nonlinear integrable optics as a route to high intensity
TRANSCRIPT
Nonlinear Integrable Optics as a Route to High Intensity Accelerators
Stephen D. Webb
RadiaSoft, LLC., Boulder, CO
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
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