JLAB-TN-02-026
24 July 2002
A Compact Mirror-Bend-Achromat-Based Energy Recovery
Transport System for an FEL Driver
David Douglas
Abstract
We present a conceptual design for a compact mirror-bend-achromat-based transport
system intended for use in an FEL driver linac. Analysis of the design suggests it can
energy-recover (with energy compression) beams with very large momentum spreads
(~15-35%) if harmonic linearization is provided by the RF system.
Introduction
The mirror-bend achromat [1] is a geometrically simple beam optics module imposing a
momentum-independent and linearly compactional deflection on a charged particle beam.
In this note, we describe the use of such an achromat as the basis of an energy recovery
transport system in an FEL driver. Design requirements are as follows:
1) the system must bend a beam with potentially large relative momentum spread
through 180o at energies of order several tens to a few hundred of MeV (“must”)
2) the system must support longitudinal matching insofar as necessary to allow energy
compression during energy recovery (“must”)
a) path length adjustability is implied (“should”)
b) momentum compaction must be provided to allow energy compression (“must”)
c) RF waveform curvature correction is required to ensure adequate control of
energy spread during/after energy recovery (“must”)
3) beam sizes must be controlled to avoid beam loss, despite potentially large relative
momentum spread (tens of percent) and emittance (tens of mm-mrad, normalized)
(“must”)
a) the arc must provide dispersion management for a beam with large momentum
spread
b) beam envelopes and dispersion (transfer matrix elements) should remain small
throughout the transport
c) betatron matching from the FEL and to the linac are desirable as a means of spot
size and chromatic aberration management (“like”)
4) the system must be compact (“must”)
Design Tradeoffs
The primary design choices for an energy recovery transport system are the selection of
RF frequency and the selection of a recirculation arc design. The former dictates the
1 of 26
JLAB-TN-02-026
24 July 2002
gross compactional properties of the latter, and thus constrains the methods used for
dispersion management. These, together with the choice of arc design, constrain the
management of the system betatron properties.
Here, we examine the potential utility of the mirror-bend achromat [1] as a basis for FEL
driver recirculation and energy recovery transport systems. We will, in particular, look at
the simplest and most compact implementation of a 180o mirror bend achromat – a pair
of 90o bends. The arc geometric concept is presented in Figure 1, wherein two of 90o
dipoles are symmetrically positioned around the system symmetry line. The entry angle
of the first bend and exit angle of the second bend are adjusted to optimize betatron
behavior; the exit angle of the first and entry angle of the second are constrained by
mirror-bend geometry to be –45o.
Figure 1: Mirror-bend achromat configuration.
Though possessed of very large momentum acceptance, mirror-bend achromats provide
little design and operational flexibility in betatron and dispersion management. They are
“completely” achromatic – the exit orbit is, by geometric construction, momentum
independent – and linearly compactional – the path length depends only linearly on
momentum offset. The simple system configuration provides only a limited number of
parameters for optimization. The dispersion at the symmetry point and the momentum
compaction are defined by the bend angle and bend radius as follows [1].
2 of 26
JLAB-TN-02-026
24 July 2002
x =sintan
l (sinp/p(sin
For 90o bends, these reduce to the following.
x =
M56 (/21(
In this geometry, the “interior” pole faces must be rotated by 45o (in the horizontally
focusing direction) to generate the mirror geometry. The bend radius, the entry pole face
rotation of the first dipole, the exit pole face rotation of the second dipole, and the bendto-bend separation are thus the only parameters available for optimization. The lower
limit of first of these is typically set by both the dipole field required to bend a beam at a
particular energy and the fact that smaller bend radii correspond to stronger focusing –
and thus aggravate the betatron matching problem imposed by the large pole face angles
used in the mirror bend configuration. A lower limit on momentum compaction is thereby
specified.
This combination of limited minimum compaction and inherently linear behavior
constrains the choice of RF frequency and operating phase. The compactional transport
matrix elements required to provide energy compression during energy recovery through
a linac operating at wavelength RF with phase offset  are as follows [2].
M 56  
 RF
2
 E0  1


 E linac  sin  0
1  2
T566   
2   RF
W5666

cos  0
M 56 2
sin  0

 1 1 cos 2  0  2
  

2
 6 2 sin  0   RF
 2
U 56666  
  RF
2

 M 56 3

3

 M 56 4 , etc.

Here, E0 is the full beam energy and Elinac the linac energy gain
Because the momentum compaction (M56) can be no smaller than the value permitted by
the bend radius (limited by the available field/beam energy and tolerance for focal insult
from dipole body and edges), the ratio of the RF wavelength and (the sine of) the
operating phase offset is limited as well. Typically  will be of order 10o or so – that is,
about 1/(2 radians. Noting (for modest injection energy) E0 ~ Elinac, we see that RF
3 of 26
JLAB-TN-02-026
24 July 2002
must be of order M56 (which is, in turn, of order the bend radius). Some flexibility is of
course allowed by the specific choice of operating phase . However the choice of phase
is driven not only by the desirability of a particularly RF wavelength, but also by the
desire to utilize the available gradient (larger phase offset provides less acceleration) and
by the impact of RF curvature on the energy recovery process. We could, for example, go
to RF ~ ½ M56 by halving the phase offset, but inspection of the above expressions shows
T566 ~ RF/sin3~ (RF/) (1/2)
Thus, moving to half the RF wavelength by halving the operating phase will fourfold the
effect of the RF curvature. Operating closer to crest will simply cause curvature effects to
be more dramatic and can adversely affect the energy recovery process.
