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Acknowledgements



Many thanks to Sasha Valishev (FNAL) for help and
discussions.

2

3

Report at HEAC 1971

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.
Single particle motion can be unstable due to resonant
perturbations (magnet imperfections and non
-
linear elements);

II.
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.


A 2
-
D Hamiltonian


will be called “
integrable
” if it has at least two
conserved functionally
-
independent quantities
(analytic functions of
x, y, p
x
, p
y
, s
) in involution.



A 1
-
D time
-
independent Hamiltonian is
integrable
.

7



s
p
p
y
x
H
H
y
x
,
,
,
,

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 constant
-
gradient (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


)
(
)
(
0
)
(
0
)
(
,
,
s
K
C
s
K
y
s
K
y
x
s
K
x
y
x
y
x
y
x













s
is “time”

9

--

piecewise constant


alternating
-
sign functions

Also applicable

to Linacs

Courant
-
Snyder Invariant


Courant and Snyder found a conserved quantity:

10

y
x
z
z
s
K
z
or



,
0
)
(
"



Equation of motion for

betatron oscillations



3
1
)
(

where






s
K
--

auxiliary (Ermakov) equation




















2
2
)
(
2
)
(
)
(
2
1
z
s
z
s
z
s
I



Normalized variables


Start with a time
-
dependent
Hamilatonian
:




Introduce new (canonical) variables:



--

new “time”



Time
-
independent Hamiltonian:




Thus,
betatron

oscillations are linear; all particles
oscillate with the same frequency!


,
)
(
2
)
(
)
(
,
)
(
s
z
s
s
p
p
s
z
z
N
N











2
2
2
1
N
N
z
p
H





1


11



2
2
)
(
2
1
z
s
K
p
H


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:

14

)
sin(
)
sin(
0
2
0
t
a
z
z






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
):



There exists one conserved
quantity (the total energy):



For energies
E

> 0.125 trajectories
become chaotic


Same nature as Poincare’s

homoclinic

tangle”

17



3
2
2
2
3
1
2
1
)
,
(
y
y
x
y
x
y
x
U




)
,
(
)
(
2
1
2
2
y
x
U
p
p
E
y
x



B
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
.:




It has two conserved quantities (integrals of motion),
E
x

and
E
y

.


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 non
-
resonance 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

)
(
2
1
)
(
2
1
2
2
2
2
y
x
p
p
H
y
x




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
.



All trajectories with total
energies


are bound.


However, for
E

>
0.125
trajectories become chaotic.

20



3
2
2
2
3
1
2
1
)
,
(
y
y
x
y
x
y
x
U




B
.1667
0


)
,
(
)
(
2
1
2
2




y
x
U
p
p
E
y
x
E = 0.113, trajectory projections

21

0.2

0.1

0
0.1
0.2
0.2

0.1

0
0.1
0.2
px
i
x
i
1

0.5

0
0.5
0.6

0.4

0.2

0
0.2
0.4
0.6
py
i
y
i
0.2

0.1

0
0.1
0.2
1

0.5

0
0.5
1
y
i
x
i
x
-

p
x

y
-

p
y

x


y

22

1

0.5

0
0.5
1
0.6

0.4

0.2

0
0.2
0.4
0.6
px
i
1

1
x
i
1

0.5

0
0.5
1

0.5

0
0.5
1
py
i
y
i
1

0.5

0
0.5
1
1

0.5

0
0.5
1
y
i
x
i
E = 0.144, trajectory projections

x
-

p
x

y
-

p
y

x


y

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

enough tune space to operate it.


However, it was still not fully
understood at the time of the first
colliders (1960)…

23

octupole

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.

24

CBX layout

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
10
12

protons/pulse
observed (1
st

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:



Tune spread is unlimited

-----------------------------------------

In a 2
-
D system:



Tune spread (in both x and y) is
limited to
~
12%

26

4
2
2
4
1
2
1
2
1
x
x
p
H
x





2
2
4
4
2
2
2
2
6
4
1
)
(
2
1
)
(
2
1
y
x
y
x
y
x
p
p
H
y
x







0
1
2
3
4
5
0
0.5
1
1.5
Energy
Freq.
f
W
(
)
W
B
1
-
D freq.

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

27

octupole

1 octupole in a linear 2
-
D lattice

28

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”)


McMillan nonlinear optics


In 1967 E. McMillan published a paper





Final report in 1971. This is what later became known
as the “McMillan mapping”:

29

)
(
1
1
i
i
i
i
i
x
f
x
p
p
x






C
Bx
Ax
Dx
Bx
x
f





2
2
)
(




const

2
2
2
2
2
2






Dxp
p
x
C
xp
p
x
B
p
Ax
If
A = B
=
0

one obtains the Courant
-
Snyder invariant

McMillan 1D mapping


At small
x
:



Linear matrix: Bare tune:



At large
x:



Linear matrix: Tune:
0.25



Thus, a tune spread of 50
-
100% is


possible!

