The Physics of
High-Brightness Sources
SSSEPB 2015
The Physics of High Brightness Sources
D. H. Dowell
David H. Dowell
SLAC
1
Boeing thermionic gun w/
subharmonic bunchers
PITZ gun
ca. 1980:
after 1985:
Sheffield et al.
Carlsten et al.
S.V. Benson, J. Schultz,
B.A, Hooper, R. Crane
and J.M.J. Madey,
NIM A272(1988)22-28
The Physics of High Brightness Sources
D. H. Dowell
< 0.5 meter
2
The Photocathode RF Gun System
•Introduction – will concentrate on the RF gun
•Field equations and eqns. of motion for photocathode guns
•Intrinsic emittance and QE
•Space charge limited emission
•Simple optical model & RF emittance
•Emittance compensation and matching
•Solenoid aberrations
•“Beam blowout” dynamics & 3rd order space charge
The Physics of High Brightness Sources
D. H. Dowell
Basic components of the photocathode RF gun system
3
Maxwell’s Equations
After 150 years, Maxwell’s
equations still describe all the
physics of photoinjectors!
Faraday’s Law (1831)
𝛻×𝐸 =−
𝜕𝐵
𝜕𝑡
Ampere’s Law (1826)
𝛻 × 𝐵 = 𝜇0 𝐽 +
Gauss’s Laws for electric and
magnetic fields
𝛻∙𝐸 =
𝜌
𝜖0
𝛻∙𝐵 =0
1 𝜕𝐸
𝑐 2 𝜕𝑡
The current and charge densities obey the continuity equation:
𝛻∙𝐽+
𝜕𝜌
=0
𝜕𝑡
where 𝐽 is the current density
The Physics of High Brightness Sources
D. H. Dowell
150 Years of Maxwell’s equations,
Science, 10 July 2015, Vol 349, pp.136-137
4
Special Properties of Maxwell’s Equations
There are useful observations to be made about the fields satisfying Maxwell’s Eqns.
Gauss’ law (in differential form and cylindrical coordinates ) in a charge free-region becomes
1 𝜕 𝑟𝐸𝑟
1 𝜕𝐸𝜃 𝜕𝐸𝑧
+
+
=0
𝑟 𝜕𝑟
𝑟 𝜕𝜃
𝜕𝑧
𝛻 ∙ 𝐸 =0
With cylindrical symmetry,
𝜕𝐸𝜃
𝜕𝜃
= 0, and
𝜕𝐸𝑟 𝐸𝑟
𝜕𝐸𝑧
+
=−
𝜕𝑟
𝑟
𝜕𝑧
r
𝑑
𝜕𝐸𝑧
𝛾𝑚𝑟 = 𝑒𝐸𝑟 = −𝑒
𝑟
𝑑𝑡
𝜕𝑧
𝑟=−
𝑒 𝜕𝐸𝑧
𝑟
𝛾𝑚 𝜕𝑧
∴The fringe of the longitudinal field leads to a radial fringe field.
𝜕𝐸𝑧
radial acceleration ∝ 𝐸𝑧 -divergence
The linear part of the radial
𝜕𝐸
field will be focusing if 𝑒
> 0 and electrons get focused to the axis. Electrons are defocused when 𝑒 𝑧 < 0.
𝜕𝑧
𝜕𝑧
When the field is accelerating the electron, 𝑒𝐸𝑧 > 0 making the z-derivative of 𝑒𝐸𝑧 is positive for an electron
entering from a zero field region. Hence the electron is focused when entering an accelerating field.
Correspondingly, the sign of the derivative reverses when the electron exits from high field to zero field. Which
defocuses the electrons.
This relation between the derivative of the longitudinal field and the radial field is expressed more generally by
the Panofsky-Wenzel theorem (Panofsky and Wenzel, Rev. Sci. Instrum., 27, 976(1956)). See also Wangler, RF
Linear Accelerators, pp. 166-167. This theorem has several important implications concerning the photocathode
injector. For example, in pure TE-modes, Ez and its derivatives are zero and therefore produces no transverse
momentum kick.
The Physics of High Brightness Sources
D. H. Dowell
𝜕𝐸
Assume that the field is uniform across the aperture, then 𝑟 = 0
𝜕𝑟
and the Lorentz eqn. can be written just as the z-partial derivative of Ez :
5
Lorentz Force:
𝐹 = 𝑒 𝐸 + 𝑐𝛽 × 𝐵
The Lorentz force and Maxwell’s equations together give the relativistic equation of
motion (sort of a relativistic Newton’s eqn.):
𝐹=
𝑑
𝛾𝑚𝑐 𝛽 = 𝑒 𝐸 + 𝑐 𝛽 × 𝐵
𝑑𝑡
𝒅
𝜸𝒎𝒙 = 𝜸𝒎𝒙 + 𝜸𝒎𝒙 = 𝒆 𝑬𝒙 + 𝒚𝑩𝒛 − 𝒛𝑩𝒚
𝒅𝒕
𝒅
𝜸𝒎𝒚 = 𝜸𝒎𝒚 + 𝜸𝒎𝒚 = 𝒆 𝑬𝒚 + 𝒛𝑩𝒙 − 𝒙𝑩𝒛
𝒅𝒕
𝛾=
𝛾=
1
1 − 𝛽2
𝑑𝛾
𝑑𝛾
=
𝑧 = 𝛾 ′𝑧
𝑑𝑡
𝑑𝑧
𝐸
1853-1928
1902 Nobel Prize
Longitudinal Dynamics
•
•
•
Extra force due to relativistic acceleration, 𝛾 > 0,
and force is proportional to velocity.
Transverse Dynamics
For the transverse dynamics, the force decreases as 1 𝛾 vs. 1 𝛾3 for the longitudinal dynamics.
The longitudinal acceleration couples the transverse and longitudinal velocities => transverse focusing during acceleration.
𝑒𝐸
Matched beam for 𝑥 = 0 => 𝑚𝑥 − 𝛾 ′ 𝑥 𝑧 = 0, esp. if 𝐸𝑥 is the defocusing space charge force
The Physics of High Brightness Sources
D. H. Dowell
𝒅
𝜸𝒎𝒛 = 𝜸𝒎𝒛 + 𝜸𝒎𝒛 = 𝒆 𝑬𝒛 + 𝒙𝑩𝒚 − 𝒚𝑩𝒙
𝒅𝒕
where 𝛾 ′ = 𝑚𝑐𝑧2
6
𝑒𝐸
Estimate of the transverse focusing due to longitudinal acceleration only, 𝛾 ′ = 𝑚𝑐𝑧2 ,
without any transverse fields, 𝐸𝑥 = 0 :
1𝑑 2
𝑥 = −𝛾′𝑧𝑥 2
2 𝑑𝑡
multiply both sizes by 𝑥 to get
Re-arranging terms gives:
1 𝑑 𝑥2
2 𝑥2
Integrating and solving for 𝑥 𝑧 :
𝑥
𝑑 𝑥
= −𝛾 ′
𝑥0 𝑥
= −𝛾 ′ 𝑑𝑧
𝑥 𝑧 = 𝑥0 𝑒 −𝛾
𝑧
𝑑𝑧
0
′𝑧
The relativistic 𝑥 𝑧-term can provide significant focusing near a cathode at high accelerating
field. The figure compares focusing at cathode fields of 20 MV/m and 50 MV/m. This 𝛾′focusing helps to counteract the space charge defocusing forces during initial stages of the
electron bunch’s formation by the laser and its acceleration from rest.
𝛾 ′ = 40 (~20 MV/m)
𝑥
𝑥0
𝛾 ′ = 100 (~50 MV/m)
The Physics of High Brightness Sources
D. H. Dowell
𝑥 = −𝛾′𝑥 𝑧,
7
z (mm)
This is one reason why a high cathode field allows higher charge/peak current operation.
Basic Cavity Shapes
of Transverse Mode
RF Guns
𝑅𝑐𝑎𝑣𝑖𝑡𝑦
𝒍𝒄𝒂𝒗𝒊𝒕𝒚
~8 cm
Most RF guns use the TM011 mode
whose non-zero field components are
(see Wangler, p30):
𝐸𝑧 = 𝐸0 𝐽0 𝑘𝑟 𝑟 cos 𝑘𝑧 𝑧 exp 𝑖 𝜔𝑡 + 𝜙0
𝑘𝑧
𝐸𝑟 = − 𝐸0 𝐽′ 0 𝑘𝑟 𝑟 sin 𝑘𝑧 𝑧 exp 𝑖 𝜔𝑡 + 𝜙0
𝑘𝑟
𝑖𝑘𝑧
𝐵𝜃 = −
𝐸0 𝐽′ 0 𝑘𝑟 𝑟 cos 𝑘𝑧 𝑧 exp 𝑖 𝜔𝑡 + 𝜙0
𝑘𝑟 𝑐
where 𝑘𝑟 =
2.405
𝑅𝑐𝑎𝑣𝑖𝑡𝑦
and 𝑘𝑧 =
𝜋
The Physics of High Brightness Sources
D. H. Dowell
Field labels for Pillbox Cavity
Tmnp
m: rotational asymmetry => quadrupole fields
n: Radial dependence of field => r3 - nonlinearities
p: Longitudinal mode of cavity => RF emittance
𝑙𝑐𝑎𝑣𝑖𝑡𝑦
𝑘 2 = 𝑘𝑟 2 + 𝑘𝑧 2 and 𝑘 =
𝜔
𝑐
Comparison of E&M equations
with Superfish is pretty good (to first order)!
8
Synchronous phase and bunch compression*
Energy gain in first half cell
Bunch compression factor for 𝛼 = 1.8
𝛼=
𝑒𝐸𝑝𝑒𝑎𝑘 𝜆𝑟𝑓
4𝜋𝑚𝑐 2
The Physics of High Brightness Sources
D. H. Dowell
The beam asymptotically approaches the synchronous phase, 𝜙𝑠𝑦𝑛𝑐 ,
as it is accelerated from rest to c from the cathode. This is the phase
slip between the bunch and the light-velocity rf fields while the
bunch is non-relativistic.
9
*See K. Floettmann, RF-induced beam dynamics in
rf guns and accelerator cavities, Phys. Rev . ST
Accel. Beams 18, 064801(2015).
Optical properties of the gun’s RF field
Maxwell’s eqns. relate the momentum kicks of the radial electric field to the z- and t-derivatives of the
longitudinal electric field:
𝐸𝑟 = −
𝑟 𝜕
2 𝜕𝑧
𝐸𝑧 and 𝑐𝐵𝜃 =
𝑟 𝜕
2𝑐 𝜕𝑡
𝐸𝑧
See K-J. Kim, NIM A275(1989)201-218 and references therein
The radial momentum kick is then
∆𝑝𝑟 = 𝑒
𝐸𝑟
𝑑𝑧
𝑒
=−
𝛽𝑐
2
𝑟 𝜕𝐸𝑧
𝑑𝑧
𝛽𝑐 𝜕𝑧
𝐸𝑧 = 𝜃 𝑧𝑓 − 𝑧 𝐸0 sin 𝜙
𝐸𝑧 (𝑧)
zf
z
The derivative of the Heaviside step function is
a delta function: 𝜕𝐸
𝑧
= −δ 𝑧𝑓 − 𝑧 𝐸0 sin 𝜙
𝜕𝑧
∆𝑝𝑟
𝑒 𝑟
𝑒𝐸0 sin 𝜙𝑒
𝑟
=−
δ 𝑧𝑓 − 𝑧 𝐸0 sin 𝜙 𝑑𝑧 =
𝑟
=
−
𝛾𝑚𝑐
2 𝛽𝑐
2𝛽𝛾𝑚𝑐 2
𝑓𝑟𝑓
The gun’s optical strength is dominantly due to the defocus at the
exit of the last cell, or at the exit of any strong electric field,
including DC. The defocusing strength is strong in a high-field gun.
E.g., at 𝑒𝐸0 sin 𝜙𝑒 = 100MV/m the focal length is only -12 cm!
∴
1
𝑒𝐸0 sin 𝜙𝑒
=−
𝑓𝑟𝑓
2𝛽𝛾𝑚𝑐 2
The Physics of High Brightness Sources
D. H. Dowell
Assume the RF field vs. z is a constant step function over the gun’s length. This looks like:
10
Implication of the gun’s defocus: RF Emittance*
2 mc 2
f rf  
eE0 sin e
*See K-J. Kim, NIM A275(1989)201-218
Phase-dependent focal strength
10
d
x   
d e
 1 

