USPAS 2005 Recirculated and Energy Recovered Linacs 1&+(66���/(33&+(66���/(33
USPAS Course on
Recirculated and Energy Recovered Linacs
I. V. Bazarov
Cornell University
D.R. Douglas, G. A. Krafft, and L. Merminga
Jefferson Lab
Computer Class: Linear Optics in JLAB IRFEL,
Longitudinal gymnastics, and Beam Break-up modeling
USPAS 2005 Recirculated and Energy Recovered Linacs 2&+(66���/(33&+(66���/(33
JLAB IRFEL
Spreadsheet model of JLAB IRFEL includes:
- full first-order optics
- longitudinal phase space visualization
- beam break-up simulations
USPAS 2005 Recirculated and Energy Recovered Linacs 3&+(66���/(33&+(66���/(33
Obtaining the lesson
Download the spreadsheet and bbu code to a single write-
enabled directory from
http://www.lns.cornell.edu/~ib38/uspas05/irfel/
Make sure macros are enabled. Start the spreadsheet (note: it
may take a while to initialize all formulas).
USPAS 2005 Recirculated and Energy Recovered Linacs 4&+(66���/(33&+(66���/(33
Spreadsheet organizationThe spreadsheet is organized into three main parts
---> elements
matrices
products
twiss
---> bbu
bbu_latfile
bbu_homs
bbu_param
---> long_phase_space
R56
z
pz
layout and lattice control
beam breakup simulation
longitudinal phase space
USPAS 2005 Recirculated and Energy Recovered Linacs 5&+(66���/(33&+(66���/(33
Organization: layout and lattice
---> elements sheet contains optics controls
USPAS 2005 Recirculated and Energy Recovered Linacs 6&+(66���/(33&+(66���/(33
Spreadsheet: beam break-up
---> bbu controls execution of beam break-up code
USPAS 2005 Recirculated and Energy Recovered Linacs 7&+(66���/(33&+(66���/(33
Spreadsheet: longitudinal phase space
---> long_phase_space tracks a bunch in the long. p.s.
USPAS 2005 Recirculated and Energy Recovered Linacs 8&+(66���/(33&+(66���/(33
Longitudinal Phase Space Manipulations
USPAS 2005 Recirculated and Energy Recovered Linacs 9&+(66���/(33&+(66���/(33
A brief overview of longitudinal dynamics
• first- and second-order correlation in longitudinal phase
space
• second-order momentum compaction
• requirements for energy recovery
USPAS 2005 Recirculated and Energy Recovered Linacs 10&+(66���/(33&+(66���/(33
First- and second-order correlations
2
02
1ll δδ βαδδ ++≅
�+
∂∂
+
∂∂
+=
==
2
0
2
2
0
0!2
1l
ll
lll
δδ
δδ
l
42
21222
0 ll σβσασσ δδδδ ++= 42
212
- 0 lll σβσσε δδδ +=
ϕβδ cos2
RF
final
linac kE
E
−=ϕαδ sinRF
final
linac kE
E
−=
GHz 1.5for m 5.31/2 1-
RFRFk == λπ
USPAS 2005 Recirculated and Energy Recovered Linacs 11&+(66���/(33&+(66���/(33
After acceleration
after the main linac:
assuming large Efinal/Einjection and small energy spread
energy spread: longitudinal emittance:
for
for
lσασ δδ ≈
3
2
1- ll σβε δδ ≈
ϕαδ RFk−≈
2
RFk−≈δβlRFk σϕ
2
1>
2
2
1lσβσ δδ ≈ lRFk σϕ
2
1<
USPAS 2005 Recirculated and Energy Recovered Linacs 12&+(66���/(33&+(66���/(33
Longitudinal transform
2
56656 δδ TRll ++=∗
δδ =∗ ⇒ δ
δ
δ αα
α
561 R+=∗
3
56
3
566
)1(
2
δδδ
δ ααβ
βR
T
+−
=∗
momentum compaction (times the path length):
∫= dsR
ρη
56
second-order momentum compaction:
∫
′++= dsT
22
22)2(
566
η
ρη
ρη
∫ ′+′++= dsyxxL222)/1( ρ
δ
l
USPAS 2005 Recirculated and Energy Recovered Linacs 13&+(66���/(33&+(66���/(33
Compression
for maximum compression need
ϕαδ RFkR
1156 ≈−=
for maximum compression need
335662
1
2 ϕαβ
δδ
RFkT ≈=
δ
l
actual (absolute) value of can be smaller566T
22222,566 22 ϕσσ
σασ
δ
σ
lRF
comp
l
l
comp
l
kT comp
l≈=∆
no is needed beyond a certain off-crest phase angle566T
2
2
0566
RF
comp
l
lT
k
σσ
ϕϕ =>
=
USPAS 2005 Recirculated and Energy Recovered Linacs 14&+(66���/(33&+(66���/(33
Achieving the right values of R56
trim quads
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-30 -20 -10 0 10 20 30
R56 [cm]
qu
ad
s s
tre
ng
th [
1/m
**2
]
Q1
Q2
∫=
2
1
56 dsR x
ρη
Q1Q2
Adjustable R56
USPAS 2005 Recirculated and Energy Recovered Linacs 15&+(66���/(33&+(66���/(33
Bunch length in the Arcs:
For off-crest of several deg:
0
0.5
1
1.5
2
2.5
3
3.5
4
0 2 4 6 8 10 12 14 16
position [m]
rms
bu
nc
h l
en
gth
[m
m]
phi = -10
phi = 0
phi = 10
2nd term
R56 = 14.4 cm
(trim quads off)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 5 10 15
position [m]
R5
6 [
m],
eta
x [
m]
R56
eta x
2
2
56
2
0,
2
, msecond_ter1 +
∂∂
+= Rl
ltl
δσσ
156 ≥
∂∂
Rl
δ
eta x
R56
2nd term
phi = 10° phi = -10°
on crest
Bunch length in the Bates’
USPAS 2005 Recirculated and Energy Recovered Linacs 16&+(66���/(33&+(66���/(33
∫
′++= dsT
22
22)2(
566
η
ρη
ρη
ηηηηηηηηη 22
212
1
32
221
1)2()2( 2)2()( hhhhkhkkhsK +′′+′+++−+−=+′′
Achieving the right values of T566
changing sextupoles
strength in the Arc…
sextupoles
USPAS 2005 Recirculated and Energy Recovered Linacs 17&+(66���/(33&+(66���/(33
Energy recovery
ll∆=∆ δβδ
l
δ
δσ56~ Rl∆ 2
566~ δσTl∆dumpδσδ ~∆
3256 ~l
dumpR
σβσ
δδ
53566 ~l
dumpT
σβσ
δδ
1
2
3
General rule of thumb for successful energy recovery is having
the full recirculating arc isochronous to first and second order (R56
= T566 = 0).
