emittance compensation theory and experimental results in hb photoinjectors

Emittance compensation theoryand experimental results in HB photoinjectors

Chun-xi WangPhysicist, 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

2Emittance compensation theory in HB photoinjectors, invited talk at HBEB2009

“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.)

solutions / pictures


Motivation: high-brightness photoinjectors are critical (examples)

X-ray FELs

ERL-based 4th generation light sources

Others: ERL-based electron cooling of RHIC, ILC, …

LCLS

Impact on a 1.5 A SASE FEL [Kim et al. whitepaper on bright e-beam, ANL/APS/LS-305]

3m

0.1m

1m

0.3m

APS

10m

Impact on ERL upgrade at APS[Borland et al. whitepaper on ERL]

XFEL-O demands 0.1m @ 40pC, 1MHz[Kim et al. PRL100, 244802(2008)]


h

rf photoinjector layout (examples)

TTF-FEL II and TESLA-FEL RF Gun @ L-band800 s pulses @ 5 Hz (multi bunch trains @ 1-10 MHz to fill the TESLA SC Linac)

UCLA/SLAC/BNLS-band next gen. RF Gun[Serafini, Joint Accelerator School, 2002]


Transverse emittance evolution in a compensated injector

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10Z_[m]

GunLinac

rms beam size [mm]rms norm. emittance [um]

-0.04

-0.02

0

0.02

0.04

0 0.001 0.002 0.003 0.004 0.005 0.006

z=0.23891

Pr

R [m]

-0.05

0

0.05

0 0.0008 0.0016 0.0024 0.0032 0.004

z=1.5P

r

R [m]

-0.04

-0.02

0

0.02

0.04

0 0.0008 0.0016 0.0024 0.0032 0.004

z=10

pr_[

rad]

R_[m]

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

z=0.23891

Rs

[m]

Zs-Zb [m]

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

Z=10

Rs

[m]

Zs-Zb [m]

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

z=1.5

Rs

[m]

Zs-Zb [m]

Final emittance = 0.4 m

Matching onto the Local Emittance Max.This brings to Ferrario’s working point, adopted by LCLS and TTF-FEL II

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

[Serafini, Joint Accelerator School, 2002]


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

x

px

projected

slice

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)]


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)]



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.




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)]

low charge




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.


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


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


Hamiltonian suitable for perturbative analysis (I)

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

[Wang, PRE74 (2006)]


Hamiltonian suitable for perturbative analysis (II)

Linear Hamiltonian

3rd order Hamiltonian

The effects of this chromatic term is significant


Linear forces and linear Hamiltonian

TM01 rf field

Solenoid field

Average space-charge field

Linear Hamiltonian0 in Larmor frame


Pseudofocusing and rf focusing/defocusing

Pseudofocusing

rf focusing/defocusing

Total linear rf focusing strength

ponderomotive focusing

important at low energy

[Hartman & Rosenzweig, PRE47,2031 (1993)][Rosenzweig & Serafini, PRE49,1599 (1995)]

[P. Lapostolle et al, (1994)]

.

Lorentz force


Focusing strengths in optimized SPARC injector


Emittance compensation in optimized SPARC injector

Emittance compensation in space-charge regime [Carlsten 1989],with newly found criteria [Wang et al. 2007]

Invariant envelope in constant acceleration/focusing channelpractical matching condition [Serafini et al. 1997, Wang 2006]double minima in drift [Ferrario 2000, PRL2007]

Transition from space-charge regime to emittance regimeuniversal envelope equation. [Wang 2009]

[m

]


Envelope equation, beam emittance

envelope emittance

envelope equation bunch emittance

linearity

x

px

Not a quadratic sum with thermal emittance


-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

Beam envelope equationIn 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)]


Invariant-envelope/equilibrium solution: generalized Brillouin flow

Invariant-envelope [Serafini & Rosenzweig (1997); Wang (2006)]

Envelope Hamiltonian

“Laminarity parameters”

Transition energy

0

0> 102

a practical matching condition

min @


Small envelope oscillations around equilibrium

Equation of motion for small oscillation

Propagation of small deviations ( )

,[Rosenzweig & Serafini, PRE49,1599 (1995)]

Independent of slice perveance!

as


Emittance evolution around equilibrium in a booster

Emittance evolution

Final emittancebooster entrance

0

the reality is more complicated


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)]


Envelope equation for general perturbative treatment

Using small deviations

to reorganize the envelop equation

as


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


First-order driving terms

Space-charge effect: perveance deviations among slices

Chromatic effect:

space-charge

chromatic

space-charge

chromatic

(s-dependence) (slice-dependence)

s[m]

slice#

[Wang, PRE74 (2006)]


Emittance evolution

Effects of the first-order driving terms

w/ s.c.

w/ chrom

s [m]

[

m]


Emittance computation formula

Hard to analytically compute

Assuming a general linear expansion & uncorrelated variations

Emittance can be computed as

=


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

=residuals absorbed into ,’ 0 0


booster entrance

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

booster entrance

= 0


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


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

w/ const. focusing


Invariant-envelope evolution in linac


Emittance evolution inside booster / linac Under constant focusing (= 0)

perturbation around invariant envelope

exact

linear perturbation around W

approx. relative emittance

exact relative emittance


Emittance evolution in linacs -- SPARC example

HOMDYN simulation vs. universal envelope (continued with the same linac)

HOMDYNuniversal envelope computation

thermal emittance quadratically included

thermal emittance correctly included


Summary

Emittance compensation in space-charge regime [Carlsten 1989],with newly found criteria [Wang et al. 2007]

Invariant envelope in constant acceleration/focusing channelpractical matching condition [Serafini et al. 1997, Wang 2006]double minima in drift [Ferrario 2000, PRL2007]

Transition from space-charge regime to emittance regimeuniversal envelope equation. [Wang 2009]

[m

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emittance compensation theory and experimental results in hb photoinjectors

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