x e
f 
 rf 
x' (mR)
5
eE cos 
 x  0 2 e  x 
2mc
Phasedependent
divergence
 e = 0 deg
0
5
definition of normalized emittance:
 e =90
deg
10
𝜖𝑛 = 𝛽𝛾
𝑥 2 𝑥′2 − 𝑥𝑥′
2
≅ 𝛽𝛾𝜎𝑥 𝜎𝑥′
1.5
1
0.5
0
0.5
1
1.5
𝜖𝑟𝑓,1 =
The Physics of High Brightness Sources
D. H. Dowell
x (mm)
𝑒𝐸0 cos 𝜙𝑒
𝜎𝑥 2 𝜎𝜙
2
2𝑚𝑐
This is the linear part of the emittance.
The non-linear part due to the RF curvature
is 2nd order in the phase spread of the bunch,
𝜖𝑟𝑓,2 =
𝑒𝐸0 sin 𝜙𝑒
2 2𝑚𝑐 2
2
𝜎𝑥 𝜎𝜙
2
High order emittance scales as the bunch length squared >>
a common feature of many emittance sources.
Normalized emittance (microns)
10
1st + 2nd
1st-order rf
emittance
1
2nd-order rf
emittance
0.1
The total rf emittance is the square-root of sum of squares:
𝜖𝑟𝑓,1+2
𝑒𝐸0
𝜙𝑒 2
2
2
=
𝜎𝑥 𝜎𝜙 𝑐𝑜𝑠 𝜙𝑒 +
𝑠𝑖𝑛2 𝜙𝑒
2
2𝑚𝑐
2
0
20
40
60
80
100
120
Exit phase (deg)
RF emittance can be eliminated in a
2-frequency gun. See D.H. Dowell et al., PAC0?
11
1
 1 L2   1



