Weak vs. Strong Focusing - Todd...

USPAS: Lecture on Strong Focusing Theory Vasiliy Morozov, January 2015 -- 1 --

Weak vs. Strong Focusing• Weak focusing rings tend to have beams with large transverse dimensions• 69,000 ton yoke of Dubna’s 10 GeV Synchrophasotron vs. NICA booster


Strong Focusing• Using alternating focusing and defocusing lenses can provide stable transverse dynamics

with stronger focusing and therefore higher betatron tunes.• Lattice can be built using combined-function magnets with alternating gradients (AGS,

Serpukhov) or separate dipole and quadrupole magnets (separated-function lattices, e.g. RHIC).

Horizontal plane:

Vertical plane:


Stability of Linear Motion• The transfer matrix of a periodic cell can be constructed by successively multiplying the

matrices of its constituting elements: dipoles, quadrupoles, and drifts. It could be the matrix of the whole ring or of the periodic section of the ring.

• Let us examine its eigenvalues to learn about the stability of the linear motion.Characteristic polynomial of an nn matrix:

Characteristic equation:

1 0z zM

a bM

c d

10

( ) de (t( ) )n

iM i i

ii

n

AP M I

101

(0) det( ) , tr( ), 1n

M i ni

AA P M A M

( ) 0MP


Stability of Linear Motion• Characteristic equation of a 22 matrix:

• Since

• Either both roots are real or

then

• The roots of the characteristic equation

2 2tr( ) 1 ( ) 1 0M a b

1 2det( ) 1M

12

1

21*

* *2 2

221 2 1

1 1 1

2 2

1 11 1 1 1tr( ) tr( ) 1, tr( ) tr( ) 12 2 2 2

M M M M


Liapunov Stability• A point is called a fixed point of the map if . A system is said to be Liapunov

stable about a fixed point if for each there exists a such that if then for all .

• I.e. there is some neighborhood about the fixed point which remains bounded for all time.• For the linear transfer matrix

• Cases with degenerate eigenvalues

are either not bounded such as

or bounded but unstable (an integer or half-integer betatron tune).• Cases with non-generate eigenvalues but with zero off-diagonal terms, i.e.

are also obviously unstable, e.g.

0x M 0 0Mx x

0x 0 0 0x x

0n nM x M x 0 n

0

00

x

1 2 1 21, 1

1 1 0drift: , thin quad:

0 1 1/ 1l

f

0b c

1.1 0 0.9 0 or

0 1/1.1 0 1/ 0.9


Liapunov Stability• Assuming eigenvalues are not degenerate and we can find two linearly independent

eigenvectors (case of and can be considered similarly)

• Expand a vector in terms of the eigenvectors

• The trajectory deviation after n turns

Since the motion can be unstable if either

0b 0c 0b

1,2

1,2 1,1,

2 ,2

1 2

=b ab bx ba b

x cb dx xc d

21,2 1, ax

1 21 2,

a ab b

1 2x A B

2 2 2 20 1 1 1 1| || | | || |n n n n n nnD B A BM x M x A

12

1

| | 1i


Liapunov Stability• If

then

• If

• If

• corresponds to the case of generate eigenvalues

| tr( ) | 2M tr( ) 2cos for some real angle M

21,2

1 tr( ) [tr( )] sin4 cos2

ii eM M

1 212 21| || | | || | | | | | bounded for all n nnD A B A B n

tr( ) 2M tr( ) 2cosh for some real 0M

1,2 cosh sinh e

1~ ) | | goes to infinit( y as nn AeD n

tr( ) 2M tr( ) 2cosh for some real 0M

1,2 cos si hh n e

2~ ) | | goes to infinit( y as nn BeD n

tr( ) 2M


Parameterization of Stable Matrix• Consider the case of stable motion

• The diagonal elements can be written as

• Since and we can define

• Partition as

2 tr( ) 2cos 2M

a bM

c d

cos , cos where ( ) / 2a k d k k a d 2 2 2 2det( ) cos sin1bc M kad k

| cos | 1, sin 0 sin where is realk

2 2(1 )sinbc bc

2

sin1 sin sin

b

c


Matrix Parameterization• A stable transfer matrix can be written as

• Since the sign of is ambiguous, we may choose (and therefore ) to be positive.

