theory and practice of free-electron lasersuspas.fnal.gov/materials/09unm/day 4.pdf · us particle...

44--11LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009

University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM

Theory and Practice ofTheory and Practice ofFreeFree--Electron LasersElectron Lasers

Particle Accelerator SchoolParticle Accelerator SchoolDay 4Day 4

DinhDinh Nguyen, Steven RussellNguyen, Steven Russell& Nathan Moody& Nathan Moody

Los Alamos National LaboratoryLos Alamos National Laboratory



Course Content

1. Introduction to Free-Electron Lasers2. Basics of Relativistic Dynamics3. One-dimensional Theory of FEL4. Optical Architectures5. Wigglers6. RF Linear Accelerators7. Electron Injectors

Chapter

ChapterChapter

ChapterChapter

Chapter

Chapter

Day 1

Day 2

Day 3

Day 4



Chapter 6RF Linear Accelerators

(continued)



RF Linear Accelerators (Linac)

• Introduction to RF Linac• Properties of RF Cavities• Coupled Cavity Linac• Superconducting RF Linac• Energy Recovery Linac



Superconducting RF Linacs

• RF Superconductivity• Meissner effect• Surface resistance• Temperature dependence

• Superconducting cavities• Cavity designs• Cavity Q versus Eacc



Superconductivity

Coopers pair: Two electrons with anti-parallel momenta and spins attracting each other via exchange of lattice phonons.

Bose condensation: Overlapping Coopers pairs condensing into a coherent superconducting state at low temperature.

Critical temperature, Tc: Temperature below which Coopers pairs form with binding energy ~10-4 –10-3 eV.

Discovery of superconductivity by Kamerlingh-Onnes and Holst in 1911

Courtesy of A. Gurevich



Meissner Effect and Critical Field

• Magnetic field is expelled from a superconductor at T < Tc

• Superconductivity is destroyed at H > Hc (critical magnetic field)

• Empirical formula for Hc

• For Nb (type II superconductor)– Tc is 9.2K– 0Hc2(0) is 0.24 T– At 2K, 0Hc is 0.22 T

2

2 (0) 1c cc

TH HT

æ öæ ö ÷ç ÷ç ÷ç ÷= - ÷çç ÷ ÷çç ÷ç ÷è ø ÷çè ø

Courtesy of A. Gurevich



Surface Resistance of Niobium

24 1 17.672 10 exp

1.5S residualfR RGHz T T

Niobium surface resistance at low temperature has two components: BCS (Bardeen-Cooper-Schrieffer) resistance and residual resistance

BCS resistance scales with f 2. SRF accelerators have lower BCS resistance (loss) at low frequency. To minimize BCS resistance, SRF cavities with f > 500 MHz have to be cooled to 2K.

Residual resistance depends on material purity and trapped magnetic field. The earth magnetic field has to be shielded with mu metal around the cavities.

• High-purity niobium sheet metals have low residual resistance (1 – 10 n)

• Residual resistance = 3.75 nT (f / GHz)0.5 so trapped magnetic field (trapped flux) should be avoided to minimize residual resistance.



Surface Resistance vs. 1/T

Tc / T

1 2 3 4 5 6 7

Geometric factor G (= Q0RS) is about 270 Nb at 4.2K at 1.3 GHz RBCS > 500 n RS > 500 n Q0 < 5 x 108

Nb at 2K at 1.3 GHz RBCS = 7 n RS = 15 n Q0 = 2 x 1010

RBCS ~

Rres

1expT

2K

4.2K



Elliptical Cavity Designs

Optimized TESLA Optimized low-loss Optimized re-entrantTM010G*R/Q 30840 37970 38380

Ep / Ea 1.98 2.36 2.3Bp / Ea 4.15 mT/(MV/m) 3.61 mT/(MV/m) 3.57 mT/(MV/m)kCC 1.9 1.52 1.56HOMk┴ 0.23 0.38 0.38k║ 1.46 1.72 1.75



Peak Fields to Average Gradient

High electric fields cause field emission → Q degradation

High magnetic fields cause RF heating → quench

Highest Ep

Ep/Ea ~ 2

Highest Bp

Bp/Ea ~ 4 mT/(MV/m)

Peak electric field Peak magnetic field

Courtesy of J. Sekutowics



Cavity Q vs. Gradient

Eacc (MV/m)

Multipacting

Field emission

Q disease

Quench

Q slope



Unloaded Q vs Gradient ofa TESLA 1.3 GHz SRF Cavity

Eacc (MV/m)

Courtesy of D. Reschke



Energy Recovery Linac (ERL)

