The RF Photoinjector Gun: Microwave...
Transcript of The RF Photoinjector Gun: Microwave...
The RF Photoinjector Gun:
Microwave point-of-view
J. Rosenzweig
UCLA Dept. of Physics & Astronomy
RF gun as microwave device
• Photoinjector guns are high gradient standing wavedevices
– Beam dynamics complex
– Space-charge and RF are large effects
• Need to understand (a lot!):– Cavity resonances
– Coupled cavity systems
– Power dissipation
– External coupling
– Time-domain response
– Measurement and tuning procedures
• Reference: Chapter 7 Fundamentals of Beam Physics
The photoinjector gun
Neptune RF gun with cathode plate removed
SPARC gun and solenoid
The 1.6 cell rf gun geometry
• Approximate with cylindrical symmetry
• -mode, full ( /2) cell with 0.6 cathode cell
• How to calculate resonant frequencies?
Cathode plane
0.6 cell Full cell
Cavity resonances: the pill-box model
Lz
Rc
• Pill-box model approximates
cylindrical cavities
• Resonances from Helmholtz
equation analysis
• Look for TM (axial field)
modes
• Conducting BCs
1
kz, n
2+
2
c2
˜ R = 0
Pill-box fields
• No longitudinal dependencein fundamental
• Electromagnetic fields
• Stored energy
0,1
2.405c
Rc
Ez( ) = E0J0 k ,0( )
H ( ) = 0
k ,0
E0J1 k ,0( ) = c 0E0J1 k ,0( )
UEM = 14 0LzE0
2 J02 k , 0( ) + J1
2 k , 0( )[ ] d0
RC
= 12 0LzE0
2Rc2J1
2 k , 0RC( )
kz,0 = 0
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
EZER
EL
EC
TR
IC F
IEL
D A
MP
LIT
UD
E
k /2.405
A circuit-model view
• Lumped circuitelements may beassigned: L, C, and R.
• Resonant frequency
• Tuning by changinginductance,capacitance
• Power dissipation bysurface current (H)
Contours of constantflux in 0.6 cell of gun
1
LC
Cavity shape and fields
• Irises needed for beam passage and RF coupling
• Fields near axis (in iris region) may be better representedby spatial harmonics
• Higher (no speed of light) harmonics have nonlinear(modified Bessel function) dependence on .
– Energy spread
– Nonlinear transverse RF forces
• Avoid re-entrant nose-cones, etc. Circle-arc sections areclose to optimum.
Ez( ,z ,t) = E0 Im ann=
exp i kn, zz t( )[ ]I0 k , n[ ] k , n = kn, z2 / c( )
2
Cavity coupling: simple model
• Circuit model allows simplederivation of mode frequencies
• Ignore resistance in walls (doesnot affect frequency much)
• Electric coupling is capacitive
• Off-axis slots can givemagnetic coupling (effectivelyreverses sign in )
• Solve eigenvalue/eigenmodeequations
(b)L
C
LC
CC
z
Electric
coupling
(a)
d2I1dt2
+ 02 1 c( )I1 = c 0
2I2
d2I2dt2
+ 02 1 c( )I2 = c 0
2I1
02 1 c( ) 2
c 02
c 02
02 1 c( ) 2
i = 0
Coupled cavity modes
• Secular equation
• Two eigenvalues
• Corresponding to eigenvectors
• Lower frequency when cells are in phase (0-mode)
• The higher frequency mode is the -mode, wherethe excitation is 180 degrees out of phase in thetwo cavity cells
– Higher frequency due to less near-axis fields
– Lower capacitance
= 0 (0 - mode) and = 0 1+ 2 c ( - mode)
4 2 02 1 c( ) 2 + 0
4 1 2 c( ) = 0
i =1
2
±1
1
Finding frequencies of cells
• Real geometry, useSUPERFISH simulation
• In simulation detune theopposing cell
• 0.6 cell has frequency of2854.01 MHz
• Full cell is slightly higher:2854.60 MHz due to lower C
• These are the geometrieswhich produce “field”balance
Full cell
0.6 cell
The coupled mode frequencies
• The -mode frequency is
2856.0 MHz as expected
• This mode needs to bebalanced (equal fields)
• Zero mode frequency is2852.66 MHz. Note thatfield arrows do not reverse
• The frequency separation is~3.34 MHz according tocalculation– This is sensitive to the iris
geometry as built
Measurement and tuning of
frequencies• Frequency response can be measured on a network analyzer
– Calibrated phase and amplitude response, calibrated frequency
• Two measurements: S11 (reflection) S21 (transmission)
• Resonance frequencies of individual cells and coupled modes
– Full cell. Remove cathode to detune 0.6 cell.
– 0.6 cell. S11 through (detuning) probe in full cell.
