Emittance compensation theory and experimental results in HB photoinjectors

1. Emittance compensation theoryand experimental results in HB photoinjectors Chun-xi Wang Physicist, Advanced Photon Source (ANL) Invited talk at ICFA workshop on the Physics and Applications of High-brightness Electron Beams 2009 Nov. 18, 2009 Sincerely thank the organizers (Massimo and James) for the invitation and unwavering support

2. solutions / pictures “We physicists love simple problems. So much so that our immediate reaction to complex puzzles is to keep staring at them until some simple picture suggests itself.” [T. Mcleish, August issue of Physics Today] I have been staring at photoinjector dynamics for a few years. Here I will share what I have seen so far. (a luxury for beam physics.)

3. Motivation: high-brightness photoinjectors are critical (examples) • X-ray FELs • ERL-based 4th generation light sources • Others: ERL-based electron cooling of RHIC, ILC, … Impact on ERL upgrade at APS [Borland et al. whitepaper on ERL] Impact on a 1.5 A SASE FEL [Kim et al. whitepaper on bright e-beam, ANL/APS/LS-305] 0.1mm 0.3mm LCLS 1mm APS XFEL-O demands 0.1mm @ 40pC, 1MHz [Kim et al. PRL100, 244802(2008)] 3mm 10mm

4. hn rf photoinjector layout (examples) UCLA/SLAC/BNL S-band next gen. RF Gun [Serafini, Joint Accelerator School, 2002] TTF-FEL II and TESLA-FEL RF Gun @ L-band 800 ms pulses @ 5 Hz (multi bunch trains @ 1-10 MHz to fill the TESLA SC Linac)

5. Matching onto the Local Emittance Max. This brings to Ferrario’s working point, adopted by LCLS and TTF-FEL II Final emittance = 0.4 mm Transverse emittance evolution in a compensated injector [Serafini, Joint Accelerator School, 2002] S-band photoinjector up to 150 MeV, HOMDYN simulation (RF Gun + 2 Traveling Wave Structures) Q=1nC, L=10ps, R=1 mm, Epeak=140 MV/m, TW Eacc = 25 MV/m

6. px projected slice x Emittance compensation --- theoretical developments • Oscillation and growth of projected emittance • linear space charge dominate, nonlinearities are small, slice emittance is preserved • but projected bunch emittance oscillates and grows due to different space-charge defocusing among slices and chromatic effects and so on • emittance compensation is a the cure [Carlsten, NIM A285,313(1989)] • Emittance compensation is critical to achieve high-brightness proper focusing can recover the projected emittance • First beam-envelope theory [Serafini & Rosenzweig, PRE55,7565 (1997)] • Recent efforts [C.-x. Wang, NIM A557, 94 (2006)] [C.-x. Wang, PRE 74, 046502 (2006)] [C.-x. Wang, K.-J. Kim, M. Ferrario, A. Wang, PRST-AB 10, 104201 (2007)] [C.-x. Wang, PRST-AB (2009)] • Simulations are still the workhorse for design Many other works can’t be covered here, e.g., [X. He, C. Tang, W. Huang, Y. Lin, NIM A560,197 (2006)] Orbit-theory approach: [K.-J. Kim, NIM A275, 201 (1989)] [Z. Huang, Y. Ding, J. Qiang, NIM A593, 148 (2008)]

7. Emittance compensation --- experiments • First observation through slice emittance measurement [X. Qiu, K. Batchelor, I. Ben-Zvi, X-J. Wang, PRL 76(20) 3723 (1996)]

8. Emittance compensation --- experiments • First observation through slice emittance measurement [X. Qiu, K. Batchelor, I. Ben-Zvi, X-J. Wang, PRL 76(20) 3723 (1996)] • Direct measurement of the double emittance minimum using emittance meter [M. Ferrario et. al., PRL 99, 234801 (2007)] [M. Ferrario et. al., SLAC-Pub-8400 (2000)]

9. Emittance compensation --- experiments • First observation through slice emittance measurement [X. Qiu, K. Batchelor, I. Ben-Zvi, X-J. Wang, PRL 76(20) 3723 (1996)] • Direct measurement of the double emittance minimum [M. Ferrario et. al., PRL 99, 234801 (2007)] [M. Ferrario et. al., SLAC-Pub-8400 (2000)] • Success of LCLS, SPARC, and other high-brightness injectors [R. Akre, D. Dowell, et. al., PR ST-AB 11,030703 (2008)] [Y. Ding, et. al., PRL 102, 254801 (2009)] [A. Cianchi, et. al., PR ST-AB 11, 032801 (2008)] Emittance compensation works well in recovering emittance degradation due to linear space-charge forces. Performance starts to be limited by thermal emittance, nonlinear space charges, etc.

