Beam Transfer Lines• Distinctions between transfer lines and circular machines• Linking machines together• Trajectory correction• Trajectory correction• Emittance and mismatch measurement• Blow-up from steering errors, optics mismatch and thin screens• Phase-plane exchange

Brennan GoddardCERN

Injection, extraction and transfer

CERN Complex• An accelerator has limited dynamic range.

• Chain of stages needed to h hi hreach high energy

• Periodic re-filling of storage rings, like LHCExternal experiments like• External experiments, like CNGS

Transfer lines transport thebeam between accelerators,and onto targets, dumps,instruments etc.

LHC: Large Hadron ColliderSPS: Super Proton SynchrotronAD: Antiproton DeceleratorISOLDE: Isotope Separator Online DevicePSB: Proton Synchrotron BoosterPS P t S h tPS: Proton SynchrotronLINAC: LINear AcceleratorLEIR: Low Energy RingCNGS: CERN Neutrino to Gran Sasso

Normalised phase space• Transform real transverse coordinates x, x’ by

⎥⎦

⎤⎢⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡⋅=⎥

⎦

⎤⎢⎣

⎡⋅=⎥

⎦

⎤⎢⎣

⎡'

011' x

xxx

SSS βαβN

'XX

1 xβ

⋅=X

⎦⎣⎦⎣⎦⎣⎦⎣ SSS ββ

'1 xx

S

βα

β

+'X 'xx SSS

βαβ

+⋅=X

Normalised phase space

Real phase space Normalised phase space

1

γαε−x’

ε='X

'X

γε 1

βε 1

γε=max'x

ε=maxX

⇒x

β

Area = πεArea = πε

⇒X

βε=maxxε=maxX

22 ''2 xxxx ⋅+⋅⋅+⋅= βαγε 22 'XX +=ε

General transport

y

s

Beam transport: moving from s1 to s2 through n elements, each with transfer matrix Mi

xx

y ss

1

2

⎥⎤

⎢⎡

⋅⎥⎤

⎢⎡

=⎥⎤

⎢⎡

⋅=⎥⎤

⎢⎡

21'2 xSCxx

M

1

∏=n

21 MM⎥⎦

⎢⎣

⎥⎦

⎢⎣

⎥⎦

⎢⎣

⎥⎦

⎢⎣

→ ''''21'2 xSCxx

M ∏=

→i

n1

21 MM

( )

( ) ( )[ ] ( )⎥⎥⎥⎤

⎢⎢⎢⎡

ΔΔΔΔ

ΔΔ+Δ=→ β

μββμαμββ

ii1

sinsincos 2111

2

21MTwiss parameterisation

( ) ( )[ ] ( )⎥⎥⎦⎢

⎢⎣

Δ−ΔΔ+−Δ− μαμββμααμααββ sincossin1cos1

22

12121

21

Circular Machine

Circumference = L

( ) ⎥⎤

⎢⎡ +

==QQQ πβπαπ

12sin2sin2cos

2MMOne turn ( ) ⎥⎥⎦⎢

⎢⎣

−+−== →→ QQQL παππαβ 2sin2cos2sin11 2021 MMOne turn

• The solution is periodic p• Periodicity condition for one turn (closed ring) imposes α1=α2, β1=β2, D1=D2

• This condition uniquely determines α(s), β(s), μ(s), D(s) around the whole ring

Circular Machine

• Periodicity of the structure leads to regular motion– Map single particle coordinates on each turn at any location

– Describes an ellipse in phase space, defined by one set of α and βvalues ⇒ Matched Ellipse (for this location)

x’

βα 2

max1' += ax βαγ 22 ''2 ⋅+⋅⋅+⋅= xxxxa

xArea = πa β

αγ21+=

βax =max

Circular Machine

• For a location with matched ellipse (α, β), an injected beam of emittance ε, characterised by a different ellipse (α*, β*) generates (via filamentation) a large ellipse with the original α β but larger ε(via filamentation) a large ellipse with the original α, β, but larger ε

x’

α, β

x’

α, β

xx

Turn 1 Turn 2

After filamentation

εο, α∗ , β∗

After filamentation

εο, α∗ , β∗

ε > εο, α, βε > εο, α, βTurn 3 Turn n>>1

Matched ellipse determines beam shape

Transfer line

⎥⎦

⎤⎢⎣

⎡⋅=⎥

⎦

⎤⎢⎣

⎡→ '21'

