iota integrable optics test accelerator at fermilab sergei nagaitsev may 21, 2012 ipac 2012, new...
DESCRIPTION
3 Strong Focusing – our standard method since 1952 s is “time” 3 -- piecewise constant alternating-sign functions Particle undergoes betatron oscillations Christofilos (1949); Courant, Livingston and Snyder (1952)TRANSCRIPT
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IOTA – Integrable Optics Test Accelerator at Fermilab
Sergei NagaitsevMay 21, 2012
IPAC 2012, New Orleans
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Collaborative effort Fermilab: S. Nagaitsev, A. Valishev SNS: V. Danilov Budker INP: D. Shatilov BNL: H. Witte JAI (Oxford) Tech X Special thanks to E. Forest
List of posters at IPAC2012: TUPPC090 “Beam Physics of Integrable Optics Test Accelerator at
Fermilab” TUEPPB003 “Motion Stability in 2D Nonlinear Accelerator Lattice
Integrable in Polar Coordinates” WEPPR012 “Simulating High-Intensity Proton Beams in Nonlinear Lattices
with PyORBIT”
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IOTA - Nagaitsev 3
Strong Focusing – our standard method since 1952
)()(
0)(0)(
,, sKCsK
ysKyxsKx
yxyx
y
x
s is “time”
3
-- piecewise constant alternating-sign functions
Particle undergoes betatron oscillations
Christofilos (1949); Courant, Livingston and Snyder (1952)
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Strong focusing
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Focusing fields must satisfy Maxwell equations in vacuum
For stationary fields: focusing in one plane while defocusing in another quadrupole:
However, alternating quadrupoles results in effective focusing in both planes
( , , ) 0x y z
2 2( , )x y x y
Specifics of accelerator focusing:
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Courant-Snyder invariant
yxzzsKz
or ,0)("
Equation of motion for
betatron oscillations
3
1)( where
sK
22 )(
2)(
)(21 zszsz
sI
Invariant (integral) of motion,a conserved qty.
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Linear oscillations
Normalized variables
In this variables the motion is a linear oscillator
Thus, betatron oscillations are linear; all particles oscillate with the same frequency!
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,)(2
)()(
,)(
szsspp
szz
N
N
1( )s
-- new time variable
22
2 0nn
d zz
d
yyxx IIH
12 n nI p dz
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Our goal
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Our goal is to create a practical nonlinear focusing system with a large frequency spread and stable particle motion.
Benefits: Increased Landau damping Improved stability to perturbations Resonance detuning
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Octupoles and tune spread
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Typical phase space portrait:1. Regular orbits at small amplitudes2. Resonant islands + chaos at larger amplitudes;
In all machines there is a trade-off betweenLandau damping and dynamic aperture (resonances).
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Linear vs nonlinear Accelerators are linear systems by
design (freq. is independent of amplitude).
In accelerators, nonlinearities are unavoidable (SC, beam-beam) and some are useful (Landau damping).
All nonlinearities (in present rings) lead to resonances and dynamic aperture limits.
Are there “magic” nonlinearities that create large spread and zero resonance strength?
The answer is – yes (we call them “integrable”) 3D: IOTA - Nagaitsev 9
VEPP2000
mlk yx
1 2 3( , , )H F J J J
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Ideal Penning trap
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The ideal Penning trap is a LINEAR integrable and system It is a linear 3-d oscillator
1 1 2 2 3 3H J J J
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Kepler problem – nonlinear integrable system
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Kepler problem:
In spherical coordinates:
Example of this system: the Solar system
kVr
2
22 r
mkHJ J J
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Nonlinear systems can be more stable! 1D systems: non-linear oscillations can remain
stable under the influence of periodic external force perturbation. Example:
2D: The resonant conditions
are valid only for certain amplitudes. Nekhoroshev’s condition guaranties detuning from resonance and, thus, stability.
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)sin()sin( 020 tazz
1 1 2 2 1 2( , ) ( , )k J J l J J m
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Example of a “good” nonlinear system Suppose that
This would be a nonlinear equivalent to strong focusing
We do NOT know how realize this particular example in practice!