• Energy Recovery Linac• Benefits and Risks• Same Cavity versus Different Cavity• Voltage Phase Diagram

• Instabilities• RF Instabilities• Beam Break-up Instabilities

• HOM Damping



Benefits and Risks of ERL

• Reduce RF power consumption (Increase wall-plug efficiency)

• Achieve much higher electron beam power than input RF power

• Susceptible to instabilities– RF instabilities– Beam break-up instabilities

• Difficult to recover energy due to large energy spread in spent beams

Pin

Pout

Pb

Benefits Risks

FEL interaction



Example of ERL: CEBAF



Same Cavity vs Different CavitySame cavity energy recovery

Accelerated electron bunches

Decelerated electron bunchesRF wave

Different cavity energy recovery

Extracted RF power



• Interaction between cavity power and beam

• Fluctuations in cavity field cause

Beam loss

Phase errors

FEL power fluctuations

• All three of the above can increase beam energy fluctuations, resulting in additional beam loss, phase errors or FEL fluctuations, hence instabilities.

RF Instabilities

180o Bend

Beam DumpSuperconducting Accelerator

Ig

Po + Po

Ib + Ib

Vc + Vc

( ) ( )0 01 tan2 2

cc g b

L L

dV Ri V I Idt Q Q

w w+ - Y = -



Beam Break-up Instabilities (BBU)• An electron traversing an accelerating cavity off axis will excite higher

order modes (HOM), some of which are dipole modes, of the cavity

• The coupling between the beam and the dipole mode is proportional to the beam current and the transverse position offset

• The dipole modes could deflect a following particle to a larger amplitude, thereby increasing the coupling to and the strength of the dipole modes which would deflect following bunches further.

Single-bunch BBU Multi-bunch BBU(head to tail kick) (deflection and beam loss)



Higher Order Modes (HOM)

)cos()sin()(10 tkrcJBEz

)sin()sin()(

)sin()cos()(

'10

10

tkrJBB

tkrkrJBBr

ack 833.32

Recall the fundamental k = 2.405/a → fdipole ~ 1.5 ffundamental

x

y

r z

2az



Single-pass Regenerative BBU

The on-axis magnetic field of the TM011 mode can impart transverse momentum to an electron. By the end of the cavity the beam is off-axis and can exchange energy with the axial electric field. The power of TM011 mode grows.

ffundamental = 1.5 GHzfd ≈ 1.5 ffundamental ≈ 2 GHzd = 15 cmVbeam = 10 MVRd/Q = 100 QL = 105

Ithreshold ≈ 240 mA

2 ( )d beam

thresholdd L

VIR Q Qlpæ ö÷ç» ÷ç ÷çè ø

Threshold current for single-pass BBU

Courtesy of T. Smith



Regenerative Two-pass BBU

)sin()( 12 rdLd

beamdthreshold TMQQR

VI

cavitydd LrR

With the same cavity as before, and assuming M12 = 1 meter, the BBU threshold is now only 12 mA, 20 times lower than the single-pass BBU

Dipole magnetic field gives the beam transverse momentum (px)

Orbit matrix converts px to x = M12px/pz

Off-axis electric field decelerates the beam, adding energy to the cavity and increasing the dipole fields.

To beam dump

Threshold current for two-pass BBU

Courtesy of T. Smith



HOM Damping

Ferrite absorber tiles

)sin()( 12 rdLd

beamdthreshold TMQQR

VI

Design a cavity with low R/Q ratios for the HOM. Then, couple the HOM into absorbers to reduce their loaded Q, thereby increasing the BBU threshold.

Single-cell SRF cavity with HOM couplers

Threshold current for two-pass BBU



Chapter 7Electron Injectors



Typical Injector Performance

months

4.6266

Cu

5

1.2

<0.001

1000

5000

100

Pulsed NCRF

4.6266

2.3530

2.3530

Photon energy (eV)Photon wavelength (nm)

<13210Average current (mA)

31010Transverse emittance(mm-mrad or m)

152050Bunch length(ps)

Cs2TeK2CsSbGaAsPhotocathodes

hours

7000

2000

26

High-duty NCRF

monthsdaysPhotocathode lifetime

200135Bunch charge (pC)

2100350Energy (keV)

13.56Gradient (MV/m)

SRF InjectorDC Gun



Photo-emission in aSemiconductor Photocathode

transport emission

Eg

EA Electron Affinity

Valence Band

Vacuum

Ef

Conduction Band

hexcitation

Bulk

EV is reduced by the applied field

Electrons are excited by the laser from the valence band to the conduction band. During their transport from the bulk to the surface, the electrons undergo electron-phonon collisions and lose energy. Photoemission occurs as a tunneling process through the potential barrier at the solid-vacuum interface. The barrier height is determined by the electron affinity and the barrier width is determined by the applied electric field.

