1. 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

2. 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

3. 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

4. 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

5. projected emittances |R| = 1 Propagate Beam…

6. Rewrite emittances… …some details

7. Introduce Symplectic Condition…

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

9. Introduce Transverse RF

10. Effects of Transverse RF

11. R  R  R k R 56  L Chicane with RF

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

13. Final Emittance Relations

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

15. 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

16. 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

17. 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 

18. 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)

19. RF Deflector (cylindrical) f = 11.424 GHz 2a  11.2 mm (iris diameter) 2b  29.1 mm (cell diameter) L  0.376 m (43 cells) Q  5300 v g /c   0.0193 V 0  7 MV P 0  14 MW f = 11.424 GHz 2a  11.2 mm (iris diameter) 2b  29.1 mm (cell diameter) L  0.376 m (43 cells) Q  5300 v g /c   0.0193 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, 1963-5) (H. Hahn, BNL, 1962-3, Y. Garauit, Orsay, 1962) (H. Hahn, BNL, 1962-3, Y. Garauit, Orsay, 1962)

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

21. 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

22. 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

23.  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