2

2

xx

xx

MOne pass:

⎥⎦

⎤⎢⎣

⎡'1

xx

⎥⎦

⎤⎢⎣

⎡'2

2

xx

⎤⎡ β

⎦⎣ 1x ⎦⎣ 2x

( )

( ) ( )[ ] ( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Δ−ΔΔ+−Δ−

ΔΔ+Δ=→

μαμββμααμααββ

μββμαμββ

sincossin1cos1

sinsincos

22

12121

21

2111

2

21M

⎦⎣

• No periodic condition existsp• The Twiss parameters are simply propagated from beginning to end of line• At any point in line, α(s) β(s) are functions of α1 β1

Transfer line• On a single pass there is no regular motion

– Map single particle coordinates at entrance and exit.

– Infinite number of equally valid possible starting ellipses for single particleInfinite number of equally valid possible starting ellipses for single particle……transported to infinite number of final ellipses…

L→0Mx’

x’

α1, β1

α∗ β∗α2, β2

⎥⎤

⎢⎡ 2x

⎥⎦

⎤⎢⎣

⎡'1

1

xx

x

α∗1, β∗1 ⎥⎦

⎢⎣

'2x

x xTransfer Line

βα∗2, β∗2Entry Exit

Transfer Line• Initial α, β defined for transfer line by beam shape at entrance

x’ x’α∗, β∗x’ x’α∗, β∗

α, βα, β

x xx x

Gaussian beamNon-Gaussian beam(e.g. slow extracted)

Gaussian beamNon-Gaussian beam(e.g. slow extracted)

• Propagation of this beam ellipse depends on line elementsPropagation of this beam ellipse depends on line elements

• A transfer line optics is different for different input beams

Transfer Line

350

• The optics functions in the line depend on the initial values

Design β functions in a transfer line

200

250

300

[m]

- Design βx functions in a transfer line− βx functions with different initial conditions

50

100

150

Bet

aX [

1500 2000 2500 30000

50

S [m]

• Same considerations are true for Dispersion function:Same considerations are true for Dispersion function:– Dispersion in ring defined by periodic solution → ring elements

– Dispersion in line defined by initial D and D’ and line elements

Transfer Line

• Another difference….unlike a circular ring, a change of an element in a line affects only the downstream Twiss values (including di i )

300

350

dispersion)- Unperturbed βx functions in a transfer line− βx functions with modification of one quadrupole strength

150

200

250

Bet

aX [m

]

10% change in this QF strength

1500 2000 2500 30000

50

100

1500 2000 2500 3000S [m]

Linking Machines

• Beams have to be transported from extraction of one machine to injection of next machine– Trajectories must be matched, ideally in all 6 geometric degrees of freedom

(x,y,z,θ,φ,ψ)

• Other important constraints can includeOther important constraints can include– Minimum bend radius, maximum quadrupole gradient, magnet aperture,

cost, geology

Linking Machines

Extraction

Matched Twiss at extraction propagated to matched Twiss at injection

Transferα1x, β1x , α1y, β1yαx(s), βx(s) , αy(s), βy(s)

s

α2x, β2x , α2y, β2y

Injection

⎥⎤

⎢⎡

⎥⎤

⎢⎡ −

⎥⎤

⎢⎡ 1

222

''''2 ββ

SSSCCSCCSCSC

The Twiss parameters can be propagated when the transfer matrix M is known

⎥⎤

⎢⎡

⎥⎤

⎢⎡

⎥⎤

⎢⎡

⎥⎤

⎢⎡ 2 xSCxx

M⎥⎥⎥

⎦⎢⎢⎢

⎣

⋅⎥⎥⎥

⎦⎢⎢⎢

⎣ −−+−=

⎥⎥⎥

⎦⎢⎢⎢

⎣ 1

122

2

2

'''2'''''

γα

γα

SSCCSSSCCSCC⎥

⎦⎢⎣

⋅⎥⎦

⎢⎣

=⎥⎦

⎢⎣

⋅=⎥⎦

⎢⎣

→ ''''21'2

2

xSCxxM

Linking Machines• Linking the optics is a complicated process

– Parameters at start of line have to be propagated to matched parameters at the end of the line

– Need to “match” 8 variables (αx βx Dx D’x and αy βy Dy D’y)