Surface



DC Injector

Courtesy of C. Hernandez-Garcia

Jefferson Lab DC Gun



DC Injector and SRF Booster Cavity

GaAs photocathode

DC Gun

SRF Booster

- Beam Voltage

Distance from Cathode

Laser Beam

Electron Beam

Cathode-Vc

Anode0 V

Vc

Voltage gain in boosterBeam voltage loss due to fringe field



Space Charge Effects

L

2R

Longitudinal space charge field

Radial space charge field

L/2-L/2( )

( ) ( )0

0

/2

/20 2

0

L

Lz

d d

Er

z

z

r z z r z z

ze p

-

-

=ò ò

Ele

ctric

fiel

d

Ele

ctric

fiel

d

Radius

( )( )

0

00 2

0 0

r

r

r drE r

r

r

e p=ò

L/2

R

2R



Beam Emittance with DC Guns

x

x’

1γγlogπREεQ0.038ε 2

00xn,

final

initial

0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.5

0.6

Z (mm)

Incl. thermal emittance

Excl. thermal emittance

Bunch charge

Electric field Emission radius



Envelope Equation in Drift

2

3 3 2 2 30

0TII

Envelope equation for electron beams in a drift (no acceleration or focusing)

The beam envelope equation describes the forces acting on the beam envelope radius. The signs of both terms on the left hand side (other than the second derivative term) are negative, indicating the beam sees defocusing forces. The second term is the space-charge induced expansion. The third term is the thermal emittance expansion. Note the use of because the low-energy beams are not quite relativistic in the drift.

2

2

ddz

c

Alfven current (17 kA)

rms radius

Peak current Thermal emittance

0 2TkTmc

e s=



GaAs Photoemission

The quantum efficiency (QE) of GaAs cathode is enhanced by adding a monolayer of cesium to the surface. Cesium reduces the effective electron affinity by lowering the vacuum energy level to below the bulk energy level. Cesiated GaAs is thus a negative electron affinity (NEA) photocathode.



Schematics of the JLab DC Gun

ElectrodesVacuum chamber

Field emission from the support tube causes charging of ceramic leading to punch-through

Corona shield

High voltage feedNEG pumps

RGA, extractor gaugeand leak valve

Photocathode

GaAs wafer is activated into an NEA photocathode with Cs deposition from inside the ball cathode

Courtesy of C. Sinclair



High-current Operation




Ion Back-bombardment

Damages on GaAs wafer after a high-current run

Crystal structure damage to the electrostatic center

QE of a new GaAsphotocathode is 5-7%

QE of a used GaAscathode drops to 1%

Illuminated area by drive laser light

Illuminated area by drive laser light

Positive ions accelerated by the electrostatic field strike the cathode surface causing a drop in quantum efficiency and damages to the crystal structure.




Normal-Conducting RF Injector

LCLS S-band NCRF Gun

Courtesy of D. Dowell



Schematic of an NCRF Injector

modelocked laser

klystron

injector

RF signal

harmoniccrystal

beamexpander

aperture

lens

solenoidcathode

RF coupling

circulator

electron beam

Electrons are generated by the laser pulses in synchrony with the accelerating RF field in an RF injector with n+½ cells. The high gradients produce electrons with relativistic energy at the injector exit.



Image Charge Effects

+

+

+

+

+

+

+

+

+

Cathode Vacuum

20

ICqEre p

=

Image charge field

0 5 10 11 1 10 10 1.5 10 10 2 10 100

7.5 105

1.5 106

2.25 106

3 106

3.75 106

4.5 106

5.25 106

6 106

E t( )

t

0 5 10 11 1 10 10 1.5 10 10 2 10 100

7.5 105

1.5 106

2.25 106

3 106

3.75 106

4.5 106

5.25 106

6 106

E t( )

t

Elec

tric

fiel

d (V

/m)

Time (s)

Elec

tric

fiel

d (V

/m)

Time (s)

For 1 nC and 0.5 cm radius, the image charge field is 1 MV/m. The applied RF field at launch phase has to exceed the image charge field.

