necessary
to consider higher than third-order contributions for single-pass transport. For
88
the Neptune dogleg compressor, the third-order effects are negligible. General
relations for the first- and second-order contributions (R56 and T566 ) are derived
explicitly in the following section.
89
1 l 0 0 0 0
0 1 0 0 0 0
0 0 1 l 0 0
D(l) =
;
(3.10)
0 0 0 1 0 0
2
0 0 0 0 1 l /γ0
0 0 0 0 0 1
Let Y represent the linear matrix for a combination of quadrupoles and drifts.
The total first order transport matrix for a dogleg can then be written R = BY B̃,
where B = B(θ, ρ), B̃ = B(−θ, −ρ) and Y has the form
Y11 Y12 0 0 0 0
Y21 Y22 0 0 0 0
0 0 Y33 Y34 0 0
Y =
;
(3.11)
0 0 Y 43 Y 44 0 0
0 0 0 0 1 Y56
0 0 0 0 0 1
The resultant horizontal dispersion function and its derivative (elements R16 and
R26 of the total transport matrix) obtained by matrix multiplication are then
given by
The longitudinal dispersion element may then be written in terms of these func-
tions as follows:
2θρ
R56 = Y56 + − 2θρ + (2ρ − R16 ) sin θ + R26 ρ(1 − cos θ). (3.13)
γ02
90
Noting that Y56 + 2ρθ/γ02 = ∆s/γ02 , where ∆s is the total path length, we see that
if the quadrupoles are effectively utilized to eliminate the horizontal dispersion
terms (R16 , R26 → 0) then Eq. (3.13) reduces to
∆s
R56 = − 2ρ(θ − sin θ) (3.14)
γ02
Noting that according to Eq. (2.69), the longitudinal dispersion element is related
to the momentum compaction via R56 = ηx = ∆s[(1/γ02 ) − αc ] we see that
Eq. (3.14) is consistent with the example of Eq. (1.53) given earlier under the
replacements ρ → R0 and θ → π/4. Furthermore, we see that for a relativistic
beam the drift dispersion term ∆s/γ02 is close to zero and so the value of R56 is
inherently negative and that of αc is positive.
91
where [Y B̃]ijk = Yil B̃ljk + Yilm B̃lj B̃mk denotes the second-order matrix for the
first two successive elements B̃ and Y and there are implied sums on repeated
indices. This produces a set of equations for the elements Tijk in terms of ρ,
θ, Yij , and Yijk . Using these expressions, which are algebraically cumbersome
and which we will therefore neglect to write out explicitly, the equation for the
longitudinal dispersion element T566 may be expressed as a linear combination of
the expressions for the other matrix elements as follows:
X
T566 = 4ρ sin2 (θ/2) cos(θ/2) + a16 R16 + a26 R26 + ai6k Ti6k , (3.16)
W
W = {(i, k) : (i, k) = (1, 1), (1, 2), (1, 6), (2, 1), (2, 2), (2, 6), (5, 1), (5, 2)} (3.17)
and
In the limits where R16 , R26 → 0, we find that Eq. (3.16) reduces to
X
T566 = 4ρ sin2 (θ/2) cos(θ/2) + ai6k Ti6k (3.19)
W
Recall from Table 2.2 that the T566 element is zero (at least in the relativistic
limit) for all of the constituent components (i.e. quads, drifts, bends, sextupoles).
It therefore arises in this system by virtue of the interaction of other nonlinear
correlations which form between the transverse and longitudinal coordinates and
92
the momentum error. These correlations are embodied by the nonlinear disper-
sion terms Ti6k which appear in Eq. (3.19). For a chirped electron beam, such as
that proposed in Section 1.5.2 as being necessary for the ramped beam mecha-
nism, the beam has a pre-existing correlation between z and δ. Consequently, the
correlations between the transverse coordinates and δ are inherently coupled to z
as well. The importance of the term T566 in the general longitudinal transforma-
tion of Eq. (3.6) will be made clear in the following sections, where it is found to
produce a profound distortion of the longitudinal phase space. This distortion,
if left uncorrected, effectively destroys the ramped shape of the electron bunches
that is predicted by the linear theory. As will be seen, the implementation of
sextupoles can be used to remedy this situation.
The basic mechanism proposed in Section 1.5.2 for generating ramped electron
bunches required an electron bunch that is initially positively chirped in energy
with higher-energy particles at the head of the bunch. A plot of the longitudinal
trace space distribution of such a beam is shown in Fig. 3.2(a). This distri-
bution was produced by a simulation of the UCLA Neptune photoinjector and
linac, using the particle tracking code PARMELA, with a simulated beam charge
of 600 pC and a beam energy of 11.8 MeV. The energy chirp was produced by
setting the simulated injection phase in the linac to 22 degrees behind the phase
which corresponds to the peak acceleration. In Fig. 3.2(b) we impose a simple
∂z
linear transformation of the form z → z + ∂δ
δ upon the longitudinal coordinate,
where we have chosen ∂z/∂δ = −5 cm. Since the transformative term is negative
(i.e. ∂z/∂δ < 0) particles at the head of the bunch, for which δ > 0, are trans-
ported backward within the bunch and particles in the tail, for which δ < 0, are
93
Figure 3.2: Artificial manipulation of a chirped energy distribution (a)
by a linear transformation with ∂z/∂δ < 0 to generate a ramped bunch
(c). In (b) a quadratic term has been added to the transformation.
transported forward.
94
in Fig. 3.2. That is, the longitudinal transformation for a dogleg compressor is
dominated by the dispersion term R56 = (∂z/∂δ)δ→0 , which is inherently negative
according to Eq. (3.14). However, with significant energy spread, the presence of
the higher-order longitudinal dispersion terms in the transformation of Eq. (3.7)
for a dogleg must be taken into consideration. Including a term that is quadratic
in the momentum error in the transformation, corresponding to the second-order
term T566 in Eq. (3.7), produces the plot shown in Fig. 3.2(b). The longitudinal
phase space in this plot is severely distorted by the quadratic correlation between
z and δ, destroying the ramped profile. As we will see in Section 3.2.4 the value
T566 = - 2 m used in Fig. 3.2(b) is close to the actual second-order longitudinal
dispersion of the UCLA Neptune dogleg. Consquently, the second-order term
T566 is of particular concern, and thus its elimination by the use of sextupole
magnets will be considered in the next section.
In order to linearize the longitudinal transport of the dogleg and thereby mimmic
the mathematical transformation of Fig. 3.2, we must eliminate the second-order
longitudinal dispersion term T566 in Eq. 3.7. The obvious method for doing this
is to use sextupole magnets, which are inherently second-order in their effects,
and have no first-order (linear) matrix elements. They may thereby be employed
to manipulate the second-order properties of the beamline with no effect upon
its optics to linear order. Higher than second-order corrections may in principle
be implemented (to eliminate U5666 for example), by use of octupole or higher
multipole magnets, but for single-pass transport this is rarely necessary.
95
each other only by quads and drifts. Although the same final result may be
obtained without them, these assumptions will greatly simplify our calculation.
Let the two sextupoles, of strengths κ and ακ, respectively, and of equal length
d, be denoted by the symbols S and S̃ and the intervening system of quads and
drifts by H. We then decompose the first- and second-order representations of
Y as Yij = Sik Hkl S̃lj and Yijk = Sil [H S̃]ljk + Silm [H S̃]lj [H S̃]mk , where [H S̃]ijk ≡
Hil S̃ljk + Hilm S̃lj S̃mk . Multiplying the linear matrices out explicitly and imposing
the requirements
H11 H12
R16 = 0, R26 = 0, det = 1, (3.20)
H21 H22
H11 = H22 .
d θ
T566 = 2 sin2 (θ/2)(A0 ρ sin θ + A+ + A− cos θ) − sin2 (3.22)
4 2
θ θ
× 4|C0 |2 2ReC0 − d cos sin + 8ρ3 cos θ(1 − sin θ) (1 − α)κ,
2 2
± ρ2 (H261 + H511 ),
96
With the additional associations
The linear dependence on κ is a reflection of the fact that the second-order matrix
elements for a sextupole are proportional to the field strength. The quantities
A and C are algebraic functions of θ and ρ, as well as the drift lengths and
quadrupole focal lengths. The general form of Eq. (3.25) is valid for any dogleg
with 2 symmetrically placed sextupoles, although the exact functional depen-
dences of A and C may vary with sextupole placement.
If the goal is to eliminate T566 altogether, then (i) to avoid asymptotic be-
havior, the value of α (the ratio of the two sextupole field strengths) should not
approach unity, and (ii) in order to minimize κ the quantity C(1 − α) should be
large and therefore α should be negative. A simple choice in agreement with these
requirements is α = −1, corresponding to sextupole fields equal in magnitude but
of opposite polarity. As a rule, the minimum number of sextupoles needed is equal
to the number of second-order matrix elements one wishes to eliminate. There-
fore α = 0 is also a possibility, although the elimination of one sextupole would
disrupt the optical symmetry and require the remaining one to have double the
field strength. Minimization of the required sextupole fields, through appropriate
placement of the correcting magnets, is desirable from the standpoint of prevent-
ing the inadvertent introduction of strong second-order geometrical effects as well
97
as third-order chromatic effects. From a heuristic perspective, the sextupole ma-
nipulation of the T566 amounts to a correction of chromatic errors introduced
by the horizontally focusing lenses. This connection is made in greater detail in
Section 3.2.4.
The sextupole correction of T566 in this system often has the added effect of
minimizing the horizontal emittance growth, due to the coupling of T566 to the
second-order horizontal dispersion discussed above. For a beam of large energy
spread and small transverse emittance, the nonlinear emittance growth is dom-
inated by the second-order horizontal dispersion elements T166 and T266 . To
demonstrate this, note that to second order, the matrix of second moments Σ
transforms according to
Z
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
Σjk = (Rjl xl + Tjlm xl xm )(Rkn xn + Tknp xn xp )f (x)det(J)d6 x, (3.26)
with δjk representing the Kronecker delta. Writing Eq. (3.26) in the bracket
notation, we have
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
Σjk = Rjl Rkn hxl xn i + 2Tknp Rjl hxl xn xp i + Tjlm Tknp hxl xm xn xp i, (3.28)
R ◦ ◦
where there is an implied sum on repeated indices and h. . .i ≡ . . . f (x)det(J)d6 x.
Now assume the beam distribution function to be uncoupled between the three
trace space planes, to have vanishing third moments, and unit Jacobian determi-
nant. Then the upper left 2 × 2 submatrix of Eq. (3.28) takes the form
◦
Σx = Mx Σx MxT + σδ2dd T + hδ 4 iD
DD T + Σgeo , (3.29)
98
◦
where D = (T166 , T266 ) is the second-order dispersion vector, and d , Mx , and Σx
are the first-order horizontal dispersion vector, the 2 × 2 linear transpor matrix
for the (x, x0 ) trace plane, and the initial 2 × 2 matrix of second moments, re-
spectively, as they were defined in Eqs. (2.87), (2.88), and (2.92). The matrix
Σgeo is the contribution from second-order geometrical terms. Equation (3.29)
may be regarded as a second-order extension of Eq. (2.108). For a beam of
small intitial emittance (with well-controlled beam sizes and angles) and large
energy spread, Eq. (3.29) is dominated by the dispersion terms, and we can set
Σgeo << hδ 4 iD
DD T . Furthermore, if the beam distribution in the z trace space
plane can be approximated by a rotated bi-Gaussian in z and δ, then hδ 4 i = 3σδ4 .
With these approximations, insertion of Eq. (3.29) into the definition of the
transverse emittance, Eq. (2.89), we have that the final emittance x,f is given
by q
◦
x,f ≈ det(Mx Σx MxT + σδ2ddT + 3σδ4DD T ). (3.30)
The first of the three terms inside the determinant in Eq. (3.30) is the contri-
bution from the initial emittance, which would be invariant if the transformation
were governed solely by the linear matrix Mx . Consequently, if the first-order
horizontal dispersion and its derivative are eliminated in accordance with the
discussion surrounding Eqs. (3.14) and (3.19), then d → 0 and the emittance
growth described by Eq. (3.30) is dominated by the third term inside the paren-
theses. The coupling of longitudinal to horizontal dispersion is such that for the
sextupole configuration described in Section 3.1.5 the values of T166 and T266 tend
to be reduced under the sextupole correction of T566 . Consequently, in many cases
sextupole correction of the longitudinal dispersion also has a reducing effect upon
the transverse emittance.
99
3.2 The UCLA Neptune Dogleg Compressor
Having derived a general theoretical treatment of the dominant first- and second-
order optics pertinent to the shaping of the longitudinal trace space using a dog-
leg, we proceed to apply these results to the specific design of the UCLA Neptune
dogleg compressor. In particular, we will derive the matching conditions on the
initial transverse Twiss parameters of the beam, as a function of the quadrupole
focal lengths, which are additionally constrained to produce a horizontally disper-
sionless beamline (to first order). We then use particle tracking codes to simulate
the compressor and verify our analytical predictions.
The layout of the UCLA Neptune dogleg compressor, which has been dubbed the
“S-Bahn” after a train system in Germany, is shown in Fig. 3.3 using the same
iconography as Fig. 3.1, but with labels indicating the quadrupole spacings L1 ,
L2 , L3 , the quadrupole focal lengths f1 , f2 , and the sextupole field strengths κ and
ακ. A more detailed description of the actual beamline hardware will be provided
in Chapter 5. Furthermore, details such as the finite lengths of the quadrupole
magnets and the fact that each of the 45 degree bends is actually composed of two
successive dipole magnets will be taken into account in the simulations, but will
be ignored for the purposes of the simplified linear analytical treatment which
we will now undertake. For our present purpose (namely, describing the linear
beam optics and parameter space of this compressor) the simple representation
of Fig. 3.3 is sufficient. For the sake of consistency with the notation of Eq.
(3.2), the initial, midpoint, and final s-positions are labeled (s = 0, s = ŝ, and
s = ∆s). The bend angle is 45 degrees, or θ = π/4, with an effective bend radius
of approximately ρ = 32 cm. The quad spacing lengths are L1 = 50 cm, L2 = 40
100
Figure 3.3: Cartoon drawing of Neptune “S-Bahn” dogleg.
101
Figure 3.4: Plot of the equation R̂16 (f1 , f2 ) = 0.
Section 3.1.5.
