Nonlinear Integrable Optics as a Route to High Intensity Accelerators

Stephen D. Webb

RadiaSoft, LLC., Boulder, CO

swebb@radiasoft.net

7 September, 2015

Oxford University

Boulder, Colorado USA – www.radiasoft.net 1

Hadron Accelerators at the Intensity Frontier

2

Input for “Snowmass – Capabilities”, WG3 “Intensity Frontier”

Issues and R&D Required for the Intensity Frontier Accelerators V.Shiltsev, S.Henderson, P.Hurh, I.Kourbanis, V.Lebedev (Fermilab)

Operation, upgrade and development of accelerators for Intensity Frontier face formidable

challenges in order to satisfy both the near-term and long-term Particle Physics program. The near-term

program continuing throughout this decade includes the long-baseline neutrino experiments and a muon

program focused on precision/rare processes. It requires:

Double the beam power capability of the Booster

Double the beam power capability of the Main Injector

Build-out the muon campus infrastructure and capability based on the 8 GeV proton source.

The long-term needs of the Intensity Frontier community are expected to be based on the following

experiments:

long-baseline neutrino experiments to unravel neutrino sector, CP-violation, etc.;

and rare and precision measurements of muons, kaons, neutrons to probe mass-scales beyond

LHC.

Both types of experiments will require MW-class beams. The Project-X construction is expected to

address these challenges. The Project X represents a modern, flexible, Multi-MW proton accelerator (see

Fig.1).

Fig.1: Accelerator beam power landscape (present-future)

from 2013 Snowmass WG3, “Issues and R&D Required for the Intensity Frontier Accelerators” arXiv:1305.6917v1 [physics.acc-ph]

Beam Halo

3

4

Strong Head-Tail Instability

W0(z)

N >8�0C!�!s

⇡r0W0c2

5

Strong Head-Tail Instability

Parametric Resonances & Tune Spread

6

r + q2✓r � L2a4

r3

◆= �

a2r

✓1� a2

r2

◆⇥(r � a) + 2

✏

a2r cos(pt)⇥(a� r)

| {z }parametric resonance

2

✏

a2r cos(pt)⇥(a� r) 7! 2

✏

a2r⇥(a� r)

Z p

0dp0%(p0) cos(p0t)

⇠ sin(p0t)

�pt

Nonlinear decoherence — frequency spreads in ensembles which suppress the forcing terms in parametric resonances. This is logically distinct from Landau damping, but has a similar effect.

7

Conventional Nonlinear Schemes Induce Chaos

eµ02

:�p

2+2↵px+�x

2 :

eS4 :x

4 :

nµ0

2⇡6= `

Perturbative single turn map

A�1eµ0 :J

:AeS4 :x

4 : 7! A0�1eµi :J : +S

04 :

J

2 : +O(S24)A0

Near a resonance…

Tune spread w/ amplitude

A0= exp

:

4X

a+b=0

Ca,b1

1� ei(a�b)µ0ra+r

b� :

!A

r± =pJe±i�

Famous “small denominators”

8

Conventional Nonlinear Schemes Induce Chaos

n⇥ ⌫x

+m⇥ ⌫y

= `Resonance Condition

9

Conventional Nonlinear Schemes Induce Chaos

n⇥ ⌫x

+m⇥ ⌫y

= `Resonance Condition

10

Nonlinear Integrable Optics

11

Elliptic Potential Properties

12

Lattice Design Requirements

t(s)

s

exp{

− : p2

2+ ti

1−δUi(x, y):∆s

}

H = µx

(1� Cx

(�))1

2

�p2x

+ x2�+

µy

(1� Cy

(�))1

2

�p2y

+ y2�+

t

Z`drift

0ds0V (x� �(⌘/p�

x

), y)

13

Preliminary Results — Halo Suppression

14

Preliminary Results — Phase Space Diffusion

15

Future Work

Injection matching

Quadrupole wake fields and integrability

Synchrotron oscillations & transverse-longitudinal chromatic coupling

…

16

Injection Matching

Matched beam is very not Gaussian

Possible schemes:

• Phase space painting

• Adiabatic up-ramp

• Adiabatic down-ramp

17

Quadrupole wake fields & integrability

H = µx

(1� Cx

(�))1

2

�p2x

+ x2�+

µy

(1� Cy

(�))1

2

�p2y

+ y2�+

t

Z`drift

0ds0V (x� �(⌘/p�

x

), y)

+hQx

ix2 + hQy

iy2

18

Synchrotron Oscillations

H = µx

(1� Cx

(�))1

2

�p2x

+ x2�+

µy

(1� Cy

(�))1

2

�p2y

+ y2�+

t

Z`drift

0ds0V (x� �(⌘/p�

x

), y)

+Urf(�)

19

Thank you for your attention

Stephen D. Webb

RadiaSoft, LLC., Boulder, CO

swebb@radiasoft.net

This work is supported in part by US DOE Office of Science, Office of High Energy Physics under SBIR award DE-SC0011340

Nonlinear Integrable Optics as a Route to High Intensity Accelerators

7 September, 2015

Oxford University

When are sextupoles optically transparent?