If the bunch is long at reinjection for energy recovery (as it will be for large incident
momentum spread and/or large M56), the residual energy spread at the dump will be
dominated by RF curvature (torsion, …) effects unless either a) the transport system
supplies the T566 (W5666, …) value specified above or b) the linac has an RF harmonic
linearization system [3]. Unlike in arcs of the JLab FEL driver accelerators, RF
waveform curvature and torsion correction (more accurately, precompensation) cannot be
performed in a mirror bend achromat – it’s a completely linear system – implying the
correction must be done with an RF linearizer. This, in turn, implies an aperture
constraint – the harmonic cavities cannot be “too high” frequency to stuff an intense
beam through… Hence, we are motivated to move to lower frequencies and perhaps
operate farther from crest, where curvature is further alleviated.
This solution represents a shift in design paradigm towards a view inspired by input from
Brau and Smith at the Spring 2002 FEL semiannual – wherein a simple transport system
is used and RF effects “locally” corrected by RF compensation. In this approach, energy
acceptance and limits on the recoverable, compressible energy spread are, through the use
of mirror-bend achromatic transport, no longer imposed by the transport system, but
instead are due to constraints stemming from longitudinal and transverse acceptance in
the RF system. The machine will have a simple linear energy recovery transport, use a
moderate M56 value and operate relatively long wavelength RF with harmonic curvature
correction somewhat farther off-crest than was done in the JLab machines.
System Description
A mirror-bend-achromat-based energy recovery system has been designed. It assumes
use of dipoles with maximum field ~20 kG, which gives ~0.3 m bend radius at 200 MeV.
The resulting momentum compaction (M56) is ~0.4 m; this corresponds to use of 750
MHz RF with energy recovery 10o before trough (acceleration 10o before crest), or 500
MHz RF with 15o offset. We based the design on the 750 MHz choice, largely because
optics for a ~200 MeV SRF linac (based on 10 five cell cavities with effective length 1 m
and nominal gradient of 20 MV/m) were available by simple scaling of CEBAF 5 cell
cavity results [4]. The sequential list of beam optics modules is as follows:
4 of 26
JLAB-TN-02-026
24 July 2002
1) a four-quad telescope (a pair of doublets), matching beam envelopes extracted
from an FEL similar to the IR Upgrade (x=3 m, x=0, y=6 m, y=-1) to values
that propagate more or less nicely through
2) a mirror bend achromat based on a pair of 90o dipoles with bend radius of order
0.3 m, followed by
3) a four-quad telescope (uniformly spaced), matching beam to values roughly
optimized for energy recovery transport (x,y=4 m, x,y=1/3) through
4) a 750 MHz 10 five-cell-cavity SRF linac, operating 10o ahead of trough, energy
recovering from ~200 MeV to 7 MeV.
The system configuration is sketched in Figure 2. A layout is presented in Figure 3. It is
very compact (somewhat less than 1.2 m wide). The bend radius was fit to control the
momentum compaction and thus to minimize the exhaust energy spread (after energy
recovery, as simulated by DIMAD). Dipole pole face angles and quad strengths were
varied to set reinjection beam envelopes and envelopes at the center of the achromat
(these selected to provide optimized chromatic performance).
As noted above, curvature correction is probably best accomplished through the use of a
3rd harmonic RF linearizer. This is not implemented in the present design; conceptually, a
set of 2250 MHz cavities (likely a pair or two) at the midpoint of the linac could provide
adequate compensation (see the following performance discussion). Path length
adjustment (for pass-to-pass phase control) can be provided either in the linac-to-wiggler
transport (not the subject of this note) or by gross motion of the achromat. The dipole
pair, could for example, be mounted on a carriage or table and moved (much as the Bates
-bends are moved for phasing), Orbit steering before, after, and at the midpoint of the
achromat could provide finer resolution and/or magnetically controlled path length
adjustment using trim magnets over at least a limited range.
injector
dump
750 MHz cryomodule
transport to wiggler
wiggler
location
Figure 2: System schematic for energy recovery transport. Regions under consideration in
this note are in solid/black; regions to be addressed elsewhere are in dashed/gray.
Figure 3: Layout.
5 of 26
JLAB-TN-02-026
24 July 2002
Performance
The performance of a large-momentum-acceptance transport system in an energy
recovering linac is characterized by
1) beam envelope/matrix element control,
2) dispersion management over a broad momentum range,
3) control of chromatic- and geometric-aberration-driven dilution of the phase space
(aberration-induced mismatch or phase space distortion), and
4) longitudinal phase space preservation during energy compression and recovery.
Beam Envelope/Matrix Element Control – High performance energy recovering systems
must limit beam envelope function and matrix element values so as to ensure adequate
halo control and minimize error sensitivities [5]. In the JLab IR Demo and Upgrade FEL,
this suggested the minimum apertures of 5 – 7.5 cm should see only beam at envelopes of
order a few 10s of meters. In a very high power system such as this one is intended to be,
the use of relatively low RF frequency provides larger potential working aperture – by as
much a factor of two – but the desire for order of magnitude higher currents suggests that
maximum beam envelopes must be limited to the order of a few to several meters, rather
than a few tens of meters.
Beam envelope functions are shown in Figure 4. The “few meter” criterion is met – the
maxima are just under 10 m in the small aperture portion of the machine– implying good
halo control and low error sensitivity.
25
20
15
beta x
beta y
10*eta x
10
5
0
0
5
10
15
20
25
30
Figure 4: Beam envelope functions from end of wiggler insertion to end of linac for
energy recovery transport (from ~200 MeV to 7 MeV).
