Acknowledgements
 Many thanks to Sasha Valishev (FNAL) for help and
discussions.
2
Report at HEAC 1971
3
How to make the beam stable?
 Landau damping – the beam’s “immune system”. It is
related to the spread of betatron oscillation
frequencies. The larger the spread, the more stable the
beam is against collective instabilities.
 External damping (feed-back) system – presently the
most commonly used mechanism to keep the beam
stable.
 Can not be used for some instabilities (head-tail)
 Noise
 Difficult in linacs
4
Most accelerators rely on both
 LHC
 Has a transverse feedback system
 Has 336 Landau Damping Octupoles
 Provide tune spread of 0.001 at 1-sigma at injection
 Lyn Evans: “The ultimate panacea for beam instabilities is
Landau damping where the tune spread in the beam is
large enough to stop it from oscillating coherently. ”
 Tevatron, Recycler, MI, RHIC etc.
 In all machines there is a trade-off between Landau
damping and dynamic aperture.
5
Today’s talk will be about…
 … How to improve beam’s immune system (Landau damping
through betatron frequency spread)
 Tune spread not ~0.001 but 10-50%
What can be wrong with the immune
system?
 The main feature of all present accelerators – particles have
nearly identical betatron frequencies (tunes) by design. This
results in two problems:
I.
II.
Single particle motion can be unstable due to resonant
perturbations (magnet imperfections and non-linear elements);
Landau damping of instabilities is suppressed because the
frequency spread is small.
6
Preliminaries
 I will discuss the 2-D transverse beam dynamics. I’ll
ignore energy spread effects, but it can be included.
 The longitudinal coordinate, s, is equivalent to the
time coordinate.
H  H x, y, p x , p y , s 
 A 2-D Hamiltonian
will be called “integrable” if it has at least two
conserved functionally-independent quantities
(analytic functions of x, y, px, py, s) in involution.
 A 1-D time-independent Hamiltonian is integrable.
7
A bit of history: single particle
stability
 Strong focusing or alternating-gradient focusing was
first conceived by Christofilos in 1949 but not
published , and was later independently invented in
1952 at BNL (Courant, Livingston, Snyder).
 they discovered that the frequency of the particle
oscillations about the central orbit was higher, and the
wavelengths were shorter than in the previous constantgradient (weak) focusing magnets. The amplitude of
particle oscillations about the central orbit was thus
correspondingly smaller, and the magnets and the
synchrotron vacuum chambers could be made smaller—
a savings in cost and accelerator size.
8
Strong focusing
Also applicable
to Linacs
 x   K x ( s ) x  0
 
 y  K y (s) y  0
K x, y (s  C )  K x, y (s)
-- piecewise constant
alternating-sign functions
s is “time”
9
Courant-Snyder Invariant
z" K ( s ) z  0,
z  x or y
Equation of motion for
betatron oscillations
 Courant and Snyder found a conserved quantity:
2
1  2   ( s)
 
I
z 
z   ( s) z  
2 ( s) 
 2
 
where
    K (s)

 
1
3
-- auxiliary (Ermakov) equation
10
Normalized variables
 Start with a time-dependent Hamilatonian:
H

1 2
p  K (s) z 2
2

 Introduce new (canonical) variables: z
zN 

1

-- new “time”
 (s)
,
p N  p  (s) 
 Time-independent Hamiltonian:

1 2
H  p N  z N2
2
 ( s ) z
,
2  (s)

 Thus, betatron oscillations are linear; all particles
oscillate with the same frequency!
11
First synchrotrons
 In late 1953 R. Wilson has constructed the first electron AG
synchrotron at Cornell (by re-machining the pole pieces of
a weakly-focusing synchrotron).
 In 1955 CERN and BNL started construction of PS and AGS.
 1954: ITEP (Moscow) decides to build a strong-focusing
7-GeV proton synchrotron.