Under the assumption of optical symmetry, the requirement that the beamline
be nondispersive [condition (ii) of Eq. (3.1)] was found in Section 3.1.1 to reduce
to Eq. (3.3), namely R̂16 = ηx (ŝ) = 0. The transport matrix from the entrance
of the first bend to the midpoint of the S-Bahn, written in terms of the linear
matrices for a bend, a thin lens, and a drift as given by Eqs. (3.8) - (3.10), reads
Setting element (1,6) of this matrix to zero produces an algebraic equation relat-
ing the parameters L1 , L2 , L3 , f1 , f2 , θ, and ρ. Among these, the focal lengths f1
and f2 are the obvious free parameters, the others having the fixed values given
in the previous section. The condition R̂16 = 0 thus describes a curve in the space
of the free parameters f1 and f2 . The plot of this curve is shown in Fig. 3.4.
Note that the values on the horizontal axis are negative because f2 represents
102
the focal length of the horizontally defocusing lens. Each point on the curve in
Fig. 3.4 represents a possible operating point. However, the choice of (f1 , f2 )
affects the values of the initial Twiss parameters which are needed in order to
produce a waist at the midpoint, and thereby symmetrize the beam envelope.
Since large initial Twiss parameter values are actually more difficult to produce
out of the photoinjector, we will search for a point on the f1 -f2 curve which
minimizes the initial Twiss parameter values. This analysis is performed in the
following section.
The last of the three conditions in Eq. (3.1) requires a beam waist at the mid-
point of the dogleg. This condition, in conjunction with the optical symmetry
requirement, symmetrizes the beam envelopes σx (s) and σy (s) about the midpoint
(s = ŝ). While this is not strictly necessary for the beam-ramping mechanism
to work, it is important from the standpoint of controlling the transverse beam
size. The corresponding mathematical relation written in terms of the midpoint
Σ matrices, Eq. (3.4), amounts to a constraint upon the initial Twiss parameters
of the beam at the dogleg entrance, which are in turn related to the midpoint
sigma matrix via the usual linear transport relation, Eq. (3.5).
Since the dogleg has been constrained to have zero horizontal dispersion at the
midpoint, we can equate the Σ-matrices with the Twiss parameter W-matrices
◦ ◦
for the RMS ellipse at both the entrance and midpoint. That is, Σ = W and
Σ̂ = Ŵ . Combining these with the midpoint waist condition of Eq. (3.4) we have
103
that
◦ ◦
◦ ◦
◦ βx −αx ◦ βy −αy
Σx = x ◦ ◦
, Σy = y
◦ ◦
, (3.32)
−αx γx −αy γy
β̂x 0 β̂y 0
Σ̂x = x . , Σ̂y = y . . (3.33)
0 1 β̂x 0 1 β̂y
◦
Relating (3.32) and (3.33) via Σ̂µ = M̂µ Σµ M̂µT produces 8 separate relations,
although half of them are not linearly independent. Eliminating the redundant
equations leaves the following:
◦ ◦ ◦
2 2
β x R̂11 − 2αx R̂11 R̂12 + γ x R̂12 = β̂x ,
◦ ◦ ◦
2 2
β y R̂33 − 2αy R̂33 R̂34 + γ y R̂34 = β̂y ,
◦ ◦ ◦ ◦
R̂11 (β x R̂21 − αx R̂22 ) + R̂12 (γ x R̂22 − αx R̂21 ) = 0, (3.34)
◦ ◦ ◦ ◦
R̂33 (β y R̂43 − αy R̂44 ) + R̂34 (γ y R̂44 − αy R̂43 ) = 0.
◦
These constitute four equations in six unknowns, the unknowns being αx,y ,
◦ ◦
β x,y , and β̂x,y . Furthermore, the equations are uncoupled between x and y, so αx
◦ ◦ ◦
and β x can separately be expressed as functions only of β̂x ; and αy and β y can
separately be expressed as functions only of β̂y . Due to the dependence of the
matrix R̂ on the focal lengths of the quadrupoles, a different set of such solutions
exists for every point (f1 , f2 ) on the curve in Fig. 3.4. Solutions corresponding
to three such points are shown in Fig. 3.5(a), (b), and (c).
In Fig. 3.5, the initial x Twiss parameters are plotted against β̂x in the first
column and the initial y Twiss parameters are plotted against β̂y in the second
column. The beta functions are in units of meters. The plots in the third column
show a reproduction of Fig. 3.4 with a dot marking the coordinates of the point
along the f1 -f2 curve for which the Twiss parameter equations were solved. Note
104
Figure 3.5: Allowed Twiss parameters for three different values of
(f1 , f2 ) corresponding to points on the curve in Fig 3.4: (a) f1 = 0.267
m, f2 = -0.42 m; (b) f1 = 0.269 m, f2 = -0.63 m; (c) f1 = 0.267 m, f2
= -0.82 m.
105
that the x-parameters in the first column are relatively insensitive to the choice
of f1 and f2 as compared with the y-parameters. Thus, it is good that these
quads are relatively unconstrained.
◦
Also, all solutions for the Twiss parameters αx,y are positive, and therefore
the incoming beam must be convergent in both dimensions. To minimize the
vertical spot size at the midpoint of the compressor and to keep the initial Twiss
parameters near the values achievable with the Neptune photoinjector and linac
(which tend to be on the order of α ≈ 1, β ≈ 1 m, we pick the solution for (f1 , f2 )
◦ ◦
which brings the αy , β y curves as close to the origin as possible. This solution
corresponds, in fact, to the plot in part (b) of Fig. 3.5, for which f2 = −0.63 m
and f1 = 0.269 m. Optimal coupling of the electron beam into the compressor
then requires matching its Twiss parameters to a set of values which lie on the
first two plots of Fig. 3.5(b). The experimental procedure for accomplishing this
matching will be discussed in Chapter 5.
For the choice of parameter values corresponding to the point (f1 , f2 ) in Fig.
3.5(b), we can determine various quantities of interest analytically by using trans-
port matrix theory. The linear R56 as given by Eq. (3.14), for example, is
For a given location of the sextupole magnets, we can also obtain an explicit
form of the second-order longitudinal dispersion corresponding to Eq. (3.25). For
the dogleg layout shown in Fig. 3.3, with the sextupoles positioned a distance h
= 10 cm inside the outermost pair of lenses and having lengths of 5 cm, the T566
106
Figure 3.6: Values of κ required to eliminate T566 as a function of
sextupole position s up to the midpoint.
Here we have assumed α = −1, so that the two sextupole magnets have equal
field strengths but opposite polarities.
For this configuration, the T566 therefore has a value of −2.11 m in the absence
of sextupole correction, but can be forced to zero at a sextupole field strength
of κ = 1204 m−3 . Our choice of location for the sextupole magnets is informed
by the observation that the nonlinear horizontal chromaticity (i.e. T261 ) of the
horizontally focusing lenses amplifies the second-order longitudinal dispersion by
coupling it to the linear horizontal dispersion of the dipoles. The T566 then grows
linearly in the drift section after each lens, due to the inherent nonlinear x-z
correlation introduced by the drifts. This correlation is due to deviation from
the paraxial approximation, which is a second-order effect. Consequently the
sextupole correctors should be located close to the horizontally focusing lenses,
so that the chromatic couplings produced by them can be counteracted before
107
they have a chance to grow in the subsequent drift sections. This is demonstrated
in Fig. 3.6 where the value of κ needed to cancel the T566 as a function of the
position s of the sextupole magnet from the entrance of the first dipole up to the
midpoint of the compressor. The other sextupole is assumed to be symmetrically
located with field strength −κ. Values of κ are plotted only in regions where a
sextupole 5 cm long (since this is the actual physical length of the sextupoles
used on the Neptune dogleg) could reasonably be placed. Note that the vertical
scale is logarithmic, so the plot indicates that the required field strength grows
by orders of magnitude as the sexupole position moves further from the vecinity
of the outermost lens (marked by the second dashed vertical line).
The location for the sextupoles previously assumed (10 cm inside the outer-
most quads) therefore reflects a compromise between the desire to locate them as
close to the quadrupoles as possible and the spatial constraints imposed by the
hardware and vacuum system. For this choice of sextupole location, the choice
of values of T566 as well as the horizontal second-order dispersion terms T166 and
T266 are plotted against κ in Fig. 3.7. We see that at the sextupole field strength
where the T566 vanishes (around 1200 m−3 ), the nonlinear horizontal dispersion
terms are simultaneously inverted in sign and reduced in magnitude by nearly a
factor of 3. Consequently, in accordance with the nonlinear emittance formula,
Eq. (3.30), the sextupole correction should also have a reducing effect on the
horizontal emittance growth in the compressor. This prediction is confirmed by
the simulation results presented in the following section.
The primary codes used in this dissertation for simulating the electron beam
dynamics in the Neptune beamline and dogleg compressor are PARMELA (ver-
108
Figure 3.7: Second-order dispersion terms as functions of sextupole
field strength.
sion UCLA-PARMELA 2.1) and ELEGANT (version 15.1.1) [58, 53]. Both of
these are particle tracking codes, which treat the beam as a collection of parti-
cles, whose position and momentum are calculated individually at various points
along the beamline. In order to reduce computation time, only a small number
N , typically much less than the number of actual particles Nb = Q/e being sim-
ulated are tracked. Thus for purposes of space-charge calculations, each of these
N macroparticles has a charge of Q/N .
109
azimuthally symmetric. ELEGANT, on the other hand, uses transport matrices
(up to third order) to track the macroparticle coordinates in trace space and
propagate them through a series of user-specified beamline components. Since
this approach neglects the space-charge forces, it is valid only at ultrarelativistic
energies (since the space-charge force in the lab frame scales as 1/γ02 ). In the sim-
ulations conducted for this thesis, a combination of PARMELA and ELEGANT
was employed in most cases. PARMELA was used to model the beam dynamics
in the photoinjector and linear accelerator. The macroparticle coordinates of the
beam were then extracted and used as input for an ELEGANT simulation of the
dogleg compressor. In addition, the PARMELA model was extended to include
the dogleg as well, so that the effects of space-charge could be gauged separately.
The longitudinal trace space distributions at the exit of the compressor, pre-
dicted from simulations using PARMELA with 10,000 macroparticles, are shown
without and with sextupole correction in Figs. 3.8(b) and 3.8(c), respectively.
The S-shaped distribution in Fig. 3.8(b) is evidence of the quadratic momentum
dependence of the z transformation produced by the second-order T566 contribu-
tion in Eq. (3.7). Note the similarity between Figs. 3.8 and 3.2. When sextupole
fields are utilized in accordance with the description of Section 3.1.5 to eliminate
this contribution, the resulting distribution [Fig. 3.8(c)] is found to correspond
very closely to that produced by a purely linear R56 transformation, such as that
shown in Fig. 3.2. The resulting current profile exhibits a sharp drop in current
at the back of the bunch, where the distribution begins to turn over on itself,
preceded by an approximately linear ramp of the sort described in Section 1.5.1
as being ideal for generating large-amplitude transformer ratios in a wakefield
accelerator.
It has been observed recently [17] that space-charge driven transverse phase
110
Figure 3.8: Plots of the z trace space and current profile from
PARMELA simulations showing (a) the beam at the entrance of the
dogleg compressor, and the same beam at the end (b) without sex-
tupole correction and (c) with sextupole correction.
space bifurcation and accompanying emittance growth are potential hazards en-
countered in low-energy (12-14 MeV) compression at Neptune. To gauge the
tranverse effects arising separately from non-linearities and space-charge forces,
ELEGANT and PARMELA simulations were employed to calculate the normal-
ized transverse emittance x,N = γ0 β0 x of the beam. The ELEGANT simulation,
with sextupoles turned off, predicts an emittance growth in the Neptune dogleg
due to nonlinear effects of ∆x,N = 13 mm mrad over the initial value of 5 mm
mrad at the entrance. This is consistent with the approximation of Eq. (3.30)
which gives ∆x,N ≈ 12 mm mrad.
111
the ELEGANT prediction of ∆x,N = 1.7 mm mrad. With the space-charge
routine turned on, PARMELA predicts a total emittance growth of ∆x,N = 11.6
mm mrad, for a 300 pC beam, indicating a significant additional contribution
from space-charge forces. These results lie in the intermediate range of ∆x,N
values measured in Ref. [17] and do not show evidence of the sort of phase
space bifurcation reported there. The predicted growth in transverse emittance,
however, imposes restrictions upon the focusability of the beam, requiring sharper
focusing angles and higher gradient quadrupole to match the beam into a PWFA.
A planned future experiment at the Neptune Laboratory to focus the beam using
high-gradient permanent magnet quadrupoles will be discussed in Chapter 6.
112
CHAPTER 4
4.1 Introduction
113
minimal Kramers-Kroenig phase).
The intensity profile of the radiation pulse could alternately be measured di-
rectly with a streak camera. This is a device that uses a photocathode to convert
the radiation into electrons by photoemission, accelerates the electrons in a cath-
ode ray tube, and then “streaks” them transversely by deflecting them with a DC
electric field. The deflected electrons then impinge upon a phosphorescent sur-
face or detector. There is currently an operational streak camera in the Neptune
laboratory, but its resolution is on the order of 1 to 2 ps, which is inadequate for
our purposes.
114
Figure 4.1: Illustration of current profile measurement using a deflect-
ing cavity.
also generally operated at or near the zero-crossing of the RF so that the momen-
tum kick imparted to the beam varies linearly with time and the transverse kicks
imparted to the head and tail of the beam are of opposite sign. Consequently,
in the drift following the deflecting cavity, the beam gradually elongates along
the coordinate corresponding to the deflection axis of the cavity. The temporal
structure of the beam thus becomes correlated with one of its transverse coordi-
nates and can be imaged on a simple phosphorescent screen inserted in the beam
path. This procedure is illustrated in Fig. 4.1.
115
length diagnostic was noted soon after [61]. So, the deflecting cavity is a rela-
tively old technology. However, the design of the structure built for the UCLA
Neptune beamline incorporates a number of unique features, including a mod-
ular o-ring-based assembly, high operating RF frequency, and low input power
requirements. In the present chapter we will describe the design process for this
cavity using a bottom-up approach, starting with the basic design parameters and
the analytical formulas for a single-cell pillbox cavity, and then extending these
results to describe multi-cell structures. We then discuss the use of computer
simulations to finalize the geometry of the cavity design, as well as the hardware
details associated with its construction, testing, and installation.