30

C
Bx
Ax
Dx
Bx
x
f





2
2
)
(
x
C
D
x
f


)
(










C
D
1
1
0







C
D
2
acos
2
1

0
)
(

x
f









0
1
1
0
4

2

0
2
4
4

2

0
2
4
p
n
k

x
n
k

A
=1,
B

= 0,
C
= 1
,
D = 2

What about 2D optics?


How to extend McMillan mapping into 2
-
D?


Danilov
,
Perevedentsev

found two 2
-
D examples:


Round beam:
xp
y

-

yp
x

= const

1.
Radial McMillan kick:
r
/(1 +
r
2
)
--

Can be realized with
an “Electron lens” or in beam
-
beam head
-
on collisions

2.
Radial McMillan kick:
r
/(1
-

r
2
)
--

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



Then,



Make beam size small in one
direction in the non
-
linear lens
(by making large ratio of beta
-
functions)

32







1
ln
Re
)
,
(
2



iy
x
y
x
V
B
1
2
)
0
,
(
2


x
x
x
f
Danilov’s approximate solution

33

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% .


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)

35

40
0
Sun Apr 25 20:48:31 2010 OptiM - MAIN: - C:\Documents and Settings\nsergei\My Documents\Papers\Invariants\Round

20
0
5
0
BETA_X&Y[m]
DISP_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)

Special time
-
dependent potential


Let’s consider a Hamiltonian



where
V(x,y,s)
satisfies the Laplace equation in 2d:



In normalized variables we will have:




)
,
,
(
2
2
2
2
2
2
2
2
s
y
x
V
y
x
s
K
p
p
H
y
x
















)
(
,
)
(
,
)
(
)
(
2
2
2
2
2
2







s
y
x
V
y
x
p
p
H
N
N
N
N
yN
xN
N





36

0
)
,
(
)
,
,
(




y
x
V
s
y
x
V




s
0
)
s
(
d
)
(


s
s
Where new “time” variable is

Three main ideas

1.
Chose the potential to be time
-
independent in new
variables



2.
Element of periodicity




3.
Find potentials
U(x, y)

with the second integral of
motion


)
,
(
2
2
2
2
2
2
N
N
N
N
yN
xN
N
y
x
U
y
x
p
p
H





s
L

β
(
s
)
T
insert

















1
0
0
0
1
0
0
0
0
1
0
0
0
1
k
k



2
2
1
1
)
(











Lk
s
L
sk
L
s

37

Test lattice (for an 8
-
GeV beam)

38

Six quadrupoles

provide a phase advance

of
π

Linear tune: 0.5


1.0









1
0
1
k
Examples of time
-
independent

Hamiltonians


Quadrupole














2
2
2
,
,
y
x
s
q
s
y
x
V






2
2
)
,
(
N
N
N
N
y
x
q
y
x
U


39

0
0.2
0.4
0.6
0.8
0
0.5
1
1.5
2

s
(
)
q
s
(
)
s
β
(s)

quadrupole

amplitude



2
2
2
2
2
2
2
2
N
N
N
N
yN
xN
N
y
x
q
y
x
p
p
H






L

Tunes:

)
1
(
)
1
(
2
0
2
2
0
2
q
q
y
x








Tune spread: zero

Examples of time
-
independent

Hamiltonians


Octupole


40

















2
3
4
4
,
,
2
2
4
4
3
y
x
y
x
s
s
y
x
V














2
3
4
4
2
2
4
4
N
N
N
N
x
y
y
x
U








This Hamiltonian is NOT
integrable

Tune spread (in both x and y) is
limited to
~
12%



2
2
4
4
2
2
2
2
6
4
)
(
2
1
)
(
2
1
y
x
y
x
k
y
x
p
p
H
y
x







B
Tracking with octupoles

41

Tracking with a test lattice

10,000 particles for 10,000 turns

No lost particles.


50 octupoles in a 10
-
m long drift space

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







)
,
(
2
2
2
2
2
2
y
x
U
y
x
p
p
H
y
x





)
,
(
2
2
y
x
D
Cp
p
Bp
Ap
I
y
y
x
x





,
,
2
,
2
2
2
ax
C
axy
B
c
ay
A





42

Second integral:

Darboux equation (1901)


Let
a


0

and
c


0
, then we will take
a

=
1



General solution






ξ
: [1, ∞],
η

: [
-
1, 1],
f
and
g

arbitrary functions






0
3
3
2
2
2







y
x
xy
yy
xx
xU
yU
U
c
x
y
U
U
xy

2
2
)
(
)
(
)
,
(







g
f
y
x
U









c
y
c
x
y
c
x
c
y
c
x
y
c
x
2
2
2
2
2
2
2
2
2
2














43

The second integral


The 2
nd

integral




Example:






2
2
2
2
2
2
2
2
)
(
)
(
2
,
,
,












g
f
c
p
c
yp
xp
p
p
y
x
I
x
x
y
y
x



2
2
2
1
)
,
(
y
x
y
x
U





1
2
)
(
2
2
2
1





c
f


2
2
2
1
1
2
)
(





c
g





2
2
2
2
2
,
,
,
x
c
p
c
yp
xp
p
p
y
x
I
x
x
y
y
x




44

Laplace equation


Now we look for potentials that also satisfy the Laplace
equation (in addition to the Darboux equation):




We found a family with 4 free parameters (b, c, d, t):


0


yy
xx
U
U

















acos
1
)
(
acosh
1
)
(
2
2
2
2
t
b
g
t
d
f







2
2
)
(
)
(
)
,
(







g
f
y
x
U
45

The Hamiltonian


So, we found the integrable Laplace potentials…







2
2
2
2
2
2
2
2
)
(
)
(
2
2
2
2
,
,
,











g
f
y
x
p
p
p
p
y
x
H
y
x
y
x
F
F
F
c
=1,
d
=1,
b=t
=0

c
=1,
b
=1,
d=t
=0

c
=1,
t
=1,
d=b
=0

→ 1 at
x,y
→ ∞

→ 0 at
x,y
→ ∞

→ ln(
r
) at
x,y
→ ∞

46

The
integrable

Hamiltonian (elliptic
coordinates)

47

B
U(x,y):
c
=1,
t
=1,
d=
0,

)
,
(
2
2
2
2
2
2
y
x
kU
y
x
p
p
H
y
x
































...
315
128
35
16
15
8
3
2
Re
)
,
(
10
8
6
4
2
iy
x
iy
x
iy
x
iy
x
iy
x
y
x
U
Multipole expansion

|k| <
0.5 to provide linear stability for small

amplitudes

Tune spread:

)
1
(
)
1
(
2
0
2
2
0
2
k
k
y
x








at small amplitudes

2
0
2
2
0
2






y
x
at large amplitudes

Max. ~100% in
y

and ~40% in
x

2



b
Example of trajectories


25 nonlinear lenses in a drift space


Small amplitude trajectories

48

0.04

0.02

0
0.02
0.04
0.04

0.02

0
0.02
0.04
px
j j
x
j j
0.04

0.02

0
0.02
0.04
0.04

0.02

0
0.02
0.04
py
j j
y
j j
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
f1
j
f2
j
f3
j

j
bare tune

y

x

Large amplitudes

49

2

1

0
1
2
2

1

0
1
2
px
j j
x
j j
2

1

0
1
2
2

1

0
1
2
py
j j
y
j j
2

1

0
1
2
2

1

0
1
2
y
j j
x
j j
0.5
0.6
0.7
0.8
0.9
1
0
10
20
30
40
f1
j
f2
j
f3
j

j
y

x

bare tune

More examples of trajectories


Trajectory encircles the singularities (
y
=0
, x=
±
1)


2
)
(

and

,
1
2
)
(
4
2
4











g
f
c
=1,
d
=1,
b=t
=0

50

Normalized coordinates:

Polar coordinates


Let
a


0

and
c

=
0


2
2
2
2
)
(
)
(
2
2
2
r
g
r
f
r
p
p
H
y
x
















2
cos
2
sin

)
(
ln

)
(
t
b
g
r
d
r
f






)
(
2
2

g
yp
xp
I
x
y



51

Two types of trajectories


d =
-
1
, b =
0
,

and

t
= 0.1











2
cos
2
sin

)
(
ln

)
(
t
b
g
r
d
r
f



52

Parabolic coordinates


a = 0


Not considered by Darboux (but considered by Landau)





Equation for potential:

)
,
(
2
2
y
x
D
Cp
p
Bp
Ap
I
y
y
x
x





x
C
y
B
A




0



0
3
2




xx
yy
y
xy
U
U
y
U
xU
53

Parabolic coordinates


Solution




The only parabolic solution is
y
2

+ 4
x
2




Potentials that satisfy the Laplace equation




.
2
)
(
)
(
)
(
)
(
,
2
)
(
)
(
)
,
(
r
x
r
g
x
r
x
r
f
x
r
p
xp
yp
I
r
x
r
g
x
r
f
y
x
U
y
y
x













r
x
r
g
x
r
f
y
x
r
p
p
H
y
x
2
)
(
)
(
)
(
10
3
2
2
2
2
2
2
2
2
2
2














2
2
2
2
)
(
)
(
x
r
t
x
r
d
x
r
g
x
r
t
x
r
b
x
r
f










54

Example of trajectories


Trajectories never encircle the singularity

F


r
x
r
y
x
y
x
U
2
1
.
0
)
(
10
3
,
2
2




55

Summary


The lattice can be realized by both






and




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










1
0
1
,
k
M
y
x










1
0
1
,
k
M
y
x
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:



Ermakov invariant is what we now call Courant
-
Snyder
invariant in accelerator physics

59

0
)
(




z
s
K
z
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