= 120

= 240

( )0 0sinzE E tw f= +

Applied RF field



Electric and Magnetic Fields

Ez

Bs

Bbc

Bucking coil (to null out magnetic field at cathode)

Main solenoid

2½ cell gun

Electric Field

Magnetic Field



Beam Envelope Equation

2

23 3 2 2 2 3

0

0TB r z

I ek E v BI mc

Acceleration damping

Solenoidalfocusing

Space charge expansion

RF focusing due to azimuthal magnetic field

Electron beam envelope equation in an RF injector

The beam envelope equation describes the forces acting on the beam envelope radius. If the signs of the terms on the left hand side are positive, the beam sees focusing forces. If the signs of the terms on the left hand side are negative, the beam sees defocusing forces.

RF focusing & defocusing due to radial electric field

Thermal emittance



Emittance Growth in RF Injectors

• Space charge Space charge is reduced by using large radius, high gradient, and long electron bunch.

• RF RF effects are reduced with small radius, solenoid focusing, short electron bunch, and low frequency.

• Thermal Reduced with small radius, semiconductor cathodes

Cathode

Space charge defocusing

Solenoidfocusing

53,

z

rA

scn

I

I

Space-charge induced emittance

RF induced emittance

Thermal emittance

, 2n thermal rkTmc

e s=

222, zrRFRFn k

RF defocusing

RF focusing

Beamenvelope



Emittance vs Bunch Charge and Current

0 100 200 3000

10

20

30

40

200

400

600

800

1,000

1,200

Micropulse Current (A)

Normalized,rmsEmittancemm-mrad)

1 nC 5 nC

UnnormalizedrmsEmittance(nm)

APEX datawith compensation

Theory without compensation

Nor

mal

ized

rms

emitt

ance

(m

)

KJ Kim’s Theory(no emittance compensation)

LANL data withemittance compensation

Micropulse peak current ()

The experimentally measured emittance is better than theoretical predictions because KJ Kim’s theory does not account for emittance compensation effects. Note the nonlinear increase in normalized emittance with charge.

Courtesy of P. O’Shea



Beam Dynamics in an RF Injectorwith Magnetic Focusing

-1

-0.5

0

0.5

1

Ez

Bz




-1

-0.5

0

0.5

1

Ez

Bz



Single-Particle Equation of Motion

50 0 50 100 150 200 250 300 350 4001

0

1

Field phase [deg]

Nor

mal

ized

ele

ctric

fiel

d

50 0 50 100 150 200 250 300 350 4001

0

1

Field phase [deg]

Nor

mal

ized

ele

ctric

fiel

d

( ) ( )0 0cos sin( )E t E kz tw f= +

Launch

phase at½ cell exit

On-axis accelerating field

Phase evolution

( )sin sin 2ddg

a f f zz

é ù= + +ë û

Energy evolution

Dimensionless gradient

02

02eEkm c

a=

Change independent variablekzz =

10 0

1

2 1

t kzc

f w f b z z fg

bu g

-

-

= - + = - +

= =-

2

RF

k pl

=RF wavenumber

21

1ddf gz g= -

-



Solution to Single-particle EOMApproximate solution for energy

( )20

0

1 1 12 sin

f g g fa f

é ù= - - - +ê úë û

01 2 sinkzg a f= +

Plug the approximate into the phase equation and integrate

Plug the phase solution into the energy equation (without 2nd term) and integrate

( )( )011 sin cos cos 22

kz kzg a f f fé ùê ú= + + - +ê úë û

To minimize emittance, select the launch phase such that the beam exits the first cell at

( )0 01sin

2p f f

a- =

Numerical examples for LCLSE0 = 100 MV/mk = = (10.5 cm)= 1.635

Optimum launch phase0 = 6o

Energy at first cell exitE = 2.1 MeV

02

02eEkm c

a=where



Energy Gain

Numerical calculations yield a beam energy that is higher than KJ Kim’s analytical predictions. The beam is relativistic at the end of the 1st cell.

Courtesy of J. Schmerge



Energy and Exit Phase vs Launch Phase

The beam energy at the exit of the LCLS 1½-cell gun is linearly proportional to the cavity gradient. The beam energy is almost independent of launch phase up to a launch phase of 50o.




Emittance Compensation(Carlsten’s) Theory

Electron bunch is sliced into N axial slices, each having its own phase space ellipse

headtail

Cathode Before solenoid

At solenoidAfter solenoid, no RF effects

After solenoid, with RF effects

x’

x

x’

x

x’

x

x’

x

x’

x

Projected emittance



Slice Emittance

Energy/Time Axis (pixels)

Head

High Energy Low Energy

Tail

Converging beam (underfocused)

Hor

izon

tal B

eam

Siz

e (p

ixel

s)




Measured Slice Emittance & Current

0.0

1.0

2.0

3.0

4.0

5.0

6.0

-4.0 -2.0 0.0 2.0 4.0

Time (ps)Projected values displayed at t=0

n (

m)

0.010.020.030.040.050.060.070.080.090.0

-4.0 -2.0 0.0 2.0 4.0

Time (ps)Projected values displayed at t=0

I (A

)

Courtesy of J. SchmergeSlice Normalized Emittance Slice Peak Current

Projected Emittance

The slice emittance is approximately one-half the projected emittance (emittance of all slices projected onto x’-x phase space). The smaller slice emittance translates into better FEL performance (e.g. shorter gain length).