116
where [62]
eV0 p eV0 p 2π
Q̂ = βd βs sin ψ0 sin ∆ ; M̂ = βd βs cos ψ0 sin ∆. (4.2)
p0 c p0 c λ
and y0 and z0 denote the coordinates of the particle at the location of the final
screen when the deflecting voltage is zero. Here, βd is the beta function at the
deflector, βs is the beta function at the screen, ∆ is the betatron phase advance
from deflector to screen, ψ0 is the injection RF phase (i.e. the phase of the bunch
centroid), and λ is the RF wavelength. The quantity Q̂ represents a vertical
deflection of the beam centroid. If the beam is phased at the so-called RF zero
crossing (ψ0 = 0) then the bunch centroid sees zero net deflection and Q̂ → 0. The
quantity M̂ represents the vertical expansion coefficient of the beam following the
drift. Thus, the contribution to the vertical RMS beam size due to the deflection
is σdef = M̂ σz . The total RMS beam size at the screen is then
q
σy = σ02 + σdef
2
, (4.3)
where σ0 is the RMS spot size at the screen when the cavity is turned off. In
order to resolve the time structure of the bunch from the vertical streak of the
beam as seen at the screen, we require that the contribution to the vertical spot
size due to the z-correlated deflection exceed the nominal spot size of the beam
with the deflecting voltage turned off by about a factor of two: σdef > 2σ0 . This
translates into a condition on the transverse voltage V0 required to resolve the
time structure of the beam, which we may write as
cU/e
V0 > Vmin = σ0 , (4.4)
σz Lπf
√
where we have used the fact that the quantity βd βs sin ∆ is equal to the drift
length L between the deflector and the screen, and have denoted the relativistic
beam energy by U = γ0 mc2 ≈ p0 c. The rule-of-thumb expression in Eq. (4.4)
117
for the minimum deflecting voltage will be useful in developing the basic design
parameters of the deflecting cavity. We can see immediately from Eq. (4.4) that
in order to minimize the needed deflecting voltage, it is beneficial to have a small
initial spot size σ0 as well as a high RF frequency. To achieve the former, it
is customary to locate the screen at a minimum of the beta function (i.e. at a
focus). To take advantage of the inverse relation between Vmin and f we will
choose a high-frequency (X-band) RF power source for the Neptune deflecting
cavity.
Let us begin by deriving the equations for the electromagnetic fields in a cylin-
drical waveguide of radius b. Under the assumption of a travelling-wave z-
dependence to the fields:
E(r) E(r, φ)
= ei(kz−ωt) (4.5)
B(r) B(r, φ)
118
where
√ ω
κ2 = k02 − k 2 ; k0 = µ (4.7)
c
Applying Maxwell’s equations, it is found that the transverse components of
the fields can be obtained directly from the solutions for the longitudinal compo-
nents. Since the boundary conditions on Ez and Bz are different, the eigenvalues
are in general different. The fields thus naturally divide themselves into two
distinct categories, with the longitudinal component of either the magnetic or
electric field being identically zero. The former case is called transverse magnetic
(TM) and the latter transverse electric (TE):
Bz = 0 ; Ez (r = b) = 0 ; TM
(4.8)
Ez = 0 ; ∂r Bz (r = b) = 0 ; TE.
∇2t + κ2 Ψ = 0
(4.9)
where
1 xmn √
κmn = ≡ µωmn (4.11)
b x0mn
with the upper (lower) line corresponding to TM (TE) modes. Here ωmn is
the resonance frequency corresponding to the eigenvalue κmn of the Helmholtz
equation. For each value of κmn (and therefore of ωmn ) one obtains a normal
mode solution for the longitudinal wave-number k,
q p
k = k02 − κ2mn = µ (ω 2 − ωmn
2 ) (4.12)
119
Consequently, if the driving frequency is above the cutoff (ω > ωmn ), then
k is real and waves of mode (m, n) can propagate in the waveguide. Otherwise,
k is imaginary and so waves in that mode are damped. The case of a resonant
cavity of length ` corresponds to a waveguide with transverse conducting plates
introduced at its ends. Imposing this restriction, the z-dependences of the fields
become those appropriate for a standing wave. Applying the boundary conditions
at z = 0, ` we have that
Ez cos (kp z) (TM)
= Ψ(r, φ) (4.13)
B sin (k z) (TE)
z p
where Ψ is the corresponding solution for the infinite waveguide of the same
cross-section, and kp is the (now quantized) longitudinal wave-number, with p
being the longitudinal mode number:
pπ
kp = (4.14)
`
The oscillation freqency ω is now restricted to the discrete set of normal mode
frequencies ωmnp :
1 q 2
ωmnp = √ κmn + kp2 (4.15)
µ
The transverse fields are obtained from the longitudinal solutions via (for TM
modes),
kp iωmnp
Et = − 2
sin (kp z) ∇ t Ψ , Bt = cos (kp z) ẑ × ∇ t Ψ , (4.16)
κmn cκ2mn
iµωmnp kp
Et = − sin (k p z) ẑ × ∇ t Ψ , Bt = cos (kp z) ∇ t Ψ . (4.17)
cκ2mn κ2mn
120
For a cylindrical cavity, these expressions yield, for the TM modes,
E0 kp 0
Er = − Jm (κmn r) eimφ sin (kp z) ,
κmn
iE0 kp m
Eφ = − 2 Jm (κmn r) eimφ sin (kp z) ,
κmn r
Ez = E0 Jm (κmn r) eimφ cos (kp z) , (4.18)
E0 ωmnp m
Br = 2
Jm (κmn r) eimφ cos (kp z) ,
cκmn r
iE0 ωmnp 0
Bφ = Jm (κmn r) eimφ cos (kp z) ,
cκmn
Bz = 0 ;
The general solution for the fields in a cylindrical pillbox cavity can be rep-
resented by a superposition of the TE and TM modes in the usual way. The
lowest-order mode with transverse fields capable of imparting a net transverse
deflection to an electron traveling along the axis of the structure is the TM mode
with m = 1, n = 1, and p = 0, which we denote T M110 . The fields near the
center of the cells in a multi-cell deflecting structure closely resemble those of
the T M110 mode. In the next section, we therefore derive the fields of this mode
explicitly and obtain the limiting forms near the axis.
121
4.3.2 On-Axis Fields for the Deflecting Mode
For acceleration, RF cavities are operated in the TM010 mode. In this mode, the
only surviving field components are
r
0
Ez = E0 J0 (κ01 r) , Bφ = iE0 J (κ01 r) . (4.20)
µ 0
For a deflecting cavity, we would like to have zero longitudinal electric field
and transverse fields on-axis which tend to produce a net transverse momentum
kick. The lowest-order TM dipole mode for this purpose is TM110 . From the
results of the previous section, the fields for this mode have the forms
Ez = E0 J1 (κ11 r) eiφ ;
E0 ω110 1
Br = J1 (κ11 r) eiφ ; (4.21)
cκ211 r
iE0 ω 0
Bφ = J (κ11 r) eiφ ;
cκ11 1
J1 (κr) iφ
Ez = E0 J1 (κr)eiφ , Br = E e , Bφ = iEJ10 (κr)eiφ . (4.22)
κr
All other field components vanish. Note that there is no transverse electric field
in this mode. Therefore any transverse deflection of the beam would have to be
due entirely to magnetic forces. (Note that in the multi-cell structure this is not
the case, as there is a nonzero transverse electric field in the iris region between
cells). For small values of the argument, we can expand the 1st Bessel function
and its derivative as follows:
J1 (ξ) 1 ξ2 ξ4 1 3ξ 2 5ξ 4
= − + + ... ; J10 (ξ) = − + + .... (4.23)
ξ 2 16 384 2 16 384
122
We then have (near the axis) the expressions
E0 E E
Ez = κreiφ , Br = eiφ ; Bφ = i eiφ . (4.24)
2 2 2
or
E0 E E
Ez = κreiφ ; Bx = ; By = i . (4.26)
2 2 2
So at r = 0, the electric field vanishes and we have only the x and y magnetic
field components, which are out of phase by 90 degrees. Consequently, the on-
axis magnetic field vector is circularly polarized (i.e. rotates about the z axis
in time). This is to be expected from the cylindrical symmetry of the structure,
which prefers no particular transverse direction. In practice, a small asymmetry
may be introduced which couples more strongly to one transverse direction or the
other. This produces a mode-splitting whereby the two linear polarizations (x
and y) have different resonant frequencies. The desired polarization direction is
then selected by setting the driving RF frequency to match the correct resonant
value. The resulting fields of the separated polarizations (for deflections along
the x and y axes respectively) read:
E0 E
x: Ez = κr cos φ ; Bx = 0 ; By = (4.27)
2 2
E0 E
y: Ez = κr sin φ ; Bx = ; By = 0. (4.28)
2 2
As we will see in Section 4.4, the technique used to separate the two polarizations
in the multi-cell deflecting cavity design for the Neptune laboratory will consist
of a pair of small holes penetrating the wall between adjacent cells.
123
4.3.3 The Transverse Cavity Voltage
e ` e `
Z Z
∆pt ' − {Et + (v × B)t } dz = −i ∇ t Ez dz (4.29)
c 0 ω 0
where ` is the length of the cell and ∇ t denotes the transverse gradient operator.
Equation (4.29) is known as the Panofsky-Wenzel theorem [63]. For a vertical (y)
deflection in the TM110 dipole mode, we have that the longitudinal electric field
is given by Eq. (4.28). Taking the transverse gradient of that expression gives us
E0 κ iωt
∇ t Ez = e ŷ (4.30)
2
where we have reinserted the explicit harmonic time dependence. Inserting this
into Eq. (4.29) and taking the real part, the transverse momentum kick reads
" #
e `/2
Z `/2
ω (z − z0 )
Z
e E0 κ
∆pt = Re −i ∇ t Ez dz = ŷ sin dz (4.31)
ω −`/2 ω 2 −`/2 c
z0 ω `ω
ψ0 = − , ∆ψ = (4.32)
c c
Here we have used the fact that (for the T M110 mode) κ = k0 = ω/c. Note
that the value of ∆ψ is determined by the frequency and length of the cavity.
However, the value of ψ0 represents the phase arrival time of the particle at the
center of the cavity. If the design particle arrives at the phase ψ0 =0 then it
124
experiences no momentum kick and hence there is no offset of the beam centroid.
Particles which arrive before or after the design particle, however will experience
a transverse momentum kick that is approximately proportional to the z-position
of the particle within the bunch (since for small arguments sin ψ0 ' ψ0 ).
Let us define the voltage VT to be the voltage over which a particle would
acquire the transverse momentum whose magnitude was determined above:
e
∆pt = VT . (4.34)
c
Hence,
cE0
VT = sin(ψ0 )sin(∆ψ/2). (4.35)
ω
This may alternately be considered the real part of the voltage phasor
ṼT = V0 ei ψ0 , (4.36)
λ 3λ 5λ
`= , , , ... (4.38)
2 2 2
Note that Eq. (4.37) may alternately be written (see e.g. Ref. [64])
E0 ` λ
V0 = T ; where T ≡ sin(∆ψ/2),
2 π`
125
Power Coupling to the Cavity
Ptot = Pw + Pc (4.39)
where Pw is the power lost in the wall of the cavity and Pc is the power that
escapes by returning out through the coupling slot. We can define the external,
resonant, and loaded quality factors:
ω0 U ω0 U ω0 U
Qe = ; Q0 = ; QL = (4.40)
Pc Pw Ptot
From these relations we then see that the quality factors satisfy the reciprocal
addition rule
1 1 1
= + (4.41)
QL Q0 Qe
The coupling of the power into the cavity may then be characterized by a
coupling parameter βc which is defined by
Qe Pw
βc = = (4.42)
Q0 Pc
Consequently, βc =1 corresponds to the case Pc =Pw . This indicates that the
stored energy is emitted by the coupler at the same rate that it is absorbed by
the cavity walls. If the cavity is instead driven in a steady state at its reso-
nance frequency, then a fraction |Γ|2 of the power Pin that is incident upon the
input coupler from the external waveguide is reflected, where Γ is the complex
reflectance. The power Pw transmitted into the cavity is then
(1 − βc )/(1 + βc ) , βc ≤ 1
2
Pw = Pin (1 − |Γ| ) , where |Γ| = . (4.43)
(β − 1)/(β + 1) , β ≥ 1
c c c
126
The above relation between Γ and βc is derived in Ref. [65], among others.
Consequently, with a coupling βc of unity, when the cavity is driven in steady
state all input power is transmitted into the cavity with zero reflections. It can be
shown that under these conditions the stored energy in the cavity is maximized
and the cavity acts like a perfectly matched load. In microwave circuit theory, the
load mismatch of a terminated line is measured by the so-called voltage standing
wave ratio or VSWR, which is related to the magnitude of the reflectance Γ by
VSWR = (1+|Γ|)/(1−|Γ|) [66]. Hence, the coupling parameter may be expressed
as
1/VSWR , (undercoupled)
βc = VSWR , (overcoupled) . (4.44)
1 , (matched)
This expression is useful, because it is common practice in the accelerator com-
munity to express the coupling in terms of βc , but the VSWR is what is generally
measured empirically.
We wish to relate the power requirement for the TM110 mode of a pillbox cavity to
the quality factor Q0 and the deflecting voltage V0 . The square of the deflecting
voltage and the power are generally expressed relative to each other by defining a
shunt impedance R̃s in analogy with circuit theory. The quality factor and shunt
impedance are given by
U V02
Q0 = ω0 , R̃s = , (4.45)
Pw Pw
where ω0 is the resonant frequency of the given mode, U is the time-averaged
energy stored in the cavity, and Pw is the power loss in the walls of the cavity.
127
[67]
( 2 ) Z
`
pπ 1 p 6= 0
U= 1+ |Ψ|2 dS × (4.46)
4 κmn ` 2 p=0
S
where S is the cross-section of the cavity and Ψ is the solution to the Eq. (4.6).
