Max Cornacchia, Paul Emma Stanford Linear Accelerator Center Max Cornacchia, Paul Emma Stanford Linear Accelerator Center  Proposed by M. Cornacchia (Nov. 2001)  Analysis taken from similar x-y coupling work by W. Spence and P. Emma  Motivation to reduce transverse and increase longitudinal emittance  faster SASE lasing and less CSR micro- bunching in compressors Transverse to Longitudinal Emittance Exchange

transverse emittance: energy spread:   < 1  m at 1 Å, 15 GeV   < 0.05% at I pk = 4 kA, K  4, u  3 cm, … We need  x < 1  m, but  z     z Can we reduce  x at the expense of  z ? RF gun produces  x ~  z ~ few  m SASE FEL needs very bright electron beam… < 300  m

Emittance Exchange Concept Electric and magnetic fields  k x0z0x0z0 x0z0x0z0 transverse RF in a chicane…  Particle at position x in cavity gets acceleration:   kx Must include magnetic field and calculate emittance in both planes… ? ?  This energy deviation  in chicane causes position change:  x =   k = 1  Choose k to cancel initial position:  x   kx  x   k = 1

Characterize Initial Beam Initial uncoupled 4  4 beam covariance matrix, with  = (1+  2 )/  Use  z and  z to describe longitudinal phase space, same as  x and  x in transverse

projected emittances |R| = 1 Propagate Beam…

Rewrite emittances… …some details

Introduce Symplectic Condition…

Final Emittance Relations equal emittances remain equal (i.e., if  x 0 =  z 0 then  x =  z )

Introduce Transverse RF

Effects of Transverse RF

R  R  R k R 56  L Chicane with RF

Full System Transport Matrix |A| = 0   k = 1

Final Emittance Relations

Numerical Example non-trivialnon-trivial get nearly complete emittance exchange

Phase 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 compressed:  z  18  m get large  x,  x

Normalized Phase Space x, x before z,  before  x = 5  m  z = 1  m x, x after z,  after  x = 1  m  z = 5  m

Final Bunch Length and Energy Spread  k = 1,  x =  z = 0  k = 1,  x =  z = 0 2 nd -order  x growth approximated by 2 nd -order dispersion… use small  x and large 

zzzz zzzz   zzzz zzzz   Bunch Compression 200  m  20  m Using transverse RF (all in last bend) Use standard energy ‘chirp’ (2 nd & 3 rd bends)

RF Deflector (cylindrical) f = GHz 2a  11.2 mm (iris diameter) 2b  29.1 mm (cell diameter) L  m (43 cells) Q  5300 v g /c   V 0  7 MV P 0  14 MW f = GHz 2a  11.2 mm (iris diameter) 2b  29.1 mm (cell diameter) L  m (43 cells) Q  5300 v g /c   V 0  7 MV P 0  14 MW R. Miller TM 11 -like mode “get aberration-free deflection from this mode” (G. Loew, et. al., SLAC, ) (H. Hahn, BNL, , Y. Garauit, Orsay, 1962) (H. Hahn, BNL, , Y. Garauit, Orsay, 1962)

Cavity ‘Thick-Lens’ Effect B y ~  t add ‘chirp’ to compensate ‘thick-lens’ l initial ‘chirp’ tail head tail head

Tracking with Thick-Lens and Chirp z  0z  0z  0z  0  x = 5  m  z = 1  m  x = 1  m  z = 5  m l = 0.4 m same as thin- lens cavity

k k tttt tttt xxxx xxxx  - tron oscillations ~disappear  - tron osc’s started from  t error Unusual System Characteristics

 System potentially reduces transverse emittance  Also increases longitudinal emittance, possibly damping the CSR instability  Bunch length is compressed (all in last bend)  Moves injector challenge to longitudinal emittance Summary  Must avoid CSR energy spread increase in 1 st bends  Scheme may have other uses not yet considered