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