For the TM110 mode, the solution is given by
Hence,
Z Z 2π Z b
2
|Ψ| dS = E02 J12 (κr)r cos2 (φ)dr dφ (4.48)
0 0
S
E02 a2 π 2
= J1 (κb) − J0 (κb)J2 (κb)
2
Noting that κb = x11 = 3.832 is the first zero of J1 and that J2 (x11 )=-J0 (x11 )=0.403
we have that the stored energy is given by
` 2
U = E02 πb2 J (κb). (4.49)
4 2
The power loss for TM modes is given by [67]
(
2 )
ξ
Z
pπ C` |Ψ|2 dS
Pw = 1+ 1 + (4.50)
σδµ κmn ` 2ξ 4A
S
where the upper (lower) line is for p 6= 0 (p = 0), and µ are the dielectric
constant and magnetic permeability, ξ is a geometrical factor of order unity, and
δ and σ are the skin depth and conductivity of the conducting boundary. So for
the TM110 mode, we have (ξ=1 for TM modes in a circular cavity)
`
Pw = 1+ E02 πb2 J22 (κb) (4.51)
2σδµ a
This expression for the power may be recast in terms of the deflecting voltage V0
by using Eq. (4.37) to replace E0 :
ω 2 a2 (1 + `/b) 2
Pw = π J (x11 ) V02 . (4.52)
2σδµ c2 sin2 [∆ψ/2] 2
128
Table 4.1: Comparison of Pillbox Parameters for Different Frequencies
Parameter Units S-Band X-Band X-Band
f GHz 2.856 11.424 9.6
λ cm 10.4 2.62 3.1
` cm 5.2 1.31 1.56
b cm 6.4 1.6 1.9
δ µm 1.22 0.614 0.670
Q0 - 23473 11735 12803
R̃s kΩ 1505 752 821
Vmin kV 593 148 176
V0 kV 1780 444 529
Pw kW 2103 263 341
Using the relations for the power loss and stored energy, Eqs. (4.49), (4.51)
and (4.52), we can now write explicit expressions for the quality factor and the
shunt impedance:
µ ` 1
Q0 = (4.53)
µc δ 1 + b`
129
with a power level a factor of 9 below those given in Table 4.1. Note that in this
regard one must consider losses in the waveguides when specifying the needed
power levels from the RF sources.
130
central iris through which the electron beam may pass.
A structure of this type may also be regarded as a waveguide loaded with pe-
riodically spaced disks (hence it is sometimes referred to as a disk-loaded waveg-
uide). A traveling wave propagating in a simple waveguide (i.e. in the absence
of loaded disks) must have a frequency above the cutoff, and therefore its phase
velocity vφ = ω/k is greater than c, as required by Eq. (4.12). This would make
it impossible for an ultrarelativistic beam to remain in phase with the wave. The
inclusion of the periodic disks effectively slows the phase velocity of the propa-
gating wave to that of light.
Due to the periodicity of the structure the electromagnetic fields can be made
to satisfy the Flouquet condition, whereby the field at the center of one cell is
equal to that at the center of another cell located an integer number of cell-
spacings downstream. This may be written E(z) = E(z + jd) where j is an
integer and d is the distance from the center of one cell to the next. For a
finite number N of cells, the allowed Flouquet phase shifts are then given by
∆φn = π(n − 1)/(N − 1) where n = 1, 2, 3, . . . , N . These discrete values for
the cell-to-cell phase shift correspond to an N -fold degeneracy of the single-cell
cavity mode produced by the coupling of the cells, with a resultant splitting
of the single-cell resonant frequency into a set of N distinct mode frequencies.
These modes are analogous to the normal modes of a chain of coupled oscillators.
In fact, the mode behavior and dispersion relation of a multi-cell cavity can be
calculated using a lumped-element circuit model of the structure as a chain of
coupled resonators. This methodology is outlined in the following section.
131
Figure 4.2: Equivalent circuit diagram for a chain of coupled cavities.
A fairly general circuit model for multicell cavities is given by Nagle, et al. [68].
This model is generalized by Yi, et al. [69], who also provide a block diagram
of a possible C+ algorithm for solving the equations. In this model, a multicell
RF cavity is represented by a circuit of the form shown in Fig. 4.2. The figure is
shown with the input power coupler located on the p’th cell, although additional
couplers could be included as well.
132
Defining the quantities
p Ln 1
Xn = Ln In , Qn = ωn , ωn = √ , (4.56)
Rn 2π Ln Cn
A · X = V, (4.57)
where
a1 κ1 0 ··· 0 X1 V1
κ1 a2 κ2 ··· 0 X2 V2
A= ··· , X = X3 , V = . (4.58)
0 κ2 a3 0 V3
.. .. .. .. .. .. ..
. . . . . . .
0 0 0 κN −1 aN XN VN
ωn2 ωn (1 + βn )
an = 1 − 2
−i . (4.59)
ω ωQn
133
Figure 4.3: Example plots of the dispersion relations for (a) a for-
ward-wave and (b) and backward-wave 9-cell structure.
f0
fn = p ; n = 1, 2, 3, . . . , N. (4.61)
1 + 2κ cos(∆φn )
The width of the passband and the frequency of the π-mode are then given by
1 1 , fN = √ f0
∆f = f0 √
−√ . (4.62)
1 − 2κ 1 + 2κ 1 − 2κ
134
A multi-cell cavity with a dispersion relation of this form is therefore called a
backward-wave structure. We make this distinction because the strong magnetic
component to the deflecting voltage in a multi-cell deflecting cavity tends to
produce a predominantly inductive cell-to-cell coupling. And in fact our final
design for the 9.6 GHz cavity will be a backward-wave device.
Generalizing from the usual results for multi-cell accelerating cavities (see for
example Ref. [70]), the transverse deflecting voltage is related to the input power
by a relation of the form
p
V0 = K P0 r̃s L (4.63)
where L is the total length of the structure, r̃s is the shunt impedance per unit
length, and K has the following forms for traveling waves (TW) and standing
135
Figure 4.4: Examples of a standing-wave (a) and a traveling-wave (b)
multi-cell structure with power vs. frequency plots.
and τ is the attenuation factor for traveling waves in the structure, given by
Lω L vg Q 0
τ= = ; where δ = (4.65)
vg Q 0 δ ω
Here vg is the traveling wave group velocity. Below we plot K for both cases
as functions of τ . We see that for τ < 1 the standing wave structure produces
a stronger deflection. The value for τ which maximizes the value of K for the
traveling wave case is approximately τ =1.26. At this value of τ , we can see
from the plot that the two curves are fairly close together and are approximately
Kmax '0.9.
136
Figure 4.5: Plot of K vs. τ for standing wave (dashed) and traveling
wave structures (solid).
V02
q
Pw
V0 ' Pcav nR̃s =⇒ Pcav ' = (4.66)
nR̃s n
The required input power Pcav is therefore approximately the single-cell value
divided by the number of cells. The primary benefit of the TW over the SW
structure is that a TW structure can maintain a larger mode separation for a
longer cavity length. So, for very long cavities (> 10 cells), a TW structure is
preferred. However, fewer cells are generally desirable in order to improve the
field flatness and avoid trapped modes. Since standing wave cavities tend to have
only a few cells, power can be coupled into them only at a few discrete frequency
values, corresponding to the normal mode frequencies in Eq. (4.61), as shown
in Fig. 4.4(a). For traveling wave structures, with tens (or hundreds) of cells,
these discrete resonances begin to overlap and thereby create a semi-continuous
137
passband, which is illustrated in Fig. 4.4(b). The presence of this continuous
passband for the traveling wave case is also aided by the fact that the output
coupler is generally terminated with a matched load, which causes the structure
to appear infinitely long with respect to the power coupling at the input. Since
the single-cell power requirement (341 kW) for the 9.6 GHz case, as indicated
in Table 4.1, exceeds the peak power of the available source (50 kW) by only a
factor of 7, a standing wave cavity is feasible for this design.
If the initial spot size is neglected then the time resolution is limited by
the deflecting voltage and the resolution of the screen and camera optics used to
138
Figure 4.6: Time resolution vs. power for n pillbox cells in series.
digitize the final y-distribution. Taking the cavity to consist of n identical pillbox
cells in series, we will approximate the total shunt impedance by nR̃s where R̃s is
the shunt impedance per cell as given by Eq. (4.54). The total deflecting voltage
is thus approximately
q
V0 ≈ nR̃s Pcav (4.68)
where Pcav is the total input power required. Thus, for a fixed value of the
deflecting voltage, the input power scales as Pcav ≈ Pw /n where Pw is the single-
cell input power as given by Eq. (4.52). Now, let ∆x represent the distance on
the final deflector screen corresponding to a single pixel in the digitized image, as
viewed using a camera and whatever collecting optics are present. The smallest
beam size ∆z = c∆t which can be resolved then is the value which results in a
deflection ∆y = M̂ ∆z. Hence, the best achievable time resolution in this case is
∆yU/d
∆t = p . (4.69)
Lπf R̃s nPcav
Equation (4.69) is plotted in Fig. 4.6 as a function of total power Pcav for dif-
139
ferent numbers n of cells, for an RF frequency of 9.6 GHz and a screen resolution
of ∆y = 30 µm (a typical value for the camera and screen setups in the Neptune
lab). Only odd values of n are considered because a standing-wave center-fed
cavity can by definition have only an odd number of cells. As was noted in Sec-
tion 4.3.4, the maximum power available from the X-band power source is 50 kW.
Furthermore, from Table 4.1 we see that the single-cell power required at 9.6 GHz
to obtain a suitable deflecting voltage was approximately 7 times the available
peak power, suggesting that the cavity should be designed to incorporate at least
7 cells. However, in order to further reduce the power requirements of the cavity
and to ensure that there would be sufficient deflecting voltage to make the desired
measurements, a 9-cell design was decided upon. The commercial RF modeling
code HFSS 9.2 was then used to finalize the design, and the particle tracking code
ELEGANT was used to simulate the effect of the cavity on the electron beam.
The computer-aided design of the UCLA Neptune deflecting cavity was a three-
year endeavor, which proceeded simultaneously with the construction and testing
of two prototypes (as well as the final cavity), included two changes in the design
frequency of the structure (due to unexpected hardware limitations of the RF
power source), and involved experimentation with a number of design features
that were later abandoned or significantly modified.
However, our goal here is to present the basic methodology of the computer-
aided design process in a compact and thematically (rather than chronologically)
organized fashion. We will therefore overview the computer modeling of the
final cavity design in a bottom-up approach, neglecting to mention a variety
of interceding design changes and modifications that would likely serve only to
140
Table 4.2: Basic Design Parameters for Deflecting Cavity
Frequency 9.59616 GHz
Flouquet mode π-mode
Number of cells 9
Peak input power 50 kW
Coupler type central coupler
Deflecting Voltage 529 kV
distract the reader. We will begin with a description of the eigenmode design and
definition of the main geometrical parameters, and then proceed to the simulation
of more advanced design features, such as the input coupler and the inclusion of
holes in the walls between cells to separate the orthogonal polarization modes
discussed at the end of Section 4.3.2.
For the purposes of the following discussions, we will take as our starting
design parameters those which are shown in Table 4.2. These are in fact the basic
design characteristics of the UCLA Neptune deflecting cavity, and are consistent
with the discussions and justifications provided by our comments in Section 4.4.
The deflecting voltage is that given in Table 4.1. The frequency shown (which
differs slightly from the previously assumed 9.6 GHz) is derived from the 252nd
harmonic of the drive laser oscillator frequency (38.08 MHz) since this was used as
the reference RF signal. The reasoning for this frequency choice will be explained
in more detail in Section 4.6.1.
The code used to design the 9-cell deflecting cavity for the Neptune Laboratory
was the commercial RF modeling software package HFSS version 9.2 by Ansoft
141
Corporation. This software uses the finite element method (FEM) to calculate
harmonic electric and magnetic fields in a high-frequency electromagnetic struc-
ture. The geometry of the electromagnetic structure to be modeled is created
using a computer-aided design or CAD-like graphical user interface. The adap-
tive grid on which the solution is performed is then generated automatically
based upon the user-specified geometry, although the fineness of the mesh may
be specified by the user on any or all of the surfaces in the model. Solutions
may be performed using an eigenvalue solver, which computes the eigenfrequen-
cies and associated eigenmode fields in the structure, or using a excitation-based
driven modal solver whereby the user defines an excitation on one or more port
boundaries and then the resulting fields are calculated inside the structure.
Both solution methods are highly valuable in the design process. The first
permits a quick way of tracking the behavior of the eigenmode frequencies under
modifications of the cavity geometry, while the second provides a more complete
but computationally intensive solution of the fields and impedance properties
of the structure, including the transmittance and reflectance at the port(s). In
addition, idealized boundary conditions (perfect electric, perfect magnetic) may
be imposed on surfaces, different materials can be specified for the composition
of the cavity and its interior volume, and symmetry planes may be introduced to
reduce the computation time.
Prior to designing the input power coupler, the basic mode structure of the cavity
was explored using the eigenmode solver. The cavity geometry is shown in Fig.
4.7, which depicts a cross-section of the half-cavity in the x-z plane. Only a
quarter of the structure is modeled in the simulation, as symmetry planes have
142
Figure 4.7: Geometry of the half-structure for the eigenmode solver.
Note that a beam pipe of length Lp has been included, as well as circular
irises of radius a and width t connecting the individual cells. The radii on the
edges of the irises are set to t/2 so that the surface has no sharp edges. The iris
radii were chosen to be sufficiently large to provide an adequate aperture for the
electron beam. The value of the cell-to-cell spacing d is constrained by the RF
wavelength of the design frequency and the Flouquet phase shift of ∆φ = π to
satisfy d = λ/2 = 15.62 mm (for a π-mode frequency of 9.59616 GHz). With all
of the cell radii set to the same value b1 = b2 = b3 = b4 = b5 = b, the simulated
eigenmode frequencies should be approximated by the dispersion relation given
in Eq. (4.61). This comparison is made in Fig. 4.8. The cavity dimensions used
in this simulation are shown in Table 4.3, where the simulated eigenmodes are
plotted in superposition with the dispersion curve for a cell-to-cell coupling of
143
Figure 4.8: Comparison of simulated eigenfrequencies (dots) with the
curve predicted by Eq. (4.61).
κ = −0.014. The value of the cell radius b has been chosen so that the π-mode
frequency is equal to the desired 9.6 GHz.
144
Flouquet modes are ∆φn = π(n − 1)/8, we have then that the supressed modes
are those corresponding to n = 2, 4, 6, and 8 or ∆φ = π/8, 3π/8, 5π/8, and 7π/8.
The mode suppression imposed by the boundary condition on the midplane is not
unphysical, however, because the eventual inclusion of a central power coupler on
the middle cell of the cavity will produce the identical form of mode supression,
as we will see in Section 4.6.3.