Invariant Envelope(Serafini-Rosenzweig’s) Theory

Perturbed trajectories oscillate around the equilibrium with the same frequency but with different amplitudes.

Invariant envelope corresponds to the matched beam envelope

Match the beam to invariant envelope (however, this is not possible inside an RF gun).

0

23inv

II

sg g

=¢

0

1 23inv

II

sg g

¢ =



Invariant Envelope Oscillation in Linac

Acceleration damps the plasma oscillation resulting in a steady-state emittance. The beam transitions from a space charge dominated beam at the linac entrance to emittance dominated beam at the exit. The transition energy is given above.

2 22

3 2 3

1

2 20n

RFk

Electron beam envelope equation in the booster linac

1 4ˆ3SC

Invariant envelope for space charge dominated beam

where

0

II

Invariant envelope for emittance dominated beam

2ˆ3

nemit

Transition energy0

23 n

II



Ferrario’s Working Point

The beam is matched into the booster whose entrance is located between two emittance minima. Plasma oscillation is damped and the beam emittance asymptotically approaches an even lower emittance at very large z.

Linac entrance

0

1

2

3

4

5

6

0 5 10 15

HBUNCH.OUT

sigma_x_[mm]enx_[um]

sigm

a_x_

[mm

]

z_[m]

nn[mm[mm--mrad]mrad]

Z [m]

Simulation for an L-band photoinjector

Courtesy of M. Ferrario

0

23inv

entrance

II

sg g

=¢

30

23inv

II

sg

¢ =-

Matched beamrms radius

Matched beamconverging angle



Superconducting RF Injectors

Laser

PhotocathodeNiobium Cavity

e-

Courtesy of J. Teichert

Rossendorf SRF Gun



Pros and Cons of SRF InjectorsPros

• Potentially high gradients in cw operation

• Potentially high quality electron beams thanks to high gradients

• Very efficient use of RF in cw operation

• Good vacuum (cryo pumping)

Cons

• Presence of cathodes in the SRF cavity reduces the cavity Q

• Unable to accept solenoid fields inside the injector to perform emittance compensation

• If low-QE superconducting cathodes are used, high-power UV lasers can warm SRF surfaces

• If NEA cathodes are used, cesium contamination may increase risks of multipacting and field emission



Options for SRF Injectors

• Number of cells– ½ cell SRF injector– n+½ cell SRF injector

• Cavity shapes– Triangular– Low-loss– Re-entrant– Quarter-wave

• Cathodes– Normal-conducting cathodes– Superconducting cathodes

• Lead• Niobium

1½-Cell Cavity



Cathode Isolation Choke JointsRF choke joint Quarter-wave choke joint

Two methods exist for electrically and thermally isolating the photocathode from the SRF cavity. Both methods involve setting up a standing wave null at the cathode to prevent RF from leaking out of the SRF cavity, hence the term choke joints or choke filters. The RF choke has been tested on the Rossendorf injector.



Emittance Compensation witha Magnetic (Higher Order) Mode

Magnetic higher order modes can be excited by the fundamental in an SRF cavity and provide axial magnetic field for performing emittance compensation.

Courtesy of J. Teichert



Emittance Compensation withan External Solenoid

A superconducting solenoid electromagnet can be placed inside the cryostat to provide the axial field for emittance compensation. The electromagnet must be turned on after the SRF cavity has been cooled to below the critical temperature.



References

RF Superconductivity H. Padamsee

Handbook of Accelerator Physics and Engineering Alex Chao

“RF and space charge effects in RF electron guns” K.J. KimNucl. Instr. Meth. Phys. A V. 275 (1989) 201-218

Books and Article

URL

SRF07 Workshop http://web5.pku.edu.cn/srf2007/proceedings.html

ICFA Newsletter http://www-bd.fnal.gov/icfabd/Newsletter46.pdf

High-power Workshop http://home.physics.ucla.edu/power/indux.html

theory and practice of free-electron lasersuspas.fnal.gov/materials/09unm/day 4.pdf · us particle...

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