Max Cornacchia, Paul EmmaStanford Linear Accelerator CenterMax Cornacchia, Paul Emma

Stanford Linear Accelerator Center

Proposed by M. Cornacchia (Nov. 2001)Proposed by M. Cornacchia (Nov. 2001) Analysis taken from similar Analysis taken from similar x-yx-y coupling coupling

work by W. Spence and P. Emmawork by W. Spence and P. Emma Motivation to reduce transverse and Motivation to reduce transverse and

increase longitudinal emittance increase longitudinal emittance faster faster SASE lasing and less CSR micro-SASE lasing and less CSR micro-bunching in compressorsbunching in compressors

Transverse to Longitudinal Transverse to Longitudinal Emittance ExchangeEmittance Exchange

transverse emittance:transverse emittance:

energy spread:energy spread:

< 1 m at 1 Å, 15 GeV < 1 m at 1 Å, 15 GeV

< 0.05% at Ipk = 4 kA, K 4, u 3 cm, … < 0.05% at Ipk = 4 kA, K 4, u 3 cm, …

We need x < 1 m , but z z

We need x < 1 m , but z z

Can we reduce x at the expense of z ?Can we reduce x at the expense of z ?

RF gun produces x ~ z ~ few mRF gun produces x ~ z ~ few m

SASE FEL needs very bright electron beam…SASE FEL needs very bright electron beam…

< 300 m< 300 m

Emittance Exchange ConceptEmittance Exchange Concept

Electric and magnetic fieldsElectric and magnetic fieldskk

x0

z0

x0

z0

transverse RF in a chicane…transverse RF in a chicane…

Particle at position x in cavity gets acceleration: kx Particle at position x in cavity gets acceleration: kx

Must include magnetic field and calculate emittance in both planes…Must include magnetic field and calculate emittance in both planes…

??

This energy deviation in chicane causes position change: x = This energy deviation in chicane causes position change: x = Choose k to cancel initial position: x kxx kk = 1 = 1 Choose k to cancel initial position: x kxx kk = 1 = 1

Characterize Initial BeamCharacterize Initial BeamInitial uncoupled 44 beam covariance matrix, with = (1+2)/

Initial uncoupled 44 beam covariance matrix, with = (1+2)/

Use z and z to describe longitudinal phase space, same as x and x in transverse

Use z and z to describe longitudinal phase space, same as x and x in transverse

projected emittancesprojected emittances

||RR| = 1| = 1||RR| = 1| = 1

Propagate Beam…Propagate Beam…

Rewrite emittances…Rewrite emittances…

…some details…some details

Introduce Symplectic Condition…Introduce Symplectic Condition…

Final Emittance RelationsFinal Emittance Relations

equal emittances remain equalequal emittances remain equal (i.e., if(i.e., if xx00

= = zz00 thenthen xx = = zz))

equal emittances remain equalequal emittances remain equal (i.e., if(i.e., if xx00

= = zz00 thenthen xx = = zz))

Introduce Transverse RFIntroduce Transverse RF

Effects of Transverse RFEffects of Transverse RF

RR RR

RRkk

RR5656

LL

Chicane with RFChicane with RF

Full System Transport MatrixFull System Transport Matrix

||AA| = 0 | = 0 kk = 1 = 1

Final Emittance RelationsFinal Emittance Relations

Numerical ExampleNumerical Example

non-trivialnon-trivialnon-trivialnon-trivial

get nearly complete emittance exchangeget nearly complete emittance exchangeget nearly complete emittance exchangeget nearly complete emittance exchange

