IRPSS: A Green's Function Approach to Simulating Photoinjectors

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Nov 16, 2013 (3 years and 8 months ago)

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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