Note also from Fig. 4.8 that the dispersion curve corresponds to that of a
backward-wave structure, as the group velocity is negative (except at ∆φ = 0, π
where it is zero). It may therefore be compared with the plot of Fig. 4.3(b).
However, there is a relative offset in frequency between the two plots due to the
fact that the monocell frequency was set to 9.6 GHz in Fig. 4.3. In the present
simulation, the monocell frequency has been effectively increased by reducing the
cell diameter b, in order to match the lowest-frequency (π) mode to the desired
9.6 GHz.
Power is coupled into the cavity by way of a coupling aperture introduced into
one or more cells, which connects the interior volume of the cavity to that of
an external waveguide which carries RF power from the generator. For a 9-cell
standing-wave structure, a single coupling iris on the center cell is the simplest
solution. In principle, the coupler could be located elsewhere but this would
disrupt the cavity symmetry, making the design and simulation of the structure
unnecessarily difficult. Figure 4.9 shows the simulation geometry for a coupling
iris of width 2w, length 2l, corner radius r, and depth db located on the center cell
of the cavity. Note that because of the symmetry boundaries in the simulation,
only a quarter of the structure is shown. Consequently, the dimensions l and w
145
Figure 4.9: Geometry of the input coupler design for the HFSS simu-
lation.
shown in Fig. 4.9 correspond to the half-length and half-width of the coupling iris
respectively. The cross-sectional dimensions of the rectangular waveguide section
are set to correspond to those of the standard size of waveguide (WR-90) for this
frequency range: 0.4”×0.9”, or 10.16 mm×22.86 mm.
If only the diameter of the center cell (cell 5) is reduced in order to compensate
for the lowering of the resonant π-mode frequency produced by the introduction
of the coupling iris, there is a marked fall-off in the field strength progressing
146
Table 4.4: Cavity and coupling iris dimensions for the field-balanced driven modal
simulation.
from the central cell out towards the end-cells. In order to create a more uniform
field balance, the impedances of the cells must be gradually increased as one
progresses from cell 5 towards cell 1 (and likewise from cell 5 towards cell 9).
This may be accomplished by increasing the cell diameters in a tapered fashion
from the central cell outward. However, in order to simplify the design process,
the radii of all cells were reduced (with the same values) and then the radii of cells
1 and 3 were varied to balance out the fields. This process was iterated many
times with small adjustments in the width of the coupling iris l to eventually
produce the values shown in Table 4.4.
147
Figure 4.10: Plots of axial (a) electric field, (b) magnetic field, and (c)
field gradient as functions of position along the cavity axis.
The simulated transverse electric and magnetic field components along the
axis of the structure, corresponding with the cavity and iris dimensions of Table
4.4 and an input power of 50 kW, are shown in Fig. 4.10(a) and (b) respectively.
Note that the on-axis fields Ex (z) and By (z) in these plots are out of phase by
π/2. Thus, the transverse force experienced by an electron traveling along the
axis of the structure is given by
This force is plotted in Fig. 4.10(c). Note that because of the π/2 phase difference
between the electric and magnetic fields the electric fields in the iris actually
contribute to the transverse deflection. Integrating under the curve in Fig. 4.10(c)
gives us the simulated transverse deflecting voltage. The resulting value of V0 , as
well as the simulated coupling βc , π-mode frequency, and total shunt impedance
are shown in Table 4.5. Note that the corner radius of the coupling iris slot
in these studies was taken to be zero. The case of a nonzero corner radius is
considered in the next section.
148
Table 4.5: Simulated cavity parameters for optimimized cavity and coupling iris
dimensions.
In principle, the presence of the central coupler breaks the azimuthal symmetry
of the cavity and thereby creates a preferred polarization of the deflecting fields
(the direction of the deflecting force being along the transverse symmetry line
of the coupler). However, the presence of small asymmetrical machining errors
in the construction of the cavity can cause a rotation of the polarization vector
from cell-to-cell. Such a rotation would cause the direction of the transverse
deflecting force seen by the beam to change as it travels through the cavity,
thereby producing a distortion of the streaked image on the downstream profile
monitor.
149
Figure 4.11: Plots (a) electric and magnetic field magnitudes in the
transverse center plane of a cell, (b) the rod geometry at a junction
between cells, and (c) the hole geometry.
desired one. The resonant frequency of the undesired mode is thereby shifted so
that it is no longer excited by the driving RF. Commonly employed techniques
for this include the introduction of carefully placed rods, holes, or grooves into
the interior of the structure, in locations where the resulting perturbation of the
conducting boundary is strongly coupled to the fields of the undesired polarization
mode, but not to the desired one.
For the present case, we studied the use of both rods and holes, symmetrically
placed on either side of the irises between adjacent cells at a distance Rp from the
axis and having radius rp . The HFSS model geometries for a single monocell with
holes and rods respectively are depicted in Fig. 4.11(b) and (c). The placement
of the holes and rods as shown in Fig. 4.11 corresponds to a perturbation of
the undesired mode. This is seen by observing the field intensity plots in part
(a) of Fig. 4.11, which shows that the depicted hole/rod positions are close to
magnetic field maxima, and therefore their presence will strongly perturb the
monocell frequency. Rotating the the azimuthal positions of the holes or rods by
90 degrees about the z-axis would place them on a magnetic field null, where the
150
Figure 4.12: Plots of resonant pi-mode frequency for a pair of (a) holes
and (b) rods of radius 2 mm, as a function of radial position. The
curves are polynomial fits, with the dashed curves representing the
desired polarization mode, and the solid curves the undesired mode.
frequency perturbation would be less. This scenario would then represent the
effect of the holes or rods on the desired polarization mode.
The π-mode frequencies for the two polarizations were simulated in this man-
ner using the eigenmode solver for both holes and rods of a fixed diameter of rp
= 2 mm, as the radial position Rp was varied from 10 to 14 mm. The results are
shown in Fig. 4.12 for (a) holes and (b) rods. From these plots we can see that
the perturbative effect of the rods on the monocell frequency is greater than that
of the holes by 2 orders of magnitude (i.e. on the order of hundreds of MHz as
opposed to a few MHz).
151
Figure 4.13: Plots of resonant pi-mode frequency for a pair of holes
located a distance 14 mm from the axis as a function of hole radius.
The dashed (solid) curve is for the desired (undesired) polarization
mode.
choose the outermost position (Rp = 14 mm) and examine variations in mode
frequency as the hole diameter rp is varied. The resulting eigenmode simulation
results are shown in Fig. 4.13.
Note that the curves in Fig. 4.13 are interpolations of the simulation data. For
a pair of 2 mm diameter holes (i.e. rp = 1 mm) the simulated mode separation is
0.94 MHz, which is the approximate width of the π-mode resonance. We therefore
chose this as the design value for the polarization separation holes. Note that the
simulations which produced the plots in Figs. 4.12 and 4.13 were conducted for
a single monocell using the eigenvalue solver.
The polarization holes as well as the effect of nonzero corner radii on the
coupling iris were then incorporated into the full 3D model using the driven
modal solver with an excitation on the input waveguide port. The iris corners
were given nonzero radii because it was decided that sharp corners on the input
coupler should be avoided to prevent electrical breakdown in the actual cavity,
and because it would simplify the process of tuning the input coupling of the
152
Figure 4.14: Plots of simulated coupling beta for two different coupling
iris radii vs the half-width of the coupler.
cavity. Simulations were conducted for two different corner radius values r =
0.075” = 1.905 mm and r = 0.0625” = 1.5875 mm as the half-width l of the
iris was varied. The simulated coupling beta values are shown in Fig.4.14. The
point corresponding to the final design model is marked by a circle, corresponding
to r = 1.5875 mm and l = 4.2754 mm. The simulated resonant frequency for
these dimensions was found to be 9.6006 GHz. The cavity was intentionally
undercoupled by design, because the tolerances specified on the machining of the
cavity cell diameters was always set to negative values so that their frequencies
would come out slightly high (by a few MHz). This way, the final frequency tuning
could be accomplished by raising and regulating the temperature of the cavity.
However, the increase in cavity temperature would also increase the coupling beta
by a few percent. Thus the nominal beta value was set by design to slightly less
than unity.
153
4.6 Hardware and Construction
The UCLA Neptune deflecting cavity was constructed with the goal of providing
a robust, functional diagnostic for the Neptune laboratory which could be easily
moved to other locations (either on the Neptune beamline or elsewhere) and
which would be simple to tune, assemble, and install. The goals of simplicity of
installation and transportation are aided by the choice of a high frequency (and
hence small wavelength) compact portable power source. In the following sections
we outline the layout of the RF system used to drive the cavity, its mechanical
construction, and the experimental results of its testing and tuning.
The ultimate source of the RF for all of the cavities on the UCLA Neptune
beamline (gun, linac, deflector) is a 38.08 MHz signal generated by the crystal
oscillator in the mode-locker driver for the Nd:YAG photocathode drive laser
(model Antares 76-S by Coherent). Phase-locked dielectric resonating oscillators
(PLDRO’s) are used to frequency multiply this base signal by 75 and 252 to gen-
erate the 2.856 GHz and 9.59616 GHz low-level reference signals that ultimately
drive the gun/linac and deflecting cavities respectively. The layout of the RF
system is shown as a block diagram in Fig. 4.15.
The output of the drive laser is modulated at twice the 38.08 MHz oscillator
frequency, producing a 76.16 MHz pulse train of 1064 nm infrared laser light.
Every 1/5 second, one of these pulses is selected by a regenerative amplifier to
be amplified (by 106 ) and upshifted twice in frequency, eventually producing a
75 µJ pulse of (266 nm) UV energy that is used to drive the photocathode. The
laser pulse train (and hence the photoinjector beam) phase is maintained relative
154
Figure 4.15: Block diagram of the RF system layout for the UCLA
Neptune photoinjector, linac, and deflecting cavity.
Since the low-level RF signals for the various cavities are harmonics of the
base frequency of 38.08 MHz, they are phase-locked with respect to the laser
oscillator and therefore with respect to the electron beam. The low-level 9.59616
GHz RF signal produced by the PLDRO (and phase-locked to the 38.08 MHz
reference) is then transported on low-loss Heliax coaxial RF cable to a phase
shifter and attenuator (which can be manipulated in the control room) and then
to a 28 dB preamplifier which can provide up to 500 mW for the input of a VA-
24G klystron. The klystron unit is a turn-key rebuilt military device provided
by Radio Research Instruments. It has a built-in high-voltage power supply and
155
pulse-forming network and contains all the necessary electronics to produce up
to a 50 kW (peak power) output pulse of 1 µm duration which is synched to
an externally provided trigger pulse. The klystron can be operated at repetition
rates up to 1 kHz.
The 50 kW pulse from the VA-24G klystron is transported to the input coupler
of the deflecting cavity by a system of WR-90 waveguide. The output waveguide
port of the klystron is attached to an isolator made by Wenteq Corporation,
which acts like an RF “diode” to allow the flow of RF power only in the forward
direction and thereby prevent any reflected power from damaging the klystron.
The trigger pulse for the klystron is derived from the same chain of timing boxes
that is used to trigger the high-power RF system for the gun and linac. Hence,
both the timing of the 1 µm pulse and the phase of the 9.59616 GHz RF power
it carries are synchronized with the gun/linac RF system and with the electron
beam itself.
The final cavity design incorporates a knife-edge vacuum seal machined directly
into the mating faces of the cells. The knife-edge seal can accommodate either
a copper gasket or Viton o-ring. This design allows the cavity to be easily dis-
assembled, and negates the need for brazing, welding, or diffusion bonding of
the structure, which could warp and detune the cells. This design feature is un-
usual, but is made feasible in this case by the low peak power required (50 kW)
and by the less-than-stringent vacuum tolerances (on the order of 10−6 torr). A
computer-aided design (CAD) drawing of the structure is shown in Fig. 4.16. A
quarter of the structure is cut away to show the interior.
The cells and their interior cavities were machined on a lathe in the UCLA
156
Figure 4.16: CAD drawing of the Neptune deflector with 1/4 section
cutaway.
Physics Department machine shop. Tolerances on the inner cell diameters was
met to within approximately 0.0002” of the design values. A rectangular slot in
the top of the center cell and the polarization mode-splitting holes were machined
using electric discharge machining (EDM) by WireCut Co. of Los Angeles. A
custom-made waveguide section (shown at top in the figure) was constructed
using wire-EDM at the UCLA Department of Electrical Engineering machine
shop. This waveguide piece was brazed into the rectangular slot on the center
cell by a local brazing company. This was the only permanent bonding that was
performed on the structure.
157
such as extensive RF breakdown. As we will see, no such deleterious effects
were observed and therefore no permanent bonding of the cavity was attempted.
The material for the structure was chosen to be GlidCop AL-15, an industrial
material which has the electrical properties of copper but a tensile strength near
that of stainless steel. It consists of copper suspended in a matrix of aluminum
oxide. This material was chosen so that copper gaskets could (in principle) be
used to join the cells, and because it was hoped that the higher tensile strength
would help prevent deformation of the structure in the event that it needed to
be permanently bonded.
The electromagnetic properties of the deflecting cavity were tested using a net-
work analyzer (model HP 8719D) which is essentially a frequency sweeper that
can measure the forward, reflected, and transmitted amplitude and phase of the
RF wave on one or two ports, and can thereby calculate the complex transmit-
tances and reflectances between the ports of a passive RF device. In Fig. 4.17(a)
the square magnitude of the complex reflectance Γ (also known in transmission
matrix theory as the S11 ) at the input coupling port of the deflecting cavity is
plotted in decibels over the full frequency range of the passband of the structure,
showing all five of the resonances. As predicted by the simulation results of Sec-
tion 4.5.2, the π-mode corresponds to the first resonance, which occurs at the
lowest eigenfrequency. A closeup of the resonance for this mode is shown in part
(b). In part (c) the complex reflectance values corresponding to the frequency
range of part (b) are transposed onto the complex plane. This type of plot is
known in microwave circuit theory as a Smith Chart. The circle traced by the
resonance in this plot measures the impedance matching of the structure and
158
Figure 4.17: Plots of reflectance at the input coupler showing (a) all
five modes of the passband, (b) a closeup of the π mode, and (c) the
reflectance plot in the complex plane (i.e. Smith Chart) for the π
mode.
thereby provides an indication of the efficiency of the coupling. Since the radius
of the circle is less than unity, we deduce that the structure is undercoupled in
the sense defined by Eq. (4.44).