Dispersion Management – The mirror bend achromat, by construction, provides exact
compensation of dispersive central orbit variation to all orders and the dispersed orbit
within the achromat is linear in the momentum offset [1]. The system therefore meets, in
principle, any beam dynamic requirement. In practice, the acceptance is constrained by
6 of 26
JLAB-TN-02-026
24 July 2002
dipole geometry to a finite momentum range. Perusal of the geometry in Figure 1 reveals
that the upper momentum limit is constrained by the dipole-to-dipole spacing; sooner or
later the corners bump into each other; the lower momentum limit is similarly constrained
by the location of the “nosepiece” needed to allow space to set the exterior pole-face
rotation.
The absolute upper bound is set by where the dipole corners touch; given the simple 90o
geometry and the resulting relations stated above, this occurs at a relative momentum
offset of D/, where 2D is the on-momentum dipole-to-dipole spacing. In this design, D
= 0.25 m and  ~ 0.3 m, so the upper acceptance limit is somewhere around 75 to 80%,
though the constraints of reality may reduce this to a smaller value when a realizable
magnet design is utilized. Similarly, the lower momentum acceptance limit will be of
order (-w)/, where w is the half-width of the exterior pole (that with the pole face
rotation; see Figure 5). Assuming, for example, a 0.10 m half width, we would have
~67% low-end acceptance (though this, too, is optimistic, as we shall see below).
D
D = x (p/p)upper=(p/p)upper

- w = x (p/p)lower=(p/p)lower
w
w
Figure 5: Bend geometry associated with momentum acceptance limits.
The conclusion to be drawn here is not that of a specific acceptance value, but rather
simply that the acceptance is “large” by typical beam transport standards. In the
following, we shall in fact see that the limiting acceptance in systems of this type is in
fact no longer that imposed by the beam transport system transverse behavior, but rather
set by the longitudinal acceptance of the linac during energy recovery.
Aberration Control & Preservation of Phase Space Quality – Chromatic and geometric
aberrations can degrade phase space quality by causing variation (with momentum and
position) in the image of the initial phase space throughout, and in particular, at the end,
of the system. Very large acceptance systems are to some extent difficult to characterize
7 of 26
JLAB-TN-02-026
24 July 2002
in terms of specific aberrations inasmuch as these are coefficients in a perturbative
expansion, which may be at best slowly convergent over much of the large phase space
volume of interest. An excellent example of this is provided by the chromatic behavior of
the JLab IR Demo and Upgrade Bates-clone recirculators. A second-order matrix
transform representation of this system is violently nonsymplectic when evaluated off the
nominal momentum, but integrative (“TURTLE-mode”) tracking, element-to-element,
restores cross-terms that provide a more precise representation of the motion, and
suppresses the observed phase space dilution. The Taylor’s-series expansion, though
“correct” to the order evaluated, is a very poor approximation to the actual motion; the
very crude “integration”, though not necessarily very accurate, provides a much better
representation of the system behavior inasmuch as it captures relevant “higher order
aberrations” and properly represents system performance. We note that the simple
geometry of this mirror achromat admits an essentially exact nonperturbative solution of
the motion, and thus provides entertaining opportunities for investigation of such issues,
one such of which is discussed below.
The moral of this story is that large acceptance systems should be treated with
considerable caution when approached with perturbation-based tools, and, when possible,
be evaluated with analytical or numerically exact models. In this regard, we note that the
geometric linearity of this system will reasonably allow its investigation using a tool such
as DIMAD (transverse positional offsets will, by and large, be modest), while the very
large momentum offsets under consideration require treatment with a more globally valid
model.
Geometric Effects – The system under consideration has no nonlinear elements (other
than the kinematic nonlinearities inherent in the bends and any error harmonics included
in the dipoles, quads, and cavities [none are, in this study]). It should therefore exhibit
very regular behavior over a large volume of phase space. That this is the case is
supported by Figure 6, which presents results of a DIMAD “line geometric aberration”
analysis. A phase space generated using the nominal matched beam envelopes at a
normalized emittance of order 400 mm-mrad (geometric emittance of 1 mm-mrad at
~200 MeV) was injected at each of 21 momentum offsets (in 1% steps between –10%
and +10%) and propagated to the reinjection point of the recirculator. As can be seen
from the figure, only modest phase space distortion is observed. The chromatic variation
is, however, rather more obvious.
Chromatic Effects – As noted above, orbit variation with momentum offset can be exactly
characterized in a mirror-bend achromat. Analysis of the chromatic variation of beam
envelopes is in this system potentially complicated by large momentum offsets, which
preclude reliance on a perturbative description, but also simplified by the geometry,
which ensures that the central orbit at all quadrupoles is at a fixed location for all
momenta and that the dipole magnet focusing is easily characterized at any momentum.
We have used both perturbative and global descriptions to assess the performance of this
design. In the design phase, the usual DIMAD “detailed chromatic analysis” was used to
gain a notional understanding of the behavior. Results are presented in Figure 7.