10. low charge Emittance compensation --- experiments • First observation through slice emittance measurement [X. Qiu, K. Batchelor, I. Ben-Zvi, X-J. Wang, PRL 76(20) 3723 (1996)] • Direct measurement of the double emittance minimum [M. Ferrario et. al., PRL 99, 234801 (2007)] [M. Ferrario et. al., SLAC-Pub-8400 (2000)] • Success of LCLS and SPARC injectors [R. Akre, D. Dowell, et. al., PR ST-AB 11, 030703 (2008)] [Y. Ding, et. al., PRL 102, 254801 (2009)] [A. Cianchi, et. al., PR ST-AB 11, 032801 (2008)] • Emittance compensation under velocity bunching [Serafini & Ferrario, AIP Conf. Proc. 581 (2001)] [M. Ferrario et. al., PAC 99 (2009)]

11. Emittance compensation --- experiments • First observation through slice emittance measurement [X. Qiu, K. Batchelor, I. Ben-Zvi, X-J. Wang, PRL 76(20) 3723 (1996)] • Direct measurement of the double emittance minimum [M. Ferrario et. al., PRL 99, 234801 (2007)] [M. Ferrario et. al., SLAC-Pub-8400 (2000)] • Success of LCLS and SPARC injectors [R. Akre, D. Dowell, et. al., PR ST-AB 11, 030703 (2008)] [Y. Ding, et. al., PRL 102, 254801 (2009)] [A. Cianchi, et. al., PR ST-AB 11, 032801 (2008)] • Emittance compensation under velocity bunching [Serafini & Ferrario, AIP Conf. Proc. 581 (2001)] [M. Ferrario et. al., PAC 99 (2009)] • Many others; apologize for the inconclusiveness.

12. Emittance compensation --- Simulation codes • Simulation are the workhorse for design • Envelope-equation based code HOMYDN[M. Ferrario, INFN] • Particle tracking codes ASTRA[K.Floetmann, DESY], PARMEALA[LANL], IMPACT-T[J. Qiang, LBNL], GPT[commercial code], TREDI, BEAMPATH, … • Code comparison [C. Limborg et. al., PAC03, 3548 (2003)] • Multi-objective optimization with parallelized particle tracking [I.V. Bazarov et. al., PR ST-AB 8, 034202 (2005)] • Need to combine particle tracking, envelop analysis, and theory It is important but not easy to analyze and quantify the limitations to beam brightness in a design simulation

13. High-brightness photoinjector dynamics are complex • Rapid acceleration from rest to relativistic • important to overcome space-charge effects • Space-charge-dominated to emittance-dominated • nonlinear space-charge force depends on details of bunch shape/laser pulse • emittance compensation is critical to overcome emittance blowup due to linear space-charge force (and more) • image-charge force near cathodes is significant • Time-dependent rf force (acceleration and focusing) • ponderomotive focusing is important and has chromatic effects • certain defocusing close to cathode • rf curvature creates nonlinearity and limits bunch length • Solenoid focusing with large fringe field • main knob for emittance compensation, chromatic effect is significant • Intrinsically nonlinear problems • forces are nonlinear, especially space-charge force • envelope equations are nonlinear,even for linear forces

14. Hamiltonian suitable for perturbative analysis (I) [Wang, PRE74 (2006)] • Starting with • Required features • suitable for perturbation • allow rapid acceleration from non-relativistic to relativistic • mostly decoupled / solvable form • Coordinate systems • use derivations from reference particle as dynamical variables for perturbation • use s as the explicit independent variable for convenience, but still use time t as implicit independent variable for calculating space-charge forces, and thus use (z, p ) instead of (t, -E) as longitudinal variables • use reduced-coordinates to decouple (x, px) etc. z

15. Hamiltonian suitable for perturbative analysis (II) • Linear Hamiltonian • 3rd order Hamiltonian The effects of this chromatic term is significant

16. Linear forces and linear Hamiltonian • TM01 rf field • Solenoid field • Average space-charge field • Linear Hamiltonian 0 in Larmor frame

17. Pseudofocusing and rf focusing/defocusing Lorentz force • Pseudofocusing • rf focusing/defocusing • Total linear rf focusing strength . important at low energy [P. Lapostolle et al, (1994)] ponderomotive focusing [Hartman & Rosenzweig, PRE47,2031 (1993)] [Rosenzweig & Serafini, PRE49,1599 (1995)]

18. Focusing strengths in optimized SPARC injector

19. Emittance compensation in space-charge regime [Carlsten 1989], with newly found criteria [Wang et al. 2007] Invariant envelope in constant acceleration/focusing channel practical matching condition [Serafini et al. 1997, Wang 2006] double minima in drift [Ferrario 2000, PRL2007] Transition from space-charge regime to emittance regime universal envelope equation. [Wang 2009] Emittance compensation in optimized SPARC injector e [mm]

20. Envelope equation, beam emittance envelope emittance linearity bunch emittance envelope equation px x Not a quadratic sum with thermal emittance

