Ring Beam Experiment

Mike SyphersFNAL

A0 Photoinjector Review 2008 Aug 18

MotivationHas been examined...

- flat-beam transformation

- trans-long transformation

Using another suggestion of Derbenev, can we ...

- make the flat beam smaller in the bigger dimension?

- ameliorate space charge effects by making a ring at the cathode?

A Possible ScenarioUse photoinjector with masked laser, say, to produce “ring” on the cathode to mitigate space charge

pass through the flat beam transformer

- this produces “no emittance” in one plane, and a “ring” in the phase space of the other d.o.f.

inject into “new device” to reduce the larger dimension

perform x-z exchange if desired

... could give much flexibility to a facility

S.C. Force within a ringassume uniform density

make ring of ave. radius a0, thickness da, with same area as circle of radius a (<a0); i.e., choose:

space charge force at radius a of smaller circle is

s.c. force at radius a0+da/2 of the “ring” is

So, might expect a reduction by factor...

2!a0da = !a2

Fring

Fcircle=

2da

a= 2

da

a0

a0

a=

a2

a20

a0

a=

a

a0a

a0

x

y

F ! e2!

2"2#0

r2 " b2

r, b < r < b + da

F ! e2!

2"2#0· 2da

F ! e2!

2"2#0a

Flat Beam x-form of a “ring”

!X0 =

!

""#

x0

0y0

0

$

%%& =

!

""#

r0 cos "0

0r0 sin "0

0

$

%%& !Xs =

!

""#

1 0 0 00 1 !k 00 0 1 0k 0 0 1

$

%%& !X0 =

!

""#

x0

!k y0

y0

k x0

$

%%& .

!Xf = R!1McR !Xs R =!

22

·

!

""#

1 0 "1 00 1 0 "11 0 1 00 1 0 1

$

%%& , Mc =

!

""#

1 0 0 00 1 0 00 0 0 !0

0 0 "1/!0 0

$

%%&

!Xf =

!

""#

x0 ! y0

k(x0 + y0)00

$

%%& =

!

""#

r0(cos "0 ! sin "0)kr0(cos "0 + sin "0)

00

$

%%& =

!

""#

rc cos "c

krc sin "c

00

$

%%& .

Initial Conditions: Exiting solenoid field:

Through a skew quad channel:

Final result:

Flat Beam x-form of a “ring”

!X0 =

!

""#

x0

0y0

0

$

%%& =

!

""#

r0 cos "0

0r0 sin "0

0

$

%%& !Xs =

!

""#

1 0 0 00 1 !k 00 0 1 0k 0 0 1

$

%%& !X0 =

!

""#

x0

!k y0

y0

k x0

$

%%& .

!Xf = R!1McR !Xs R =!

22

·

!

""#

1 0 "1 00 1 0 "11 0 1 00 1 0 1

$

%%& , Mc =

!

""#

1 0 0 00 1 0 00 0 0 !0

0 0 "1/!0 0

$

%%&

!Xf =

!

""#

x0 ! y0

k(x0 + y0)00

$

%%& =

!

""#

r0(cos "0 ! sin "0)kr0(cos "0 + sin "0)

00

$

%%& =

!

""#

rc cos "c

krc sin "c

00

$

%%& .

Initial Conditions: Exiting solenoid field:

Through a skew quad channel:

Final result:

Turn “ring” into “bunch”Derbenev noted the process in NIM-A 441 (2000) 221-233.

Akin to adiabatic capture in synchrotron RF systems (h=1)

Assume a focusing channel or ring, with a repeat period of length

L that has phase advance near, but just below, 2π;

- Use octupole fields within the channel to give an amplitude dependent tune shift; adjust octupole to produce phase advance

of exactly 2π at an amplitude of a0 (which will be the radius of the initial ring distribution in phase space).

- Then, introduce a gradually increasing dipole field at the end of each repeat period (could be distributed, w/ proper phasing)

Parameterizationsmall amplitude phase advance of channel: 2π - ∆φ

phase shift with amplitude due to octupole field...

- do not want octupole field to contribute to ∆a; only ∆ψ; then, yields ∆a = 0

- define , then

dipole kick:

total kick (depicting ramped field):

resulting Hamiltonian:

!8 ! (B!!!/6)0 a30L/(B!)

(See details in Edwards/Syphers note)

!!8 =34

"#0

L

#0"8

a30

(a2 ! a20)

!2 ! B0L/(B!)

!B8 =!

B!!!(s)6

"x3 =

!!0

!

"3 !B!!!

6

"

0

x3

!2 · b, where 0 < b < 1

H = k!3!816 (r2 ! 1)2 !!2br cos !

"

k ! !"0L

"0a0

r ! a/a0

Resulting Hamiltonianfixed Points at ψ = 0 (stable), ψ = π (unstable)

for full “b”, island at r ~ 1 (i.e., a~a0) of width:

island “tune”:

- must be sufficiently small, compared with unity, for adiabatic process

w = 2!

