INTRODUCTION TO HIGH
BRIGHTNESS ELECTRON BEAMS
Wai-Keung Lau [ NSRRC Winter School on Free Electron Lasers 2019
Jan. 14th – 18th, 2019
Single Particle Dynamics in EM Fields
Relativistic dynamics
Applied EM fields are usually expressed as expanded series called paraxial
approximation
Complicated EM field distributions can only be solved by numerical methods.
Examples of simulation codes are: POISSON/SUPERFISH
High Frequency Structure Simulator (HFSS) -- http://www.ansoft.com/products/hf/hfss/
CST Microwave Studio -- http://www.cst.com/Content/Products/MWS/Overview.aspx
BvEqvmdt
d
dt
pd
0
The Lorentz factor is in general not a constant of time
Lorentz force law
What is a Charged Particle Beam?
an “ordered flow” of charged particles
all particles are moving
along the same trajectory
for a perfect beam
a random distribution
of charges
something in between
(real world)
x’
x
(xi,xi’) x’
x
a single particle in trace-space
a system of particles in trace-space
Trace Space Area Definition of Emittance
Trace space area definition of emittance
The trace-space area Ax is related to the phase-space area in x-px plane by
We can therefore useful to define normalized emittance that is independent of particle acceleration such that
However, beams with quite different distributions in trace space may have the same area!!
xdxdAxx
many authors identify the emittance as trace-space area divided by (unit: [m-rad])
[ m-rad]
xx
z
x dxdpmc
dxdpp
A
11this integral can be a
constant as long as
Liouville’s theorem
applies.
n
Let f (x,x’) be the distribution function such that . N is the total number of particles. Beam parameters can be defined accordingly as:
Nxdxdxxf ,
xdxdxxfxdxdxxfxxxx
xdxdxxfxdxdxxfxx
xdxdxxfxdxdxxfxx
xdxdxxfxdxdxxfxx
xdxdxxfxdxdxxxfx
xx
x
x
,/,
,/,
,/,
,/,
,/,
2
2
averaged beam size
averaged beam divergence
rms beam size
rms beam divergence
beam correlation
root-mean square of a set of n values is defined as:
22
2
2
1
1nrms xxx
nx
Define rms emittance as
If x and x’ are not correlated (at beam waist where the beam is neither converging nor diverging)
RMS emittance gives quantitative information on beam quality
RMS emittance gives more weight to the particles in the outer region of the trace-space area. Therefore, remove some of the outer particles will significantly improve RMS emittance without too much degradation of beam intensity.
det~ 222222 xxxxx xxxx
xxx xx ~~~
2
2
xxx
xxx
-matrix of beam distribution
Effective Emittance In a system that all forces (space charge and external forces) acting on the particles are
linear (i.e. proportional to particle displacement x from the beam axis), it is still useful to
assume an elliptical shape for the area occupied by the beam in trace space such that
We are able to define an emittance as
The relation between and the corresponding RMS quantities
are given by
mmx xxA
xmmx
Axx
xthxx ~,~,~ xmm xx ,,
xx
thm
m
xx
xx
~4
~2
~2
x
x’
Beam Brightness
Definition of beam brightness:
It is useful know that total beam current that can be confined within a 4-D trace space volume V4. We can define average brightness as
If any particle distribution whose boundary in 4-D trace space is defined by a hyperellipsoid
one finds and average brightness is
RMS brightness is then defined as:
dsd
dI
d
JB
4V
IB dsdV4
1
2
2
22
2
2
yx
yb
b
yxa
a
x
yxdsd 2/2
yx
IB
2
2
(note: 2/2 is left out)
yx
IB
~~~
Thermal Emittance of a Beam from Cathode For a beam from a thermionic cathode at temperature T, rms thermal velocity spread
is related to rms beam divergence such that
Assume Maxwellian velocity distribution from a round cathode with radius rs,
where T is the temperature of the cathode, then
On the other hand, for a beam emitted from the cathode
Thermal emittance of such beam from the cathode is given by
0, /~~ vvx thx
Tk
vvvmfvvvf
B
zyx
zyx2
exp,,
222
0
2/1
~~
m
Tkvv B
yx
2/~~sryx
0
2/1
, 2~
v
m
Tk
r
B
syx
2/1
00
0
3
0
2
2
2
1
2
~~
drrn
drrnrx
a
a
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y=exp(-X.2)
2/1
22~
mc
Tkr B
sn
the total Coulomb force acting on q by a thin volume dV
of charge Q is independent of r !!
