Anderson 2000 0146
Anderson 2000 0146
Anderson 2000 0146
external forces. We analytically study these dynamics to more standard accelerator terminology, this limit corre-
determine the conditions under which phase space wave sponds to full depression of the betatron tune by space-
breaking occurs, for coasting beams, in slab-symmetric, charge defocusing. This type of ultrarelativistic cold-fluid
as well as cylindrically symmetric, geometries. The slab- (laminar) beam analysis gives an excellent approximation
symmetric case is included mainly to allow use of ex- when one studies high brightness electron beams, but is
act and physically transparent results, which illustrate the less applicable in the context of ion beams, which are typi-
mechanisms involved in phase space wave breaking. In cally not as extremely space-charge dominated.
practice, one is nearly always concerned with cylindrically Also, the parameter h is a measure of the second-order
symmetric beams, and so we extend our discussion of this focusing, e.g., nonsynchronous rf wave [12,13] and/or so-
case to include acceleration in an rf structure. Because the lenoid focusing [4], applied to the beam as it acceler-
analysis of dynamics of this system is not tractable after ates with normalized, average (over an rf period) spatial
wave breaking has occurred, we then also employ com- rate g 0 苷 q具Ez 典兾m0 c2 . For a standing wave accelerator
putational simulations to further our understanding of the h ⬵ 1, while for a disk-loaded traveling wave accelerator
cylindrically symmetric beam physics in both the coasting it is an order of magnitude smaller [13]. If solenoid focus-
and accelerating cases. The results of this analysis show ing is also applied, h ! h 1 2b 2 , where b 苷 Bz 兾具Ez 典. It
that, in order to compensate the beam emittance within should be noted that we have adopted the ultrarelativistic
a slice, in the presence of significant nonlinearities in the limit here mainly to simplify the model of the rf focusing,
space-charge field, one must avoid matching of the beam to as well as the mathematics of the analysis. The results we
the generalized equilibria (e.g., Brillouin flow, or invariant obtain can be straightforwardly generalized to moderately
envelope), and that the optimal transport of a space-charge relativistic or even nonrelativistic beams.
dominated beam is typically not close to such equilibria. When the beam is focused by a solenoid, but not ac-
celerating, g 0 苷 0, we recover the familiar rms envelope
II. ENVELOPE DYNAMICS AND LINEAR equation
EMITTANCE COMPENSATION re l共z 兲
sr00 共z , z兲 1 kb2 sr 共z , z兲 苷 3 , (2)
The purpose of this section is to provide a review of g sr 共z , z兲
the analytical theory of emittance compensation as formu- where kb 苷 qBz 兾bgm0 c2 ⬵ qBz 兾gm0 c2 is the spatial
lated by SR in Ref. [4]. This background is needed in betatron frequency [14], which in this case is identical to
order to understand the detailed nature of the problems ad- the Larmor frequency of the particle. Equation (2) is a
dressed in this paper. The invariant envelope theory begins nonlinear differential equation with no general analytical
with the writing of the cylindrically symmetric rms enve- solution, but does have a particular equilibrium solution,
lope equation of each beam slice in the long-beam (two- s
1 re l共z 兲
dimensional) limit. This limit is reached when the beam seq 共z 兲 苷 . (3)
is highly relativistic nb ! c 共g ¿ 1兲, and even a short kb g3
pulse of particles appears elongated in the longitudinal di- This steady state envelope given by Eq. (3) corresponds
mension in its rest frame. In this limit, which is assumed to a rigid rotor equilibrium known as Brillouin flow, in
for the remainder of this section, the transverse defocusing which the beam’s canonical angular momentum is 0. The
due to space-charge forces is dependent only on the local typical way of dealing with solution of Eq. (2) is to expand
value of the current I共z 兲 苷 ql共z 兲nb ⬵ ql共z 兲c and the it to first order about its equilibrium, in the parameter
rms beam size at the particular slice in question sr 共z , z兲, dsr 苷 sr 2 seq ø seq , to obtain
and the envelope equation including acceleration is
re l共z兲
µ 0 ∂ µ 0 ∂2 dsr00 共z , z兲 1 kb2 dsr 共z , z兲 苷 2 3 2 dsr 共z , z兲 ,
g h g g seq 共z 兲
sr 共z , z兲 1
00
sr 共z , z兲 1
0
sr 共z , z兲 (4)
g共z兲 8 g共z兲
re l共z 兲 or
苷 . (1)
g共z兲3 sr 共z , z兲 dsr00 共z , z兲 1 2kb2 dsr 共z , z兲 苷 0 . (5)
Here z 苷 z 2 ct is the internal longitudinal coordinate of The general solution for small amplitude motion about
a fixed position moving beam (and thus labels a slice), z is the equilibrium associated with each beam slice is thus,
the distance along the beam propagation direction, and we assuming for simplicity that all slices are initially launched
have suppressed the thermal emittance term, which means at the same rms size sr 共z , 0兲 苷 sr0 with no rms angular
we are assuming a highly space-charge dominated beam. motion sr0 共z , 0兲 苷 0,
p
Note that this is a relatively extreme limit, where the ther- sr 共z , z兲 苷 sr0 1 关sr0 2 seq 共z 兲兴 cos共 2 kb z兲 , (6)
mal motion of particles is completely negligible, and the
with derivative
motion of the ensemble of beam particles resembles that p p
of a cold fluid— a one-component relativistic plasma. In sr0 共z , z兲 苷 2 2 kb 关sr0 2 seq 共z 兲兴 sin共 2 kb z兲 . (7)
094201-2 094201-2
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
In this case the 共sr , sr0 兲 trace space trajectory of the enve- quency. This phenomenon is illustrated in Fig. 2, which
lope is simply an ellipse whose origin is offset to 共seq , 0兲. displays the trace space area described by the three slices
The mismatched envelopes rotate about thisp offset posi- at kp z 苷 0, p兾2, 3p兾2, 2p. It can be seen that the trajec-
tion with spatial frequency (wave number) p 2 kb , which tories fan out to produce a large summed (or projected )
is equal to the plasma frequency kp 苷 4prc nb 兾b 2 g 3 emittance at kp z 苷 p兾2, 3p兾2, while to lowest order the
of the equivalent uniform density 关nb 共z 兲 苷 l共z 兲兾2psr2 兴 emittance vanishes at kp z 苷 0, 2p and also at kp z 苷 p
matched beam [6]. In the small amplitude limit, the oscil- (not shown). These emittance oscillations repeat twice
lation frequency is independent of l共z 兲. Thus every tra- every plasma oscillation, but eventually decohere due to
jectory of this form aligns in trace space twice per plasma small, higher order differences in the nonlinear plasma fre-
period, points at which the projected rms emittance ob- quency in each slice [15]. The proper execution of such
tained by summing 共具 典兲 over the ensemble of beam slices an emittance oscillation due to differential slice motion is
in this trace space, termed emittance compensation in the context of high cur-
q rent, space-charge dominated beams in rf photoinjectors.
