IRPSS: A Green’s Function Approach to
Modeling Photoinjectors
Mark Hess
Indiana University Cyclotron Facility &
Physics Department
*Supported by NSF and DOE
Electron Source
Requirements for Future
Experiments*
•
Linear collider:
I = 500 A, t
FWHM
= 8 ps,
e
n
= 10 mm

mrad ,
B
n
= 1x10
13
A/m
2
rad
2
•
SASE

FEL:
I = (180

500) A, t
FWHM
= (1

6) ps,
e
n
= (0.1

2) mm

mrad , B
n
= (1x10
13

10
17
) A/m
2
rad
2
•
Laser Wakefield Accelerators:
I = 1000 A, t
FWHM
= 0.2 ps,
e
n
= 3 mm

mrad , B
n
= 2x10
14
A/m
2
rad
2
Future experiments demand high

brightness electron
beams from photoinjectors:
*G. Suberlucq, EPAC 2004
Challenges for
Photoinjector Simulations*
•
There are two main challenges with simulations of high

brightness photoinjectors:
•
Resolution of small length/time scale space

charge fields relative to long
length/time scales of injector, e.g. 1

10 ps bunch lengths for 1.3

2.8 GHz
•
Removal of unphysical simulation effects such as numerical grid
dispersion and numerical Cherenkov effects in FDTD methods
•
Beam dynamics resolution would require FDTD longitudinal cell sizes of at most
1/10 bunch length
•
Error analysis of FDTD methods set a bound of 10 cells per characteristic
wavelength for 1% dispersion error (~100 cells per bunch length)*
•
Since bunch length ~ 1/100 of free space wavelength then 2,500+ fixed size cells
in longitudinal direction are necessary
•
In transverse direction, 4,000+ cells would be required for BNL gun simulation
(laser spot size/cavity radius=1/40)
*K. L. Schlager and J.B. Schneider, IEEE Trans. Antennas and Prop., 51, 642 (2003).
IRPSS Method for Modeling
Photoinjectors
•
We are developing a self

consistent code called IRPSS
(Indiana Rf Photocathode Source Simulator)
•
IRPSS uses time

dependent Green’s functions for
calculating electromagnetic space

charge fields
•
Green’s functions are generated by delta function
sources in space and time which enable arbitrarily
small resolution of length and time scales
•
Since electromagnetic fields are defined everywhere in
simulation space (not just on a grid), numerical grid
dispersion effects are completely removed
•
Green’s functions can be constructed to satisfy the
appropriate conductor boundary conditions
Code Development Path
2) Cathode with iris
1) Cathode
3) Cathode with irises
IRPSS can currently
simulate geometry 1)
We are developing
methods for simulating
geometry 2)
Theory
•
IRPSS solves the fields in the Lorentz Gauge
A
B
A
E
t
Field

Potential Relations
J
A
0
0
2
2
2
2
1
e
t
c
Potential

Source
Relations in
Lorentz Gauge
0
s
0

s
A
0

s
E
0
s
B
Metallic Boundary
Conditions
•
For the special case of currents in the axial direction
in an pipe with a cathode, the potentials are given by
Theory: Green’s Function
Method (Pipe w/ Cathode)*
0
,
0
,
0
cathode
z
wall
side
z
boundary
z
A
A
Boundary
Conditions:
*M. Hess and C. S. Park, submitted to PR

STAB.
t
t
t
G
d
t
d
t
t
,
,
;
,
1
,
3
0
r
r
r
r
r
e
t
J
t
t
G
d
t
d
t
A
z
A
t
z
,
,
;
,
,
3
0
r
r
r
r
r
Time

Dependent Green’s
Functions (Pipe w/ Cathode)
2
0
2
0
2
1
2
2
n
n
mn
im
mn
m
mn
m
mn
m
A
k
J
k
J
e
j
J
a
r
j
J
a
r
j
J
a
c
G
G
2
2
2
z
z
t
t
c
Solution:
Where:
and
a
is the cavity radius
IRPSS Numerical Methods
(Particle/Slice Evolution)
•
Current:
•
Particle/Slice has predetermined trajectory.
Trajectory is discretized into elements for numerical
integration.
•
Can be used to calculate the approximate effect of
space charge forces via perturbation
•
Future:
•
Trajectory will evolve within simulation with space
charge fields included. Trajectory needs to be
tracked for “sufficiently” long times in to compute
fields.
Space