 0 1  f sol
𝑓𝑒𝑓𝑓
1
0  1 0  1 L
 1 L2   1

1
  1
1  0 1    0 1 
1 

 
 f eff
 f rf

=
1
1
+
𝑓𝑟𝑓 𝑓𝑠𝑜𝑙
0 1 L

1
1  0 1 



Image of electron emission from Cu cathode in LCLS-gun
𝑟𝑣𝑠
𝑟′𝑣𝑠 =
=
L

1  2
f eff


1

 f eff
L1 L2 

f eff 

L
1 1

f eff

L1  L2 
Point-to-point imaging when zero
𝐿2
𝑓𝑒𝑓𝑓
1
−
𝑓𝑒𝑓𝑓
1−
1
𝑓𝑒𝑓𝑓
=
𝐿1 + 𝐿2 −
1−
𝐿1 𝐿2
𝑓𝑒𝑓𝑓
𝐿1
𝑓𝑒𝑓𝑓
𝑟𝑐
𝑟′𝑐
1
1
𝐿1 + 𝐿2
+
=
𝑓𝑟𝑓 𝑓𝑠𝑜𝑙
𝐿1 𝐿2
1
𝐿1 + 𝐿2 𝑒𝐸0 sin 𝜙𝑒 0.1 + 1
1
1
=
+
~
+
≈
11
+
8
=
𝑓𝑠𝑜𝑙
𝐿1 𝐿2
2𝛽𝛾𝑚𝑐 2
0.1 ∗ 1 0.12
0.05𝑚
•
•
The solenoid focal length to image the cathode is only ~5 cm with RF on!
Demonstrates beam quality
The Physics of High Brightness Sources
D. H. Dowell
1
Using the RF gun
like a PEEM!
12
Optical Model of a RF Gun
Drive Laser
2mc2
f rf  
eE0 sin e
frf ~-15 cm for 100 MV/m
which needs to
be compensated for with
solenoid
Gun
Solenoid
Defocusing and
Focusing RF Lenses
Focusing of
Gun Solenoid
Solenoid cancels defocusing of
gun RF and performs emittance
compensation and matching to
booster linac.
Solenoid has focal length of ~15 cm but is ~20 cm long=> thick lens => aberrations
The principal solenoid aberrations can be classified as :
Chromatic
Geometric
Anomalous fields
Misalignment (not discussed here)
The Physics of High Brightness Sources
D. H. Dowell
2-6 MeV
Electron Beam
13
The Physics of High Brightness Sources
D. H. Dowell
Quotes taken from P. Musumeci, SSSEPB, SLAC, July 2013
14
See Hommelhof et al.,… for ultra fast emission from needles
and 2010 USPAS class notes, Electron Injectors for 4th Generation Light Sources by D. H. Dowell
• High QE (QE>5%) at long wavelengths
• fast time response: depends upon RF frequency + bunch length, typ. < 1ps
• Uniform emission
𝑄𝑏𝑢𝑛𝑐ℎ
𝐸𝑙𝑎𝑠𝑒𝑟 𝑝𝑢𝑙𝑠𝑒
=𝑒
𝑄𝐸
ℏ𝜔
D. Dowell et al., Cathode R&D for future light sources, NIM A 622:13(2010)
The Physics of High Brightness Sources
D. H. Dowell
QE and the Drive Laser
15
Emittances Near the Cathode
Intrinsic (aka Thermal) Emittance, e intrinsic :
Cathode’s material properties (EF ,w , EG ,EA , m* ,…)
Cathode temperature, phonon spectrum
Laser photon energy, angle of incidence and polarization
Large scale space charge forces across diameter and length of bunch
Image charge (cathode complex dielectric constant) effects space charge limit
Emittance compensation
Bunch shaping (beer-can, ellipsoid) to give linear sc-forces
Rough Surface Emittance:
Electron and electric field boundary conditions important
Surface angles washout the exit cone
Coherent surface modulations enhances surface plasmons
Three principle emittance effects:
Surface tilt washes out intrinsic transverse momentum > escape angle increases
Applied field near surface has transverse component due to surface tilt
Space charge from charge density modulation due to Ex surface modulation
The Physics of High Brightness Sources
D. H. Dowell
Bunch Space Charge Emittance:
16
Photo-Electric Emission and the 3-Step Model
𝑄𝑒−𝑏𝑢𝑛𝑐ℎ = 𝑒 𝑄𝐸 𝑁𝛾 = 𝑒 𝑄𝐸
Metal
Energy
1)Photon
absorbed
Excess energy:
𝐸𝑒𝑥𝑐𝑒𝑠𝑠 = ℏ𝜔 − 𝜙𝑤
e-
3)Electrons
escape to vacuum
e-
Vacuum level, E=0
Potential barrier
due to spillout electrons
Fermi
Energy
occupied
valence
states
energy
Vacuum
2)Electrons
move to surface
e-
𝐸𝑙𝑎𝑠𝑒𝑟 𝑝𝑢𝑙𝑠𝑒
ℏ𝜔
•QE and emittance depend upon electronic
structure of the cathode
•Sum of electron spectrum yield gives QE:
•Width of electron spectrum gives intrinsic
emittance
•Both are dependent upon the density of
occupied states near the Fermi level
QE 
Optical depth
photon
number of
emitted
electrons
EF