´r 苷 具sr2 典 具sr0 2 典 2 具sr sr0 典2 , (8) This simple picture is complicated somewhat by accelera-
vanishes. This definition of emittance is identical tion, as discussed below, but essentially illustrates the rele-
to that of the standard radial rms emittance ´rms 苷 vant physics of the compensation process.
p
具r 2 典 具r 0 2 典 2 具rr 0 典2 if each slice of the beam is a line in The picture of the slice dynamics displayed in the trace
共r, r 0 兲 trace space, which connects the origin to the edge space diagrams of Figs. 1 and 2 assumes — as is true
of the slice distribution through the value 共sr , sr0 兲. This of motion originating at a cathode in an rf photoinjec-
case, which is physically realized when the beam’s density tor — that the beam expands from its initial size, exceeds
p
distribution is uniform inside of the radius 2 苷 2 sr an equilibrium value, and finally returns to its initial state.
and vanishing outside of this radius, was the subject of As this is not the most general case, a more complicated,
the envelope dynamics analyzed in Ref. [4]. but equally relevant, picture is displayed in Fig. 3, where
In the case most relevant to the emittance compensa- only two of the slices are launched with sizes below equi-
tion process, the beam is launched with a size smaller than librium, but the third has low enough line charge density
equilibrium for all portions of the beam, and the trace that, at the same initial size of the other two slices, it is
space trajectories for various slices are nested ellipses. above equilibrium. This picture displays what happens if a
This is shown in Fig. 1, which displays three elliptical beam is launched with size matched in an rms, integrated
trajectories corresponding to three different slices with beam sense, so that all slices are the same size, but due to
l1 , l2 , l3 . These ellipses are traversed, according to variations in current, some slice sizes are initially above,
our linear analysis, with the same frequency. Thus the area and others below, equilibrium. It can be seen that, while
in trace space that the points on the three ellipses describe the slice dynamics and associated emittance evolution
when connected to the trace space origin (at an instant in are in some ways different (the maximum emittance is
time), which is proportional to the emittance defined by larger in this case), the overall periodicity of the emittance
Eq. (8), oscillates with twice the mismatch oscillation fre- oscillation is the same. The most important way in which
the two situations differ is that in Fig. 1 the rms beam
σ'
r
σ'
r
kpz=π/2
λ1 λ2 λ3
σ λ1 λ2 λ3
σ r
eq1 σ
σ kpz=0,2 π r
eq2 σ
eq3
λ 1<λ 2<λ 3
kpz=3π /2
λ <λ 2 <λ 3
1
094201-3 094201-3
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
∑p µ ∂∏
σ' 11h g0
r dsr 苷 关sr0 2 sinv 兴 cos ln , (11)
2 g共z兲
so that we can write
∑p µ ∂∏
11h g0
sr 共z兲 苷 sinv 1 关sr0 2 sinv 兴 cos ln ,
kpz=π/2 2 g共z兲
(12)
λ1 λ2
σ and
σeq1 r p
λ σ 1 1 h g0
σ 3 eq2 sr0 共z兲 苷 关sr0 2 sinv 兴
eq3 2 g共z兲
∑p µ ∂∏
11h g0
λ 3< λ 1<λ 2 3 cos ln , (13)
2 g共z兲
with g0 苷 g共0兲. Thus the mismatch envelope dynamics
are not conceptually much different than in the coast-
FIG. 3. Projected trace space area described by three slice en- ing beam case, with oscillations about the particular
velopes with line charge l3 , l1 , l2 with the line charge of
slice 3 so low that sr0 . seq , shown at kp z 苷 p兾2. The emit- solution (no longer an equilibrium, but a secularly dimin-
tance evolution behavior is qualitatively the same as in Figs. 1 ishing envelope) proceeding at approximately the plasma
and 2, but with larger amplitude of oscillation. frequency. Note that the plasma frequency is no longer
a constant in this case, but monotonically diminishesp
angle sr0 is the same sign for all slices, while in Fig. 3 the with acceleration, approximately as kp ~ nb 兾g 3 ~
21 23兾2
angle of the low current slice is of opposite sign from the sinv g ~ g 21兾2 . Mismatch envelope dynamics cor-
other two. We will return to this important point below. responding to Eqs. (12) and (13) are illustrated by the
The extension of this type of motion about an equilib- normalized trace space (phase space) picture given in
rium to a system with longitudinal acceleration has been Fig. 4, which shows the dynamics of three slices corre-
considered by SR, who have analyzed the motion of such a sponding to the hierarchy of currents introduced in Fig. 3.