Charge Fields
Calculation in IRPSS
Specify z’’
i
(t)
and
s
i
(t) for
each slice
Compute E
and B due to
space

charge
Simulate
trajectories of
test particles
Current
Method :
N
i
i
i
t
z
z
t
r
t
r
1
,
,
s
N
i
i
i
i
z
t
z
z
dt
z
d
t
r
t
r
J
1
,
,
s
•
Bunches are divided into slices, each having a
transverse charge density and zero thickness
longitudinal distribution, i.e.
Computational Criteria
b
n
b
n
r
a
j
r
a
j
60
~
,
2
max
,
0
max
,
0
1. In order to resolve the transverse profile of the beam, there needs to
be “enough” radial modes
For BNL 1.6 cell gun typical mode numbers are n
max
~2000
2. The time step within the potential integrals needs to be sufficiently
small in order to resolve the oscillations of the Bessel function
integrand
c
r
t
cj
a
t
b
n
01
.
0
~
,
2
max
,
0
For BNL 1.6 cell gun this corresponds to a time step of 33 fs (5,000
time steps to model ½ cell)
Computational Criteria
3. In order to resolve the longitudinal profile of the beam it is
necessary to include a sufficient number of slices. Each slice
produces a localized peak within the bunch. A good estimate for
determining the minimum number of slices is:
b
b
slice
b
b
b
slice
r
l
N
r
l
length
bunch
l
slice
for
of
FWHM
N
10
,
0.005
0
0.005
0.01
0.015
0.02
0.075
0.085
0.095
0.105
0.115
0.125
z/
Normalized E
r
1 Slice
10 Slices
30 Slices
•
Electric field at the edge of the
beam for BNL gun w/ 10 ps bunch
compared to zero bunch length
•
As bunch length decreases slice
number decreases!
Future Status of IRPSS
Future
Method:
Specify initial
conditions of
each “ring” of
charge
At half

time
step, calculate
E and B (slice
approximation)
At next half

time step
compute
trajectories
Update
trajectory
registry
•
In the future, IRPSS will maintain a trajectory registry
which will keep track of all simulation particles (rings)
for all time which is necessary for field calculation
IRPSS Simulation Results for
Bunched Disk Beam
•
We have performed simulations of a zero thickness bunch with the
BNL 1.6 cell gun* parameters excluding the iris
•
The bunch trajectory was calculated by solving the equations of
motion for an external rf

field:
z
V
dt
z
d
t
z
k
eE
dt
dP
z
sin
)
cos(
0
Where:
E
0
=100 MV/m , f=2.856 GHz
a=0.04 m , r
b
=0.001 m ,
φ
=68
°
t
z
z
r
r
r
r
r
Q
t
r
b
b
b
2
2
2
1
2
,
t
z
z
r
r
r
r
dt
z
d
r
Q
t
r
J
b
b
b
z
2
2
2
1
2
,
Charge Density:
Current Density:
Equations
of motion:
*K. Batchelor et al, EPAC’88.
Simulation Trajectory
Bunch Trajectory (red) and light

line (blue) for BNL 1.6
Cell Photocathode Gun
Cathode
End of Full

Cell
z/
λ
=0.75
End of Half

Cell
z/
λ
=0.25
~ 4.0
~㤮9
Numerical Solution
of E
r
(C.S. Park)
t 1.0
t 2.0
t 3.0
t 4.0
Benchmark Results
v
v
Q

Q
v
v
Q

Q
•
IRPSS simulation of a
disk bunch of charge
emitted at time t = 0
from the cathode surface
moving uniformly with
speed v
•
Analytical model of two disks of
charge moving uniformly in
opposite directions for all time
and intersecting at t = 0
Benchmark Results
For times before reflection from the side wall, but
sufficiently long after the t=0 the two results (
IRPSS

Blue
,
Model

Red
) agree within <1%
Benchmark Results
For times after reflection from the side wall, the side
wall image charge reduces the simulation potentials
(
IRPSS

Blue
,
Model

Red
)
Benchmark Results
For times shortly after t=0, the results agree to within
1% up when z < t/c
–
causality constraint (
IRPSS

Blue
,
Model

Red
)
Future Plan
•
Include the effects of more complicated geometries such as irises
in IRPSS
•
Possible method is Bethe multipole moment technique
•
Update trajectories in IRPSS due to Lorentz force law
•
Continue simulations of experimental systems
–
currently working
with Argonne on simulations of AWA photoinjector
•
Explore parallelization options for IRPSS
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