JLab Accelerator Seminar, June 3, 2010 1

Twin-Helix Magnetic Channel for

Parametric-resonance Ionization Cooling

V.S. Morozov1,2, A. Afanasev2,3, Y.S. Derbenev4, R.P. Johnson2

1 Old Dominion University 2 Muons, Inc.

3 Hampton University4 Jefferson Lab

Muons, Inc.

Outline

• Concept of Parametric-resonance Ionization Cooling (PIC)

• PIC linear optics requirements

• Epicyclic channel for PIC

• Twin-helix channel for PIC

– Magnetic optics design

– G4beamline simulations

– Possible practical implementation

• Approach to compensating aberrations

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 2

LEMC Scenario

R.P. Johnson - Dec. 9, 2008 MC Design Workshop JLab 3

Muons, Inc.

Bogacz Dogbones

Scheme

New Fernow-Neuffer plot

ν = 0.325 GHzλ = 1.0 – 0.8 m

ν = 0.65 GHzλ = 0.5 – 0.3 m

ν = 1.3 GHzλ = 0.3 m

100 % @ z = 0 m

97 % @ z = 40 m91 % @ z = 49 m

89 % @ z = 129 m

84 % @ z = 129 m

84 % @ z = 303 m

• GH2 pressure = 160 atm

• 60 μm Be RF window

• E ~ 27 MV/m

• Detailed parameter will be given in later slide (slide 15)

PIC

REMEX

K. Yonehara 12/02/09 4

Goal phase space

Study2a

PIC + REMEX = factor of ~100 in luminosity critical for LEMC feasibility

PIC Concept

• Parametric resonance induced in muon cooling channel

• Muon beam naturally focused with period of free oscillations

• Wedge-shaped absorber plates combined with energy-restoring RF cavities

placed at focal points (assuming aberrations corrected)

– Ionization cooling maintains constant angular spread

– Parametric resonance causes strong beam size reduction

– Emittance exchange at wedge absorbers produces longitudinal cooling

• Resulting equilibrium transverse emittances are an order of magnitude

smaller than in conventional ionization cooling

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 5

Absorber Stabilizing RoleMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 6

No absorbers

AbsorbersOptics to restore parallel beam envelope

PIC Schematic

• Equilibrium angular spread and beam size at absorber

• Equilibrium emittance

(a factor of improvement)

w

Absorber plates Parametric resonance lenses

/8

x x

xx

xx const

2

2

3 ( 1)

2

ea

mZ

m

1

2 3a aw

3( 1)

4

en

mZ w

m

3 2 3

acc

abs

w

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 7

• Horizontal free oscillations’ period x equal to or low-integer multiple of

vertical free oscillations’ period y

• Oscillating dispersion– small at absorbers to minimize energy straggling

– non-zero at absorbers for emittance exchange

– large between focal points for compensating chromatic and spherical aberrations

• Correlated optics: correlated values of x, y and dispersion period D

– x = n y = mD , e.g. x = 2 y = 4D or x = 2 y = 2D

• Fringe-field-free design

PIC Optics RequirementsMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 8

Helical Harmonics

• Practical fringe-field-free approach

• Periodic solutions of source-free Maxwell equations in vacuum

• Harmonic of order n given by

• Total field

Muons, Inc.

0, 0B B

0

1 1

1 1 010

0

2( , , ) [ ( ) ( )]cos( [ ])

n

n n n

n n

n nnkz

BB z I nk I nk n kz

nk

0

1 1

1 1 010

0

2( , , ) [ ( ) ( )]sin( [ ])

n

n n n

n n

n nnkz

BB z I nk I nk n kz

nk

0

1 1

010

0

2( , , ) 2 ( )cos( [ ])

n

n n n

n n

z nnkz

BB z I nk n kz

nk

n

n

B B

JLab Accelerator Seminar, June 3, 2010 9

Epicyclic Channel

• Two uniform (non-Maxwellian) transverse helical fields with wave numbers

k1 and k2

• Equation of motion:

• Analytic solution under approximation kc = const (pz = const)

• Dispersion function containing two oscillating terms

• Condition for dispersion to periodically return to zero

•

1 2( ), /T c T c z zp ik p i b b k eB p c

1 2

1 2 1 2

ik z ik zb b B e B e

1 2 1 1 2 2

1 2 1 2

/ /,T

c c z c c

b b b k b kip u x iy

k k k k p k k k k

Muons, Inc.