system with Eq. (1). This equation is again nonlinear, but While the picture in Fig. 4 gives a similar schematic
also has a useful particular solution— which is no longer view of emittance oscillations as Fig. 3, it has two notable
an equilibrium, however— with which one can begin an differences with the nonaccelerating case. The first is sim-
analysis, termed the invariant envelope [4], ply that the emittance one needs to be concerned with when
s the pbeam accelerates is the normalized emittance ´r,n 苷
2 re l共z 兲 p
sinv 共z , z兲 苷 0 . (9) bg 具sr2 典 具sr0 2 典 2 具sr sr0 典2 ⬵ g 具sr2 典 具sr0 2 典 2 具sr sr0 典2 ,
g 共1 1 h兾2兲g共z兲 which is a measure of the transverse phase space area, and
is thus conserved under linear transport and acceleration.
It can be seen that the existence of this particular solution
is not dependent on external focusing, as even with h 苷 0 γσ '
(pure traveling wave, no solenoid) the state correspond- r
ing to this solution exists due to the effects of adiabatic
damping.
The invariant envelope has the unique property that the
trace space angle sr0 兾sr 苷 2g 0 兾2g is independent of
l共z 兲. Thus if one places all slices on their invariant en-
velope, they will be aligned in trace space angle and the Phase space area after 1/4 plasma oscillation
094201-4 094201-4
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
The usual “adiabatic damping” of the trace space area, in The equations of motion for the electron position for the
which the trace space angles diminish through acceleration free-expansion scenario are, under laminar flow conditions
as bg 21 , is emphasized in Fig. 4 by rescaling of the (meaning particle trajectories do not cross),
vertical axis with g (recall that we have set b 苷 1 in this Z x0
analysis). This rescaling removes the apparent damping x 00 共z兲 苷 kp0
2
F共x0 兲, F共x0 兲 苷 f共x兲 dx 苷 const,
0
of the motion and makes the diagram approximately
a correct phase space plot, in the limit b 苷 1. The (15)
second difference is that all mismatch oscillations have where the local value of the initial (spatial) plasma fre-
end points attached to a line gsr0 兾sr 苷 2g 0 兾2 instead quency in the plane of symmetry has been defined as
of sr0 苷 0. As the invariant envelope associated with
4prc nb0
the slices becomes smaller with increasing energy as 2
kp0 苷 . (16)
g 21兾2 (the ensemble of ellipses shown slides up the line b2g3
gsr0 兾sr 苷 2g 0 兾2 towards the origin), the area associated If laminarity is obeyed, the integral F共x0兲 is constant and
with the emittance not only oscillates, but secularly damps these equations have solutions dependent only on initial
as g 21兾2 . conditions,
Note that the offset phase space area described by the 共kp0z兲2
mismatch oscillations (the ellipses in Fig. 4) is actually x共x0 , z兲 苷 x0 1 F共x0 兲 . (17)
conserved, as can be seen through inspection of Eqs. (12) 2
and (13). This means that for an ensemble of slices placed The density distribution is also a simple function of
all at the same initial phase space condition, but with dif- its initial state, as conservation of probability gives
ferent l共z 兲, the set of points which makes up the section of f共共x共x0 兲, z兲兲 dx 苷 f共x0 兲dx0 or
the phase space boundary not attached to the origin form f共x0 兲 f共x0 兲
a line with varying length but no area. This ensemble f共共x共x0 兲, z兲兲 苷 dx共x0 兲
苷 .
共kp0 z兲2
(18)
line stretches and rotates about the invariant envelope of dx0 11 2 f共x0 兲
the matched slice. If the invariant envelope slice is actu- In the freely expanding case, the density distribution be-
ally present in the beam, the ensemble line passes through comes more uniform as it expands over many plasma ra-
the invariant envelope line gsr0 兾sr 苷 2g 0 兾2, and rotates dians 共kp z ¿ 1兲,
about the intersection point of these two lines. This inter- f共x0 兲 2
section is therefore a fixed point in phase space. Thus the f共共x共x0 兲兲兲 苷 共kp0 z兲2
) (19)
.