Phase Space Before and After ExchangerPhase Space Before and After Exchanger

x, x before z, before

x = 5 m z = 1 m

x, x after z, after

x = 1 m z = 5 m

bunch is also bunch is also compressed:compressed: zz 18 18 mm

bunch is also bunch is also compressed:compressed: zz 18 18 mm

get large x, xget large x, x

Normalized Phase SpaceNormalized Phase Space

xx, , xx beforebefore zz, , beforebefore

xx = 5 = 5 mm zz = 1 = 1 mm

xx, , xx afterafter zz, , afterafter

xx = 1 = 1 mm zz = 5 = 5 mm

Final Bunch Length and Energy SpreadFinal Bunch Length and Energy Spread

kk = 1, = 1,xx = = zz = 0 = 0kk = 1, = 1,xx = = zz = 0 = 0

22ndnd-order-order xx growth approximated by 2growth approximated by 2ndnd-order dispersion…-order dispersion…22ndnd-order-order xx growth approximated by 2growth approximated by 2ndnd-order dispersion…-order dispersion…

use smalluse small xx

and largeand large use smalluse small xx

and largeand large

zzzz

zzzz

Bunch CompressionBunch Compression

200 200 m m 20 20 mm200 200 m m 20 20 mm

Using Using transverse RF transverse RF (all in last bend)(all in last bend)

Using Using transverse RF transverse RF (all in last bend)(all in last bend)

Use standard Use standard energy ‘chirp’ (2energy ‘chirp’ (2ndnd & 3& 3rdrd bends) bends)

Use standard Use standard energy ‘chirp’ (2energy ‘chirp’ (2ndnd & 3& 3rdrd bends) bends)

RF Deflector (cylindrical)RF Deflector (cylindrical)f = 11.424 GHz2a 11.2 mm (iris diameter)2b 29.1 mm (cell diameter)L 0.376 m (43 cells)Q 5300vg/c 0.0193V0 7 MVP0 14 MW

f = 11.424 GHz2a 11.2 mm (iris diameter)2b 29.1 mm (cell diameter)L 0.376 m (43 cells)Q 5300vg/c 0.0193V0 7 MVP0 14 MW

R. MillerR. Miller

TM11-like modeTM11-like mode

“get aberration-free deflection from this mode”

“get aberration-free deflection from this mode”

(G. Loew, et. al., SLAC, 1963-5)(G. Loew, et. al., SLAC, 1963-5)

(H. Hahn, BNL, 1962-3,

Y. Garauit, Orsay, 1962)

(H. Hahn, BNL, 1962-3,

Y. Garauit, Orsay, 1962)

Cavity ‘Thick-Lens’ EffectCavity ‘Thick-Lens’ Effect

BByy ~ ~ tt

add ‘chirp’ to add ‘chirp’ to compensate compensate ‘thick-lens’‘thick-lens’

ll

initial ‘chirp’initial ‘chirp’

tailtail

headhead

tailtail

headhead

Tracking with Thick-Lens and ChirpTracking with Thick-Lens and Chirpzz 0 0

xx = 5 = 5 mm zz = 1 = 1 mm

xx = 1 = 1 mm zz = 5 = 5 mm

ll = 0.4 m = 0.4 mll = 0.4 m = 0.4 m

same as thin-lens cavity

same as thin-lens cavity

kk

kk

tttt

xxxx

--tron tron oscillations oscillations ~disappear~disappear

--tron tron oscillations oscillations ~disappear~disappear

--tron osc’s tron osc’s started fromstarted from tt errorerror

--tron osc’s tron osc’s started fromstarted from tt errorerror

Unusual System CharacteristicsUnusual System Characteristics

System potentially reduces transverse emittanceSystem potentially reduces transverse emittance Also increases longitudinal emittance, possibly Also increases longitudinal emittance, possibly

damping the CSR instabilitydamping the CSR instability

Bunch length is compressed (all in last bend)Bunch length is compressed (all in last bend)

Moves injector challenge to longitudinal emittanceMoves injector challenge to longitudinal emittance

SummarySummary

Must avoid CSR energy spread increase in 1Must avoid CSR energy spread increase in 1stst bends bends

Scheme may have other uses not yet consideredScheme may have other uses not yet considered