In IRFEL, the main difficulty is an additional energy spread
generated at the wiggler due to FEL interaction.
USPAS 2005 Recirculated and Energy Recovered Linacs 18&+(66���/(33&+(66���/(33
Controlling Bates’ quads
---> elements sheet, yellow region
1st Bates’ 2nd Bates
USPAS 2005 Recirculated and Energy Recovered Linacs 19&+(66���/(33&+(66���/(33
Off-crest phase, FEL energy spread and T566
---> long_phase_space sheet
1st Bates’
2nd Bates
FEL energy spread
Off-crest angle
USPAS 2005 Recirculated and Energy Recovered Linacs 20&+(66���/(33&+(66���/(33
1) Set the off-crest phase angle in the main linac to 0 and ‘turn-off’ laser
interaction. Observe how the longitudinal phase space looks throughout
the accelerator and at the beam dump.
2) Set the off-crest phase angle in the main linac to –10°. Achieve the
shortest bunch possible at the wiggler location using linear optics only
(T566 should be 0). Compare calculated R56 with the value in the model.
3) Use T566 to maximally compress the bunch at the wiggler. Compare
calculated T566 with the value in the spreadsheet. How much shorter is the
bunch length when both second- and first-order compaction is used, as
opposed to only the first-order compression? Achieve less than 150 fs rms
bunch duration.
4) ‘Turn-on’ the laser interaction (actual max. energy spread of 5 %).
Observe the longitudinal phase space at the dump. Is the beam being
successfully recovered?
Exercises
USPAS 2005 Recirculated and Energy Recovered Linacs 21&+(66���/(33&+(66���/(33
5) Adjust R56 in the second Bates’ section to minimize energy spread at
the dump. Note the smallest energy spread you were able to achieve.
6) Use T566 to minimize energy spread at the dump. Note the values of R56
and T566 of the whole recirculating arc that allowed the result. What is the
smallest energy spread you were able to achieve? Achieve less than 15 %
max energy spread at the dump.
Exercises (contd.)
USPAS 2005 Recirculated and Energy Recovered Linacs 22&+(66���/(33&+(66���/(33
Beam break-up analysis
USPAS 2005 Recirculated and Energy Recovered Linacs 23&+(66���/(33&+(66���/(33
Higher order modes
Two basic concerns:
monopole (m = 0)
TM01-likey
z
E B
x
y
high energy losses, no kick
y
z
E B
x
yTM11-like
dipole (m = 1)
kick and losses when
beam is not centered
y
z
E B
x
yTM21-like
quadrupole (m = 2)
kick, coupling and losses
when beam is not centered
•Multipass beam breakup (dipoles)
•Resonant excitation of a higher order mode (monopoles)
USPAS 2005 Recirculated and Energy Recovered Linacs 24&+(66���/(33&+(66���/(33
BBU threshold for a single dipole mode
≈ , is independent of
YES
YES
NO
YES
NO NOuse code!
USPAS 2005 Recirculated and Energy Recovered Linacs 25&+(66���/(33&+(66���/(33
Controlling BBU code
---> bbu controls execution of beam break-up code
output voltage and
coordinate at a given
cavity (must have a
HOM!)
controls length
(time) of code
execution
USPAS 2005 Recirculated and Energy Recovered Linacs 26&+(66���/(33&+(66���/(33
Controlling BBU code: HOMs
bbu_homs spreadsheet
comments are denoted by
‘!’ and are ignored; use it
to ‘disable’ HOMs
HOM data that goes
into BBU code
USPAS 2005 Recirculated and Energy Recovered Linacs 27&+(66���/(33&+(66���/(33
1) By commenting out modes in bbu_homs sheet, determine the worst mode
(the one with highest threshold). How does the threshold due to the single worst
mode compares to the situation when all modes are present?
2) Work with the worst offending mode (for faster computing speed). Slightly
change the frequency of the mode and obtain dependence of threshold vs. the
mode frequency. Plot the dependency. What is the ratio of max over min
threshold that you found in this manner? What is the frequency difference
between the two adjacent maxima?
3) Add ‘fake’ 1000 m to the recirculation length (---> bbu sheet) and repeat
the steps from 2). What is the ratio of max over min threshold in this case? What
is the frequency difference between the two adjacent maxima? Try to explain the
result.
4) Enable all modes. By changing quads in the arc (in a FODO-like section, sheet ---> elements) try to increase the threshold as much as possible. Compare
your highest threshold with others in the group. The winner gets a prize!
Exercises