11 f共x0 兲 共kp0z兲2
matched invariant envelope is a generalized fixed point in 2
the envelope phase space. This is an important observation This observation is critical, as it implies that the transport
having implications for particle motion within a slice. is “more linear,” since the space-charge defocusing for a
uniform beam becomes approximately linearly dependent
III. LAMINAR AND NONLAMINAR MOTION IN on offset,
COASTING SLAB BEAMS x 00 共z兲 苷 kp0
2
F共x0兲 艐 x兾2z 2 . (20)
As can be seen by the analysis above, the self-consistent This will in turn imply that the phase space wave-breaking
collective motion of particle beams in cylindrical symme- effects which lead to irreversible emittance growth are
try is complicated somewhat by the need to approximate mitigated, since the angle that a particle attains becomes
the solutions to the relevant differential equations. Because more linearly correlated with position,
of this, it is most instructive to begin our analysis using a
Cartesian, or slab-symmetric (sheet) beam, following the x0
2
kp0 zF共x0 兲 2
苷 1 2 2 ! , (21)
general methods introduced by Anderson in Ref. [6]. x x0 1 2 kp0z F共x0兲 z
We start this discussion by examining a freely expand-
If the phase space distribution lies along a straight line, the
ing (unfocused) laminar beam, with initial 共z 苷 0兲 density
emittance vanishes, so Eq. (21) indicates a desirable trend.
profile, infinite in the y and z dimensions, and propagating
An example of this increased distribution uniformity is
in the 1z direction,
shown in Fig. 5, where a beam with initial parabolic profile
Sb µ ∂2
nb 共x0 兲 苷 f共x0 兲 苷 nb0f共x0 兲 with f共0兲 苷 1 , (14) x
a0 f共x0 兲 苷 1 2 (22)
a
where Sb is the beam charge per unit (slab) area and a0 苷 has freely expanded for a distance kp0 z 苷 4. The profile
Sb 兾nb0 is the effective initial beam width. The case of free has become noticeably flattened during this expansion.
expansion can be considered to be the most nonequilibrium It is instructive at this point to calculate the emittance
scenario possible. It can also be thought of as forming one evolution associated with the freely expanding beam. In
portion of propagation under periodic application of thin order to do so, we consider a number of possible forms
lenses separated by drifts or free-expansion regions. of the distribution: Gaussian, parabolic, and uniform
094201-5 094201-5
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
094201-6 094201-6
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
x'
or -0.050 Fixed point
-0.20
It can be seen that wave breaking always occurs 0.0 0.20 0.40 0.60 0.80 1.0 1.2
for a sufficiently small value of f共x0 兲, i.e., portions of x
the beam found in a long continuous tail, assuming a FIG. 6. (Color) Trace space picture of slab symmetric beam at
monotonically decreasing function f共x0 兲. Quantitatively, wave breaking onset 共kb z 苷 p兾2兲, for the case of kb2 兾kp0
2
苷
Eq. (31) states that wave breaking eventually occurs for 2兾3, showing two fixed points with opposing direction of
all f共x0 兲 , kb2 兾2kp0
2
, with the most interior value of x0 rotation.
undergoing wave breaking at kb z 苷 p (for distributions
which smoothly approach zero, wave breaking begins in
these tails at kb z 苷 p兾2). It is also apparent that wave and our present analysis breaks down. Assuming a cold
breaking can be avoided by a combination of removal of beam initially at a waist 共x00 苷 0兲 the emittance evolution
the distribution tails, so that f共x0 兲 discontinuously goes is found to be
to zero at a hard-edge beam boundary, and by making the
ratio kb2 兾kp0
2
become small. When this ratio is near unity, 2
kp0
the beam is closely “matched” to the external focusing, ´苷a s0 jsin共kb z兲j , (32)
kb
and when the ratio is much smaller than unity the beam is
mismatched, with the focusing being too weak to control where, again, a is a constant depending on the form of the
the beam distribution at its launch size. An alternative 2
initial distribution and the factor kp0 s0 ~ Sb has no de-
way of understanding wave breaking is to note that the pendence on initial beam size. We note from this that the
equilibrium beam size xeq associated with the initial predicted maximum emittance occurs at kb z 苷 p兾2, as
wave-breaking position is a fixed point of the oscillation. with the correlated interslice emittance studied in Ref. [4].
On the other hand, we know that the origin in trace space It should also be emphasized that this is the same longitu-
is also a fixed point, with an opposing sense of phase dinal position that the initial wave breaking occurs in for
space rotation about it. The existence of two such fixed a distribution with a continuous tail.
points guarantees that the trace space will filament after
wave breaking and the emittance will grow irreversibly.
The trace space picture of this system is displayed 5.0
in Fig. 6.
Thus we deduce that a mismatched beam is more likely
to preserve its laminar flow, under mismatched conditions, 4.0
mismatched
which is an extension and deepening of what we have matched
learned from the case of free expansion. To emphasize 3.0
f/f(0)
094201-7 094201-7
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
frame) 0.80
p
r共r0 , z兲 苷 req 共r0 兲 1 关r0 2 req 共r0 兲兴 cos共 2 kb z兲 .
f/f(0), g/g(0)
0.60
(40)
The wave-breaking condition is again given by 0.40
∑ ∏
≠r ≠req ≠req p
苷 1 12 cos共 2 kb z兲 苷 0 , 0.20
≠r0 ≠r0 ≠r0
p (41) 0.0
≠req cos共 2 kb z兲
or 苷2 p . 0.0 0.50 1.0 1.5 2.0 2.5 3.0 3.5
≠r0 2 sin2 共kb z兾 2 兲 r/ σr
The quantity on the left-hand side of Eq. (41) can be writ- FIG. 8. A comparison of the function g共r0 兲 with Gaussian
ten as f共r0 兲.