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22 3
2 0, where is or nn n n n n
d z z z z x yd
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On the way to integrability
1) Colliding beams: a) Round beam – angular momentum conservation-
1D motion in r (Novosibirsk, 80’s, realized at VEPP2000, tune shift around 0.15 achieved);
b) Crab waist - decoupling x and y motion (P. Raimondi (2006), tune shift 0.1 achieved at DAFNE).
2) Numerical methods to eliminate resonances (e.g. J. Cary and colleagues; D. Robin, W. Wan and colleagues);
3) Exact solutions for realization– our goal. The list is presented in next slidesMajor limiting factor: fields must satisfy Maxwell eqtns.
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1-D nonlinear optics
In 1967 E. McMillan published a paper
Final report in 1971. This is what later became known as the “McMillan mapping”:
Generalizations (Danilov-Perevedentsev, 1992-1995)
)(1
1
iii
ii
xfxppx
CBxAxDxBxxf
2
2
)(
const 222222 DxppxCxppxBpAx
If A = B = 0 one obtains the Courant-Snyder invariant
4 2 0 2 4
4
2
0
2
4
pn k
xn k
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2D case with realistic fields1. The 1-D McMillan mapping was extended to 2-D round thin lens by R. McLachlan (1993) and by D-P (1995)
Round lenses can be realized only with charge distributions:
a) 1 or 2 thin lenses with radial kicks
b) Time dependent potential .
2. Approximate cases – J. Cary & colleagues - decoupling of x-y motion and use of 1D solutions;
3. Stable integrable motion without space charge in Laplace fields – the only known exact case is IOTA case (Danilov, Nagaitsev, PRSTAB 2010);
)())((
21221 ccbr
arrff
)(1rU
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Choice of nonlinear elements1)For large beam size accumulators and
boosters – external fields can produce large frequency spread at beam size amplitudes;
2) Small beam size colliders need nonlinearities on a beam-size scale. The ideal choice is colliding beam fields (like e-lens in Tevatron)
The IOTA ring can test both variants of nonlinearities.
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Advanced Superconductive Test Accelerator at Fermilab
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IOTAILC Cryomodules
Photo injector
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Layout
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Beam from linac
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IOTA schematic
pc = 150 MeV, electrons (single bunch, 10^9) ~36 m circumference 50 quadrupoles, 8 dipoles, 50-mm diam vac
chamber hor and vert kickers, 16 BPMs
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Nonlinear inserts
Injection, rf cavity
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Why electrons? Small size (~50 um), pencil beam Reasonable damping time (~1 sec) No space charge
In all experiments the electron bunch is kicked transversely to “sample” nonlinearities. We intend to measure the turn-by-turn BPM positions as well as synch light to obtain information about phase space trajectories.
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Proposed experiments We are proposing several experiments with
nonlinear lenses Based on the electron (charge column) lens
Based on electromagnets
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2 0U
2 0U
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Experiments with electron lens
For IOTA ring, we would use a 5-kG, ~1-m long solenoid
Electron beam: ~0.5 A, ~5 keV, ~1 mm radius
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Example: Tevatron electron lens
150 MeV beam
solenoid
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Experiment with a thin electron lens The system consists of a thin (L < β) nonlinear
lens (electron beam) and a linear focusing ring Axially-symmetric thin McMillan lens:
Electron lens with a special density profile The ring has the following transfer matrix
IOTA - Nagaitsev 24
2( )1
krrar
electronlens
0 0 01 0 0 0
0 0 010 0 0
cI sIsI cI
cos( )sin( )1 00 1
cs
I
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Electron lens (McMillan – type) The system is integrable. Two integrals of
motion (transverse): Angular momentum: McMillan-type integral, quadratic in momentum
For large amplitudes, the fractional tune is 0.25
For small amplitude, the electron (defocusing) lens can give a tune shift of ~-0.3
Potentially, can cross an integer resonanceIOTA - Nagaitsev 25
y xxp yp const
4 2 0 2 44
2
0
2
4
pxi
xi
4 2 0 2 44
2
0
2
4
py i
y i
Electron lens current density:
22( )
1
In rar
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Ideal McMillan round lens
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FMA fractional tunes