In Table 4.6 we list the various cavity parameters extracted from the resonance
plot of the π-mode shown in Fig. 4.17(b) and (c), including the resonant frequency
and the full-width at half-max (FWHM) of the resonance. Since the cavity is
slightly undercoupled (as it was intended to be, pursuant to the discussion of
Section 4.5.4) the coupling beta is inversely related to the VSWR. The resonant,
loaded, and external quality factors as defined in Section 4.3.3 are also listed.
To measure the field balance of the structure, a bead pull was performed.
This is a technique whereby a metallic or dielectric bead, whose dimensions are
small compared to the RF wavelength, is suspended on a wire and then pulled
along the axis of the cavity. The presence of the bead at a given location z along
the axis of the structure produces a perturbation ∆f of the resonant frequency
given by Slater’s theorem: ∆f /f = (αm |H|2 − αe |E|2 )/W0 , where f is the unper-
turbed frequency, W0 is the stored energy in the unperturbed cavity, H and E are
the magnetic and electric fields at the location of the bead (in the unperturbed
159
Table 4.6: Measured parameters of the π resonance in air at room temperature.
Parameter Value Units
fπ 9.60084 GHz
FWHM 1.51 MHz
VSWR 1.15
βc 0.870
Q0 11889
QL 6359
Qe 13672
cavity), and αm , αe are factors which depend upon the geometry and material
properties of the bead.
The plots in Figs. 4.17 and 4.18 and the corresponding parameter values in
160
Figure 4.18: Aluminum bead pull results showing (a) frequency shift
and (b) the square root of the frequency shift vs. position along the
cavity axis.
Table 4.6 were taken in air at room temperature. The frequency and coupling
were then measured as the temperature of the cavity was increased. Cavity
temperature was regulated using a heater tape with a temperature controller. The
controller was connected to a thermocouple measuring the cavity temperature and
was set to regulate the current in the heater tape in order to stabilize the cavity
at a preset temperature value. The feedback mechanism for this temperature
control was PID (proportional, integral, derivative) with the PID parameters set
automatically by the controller’s “autotune” feature. The cavity with heater tape
and thermocouple attached was then wrapped in several layers of aluminum foil
to help thermally isolate the cavity from the environment and thereby permit
it to reach thermal equilibrium more quickly. The resonant frequency of the
cavity π-mode is plotted as a function of the thermocouple temperature reading
in Fig. 4.19(a) for both in-air (red line) and in-vacuum (blue line) operation of
the cavity. For the vacuum measurements the cavity was evacuated on a test-
stand to a pressure of 1.8 × 10−6 torr using a turbo-molecular pump. The vertical
offset of approximately 2 MHz between the red and blue curves is consistent with
a simple estimate obtained by noting that the monocell resonant frequency varies
161
Figure 4.19: Plots of (a) resonant frequency and (b) reflectance as
functions of cavity temperature in air (red curves) and in vacuum
(blue curves).
√
with dielectric contant as 1/ . Substituting the value for air ( = 1.00059) gives
a shift of 0.3 % or 2.8 MHz for air vs. vacuum.
The plot in Fig. 4.19 (b) shows the corresponding reflectance values (in dB)
for both air and vacuum. The curves shown are interpolations of the data points.
They are not intended as a theoretical fit to the data, but merely as an aid in
visualization. From the linear fit to the vacuum plot in part (a) we find that the
resonant frequency matches the design value of 9.59616 GHz at a temperature
of 65 C. We see from the plots in part (b) that the cavity passes through the
optimal coupling point at a temperature near 45 C. After this point, it becomes
slightly overcoupled. Consequently, at the projected resonant temperature of 65
C we find that the reflectance value is -35 dB which corresponds to a coupling of
βc = 1.036.
After installing the deflecting cavity on the UCLA Neptune beamline, high-
power testing was performed to verify that it was operating normally and would
accept high-power RF without experiencing significant breakdown. The forward
and reflected power levels were measured on a 49 dB cross-waveguide coupler
located just before the input waveguide flange of the cavity, using a calibrated
162
Figure 4.20: Plots of deflecting cavity forward and reverse power vs
time at several different output levels of the klystron.
crystal detector. The waveforms on the forward and reverse couplers were ob-
served on an oscilloscope terminated at 1 MΩ. The captured oscilloscope traces
are reproduced in Fig. 4.20. The vertical scales have been converted into units of
kilowatts using the measured calibrations of the forward and reverse couplers and
the crystal detector. Traces are shown for three different settings of the input at-
tenuation on the X-band klystron. The sharp spikes in the reflected power trace
are typical of a resonant standing wave structure. The first spike is an initial
reflection due to the nonzero fill time of the cavity, and the second represents the
exponential leakage of stored power in the cavity through the coupler following
the sudden termination of the drive pulse.
The maximum peak power output of the klystron was found to be approxi-
mately 50 kW, which is consistent with the nominal output power quoted by the
vendor. One of the primary concerns in any high-power test of a metallic cavity
is electrical breakdown due to the strong electromagnetic fields in the structure.
The electrical plasma discharge characteristic of a breakdown event causes all
of the input power to be reflected after the point in time when the breakdown
occurs, resulting in an obvious distortion of the reflected power trace. During
the high-power testing (and subsequent operation) of the deflecting cavity no
163
breakdown events were observed even at the highest achievable power levels.
164
CHAPTER 5
165
5.1.1 Beamline
The linac is a 7+2/2 cell π-mode standing wave structure and the first fully
operational version of a plane-wave transformer (PWT). Under normal operating
conditions, roughly 9 MeV of acceleration is produced in the linac (45 MV/m
peak field), but more acceleration is possible. The design of this novel device
is discussed in Ref. [71] and summarized here. The structure is similar to a
disc-loaded linac, except that the outer wall is moved to a large radius, leaving
a gap between the disks and the wall. This gap serves as a coaxial, plane-
wave transmission line, which provides extremely strong cell-to-cell coupling and
166
Figure 5.1: Schematic of the Neptune beamline and the S-Bahn dogleg
compressor.
excellent mode separation. In the absence of support rods (which are needed to
hold the disks in place), this structure also has a very high shunt impedance Rs
and high unloaded Q, with the ratio Rs /Q approximately the same as a more
standard structure. The introduction of the four support (and water-cooling)
rods into the cavity causes enhanced RF power losses, and Rs is actually smaller
in this case than that found in a standard structure. Even so, Q is enhanced,
and the PWT has a long fill time. Both the gun and PWT are approximately
critically coupled, and the PWT barely fills during the 3.5 µs RF pulse.
The PWT was designed at UCLA and built in the Physics Deptartment ma-
chine shop. Engineering issues to be solved included effective internal cooling,
incorporating all-metal sealing surfaces, and optimization of the iris apertures
to reduce the maximum surface gradient. The continued successful operation of
the PWT, despite its unusual mode and frequency spectrum, demonstrates its
167
usefulness as an RF linac.
5.1.2 RF System
The RF system at Neptune operates at the SLAC frequency of 2.856 GHz and
relies on standard S-band technology. Synchronization with the drive laser is
ensured by using the laser modelocker output (at 38.08 MHz) as the RF clock,
after frequency multiplication by 75. About one watt of continuous low-level RF
is delivered to a pulsed (<12 µs) solid state amplifier, which in turn drives the
input of the XK-5 klystron. The klystron is pulsed using a custom modulator with
pulse length variable between 3.5 and 7 usec, achievable by varying the number of
capacitors in the modulator circuit. For normal (short-pulse) operation, a PFN
composed of 10 capacitors is charged to 41 kV and discharged using a thyratron
switch into the klystron pulse transformer, where the pulse is voltage-multiplied
by 12.
Forward and reflected power levels in the system are measured using direc-
tional couplers in different portions of the waveguide and calibrated crystal de-
tectors. RF power values can be calculated from oscilloscope traces of the crystal
detector voltages and displayed automatically in the control room.
168
5.1.3 Photoinjector Drive Laser
The photocathode drive laser illuminates the gun cathode with a pulse of ultravio-
let (266 nm) light having a minimum temporal length of about 1 psec (FWHM),
at an energy up to 100 microjoule/pulse. This is accomplished using chirped
pulse amplification and compression of a 1.064 µm pulsed mode-locked Nd:YAG
laser (using the 38.08 MHz master RF signal) amplified by a Nd:glass regener-
ative amplifier (regen) at 5 Hz. To amplify the pulse in the regen the laser is
first matched into a 500 m long optical fiber to produce pulse lengthening and
a frequency chirp. The chirped pulse is then amplified by a factor of 1 million
in the regen and sent to a grating pair where it is compressed by removing the
chirp correlation. The pulse length can be arbitrarily lengthened by detuning the
grating compressor, allowing a choice of pulse lengths for emittance optimiza-
tion. The resulting picosecond pulse is frequency up-converted using two stages
of BBO doubling crystals. The conversion efficiency is typically 10%. Because
of the many nonlinear stages in this laser configuration the pulse-to-pulse fluc-
tuation amplitude is at best 10% RMS. There are also pulse length fluctuations
associated with these effects, but they are not of great consequence operationally.
The time of arrival of the laser with respect to the RF wave is controlled by a
phase-shifters (manual and electronic) in the low level 2.856 GHz RF system.
The fluctuations in the arrival time of the laser pulse on the cathode with re-
spect to this RF signal are suppressed by use of a Lightwave electronics feedback
system on the oscillator. The fluctuations have been measured to be below one
picosecond.
169
5.1.4 Video and Trigger System
Along the beamline there is a system of CCD cameras for imaging the beam on
various insertable profile monitors. The cameras are all gen-locked to a master
video signal which originates from a master camera located in the control room,
and is then repeatedly amplified by a series of distribution amplifiers, which
provide gen-lock reference signals for the cameras. This ensures that the video
signals from all of the cameras are synchronized. The video signal from the master
camera is also the source of the 5 Hz master trigger for the regenerative amplifier
and high-power RF systems. This 5 Hz trigger is derived from the 60 Hz sync
pulse of the master video signal, which is extracted by a video sync box and
is then used as the reference for a Stanford Research Systems delay generator,
which outputs a 5 Hz pulse train to trigger the lamps and Pockel’s cells of the
regenerative amplifier.
A homemade box located in the control room reduces the repetition rate of
the master trigger from 5 Hz to 1 Hz. The output of this box is the trigger source
for the S-Band and X-Band klystrons as well as the freeze-frame which is used to
capture images from the cameras. Therefore, although the rep rate of the drive
laser is fixed at 5 Hz, that of the high-power RF system can be set to either 5 Hz
or 1 Hz, depending upon whether the homemade box is turned on or off. Under
most conditions the RF system is triggered at 1 Hz repetition, as this is sufficient
for most experimental efforts and it gives the beam operator sufficient time to
interrupt the trigger to the high-power RF in the event of a series of breakdown
events.
170
5.1.5 Control System
171
Table 5.1: Beam Parameters
Parameter Measured Value Location
Q 35.4±3.8 pC Faraday Cup
U 11.5 MeV Screen 10
σδ <0.5% Screen 10
αx -0.94±0.43 Prefocus Quads
βx 0.77±0.43 m Prefocus Quads
N 6.03±1.8 mm mrad Prefocus Quads
least 1 meter) and the beam should be highly convergent at the entrance to the
first dipole.
However, technical and spatial constraints required that the photoinjector and
upstream optics be operated in a mode with somewhat smaller beta functions and
a larger emittance. It was observed that when the beamline is operated in accor-
dance with the emittance compensation scheme described in Section 1.2.4, the
beam which it produces tends to be transversely small and well-collimated (i.e.
βx,y < 1 m and αx,y ≈ 0). In order to increase the beta functions and make
the beam more convergent, the photoinjector was operated in a somewhat non-
optimal regime (from the perspective of emittance compensation) by increasing
the solenoid field by about 20% above the value corresponding to minimum emit-
tance, and moving the first set of quadrupole magnets shown in Fig. 5.1 (dubbed
the prefocus quads) as close to the first dipole magnet as spatial constraints would
allow. The empirical values for the Twiss parameters and normalized emittance
obtained from quadrupole scans are shown in Table 5.1.
172
spread rather than to produce a chirped beam. Previous measurements have
indicated an energy spread of less than 0.5% under these operating conditions. A
stable nondispersive operating point was determined empirically by observing the
beam on the six profile monitors (Screens 5, 10, 11, 12, 13, and 14 in Fig. 5.1).
The horizontal dispersion function ηx (or R16 ) was minimized by observing the
beam centroid position at the S-Bahn midpoint (Screen 11) under a variation of
the fields of all magnetic elements on the dogleg (B1, B2, Q1, Q2) by a fractional
offset ζ from the values corresponding to the desired operating configuration.
For a beam of constant central energy, the resultant shift in the centroid position
is the same as that which would be observed due to a change in the central
momentum by a fractional amount δ = −ζ, and is therefore given by
Consequently, the first- and second-order horizontal dispersion terms R16 and
T166 can be obtained by fitting the measured centroid position data to a quadratic
in ζ. The values of T166 at the exit of the S-Bahn (Screen 13) obtained by this
method are shown in Table 5.2 for three different settings of the sextupole field
strength. Simulation values from the transport code ELEGANT are provided for
comparison. The geometrical field strength κ and ratio α correspond with the
quantities in Eq. (3.25).
173
Table 5.2: Comparison of experimental and simulated second-order dispersion
values for various sextupole field settings
Since the horizontal dispersion does not provide a diagnostic of the longitudinal
trace space, the measurements of Table 5.2 were performed using a beam with
no momentum chirp and a relatively small (<0.5%) energy spread. To obtain
information about the effect of the sextupoles on the longitudinal distribution
of the beam, the beam was then chirped in momentum by injecting it with an
RF phase offset of -28 degrees relative to the crest of the accelerating field in
the standing wave linac cavity. The bunch length was then measured at differ-
ent sextupole settings using coherent transition radiation (CTR) autocorrelation.
Transition radiation emitted by the beam at a metal foil on Screen 14 of Fig. 5.1,
oriented at 45 degrees incidence, was autocorrelated using a Martin-Puplett-type
interferometer with wire grid polarizing beam splitters [72]. The bunch length σt
was extracted from the interferograms using the time-domain fitting procedure of
Ref. [73]. The extracted values are plotted in Fig. 5.2 as a function of sextupole
strength κ. The ratio of the two sextupole fields was set to α = −1. The data
show the dependence of bunch length upon the magnitude of the sextupole cor-
rection, with an approximately twofold compression occuring near the field value
κ = 1094 m−3 .