Planar Diode with Space Charge – Child-Langmuir Law
0
2
22
dx
d
constxJ x
02
2 xexm
2/12/1
0
2
2 1
/2
me
J
dx
d
C
me
J
dx
d
2/1
2/1
0
2
/2
4
xm
eJ4/12/1
0
4/3 22
3
4
3/4
0
d
xVx 2
2/3
0
2/1
0
2
9
4
d
V
m
eJ
2
2
2/3
06 /1033.2 mAd
VJ
x
e-
=0
=V0
cathode
C=0 under the boundary conditions: =0 and
d/dx=0 at x=0; the condition d/dx=~x1/3=0 at
x=0 implies that the special case electric field
at cathode surface is null (a steady state
solution).
Emission of electrons
from cathodes:
1. Thermionic
emission
2. Secondary
emission
3. Photo-emission
with
Child-Langmuir law
Electrostatic repulsion forces between
electrons tend to diverge the beam
The current density required in the
electron beam is normally far greater
than the emission density of the cathode
Optimum angle for parallel beam is
referred to as Pierce electrodes
Conical diode is needed for convergent
flow
Defocusing effect of anode aperture has
to be considered a simple diode gun 140 kV Electron Gun System for TLS
r
z focusing continuous beam
For a given focusing channel, the
size of beam waist is in general
determined by beam emittance,
space charge forces etc..
beam expansion due
to space charge
propagation of a continuous beam and a bunched beam in drift space
change of particle
distribution in phase
space due to space
charge (transverse
beam size, bunch
length, divergence,
energy spread etc..)
A Cylindrical Beam in a Strong Magnetic Field
Particles enter the drift tube with kinetic energy qb.
A new potential will be setup in the drift tube which will reduce the K.E. of
the particles according to energy conservation law.
As the potential is strong enough, K.E. of particles completely convert into
potential energy. There exists a beam current limit!!
b
0
(z)
rqrqTrT eb
E-BEAM
- + b
DRIFT TUBE
STRONG MAGNETIC FIELD
a b CATHODE
a
b
c
Ie ln21
40
0
electric fields of a uniformly moving charged particle
v = 0 v < c v = c
1/
Er 2q(z-ct)/r 0ˆ BzEeF
v c
B 2q(z-ct)/r
16
Paraxial Approximation of Axisymmetric
Electric and Magnetic Fields General assumptions:
Consider only static electric and magnetic fields at this point.
That is, no time-varying fields and displacement currents are excluded.
Electric and magnetic fields for axisymmetric lenses in a beam focusing system do not usually have azimuthal components (i.e. ).
Other field components have no azimuthal dependence
In paraxial approx., fields are calculated at small radii from the system axis with the assumption that the field vectors make small angles with the axis
0
0
B
E
0
rz EE
zr EE zr BB
0
0
E
B
0
rz BB
0;0 BE
17
The axial and radial fields in paraxial
approximation are:
4
44
2
22
3
33
644,
162,
z
Er
z
ErEzrE
z
Er
z
ErzrE
z
r
4
44
2
22
3
33
644,
162,
z
Br
z
BrBzrB
z
Br
z
BrzrB
z
r
(1)
(2)
(3)
(4)
18
Derive linear equations of particle motion in which only terms up to first order in r and r’=dr/dz are considered
Assumptions: ◦ Cylindrical beam; self-fields are neglected ◦ Particle trajectories remain close to the axis. That is, r <<
b. And b is the solenoid or radii of electrodes that produce the electric and magnetic fields. This also implies that the slopes of the particle trajectories remain small (i.e., or ).
◦ Azimuthal velocity must remains very small compared to the axial velocity (i.e., ). Thus, in this linear approximation, we have
zr
vzr
vrrvz 2/12222
center line
focusing solenoid
B fields
r z-axis
particle trajectory
b
1r
19
To describe the motion of a charged particle that moves close to the beam
axis, ‘paraxial ray equation’ can be derived
This equation is not a linear differential equation and it is not very useful in
practice. However, the contribution from the last term can be neglected for a
beam without angular momentum or if the particle with constant angular
momentum, we can choose a rotating reference frame in which the last term
can be avoided. Especially when we are interested only on the evolution of
beam envelop rm along z
022 32222
22
22
rcm
pr
mc
qBrrr
022
2
22
mmmm r
mc
qBrrr
For a simple beam moving at constant energy in a field free region, its
envelop diverges at a constant rate along the beam axis z (i.e. r’m = const.).