• and are often called the Twiss parameters.

coscos

cos s

sin sinsin sin

1 00 1

cos sin

in

J

I

M

J

2

22

2

sin

det( ) 1

00

cos J

J

J I

M I J e

, ,


FODO Cell Example• One of the simplest periodic building blocks


FODO Cell Example• Let’s examine its transport matrix for horizontal motion

coscos

0.8250051897 16.062255530.1559954483 1.82500

sin sinsin s

1

in

5 9

M

1

11

12

21

1 1tr( ) (2 2

cos (0.5) / 3, / (2 ) 1/ 6

sin 3 / 2

( cos ) / sin 3 / 2) 1.53

/ sin 3 / 2) 18.5

cos 0.8250051897 1.82500519) 0.5

( 0.8250051897 0.5) / (

16.06225553 / (

0.1559954

5

/ sin 4

x

x x x

x

x x x

x x

x x

M

Q

M

M

M

3 / 2)83 / 8( 0.1


Analytic Approach• Equations of motion

2 2

0

11 1 ( ) 12

sqA xH x yp

,dx H dx Hds x ds x

0

1sAdx H qxds x p x

1 1, 1 / 1 /

sx y

sA AB Bx y x x

0 20 0

1 11 ( )( )

y yy x

B Bq x qx B x x k s xp x p x s


Analytic Approach• Equations of motion

where

( )( )

( ) 0

x

y

x k s xs

y k s y

20

0

1( )

( )

yx

yy

Bqk sp xBqk s

p x


Hill’s Equation• Let us first consider the general homogeneous equation

with

• According to Floquet’s theorem, solutions have the form

where is the amplitude function periodic with the same period andis a non-periodic phase of the oscillations.

( ) 0z k s z

( ) ( )k s L k s

( ) ( ) ( )( ) ( ) cos ( )( ) ( )sin ( )

A B

A

B

z s z s z sz s Aw s sz s Bw s s

( )w s L( )s


Hill’s Equation• Substituting one of the solutions into the Hill’s equation

• This gives

2

2

( ) cos sin

( ) cos 2 sin sin cos

[( (2 )sin)cos ] 0

A

A

z s Aw Aw

z s Aw Aw Aw Aw

A kww w w w

2

2

3

20

10 1

0

/w ww

kw

w kw

w w

w


General Solution• The general solution is

• Initial conditions at

( ) cos sinsin cos( ) ( cos ) ( sin )

cos sin( , )sin coscos sin

z s Aw Bw

z s A w B ww w

w wz A AF w

z B Bw ww w

0s s

0 010 0 0 0

0 0

010 0

0

10 0

( , ) ( , )

( , ) ( , ) ( , )

( ) ( , ) ( , )

z zA AF w F w

z zB Bzz A

F w F w F wzz B

M s F w F w


General Solution

00 0 0 0

010 0

00 0 0 0

0

10 0

0 0

cos sin( , ) sin coscos sin

det( ) 1cossin sin

( , )sincos cos

( , ) ( , )cosincos sin

sin coscos sin

w wF w

w ww w

F

w ww

F ww w

w

M F w F w

ww w

w ww w

00 0

0

00 0 0 0

0

s sin

sincos cos

ww

w ww


General Solution

0 0 00

0 0

0 0 0 00 0

0 0

00 0

( ) ( )( ) ( )( )( ) cos[ ( ) ] ( ) sin[ ( ) ]

( ) ( ) sin[ ( ) ]

1 ( ) ( ) ( )( ) sin[ ( ) ] cos[ ( ) ]( ) ( )

( ) cos[ ( ) ] ( )sin[( )

a s b sM

c s d sw sa s s w s w sw

b s w s w s

w s w w s w w w sc s s sw s w w s w

wd s s w w sw s

0( ) ]s


General Solution• Defining

• is called the betatron function, is called the betatron phase, and is sometimes called a correlation function. Since is periodic, both and are also periodic with the same period.