174
Figure 5.2: CTR autocorrelator measurements of electron bunch
length as a function of sextupole field strength, with superimposed
theoretical result (dashed line) obtained from PARMELA/ELEGANT
simulation combined with an autocorrelation algorithm.
It should be noted that, due to both the limited frequency bandwidth of the
autocorrelator apparatus and the nature of the fitting procedure used to extract
the pulse length from the data (which assumes a Gaussian current profile), for a
beam whose temporal profile is asymmetric, the value of σt obtained from the in-
terferogram is more closely connected with the FWHM than with the RMS width
of the distribution. Consequently, we have found that obtaining a theoretical
prediction to complement the data of Fig. 5.2 involves a somewhat complicated
computational procedure, the result of which is superimposed as a dashed curve.
To produce this theoretical curve, first the creation and transport of the beam
in the accelerating sectio were simulated using the tracking code PARMELA. This
detailed simulation employed 5000 macroparticles, whose initial temporal profile
(inherited from the laser pulse) was modulated in a way consistent with observa-
tions of the energy modulation of the beam, and a -28 degree phase offset in the
linac, producing a chirped beam. The set of output 6D trace space coordinates
175
Figure 5.3: Longitudinal phase space plots and density profiles ob-
tained from ELEGANT results corresponding to the sextupole set-
tings of the (a) first, (b) third, (c) fourth, and (d) sixth data points
in Fig. 5.2 respectively, illustrating the progression of the phase space
compression and decompression.
obtained from PARMELA was then used as the input beam for an ELEGANT
simulation of the dogleg section, including a truncation of outlying particles con-
sistent with the observed 60% electron transmission efficiency through the device.
The longitudinal (z) coordinates of the particles were extracted from the ELE-
GANT simulation at the location corresponding to Screen 14, where the CTR
foil was inserted. An algorithm was used to reconstruct from the extracted z co-
ordinates the predicted autocorrelation function, including appropriate filtering
of the frequency content due to diffraction, collection, and transport efficiency
effects.
The simulated autocorrelation function was then subjected to the same fitting
procedure that was used to extract σt from the empirical interferograms, yielding
values which produce the dashed curve in Fig. 5.2. These simulation results
suggest that the observed compression and decompression results from a “folding
over” of the longitudinal trace space due to the quadratic T566 dependence in
Eq. (3.7), where particles of both high and low energy begin to occupy the
176
same longitudinal position within the bunch. This scenario is illustrated by the
trace space plots in Fig. 5.3. The maximum compression [Fig. 5.3(b)] occurs
at the sextupole field value where this folding over begins to change direction in
z, corresponding to the point at which the second-order term T566 changes sign.
The discrepancy between theory and data near the fourth data point in Fig. 5.2
appears to be due to the sensitivity of the theoretical autocorrelation algorithm
to the sharp spikes in the temporal distribution displayed in Figs. 5.3(c) and
5.3(d).
177
5.3 Deflecing Cavity Longitudinal Profile Measurements
As was seen in Fig. 5.1, the dogleg section is followed by a triplet of quadrupole
magnets (dubbed the final focus quads) which are used to help transport the
emerging electron beam through the 1 cm aperture of the deflecting cavity and
focus it on a profile monitor located 28 cm from the exit of the deflecting cavity.
A cartoon diagram of the setup is shown in Fig. 5.4 for reference. The screen
on which the final beam profile is imaged (Screen 15) consists of a 1”-diameter
Cesium-doped yttrium-aluminum garnet (YAG) crystal, which is mounted per-
pendicular to the beam path and followed by a 45 degree mirror. This is the same
basic configuration used for all of the profile monitors on the Neptune beamline
except that a larger diameter crystal was used in this case, to provide an ade-
quate aperture for the deflected (i.e. streaked) electron beam. A CCD camera
was used to capture the reflected image of the beam profile from the back side of
the YAG crystal.
The mount for the YAG crystal incorporates a built-in Faraday cup for mea-
suring the final beam charge. The center of the Faraday cup is vertically displaced
by 2 inches from the center of the YAG screen, as seen in Fig. 5.5. The entire
mount is then attached to a pneumatic actuator with a 2” stroke, so that either
the YAG or the Farday cup can be selectively inserted into the beam path.
178
Figure 5.4: Block diagram of the experimental setup for the deflecting
cavity measurements.
Figure 5.5: Schematic of the combined beam dump and YAG profile
monitor built as a final diagnostic device.
179
deflecting cavity was traced onto a sheet of paper placed on the wall past the end
of the beamline. In addition, the laser position was checked at two locations on
the final vacuum chamber to verify that it coincided with the center of the beam
pipe. Due to small variations in the actuator assemblies for Screens 13 and 14
(such as minor differences in the lengths of the custom-made extension pieces for
the actuator arms and issues such as whether copper or viton o-rings were used)
it was found that the centers of the YAG crystals deviated from the center of the
beamline by as much as 2 to 3 mm. Thus, the vector trajectory of the electron
beam was adjusted, using steering magnets and the trim coils on the final pair of
dipole magnets, to coincide on these screens with the measured position of the
alignment laser rather than with the geometrical centers of the screens.
Recalling that the y − z correlation was given by Eq. (4.1), we see that the
position of the centroid is described by y = Q̂ where Q̂ is the centroid offset in
Eq. (4.2), which varies sinusoidally with phase ψ0 . The maximum deflection is
180
Figure 5.6: Images of the deflected beam at two different phases and
measured on-screen deflection distance y plotted against the RF phase.
A sinusoidal function is fitted to the data (solid curve).
given by the amplitude Q̂max = eV0 L/p0 c. The sinusoidal fit in Fig. 5.6 gives an
amplitude value of Q̂max = 5.51±0.21 mm. Hence, at the measured beam energy
of 11.73 MeV and a drift length of 27.9 cm, the predicted deflecting voltage is
V0 = 232 ± 9.6 kV. This value is compared with those calculated from the RF
power levels measured at two different internal oscilloscope impedance settings
(50 Ω vs 1 MΩ) in Table 5.3. The RF power going into the cavity was measured on
the oscilloscope using a crystal detector attached to the forward power coupler on
the deflecting cavity. Since the impedance of the oscilloscope alters the amplitude
of the power signal, separate power-to-signal calibrations were performed for the
two impedance settings. The deflecting voltage was then calculated using the
p
formula V0 = Pcav R̃s where we take the shunt impedance value R̃s = 5.6 MΩ
predicted by the HFSS simulations.
Note that the deflecting voltage values obtained from the RF measurements
are consistent with each other but differ from the phase scan result by about 14%.
Since the phase scan is a more direct method and has fewer potential sources of
181
Table 5.3: Comparison of deflecting voltage calculated from phase scan and RF
power measurements.
measurement and systematic error, this value is more reliable. The magnitude of
this discrepancy lies within the random errors in the RF measurements, which
are large because of the large (49 dB) attenuation of the forward power coupler,
which makes power values calculated from the coupled signal extremely sensitive
to any measurement error in the attenuation value of the coupler itself. Some
additional systematic error may also be present due to the fact that the simulated
value of the shunt impedance was used in the RF calculations.
Prior to running a chirped beam through the dogleg and the deflector, the beam
was run near the optimal linac phase for minumum energy spread in order to op-
timize the transport and measure the longitudinal structure of the uncompressed
pulse. An example trace of the deflector streak produced by such a beam is shown
182
Figure 5.7: False color plots of uncompressed electron beam with (a)
deflecting cavity turned off, (b) deflecting cavity turned on, as well as
(c) the current profile reconstruction of the image in part (b).
in Fig. 5.7. In this case, the beam energy was 13.14 MeV and the final beam
charge measured on the Faraday cup was 270 pC.
The image in Fig. 5.7(a) shows the focused beam with the deflecting cavity
turned off. When the deflecting cavity is turned on at an input power of 40.6 kW
the beam is streaked along the vertical axis producing the image in Fig. 5.7(b).
The contour plots in parts (a) and (b) of Fig. 5.7 are false-color reconstructions
of the captured black-and-white CCD camera images on the final large-aperture
YAG screen located after the deflecting cavity (i.e. Screen 15). The bit count
on each pixel of the captured CCD image was assumed to be proportional to the
number of electrons hitting the YAG within the geometrical area corresponding
to the size of one pixel. Then 5 × 5 blocks of neighboring pixels were averaged
together and interpolated in order to smooth over the noise in the video signal.
The background has been subtracted out by sampling a region well outside the
area where the beam is located and then subtracting the average bit count per
pixel from the whole image. The horizontal and vertical axes have been scaled to
183
units of mm based upon the calibrated pixel count per unit length of the imaged
object for the camera and optical setup used.
The plot in Fig. 5.7(c) shows the reconstructed current profile based upon
the streaked image in part (b) and the measured charge of the beam. To produce
this reconstruction, the bit counts per averaged 5 × 5 block used to produce the
interpolated (i.e. smoothed) false-color plot in part (b) were projected on the
vertical (y)axis by summing over horizontal rows. The vertical axis was then
rescaled to units of time by virtue of Eq. (4.67). Note that there is some ambi-
guity as to which end of the plot corresponds to the head of the beam as opposed
to the tail. This is due to the fact that there are two zero-crossings of the RF
per period, separated in phase by 180 degrees, and with the sign of the deflection
correspondingly reversed for the head relative to the tail. However, based upon
the results for the chirped beam, which will be presented in Section 5.3.4, the
locations of the head and tail have been inferred here and the directionality of
the horizontal scale on the current profile plot has been adjusted appropriately.
Thus the plot indicates an uncompressed beam profile with a sharp initial spike
followed by a gradually decaying tail.
184
Figure 5.8: Plots showing (a) the autocorrelation function of the recon-
structed electron bunch profile in Fig. 5.7(c) and (b) the normalized
autocorrelation of the drive laser obtained by second-harmonic-gener-
ation interferometry.
function of the reconstructed profile of the electron bunch from Fig. 5.7(c) is
shown in Fig. 5.8(a), having a FWHM of 10 ps. The discrepancy is partially
resolved by noting that the laser autocorrelation measurement is taken prior to
two stages of frequency up-conversion by colinear SHG in a pair of nonlinear
crystals. Since the SHG pulse varies with the square of the intensity of the input
√
laser pulse there is an approximate 2 reduction in pulse length corresponding
to each stage of frequency conversion. Consequently, the drive laser pulse is ex-
pected to be shortened by approximately a factor of 2 by the time it reaches the
photoinjector cathode. Note that there is also additional structure in the wings
of the autocorrelation in Fig. 5.8(a) as compared with (b), which is due to the
highly structured tail seen in Fig. 5.7.
The profile of Fig. 5.7 is typical of streaks taken of the uncompressed beam,
including the large degree of structure in the tail. In some streaks, this structure
is more pronounced, as seen in Fig. 5.9. The origin of this structure is not fully
185
Figure 5.9: False color streak images with (a) deflecting cavity turned
off, (b) deflecting cavity turned on, as well as (c) the current profile
reconstruction of the image in part (b) showing structure in the tail
region.
understood. One possibility is that there is additional structure in the tail of the
drive laser pulse in the IR which varies from shot-to-shot, and is not apparent in
the wings of the laser autocorrelation of Fig. 5.8(b) because the data has been
averaged over many shots. Any such structure in the tail of the laser pulse could
subsequently be amplified by the frequency doubling mechanism in the nonlinear
crystals due to the previously mentioned dependence of the SHG on the square
amplitude of the input pulse.
186
Figure 5.10: Measured electron beam charge vs gun RF phase with
superimposed theory curve from Eq. 5.2 and corresponding fitting
parameters.
component for phases exceeding this value, in order to account for the loss of
charge resulting from radial blowout of the beam at large phase values. This decay
has associated fitting parameters α and τ which represent the fractional phase
offset of the falloff region from π/2 and the exponential decay rate respectively.
The quantity ηq is the effective quantum efficiency of the cathode, given by
r 2
hν W e
ηq (ψ0 ) = − + βE0 sin ψ0 (5.3)
e e 4π0
where hν is the drive laser photon energy, W is the cathode work function, β is
an empirical field enhancement factor, and E0 is the electric field amplitude at
the cathode.
187
The resulting fit is superimposed as a solid curve on the plot in Fig. 5.10 alongside
the corresponding fit parameters. The fitted σ` value of 5 ps is consistent with
the RMS width of the pulse in Fig. 5.7, which was 5.9 ps. However, it should be
noted that the region of the theoretical curve in Fig. 5.10 that is most sensitive to
the laser pulse width is the rising edge near zero phase, where the data is distorted
by the presence of the secondary pulse, making the fitted value of this parameter
less reliable than the others. Nevertheless, the same fit parameters are found to
produce a good fit to the pedestal as well with a simple offset in phase and an
adjustment of the amplitude factor A. This suggests that the secondary beam
responsible for the pedestal is a ghost image of the primary pulse but offset in
phase by 50 degrees. The most likely explanation for the presence of the pedestal
is that there is a pre- or post-pulse from the regenerative amplifier due to the
nonzero contrast ratio of the optical cavity. This pulse is would then be separated
from the main one by about 7 ns and would therefore be injected many RF cycles
later and at a different RF phase relative to the peak of the RF. The relative
amplitudes of the main pulse and the pedestal in Fig. 5.10 are consistent with
the typical 1:10 contrast ratio for the regenerative amplifier.
Deflecting cavity data for an electron beam that was chirped in energy by running
the RF phase in the linac backward of crest by approximately 20 degrees is shown
in Fig. 5.11. Experimental parameters corresponding to the data of Fig. 5.11 are
shown in Table 5.4. The sextupole magnets were set to have opposite polarities,
corresponding to α = −1 where α is the field ratio parameter appearing in Eq.
(3.25).
As seen in Fig. 5.11(a), when the sextupoles are turned off the current profile
188
Figure 5.11: Deflecting cavity streaks and current profile reconstruc-
tions of an (initially) chirped electron beam for five different sex-
tupole field values, with a sextupole field ratio α = −1: (a) κ = 0, (b)
κ = 547 m−3 , (c) κ = 1094 m−3 , (d) κ = 1641 m−3 , and (e) κ = 2188 m−3 .
