uspas course on recirculated and energy recovered linacsib38/uspas05/computer2.pdf · &+(66...

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


JLAB IRFEL

Spreadsheet model of JLAB IRFEL includes:

- full first-order optics

- longitudinal phase space visualization

- beam break-up simulations


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).


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


Organization: layout and lattice

---> elements sheet contains optics controls


Spreadsheet: beam break-up

---> bbu controls execution of beam break-up code


Spreadsheet: longitudinal phase space

---> long_phase_space tracks a bunch in the long. p.s.


Longitudinal Phase Space Manipulations


A brief overview of longitudinal dynamics

• first- and second-order correlation in longitudinal phase

space

• second-order momentum compaction

• requirements for energy recovery


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


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<


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


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 =>

=


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


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’


∫

′++= 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


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.


Controlling Bates’ quads

---> elements sheet, yellow region

1st Bates’ 2nd Bates


Off-crest phase, FEL energy spread and T566

---> long_phase_space sheet

1st Bates’

2nd Bates

FEL energy spread

Off-crest angle


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


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.)


Beam break-up analysis


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)


BBU threshold for a single dipole mode

≈ , is independent of

YES

YES

NO

YES

NO NOuse code!


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


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


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

uspas course on recirculated and energy recovered linacsib38/uspas05/computer2.pdf · &+(66...

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