094201-8 094201-8
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
TABLE II. Values of the form factor a for various initial cylin- and the calculation of the space-charge force is straight-
drical beam slice distribution types. forward. Because the use of Riccian particles provides
Profile a very smooth beam densities, it is not feasible to simulate
a perfectly hard-edged distribution. In the context of this
Gaussian 0.141 study, that drawback is not very important since we are
Parabolic 0.065 interested in studying distributions that will strongly wave
Flattop 0
break.
We found in the case of the slab beam expanding under
This is in contrast to the equivalent condition found in the its space-charge force that there was no wave breaking for
slab beam case, any type of distribution. Equation (34) tells us that this is
2 not the case for a freely expanding cylindrical beam if the
kp0
f共 y0 兲 ⬵ 1 , (49) initial distribution function falls off, so that the integral of
2kb2 the charge density does not increase proportionally with r.
which has the stronger quadratic dependence on the mis- In this case we expect wave breaking and look to simu-
match parameter kp0 兾kb . lations for understanding of the beam behavior after wave
As the linear dynamics of the axisymmetric beam have breaking occurs.
been seen to be formally quite similar to those of the slab The emittance evolution of the freely expanding beam
beam, it is not surprising that the emittance evolution is shows the effects of wave breaking. As in the focusing
similar as well. Given the same initial conditions as as- channel, the emittance increases to a maximum at lp0兾4
sumed in the slab case, we find the emittance to be of the where wave breaking occurs. While the beam contin-
same form as well, ues to expand, the particles in the vicinity of the initial
p wave-breaking point (where the maximum outward force is
´ 苷 as02 kp0 jsin共 2 kb z兲j . (50) found) effectively rotate about this outward-moving point.
Here a is again a form factor, defined as in the previous This rotation causes the tail particles to “tuck under” in
section. The numerical values of a found for the cylindri- phase space in a distance a bit longer than the initial plasma
cally symmetric case are shown in Table II. We will see half-wavelength (the plasma frequency is not constant, but
that Eq. (50) provides a very accurate description of the decreases as the beam expands), as would be expected, and
emittance evolution up until wave breaking. Note that the the emittance decreases during this initial rotation. The
emittance in Eq. (50) is in fact linearly dependent on s0 , emittance growth is not perfectly compensated by this non-
as kp0 ~ s021 . linear effect, however, and the emittance reaches a local
minimum. After that, ´ becomes simply proportional to
s as the beam continues to expand. Examination of the
V. SIMULATION OF COASTING CYLINDRICAL
beam phase space evolution, shown in Figs. 9 and 10, il-
BEAMS
lustrate this process.
The analytical treatments of intraslice transverse space We note from Fig. 10(b) that this tuck under effect on
charge detailed above are limited to the laminar flow the emittance occurs only after the rms beam size has
regime and, in the case of cylindrical beams, are only grown substantially (recall that kp0z . p, and the beam
approximate. They do, however, predict where wave has had a large distance in which to expand), as the emit-
breaking will occur, and that it can be minimized or tance minimum occurs when s兾s0 艐 8.5.
avoided by mismatching the beam-focusing channel
system. In order to test these predictions and examine 20 10
the behavior of a beam slice after wave breaking, we use Emittance [Arbitrary Units]
self-consistent simulations, using a one-dimensional time 15 7.5
dependent code called NORSE, that follow the evolution
of the beam using the space-charge force of Eq. (34). To
10 5
0
σ /σ
094201-9 094201-9
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
0.10 1 0 6
0.060
0
σ/σ
5.0 3
0.040
2.5 1.5
0.020
0.0 0.0 0
0.0 1.0 2.0 3.0 4.0 5.0 0.00 0.29 0.58 0.87 1.16 1.45
(a) r(mm) Ζ/λ
0.090
linear field-derived emittance— is understandable in a
number of different ways. If the dynamics being described
0.060
were only the linear slice dynamics, Figs. 2 and 3 illustrate
that the emittance performance would be qualitatively
0.030 the same in Figs. 11 and 12. They are not, however, and
this is because of the strong wave breaking induced in
0.0 the intraslice dynamics by the beam being too close to
0.0 5.0 1 0 15
(b) r [mm] equilibrium. In other words, the existence of the off-
origin moving “fixed point” in trace space gives rise to
FIG. 10. Trace space plots of a freely expanding, initially wave breaking, trace space filamentation, and associated
Gaussian beam at the initial emittance (a) maximum and
irreversible emittance growth. O’Shea has identified
( b) minimum.