EF 
W
EDOS ( E )dE
 
n 
Direction normal to surface
*D. H. Dowell, K.K. King, R.E Kirby and J.F. Schmerge, PRST-AB 9, 063502 (2006)
The Physics of High Brightness Sources
D. H. Dowell
Photoelectric emission from a metal given by Spicer’s
3-step model:
1. Photon absorption by the electron
2. Electron transport to the surface
3. Escape through the barrier
  eff
17
Refraction of electrons at the cathode-vacuum boundary:
Snell’s law for electrons
Conservation of transverse momentum at
the cathode-vacuum boundary:
pxin  pxout
in
ptotal
 2m  E   
out
ptotal
 2m  E    EF  W 
Refraction law for electrons:
To escape electron longitudinal momentum needs to be
greater than barrier height:
pzin  2m  EF  eff 
2m  E    cos in  2m  EF  eff 
Outside angle is 90 deg at inmax which is typically ~10 deg.
sin out

sin in
n
E 
 in
E    EF  eff nout
Maximum internal angle for electron with
energy E which can escape:
The Physics of High Brightness Sources
D. H. Dowell
in
out
ptotal
sin in  ptotal
sin out
18
cos inmax  E  
EF  eff
E 
Elements of the Three-Step Photoemission Model
Fermi-Dirac distribution at 300degK
f FD ( E ) 
1
1  e ( E  E F ) / k BT
Electrons lose energy
by scattering, assume
e-e scattering
dominates,
Fe-e is the probability
the electron makes it to the
surface without scattering
h
eff
heff
1
Bound electrons
.5
0
ptotal  2m( E   )

p
pnormal  2m( E  ) cos
Emitted electrons
cos  max 
E
0
5
EF
EF+eff-h
Energy (eV)
E+h
EF  eff
p

ptotal
E  
10E +
F
eff EF+h

QE ( )  (1  R( ))
2
pnormal
 EF  eff
2m
Escape criterion:
eff     schottky
h
.5
Step 3:
Escape over barrier
Step 2:
Transport to surface
EF 


eff
dE N ( E   )(1  f FD ( E   )) N ( E ) f FD ( E )
 
  dE
EF 
d (cos  )F


ee
cos

2
1
max
( E ,  ,  )  d
(E)
The Physics of High Brightness Sources
D. H. Dowell
Step 1: Absorption of photon
0
1
2
1
0
N ( E   )(1  f FD ( E   )) N ( E ) f FD ( E )  d (cos  )  d
19
QE for a good metal, like Cu
Step 3: Escape over the barrier and integrate
up to max escape angle
1
R is the reflectivity
eff is the effective work function
eff  W  Schottky
EF
QE ( )  1  R( )  Fe e  
Step 1: Optical Reflectivity
~40% for metals
~10% for semi-conductors
Optical Absorption Depth
~120 angstroms
Fraction ~ 0.6 to 0.9
Step 2: Transport to Surface
e-e scattering (esp. for metals)
~30 angstroms for Cu
e-phonon scattering (semiconductors)
Fraction ~ 0.2
QE ~ 0.5*0.2*0.04*0.01*1 = 4x10-5


EF 
eff
EF
dE
 
  dE
EF 

d (cos  )
EF eff
2
 d
E 
1
0
2
1
0
 d (cos  )  d 
•Azimuthally
isotropic
emission
Fraction =1
•Fraction of electrons
within max internal
angle for escape,
Fraction ~0.01
•Sum over the fraction of
occupied states which are
excited with enough energy
to escape,
Fraction ~0.04
D. H. Dowell et al., PRST‐AB 9, 063502 (2006)
The Physics of High Brightness Sources
D. H. Dowell
E is the electron energy
EF is the Fermi Energy
20
Performing the integrals gives the QE:
e e
2
2

Eexcess
(1  R(  ))
 


8eff ( EF  eff )

1  opt
Vacuum
Level
e e
ℏ𝜔
𝐸𝑒𝑥𝑒𝑠𝑠
𝑬𝒆𝒙𝒄𝒆𝒔𝒔
𝟐
Intrinsic emittance can be lowered from 0.5 to
<0.35 microns/mm-rms, if excess energy is reduced
from 0.4 to <0.2 eV. Unfortunately the QE goes to
zero faster than the emittance!
The Physics of High Brightness Sources
D. H. Dowell
EF  eff
(1  R(  )) ( EF   ) 
QE 
1 

2  
EF  
1  opt
21
22
P. Musumeci, 2013 SSSEPB
The Physics of High Brightness Sources
D. H. Dowell
Vacuum
-e
q’
e
xd
x
Potentials Near the Surface
eschottky  e
eE0
 0.0379 E MV / meV
40
eschottky  0.379eV @100MV / m
The Physics of High Brightness Sources
D. H. Dowell
Cathode
23
RF processing and machining processes can produce rough surfaces which increase
field emission and emittance very close to the cathode
Field enhancement factor
for various geometries
E  E0
The Physics of High Brightness Sources
D. H. Dowell
Cathode damage during RF processing
and beam operation
24
“High Voltage Vacuum Insulation, Basic
Concepts and Technological Practice,”
Ed. Rod Latham, Academic Press 1995
Emittance Due to a Tilted Surface
z
z
p x ,out
~
p x ,out
T
x
pout  2mE    EF  eff 
out
Vacuum
in
Continuity of transverse momentum at surface:
~
p x ,in  ~
p x ,out
Metal
pin  2mE   
p x ,in
sin  out
p
 in 
sin  in
pout
E  
E    E F  eff
p x ,in
Max angle of incidence:
p x ,out 
p out
2
~
sin  out cos T  cos  out sin T 
pout
cos  out sin T
2
sin  max 
E    EF  eff
The Physics of High Brightness Sources
D. H. Dowell
x
E  
Surface momentum
proportional to intrinsic
momentum, pout cosout
25
26
The Physics of High Brightness Sources
D. H. Dowell
Surface Emittance Due to an Applied Electric Field + …
The transverse momentum due to an applied electric field is given by the integral of the
acceleration along the electron’s trajectory, (x(t),z(t)).
[D. J. Bradley et al., J. Phys. D: Appl. Phys., Vol. 10, 1977, pp. 111-125.],
t
p x, field  eE0 an k n  sin k n x(t )e kn z (t ) dt
At large distances from the surface, z > a few n , the transverse field vanishes and transverse
momentum gained in the field becomes constant. The variance of this transverse momentum,
px,field , gives the emittance due to the applied field [See D. Xiang et al., Proceedings of PAC07,
pp. 1049-1051],
 field
 2 an2eE0

x
2n mc2
n 
2
kn
This is the emittance only due to the transverse component of the applied field. Include the
cross terms between the intrinsic, tilt and applied field momenta and compute the variance of
the transverse momentum to get the total intrinsic, tilted-surface and high field emittance,
 intrinsic tilt  field   x
D. H. Dowell -- P3 Workshop
    a k      


