theory and practice of free-electron lasersuspas.fnal.gov/materials/09unm/day 4.pdf · us particle...
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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
44--22LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--33LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Chapter 6RF Linear Accelerators
(continued)
44--44LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
RF Linear Accelerators (Linac)
• Introduction to RF Linac• Properties of RF Cavities• Coupled Cavity Linac• Superconducting RF Linac• Energy Recovery Linac
44--55LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Superconducting RF Linacs
• RF Superconductivity• Meissner effect• Surface resistance• Temperature dependence
• Superconducting cavities• Cavity designs• Cavity Q versus Eacc
44--66LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--77LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--88LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
44--99LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--1010LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--1111LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--1212LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Cavity Q vs. Gradient
Eacc (MV/m)
Multipacting
Field emission
Q disease
Quench
Q slope
44--1313LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Unloaded Q vs Gradient ofa TESLA 1.3 GHz SRF Cavity
Eacc (MV/m)
Courtesy of D. Reschke
44--1414LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--1515LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--1616LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Example of ERL: CEBAF
44--1717LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Same Cavity vs Different CavitySame cavity energy recovery
Accelerated electron bunches
Decelerated electron bunchesRF wave
Different cavity energy recovery
Extracted RF power
44--1818LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
• 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 = -
44--1919LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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)
44--2020LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--2121LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--2222LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--2323LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--2424LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Chapter 7Electron Injectors
44--2525LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--2626LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--2727LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
DC Injector
Courtesy of C. Hernandez-Garcia
Jefferson Lab DC Gun
44--2828LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--2929LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--3030LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--3131LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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=
44--3232LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
44--3333LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--3434LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
High-current Operation
Courtesy of C. Hernandez-Garcia
44--3535LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
Courtesy of C. Hernandez-Garcia
44--3636LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Normal-Conducting RF Injector
LCLS S-band NCRF Gun
Courtesy of D. Dowell
44--3737LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
44--3838LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--3939LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--4040LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--4141LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--4242LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--4343LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--4444LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--4545LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--4646LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--4747LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--4848LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--4949LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--5050LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--5151LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Beam Dynamics in an RF Injectorwith Magnetic Focusing
-1
-0.5
0
0.5
1
Ez
Bz
44--5252LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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= -
-
44--5353LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--5454LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--5555LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
Courtesy of J. Schmerge
44--5656LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--5757LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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)
Courtesy of J. Schmerge
44--5858LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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).
44--5959LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
¢ =
44--6060LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
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University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--6262LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Superconducting RF Injectors
Laser
PhotocathodeNiobium Cavity
e-
Courtesy of J. Teichert
Rossendorf SRF Gun
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University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
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University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
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University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
44--6666LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
44--6767LA-UR 09-03377US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
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University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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