Yuri F. Orlov recalls: “In 1954 G. Budker gave several seminars
there. At these seminars he predicted that the combination of
a big betatron frequency with even a small nonlinearity would
result in stochasticity of betatron oscillations. ”
12
Yuri Orlov
 Professor of Physics, Cornell
 In 1954 he was asked to check Budker’s serious
predictions.
 He writes: “… I analyzed all reasonable linear and
nonlinear resonances with tune-shifting nonlinearities
and obtained well-defined areas of stability between
and below resonances and the corresponding
tolerances.”
 Work published in 1955.
13
Concerns about resonances
 So, at the time of AGS and CERN PS construction (1955-
60) the danger of linear betatron oscillations were
appreciated but not yet fully understood.
 Installed (~10) octupoles to “detune” particles from
resonances. Octupoles were never used for this purpose.
 Initial research on non-linear resonances (Chirikov, 1959)
indicated that non-linear oscillations could remain stable
under the influence of periodic external force
perturbation:
z   2 sin( z )  a sin(  t )
0
0
14
First non-linear accelerator
proposals
 In a series of reports 1962-65 Yuri Orlov has proposed
to use non-linear focusing as an alternative to strong
(linear) focusing.
 Final report (1965):
15
Henon-Heiles paper (1964)
 First general paper on appearance of chaos in a 2-d
Hamiltonian system.
16
Henon-Heiles model
 Considered a simple 2-d potential (linear focusing plus
a sextupole):
U ( x, y ) 


1 2
1
x  y2  x2 y  y3
2
3
 There exists one conserved
quantity (the total energy):
E
1 2
( p x  p y2 )  U ( x, y )
2
 For energies E > 0.125 trajectories
become chaotic
 Same nature as Poincare’s
“homoclinic tangle”
B
17
KAM theory
 Deals with the time evolution of a conservative dynamical system




under a small perturbation.
Developed by Kolmogorov, Arnold, Moser (1954-63).
Suppose one starts with an integrable 2-d Hamiltonian, eg.:
1 2
1 2
2
H  ( px  p y )  (x  y 2 )
2
2
It has two conserved quantities (integrals of motion), Ex and Ey .
The KAM theory states that if the system is subjected to a weak
nonlinear perturbation, some of periodic orbits survive, while
others are destroyed. The ones that survive are those that have
“sufficiently irrational” frequencies (this is known as the nonresonance condition). The KAM theory specifies quantitatively
what level of perturbation can be applied for this to be true. An
important consequence of the KAM theory is that for a large set of
initial conditions the motion remains perpetually quasiperiodic.
18
KAM theory
 M. Henon writes (in 1988):
 It explained the results Orlov obtained in 1955.
 And it also explained why Orlov’s nonlinear focusing
can not work.
19
KAM for Henon-Heiles potential
 This potential can be viewed as resulting from adding
a perturbation to the separable (integrable) harmonic
potential. It is non-integrable.
U ( x, y ) 


1 2
1
x  y2  x2 y  y3
2
3
 All trajectories with total
energies E  1 ( p 2  p 2 )  U ( x, y)
x
y
2
are bound.
 However, for E > 0.125
trajectories become chaotic.
B
 0.1667
20
E = 0.113, trajectory projections
0.2
0.6
0.4
0.1
0.2
x - px
pxi
py i
0
y - py
0
 0.2
 0.1
 0.4
 0.2
 0.2
 0.1
0
0.1
 0.6
1
0.2
 0.5
0
xi
0.5
yi
1
0.5
x–y
yi
0
 0.5
1
 0.2
 0.1
0
xi
0.1
0.2
21
E = 0.144, trajectory projections
0.6 1
1
1
0.4
0.5
0.2
x - px
pxi
py i
0
y - py
0
 0.2
 0.5
 0.4
 0.6
1
 0.5
0
0.5
1
1
1
 0.5
0
xi
0.5
yi
1
0.5
x–y
yi
0
 0.5
1
1
 0.5
0
xi
0.5
1
22
Accelerators and KAM theory
 Unlike Henon-Heiles potential, the nonlinearities in
accelerators are not distributed uniformly around the
ring. They are s-(time)-dependent and periodic (in
rings)! … And non-integrable (in general).
 Luckily, an ideal accelerator is an integrable system
and small enough non-linearities still leave
octupole
enough tune space to operate it.
 However, it was still not fully
understood at the time of the first
colliders (1960)…
23
First storage ring colliders
 First 3 colliders, AdA (1960), Princeton-Stanford CBX
(1962) and VEP-1 (1963), were all weakly-focusing
machines.
 This might reflect the concern designers had for the
long-term particle stability in a strongly-focusing
storage ring.
CBX layout
24
Octupoles and sextupoles enter
 1965, Priceton-Stanford CBX: First mention of an 8-pole
magnet
 Observed vertical resistive wall instability
 With octupoles, increased beam current from ~5 mA to 500 mA
 CERN PS: In 1959 had 10 octupoles; not used until 1968
 At 1012 protons/pulse observed (1st time) head-tail
instability. Octupoles helped.
 Once understood, chromaticity jump at transition was
developed using sextupoles.
 More instabilities were discovered; helped by octupoles and
by feedback.
25
Tune spread from an octupole
potential
In a 1-D system:
Freq.
1.5
H
f ( W) 1
1-D freq.
0.5
0
0
1
2
3
W
Energy
4
5
1 2 1 2 1 4
px  x  x
2
2
4
Tune spread is unlimited
----------------------------------------In a 2-D system:

1 2
1 2
1 4
2
2
H  ( px  p y )  ( x  y )  x  y 4  6 x 2 y 2
2
2
4
Tune spread (in both x and y) is
limited to ~12%
B
26

Tune spread from a single
octupole in a linear latice
 Tune spread depends on a linear tune location
 1-D system:
 Theoretical max. spread is 0.125
 2-D system:
 Max. spread < 0.05
octupole
27
1 octupole in a linear 2-D lattice
Typical phase space portrait:
1. Regular orbits at small amplitudes
2. Resonant islands + chaos at larger
amplitudes;
Are there “magic” nonlinearities that
create large spread and zero resonance
strength?
The answer is – yes
(we call them “integrable”)
28
McMillan nonlinear optics
 In 1967 E. McMillan published a paper
 Final report in 1971. This is what later became known
as the “McMillan mapping”:
xi  pi 1
pi   xi 1  f ( xi )
Bx 2  Dx
f ( x)   2
Ax  Bx  C

 

Ax2 p 2  B x 2 p  xp2  C x 2  p 2  Dxp  const
If A = B = 0 one obtains the Courant-Snyder invariant
29
McMillan 1D mapping
 At small x:
D
f ( x)   x
C
Linear matrix:
 At large x:
1 
0

D
 1  
C

Bx 2  Dx
f ( x)   2
Ax  Bx  C
1
 D 
acos 

2
2
C


Bare tune:
f ( x)  0
A=1, B = 0, C = 1, D = 2
4
Linear matrix:
 0 1



1
0


Tune: 0.25
2
p n k 0
 Thus, a tune spread of 50-100% is
2
possible!
4
4
2
0
xn k
2
4
30
What about 2D optics?
 How to extend McMillan mapping into 2-D?
 Danilov, Perevedentsev found two 2-D examples:
 Round beam: xpy - ypx = const
1. Radial McMillan kick: r/(1 + r2) -- Can be realized with
an “Electron lens” or in beam-beam head-on collisions
2. Radial McMillan kick: r/(1 - r2) -- Can be realized with
solenoids (may be useful for linacs)
 In general, the problem is that the Laplace equation
couples x and y fields of the non-linear thin lens
31
Danilov’s approximate solution
 McMillan 1-D kick can be obtained by using
 

V ( x, y )  Re ln  x  iy   1
2
 Then,
f ( x,0) 
2x
x2 1
 Make beam size small in one
direction in the non-linear lens
(by making large ratio of betafunctions)
B
32
Danilov’s approximate solution
FODO lattice,
0.25,0.75 bare tunes
2 nonlinear, 4 linear
lenses.
For beta ratio > 50,
nearly regular
decoupled motion
Tune spread is
around 30% .
33
Summary thus far
 In all present machines there is a trade-off between Landau
damping and dynamic aperture.
 J. Cary et al. has studied how to increase dynamic
aperture by eliminating resonances.
 The problem in 2-D is that the fields of non-linear
elements are coupled by the Laplace equation.
 There exist exact 1-D and approximate 2-D non-linear
accelerator lattices with 30-50% betatron tune spreads.
34
New approach
 See: http://arxiv.org/abs/1003.0644
 The new approach is based on using the time-
independent potentials.
 Start with a round axially-symmetric beam (FOFO)
OptiM - MAIN: - C:\Documents and Settings\nsergei\My Documents\Papers\Invariants\Round
V(x,y,s)
V(x,y,s)
V(x,y,s)
0
V(x,y,s)
0
V(x,y,s)
DISP_X&Y[m]
BETA_X&Y[m]
5
20
Sun Apr 25 20:48:31 2010
0
BETA_X
BETA_Y
DISP_X
DISP_Y
40
35
Special time-dependent potential
 Let’s consider a Hamiltonian
2
 x2 y2 
p x2 p y
  V ( x, y, s)
H