– Maximum β and D values are imposed by magnet apertures

Other constraints can exist– Other constraints can exist • phase conditions for collimators,

• insertions for special equipment like stripping foils

– Need to use a number of independently powered (“matching”) quadrupoles

– Matching with computer codes and relying on mixture of theory, i i t iti t i l dexperience, intuition, trial and error, …

Linking MachinesF l t f li i lif th bl b d i i th• For long transfer lines we can simplify the problem by designing the line in separate sections– Regular central section – e.g. FODO or doublet, with quads at regular

i ( b di di l ) ith t d i ispacing, (+ bending dipoles), with magnets powered in series

– Initial and final matching sections – independently powered quadrupoles, with sometimes irregular spacing.

150

200

250

300

β [m

]

BETXBETY

0

50

100

0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000S [m]Regular lattice (FODO)

SPS LHC(elements all powered in serieswith same strengths)

Final matching

section

Initial matching

section

SPS to LHC Transfer Line

Extractionpoint

Injectionpoint

Trajectory correction

• Magnet misalignments, field and powering errors cause the trajectory to deviate from the design

• Use small independently powered dipole magnets (correctors) toUse small independently powered dipole magnets (correctors) to steer the beam

• Measure the response using monitors (pick-ups) downstream of the corrector (π/2 3π/2 )corrector (π/2, 3π/2, …)

Corrector dipole Pickup Trajectory

QF QF

π/2

QF

QD QD

QF

• Horizontal and vertical elements are separated

• H-correctors and pick-ups located at F-quadrupoles (large βx )

V t d i k l t d t D d l (l β )• V-correctors and pick-ups located at D-quadrupoles (large βy)

Trajectory correction

• Global correction can be used which attempts to minimise the RMS offsets at the BPMs, using all or some of the available corrector magnets.

• Steering in matching sections, extraction and injection region requires particular care

D and β functions can be large → bigger beam size– D and β functions can be large → bigger beam size

– Often very limited in aperture

– Injection offsets can be detrimental for performance

Trajectory correction

Uncorrected trajectory.

y growing as a result f d i thof random errors in the

line.

The RMS at the BPMs is 3.4 mm, and ymax is , ymax12.0mm

Corrected trajectory.

The RMS at the BPMs is 0.3mm and ymax is 1mm

Trajectory correction

• Sensitivity to BPM errors is an important issue– If the BPM phase sampling is poor, the loss of a few key BPMs can

allow a very bad trajectory, while all the monitor readings are ~zero

Correction with some monitors disabled

y j y, g

With poor BPM phase sampling the correction algorithm produces a trajectory with 185mmtrajectory with 185mm ymax

Note the change of vertical scale

Steering (dipole) errors

• Precise delivery of the beam is important.– To avoid injection oscillations and emittance growth in rings

– For stability on secondary particle production targetsFor stability on secondary particle production targets

• Convenient to express injection error in σ (includes x and x’ errors)

Δ [ ] √ √ 2 2 β 2

Δa

Δa [σ] = √((X2+X’2)/ε) = √((γx2 + 2αxx’+ βx’2)/ε)

X

'X

Septum

XBumpermagnets kicker Mis-steered

injected beam

Steering (dipole) errors

• Static effects (e.g. from errors in alignment, field, calibration, …) are dealt with by trajectory correction (steering).

B t there are also d namic effects from• But there are also dynamic effects, from:– Power supply ripples

– Temperature variations

– Non-trapezoidal kicker waveforms

• These dynamic effects produce a variable injection offset which can vary from batch to batch or even within a batchvary from batch to batch, or even within a batch.

• An injection damper system is used to minimise effect on emittance

Blow-up from steering error• Consider a collection of particles with amplitudes A

• The beam can be injected with a error in angle and position.

• For an injection error Δa (in units of sigma = √βε) the mis-injected• For an injection error Δay (in units of sigma = √βε) the mis-injected beam is offset in normalised phase space by d = Δax√ε

'X Mi i j t dMatched X Misinjectedbeam

Matchedparticles

X

A

d

Blow-up from steering error• The new particle coordinates in normalised phase space are

θcosnew dXX 0 +=

θsinnew dXX '0

' +=

Mi i j t dMatched 'X

• For a general particle distribution, where A denotes amplitude in

Misinjectedbeam

Matchedparticles

X

'222 XXA +=

where A denotes amplitude in normalised phase space A

θ X

2/2A=ε

d

Blow-up from steering errorS if l i th di t

( ) ( )'00

'222 dXdXXXA +++=+=22

θθ sincosnewnewnew

• So if we plug in the new coordinates….