• We note

2

0

( ) ( )1( ) ( ) ( )2

( ) ( )

s w s

s w s w s

s s

0 0 0 00 0

0 0 0 0

0 0 0 0

0 0

www w ww

w w ww www w w w ww

ww

( )s ( )s ( )s( )w s ( )s ( )s


General Solution

• Note

• Matrix for one period

0 00

0 0 0

0

( )[cos ( ) ( )] ( ) ( )

( )[ ( ) ]cos ( ) [1 ( )]

sin si

sin ( ) [cos ( ) ( ) ( )]( )

n

i(

n)

s

s s s s s

M ss s s s s s s

ss

0

0 0

0 0 0 0

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( )

( ) ( )( )

(

( ) ( )

) ( )

z s a s z b s zz s a s z b s z c s z d s z c s a s d s b s

z s M s z

a s b sM s

a s b s

2

cos( ) 1 sin ( )

s

c

in

os

sin

sinM L

s


Betatron Tunes• Betatron phase advance over one period

• The number of oscillations per turn is called a betatron tune

• The betatron tunes are global parameters of the ring, i.e. the phase advance does not depend on the starting point of the integration.

• In an uncoupled case, the fraction parts of the tunes can be readily obtained from the diagonal 22 blocks of the transfer matrix:

21 1

( )

s L s L s L

s s s

ds ds dssw

,,

,

12 2

1x yx y

x y

Q ds

12

,

2,tr( )1 cos

2 2x y

x y

Mq


Courant-Snyder Invariant• The particle trajectory can be written as

where

0 0 0 00

0 00 0 0

0 0

0 0 0

( )( ) [cos ( ) ( )] ( ) ( )sin sin

cos sin

[cos cos sin cos[

( ) ( ) ( )

( ) ( ) sin )] ( ) ( ) ](

sz s s s z s s z

s s z s

W s s s W

z z

s s

0 0 0 0 0 0 00 0

0 00 0 0

0 0

22 20

0 0 0 0 0 0 0 0

2 2

2 20 0 0 0

00

/ /sincos ,

1 2 2

zW W

W z

z z

z

z

z

z z z zz z z


Courant-Snyder Invariant• We can show that this quantity is conserved along the orbit

2

2

2

22

2 2

2

2 22

1 1cos sin cos sin2

cos sin

cos sin

cos sin

cos sin cos sin

[( 2

cos

cos

cos

2

cos 2 cos

)cos 2 2(

z W

W

zz W

zz

W W

z

z W W

W

Wz

z zW

W

2) sinco n ]s si W


Courant-Snyder Invariant• An invariant of motion called Courant-Snyder invariant

• This equation describes a skew ellipse in the phase space with an area of

• For a periodic lattice, the particle lies on an ellipse determined by its initial conditions.• Since the Twiss parameters are independent of initial conditions, there is a family of similar

inscribed ellipses corresponding to different values of .

2 2 2 20 0 0 0 0 0 02 2W z z zzz z z z

W

W


Emittance• A particle inside a certain ellipse has a phase space trajectory that stays inside the ellipse on

successive turns since it is also an inscribed non-intersecting ellipse.• Similarly, any particle outside an ellipse stays outside.• The phase space region contained within the largest ellipse than an accelerator is able to

transport is called the acceptance or admittance.• It is convenient to choose an ellipse

containing a certain fraction of the beam, e.g. 39% (1 or rms), 90% (4.6), or 95% (6). The ellipse will always contain the same fraction of the beam. This is another manifestation of Liouville’s theorem.

• The area of the selected ellipse is called the emittance. It is quoted with the factor of explicitly written out.

2 22 zzz z


Beam Envelope• The trajectory of a particle is given by

• The transverse amplitude is

• The rms envelope of the beam is

Thus, can be interpreted as the beam envelope function specifying the transverse size of the beam as a function of . Of course, the horizontal and vertical beam sizes are determined by their respective betatron functions and emittances.

0( ) cos(( ) ( ) )z s W s s

( )maxz W s

( )ss

( )rmsrmsz s


Adiabatic Invariants• Let us find an invariant of motion preserved during acceleration. The rate of energy change

during acceleration is slow on the time scale of betatron oscillations. An adiabatic approximation applies, i.e. we can assume a slow approximately uniform acceleration of particles with no heating of the beam in its rest frame.