 K s  
2
2
2 
 2
where V(x,y,s) satisfies the Laplace equation in 2d:
V ( x, y, s )  V ( x, y )  0
 In normalized variables we will have:
HN 
2
2
p xN
 p yN
2

x N2  y N2

  ( )V x N  ( ) , y N  ( ) , s( )
2

ds 
 (s)
0
s
Where new “time” variable is
 (s)  
36
Three main ideas
1.
Chose the potential to be time-independent in new
2
2
p xN
 p yN
variables
x N2  y N2
HN 
2

 U (xN , y N )
2
β(s)
T insert
2. Element of periodicity
 (s) 
 1 0 0 0


  k 1 0 0
 0 0 1 0


 0 0  k 1


L  sk L  s 
 Lk 
1  1 

2 

2
L
s
3. Find potentials U(x, y) with the second integral of
motion
37
Test lattice (for an 8-GeV beam)
Six quadrupoles
provide a phase advance
of π
Linear tune: 0.5 – 1.0
 1 0 
 k  1


38
Examples of time-independent
Hamiltonians
V  x, y , s  
 Quadrupole
q
 s 2
x
2
 y2 

U ( x N , y N )  q x N2  y N2
HN 
2
quadrupole
amplitude
1.5
q( s)
1
β(s)
0
0.2
0.4
0.6
s
2
x N2  y N2

 q x N2  y N2
2

 x2   02 (1  q)
 y2   02 (1  q)
Tune spread: zero
0.5
0
2
2
p xN
 p yN
Tunes:
 ( s)

0.8
L
39

Examples of time-independent
Hamiltonians
 Octupole

 x 4 y 4 3x 2 y 2 
 

V x, y, s  

3 
4
2 
 s   4
 x N4 y N4 3 y N2 x N2
U   


4
2
 4





1 2
1 2
k 4
2
2
H  ( px  p y )  ( x  y )  x  y 4  6 x 2 y 2
2
2
4
This Hamiltonian is NOT integrable
Tune spread (in both x and y) is
limited to ~12%
B
40

Tracking with octupoles
Tracking with a test lattice
10,000 particles for 10,000 turns
No lost particles.
50 octupoles in a 10-m long drift space
41
Integrable 2-D Hamiltonians
 Look for second integrals quadratic in momentum
 All such potentials are separable in some variables
(cartesian, polar, elliptic, parabolic)
 First comprehensive study by Gaston Darboux (1901)
 So, we are looking for integrable potentials such that
H
Second integral:
p x2  p y2
2
x2  y2

 U ( x, y )
2
I  Apx2  Bp x p y  Cpy2  D( x, y)
A  ay 2  c 2 ,
B  2axy,
C  ax 2 ,
42
Darboux equation (1901)
 Let a ≠ 0 and c ≠ 0, then we will take a = 1
xyU xx  U yy   y 2  x 2  c 2 U xy  3 yU x  3xU y  0


 General solution
f ( )  g ( )
U ( x, y ) 
 2  2

x  c 2  y 2  x  c 2  y 2
2c
x  c 2  y 2 

2c
x  c 2  y 2
ξ : [1, ∞], η : [-1, 1], f and g arbitrary functions
43
The second integral
 The 2nd integral
2
2
f
(

)


g
(

)

2
I x, y, p x , p y   xpy  yp x   c 2 p x2  2c 2
 2  2

1 2
U ( x, y )  x  y 2
2
 Example:


c2 2 2
f1 ( )     1
2


c2 2
g1 ( )   1   2
2

I x, y, p x , p y   xpy  yp x   c 2 p x2  c 2 x 2
2
44
Laplace equation
 Now we look for potentials that also satisfy the Laplace
equation (in addition to the Darboux equation):
U xx  U yy  0
 We found a family with 4 free parameters (b, c, d, t):
f 2 ( )    2  1d  t acosh  
g 2 ( )   1   2 b  t acos 
U ( x, y ) 
f ( )  g ( )
 2  2
45
The Hamiltonian
 So, we found the integrable Laplace potentials…
2
p x2 p y x 2 y 2 f 2 ( )  g 2 ( )
H x, y, p x , p y  




2
2
2
2
 2  2
F
→ 1 at x,y → ∞
→ 0 at x,y → ∞
→ ln(r) at x,y → ∞
c=1, d=1, b=t=0
c=1, b=1, d=t=0
c=1, t=1, d=b=0
46
F
F
The integrable Hamiltonian (elliptic
coordinates)
H
p x2  p y2
2
x2  y2

 kU ( x, y )
2
Multipole expansion
2
8
16
128

2
4
6
8
x  iy 10  ...
U ( x, y )  Re x  iy   x  iy    x  iy    x  iy  
3
15
35
315