( )

( ) 2'00'2002

2'00'200

dXXDXXA

dXXdXX

++++=

++++=

2

2

2

2

θθ

θθ

sincos

sincos

new ( )

( ) 2'00

0000

dXXD +++= 0 22 θθε sincos

new

0 0

2d+= 02ε

Giving for the emittance increase

2/2/ 022 dA εε +== newnew

• Giving for the emittance increase

( )2/102aΔε +=

Blow-up from steering error

A numerical example….

Consider an offset Δa of 0.5 sigma for ginjected beam

Misinjected beam( )0 2/1 Δεε += 2anew

'X

0ε1.125=X

0.5√ε√ε

X

MatchedBeam

Blow-up from betatron mismatch

• Optical errors occur in transfer line and ring, such that the beam can be injected with a mismatch.

• Filamentation will produce an emittance increase.

• In normalised phase space, consider Mismatchedbeam'Xp p ,

the matched beam as a circle, and the mismatched beam as an ellipse.

beamX

X

Matchedbeam

Blow-up from betatron mismatch

[ ])sin()cos('),sin( 2222222 ooo xx φφαφφβεφφβε +−+=+=

General betatron motion – coordinates of particles on mismtached ellipse

⎤⎡⎤⎡⎤⎡ 2011 xX

applying the normalising transformation for the matched beam (subscript 1)

⎥⎦

⎤⎢⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡⋅=⎥

⎦

⎤⎢⎣

⎡

2

2

111 '011

xx

βαβ22'X

X

an ellipse is obtained in normalised phase space

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

2

121

1

2

1

22

2

121

1

2

2

12 2ββαα

ββ

ββ

ββαα

ββ

ββ

2222

22 'XX'XXA

an ellipse is obtained in normalised phase space

2121212 ⎟⎞

⎜⎛

⎟⎞

⎜⎛− ββββ

βββ

characterised by γnew, βnew and αnew, where

2

121

1

2

2

1

1

2

2

121

1

2⎟⎟⎠

⎜⎜⎝

−+==⎟⎟⎠

⎜⎜⎝

−=ββ

ααββ

ββ

γββ

βββ

ααββ

α newnewnew , ,

Blow-up from betatron mismatch

Mismatched( ) ( )112

112

−−+=−++= HHAbHHAa ,

From the general ellipse properties

'Xbeam

b

22

where

X

ab

( )

⎟⎞

⎜⎛

+⎟⎟⎞

⎜⎜⎛

+

+=

2

2

1211

21

ββααββ

βγ newnewH

AX

⎟⎟

⎠⎜⎜

⎝+⎟⎟

⎠⎜⎜⎝

−+=1

2

2

121

1

2

2

1

2 ββαα

ββ A

λAa

MatchedBeam generally

giving

( ) ( )112

11112

1 −−+=−++= HHHHλ

λ ,

λλ

⋅==

AA

ba

)cos(1),sin( 1new1new φφλ

φφλ +=+⋅= A'XAX

Blow-up from betatron mismatch

)(cos1)(sin 2202

220

2211newnew φφ

λφφλ +++⋅=+= AA'XXA 22

new

We can evaluate the square of the distance of a particle from the origin as

The new emittance is the average over all phases

⎟⎟⎠

⎞⎜⎜⎝

⎛ +++== 22

22 )(cos1)(sin21

21 φφ

λφφλε 11new

20

20

2 AAAnew

⎟⎟⎠

⎞⎜⎜⎝

⎛ +++=

⎟⎠

⎜⎝

22

22

2

)(cos1)(sin21

22

φφλ

φφλ

λ

1120A

0.5 0.5

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=2

20

121

λλε

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+==⎟

⎠⎞

⎜⎝⎛ += 2

2

121

21002

20 2

1121

ββ

ββαα

ββ

ββεε

λλεε Hnew

If we’re feeling diligent, we can substitute back for λ to give

⎟⎠

⎜⎝ ⎠⎝⎠⎝ 121222 ββββλ

where subscript 1 refers to matched ellipse, 2 to mismatched ellipse.