8 of 26
JLAB-TN-02-026
24 July 2002
yy' phase space, -10% to +10% , eps=1 mm-mrad (geometric)
xx' phase space, -10% to +10% , eps=1 mm-mrad (geometric)
0.0020
0.0025
10%
10%
9%
9%
0.0020
8%
0.0015
8%
7%
6%
0.0010
5%
3%
-0.004
-0.003
-0.002
0.001
0.002
0.003
0.004
0.005
0%
y' (rad)
x' (rad)
-0.005
1%
-1%
-0.0005
4%
3%
-0.005
-0.004
-0.003
-0.002
0.0000
-0.001 0.000
2%
1%
0.001
0.002
0.003
0.004
0.005
-0.0005
-2%
-3%
-0.0010
5%
0.0005
2%
0.0000
-0.001 0.000
6%
0.0010
4%
0.0005
7%
0.0015
-7%
-0.0020
x (m)
-10%
-7%
-8%
-0.0025
y (m)
Figure 6: Results of DIMAD “line geometric aberration analysis” modeling from
wiggler insertion to reinjection for beam of emittance 1 mm-mrad (geometric),
equivalent to ~400 mm-mrad normalized at ~200 MeV at each of 21 momentum offsets
(1% steps between –10% and +10%). Only modest phase space distortion is seen;
chromatic variations are apparent (especially horizontally, for low momentum cases).
When acceptable performance was achieved, an “exact” Excel model was built. Within
the mirror bend achromat, the orbits are readily described at any momentum
(()=0(1+), =/2), and the dipole entry/exit angles are constant. The focusing around
any off-momentum orbit is thus that imposed by a bend of radius (), angle /2, with the
nominal entry and exit angles. The bend-to-bend drift length is readily characterized as a
function of momentum (D()=D0-x); see, for example, Figure 5). The orbit outside the
achromat is momentum independent; all energies go through the centers of the quads,
which focus with a focal length f()=f0(1+) and have, of course, fixed length. The
matrices for motion around each element are thus readily evaluated at any momentum.
These results may be used to propagate an initial phase space and obtain its image
throughout the system at any momentum offset. Results are given in Figure 8, and show
that beam envelopes are very regular over a large momentum range.
Two features of the chromatic behavior are of interest. First, “acceptance”, broadly
defined, is large. The envelopes mismatch in one transverse plane or the other by only a
factor of four (spot sizes change by only a factor of two) as the momentum varies from –
8% to +64% or so. This implies an (albeit asymmetric) 80% acceptance. Secondly, the
acceptance is asymmetric. Recall from the discussion above that the upper limit imposed
by the dipole physical geometry topped out at approximately +70%. This is not matched
either physically or beam dynamically on the low end. The dipole “nose” needed to
provide space for the required pole face rotation precludes arbitrarily large low end
acceptance, and, as the bend radius drops with decreasing momentum, the quadrupole
focusing and dipole edge focusing becomes rather ferocious, generating increasing
mismatch and beam envelope blow-up. This asymmetry need not, however, impose
severe operational constraints. The energy recovery transport can be designed and set up
9 of 26
-5%
-6%
-0.0020
-8%
-9%
-3%
-4%
-0.0015
-6%
-0.0015
-1%
-2%
-0.0010
-4%
-5%
0%
-9%
-10%
JLAB-TN-02-026
24 July 2002
for comparatively low beam energy, but operated with higher energy beam. For example,
if set up for 100 MeV, beam with energy 130 MeV could be transported and ±40 MeV
acceptance on either side of the 130 MeV set-point (90 to 170 MeV) would be available.
This is no different in principle than the systematic offset used in the IR Demo endloops
– it is just an order of magnitude larger!
The DIMAD analysis was performed with subdivision of each quadrupole into five
segments to capture cross-terms (subdivision 25 times changes the answer only a little).
The accuracy of the resulting description is benchmarked by Figure 9, which compares
the DIMAD description to the corresponding Excel numerically exact calculation. The
agreement is quite good, particularly from the perspective that each DIMAD sub-element
is seeing only a linear chromatic shift in focusing strength. Another amusing aspect of
this comparison will be discussed below.
perturbative momentum scan (DIMAD)
5
4
variation
3
dbetax/betax
2
dalphax
dbetay/betay
1
dalphay
0
-0.125
-0.1
-0.075 -0.05 -0.025
0
0.025
0.05
0.075
0.1
0.125
-1
-2
dp/p
Figure 7: Perturbative (DIMAD “detailed chromatic analysis”) momentum scan results.
Figure 8a: Exact (Excel model) momentum scan results for x(s,).
10 of 26
JLAB-TN-02-026
24 July 2002
Figure 8b: Exact (Excel model) momentum scan results for y(s,).
comparison of perturbative and exact momentum scans
4
3
3
variation
2
dbetax/betax DIMAD
dbetax/betax exact
2
dbetay/betay DIMAD
dbetay/betay exact
1
1
0
-0.13
-0.1 -0.08 -0.05 -0.03
0
0.025 0.05 0.075
0.1
0.125
-1
dp/p
Figure 9: Comparison of perturbative and exact results.
Longitudinal Phase Space Management – Scenarios for control of the longitudinal phase
space must provide matching for (at least) three independent parameters:
1) RF phase: path length must be controlled to return the beam to the module a
phase appropriate to energy recover. This will typically be 180o away from the
accelerating phase (and might possibly have to be, to ensure RF drive system
stability).
2) linear momentum compaction: operating phase, energy profile, RF frequency
(wavelength), and recirculator FEL-to-linac compaction must match so as to
ensure energy compression during energy recovery
11 of 26
JLAB-TN-02-026
24 July 2002
3) recirculator nonlinear compaction: “Arc T566 (W5666, …)” must be consistent
with the RF waveform curvature (torsion, …) so as to ensure complete energy
compression over the full bunch length and to avoid generation of “tails and
halo” at the dump.
As noted above, phasing can be provided by path length adjustment in the linac-to-FEL
transport, gross motion of the mirror-bend achromat, and/or by steering within the
achromat. Linear momentum compaction in a mirror-bend achromat is dictated solely by
bend radius and angle. The desire for a compact, single mirror-bend system with good
betatron properties drives the compaction to a value of order 0.4 m, which, in turn,
dictates the use of RF with a frequency in the vicinity of 750 MHz.