irreversible emittance growth of this type with an increase
While the drifting beam is instructive, we are interested in the entropy which, we note, is also equivalent to loss
in beam transport involving focusing elements. We pro- of order or information in the system. In the case of
ceed again as before by examining two cases: periodic Fig. 11, the emittance increase due to field nonlinearities
thin lenses separated by drifts and a focusing channel. In is reversed (compensated) and the information about the
the case of thin lens focusing we can directly apply the beam’s initial state is preserved. An excellent illustration
result of the drifting beam. We find that, for a given trans- of this phenomenon is given in Fig. 13, which shows the
port length, fewer lenses and larger beam size oscillations beam distribution in r at three points in the propagation
will produce a better emittance at the end of the transport
line provided that the beam makes an integer number of 2.0 6
1.0 3
094201-10 094201-10
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
700 1.2
Density [Macroparticles/mm ]
600
2
1.0
200 0.40
Simulation
Theory
100 0.20
0 0.0
0.0 0.20 0.40 0.60 0.80 1.0 1.2 0.0 5.0×10
2 3
1.0 × 10 1.5 × 10
3
2.0× 10
3
2.5 ×10
3
(a)
r [mm] Z [mm]
12
600
2
094201-11 094201-11
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
VI. LAMINAR AND NONLAMINAR MOTION IN envelope. This situation is slightly different from that of
ACCELERATING CYLINDRICAL BEAMS coasting beams because we are required to reference s 0 to
the nonstationary particular solution
In the case of a beam accelerating under the influence s
of radio frequency fields, the paraxial equation of motion 4 re l共r0 兲
for a particle in a laminar flow condition now contains rp 共r0 , z兲 苷 0
g 共2 1 h兲g共z兲
terms arising from adiabatic damping and ponderomotive s
(alternating transverse gradient) forces, k̄p 共r0 兲 1 g0
µ 0 ∂ µ 0 ∂2 ⬅ r0 . (52)
g h g 2re l共r0 兲 kb 2 1 h g共z兲
r 共z兲 1
00
r 共z兲 1
0
r共z兲 苷 , p
g共z兲 8 g共z兲 g共z兲3 r In Eq. (52) we have identified kb 苷 g 0 兾 8 g, and can
(51)
see that the particular solution is again proportional to the
which is again a nonlinear equation without an analyti- initial ratio k̄p 共r0 兲兾kb . We can again proceed to linearize
cal solution. In this system, there is also an equilibrium- Eq. (52) about these particular solutions to obtain
like particular solution to Eq. (51), which is analogous to µ 0∂ µ ∂
00 g 0 1 1 h g0 2
the invariant envelope above, corresponding to each value dr 1 dr 1 dr 苷 0 , (53)
g 4 g
of r0 .
As in previous sections, we proceed by finding an ana- where dr 苷 r 2 rp . This equation has a general form
lytical formula for the emittance of a “matched” beam. In of solution similar to that given by Eqs. (12) and (13).
the case of an accelerating beam, we mean matched in the Therefore, we can solve Eq. (51) to find the single particle
sense that the rms size of the beam follows the invariant motion, yielding
∑p µ ∂∏ ∑p µ ∂∏
11h g0 1 11h g0
r共r0 , z兲 苷 rp 共r0 , z兲 1 关r0 2 rp0共r0 兲兴 cos ln 1 p 关r0 2 rp0 共r0 兲兴 sin ln ,
2 g 11h 2 g
(54)
094201-12 094201-12
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
CYLINDRICAL BEAMS
1.2
In this section we study the behavior of an initially
parabolic profile accelerating beam matched in the rms
1
sense to the invariant envelope and compare simulation to
≠´2n
苷0
≠z
µ ∂ ∑p µ ∂∏ Ω ∑ p µ ∂∏ ∑p µ ∂∏æ
16a 2 re lb 2 11h g0 11h g0 p 11h g0
苷2 sin ln sin ln 1 1 1 h cos ln ,
共1 1 h兲pg0 g 0 g 2 2 g 2 g 2 g
(59)
094201-13 094201-13
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
-0.02
Laboratory [17] for acceleration after the gun. In this case,
we have ´n,max 苷 2.75a, which produces a more tolerable
-0.025
margin for emittance due to nonlinearities and wave
breaking.
-0.03
094201-14 094201-14
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
1.0
PWT linacs
High gradient RF
photocathode gun Focusing solenoid
0.0
0 50 100 150 200 250 300 350 400
z (cm)
FIG. 18. (Color) Results of a PARMELA simulation of a uniform beam emitted at the cathode in the LCLS photoinjector design, with
a 1.6 cell rf gun followed by an emittance-compensating solenoid and two PWT standing wave sections.
oscillation) are nearly identical in this optimized design. cates into two populations. One population is composed
This fact points to the close relationship that interslice of slices whose dynamics were described in Sec. II, and
(linear) and intraslice (nonlinear) emittance oscillations has essentially no wave breaking associated with it, and
have with each other— they are both governed by the another in which beam particles wave break near the trace
same oscillation frequency but, in contrast to the intraslice space origin (not off axis, as our analysis in preceding sec-
emittance growth, the interslice dynamics are always tions has examined).
reversible. These populations can also be observed in the 共x, y兲 spa-
The 共x, z兲 spatial profile of the 10 000 simulation par- tial distribution shown in Fig. 20(a) and the phase space
ticles at the z 苷 4 m point is shown in Fig. 19. It can be distribution displayed in Fig. 21(a). In Fig. 20(a) the bi-
seen that, while the core of the beam is well behaved in furcated population produces a slight beam halo, while in
terms of the different z slices ending up in the same con- Fig. 21(a) one can directly see the bifurcation as distinct
figuration, the leading and trailing beam edges display very lengths of the slices in phase space. These points are em-
different behavior. This is due to the fact that the transverse phasized by Figs. 20( b) and 21( b), in which the 共x, y兲 spa-
space-charge forces drop dramatically in these longitudi- tial and phase space distributions are shown for the beam
nal “tail” regions, and the particles in these slices do not population located only within dz 苷 60.1 mm from the
focus to space-charge dominated waists, but actually cross beam longitudinal center. In Fig. 20( b), the beam halo es-
the beam axis. This is clearly an example of nonlaminar sentially disappears when this cut is made. In Fig. 21(b),
flow, and, as a result, the total transverse phase space bifur- one sees a very interesting situation— even though the lon-
gitudinal tails have gone bifurcated, they are realigned in
phase space with the beam cores. The difference between
the bifurcated and unbifurcated populations is simply that
the length in phase space is larger for the bifurcated popu-
lation in the longitudinal tails.