1 2

2 2

1 2( ) ( )c c

b buD p

p k k k k

2

221 2

1 1

,c

c

k kBk k

B k k

JLab Accelerator Seminar, June 3, 2010 10

1 2 2 1 0 0/ 2 / 9, [2 cos( ) sin( )], sin( )ck k k k B B u u i kz kz D u kz

Oscillating Dispersion

p→p+Δp

1 20, 0 (HCC)B B 1 2

1 2

0, 0 (Epicyclic)

/ 2c

B B

k k k

aberration correction

absorbers

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 11

HCC-Based Approach

to Designing Correlated Optics

• Since there is no exact analytic solution for two Maxwellian helices,

start with single helix considering second helix perturbation

• Using available analytic solution for dynamics in single Maxwellian helix,

adjust desired free-oscillation period ratio - /+ = 1 or 2 for primary helix

• By choosing wave number k2 of second helix, set dispersion oscillation period

D = |2/(k2-k1)| such that + /D = 2

• Adjust strength of second helix to create oscillating dispersion

• Iteratively adjust - /+ and + /D by changing helices’ parameters until

correlated optics is achieved

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 12

Single Helix

• Equilibrium condition

• Orbit stability condition

• Betatron tunes

• For given r = Q+/Q- , one can solve for ∂b/∂a if

2, , 1

1 1

cT

z

kpb qq

B q p k

22

1 2

2

1ˆ0 ( ) 14 1

qG q g D R

2 2 2 3/2

1

2 2

(1 ) (1 )ˆ ,1

q bD g g

pk a

Muons, Inc.

2 2Q R R G

42 2 2 2 2 2 2 2 2 2 2( 1) [2(1 ) ( 1) ] ( )( 1) 0

4r q r r q q r r q

JLab Accelerator Seminar, June 3, 2010 13

• No solution for Q+ = Q-

• Two solution regions for Q+ = 2Q-

– | | << 1, -2 < q < -1, B2/B1 ~ 1

– | | >> 1

Choose:

= -5.4

q = -1.54

Bsol = 2 T

bd = -0.154 T

bq = 0.065 T/m

kc = 32.9 m-1

k = -61.0 m-1

Q- = 0.464, Q+ = 0.929

B2/B1 ~ 0.04

Adjusting Betatron TunesMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 14

• From one-period linear transfer matrix in terms of canonical coordinates

• Track particle over many periodsand take Fourier transform of coordinate vector component

1 1

11 1

2 2

2 21

1 2

0, ,

0

1 1cos(2 ) tr , cos(2 ) tr

2 2

n n

u uu u A E A

M M X M X Mu u F B Bu u

Q A Q B

Determining Betatron Tunes

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 15

Finding Periodic Orbit

• No exact analytic solution in case of two helices

• Stable periodic orbit does not always exit

• Begin with single helix where stable periodic orbit is known to exist

• Use one or combination of the following to find periodic orbit when second

helix is present

– Adiabatically increase strength of second helix while tracking orbit

– Use “friction” force making particle trajectory converge to periodic orbit

– Increase second helix’s strength from zero in small steps while iteratively

determining periodic orbit on each step by locating fixed point in phase space

Muons, Inc.

[ ]( )

z

z

p e B Q Q ee

JLab Accelerator Seminar, June 3, 2010 16

Adiabatic Turn On

of Secondary Helix

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 17

Friction Force Approach• Introduce effective “friction” force

– total energy conserved while phase volume is not

– analogous to cooling

– all trajectories converge towards periodic orbit

– aids in finding periodic orbit when analytic solution is not available

[ ]( )

z

z

p e B Q Q ee

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 18

Dispersion in Epicyclic Channel

Muons, Inc.

Second helix strength

|D| = const |D| oscillates not reaching 0 |D| oscillates reaching 0

JLab Accelerator Seminar, June 3, 2010 19

Periodic Orbit

in Epicyclic Channel

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 20

• Problem with dynamic aperture

• Consider two dipole (n = 1) harmonics with b1= b2 and k1 = -k2 = 2/

• Vertical field only in horizontal plane Periodic orbit in horizontal plane

• Horizontal and vertical motion uncoupled

• Region of stable transverse motion in both planes• D = x = 2y = 4 x = 0.25, y = 0.5

Twin HelixMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 21

Periodic Orbit and TunesMuons, Inc.