20

• Lie operator approach

• Off-momentum particles do not cancel exactly because θ is energy-dependent. This is the basis of chromaticity correction.

M = e�Sn :zn :e� :h2 :e�Sn :zn :

M = e� :h2 :exp(�Sn :e

:h2 :zn :) exp(�Sn :zn:

)

M = A�1R exp(�Sn :R(z)n :) exp(�Sn :zn:

)

R / �1, ✓ = (2n+ 1)⇡ =) exp(�Sn :R(z)n :) exp(�Sn :zn:

)A = 1

e� :h2 : = A�1 R(✓)| {z }pure rotation

A

• Pictorial approach (design momentum)

When are sextupoles optically transparent?

21

R(⇡)

x

p

• Pictorial approach (off-momentum)

When are sextupoles optically transparent?

22

x

pR(⇡ � C�)

The horror…

23

M =

0

@N/2Y

i=0

exp

⇢� :

p

2

2

+ tiVi(x, y):�s

�1

Ae

� :h0 :

0

@NY

i=N/2

exp

⇢� :

p

2

2

+ tiVi(x, y):�s

�1

A

=

0

@

N/2Y

i=0

exp

n

�ti :e�(i+1/2) :p2/2:�sVi(x, y):�s

o

1

A �

e

� :p2/2: /2e

� :h0 :e

� :p2/2: /2| {z }

e� :h2 :

�

0

@

NY

i=N/2

exp

n

�ti :e(i+1/2) :p2/2:�sVi(x, y):�s

o

1

A

… the horror

24

e� :h2 : = Ae� :h2 :A�1 Normalized coördinates

A�1 =

0

BBBBBB@

1/p�

x

0 0 0 0 �⌘/p�

x

↵

x/p�

x

p�x

0 0 0 �↵

x

⌘+�

x

⌘

0/p�

x

0 0 1/p

�

y

0 0 00 0 ↵

x/p

�

y

p�y

0 0⌘0 ⌘ 0 0 1 00 0 0 0 0 1

1

CCCCCCA

Courant-Snyder Parameterization

M =

0

@

N/2Y

i=0

exp

n

�ti :e�(i+1/2) :p2/2:�sVi(x, y):�s

o

1

A �

⇣

Ae

� :h2 :A�1⌘

�0

@

NY

i=N/2

exp

n

�ti :e(i+1/2) :p2/2:�sVi(x, y):�s

o

1

A

25

A�1exp

n

�ti :e(i+1/2) :p2 : 2Vi(x, y):�s

o

=

A�1exp

n

�ti :e(i+1/2) :p2 : 2Vi(x, y):�s

o

AA�1=

A�1e

�(i+1/2) :p2/2:�s| {z }

A�1i

exp {�ti :Vi(x, y):�s} e(i+1/2) :p2/2:�sA| {z }

Ai

Vi(x, y) = Vi

⇣A

(i)0 (x, y)

⌘

A(i)0 =

0

BB@

1/p�

x

0 0 0↵

x/p�

x

p�x

0 00 0 1/

p�

y

00 0 ↵

x/p

�

y

p�y

1

CCA

The Danilov-Nagaitsev potential normalizing trick as follows:

26

A�1e

�(i+1/2) :p2/2:�s

exp {�t

i

:Vi

(x, y)

:

�s} e(i+1/2) :p2/2:�sA =

exp

⇢�t

:V✓x� �

⌘p�

x

, y

◆:

�s

�

h2 =µ0

2

⇥(1� C

x

�)�p

2x

+ x

2�+ (1� C

y

�)�p

2y

+ y

2�⇤

Final transfer map in normalized coordinates

0

@N/2Y

i=0

exp

⇢� :

p

2

2

+ tiVi(x, y):�s

�1

Ae

� :h0 :

0

@NY

i=N/2

exp

⇢� :

p

2

2

+ tiVi(x, y):�s

�1

A=

A exp

(X

i

�(1� �)t

:V✓x� �

⌘ip�i

, y

◆:

)e

� :h2 :exp

(X

i

�(1� �)t

:V✓x� �

⌘ip�i

, y

◆:

)A�1