This dynamical picture changes significantly if one in-
jects a beam with a radially nonuniform current profile
( but same rms beam dimensions), as is the case in the
simulation results shown in Fig. 22. Here a radial Gauss-
ian profile, cutoff at r 苷 s, is introduced at the cathode.
While the rms beam envelope does not change appreciably
with a nonuniform injected beam, as is well known in the
theory of space-charge dominated beams [10], the phase
space dynamics reveal many changes from the uniform
beam case, as can be seen by examining the rms emittance
FIG. 19. (Color) Spatial 共x, z兲 distribution at the end of the evolution in Fig. 22. As expected, the emittance grows
PARMELA simulation shown in Fig. 18. significantly when nonlinearities in the space-charge field
094201-15 094201-15
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
0.10 0.030
(a) (a)
0.020
0.010
0.050
x' (rad)
0.0
x (cm)
-0.0100
0.0
-0.020
-0.030
-0.050 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
x (cm)
0.030
-0.10 (b)
0.020
-0.1 -0.05 0 0.05 0.1
y (cm) 0.010
x' (rad)
0.10 0.0
(b)
-0.0100
0.050 -0.020
-0.030
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
x (cm)
x (cm)
0.0
FIG. 21. (Color) (a) Phase space distribution at the end of the
PARMELA simulation shown in Fig. 18. (b) Phase space distri-
bution at the end of the PARMELA simulation shown in Fig. 18,
with a cut made on the distribution at dz 苷 60.1 mm to remove
-0.050 longitudinal tails.
094201-16 094201-16
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
5.0
rms beam size (nonuniform beam)
2.0
0.0
0 50 100 150 200 250 300 350 400
z (cm)
FIG. 22. (Color) Results of a PARMELA simulation of a radially nonuniform beam (Gaussian, cutoff at r 苷 s) emitted at the cathode
for the same accelerator conditions as in Fig. 18.
(purely radial) dynamics analysis to be surprisingly more The core of the beam, which is inside of the initial wave-
applicable to modeling of finite length photoinjector breaking point, is well behaved, laminar, and aligned to
beams than might be deduced from the purely uniform near the x axis. On the other hand, the particles which be-
beam case. gin outside of the initial wave-breaking point fold twice:
The phase space pictures shown in Fig. 25 display first these particles “tuck” under the core of the distribu-
again a bifurcation of particle populations, along with the tion, and then the ones at largest initial x move quickly
long halo of particles at large amplitude in phase space. across the x 苷 0 axis to large amplitude and positive
This bifurcation is essentially not due to differential slice 共x, x 0 兲 correlation. The particles at intermediate initial
motion in this case, but is due to wave breaking. The values of x between these two cases are almost, but not
components of the beam which have not undergone wave quite, prevented by the space-charge forces from cross-
breaking are localized within jx 0 j , 10 mrad, while the ing the x 苷 0 axis, and end up nearly aligned to this 共x 0 兲
components which have undergone wave breaking display axis. Thus the characteristic “cross” shape within a large
large angles and offsets in x. amplitude halo seen in Fig. 25 is formed. This elaborate
The phase space distributions shown in Fig. 25 are com- phase space picture of course implies that a large amount of
plicated pictures, which result from two separate folding irreversible emittance growth has occurred. The differ-
bifurcations of the transverse distribution within a slice. ences between Figs. 21 and 25 illustrate well one of the
main points of this paper— the onset of wave breaking and
emittance growth which results from matching of a beam
to a generalized transverse equilibrium.
0.3
We also note that the emittance as shown in Fig. 22 is
0.2
much larger than the sum of the emittance in the uniform
beam case with the estimate of Eq. (62) added to it. The
0.1 extra component of the emittance given in Eq. (62) is due
only to wave breaking occurring after entrance into the
x (cm)
0 first PWT linac and ignores the emittance one may expect
from the initial beam oscillation from the cathode to this
-0.1
point. We have not provided an emittance estimate from
-0.2
the nonlinear intraslice dynamics during this oscillation.
This is in part because our previous analyses do not allow
-0.3 a good modeling of cylindrically symmetric beams so far
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 from equilibrium, and also because the transverse fields of
z (cm)
the beam very near to the cathode are not of simple quasi-
FIG. 23. (Color) Spatial 共x, z兲 distribution at the end of the one-dimensional form, as is also assumed in our previous
PARMELA simulation shown in Fig. 22 analyses.
094201-17 094201-17
PRST-AB 3 S. G. ANDERSON AND J. B. ROSENZWEIG 094201 (2000)
0.3 0.3
(a) (b)
0.2 0.2
0.1 0.1
y (cm)
y (cm)
0 0
-0.1 -0.1
-0.2 -0.2
-0.3 -0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
y (cm) x (cm)
FIG. 24. (Color) (a) Spatial 共x, y兲 distribution at the end of the PARMELA simulation shown in Fig. 22. (b) Spatial 共x, y兲 distribution
at the end of the PARMELA simulation shown in Fig. 22, with a cut made on the distribution at dz 苷 60.1 cm to remove longi-
tudinal tails.