• Tuning via Slater’s theorem/circuit model guide
0
0
=Vc
UEM
12 0
r E 2 1
2 µ 0
r H 2[ ]
Measurement of fields
• Use so-called “bead-pull” technique
– “Bead drop in gun, beam tube pointed upwards…
• Metallic or dielectric bead (on optical fiber)
• Metallic bead on-axis gives negative frequency shift (electric fieldenergy displaced)
– No magnetic effects on-axis for accelerating mode
• More complex if one has magnetic fields (deflector)
Ez
Field balance and mode
separation• Bead-pull technique is invasive
• Calibrate mode separation
– Slightly different than coupled
cavity model…
• Field balance possible in fast
cathode swap
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 2 4 6 8 10
Neptune Gun Bead Pull
DataSuperfish balancedSuperfish x 1.04 in 0.6 cell
Fiel
d am
plitu
de s
quar
ed
z [cm]
New Neptune gun field balance from bead pull
0
0.5
1
1.5
2
2.5
3
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6
Fie
ld B
alan
ce (
0.6
/Fu
ll)
Mode splitting (MHz)
Other parameterizations
0
0.5
1
1.5
2
2.5
3
-4 -3 -2 -1 0 1 2
Fie
ld b
alan
ce (
0.6
/fu
ll)
0.6 cell - full cell frequency difference (MHz)
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
-4 -3 -2 -1 0 1 2
Mo
de
spli
ttin
g (
MH
z)
Field
balance
0.6 cell - full cell frequency difference (MHz)
• Fundamental problem:the field balance is adouble valued functionof the mode splitting
• Full cell frequency istuned by insertable(magnetic) tuners(in=higher)
• If you find minimumsplitting, make full cellhigher in frequency
Final tuning
• Do not want the full-cell tuners re-entrant (breakdown)
• Cathode deformation using “tuning nut”
• Temperature tuning=44 kHz/degree
• Almost perfect balance between atmospheric index and20° C above room temperature operation
TemperaturetuningCapacitive
tuning thru deformation
Power dissipation in walls
• Wave equation in conductors, harmonic solution
• Complex wavenumber into wall normal, givesskin-depth
• Power is lost in a narrow layer (skin-depth) of thewall by surface current excitation
dP
dA=
Ks2
4 s c
=Ks2
4
µ 0
2 c
=Ks2
2Rs , Rs
1
2
µ 0
2 c
Ks =
r H || = µ 0
r B ||
Surface resistivity
r 2 µ 0 c t
µ 0
2
t2
r E r B
= 0 k2 + i µ 0 c + µ 0
2 = 0
k =µ0 c
21+ i( ) s Im k[ ]
1=
2
µ 0 c
Power dissipation and Q
• Total power dissipated in walls (pill-box example)
• Internal quality factor (pill box)
• For ~2856 MHz (S-band), Q~12,000
• Other useful interpretations of Q
– Exponential response in time domain
– Frequency response half-width
QUEM
P=Z02Rs
2.405LzRc + Lz( )
P = Rs c 0E0( )2RcLzJ1
2 k , 0Rc( ) + 2 J12 k , 0( ) d
0
Rc
= Rs c 0E0( )2RcJ1
2 k , 0Rc( ) Lz + Rc[ ]
Z0 = 377
Frequency domain picture
10-6
10-5
0.0001
0.001
0.01
0.1
1
0 0.5 1 1.5 2
exactresonant approximation
||I()||
/0
• Resonant widthproportional toinverse of power loss
• Allows definition oflumped resistance
• Can measure withnetwork analyzer
• Be careful aboutexternal couplingQ =
1/ 2
=L
R
Cavity filling, emptying, and
external coupling
• Exponential response of
cavity voltage
• Controlled by loaded QL
• Energy extraction by
– Radiation into waveguide
– Beam acceleration (high
average current systems)
0
0.2
0.4
0.6
0.8
1
1.2
-5 0 5 10 15 20
no beam with beam
||Vc||/||V
f||
t/f
Vc t( )2 c
1+ c
VF sin 0 t t0( )[ ]( ) 1 exp 0
2QL
t t0( )
VSWR and
• Measure reverse and forward
voltage
– Time domain
– NWA Smith chart
• Calculate VSWR
• We want critical coupling
• This is rc=1 on lower right
• Loaded Q
c =1
QL =Q
1+ c
=Q
2
VSWRVF + VRVF VR
=c , c >1
c1, c <1.
Temporal response of the cavity
• Standing wave cavity fillsexponentially
• Gradual matching ofreflected and radiate power(E2) from input coupler– Reflected wave from input
coupler=re-radiated wave
• In steady-state, all powergoes into cavity (criticalcoupling) so reflectedpower is eventuallycancelled
0.0
0.20
0.40
0.60
0.80
1.0
0 1 2 3 4
Cavity PowerReflected power
Rel
ativ
e po
wer
t/f
E 1 expt
2Q
1 exp
t
f
Voltage, power and acceleration
• The square of the cavity voltage isproportional to the shunt impedance
• The accelerating field is The square of thecavity voltage is proportional to the shuntimpedance per unit length
• The 1.6 cell gun S-band cavity has
Z s =dP
dzE02
Pc =Uc
Q0
=L Ic
2
L /R= Ic
2R =
Vc2
R=Vc2
2R=Vc2
Zs
Z s 40 M /m
The 1.6 cell RF gun
• The design power for the Neptune gun isP~6 MW
• The “accelerating” length of the gun isLg=0.0845 m
• The average accelerating field is
• The peak on-axis accelerating field is ~twice the average, or over 100 MV/m
E0 = P Z s
Lg
= 53 MV/m