The third constraint has perhaps the greatest impact. Mirror bend achromats are
completely linear, and thus do not provide ready means for compensation of RF
waveform nonlinearities. As discussed earlier, this suggests that RF be of lower
frequency and operate farther off crest, and, further, that a harmonic RF linearizer be
used. We note that such a system must be at an odd multiple of the fundamental to ensure
that it will energy recover – even multiple harmonics (first, third, … at two, four, …
times the fundamental frequency) will be at a full wavelength phase offset when the
fundamental is at a half wavelength delay after recirculation. Assuming further that the
linearizer will be an SRF system as well, our attention will be limited to only the lowest
allowed harmonic (the second – three times the fundamental frequency) inasmuch as
cavity apertures become small at higher frequencies and low-loss transmission of high
currents could therefore become very difficult.
Figure 10 illustrates RF linearization using a simple waveform construction. The curves,
generated with the following expression, represent the energy E recovered by a linac as a
function of phase  at a fundamental frequency with a second harmonic (three times the
fundamental) linearization of amplitude A2 at phase offset.
E   S cos  A2 cos3  2 
The overall scale factor S is included to match the linearized waveform amplitude (shown
in blue) to that of the fundamental (Efundamental()=cos , shown in pink) and its linear
approximation (Elinear= cos 0 - sin 0 ( -0), shown in red) at a selected fundamental
phase offset (“operating phase”) . These expressions are evaluated on the phase interval
(-90o, 360o) for operating phases  0 of –10o (corresponding to 750 MHz RF at the
available compaction of 0.4 m, Figures 10a and b) and –15o (500 MHz at the same
compaction, Figures 10c and d).
12 of 26
JLAB-TN-02-026
24 July 2002
Figure 10: Linearization of RF waveform at operating phase offsets of –10o and –15o.
Figure 10a: 0 = –10o (750 MHz), S = 1.143,
A2=0.125, 2 = 153.4o.
Figure 10b: 0 = –15o (500 MHz), S = 1.145,
A2=0.125, 2 = 140o.
1.5
1.5
1
1
0.5
0.5
RF amplitude
0
-180
-90
0
90
180
270
360
linearized
harmonic
-90
0
-0.5
-1
-1
-1.5
-1.5
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
energy discrepancy
0
90
180
270
360
90
180
270
360
linearized
harmonic
Figure 10d: Deviation of linearized waveform from
linearity at –15o operating phase.
0.02
-90
fundmental
0
-180
-0.5
Figure 10b: Deviation of linearized waveform from
linearity at –10o operating phase.
-180
RF amplitude
fundmental
0
-180
-90
energy discrepancy
0
-0.005
-0.005
-0.01
-0.01
-0.015
-0.015
-0.02
-0.02
90
180
270
360
The amplitude and phase offset of the second harmonic (shown in purple) were chosen to
control the deviation of the synthesized waveform from linearity (Figures 10b and d,
which show E()-Elinear()), keeping it to order ¼ 10-2 of the fundamental amplitude.
In practice, this suggests a 200 MeV linac could be linearized to within ½ MeV after
energy recovery – likely within the acceptance of an aggressively designed beam dump.
At both operating phases, linearization holds the deviation within the specified limits
over a range of about 50o in fundamental RF phase. This phase interval  translates to a
“recovered energy acceptance” via the momentum compaction and RF wavelength.
l
M p / p 

 bunch  56
o
 RF
 RF
360
When evaluated for the two cases under consideration, this gives about 15% relative
momentum acceptance at 750 MHz, and 22.5% at 500 MHz.
13 of 26
JLAB-TN-02-026
24 July 2002
Two aspects of this result are noteworthy. First, lower frequency accepts larger
momentum spread. Second, the transport system does not impose the acceptance limit,
RF phase synchronism does. One could argue that the dump imposes the limit, but even if
a dump with arbitrarily large acceptance were available the bunch length would exceed
the RF wavelength for large enough momentum spread. For example, the mirror-bend
achromat described above could potentially transport an FEL exhaust drive beam with
~70% relative momentum acceptance. Given the 0.4 m compaction, this will produce a
bunch of length of 0.4 m  0.7 = 0.28 m. This exceeds half the wavelength at 750 MHz;
even if linearization were performed using arbitrarily high harmonics, some portion of
the beam (the head and tail 4 cm) could not be recovered. As with beam transport
systems, there are limits to linearization as well!
Results for more aggressive design parameters are shown in Figure 11, wherein the
linearity specification was relaxed to 2 10-2 and the parameters adjusted accordingly.
For 200 MeV initial energy, this yields ~8 MeV energy spread at the dump; the phase
acceptance grows to about 90o for either operating point, suggesting (by the relation
given above) a 25% relative momentum phase acceptance at 750 MHz and 37.5% at 500
MHz, both still well within the transport system acceptance. Superposition of a fourth
harmonic (5fundamental) term could improve these results, but at the cost of
accommodating the small apertures of a high frequency cavity. Through perhaps tolerable
at 500 MHz (fourth harmonic of 2500 MHz), this likely becomes a problem at 750 MHz
where the fourth harmonic is 3750 MHz.
Figure 11: “Relaxed” linearization of RF waveform (deviations at 210-2).
Figure 11a: 0 = –10o (750 MHz), S = 1.178,
A2=0.165, 2 = 153.4o.
Figure 11b: 0 = –15o (500 MHz), S = 1.178,
A2=0.164, 2 = 140o.