0.060 0.060
(a) (b)
0.040 0.040
0.020 0.020
x' (rad)
x' (rad)
0.0 0.0
-0.020 -0.020
-0.040 -0.040
-0.060 -0.060
- 0 . 2 -0.15 - 0 . 1 -0.05 0 0.05 0.1 0.15 0.2 - 0 . 2 -0.15 - 0 . 1 -0.05 0 0.05 0.1 0.15 0.2
x (cm) x (cm)
FIG. 25. (Color) (a) Phase space distribution at the end of the PARMELA simulation shown in Fig. 22. (b) Phase space distribution
at the end of the PARMELA simulation shown in Fig. 22, with a cut made on the distribution at dz 苷 60.1 cm to remove longi-
tudinal tails.
IX. CONCLUSIONS negligible nonlinearities in the field for a short time, and
is then accelerated and focused (to the same radius as at
In this paper, we have explored the consequences— the cathode) to produce a beam which is appropriate for
wave breaking and associate emittance growth—of the injection into the linac. This example illustrates one of
choice of beam envelope trajectory, i.e., the degree to the conclusions of our work— emittance compensation in
which a beam is matched to a generalized equilibrium. cylindrically symmetric systems works well when the os-
In cases where the nonlinearity of the field is tolerable cillation has a large amplitude, far from a matched con-
(as in the perfectly uniform beam simulated in Sec. VIII), dition. This conclusion is applicable to transport in a rel-
running the beam essentially on the invariant envelope in atively low energy beam line (e.g., 20 MeV in the Tesla
a booster linac works well, as predicted by the analysis Test Facility [20]) placed after an emittance compensated
of SR. In the example of such a case (Figs. 18 –21), it photoinjector.
can be seen that very few nonuniformities are introduced On the other hand, when a moderately nonuniform
into the distribution by the initial emittance oscillation, in beam is injected at the cathode, the initial emittance
which the beam leaves the cathode, experiences some non- compensation is degraded and the “second compensation”
094201-18 094201-18
PRST-AB 3 NONEQUILIBRIUM TRANSVERSE MOTION AND … 094201 (2000)
achieved by matching to the invariant envelope is almost [1] K. J. Kim, Nucl. Instrum. Methods Phys. Res., Sect. A 275,
eliminated —in the uniform beam, the emittance dimin- 201 (1989).
ishes by 60% in the second compensation, and in the [2] B. E. Carlsten, Nucl. Instrum. Methods Phys. Res., Sect. A
nonuniform beam case it is diminished by only 20%. The 285, 313 (1989).
minimum emittance associated with the process illustrated [3] X. Qiu et al., Phys. Rev. Lett. 76, 3723 (1996).
[4] Luca Serafini and J. B. Rosenzweig, Phys. Rev. E 55, 7565
by this example is given by Eq. (62), which serves a useful
(1997).
guide to estimation of the best performance possible for a [5] L. Brillouin, Phys. Rev. 67, 260 (1945).
given injector configuration. [6] O. A. Anderson, Part. Accel. 21, 197 (1987).
In conclusion, in this work we have attempted to unify [7] T. P. Wangler et al., IEEE Trans. Nucl. Sci. 32, 2196
the microscopic concepts of linear emittance compensa- (1985).
tion, which arise in high brightness electron beam physics, [8] I. Hoffman and J. Struckmeier, Part. Accel. 21, 69
and nonlinear wave breaking, which has had an impact (1987).
on the understanding of ion beam physics, showing their [9] Patrick G. O’Shea, Phys. Rev. E 57, 1081 (1998).
relationship to one another in the context of high bright- [10] Martin Reiser, Theory and Design of Charged Particle
ness photoinjectors. We have provided both a qualita- Beams (Wiley, New York, 1994).
tive picture of extremely space-charge dominated beam [11] T. P. Wangler, RF Linear Accelerators (Wiley, New York,
1998).
dynamics in radially nonuniform beams and quantitative
[12] S. C. Hartman and J. B. Rosenzweig, Phys. Rev. E 47, 2031
predictions concerning the irreversible emittance expected (1993).
to arise due to wave breaking and subsequent phase space [13] J. B. Rosenzweig and L. Serafini, Phys. Rev. E 49, 1599
filamentation. This understanding aids in the classifica- (1994).
tion of global characteristics of beam distributions, such as [14] P. Lapostolle, IEEE Trans. Nucl. Sci. 18, 1101 (1971).
nonlinear field energy and entropy, which have been orig- [15] Bruce E. Carlsten, Phys. Rev. E 60, 2280 (1999).
inally introduced in the field of intense ion beams. While [16] L. Serafini (private communication).
it is clear that this work has imported some valuable con- [17] J. B. Rosenzweig et al., Nucl. Instrum. Methods Phys. Res.,
cepts from this field into the study of the high brightness Sect. A 410, 532 (1998).
electron beams, it is not yet clear that our results are of [18] J. Arthur et al., SLAC Report No. SLAC-R-0521, 1998.
other than conceptual significance to the study of intense [19] M. Ferrario et al., in Proceedings of the ICFA Advanced
Accelerator Workshop on the Physics of High Brightness
ion beam transport. This question will be left to further
Beams, Los Angeles, 1999 (World Scientific, Singapore,
investigations. 2000).
[20] E. Colby et al., in Proceedings of the 1997 Particle Accel-
ACKNOWLEDGMENT erator Conference, Vancouver, BC (IEEE, Piscataway, NJ,
The authors wish to thank Luca Serafini for his guid- 1998).
ance in the computational aspects of this work as well his
interest and comments.
094201-19 094201-19