• The adiabatic approximation allows use of the Poincaré-Cartan integral invariant:

where and are canonically conjugate coordinates and the integration is over one period of the betatron oscillations.

• For horizontal motion:

const.I pdq p q

( )x

q xds dxp mx m m x pxdt ds


Adiabatic Invariants• The horizontal invariant becomes

where we found a quantity invariant under acceleration called the normalized emittance:

• Similarly, for the vertical motion:

• If the energy is adiabatically increased the area of the ellipse remains constant while the area of the ellipse shrinks. This effect is called adiabatic damping.

*x x x xI p x dx p mc mc

*x x

*

*

y y y y

yy

I p y dy p mc mc

( , )xx p( , )x x


Dispersion• To account for momentum deviation, consider the horizontal equation of motion with the

inhomogeneous term:

• The solution may be written as

where and are solutions of the homogeneous equation and is a particular solution of the inhomogeneous equation with having the initial conditions

• The slope is given by

• The constant-energy trajectory equation in the matrix form is then

/ ( )/

( )x k s x sp p

0 0 0( )( ( () ))x S s xC s ss x D

(0) 1 (cosine-line component)(0) 0 (sin

(0)e-like component)

(0) (0)(0

0)

CC

SS

D D

( )C s ( )S s ( )D s1

0 0 0( ) () )( ( ) S sx s C s D sx x

0

0

0

( ) ( ) ( )( ) ( ) ( )0 0 1

x C s S s D s xx C s S s D s x


Dispersion• Consider the eigenvalues of a 1-turn matrix. The characteristic equation is

The eigenvalues of the 22 matrix were obtained before

and

• Let us write the eigenvector corresponding to the 3rd eigenvalue as

• The periodic function is called the dispersion function

where the matrix elements are evaluated for the periodic cell starting at position .

3 3 2 2) )(1 )( 0(P P

( ) ( ) ( )( ) ( ) ( )

1 0 0 1 1

C s S s D sC s S s D s

2

2

1,2 2 221 1tr( ) tr( ) 12 2

iM M e

3 1

1

s


Dispersion• Solving for and

• The dispersion, nominally, is the closed orbit for a particle with in linear approximation. Of course, the approximation and therefore the dispersion are only applicable when .

1

22

( ) ( ) ( )( ) ( )

1 ( ) ( ) ( )( ) 1 ( )

1 ( ) ( )det 1 ( ) ( ) ( ) ( ) ( ) ( )

( )

( )

( )

1 ( )

2 tr(

C s S s D sC s S s

C s S s D sC s S s

C s S sC S S s C S S s S s C s

C s S s

M

D s

D s

) 2(1 cos1 ( ) ( ) ( ) 1 ( ) ( ) ( )1

( ) 1 ( ) ( ) 1 ( )2(1 cos[1 ( )] ( ) ( )1

( ) ( ) [1 ( )]2

)

( ) ( ))( )(( cos )1 )

C s S s D s S s S s D sC s S s C s C s

S sD s

D s S sC s D s

D sC

D s

D ss

1 1


Dispersion• Note that the periodic dispersion function is not the same as the matrix

element .• is the inhomogeneous part of the trajectory due to the dispersion.• The total horizontal trajectory component consists of a part due to the betatron oscillations

and the part due to the dispersion:

• Since the two components are independent, their contributions to the total beam size must be added in quadrature:

using rms values of the involved quantities.

( ) /s dx d /( )s xD

) )( (x s s

( ) ( ) ( ) ( ) ( )x s x s x s x s s

2 2 2( ) ( ) ] ( ) ( ) ]( ) [ [tot rmss s s s s


Momentum Compaction• The momentum compaction: relative change of circumference per unit change in relative

momentum offset

• The circumference

• The momentum compaction then becomes

//p

L Lp p

0 0

1

( ) (

,

( )

)( )

L L ds L L d

d d d ds L ds

L

x xx

x s sL ds

ds s ds

( )(

1)p

s dsL s

Weak vs. Strong Focusing - Todd...

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