20
Recall paraxial ray equation:
With
For pure magnetic field, particle energy conserved. Therefore,
With
Solenoid Magnetic Lenses
021 rzgrzgr
21
zg
22
2
222 c
zg L
0
02 rKr
22
22
2
2 cmc
qBK L
21
Since K2 is always positive and if the particle does not cross the axis inside the
lens field (i.e., r > 0), the lens is focusing.
For thin lens, r can be treated as a constant and taken out from the integral
Further, if f2 = f1, then
On the other hand, the particle is rotated by an angle
But for solenoids, integration of H field along z equals to NI (number of ampere
turns of the coil)
rdzKrrz
z2
1
2
12
dzBmc
qrr
z
z
2
1
2
2
2
dzBmc
q
r
r
f
z
z
2
1
2
2
2
1
2
1
z
zr Kdz
NImc
qr
20
22
Hard edge approximation:
1. The field in the region 0 < z < l is assumed to be uniform and zero outside
this region, where l is the effective length of the solenoid
2. D/l << 1
From energy conservation law,
dzzBlB
22
0
dzzBB
l
2
2
0
1(44)
effective length
23
Initial conditions at z = 0, (i.e. )
At the end of the solenoid,
where
The focal length
0,0 rrr
KzKrr
Kzrr
sin
cos
0
0
sin
cos
0
0
l
rr
rr
l
l
mc
qNI
mc
lqBKl
22
00
lr
r
f
l sin1
0
lKl
22
thin lens approximation
Image rotation:
l
r KlKdz0
(45)
(46)
(47)
24
Effects of a Lens on Trace Space Ellipse
and Beam Envelope
0
1 2
3
4
x’
x
Uniform Beam Model The beam is assumed to have a sharp boundary
The uniformity of charge and current densities assures that the
transverse electric and magnetic self-fields and the associated
forces are linear functions of transverse coordinates (see below)
This beam model allows us to extend the linear beam optics to
include the space charge forces.
constJ
constinside the boundary
0 J everywhere outside the boundary
a b
beam pipe charged particle beam
evolution of beam envelop
along the propagation
direction is exaggerated!!
• A axisymmetric laminar beam
• Particle trajectories obey
paraxial assumption that the
angle with the z-axis is small.
• The variation of beam radius
along z-axis is slow enough
that Ez and Br can be neglected.
Based on the above assumptions, we have J, and vz v are all
constant values across the beam.
Therefore, with denoting the charge density of the
beam, we obtain
Since we assumed the electric field has only a radial component
and by Gauss’ law,
vaI 2
0 /
va
I
a
IJJ z
20
2
for 0 r a
0 J for r > a
(1)
va
IrrEr 2
00 22
for 0 r a (2a)
vr
IEr
02 for r > a (2b)
the Lorentz force
exerted on an electron
in the beam has radial
component only and is
a linear function of r.
The magnetic field, on the other hand has only an azimuthal component,
is obtained by applying Ampere’s circuital law:
20
2 a
IrB
r
IB
20
for 0 r a
for r > a
(4)
2
2
ln21a
r
a
bVr s for 0 r a
r
bVr s ln2 for r > a
(5)
Integrate the electrostatic field along an arbitrary path from r = a to r = b,
IIaVs
30
44 00
2
0 (6)
by setting that = 0 at r = b, where The peak potential on the beam axis (r =
0) is thus . And
the maximum electric field at the beam
edge is
aI
a
VEa /60
2
abVV s ln210 0
The motion of a beam particle in such field is described by the radial force equation
BvEqrmrmdt
dzr
where we dropped the force term that is negligibly small and
because there is no external acceleration. zBqr r .const
Substitution for Er from the first equation of (2), B from the first equation of (3)
and with , , we have 2
00
c cvz
(8)
2
2
0
12
ca
qIrrm (9)
with rcdz
rdvr z
22
2
22 we have
(10) 3332
02 mca
qIrr
Define characteristic current (Alfve’n current) I0 as
q
mc
q
mcI
23
00
30
14
which is approx. 17 kA for electrons and 31 (A/Z) MA for ions of mass number A
and charge number Z.