• Two dipole helical harmonics only

• Periodic orbit determined by locating fixed point in phase space

• Betatron tunes from linear transfer matrix for canonical coordinates

• y = 1 parametric resonance use x = 0.25, y = 0.5

JLab Accelerator Seminar, June 3, 2010 22

Adjusting Correlated OpticsMuons, Inc.

• Introduce straight quad to redistribute horizontal and vertical focusing

• Down side: cannot satisfy correlated optics conditions for both charges

• Iteratively adjust Bd and By/x until correlated optics is reached

• No success applying same procedure with double helical quad

JLab Accelerator Seminar, June 3, 2010 23

Periodic OrbitMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 24

Dispersion and ChromaticityMuons, Inc.

• Dispersion:

• Chromaticity:

• Scaling pattern:

max / 0.098 mx aD p x p

/ 0.646, 0.798x x yp p

2/ , / / , , , , constd y a x x yB p B x p x D

JLab Accelerator Seminar, June 3, 2010 25

G4BL ImplementationMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 26

G4BL Simulation

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 27

G4BL Optics TestMuons, Inc.

Going In Coming Out

JLab Accelerator Seminar, June 3, 2010 28

• No absorber and no RF

• 105 100 MeV/c - through 100 periods of “twin helix” with correlated optics

• Initially parallel beam uniformly distributed with 10 10 cm square

Final Phase SpaceMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 29

Transverse Motion(relative to reference particle)

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 30

• 100 MeV/c - beam from single point with uniform 0.1 0.1 rad angular spread

3 cm

3 cm

Possible Practical ImplementationMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 31

Layer of positive-helicity

helical conductors with cosazimuthal current dependence

Layer of negative-helicity

helical conductors

Normal quad

Possible Practical ImplementationMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 32

Layer of positive-tilted loops

with cos z longitudinal current

dependence

Layer of negatively-tilted

loops

Normal quad

• Adopt existing technology?

Reverse EMittance Exchange

• Another potential application of twin helix channel

• Longitudinal emittance after PIC smaller than needed for collider

• Reverse EMittance EXchange (REMEX)

– Continue resonant regime

– Reverse wedge gradient to maximize transverse cooling decrements at the cost of antidamping in longitudinal direction

– Increase dispersion at absorber plates

– Maintain reasonable relative momentum spread by bunch stretching in the first stage and by beam acceleration in the second stage

• Another order of magnitude reduction in transverse emittance

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 33

Aberration Compensation

• Aberration compensation at beam focal points critical and most

challenging for PIC

• Take full advantage of system’s symmetry

• Compensation of 2nd-order terms with two sextupole harmonics

• Compensation of 3rd-order terms with three octupole harmonics

• Conceptually very similar to problem of aberration compensation at

collider interaction point

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 34

1 2( ) sin( ) sin(2 )s s sn s n ks n ks

1 2 3( ) cos( ) cos(2 )o o o on s n n ks n ks

Symmetry formulation of Achromatic IP(Standard Model)

*

0

2

0

*

0

2

0

2

02 dsydsnydsyDns

*

0

2

0

*

0

2

0

2

02 dsxdsnxdsxDns

02

00 dsyxns 03

0dsxns0)( 0 dsDxnDns

• “Canonical” conditions (compensation for original chromatic terms )

• Conditions connected to the betatron and 2nd order dispersion beam sizes:

These 3 conditions on sextupoles can be satisfied “automatically”, if to implement symmetry to the compensating block: symmetric and , while symmetry of and is opposite to symmetry of .

; ;

;

What is achieved with this compensation:Suppression of tune chromatic spread (usual)Suppression of intrinsic chromatic and sextupole 3d smear of beam core at star point (new)What may not have been achieved: maintaining the dynamical aperture

35Y.S. Derbenev, Rick’s Café Talk

IR Design

• Modular approach

• Utilize COSY Infinity

– calculates coefficients M(x|) of expansion of type

to arbitrary order (+++++) for each of coordinate components

• Design system such that

Muons, Inc.

, , ,, ,

( | )x M x x x y y t q

2 2

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

s s

o o

x s x s

y s y s

D s D s

n s n s

n s n s

n s n s

JLab Accelerator Seminar, June 3, 2010 36

Interaction RegionMuons, Inc.