83

!2!8

"1/2

!s !!

332!2 k2!2!8b

"1/2

Expected Phase Spaceuse as input parameters:

- given L, determine β0

- scale dimensions w/ a0

- choose ∆φ, w

- from which:

•

•

-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4

-2

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

2

!8 = 4La03!"2

0"!

!2 = 332!8w2

x/a0

!0x!/a0

Expected Phase Spaceuse as input parameters:

- given L, determine β0

- scale dimensions w/ a0

- choose ∆φ, w

- from which:

•

•

-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4

-2

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

2

!8 = 4La03!"2

0"!

!2 = 332!8w2

x/a0

!0x!/a0

w

Expected Phase Spaceuse as input parameters:

- given L, determine β0

- scale dimensions w/ a0

- choose ∆φ, w

- from which:

•

•

-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4

-2

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

2

!8 = 4La03!"2

0"!

!2 = 332!8w2

x/a0

!0x!/a0

w

Expected Phase Spaceuse as input parameters:

- given L, determine β0

- scale dimensions w/ a0

- choose ∆φ, w

- from which:

•

•

-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4

-2

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

2

!8 = 4La03!"2

0"!

!2 = 332!8w2

x/a0

!0x!/a0

w

A Device?How would one make a device?

- FODO channel -- how long?

• communications from others who have looked at this -- "process takes long time"

- use FODO ring? -- how big?

- try weak focus ring?

• easiest to quickly examine...

Numerical Simulationassume weak focusing, circumference L

- tune = √(1-n); as an example, choose n such that tune

= 15/16, ∆φ = 2π/16

take u = x/a0, v = β0 x’/a0 ; choose w = 0.5

divide ring into m sections, at the center of which an octupole kick is received:

- ∆v = -3/8 ∆φ/m u3

after m sections, s = L and dipole kick is provided:

- ∆v = 1/4 ∆φ w2 b, 0<b<1 (ramped dipole)

Result

adiabatically ramp the dipole

-2 -1 0 1 2

-2-1

01

2

u

vrev= 0

-2 -1 0 1 2

-2-1

01

2

u

v

rev= 10000

0 2000 4000 6000 8000 10000

0.0

0.2

0.4

0.6

0.8

1.0

revolution number

dip

ole

pro

file

0 2000 4000 6000 8000 10000

0.0

0.2

0.4

0.6

0.8

1.0

revolution number

u_

rms (

red

), v

_rm

s (

bla

ck)

initialdistribution

“final”distribution

rms of distribution

position (red)angle (black)

Animation

Parameters of presumed ring:

L = 2 mB0 = 0.314 T

ν = Δφ/2π = 15/16

β0 = 0.34 mpc = 30 MeVa0 = 4 mm

w = 0.5

Θ8 = 12.3 mr

Θ2 = 0.289 mrB2L = 0.03 T-mm

u = x/a0

v = β0 x’/a0

Animation

Parameters of presumed ring:

L = 2 mB0 = 0.314 T

ν = Δφ/2π = 15/16

β0 = 0.34 mpc = 30 MeVa0 = 4 mm

w = 0.5

Θ8 = 12.3 mr

Θ2 = 0.289 mrB2L = 0.03 T-mm

u = x/a0

v = β0 x’/a0

Animation

Parameters of presumed ring:

L = 2 mB0 = 0.314 T

ν = Δφ/2π = 15/16

β0 = 0.34 mpc = 30 MeVa0 = 4 mm

w = 0.5

Θ8 = 12.3 mr

Θ2 = 0.289 mrB2L = 0.03 T-mm

u = x/a0

v = β0 x’/a0

-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4

-2

-1.6

-1.2

-0.8

-0.4

0.4

0.8

1.2

1.6

2

RemarksType of device

- if channel, very very long

- if ring, worry about bunch lengthening (isochronous ring?), chromatic effects, etc.

Possible Further Studies

- distribute dipole field along channel/ring

- how far from adiabaticity can be tolerated?

With Liouville in mind, only makes sense if a “ring beam” off the cathode helps with space charge effects

RemarksCould make nice student project; “too large” (?) for one student, if large-scale ring or channel is involved; could be enough work for several students

- needs further thought -- can be scaled down?

References

[1] Ya. S. Derbenev, “Advanced optical concepts for electron cooling”, Nucl.Instr. and Meth. A 441 (2000) 221-233. See the discussion beginning at thebottom of page 226.

[2] D. A. Edwards, M. J. Syphers, “Introduction to the Physics of High EnergyAccelerators”, J. Wiley and Sons, 1993. The material used here begins onpage 116 of this textbook.

[3] Private communications from R. Brinkmann, K. Flottmann, S. Nagaitsev

[4] D.A. Edwards, M.J. Syphers, “Comments on an Experiment on TransverseCoalescing,” unpublished.