Blow-up from betatron mismatchA numerical example….consider b = 3a for the mismatched ellipse

3/ abλ

Mismatched

3/ == abλ

'X

( )22 11 λλεε +

beamThen

X

( )0

0

67.1

12

ε

λλεε

=

+=new

oεa b=3a

X

MatchedBeam

Emittance and mismatch measurement• A profile monitor is need to measure the beam size

– E.g. beam screen (luminescent) provides 2D density profile of the beam

• Profile fit gives transverse beam sizes σ• Profile fit gives transverse beam sizes σ.

• In a ring, β is ‘known’ so ε can be calculated from a single screen

Emittance and mismatch measurement

• Emittance and optics measurement in a line needs 3 profile measurements in a dispersion-free region

• Measurements of σ0 σ1 σ2 plus the two transfer matrices M01 andMeasurements of σ0,σ1,σ2, plus the two transfer matrices M01 and M12 allows determination of ε, α and β

Ms0 s2s1

M 32→M21→M

2

22

1

21

0

20

βσ

βσ

βσε ===

σ0 σ1 σ2

Emittance and mismatch measurement⎤⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−+−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0

0

0

2111

21

11111111

2111

21

1

1

1

'''2'''''

2

γαβ

γαβ

SSCCSSCSSCCC

SSCC

We have

( )

( ) ( )[ ] ( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+−−

+=⎥

⎦

⎤⎢⎣

⎡

μΔαμΔββμΔααμΔααββ

μΔββμΔαμΔββ

sincossin1cos1

sinsincos

22

12121

21

2111

2

'1

'1

11

SCSC

where

( ) ( )20

0

22

0220222

20

0

21

01102

11 1212 αβ

αββαβ

αββ ++−=++−= SSCCSSCC ,so that

⎥⎦⎢⎣ βββ 221

Using 0

2

0

220

2

0

11

20

0 βσσ

ββσσ

βε

σβ ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛== , ,

W00 21 βα =we find

( ) ( ) ( ) ( )222222 ////// SCSCSS +σσσσ( ) ( ) ( ) ( )( ) ( )2211

11222

1012202

////////

SCSCSCSCSS

−+−−= σσσσ Wwhere

Emittance and mismatch measurement

Some (more) algebra with above equations gives

( ) ( ) ( ) 4/////1 222

222

22

2020

2WW −+−= SCSCSσσβ

2 β

And finally we are in a position to evaluate ε and α0

W1 β020 βσε =

Comparing measured α β0 with expected values gives numerical

W00 2βα =

Comparing measured αo, β0 with expected values gives numerical measurement of mismatch

Blow-up from thin scattererS tt i l t ti i d i th b• Scattering elements are sometimes required in the beam– Thin beam screens (Al2O3,Ti) used to generate profiles.

– Metal windows also used to separate vacuum of transfer lines from vacuum in circular machines.

– Foils are used to strip electrons to change charge state

• The emittance of the beam increases when it passes through dueThe emittance of the beam increases when it passes through, due to multiple Coulomb scattering.

θs

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+=

radradinc

cs L

LLLZ

cMeVpmrad 10

2 log11.01]/[

1.14][β

θrms angle increase:⎠⎝ radradc p ][β

βc = v/c, p = momentum, Zinc = particle charge /e, L = target length, Lrad = radiation length

Blow-up from thin scatterer

Ellipse afterscattering

Each particles gets a random angle change θs but there is no effect on the positions at the scatterer

'X

θβ+=

=

'0

'

0

XX

XXnew

sθβ+0XXnew

After filamentation the particles have different amplitudes and the beam has

X

Matchedellipse

a larger emittance

2/2Anew=ε

Blow-up from thin scatterer+ '222 XXA

Ellipse afterfilamentation

( )2sθβ++=

+=

'020

222

XX

XXA newnewnew'X

uncorrelated( )22

22

2

2 ss

θβθβ

βθθβ

+++

+++=

''22

'0'200

XXXA

XXX

20

22

22

2

ss

ss

θβθβε

θβθβ

++=

+++=

'0

02002

X

XXXA new

0

X

Matchedellipse

202 sθβε +=

20 2 snew θβεε +=

Need to keep β small to minimise blow-up (small β means large spread in angles in beam distribution, so additional angle has small effect on distn.)