21. Beam envelope equation In high-brightness photoinjectors, electrons behave as laminar flow in both longitudinal and transverse planes. Thus, a bunch can be treated as many individual slices, each follows its own envelope equation.[Serafini & Rosenzweig (1997)] • b-function of time-dependent harmonic oscillator • Beam envelope equation governing (linear) transverse beam dynamics • self-consistent space-charge force is built into this equation • coefficients are rapidly changing • coefficients are slice-dependent, especially the perveance • nonlinear, nonautonomous ODE, notoriously hard to solve analytically • Emittance is very difficult to handle analytically

22. Invariant-envelope/equilibrium solution: generalized Brillouin flow • Invariant-envelope [Serafini & Rosenzweig (1997); Wang (2006)] • Envelope Hamiltonian • “Laminarity parameters” • Transition energy 0 0 a practical matching condition min @ . . . 2 > 10

23. Small envelope oscillations around equilibrium • Equation of motion for small oscillation • Propagation of small deviations ( ) Independent of slice perveance! asg [Rosenzweig & Serafini, PRE49,1599 (1995)] ,

24. Emittance evolution around equilibrium in a booster • Emittance evolution • Final emittance 0 booster entrance the reality is more complicated

25. Shortcomings of invariant-envelope theory • Emittance evolution far from equilibrium in the gun • no equilibrium at all in the gun (everything is time-dependent) and in the following drift space (no focusing) • envelopes are far away from equilibrium • lack of criteria for emittance compensation (despite the matching condition) • practical designs rely on simulations (with handwaving theory) • general perturbation theory and new compensation criteria • Transition from space-charge regime to the thermal regime • space-charge-dominated theory isn’t enough • no equilibrium solution away from space-charge regime (inadequate focusing) • perturbative solution around invariant envelope diverges • nonlinear effects are significant • universal envelope equation and emittance evolution during transition [C.-x. Wang, K.-J. Kim, M. Ferrario, A. Wang, PRST-AB (2007)] [C.-x. Wang, PRST-AB (2009)]

26. Envelope equation for general perturbative treatment Using small deviations to reorganize the envelop equation as

27. Perturbative envelope solution • To the first-order in small deviations, the envelope equation reduces to a simple inhomogeneous first-order ODE: • General solution for envelope deviations: • For the reference envelope • General envelope solution

28. First-order driving terms • Space-charge effect: perveance deviations among slices • Chromatic effect: [Wang, PRE74 (2006)] s[m] space-charge space-charge chromatic chromatic slice# (s-dependence) (slice-dependence)

29. Emittance evolution • Effects of the first-order driving terms w/ chrom e [mm] w/ s.c. s[m]

30. = Emittance computation formula • Hard to analytically compute • Assuming a general linear expansion & uncorrelated variations • Emittance can be computed as

31. residuals absorbed into s ,s’ 0 0 Emittance compensation – removal of slice-dependent effects • Estimation of driving term contributions (w/ major approximation) • Envelope expansion reduces to • Emittance can be approximated as = 0 to remove slice-dependent emittance growth

32. Emittance compensation criteria from cathode to booster • Criteria to minimize emittance growth from slice-dependent effects • Equivalent conditions • Equivalent integral form • It is surprisingly good = 0 booster entrance booster entrance

33. Emittance compensation inside booster / linac • Transition from space-charge regime to thermal regime • common to most high-brightness beam, but not well studied • invariant-envelope theory is limited to space-charge regime • intrinsic nonlinearity is significant and very hard to treat • magnetized beam can cross the transition at low energy • Some obvious questions • what happens to the invariant-envelope solution? • how restrictive is the matching condition (phase-space acceptance)? • is it possible to preserve the emittance across the transition? • any criteria besides matching to the invariant-envelope? • Recent findings: • universal envelope equation • solution of invariant-envelope across the transition • emittance formula that correctly includes thermal emittance

34. w/ const. focusing Universal beam envelope equation in axisymmetric linac • Scaled energy (by the transition energy) as independent variable • energy increases monotonically in linac • Scaled envelope (by the invariant envelope) as dependent variable • Envelope equation reduces to • Under linear acceleration with focusing

35. Invariant-envelope evolution in linac

36. linear perturbation around W W Emittance evolution inside booster / linac • Under constant focusing (W = 0) exact exact relative emittance perturbation around invariant envelope approx. relative emittance

37. Emittance evolution in linacs -- SPARC example HOMDYN simulation vs. universal envelope (continued with the same linac) HOMDYN universal envelope computation thermal emittance quadratically included thermal emittance correctly included

38. Emittance compensation in space-charge regime [Carlsten 1989], with newly found criteria [Wang et al. 2007] Invariant envelope in constant acceleration/focusing channel practical matching condition [Serafini et al. 1997, Wang 2006] double minima in drift [Ferrario 2000, PRL2007] Transition from space-charge regime to emittance regime universal envelope equation. [Wang 2009] Summary e [mm]