(11)
Define “generalized perveance” K such that
33
0
2
I
IK
In terms of generalized perveance, the equation of motion can be expressed as
(12)
ra
Kr
2
Under the condition of laminar flow, the trajectories of all particles are similar
and scale with the factor r/a. That is, the particle at r=a will always remain at the
boundary of the beam. Thus, by setting r = a = rm, we obtain
Krr mm
(13)
(14)
022 332
2
32222
22
22
mm
n
m
mmmmr
K
rrcm
pr
mc
qBrrr
From the paraxial ray equation and considering the effects of finite emittance and
linear space charge, we obtained the beam envelop equation:
(15)
Electric field along the axis
define electron phase with respect to rf field as
then
or
acceleration of an electron in the gun is
)sin(cos 00 tkzEEz
0 kzt
00
z
dzv
dzk
00 2 1
dzk
z
center line e-
/4 /2 /2 /2
cathode
• The electron emitter is laser driven photo-cathode
• Operating mode in each cell is TM010 – like. Coupled cavity mode is -mode.
• The cavity cells are on-axis coupled through apertures
• Lengths of cells are about /2 • The first cell is shorter because
electrons move slower there.
(1)
(2)
(3)
For initial phase spread of electron 0 , spread of electron phase after acceleration is
The rf effects on electron distribution in longitudinal phase space can be derived from the approx. expression of at the end of the n+1/2 cavity:
If we write
then the longitudinal emittance by definition becomes
in terms of and
0
0
2
0
sin2
cos1
the bunch length dz = -/k
will in general be compressed during acceleration.
(11)
cossin121 n
zzz
ppp zzz
2222
zpzp zzz
22221
k
z
(12)
(13)
(14)
(15)
at the gun exit
setting to minimize transverse emittance
expand for small ,
we obtain,
for Gaussian distribution,
...!3
12
1 32
f
cossin121n
2
sin1cos
241
1 f
rf
zk
3213 zf
rf
z k
(16)
(17)
(18)
(19)
(20)
Longitudinal emittance scales as the third power of bunch length and as the square of rf frequency.
Assuming the following longitudinal E-field,
Er, B can be found from Maxwell’s equations, that is
The radial force is given by
then the equation describing radial motion is
with
0sincos tkzzEEz
z
ErE z
r
2 t
E
c
rcB z
2
cBEeF rr
rr F
mcdt
dp 1
000 cossin
2sincos
2
1cossin
2
1
tkz
dz
zdEtkz
dz
zdEtkzzE
dt
d
cerFr
Assume the transverse deflection is small, r can be regarded as constant, integrate the equation of radial motion gives the radial impulse and assume pr = 0 at t = 0; v c near cavity exit in Cartesian coordinates, Electron trajectories in trace-space is therefore are lines with different slopes Correspond to different
f
rr dtFmc
p0
1
000 cossinsincos tkztkzkrpp ffrr
sinkrpr here is the rf phase at the cavity exit
xkxpx sin
x
x’
The normalized transverse emittance is therefore and therefore the emittance is minimized when <> = 90 such that for Gaussian distribution,
222222 sinsin xkxpxp xx
rf
x
2
22424
222 sin
4
1cos
3
1xkrf
x
2
24
2
2
xkrf
x
2
22
xkrf
x
The equation of motion of a particle accelerated by a traveling-wave
where the phase of the traveling-wave is
the phase motion is
expressing in the equation of motion in , we have
tzqEdt
d
d
dmcmc
dt
d,cos0
3
zttz
2,
1
2 c
dt
d
Since <1, increases with time. The particle falls behind the initial phase on the wave
cos
211
12
0
2 mc
qE
d
d
integration
1
1
1
12sinsin
0
2
i
ii
qE
mc
at asymptotic phase ( 1), if we want to set at the rf crest for efficient acceleration, the condition is
i
ii
qE
mc
1
12sinsin
0
2
i
ii
qE
mc
1
12sin
0
2
injected beam
asymptotic beam
E
• Injected bunch: all of the electrons distributed on a single equi-potential line the ideal case
• rf nonlinearity leads to asymmetric distribution of longitudinal phase space the limitation of compression in bunch length
• Solution injection of lower energy beam !!
• flat region covered widely
• Benefit to relieve this limitation
Generally, thermionic rf gun ~ 2 MeV ;
photocathode gun ~ 4 MeV
nonlinearity of the equi-potential line
at higher injection energy is stronger
(1) γ=10, 3 ps γ=33, 335 fs,
(2) γ=5, 3 ps γ=49, 135 fs
39 Thermionic rf gun injector is suitable with
velocity bunching method for bunch
compression.
RF Bunch Compression in an RF Linac