0 0.50
0.5
νx
νy
Linear tuneLarge amplitudes(0.3,0.2)
Small amplitudes
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Practical McMillan round lens
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All excited resonances have the form k ∙ (x + y) = mThey do not cross each other, so there are no stochastic layers and diffusion
e-lens (1 m long) is represented by 50 thin slises. Electron beam radius is 1 mm. The total lens strength (tune shift) is 0.3
FMA analysis
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Experiments with nonlinear magnetsSee: Phys. Rev. ST Accel. Beams 13, 084002 (2010)
Start with a round axially-symmetric LINEAR focusing lattice (FOFO)Add special non-linear potential V(x,y,s) such that
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0),(),,( yxVsyxV
400
Sun Apr 25 20:48:31 2010 OptiM - MAIN: - C:\Documents and Settings\nsergei\My Documents\Papers\Invariants\Round
200
50
BE
TA_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
V(x,y,s) V(x,y,s) V(x,y,s) V(x,y,s)V(x,y,s)
βx = βy
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Fake thin lens inserts
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110.0240
Tue Feb 22 14:40:15 2011 OptiM - MAIN: - C:\Users\nsergei\Documents\Papers\Invariants\Round lens\quad line1.opt
200
0
50
BE
TA_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
1 0 0 01 0 0
0 0 1 00 0 1
k
k
1 0 0 01 0 0
0 0 1 00 0 1
k
k
1 0 0 01 0 0
0 0 1 00 0 1
k
k
1 0 0 01 0 0
0 0 1 00 0 1
k
k
V(x,y,s) V(x,y,s) V(x,y,s)
Example only!
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4-Magnet Lattice (Exp. 3-4)
Equal beta-functions, Qx=5.0+0.3×4, Qy=4.0+0.3×4 Dispersion=0 in the Nonlinear Magnet Maximum Vertical amplitude in the NM=11 mm =0.015
x=y, Q=0.5 Q=0.5
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Main ideas
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1. Start with a time-dependent Hamiltonian:
2. Chose the potential to be time-independent in new variables
3. Find potentials U(x, y) with the second integral of motion and such that ΔU(x, y) = 0
),(
22
2222
NNNNyNxN
N yxUyxpp
H
)(,)(,)()(22
2222
syxVyxpp
H NNNNyNxN
N
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Nonlinear integrable lens
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2 2 2 2
( , )2 2
x yp p x yH U x y
Multipole expansion :
For c = 1|t| < 0.5 to provide linear stability for smallamplitudes
For t > 0 adds focusing in x
Small-amplitude tune s:
tt
2121
2
1
This potential has two adjustable parameters:t – strength and c – location of singularities
For |z| < c 2 4 6 82 2 4 6
2 8 16, Im ( ) ( ) ( ) ( ) ...3 15 35
tU x y x iy x iy x iy x iyc c c c
B
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Transverse forces
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-300
-200
-100
0
100
200
300
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-50
-40
-30
-20
-10
0
10
20
30
40
50
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Fx Fy
Focusing in x Defocusing in y
x y
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Nonlinear Magnet
Practical design – approximate continuously-varying potential with constant cross-section short magnets
Quadrupole component of nonlinear field
Magnet cross section
Distance to pole c2×c
V.Kashikhin8
14
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2-m longmagnet
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Examples of trajectories
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Ideal nonlinear lens A single 2-m long nonlinear lens creates a tune
spread of ~0.25.
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FMA, fractional tunes
Small amplitudes(0.91, 0.59)
Large amplitudes
0.5 1.00.5
1.0
νx
νy
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Experimental goals with nonlinear lenses Overall goal is to demonstrate the
possibility of implementing nonlinear integrable optics in a realistic accelerator design
Demonstrate a large tune shift of ~1 (with 4 lenses) without degradation of dynamic aperture minimum 0.25
Quantify effects of a non-ideal lens
Develop a practical lens design.
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Summary We have found first (practical) examples of
completely integrable non-linear optics.
We have explored these ideas with modeling and tracking simulations.
The Integrable Optics Test Accelerator (IOTA) ring is now under construction. Completion expected in 2014. Poster:TUPPC090 “Beam Physics of Integrable Optics
Test Accelerator at Fermilab” The ring can also accommodate other
Advanced Accelerator R&D experiments and/or users Current design accommodates Optical Stochastic
Cooling IOTA - Nagaitsev 39