Blow-up from charge stripping foilF LHC h i Pb53+ i t i d t Pb82+ t 4 25G V/ i• For LHC heavy ions, Pb53+ is stripped to Pb82+ at 4.25GeV/u using a 0.8mm thick Al foil, in the PS to SPS line

• Δε is minimised with low-β insertion (βxy ~5 m) in the transfer liney

• Emittance increase expected is about 8%

120TT10 optics

80

100

Stripping foil

]

beta Xbeta Y

40

60

Bet

a [m

]

0 50 100 150 200 250 3000

20

S [m]

Emittance exchange insertion• Acceptances of circular accelerators tend to be larger in horizontal

plane (bending dipole gap height small as possible)

• Several multiturn extraction process produce beams which have p pemittances which are larger in the vertical plane → larger losses

• We can overcome this by exchanging the H and V phase planes (emittance exchange)(emittance exchange)

Low energy machine After multi-turn Aft itt High energy machine

y

extraction After emittanceexchange

x

In the following, remember that the matrix is our friend…

Emittance exchangePhase-plane exchange requires a transformation of the form:

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

0

0

2423

1413

1

1

'0000

' xx

mmmm

xx

⎟⎟⎟⎟

⎠⎜⎜⎜⎜

⎝⎟⎟⎟⎟

⎠⎜⎜⎜⎜

⎝

=

⎟⎟⎟⎟

⎠⎜⎜⎜⎜

⎝ 0

0

0

4241

3231

2423

1

1

1

'0000

00

' yyx

mmmm

mm

yyx

A skew quadrupole is a normal quadrupole rotated by an angle θ.

The transfer matrix S obtained by a rotation of the normal transfer matrix Mq:1S = R-1MqR

where R is the rotation matrix ⎟⎟⎟⎞

⎜⎜⎜⎛

θθθθ

sin0cos00sin0cos

⎟⎟⎟

⎠⎜⎜⎜

⎝ −−

θθθθ

cos0sin00cos0sin

( i lf f h t R d b h ki th t i t f d t(you can convince yourself of what R does by checking that x0 is transformed to x1 = x0cosθ +y0sinθ, y0 into -x0sinθ +y0cosθ, etc.)

Emittance exchange⎞⎛

For a thin-lens approximation⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

=

10001000010001

δ

δqM

(where δ = kl = 1/f is the quadrupole strength)

So that⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

−⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

−−

== −

0cos0sinsin0cos0

0sin0cos

01000010001

0cos0sinsin0cos00sin0cos

1

θθθθ

θθδ

θθθθ

θθ

RMRS q

⎟⎟⎞

⎜⎜⎛

⎟⎟⎠

⎜⎜⎝ −⎟⎟⎠

⎜⎜⎝ −⎟⎟⎠

⎜⎜⎝

02i120001

cos0sin0100cos0sin0

θδθδ

θθδθθ

⎟⎟⎟⎟

⎠⎜⎜⎜⎜

⎝ −

=

12cos02sin010002sin12cos

θδθδ

θδθδ

N l d45º k d

For the case of θ = 45º, ⎟⎟⎟⎞

⎜⎜⎜⎛

=0100001

δS

Normal quad45º skew quad

this reduces to ⎟⎟⎟

⎠⎜⎜⎜

⎝

=

1000100

δ

S

Emittance exchangeThe transformation required can be achieved with 3 such skew quads in a lattice,

of strengths δ1, δ2, δ3, with transfer matrices S1, S2, S3

A B

Skew quad Skew quad Skew quad

δ1 δ2 δ3

The transfer matrix without the skew quads is C = B A .

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

y

x

CC

C

0000

0000

[ ]

( ) ( ) [ ]⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

−+−−

+

=xxxxxxx

xxxxxxx

x

x

φΔαφΔβφΔααφΔαα

φΔββφΔαφΔββ

sincossin1cos

sinsincos

12112

2111

2

C

and similar for C[ ]⎟⎟⎠

⎜⎜⎝

− xxxx

x

xx

φΔαφΔβββsincos 2

21

21and similar for Cy

Emittance exchangeWith the skew quads the overall matrix is M = S3B S2A S1

⎟⎟⎟⎞

⎜⎜⎜⎛

⎤⎡⎤⎡

++ 23412233121121221341211 δδδδδ ababccabc

( ) ( )