2
eff
3mc 2
n
n
2


eff
3mc 2
eE0 

2k n mc 2 
The Physics of High Brightness Sources
D. H. Dowell
0
Comparison of these three emittances for
emission from a diamond turned surface
AFM of LCLS cathode sample
LCLS gun & cathode parameters:
an  17nm
n  10microns
k n  0.628 / micron
an k n  0.011
~35 nm
Ea  57.5MV / m
eff ( E0 )  W  Schottky
Schottky  e
 intrinsic tilt  field

x
    1  a k 
2
3mc
eff
2


 intrinsic
 0.48microns / mm  rms
x
 tilt
 3.7 10 3 microns / mm  rms
x
 field
 0.13microns / mm  rms
x
n n
2
eE0
 0.29eV
40
 eE0
2


a
k

n n
2
 4k n mc
 intrinsic tilt  field
 0.49 microns / mm  rms
x
∴ The intrinsic emittance is by
far the largest contributor!
See D. H. Dowell, 2012 P3 Workshop
The Physics of High Brightness Sources
D. H. Dowell
~10 microns
  4.86eV
W  4.8eV
28
Charge Density Modulations produced by Surface Roughness can increase
the emittance due to space-charge forces
Electrons are focused and can go through a
crossover a few mm from the cathode for
emission from a sinusoidal surface
beam cross section ~6.5 mm from cathode
120
100
80
60
7000
40
6000
20
0
0
5000
2
4
6
8
10
8
10
of x-y
x Mapping
(microns)
6538
4000
6537.5
3000
6537
2000
1000
6536.5
6536
6535.5
0
-15
-10
-5
transverse position, x (microns)
0
5
6535
6534.5
2
4
6
xMapping
(microns)
of x-y
62.7
62.65
62.6
62.55
y (microns)
Due to this crossover, space charge forces and
other effects such as Boersch-scattering should be
investigated. Since this ‘surface lens’ is non-linear
it can also produce geometric aberrations and
increase the emittance. Plus the time dependence
of the RF field which changes the focus with time.
Rich area of study!
0
62.5
29
62.45
62.4
62.35
62.3
62.25
See D. H. Dowell, 2012 P3 Workshop at Cornell for space-charge effects
The Physics of High Brightness Sources
D. H. Dowell
8000
y (microns)
longitudinal position, y (microns)
modulation amplitude = 0.02 microns; spatial wavelength = 10 microns; emittance = 0.16412 microns
62.2
0
2
4
6
x (microns)
8
10
Space Charge Limit (SCL) is different for DC diode, short pulse and long bunch
photo-emission. Space-charge-limited current vs. space-charge-limited field
Space charge limited current across a DC diode,
Child-Langmuir law:
4
2e V 3/ 2
J CL   0
2
m d
Space Charge Limited Field of a Short Electron Bunch from
a Laser-driven Photocathode. Parallel plate (capacitor)
model:
 SCL   0 Eapplied
Space charge limited field of an ellipsoid near the cathode (water bag model):
Potential energy and longitudinal electric field of a
uniformly charged ellipsoid (aka. water bag) next to
the cathode The ellipsoid edge semi-axes of a and
zm, corresponding to the edge radius and half the
bunch length, respectively. [See Reiser, page 406],
𝜕𝜙𝑓𝑠
𝜌0 𝑎
𝐸𝑓𝑠,𝑧 𝜁 = −
=
𝜁 for −𝑧𝑚 ≤ 𝜁 ≤ 𝑧𝑚
𝜕𝜁
6𝜖0 𝑧𝑚
𝜌0 =
𝐸𝑓𝑠,𝑧
3 𝑄𝑏𝑢𝑛𝑐ℎ
4 𝜋𝑎2 𝑧𝑚
𝜌0
4𝜋𝑎𝑧𝑚
ℎ𝑒𝑎𝑑 =
𝑎 => 𝑄𝑠𝑐𝑙−𝑒𝑙𝑙𝑖𝑝𝑠𝑜𝑖𝑑 =
𝐸𝑎𝑝𝑝𝑙𝑖𝑒𝑑
6𝜖0
3
The Physics of High Brightness Sources
D. H. Dowell
9
30
z
Transverse Electron Beam Shape: The beam core is clipped at the SCL
eElaser

QE
•SCL truncates the beam
core to a flat, uniform
transverse distribution.
•SCL also flattens hot
spots.
below the SCL:
𝑄𝑏𝑢𝑛𝑐ℎ
𝑒𝐸𝑙𝑎𝑠𝑒𝑟
=
𝑄𝐸
ℏ𝜔
 rm2 0 Erf sin rf  QE
eElaser


e
rm2
2 r2
Space Charge Limit
of Gaussian peak
QE Limited Emission
Space Charge Limited Emission
radial distribution follows laser & QE
radial distribution saturates at the applied field
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
3
2
1
0
1
2
3
0
The Physics of High Brightness Sources
D. H. Dowell
QE scan:
Emitted charge vs
Laser pulse energy
31
3
2
1
0
1
2
3
J. Rosenzweig et al., NIM A 341(1994) 379-385
The Schottky scan: Emitted Charge vs. Laser Phase
The Schottky scan can be divided into two regions. At low rf phases the emission is space charge limited
while at higher field phases the emission becomes QE-limited. In the Schottky scan, these two phenomena
are equal at the phase m .
𝑄𝑠𝑐−𝑙𝑖𝑚𝑖𝑡 = 𝜋𝜖0 𝑅2 𝐸𝑝𝑒𝑎𝑘 𝑠𝑖𝑛𝜙
𝐸𝑙𝑎𝑠𝑒𝑟−𝑝𝑢𝑙𝑠𝑒
ℏ𝜔

