Quadrupole Transverse Beam Optics

Chris Rogers

2 June 05

Plan

1. Equations of Motion

2. Transfer Matrices

3. Beam Transport in FoDo Lattices

4. Bunch Envelopes

5. Emittance Invariant• I haven’t had time to do numerical techniques (e.g. calculating beta

function)

• I had hoped to introduce Hamiltonian dynamics but again no time.

• This is quite a mathsy approach - not many pictures

• I don’t claim expertise so apologies for mistakes!

Quadrupole Field

• Quad Field “hyperbolic”

• Transverse focusing & defocusing• Used for beam containment

0

)(

)(

1

1

zxB

zyB

Bquad

Hyperbolicpole faces

Field gradientr

B

Equations of Motion - in z• Lorentz force given by

• Constant energy in B-field => relativistic constant

)( EBvqdt

pdm

Equations of Motion - in z• Lorentz force given by

• Constant energy in B-field => relativistic constant

• Use chain rule so d/dt = vzd/dz

)( EBvqdt

pdm

yzz Bqvxmv ''2 xzz Bqvymv ''2and

dt

pd Bv

Equations of Motion - in z• Lorentz force given by

• Constant energy in B-field => relativistic constant

• Use chain rule so d/dt = vzd/dz

• Substitute for B-field to get SHM (Hill’s equation)

)( EBvqdt

pdm

yzz Bqvxmv ''2

0'' 1 xBp

qx

z

0'' 1 yBp

qy

z

and

xzz Bqvymv ''2and

zzvp

Focusing strength K (dependent on m/q)Signs!

• Recall solution of SHM

• Take e.g. K>0 solution with

Transfer Matrices 1

• Recall solution of SHM

)cosh(

)cos(

)( 0

za

yaz

za

zyK>0

K=0

K<0

)sin(' zKaKy

Transfer Matrices 1

• Recall solution of SHM

• Take e.g. K>0 solution with

• Use double angle formulae

)cosh(

)cos(

)( 0

za

yaz

za

zyK>0

K=0

K<0

)sin(' zKaKy

zKazKay coscossinsin zKKazKKay cossinsincos'

Transfer Matrices 1

• Recall solution of SHM

• Take e.g. K>0 solution with

• Use double angle formulae

)cosh(

)cos(

)( 0

za

yaz

za

zyK>0

K=0

K<0

)sin(' zKaKy

zKazKay coscossinsin zKKazKKay cossinsincos'

)0( zyK )0(' zy

K

zy )0(' )0( zy

Transfer Matrices 2

• This is tidily expressed as a matrix

• This is no coincidence– Actually, this is the first order solution of a

perturbation series– Can be seen more clearly in a Hamiltonian

treatment

• What do the matrices for K=0, K<0 look like?

)0('

)0(

cossin

sin1

cos

)('

)(

y

y

zKzKK

zKK

zK

zy

zy

0101 YMY

Transfer Matrices for other K

• Quote:

• Assumes K is constant between 0 and z– Introduce “effective length” l to deal with fringe fields

)0('

)0(

cossin

sin1

cos

)('

)(

y

y

zKzKK

zKK

zK

zy

zy

)0('

)0(

10

1

)('

)(

y

yz

zy

zy

)0('

)0(

coshsinh

sinh1

cosh

)('

)(

y

y

zKzKK

zKK

zK

zy

zy

K=0

K<0

K>0

FoDo Lattice - an example• It is possible to contain a beam transversely using

alternate focusing and defocusing magnetic quadrupoles (FoDo lattice)

• This is possible given certain constraints on the spacing and focusing strength of the quadrupoles

• We can find these constraints using certain approximations

Thin Lens Approximation

• In thin lens approximation,

• Define focusing strength

• Then

0lK

Klf

1

1cosh

cos

lK

lK

0

sinh1

sin1

lKK

lKK

fKl

lKK

lKK 1

sinh

sin

Thin Lens Transfer Matrices

• Transfer matrices become

• Should be recognisable from light optics

1

101

cossin

sin1

cos

fzKzKK

zKK

zK

1

101

coshsinh

sinh1

cosh

fzKzKK

zKK

zK

Multiple Components

• We can use the matrix formalism to deal with multiple components in a neat manner

• Say we have transport matrices M10 from z0 to z1 and M21 from z1 to z2 so

• What is transfer matrix from z0 to z2?