⎟⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜⎜

++

+⎥⎦

⎤⎢⎣

⎡++

++⎥⎦

⎤⎢⎣

⎡++

+

=

34211234332341221134134

33423422

211234333

2332212232123422

211341343

21342221

δδδδδ

δδδδδ

δδδδδδδ

δδ

cabcbaabc

cababc

abcabcabc

abc

M

( ) ( )⎟⎟⎟⎟⎟

⎠⎜⎜⎜⎜⎜

⎝+⎥

⎦

⎤⎢⎣

⎡++

++⎥

⎦

⎤⎢⎣

⎡++

+

++

3234124421124443

23312112321244312

213422144

213412113

34211234332341221134134

δδδδδδδ

δδδδδδδδ

δδδδδ

abcabcabc

abcabcabc

cabcbaabc

⎟⎟⎟⎞

⎜⎜⎜⎛

000000

2423

1413

mmmmmm

Equating the terms with our target matrix form⎟⎟⎟

⎠⎜⎜⎜

⎝ 0000

4241

3231

mmmm

a list of conditions result which must be met for phase-plane exchangea list of conditions result which must be met for phase plane exchange.

Emittance exchange

21341211

34

12

000

δδabccc

+===

( )32341244

21123433

32123422

21341211

000

δδδδδδ

abcabcabc

+=+=+=

( )( )23312112321124443

21134134321342221

00

δδδδδδδδδδ

abcabcabcabc

+++=+++=

The simplest conditions are c12 = c34 = 0.The simplest conditions are c12 c34 0.

Looking back at the matrix C, this means that Δφx and Δφy need to be integer multiples of π (i.e. the phase advance from first to last skew quad should be 180º, 360º, …))

W l h f th t th f th k d1234

33

3412

1121 ab

cab

c−=−=δδ

We also have for the strength of the skew quads3412

44

1234

2232 ab

cab

c−=−=δδ

Emittance exchangeSeveral solutions exist which give M the target form.

One of the simplest is obtained by setting all the skew quadrupole strengths the same, and putting the skew quads at symmetric locations in a 90º FODO l ttilattice

A B (=A)

δ δ δFrom symmetry A = B, and the values of α and β at all skew quads are identical.

( )⎟⎟⎞

⎜⎜⎛ + xxxxx φΔβφΔαφΔ sinsincos

with the same form for yTherefore ( ) ( )⎟⎟⎟

⎠⎜⎜⎜

⎝−

−−

==xxx

x

xxxx

φΔαφΔβφΔα

sincossin1 2BA

The matrix C is similar, but with phase advances of 2Δφ

Emittance exchange

⎞⎛ βα 00

Since we have chose a 90º FODO phase advance, Δφx = Δφy = π/2, and 2Δφx = 2Δφy = π which means we can now write down A,B and C:

( )

⎟⎟⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎜⎜⎛

−−

−

==

xx

x

xx

αβα

βα

2

001

00

BA ⎟⎟⎟⎞

⎜⎜⎜⎛

−−

=00100001

Ci.e. 180º across the insertion in both planes

( )⎟⎟⎟⎟⎟⎟

⎠⎜⎜⎜⎜⎜⎜

⎝−

−− y

y

y

yy

αβ

α

βα

2100

00BA

⎟⎟⎟

⎠⎜⎜⎜

⎝ −−

10000100 insertion in both planes

⎠⎝

sδδδδ 1321 ====we can then write down the skew lens strength as

yxs

ββ321we can then write down the skew lens strength as

12For the 90º FODO with half-cell length L, 2

1,2LL sDF ==−= δδδ

Summary• Transfer lines present interesting challenges and differences from

circular machines– No periodic condition mean optics is defined by transfer line element

strengths and by initial beam ellipse

– Matching at the extremes is subject to many constraints

– Trajectory correction is rather simple compared to circular machineTrajectory correction is rather simple compared to circular machine

– Emittance blow-up is an important consideration, and arises from several sources

Phase plane rotation is sometimes required skew quads– Phase-plane rotation is sometimes required - skew quads

Keywords for related topics• Transfer lines

– Achromat bends

– Algorithms for optics matchingAlgorithms for optics matching

– The effect of alignment and gradient errors on the trajectory and optics

– Trajectory correction algorithms

– SVD trajectory analysis

– Kick-response optics measurement techniques in transfer lines

– Optics measurements including dispersion and δp/p with >3 screensOptics measurements including dispersion and δp/p with >3 screens

– Different phase-plane exchange insertion solutions