EF  W  e eErf sin rf /( 40 )
1  R EF   
QE 
1


2 
EF  
1  opt

e e
Measured Schottky
scans agree with
expected shapes due
to SC- and QE-limits




2
The Physics of High Brightness Sources
D. H. Dowell
𝑄𝑄𝐸−𝑙𝑖𝑚𝑖𝑡 = 𝑒𝑁𝛾 QE = eQE
32
data courtesy of M. Krasilnikov
A virtual cathode forms and the electrons oscillate in a
time-dependent potential well just in front of the cathode.
The Physics of High Brightness Sources
D. H. Dowell
What happens when you go way above the SCL?
33
See: An Introduction to the Physics of Intense Charged Particle
Beams by B. Miller, Plenum Press, New York, March 1985.
The relativistic, paraxial ray and envelope equations
Begin with the radial equation of motion (Lorentz eqn.):
𝑚
𝑑
𝛾𝑟 − 𝑚𝛾𝑟𝜃 = 𝑒 𝐸𝑟 − 𝛽𝑐𝐵𝜃
𝑑𝑡
And transform from time to z as the independent variable to get the paraxial ray equation.
Reiser (p. 210) gives a modified version of the relativistic paraxial ray equation:
2
The accelerating
damping term,
discussed in slides 6-10
1
𝐾
−
=0
𝑟 3 𝑟𝑚 2
external focusing,
e.g. solenoid
𝑟𝑚 is the radius of the beam envelope
K is Lawson’s generalized perveance.
Gives the space-charge force.
angular momentum
term and x-y mixing
entrance/exit radial kick given by the
𝜕
accelerating field, E z. 𝛾 ′′ ∝ 𝐸𝑧 ∝ 𝐸𝑟
𝜕𝑧
small near the cathode, unless curved
normalized emittance term,
𝜖𝑛 = 4𝛽𝛾𝜖𝑟𝑚𝑠
And he writes the relativistic envelope equation as:
𝑟𝑚′′
1
𝑝𝜃
+ 2 𝛾 ′ 𝑟𝑚′ + 𝛾 ′′ 𝑟 + 𝑘0 2 𝑟𝑚 −
𝛽 𝛾
𝛽𝛾𝑚𝑐
The relativistic, generalized perveance is defined as 𝐾 ≡
2
1
𝜖𝑛 2 1
𝐾
−
−
=0
𝑟𝑚 3 𝛽 2 𝛾 2 𝑟𝑚 3 𝑟𝑚
2 𝐼
𝛽 3 𝛾 3 𝐼0
where 𝐼0 = 17𝑘𝐴,
characteristic current for electrons
In the non-relativistic limit the perveance gives the same 𝑉 3/2 voltage-dependence as the
Childs-Langmuir law:
𝐼
1
𝐾𝑁𝑅 = 3/2
𝑉
4𝜋𝜖0 2𝑒/𝑚 1/2
See Reiser, p. 197
The Physics of High Brightness Sources
D. H. Dowell
𝛾′
𝛾′′
𝑝𝜃
𝑟 + 2 𝑟 ′ + 2 𝑟 + 𝑘0 2 𝑟 −
𝛽 𝛾
2𝛽 𝛾
𝛽𝛾𝑚𝑐
′′
34
Brillouin flow, confined flow and other flows
Brillouin flow (1948) maintains constant beam size of 𝑟𝑚 by balancing the outward
space-charge force against the focusing of a axial magnetic field. The conditions for BF
are zero emittance, zero angular momentum and no acceleration. The ray equation for
no radial acceleration then
For magnetic focusing: 𝑘0
The magnetic field needed for
Brillouin flow is
2
𝐾
𝑟𝑚
=> 𝑟𝑚 =
𝑒𝐵
=
2𝛽𝛾𝑚𝑐
𝑒𝑐𝐵𝐵𝐹
𝐾
𝑘0
2
and with
𝐾=
2 𝐼
𝛽 3 𝛾 3 𝐼0
2𝑚𝑐 2 2 𝐼
=
𝑟𝑚
𝛽𝛾 𝐼0
∴ to confine a 100 ampere beam inside a radius of 1mm with 𝛽𝛾 = 2 (~800 KeV)
requires a magnetic field of
The Physics of High Brightness Sources
D. H. Dowell
𝑟 ′′ = 0 = 𝑘0 2 −
c𝐵𝐵𝐹 = 78 𝑀𝑉/𝑚 or 𝐵𝐵𝐹 = 0.26𝑇 = 2.6 𝑘𝐺
35
The Concept of Slices and Emittance Compensation
slice width is determined by the beam energy
due to Lorentz contraction of the longitudinal fields
v=0
2
𝛾
~ r
v~c
r
1
𝛾
field lines
inside a
bunch
Electric field lines of a point charge
2𝑧𝑚
Emittance Compensation
•The beam at the cathode begins with all slices of nearly equal peak current propagating with equilibrium radii
in Brillouin flow.
•The beam envelope equation is linearized about the Brillouin equilibrium and solved for small perturbations
about this equilibrium point.
•The solution obtained shows all slice radii and emittances oscillate with the same frequency (determined by
the invariant envelope), independent of amplitude.
•Assuming the slices are all born aligned, they will re-align at multiple locations as the beam propagates, with
the projected emittance being a local minimum at each alignment. The beam size will oscillate with the same
frequency, but shifted in phase by /2.
The Physics of High Brightness Sources
D. H. Dowell
𝜁, 𝑧, 𝛾
36
Projected Emittance Compensation*
The radial envelope equation for each slice position, zm <  < zm
0
 


 r ( )   r ( )  2
 
External focusing by
magnetic and RF fields

 n2 ( )
K ( )
 2 2 3
0
  kr r ( ) 
 r ( )    r ( )

Space charge defocusing,
K is the generalized perveance
acceleration changes
magnification of the
divergence
Generalized perveance
for each slice at 
(Lawson, p. 117)
emittance acts like a
defocusing pressure
I ( )
K ( )  3 3
  I0
2
*Serafini & Rosenzweig, Phys. Rev. E55, 1997, p.7565
I is the peak current of slice 
I0 is 17000 amps
The Physics of High Brightness Sources
D. H. Dowell
 slice
37
Projected Emittance Compensation: How to align the slices?
Beam Envelope Equation:
Assume no acceleration, zero slice emittance
 r  kr r 
K
r
I+I
I-I
0
 r   e  
Substituting into the envelope eqn. and
expanding in a Taylor series we get
   kr e 

K 
k 
   0
 e  r  e2 
K
constant terms,
set sum to zero and solve for e :
e 
Envelope eqn. for small amplitude radial
perturbations:
K
2 I 1

kr
 3 3 I 0 kr
   2
K

2
e
  0
𝑘𝑒 =
2𝐾
𝜎𝑒
sin ke z 

  1   cos ke z
   0 
k

e

  

  1   k sin k z cos k z    0 
e
e 
 e
This is known as balanced or Brillouin flow, when the
outward space charge force is countered by external
focusing, usually a magnetic solenoid. This establishes the
invariant envelope of Serafini & Rosenzweig.
The Physics of High Brightness Sources
D. H. Dowell
Consider solutions for small perturbations
from equilibrium radius:
38
Derivation of projected emittance for a slice current spread
The emittance due to a sc-lensing strength
dependence upon beam current is:
Envelope eqn. for small amplitude radial perturbations
K
 e2
  0
𝑘𝑒 =
2𝐾
𝜎𝑒
 n , sc comp   e2 I
This is the wave equation with oscillating solutions:

sin ke z 
  
ke   0 
  
 0 
 ke sin ke z cos ke z 
Using the expressions for 1/fe and the
equilibrium wave number in this eqn. gives the
projected emittance for a beam with a I rms
spread in slice current:
  1   cos ke z
   