1.5
1.5
1
1
0.5
0.5
RF amplitude
-180
-90
RF amplitude
fundmental
0
0
90
180
270
360
linearized
harmonic
fundmental
0
-180
-90
0
-0.5
-0.5
-1
-1
-1.5
-1.5
14 of 26
90
180
270
360
linearized
harmonic
JLAB-TN-02-026
24 July 2002
Figure 11b: Deviation of linearized waveform from
linearity at –10o operating phase.
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
-90
energy discrepancy
0
90
180
270
360
0
-180
-90
energy discrepancy
0
-0.005
-0.005
-0.01
-0.01
-0.015
-0.015
-0.02
-0.02
90
180
270
360
A simple simulation of energy recovery is presented in Figures 12 and 13. A short initial
phase space at 200 MeV with 40 MeV full momentum spread (Figure 12/13a), is
recovered and compressed using a mirror bend achromat with M56 = 0.4 m to rotate the
bunch (Figure 12/13b) and recover it with energy compression using a linac with
embedded second harmonic linearization (Figure 12/13c). Results are given for both
750/2250MHz (Figure 12) and 500/1500 MHz (Figure 13). As may expected from the
initial discussion (see the compaction equations on page 3), the residual energy spread is
a few times smaller at the lower frequency.
Figure 12: Phase space at 750 MHz with 2250 MHz linearization.
Figure 12a: After wiggler
Figure 12b: At reinjection
750 MHz phase space after wiggler
750 MHz phase space at reinjection
250
250
225
225
E (MeV)
E (MeV)
-180
Figure 11d: Deviation of linearized waveform from
linearity at –15o operating phase.
200
200
175
-0.0006
-0.0003
150
0.0000
175
150
0.0003
0.0006
-0.06
s (m )
-0.04
-0.02
0
s (m )
15 of 26
0.02
0.04
0.06
JLAB-TN-02-026
24 July 2002
Figure 12c: After energy recovery.
750 MHz phase space at end of linac
12
10
E (MeV)
8
6
4
2
0
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
s (m )
Figure 13: Phase space at 500 MHz with 1500 MHz linearization.
Figure 13a: After wiggler
Figure 13b: At reinjection
500 MHz phase space at reinjection
250
250
225
225
E (MeV)
E (MeV)
500 MHz phase space after wiggler
200
200
175
175
150
-0.0003
150
0
0.0003
0.0006
-0.06
-0.04
-0.02
s (m )
0
s (m )
Figure 13c: After energy recovery.
500 MHz phase space after linac
8
7.9
7.8
7.7
E (MeV)
-0.0006
7.6
7.5
7.4
7.3
7.2
7.1
-0.06
-0.04
-0.02
0
s (m )
16 of 26
0.02
0.04
0.06
0.02
0.04
0.06
JLAB-TN-02-026
24 July 2002
As a final performance check, we have made a DIMAD simulation of energy recovery.
This modeling is based on ray-tracing an initial distribution with a 25% (roughly
uniformly distributed) initial momentum spread through the beamline with the
quadrupoles subdivided as discussed above to improve the simulation of chromatic
behavior. No subdivision of the bends was performed (for reasons discussed below); the
momentum spread was selected to examine the system behavior for the “more
aggressive” linearization illustrated for 750 MHz in Figure 11a, in which a bunch
subtending up to 90o in phase could be energy-recovered.
Harmonic linearization was not included inasmuch as the DIMAD RF model of
deceleration from ~200 MeV to ~7 MeV is purely linear. Given the asymmetric betatron
acceptance (Figures 7–9 and the associated discussion), we have selected an operational
scenario in which the recirculator is run at an excitation ~10% below the nominal central
energy. The longitudinal phase space is therefore “translated 10% up the energy axis”,
matching the beam momentum distribution more closely to the focusing system
chromatic acceptance. We note that the longitudinal phase rotation performed during
energy recovery maps this translation into a longitudinal (phase) displacement. This can
be operationally compensated by changing the recirculation path length, thereby bringing
the beam centroid into synchronism with the nominal phase.
Initial phase space plots are given in Figure 14, initial distributions in Figure 15;
corresponding presentations at reinjection and after energy recovery are given in Figures
16 through 19. Behavior is generally good, with little (chromatically betatron mismatch
driven) emittance dilution observed. The simulated energy compression is quite effective,
with the initial large spread (~50 MeV) reducing to only a few hundred keV (a few
percent relative spread). It is entertaining to observe the interchange of the l and p/p
distributions as the energy compression rotates the longitudinal phase space during
transport from wiggler to the end of the linac.
We thus conclude that a mirror bend achromat and linac with an RF linearizer can
capture, compress, and recover a beam with quite large momentum spread across a bunch
that upon reinjection is several tens of RF degrees long. Figures 10–13 suggest that
momentum spread of 15–35 %, subtending from ~50o to as much as 90o in RF phase may
be manageable with an appropriate selection of RF frequency. Figures 7–9 and 14–17
indicate betatron behavior remains acceptable during this process and that the large
longitudinal phase space is controlled without undue degradation of transverse beam
quality.