U(x,y): c=1, t=1, d=0, b  

2
|k| < 0.5 to provide linear stability for small
amplitudes
Tune spread:
 x2   02 (1  k )
 y2   02 (1  k )
 x2   02
 y2   02
at small amplitudes
at large amplitudes
Max. ~100% in y and ~40% in x
B
47
Example of trajectories
 25 nonlinear lenses in a drift space
 Small amplitude trajectories
0.04
0.04
0.02
0.02
pxjj
py jj
0
0
 0.02
 0.02
 0.04
 0.04
 0.04  0.02
0
0.02
 0.04  0.02
0.04
xjj
0
0.02
0.04
y jj
0.6
y
f1j 0.4
bare tune
f2j
f3j
0.2
0
0.5
x
0.6
0.7
0.8
j
0.9
1
48
Large amplitudes
pxjj
2
2
2
1
1
1
py jj
0
1
2
2
y jj
0
1
1
0
1
1
2
2
2
0
1
0
xjj
1
2
2
2
1
y jj
0
1
2
xjj
40
bare tune
y
30
f1j
f2j
20
x
f3j
10
0
0.5
0.6
0.7
0.8
j
0.9
1
49
More examples of trajectories
f ( ) 
4
2
    1, and g ( )  
2
4
2
c=1, d=1, b=t=0
Normalized coordinates:
 Trajectory encircles the singularities (y=0, x=±1)
50
Polar coordinates
 Let a ≠ 0 and c = 0
2
p x2 p y r 2
g ( )
H

  f (r )  2
2
2
2
r
f (r )  d ln r 
g ( )  b sin 2   t cos2 
I  xpy  yp x   2 g ( )
2
51
Two types of trajectories
f (r )  d ln r 
g ( )  b sin 2   t cos2 
 d = -1, b = 0, and t = 0.1
52
Parabolic coordinates
a=0
 Not considered by Darboux (but considered by Landau)
I  Apx2  Bp x p y  Cpy2  D( x, y)
A0
B  y
Cx
 Equation for potential:
2 xU xy  3U y  y U yy  U xx   0
53
Parabolic coordinates
 Solution
f (r  x)  g (r  x)
,
2r
(r  x) f (r  x)  (r  x) g (r  x)
I   yp x  xpy  p y 
.
2r
U ( x, y ) 
 The only parabolic solution is y2 + 4x2
2
p x2 p y r 2 3 2
f ( r  x)  g 2 ( r  x)
H

  (x  y 2 )  2
2
2
2 10
2r
 Potentials that satisfy the Laplace equation
f 2 (r  x)  b r  x  t r  x 
2
g 2 (r  x)  d r  x  t r  x 
2
54
Example of trajectories
F
U  x, y  
3 2
rx
( x  y 2 )  0.1
10
2r
 Trajectories never encircle the singularity
55
Summary
 The lattice can be realized by both
and
 1 0
 1 0 
M x, y  
M x, y  



k
1
k

1




 Found nonlinear lattices (e.g. octupoles) with one integral
of motion and no resonances. Tune spread 5-10% possible.
 Found first examples of completely integrable non-linear
optics.
 Betatron tune spreads of 50% are possible.
 We used quadratic integrals but there are might other
functions.
 Beyond integrable optics: possibly there exist realizable
lattices with bound but ergodic trajectories
56
Conclusions
 Nonlinear “integrable” accelerator optics has
advanced to possible practical implementations
 Provides “infinite” Landau damping
 Potential to make an order of magnitude jump in beam
brightness and intensity
 Fermilab is in a good position to use of all these
developments for next accelerator projects
 Rings or linacs
 Could be retrofitted into existing machines
57
Extra slides
58
Vasily P. Ermakov (1845-1922)
 Professor of Mathematics, Kiev university
 “Second order differential equations: Conditions of
complete integrability”, Universita Izvestia Kiev, Series
III 9 (1880) 1–25.
 Found a solution to the equation: z   K ( s ) z  0
 Ermakov invariant is what we now call Courant-Snyder
invariant in accelerator physics
59
Joseph Liouville (1809-1882)
 Liouville (1846) observed that if the
Hamiltonian function admitted a
special form
 in some system of coordinates (u, v),
the Hamiltonian system could be
solved in quadratures. The form of the
Hamiltonian function also implies
additive separation of variables
 for the associated Hamilton-Jacobi
equation.
60