  1  
with the equilibrium wave number defined as:
K
I
 e2
1
0   0 
 1  



  
 1   ke sin ke z 1   0 
;
ke2 
 e I
sin ke z  ke z cos ke z
 I 0 I
5
To simplify the emittance calculation let’s make the
crude approx. that the solenoid focusing-sc defocusing
channel can be replaced by a thin lens such that
1
 ke sin ke z
fe
 n, 
4
I
 3 3 e2 I 0
emittance (microns)
k  2kr  2
2
e
d  1
 
dI  f e 
The Physics of High Brightness Sources
D. H. Dowell
   2
Locations where the
slices align
4
3
2
39
1
0
0
0.5
1
z (m)
1.5
2
The Ferrario Working Point: matching the low energy beam to the booster linac
In addition to compensating for the emittance from the gun, it is necessary to carefully match the beam into
a high-gradient booster accelerator to damp the emittance oscillations. The required matching condition is
referred to as the Ferrario working point* and was initially formulated for the LCLS injector and based upon
the theory of L. Serafini and J. Rosenzweig, see Phys. Rev. E55,1997, p.7565.
The working point matching condition requires the emittance to be a local maximum and the envelope to
be a waist at the entrance to the booster. The waist size is related to the strength of the RF fields and the
peak current. RF focusing aligns the slices and acceleration damps the emittance oscillations.
Assume the RF-lens at the entrance to the booster is similar to that at the gun exit with an injection phase
at crest for maximum acceleration, e=/2, so the angular kick is,
eE0
Taking the derivative gives the rf term needed for the envelope equation,
eE0
 2
    2 2     2
2 mc
2
2mc 2
since

eE 0
mc 2
with E0 the accelerating field of the booster. For a matched beam we want the focusing strength of the accelerating
field to balance the space charge defocusing force, i.e. no radial acceleration:
2
    match
2.5
Transverse Emittance (microns)
Beam Radius at Exit, rms (mm)
125 cm
S-Band TW Section
Elinac = 19 MV/m
2
Solenoid = 2080 G.
1.5
Solving gives the matched beam size:
E beam =62 MeV
E0=106 MV/m, 0 = 50 deg
E3=16 MV/m,  = 1.7 deg

I

0
2
3

2 I A  match
 matched 
1
Transverse Emittance
rms Radius
matched is the waist size at injection to the accelerator. The matched
beam emittance decreases along the accelerator due the initial focus at
the entrance and damping during acceleration.
0.5
0
0
100
200
300
z (cm)
400
1
I
  2 I A
500
*M. Ferrario et al., “HOMDYN study for the LCLS RF photoinjector”, SLAC-PUB-8400, LCLS-TN-00-04, LNF-00/004(P).
The Physics of High Brightness Sources
D. H. Dowell
 
40
Chromatic Aberration in the Emittance Compensation Solenoid
Emittance due to the momentum dependence of the solenoid’s focal length:
f is the focal length of the solenoid
p is the beam momentum
is the rms momentum spread
p
This is a general expression for the emittance produced by a thin lens when the focal length is
varied. E.g., it was used earlier to describe emittance compensation.
In the rotating frame of the beam the solenoid lens focal strength is given by
1
 K sin KL,
f sol
Chromatic emittance
(microns/mm-rms
/20 keV @6 MeV)
100
1935  1 
0.020 

6 
K
typical
2
solenoid
field, 2 kG
10
B( 0 ) eB (0)

2 Bρ0
2p
B(0) is the solenoid field
L is the solenoid effective length
Br0 is the beam magnetic rigidity
B r0 
1
p
 33.356 p  GeV / c  kG  m
e
 n,chromatic   x2 K sin KL  KL cos KL
0.1
p
p
The solenoid is a first order achromat when
0.01
0
5
10
15
 2  33.356
 0.006
SolenoidKxfield,
B(0)
(kG)
20
tan KL  KL
The Physics of High Brightness Sources
D. H. Dowell
 n,chromatic
d 1
 
 p
dp  f 
2
x
 x is the rms beam size at the solenoid
41
Chromatic Aberration: Comparison with simulation & expt.
 n ,chromatic   x2 p
d  1 


dp  f sol 
1
B(0)
 K sin KL, K 
f sol
2Br 0
 n,chromatic   K sin KL  KL cos KL
2
x
p
p
Chromatic Emiitance (microns)
1
Projected Energy Spread
measured at 250 pC, 6 MeV
Projected
Assumes 1 mm rms
beam size at solenoid
0.1
KL = 1
K = 5/m
Slice
0.01
0.001
0
5
10
15
20
25
30
The Physics of High Brightness Sources
D. H. Dowell
Comparison of Eqn. with Simulation
Energy Spread (KeV-rms)
Solenoid chromatic aberration is a significant contributor to the projected emittance. But with a slice
energy spread of 1 KeV, the slice chromatic emittance is only ~0.02 microns
42
Geometric Aberration of the Emittance Compensation Solenoid
To numerically isolate the geometrical aberration from other effects, a simulation was
performed with only the solenoid followed by a simple drift. Maxwell’s equations were used to
extrapolate the measured axial magnetic field, Bz(z), and obtain the radial fields
[see GPT: General Particle Tracer, Version 2.82, Pulsar Physics, http://www.pulsar.nl/gpt/].
post focus
“pincushion” shape
preceding focus
solenoid entrance
time=2.401e-009
time=2.3995e-009
time=1e-012
0
-1
0.004
2e-4
0.002
y(mm)
y(mm)
y(mm)
Transverse beam
distributions:
1
4e-4
0e-4
0.000
-2e-4
-0.002
-4e-4
-0.004
-2
-2
GPT
-1
0
x(mm)
1
-5e-4
2
GPT
0e-4
x(mm)
-0.005
5e-4
GPT
0.000
x(mm)
0.005
The Physics of High Brightness Sources
D. H. Dowell
2
43
The emittance due to anomalous quadrupole field at the solenoid entrance
When the quadrupole is rotated about the beam axis by angle, , with respect to a
normal quadrupole orientation, then total rotation angle becomes the sum of the
quadrupole rotation plus the beam rotation in the solenoid and the emittance
becomes
sin 2  KL   
fq
quad
solenoid
Comparison with simulation for a 50 meter focal length quadrupole followed by a strong
solenoid (focal length of ~15 cm). The emittance becomes zero when
KL    n
Adding a normal/skew quadrupole pair allows recovery of the emittance caused by this x-y
correlation.
The Physics of High Brightness Sources
D. H. Dowell
 x ,qs   x , sol y , sol
44
Summary of Emittance Contributions from the Solenoid
rms beam size from gun to linac
Emittance due to chromatic aberration:
2.0
p
1.6
Emittance due to anomalous quad field:
 n ,quad  sol ( x , sol )   x2, sol
sin 2 KL
fq
solenoid
1.8
200 pC, Rc=0.6 mm
1.4
100 pC, Rc=0.3mm
1.2
100 pC, Rc=0.6mm
1.0
10 pC, Rc=0.1mm
0.8
10 pC, Rc=0.3mm
0.6
1 pC, Rc=0.01mm
0.4
Spherical aberration emittance:
1 pC, Rc=0.10mm
0.2
0.0
 n,spherical ( x )  0.0046 x4
0
20
40
60
80
100
Distance from cathode (cm)
LCLS at 250 pC, slice emittance
Emittance (microns)
1
 n , spherical   n ,chromatic   n ,quad  sol
 n , spherical   n ,chromatic
0.1
 n, spherical
The Physics of High Brightness Sources
D. H. Dowell
p
x-rms beam size
 n,chromatic( x )   K sin KL  KL cos KL
2
x
0.01
1 10
Chromatic emittance assumes
1 KeV-rms energy spread
3
0
0.5
1
1.5
2
rms beam size (microns)
2.5
3
45
Space Charge Shaping*, sometimes known as “Beam Blowout!”
Step 1: Compute the axial potential for a
Here we derive the radial force on an electron confined parabolic charge distribution,
to a thin disk of charge. The surface charge density is
assumed to have a radial quadratic surface charge
 (r )   0 1   2 r 2
density. The quadratic factor can be adjusted to cancel
the 3rd order space charge force of the disk’s
distribution.