17 of 26
JLAB-TN-02-026
24 July 2002
Figure 14: Initial phase spaces (immediately following wiggler)
yy'
-0.002
-0.001
0.0010
0.0005
0.0005
0.0000
0.000
0.001
0.002
0.003
y' (rad)
-0.003
0.0010
-0.003
-0.002
-0.001
0.0000
0.000
-0.0005
-0.0005
-0.0010
-0.0010
0.001
0.002
0.003
0.001
0.002
0.003
y (m)
x (m)
xy
0.003
0.002
0.001
y (m)
x' (rad)
xx'
-0.003
-0.002
-0.001
0.000
0.000
-0.001
-0.002
-0.003
x (m)
18 of 26
JLAB-TN-02-026
24 July 2002
Figure 15: Initial distributions
initial x' distribution
2500
2500
2250
2250
2000
2000
1750
1750
1500
1500
Nparticle
Nparticle
initial x distribution
1250
1250
1000
1000
750
750
500
500
250
250
0
-0.003
-0.002
-0.001
0
0
0.001
0.002
0.003
-0.001
-0.0005
0
x (m)
2500
2250
2250
2000
2000
1750
1750
1500
1500
Nparticle
Nparticle
2500
1250
-0.002
1000
750
750
500
500
250
250
-0.001
0.001
0.002
0.003
0.004
-0.001
-0.0005
0
y' (rad)
initial p/p distribution
2500
2500
2250
2250
2000
2000
1750
1750
1500
1500
Nparticle
Nparticle
initial l distribution
1250
1250
1000
1000
750
750
500
500
250
250
0
-0.0002
-0.0001
0.001
0
0
y (m)
-0.0003
0.0005
1250
1000
0
-0.003
0.001
initial y' distribution
initial y distribution
-0.004
0.0005
x' (rad)
0
0
0.0001
0.0002
0.0003
l (m)
-0.1
-0.05
0
0.05
0.1
p/p (rad)
19 of 26
0.15
0.2
0.25
0.3
JLAB-TN-02-026
24 July 2002
Figure 16: Phase spaces at reinjection to linac
yy'
-0.002
-0.001
0.0010
0.0005
0.0005
0.0000
0.000
0.001
0.002
0.003
y' (rad)
-0.003
0.0010
-0.003
-0.002
-0.001
0.0000
0.000
-0.0005
-0.0005
-0.0010
-0.0010
0.001
0.002
0.003
0.001
0.002
0.003
y (m)
x (m)
xy
0.003
0.002
0.001
y (m)
x' (rad)
xx'
-0.003
-0.002
-0.001
0.000
0.000
-0.001
-0.002
-0.003
x (m)
20 of 26
JLAB-TN-02-026
24 July 2002
Figure 17: Distributions at reinjection to linac
x' distribution at reinjection
2500
2500
2250
2250
2000
2000
1750
1750
1500
1500
Nparticle
Nparticle
x distribution at reinjection
1250
1250
1000
1000
750
750
500
500
250
250
0
-0.003
-0.002
-0.001
0
0
0.001
0.002
0.003
-0.001
-0.0005
0
x (m)
2500
2500
2250
2250
2000
2000
1750
1750
1500
1500
1250
-0.002
1000
750
750
500
500
250
250
-0.001
0.001
0.002
0.003
0.004
-0.001
-0.0005
0
y' (rad)
p/p distribution at reinjection
l distribution at reinjection
2500
2500
2250
2250
2000
2000
1750
1750
1500
1500
Nparticle
Nparticle
0.001
0
0
y (m)
1250
1250
1000
1000
750
750
500
500
250
250
0
0
-0.025
0.0005
1250
1000
0
-0.003
0.001
y' distribution at reinjection
Nparticle
Nparticle
y distribution at reinjection
-0.004
0.0005
x' (rad)
0
0.025
0.05
0.075
0.1
-0.1
-0.05
0
0.05
0.1
p/p (rad)
l (m)
21 of 26
0.15
0.2
0.25
0.3
JLAB-TN-02-026
24 July 2002
Figure 18: Phases spaces after energy recovery
-0.004
-0.002
0.020
0.020
0.015
0.015
0.010
0.010
0.005
0.005
0.000
0.000
-0.005
0.002
0.004
0.006
y' (rad)
-0.006
yy'
-0.010
-0.005
0.000
0.000
-0.005
-0.010
-0.010
-0.015
-0.015
-0.020
-0.020
x (m)
0.005
0.010
0.005
0.010
y (m)
xy
0.010
0.005
y (m)
x' (rad)
xx'
-0.010
-0.005
0.000
0.000
-0.005
-0.010
x (m)
22 of 26
JLAB-TN-02-026
24 July 2002
Figure 19: Distributions after energy recovery
x' distribution after energy recovery
x distribution after energy recovery
2500
4000
2000
3000
1500
Nparticle
Nparticle
5000
2000
1000
1000
500
0
-0.01
-0.005
0
0
0.005
0.01
-0.02
-0.015
-0.01
-0.005
x (m)
0
0.005
0.01
0.015
0.02
0.015
0.02
0.015
0.02
x' (rad)
y distribution after energy recovery
y' distribution after energy recovery
5000
2500
4500
4000
2000
3500
1500
Nparticle
Nparticle
3000
2500
2000
1000
1500
1000
500
500
0
-0.01
-0.005
0
0
0.005
0.01
-0.02
-0.015
-0.01
-0.005
y (m)
0
0.005
0.01
y' (rad)
p/p distribution after energy recovery
l distribution after energy recovery
2500
2000
2000
1500
1500
Nparticle
Nparticle
2500
1000
1000
500
500
0
0
-0.025
0
0.025
0.05
0.075
0.1
-0.02
-0.015
-0.01
-0.005
0
p/p (rad)
l (m)
23 of 26
0.005
0.01
JLAB-TN-02-026
24 July 2002
Entertaining aside – By construction, the orbit following a mirror-bend achromat is
explicitly momentum independent. It is, further, readily described analytically throughout
the achromat, allowing (as was done above) an exact representation of the transfer matrix
through the beamline at any momentum offset. This provides opportunity to compare the
precision of various approximations to the exact solution. Results of such a comparison
are given in Figures 20 and 21.