189
is characterized by a sharp narrow spike at the head, followed by a shallow tail,
as was seen in the simulation predictions of Figs. 3.8(b) and 5.3(a). (Note that
since the horizontal coordinate in Fig. 5.11 is arrival time t, whereas in Figs.
3.8, 5.3 it is longitudinal position z there is a reversal of direction due to the
convention z = −ct). As the sextupole field strength is increased, the hard edge
at the head of the beam gives way to a gradual ramp followed by a sharp drop at
the tail, as seen in Fig. 5.11(e). The intermediate stages shown in (b) through
(d) demonstrate the progression of this process.
Several significant features in Fig. 5.11 should be remarked upon. One is the
significant energy correlation on the horizontal axis, producing the obvious tilt
of the streak profiles. This energy correlation was produced by slightly detun-
ing the first horizontally focusing quadrupole magnet on the dogleg section by
approximately 2%. The resulting effect is a combination of the usual residual
second-order horizontal dispersion (T166 ), which was discussed in Section 5.2.1,
combined with an induced linear dispersion due to the detuning. The resulting
effect upon the streak data is that the vertical (y) dependence on arrival time
is complemented by a horizontal (x) dependence on energy, which results in a
reconstruction of the longitudinal phase space distribution of the beam. This
permits a direct visualization of the turning over of the phase space as the non-
linear correction is implemented by the sextupole magnets. Simulations of the
beamline using ELEGANT predict that a detuning of the first quadrupole by 2%
should produce a residual horizontal dispersion of R16 = -1.1 cm and a slightly
increased longitudinal dispersion value of R56 = -5.7 cm.
The sextupole field strength (κ = 2188 m−3 ) required to produce the ramp-
shaped beam in Fig. 5.11(e) is 80% higher than what was predicted by the
simulations of Section 3.2.5 and by the empirical measurements of Section 5.2.1 to
190
be necessary in order to cancel the T566 of the dogleg. Consequently, the ramped
beam of Fig. 5.11(e) is characterized by a significant overcorrection of the second-
order longitudinal dispersion, similar to the simulated result of Fig. 5.3(d). This
overcorrection is most likely required by the fact that the uncompressed electron
bunches are not Gaussian but are asymmetrical in shape, as was seen in Figs.
5.7 and 5.9, and have a significant amount of charge concentrated at the head of
the beam.
191
Figure 5.12: Plots of the simulated asymmetric bunch used as input
for the ELEGANT results of Fig. 5.13 showing (a) longitudinal trace
space and (b) current profile.
qualitative features of the data in Fig. 5.11, including the shape of the streak
images and the higher sextupole field strength required in order to achieve a
ramp-shaped bunch. When an initially symmetric Gaussian bunch is used as the
input for these simulations (with all other simulation parameters unaltered), the
results are found to be consistent with the previously shown PARMELA results
of Section 3.2.5, which predicted that the ramp-shaped current profile should
occur at a sextupole field strength of κ = 1204 m−3 . This suggests that the
sextupole overcorrection required in the experimental run to produce ramped
beams can be largely explained by the temporal asymmetry of the initial bunch.
It also indicates that the bunch-ramping mechanism is somewhat forgiving of
such asymmetries, since ramped bunches can be obtained in spite of them simply
by adjusting the strength of the sextupoles.
192
Figure 5.13: Simulated deflecting cavity streaks and current profiles
(using ELEGANT with 10,000 macroparticles) of an (initially) chirped
Gaussian electron beam for four different sextupole field values, with
a sextupole field ratio α = −1: (a) κ = 0, (b) κ = 1094 m−3 , (c)
κ = 1641 m−3 , (d) κ = 2188 m−3 .
193
5.4 Summary of Findings
Deflector studies with uncompressed bunches indicate that the beam has an
asymmetrical (non-Gaussian) shape with a sharp peak at the head of the beam
followed by a decaying tail, and an RMS duration of approximately 6 ps. The
measured bunch length is compared with autocorrelation measurements of the
drive laser and with the bunch length extracted from fitting to a charge vs. RF
phase scan of the photoinjector, and is found to be reasonably consistent with
those results. Studies with chirped bunches indicate that ramp-shaped current
profiles can be produced. The resulting deflector streak images and reconstructed
current profiles for the compressed beam are found to be qualitatively consistent
with ELEGANT simulations, so long as the initial temporal asymmetry of the
beam is included in the simulation. However, as a result of the asymmetric initial
current profile of the beam, the sextupole field strength required to produce
194
a ramped current profile is found (in both simulation and experiment) to be
80% higher than what was predicted by simulations of beams with an initially
Gaussian current profile.
195
CHAPTER 6
Future Directions
196
In these relations, we have approximated the beam density by nb = Q/(eπσr2 σz )
and have used the definition of the normalized emittance N = γβ(σr2 /βr ), where
β is the bunch velocity normalized by the speed of light, βeq is the equilibrium
beta function given above, σz is the RMS bunch length, and Q is the bunch
charge. We can use these constraints to obtain the following expression for the
minimum required beam brightness:
2cQ
B > Bmin = . (6.4)
2N,max σz
Here we have used the definition for the transverse brightness B = 2I/2N , where
I = enb βcπσr2 is the beam current. These relations provide us with an estimate
of the required beam parameters for successfully applying the bunch shaping
technique to create an adequate drive beam for a PWFA. In the present chapter,
we discuss several future experiments planned at the Neptune laboratory as an
extension of the work comprising this dissertation, which are aimed at addressing
these issues of beam quality. We also propose a simple technique for creating a
witness bunch.
For the next stage of experiments on the Neptune dogleg beamline, an alternative
diagnostic setup will be employed, as shown in the inset in Fig. 6.1. Setup
1 corresponds with the previously conducted bunch profile measurements and
longitudinal phase space reconstruction which was discussed in Section 5.3.4.
Setup 2 will consist of a triplet of permanent magnet quadrupoles followed by a
Ce:YAG profile monitor, which will be used to obtain a high-brightness focus.
197
Figure 6.1: Cartoon graphic of the experimental beamline (not to
scale). Blue lenses, red rectangles, and yellow wedges represent
quadrupoles, sextupoles, and dipole magnets respectively. Two al-
ternate setups are shown for the final diagnostics section.
198
Using the constraints on the beam parameters imposed by Eqs. (6.2)-(6.3),
we can construct plots showing the maximum allowable beam size and emittance
as a function of plasma density n0 for a 0.5 mm long 300 pC beam. This is shown
in Fig. 6.2(a). Combining this with the minimum density of n0 = 2.8 × 1013 cm−3
required by Eq. (6.1) gives us estimated upper limits on RMS beam size and
normalized emittance of 110 µm and 50 mm mrad respectively. A corresponding
plot of minimum brightness using Eq. (6.4), shown in Fig. 6.2(b), gives a lower
limit of 250 mA/µm2 . These limits are compared with simulation results in
Section 6.1.3.
The permanent magnet quadrupoles (PMQs) to be used for the S-Bahn final fo-
cus shown in Setup 2 of Fig. 6.1 are compact, with a high magnetic field gradient,
making them useful for focusing of high-brightness space-charge dominated beams
such as those produced at the Neptune laboratory. The magnets incorporate a
hybrid iron and permanent magnet design originally developed for the nonlinear
inverse Compton scattering experiment that is currently in progress at the Nep-
tune laboratory [75]. The magnets, shown in Fig. 6.3, contain cubes of NdFeB,
surrounded by an iron yoke which serves to close the magnetic circuit. Four hy-
perbolic pole faces constructed by wire electric discharge machining (EDM) are
held against the NdFeB cubes in a quadrupole array around the geometric center
by an aluminum keeper.
199
Figure 6.3: Drawing of hybrid permanent magnet quadrupole design
(a), courtesy of A. Doyuran, and schematic of assembled triplet and
stand (b).
defocusing 2 cm long PMQ. A side view of the assembly is shown in Fig. 6.3(b)
at a reduced scale.
The PMQ focusing system described in Section 6.1.2 was simulated initially using
the matrix-based beam envelope code PowerTrace. Further studies were then
done using the particle tracking code ELEGANT to model the dogleg and final
focus sections [53]. The phase space coordinates for the particles used as the
input for the ELEGANT simulation were generated by a PARMELA simulation
of the photoinjector and linac.
200
Table 6.1: Simulated Experimental Parameters
E 13 13 MeV
Q 300 240 pC
x,N 5 41 mm mrad
y,N 5 15 mm mrad
σt 2.5 1.8 ps
σx 1 0.130 mm
σy 1 0.057 mm
final focus location of the permanent magnet quadrupole triplet of Setup 2. Fi-
nal values represent design goals based upon the PARMELA and ELEGANT
simulation results. The reduction in charge is a prediction based upon observed
transportation losses in the beamline, and the emittance growth is due primar-
ily to transverse nonlinear effects in the dogleg. Note that the predicted beam
sizes, emittance values, and brightness fall roughly within the limits set by Eqs.
(6.1)-(6.4), for applicability to plasma wake-field studies with large transformer
ratios.
In order for the ramped bunch mechanism presented in this dissertation to repre-
sent a useful technology for the wake-field accelerator, it must be compatible with
some feasible scheme for creating a witness bunch. The witness bunch would ide-
ally be a bunch of much lower charge which trails behind the main drive bunch,
201
Figure 6.4: Simulation of undercorrected beam at exit of dogleg with
collimator removed in (a) and inserted in (b), thereby producing a
ramped drive beam followed by a low-charge witness bunch.
and can therefore be accelerated by the wake-fields which are generated by it.
One technique used in the past has been to accelerate the tail of the drive bunch
itself. However, a ramped drive bunch is intended by design to have a sharp
cutoff at the tail end. We saw, however, in Fig. 3.8(b), that with the sextupole
correctors turned off, the nonlinear effects produce a significant lower-energy tail
behind the bunch, though the ramped shape is lost. A potential solution would
be to operate in a regime intermediate between the conditions represented in
Fig.3.8(b) and 3.8(c), where the sextupole magnets are turned on but at a lower
field strength, producing a beam with a ramp at the front followed by a more
tapered fall-off at the back. This situation is seen in Fig.6.4(a), which shows the
results of an ELEGANT simulation of the dogleg compressor. By inserting a 1
cm wide collimator in the x-direction, at a location in the dogleg (corresponding
to the position of the quadrupole before the final dipole in Fig. 6.1) where the
202
horizontal dispersion is large and therefore there is a strong correlation between
x and z, the tail of the beam can be truncated from the main body. As shown in
Fig. 6.4(b) this results in a ramp-shaped primary bunch followed by a separate
trailing bunch of lower charge. This scheme has the benefit of being relatively
simple, requiring only the insertion of a collimator into the beamline. However,
the resultant reduction in charge and horizontal truncation of the beam must be
taken into consideration in the design of the downstream focusing optics.
Future upgrades to the Neptune laboratory, including a new drive laser oscilla-
tor, replacement of the photoinjector, photocathode laser cleaning, and higher
RF power levels in the gun are expected to increase the bunch charge to as high
as 4 nC. It is therefore of interest to consider how the bunch shaping mech-
anism described previously scales to higher charge. Preservation of the beam
envelope under the emittance compensation mechanism in the gun and linac re-
quires that the bunch dimensions at the cathode scale with charge as Q1/3 . This
scaling is accomplished by stretching the pulse length of the photocathode drive
laser and expanding its transverse dimensions accordingly. Simulations of the
Neptune photoinjector under this scaling in UCLA-PARMELA indicate a nor-
malized emittance of x,N = y,N = 25 mm mrad at the exit of the linac with 4%
transportation losses for an initial charge of 4 nC. Since the dogleg compression
mechanism requires a chirped beam, the bunch was chirped in energy by setting
the RF phase of the linac in the simulation to a value corresponding to an in-
jection phase of 22 degrees back of crest. This chirp, and the increase in bunch
length due to the charge scaling, result in a predicted 4.5% RMS energy spread,
compared with 1.8% energy spread for the 300 pC case.
203
Table 6.2: Simulated Parameters Corresponding to Fig. 6.5 (a), (b), and (c)
Figure 6.5: Simulation of longitudinal phase space and current profiles of 4nC
beam at exit of dogleg compressor with (a) with sextupoles set to eliminate
second-order longitudinal dispersion (T566 ), (b) with sextupoles set to eliminate
second-order horizontal dispersion (T166 ), and (c) with sextupoles set as in part
(a) but with a collimator inserted to remove low-energy tail.
204
The final phase space coordinates of the particles in the PARMELA simu-
lation were then used as the input for an ELEGANT simulation of the dogleg
compressor. The results of these simulations indicated that due to the larger en-
ergy spread of the 4 nC beam, two undesirable effects became more pronounced:
(1) distortion of the longitudinal phase space by third-order longitudinal disper-
sion (U5666 in transport notation) and (2) emittance growth due to horizontal
second-order dispersion (T166 and T266 ). The first effect results in the formation
of a low-energy tail behind the beam. This is seen in Fig.6.5(a). The tail can, in
principle, be corrected by the use of octupole magnets. The second effect requires
the use of sextupole magnets and is somewhat more difficult to remedy, due to the
fact that, at least for the particular optical configuration of the Neptune dogleg,
it is impossible to simultaneously eliminate both the horizontal and longitudinal
second order dispersion (T166 and T566 respectively). Consequently, the sextupole
magnets may be used to eliminate the second order horizontal dispersion, thereby
improving the final emittance, but as a result the longitudinal dispersion becomes
nonzero and so the shape of the ramped profile is destroyed. This scenario is il-
lustrated in Fig. 6.5(b). A solution which appears to solve both problems is
to simply eliminate the tail in part (a) by collimating the beam. As it turns
out, much of the emittance growth is due to the low-energy particles contained
in this tail, and their removal improves the final emittance by a factor of two
and restores the ramped profile, as seen in Fig. 6.5(c). By using a collimator of
finite width, a small subset of the tail particles could be left as a witness bunch,
making this technique compatible with the results of the previous section. To
clarify these results, the simulated emittance and corresponding matrix element
values are provided in Table 6.2.
205
RMS beam sizes in the dogleg for the high-charge case are found to exceed the
radius of the beam pipe. Consequently, although scaling the compressor to higher
charge appears theoretically feasible, it would, in practice, be necessary to expand
the apertures of the beamline, which would require a significant redesign of the
beamline hardware.
206
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