0101 YMY 1212 YMY and

Multiple Components

• We can use the matrix formalism to deal with multiple components in a neat manner

• Say we have transport matrices M10 from z0 to z1 and M21 from z1 to z2 so

• Then– with 020010212 YMYMMY

0101 YMY 1212 YMY

102120MMM

and

Transfer matrix for FoDo Lattice

• Wrap it all together then we find that the transfer matrix for a FoDo lattice is

10

11

101

10

11

101 l

f

l

fM

FoDo

f

l

f

l

f

lf

ll

f

l

421

2

421

2

2

2

Stability Criterion• What are the requirements for beam containment - is

FoDo really stable?

• Transfer Matrix for n FoDos in series:

• For stability re quire that Mtot is finite & real for large n.

• Route is to solve the Eigenvalue equation (mathsy)

• Then use this to get a condition on f and l

n

FoDototMM

YYM

Eigenvalues of FoDo lattice• Standard way to solve matrix equation like this - take the

determinant

• Then we get a quadratic in

• Neat trick - define such that• Giving eigenvalues

– Try comparing with quadrupole transport M

0 YYM

0)()( 2112221122112 mmmmmm

2cos 2211 mm

iei sincos1

iei sincos2

Transfer Matrix ito eigenvalues

• Then we recast the transfer matrix using eigenvalues, and remaining entirely general

• Here I is the identity matrix and J is some matrix with parameters

• Then state the transfer matrix for n FoDos

sincos JIM

J

)sin()cos( nJnIM n

Stability Criterion• For stability we require cos(n), sin(n) are finite for

large n, i.e.

• Recalling the transfer matrix for the FoDo lattice, this gives

• or

2)(2

1

FoDoMtrace

14

f

l

242

12

12

2

f

l

f

l

f

l

Bunch Transport• We can also transport beam envelopes using the transfer

matrices

• Say we have a bunch with some elliptical distribution in phase space (x, x’ space)– i.e. density contours are elliptical in shape

• Ellipse can be transported using these transfer matrices

Density contour

Contour equation• General equation for an ellipse in (x,x’)

given by

• Or in matrix notation

1'

)'(

x

x

ca

abxx

11

0 XVX T

Transfer Matrices for Bunches

• We can transport this ellipse in a straight-forward manner– We have

– What will V1, the matrix at z1, look like?

xMx10

Transfer Matrices for Bunches

• We can transport this ellipse in a straight-forward manner– We have– Define the new ellipse using

– So that

xMx10

TMVMV100101

110

1

110 XMVXM T

Emittance• Define un-normalised emittance as the area

enclosed by one of these ellipses in phase space– E.g. might be ellipse at 1 rms (so-called rms

emittance)• Or ellipse that contains the entire/95%/whatever of the

bunch

• Area of the ellipse is given by

• e.g. for rms emittanceV

2222 )',()'()( xxxxV

Emittance Conservation• Claim: Emittance is constant at constant momentum

– 0=1 if |V1| = |V0|

– Use |A B| = |A| |B|

– Then requirement becomes |M10|=1

• Consider as an example

– State principle that to 1st order |M|=1 for all “linear” optics1sincoscossin

sin1

cos 22

zKzKzKzKK

zKK

zK

Normalised emittance

• Apply some acceleration along z to all particles in the bunch– Px is constant

– Pz increases

– x’=Px/Pz decreases!

• So the bunch emittance decreases– This is an example of something called

Liouville’s Theorem

– ~“Emittance is conserved in (x,Px) space”

• Define normalised emittance

m

p zn

Summary• Quadrupole field => SHM• We can transport individual particles through

linear magnetic lattices using transfer matrices• Multiple components can be strung together by

simply multiplying the transfer matrices together• We can use this to contain a beam in a FoDo

lattice• We can understand what the bunch envelope

will look like• We can derive a conserved quantity emittance