*J.D. Jackson, Classical Electrodynamics, 3rd ed., p.101-104
V ( z) 
r
dr
P
z
R
4 0V ( z )  
 ds
z2  r2
R
 2 
 (r )rdr
0
R
 0  R rdr
r 3 dr 
V ( z) 
2 


2
2
2 0  0 z 2  r 2
z

r
0

 0  2 2 1/2
 1 2 2 3/2 2 2 2 1/2 2 3  
z

R

z


 z  z  R   z    V0 ( z )  V2 ( z )


2  z  R 
2 0 
3
3 

uniform distribution
z2  r2
parabolic distribution
*Serafini, AIP Conf Proc 279, p645 1992; Luiten et al., PRL, 93, 2004,p.94802
The Physics of High Brightness Sources
D. H. Dowell
The mathematical technique* is
Step 1: Compute the electrical potential energy on the
axis of symmetry.
Step 2: Expand this potential into a power series and
multiply each term by the same order of Legendre
polynomial to obtain the potential at any point on
space.
Step 3: Take the divergence of the potential to get the
radial electric field.

46
Step 2: Expand V0 in powers of r and multiply each term by the same order of Legendre
polynomial to get the potential for a uniformly charged disk at any point P,

V0 ( , r )  0
2 0
r


1 r2
1 r4
RP
(cos

)

rP
(cos

)

P
(cos

)

P
(cos

)

0
1
2
4


2 R
8 R3


z
z
2
2
2
cos



r

r

z
where
r
r 2  z2

z
r
In the plane of the disk: z = 0 => cos q = 0 and r = r,
so the Legendre polynomials are evaluated at zero, i.e. Pn (0).
Then the potential in the plane of a uniformly charged disk is

V0 ( r )  0
2 0
R

1 r2 3 r4
 2 3
R 
4 R 8 R




Following the same procedure for the parabolic term gives the potential, V2

V2 ( r )  0 2
2 0
2
4
1 3 1
3 r
2

 R  Rr   
3
4
8
R
 




The total potential in the plane of the disk is given by the sum of these
potentials
V ( r )  V0 ( r )  V2 ( r )
The Physics of High Brightness Sources
D. H. Dowell
P
47
Step 3: Take derivative of potential to get radial field.
V ( r )  V0 ( r )  V2 ( r )
Total potential inside the disk:
Summing uniform and parabolic potentials and collecting powers of r gives
4

 0    2 R2  1 
1  2 3 1
r
6
V (r ) 
 O  r 
 R 1 
    2  2  R r  2  2  3 2 
2 0  
3  4
R 
8 R
R


V
Er ( r ) 
r
0 1 
1 
3
1  r3
5 
Er ( r ) 
   2  2  Rr   3 2  2   O  r 
2 0  2 
R 
16 
R  R

Er
r
Er is the radial space charge
field at point P in the plane of
the disk for r<R
P
R
Linear force
No emittance

 ( r )   0 1 
Disk with surface charge density:
10
 ( r )   0 1f( rx) 2r( 12 rx)
0.5
00
00
2  
If we make:
1.5

0.5
R1
rx
Non-linear force
Emittance growth
r2 
1
3R 2
Then there’s no 3rd order force
and no space charge emittance
during expansion of beam from
cathode!

3R 2 
1.5
The Physics of High Brightness Sources
D. H. Dowell
The radial space charge field is then:
48
How large are these radial fields?
1.5

 ( r )   0 1 

10
r2 
0 1 
1 
3
1  r3 
Er ( 2 , r ) 
   2  2  Rr   3 2  2  
2 0  2 
R 
16 
R  R

3R 
2
0.5
3R
00
0.5
R1
1.5
rx
For 250 pC and R=0.6 mm (LCLS-like parameters): Er  36 MV / m
This is nearly as large as the RF field( !!!) which is Epeakcosrf = 57.5 MV/m
However, even with these strong fields
they are now linear and the space charge
emittance is greatly reduced by parabolic
shaping of the transverse shape.
The Physics of High Brightness Sources
D. H. Dowell
00
1
1 0 r


Er   2   2 , r   
3R
3 0 R


For  2   1 2
49
Luiten et al., PRL, 93, 2004,p.94802
Summary
•Electron emission physics
•Quantum efficiency and intrinsic emittance computed with the 3-step model
•QE agrees but expt. emittance is 2x-larger than theory
•RF emittance
•A simple optical model, RF defocusing at exit of gun canceled by solenoid
•Time-dependent emittance: 1st and 2nd order
•Projected emittance compensation
•Balanced flow and plasma oscillations
•Matching to first linac section, Ferrario matching condition
•Solenoid aberrations
•Chromatic, projected vs. slice energy spread
•Geometric, scales with beam size to 4th power
•Anomalous quadrupole fields, recoverable emittance with normal/skew quad correctors
•Beam blowout dynamics
•Transverse shaping to eliminate space charge emittance due to Serafini and Luiten
•Begin with very short bunch, a single slice which can be modeled as a disk of charge
•Parabolic transverse shaping eliminates 3rd order space charge force
The Physics of High Brightness Sources
D. H. Dowell
•Space charge limited (SCL) emission
•Child-Langmuir Law vs. short pulse emission
•short pulse emission has higher space charge limit than C-L law
•Analysis of space charge limited emission data
•SCL flattens the transverse profile
50
The Physics of High Brightness Sources
D. H. Dowell
Thanks for your attention!
51