Figure 21 presents the central orbit and its slope as a function of momentum offset (each
nominally zero) as calculated with a DIMAD “detailed chromatic analysis” operation
using various levels of subdivision of the dipoles. As seen earlier, the linear and quadratic
(“second order TRANSPORT”) model provides the “correct” answer. However,
subdivision and ray-tracing (“TURTLE mode”) modeling, such as that done in this
DIMAD computation, accumulates cross-terms amongst the elements, producing higher
order effects. In this case, they are completely spurious and would be cancelled by the
higher-order terms in the transfer maps of the individual elements, were these not
truncated at second order. It is interesting to realize, however, that in cases with highly
symmetric geometry (such as this one), it is thus possible to compute the “missing”
higher order terms of the transfer map by simply noting they must cancel the observed
residual orbit and slope.
The convergence of the integrative tracking method is characterized by repeating the
calculation at various levels of subdivision. The result is approaching stability at 30 steps
per dipole, and little difference is seen between 100 and 300 steps per dipole. These
results suggest the method is accurate to a few millimeters over a 20% momentum
bandwidth.
3.00E-03
2.50E-03
2.00E-03
x, 10 subdivisions
x' 10 subdivisions
x, 30 subdivisions
1.50E-03
x', 30 subdivisions
x, 100 subdivisions
1.00E-03
x', 100 subdivisions
x, 300 subdivisions
5.00E-04
x', 300 subdivision
x, exact
-0.15
-0.1
0.00E+00
-0.05
0
x', exact
0.05
0.1
0.15
-5.00E-04
-1.00E-03
Figure 21: Comparision of DIMAD and exact computation of central orbit behavior as a
function of momentum.
24 of 26
JLAB-TN-02-026
24 July 2002
Figure 22 presents the analogous calculation for propagated beam envelopes. As in
Figure 9, agreement is at the few tens of percent level. Horizontal variations differ little
amongst the various DIMAD trials; vertical results degrade (but rapidly become stable)
with the use of subdivision.
3.5
3
exact dbx/bx vs d
exact dby/by vs d
2.5
DIMAD dbx/bx 10 steps
DIMAD dby/by 10 steps
2
DIMAD dbx/bx 30 steps
DIMAD dby/by 30 steps
1.5
DIMAD dbx/bx 100 steps
DIMAD dby/by 100 steps
1
DIMAD dbx/dx 300 steps
DIMAD dby/by 300 steps
0.5
DIMAD dbx/bx no subdivision
DIMAD dby/by no subdivision
0
-0.1
-0.08 -0.06 -0.04 -0.02
0
0.02
0.04
0.06
0.08
0.1
-0.5
Figure 22: Beam envelope variation with momentum as computed “exactly” and with
DIMAD integration through variously subdivided dipoles.
Conclusions and Comments
We have described a novel design paradigm for FEL driver energy recovery systems –
the use of a large acceptance, completely linear, mirror-bend-achromat-based transport –
that moves acceptance limitations from the magnetic transport (beam line) to the RF. To
ensure effective energy compression during energy recovery (via active curvature
compensation), the RF system must as a result include a third harmonic linearizer. This
motivates the use of relatively low RF fundamental frequency.
We note that this approach potentially provides better performance (larger acceptance,
better transmission, etc.) than the use of “linear RF, nonlinear transport (with passive DC
magnetic precompensation of RF curvature and torsion)”. It may not, however, be cost
effective in all circumstances. Unless relatively high FEL extraction efficiency is required
(several percent, with attendant tens of percent exhaust energy spread) the expense
associated with large low frequency RF systems and RF linearization may exceed the
cost of a conventional single-frequency machine based on a somewhat more complex,
lower performance transport system, even when nonlinear corrections are included.
Drivers designs of the type under consideration here thus are likely most applicable for
25 of 26
JLAB-TN-02-026
24 July 2002
systems intended to produce extremely high powers or function within a very small
footprint.
Acknowledgments
I would like to thank Charlie Brau and Todd Smith for stimulating this investigation with
their discussions and advice during the Spring 2002 FEL Upgrade semiannual review. I
would also like to thank Byung Yunn for his assistance with the mirror bend achromat,
and Bob Legg, George Biallas, George Neil, and Fred Dylla for useful discussions on,
and considerable encouragement regarding, this design. Mike Hughes of AES provided
tremendous assistance by discovering an error, and enabling its correction, in the 750
MHz cavity model.
References
[1]
D. Douglas, “A Nearly Isochronous Arc With Unlimited Momentum
Acceptance”, JLAB-TN-02-020, 30 May 2002.
[2]
D. Douglas, “Longitudinal Phase Space Management in the IR Upgrade FEL
Driver”, JLAB-TN-00-020, 13 September 2000.
[3]
D. H. Dowell, T. D. Hayward, and A. M. Vetter, “Magnetic Pulse Compression
Using a Third Harmonic RF Linearizer”, Proc. PAC95.
[4]
The 5-cell 1497 MHz CEBAF cavity Excel model was scaled to 748.5 MHz. The
first and final cavities were run at energy gains of 10 MeV (gradients of 10
MV/m) to alleviate over-focusing of the beam at injection and at the end of the
linac during energy recovery. The other 8 cavities were run at 20 MeV (gradients
of 20 MV/m).
[5]
D. Douglas, “Design Considerations for Recirculating and Energy Recovering
Linacs”, JLAB-TN-00-027, 13 November 2000.
26 of 26