1st-order matrix:x x’ y y’ t

M(x|x)

M(y|y)

M(x|q)

JLab Accelerator Seminar, June 3, 2010 37

Interaction RegionMuons, Inc.

23.2 mm

4.8 mmx

y

JLab Accelerator Seminar, June 3, 2010 38

IR Phase AdvanceMuons, Inc.

JLab Accelerator Seminar, June 3, 2010 39

1st and 2nd-Order AberrationsMuons, Inc.

Assume:4

10 GeV/c e , 85 , 17 , 2 km,

3.5 km, / 3 10

N N i

x y x

i

y

m m

E E

2nd-order aberrations:

M(x|xq)

M(y|yq)

1st-order aberrations:

M(x|x’)

M(y|y’)

Geometric beam size at IP due to emittance

JLab Accelerator Seminar, June 3, 2010 40

3rd-order aberrations: small

Sextupole CompensationMuons, Inc.

Make M(x|xq) = 0 and M(y|yq) = 0 by adjusting

s1 0.40 T @ 5 cm and s2 -1.15 T @ 5 cm

1s 2s 1s

JLab Accelerator Seminar, June 3, 2010 41

1st and 2nd-Order Aberrationsafter Sextupole Compensation

Muons, Inc.

Assume:4

10 GeV/c e , 85 , 17 , 2 km,

3.5 km, / 3 10

N N i

x y x

i

y

m m

E E

2nd-order aberrations:

M(x|xq)

M(y|yq)

1st-order aberrations:

M(x|x’)

M(y|y’)

Geometric beam size at IP due to emittance

M(x|xx’)

M(y|x’y)

M(y|xy’)

JLab Accelerator Seminar, June 3, 2010 42

3rd-Order Aberrationsafter Sextupole Compensation

Muons, Inc.

Assume:4

10 GeV/c e , 85 , 17 , 2 km,

3.5 km, / 3 10

N N i

x y x

i

y

m m

E E

3rd-order aberrations: M(x|x3)

M(x|x2y)

M(x|y3)

JLab Accelerator Seminar, June 3, 2010 43

Octupole CompensationMuons, Inc.

Make M(x|x3) = 0, M(y|x2y) = 0 and M(y|y3) = 0 by introducing 3 pairs of octupoles with

o1 -0.80 T @ 5 cm, o2 0.90 T @ 5 cm, and o3 -0.34 T @ 5 cm

1o1o2o 2o

3o3o

JLab Accelerator Seminar, June 3, 2010 44

1st and 2nd-Order Aberrationsafter Octupole Compensation

Muons, Inc.

Assume:4

10 GeV/c e , 85 , 17 , 2 km,

3.5 km, / 3 10

N N i

x y x

i

y

m m

E E

2nd-order aberrations:

M(x|xq)

M(y|yq)

1st-order aberrations:

M(x|x’)

M(y|y’)

Geometric beam size at IP due to emittance

M(x|xx’)

M(y|x’y)

M(y|xy’)

JLab Accelerator Seminar, June 3, 2010 45

3rd-Order Aberrationsafter Octupole Compensation

Muons, Inc.

Assume:4

10 GeV/c e , 85 , 17 , 2 km,

3.5 km, / 3 10

N N i

x y x

i

y

m m

E E

3rd-order aberrations: M(x|x3)

M(x|x2y)

M(x|y3)

4th-order aberrations: small

JLab Accelerator Seminar, June 3, 2010 46

Twin Helix SymmetryMuons, Inc.

Symmetry plane

x

JLab Accelerator Seminar, June 3, 2010 47

y

Dx

Opposite symmetries

Conclusions and Future Plans

• Designed twin-helix magnetic structure satisfying PIC requirements of

correlated linear optics

• Confirmed optics properties with GEANT4-based G4beamline simulations

• Large dynamic aperture suggested by simulations but more systematic

study is needed

• Suggested straightforward possible practical implementations

• Next study aberration compensation following a well-defined approach

(try adopting COSY Infinity?)

• Need to introduce transverse coupling for cooling decrement equalization

i.e. by slightly offsetting spatial period of one helical harmonic

Muons, Inc.

JLab Accelerator Seminar, June 3, 2010 48