irpss: a green’s function approach to modeling photoinjectors
DESCRIPTION
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*. Future experiments demand high-brightness electron beams from photoinjectors:. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/1.jpg)
IRPSS: A Green’s Function Approach to Modeling Photoinjectors
Mark Hess
Indiana University Cyclotron Facility & Physics Department
*Supported by NSF and DOE
![Page 2: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/2.jpg)
Electron Source Requirements for Future
Experiments*
•Linear collider: I = 500 A, tFWHM = 8 ps, n = 10 mm-mrad , Bn= 1x1013 A/m2rad2
•SASE-FEL: I = (180-500) A, tFWHM = (1-6) ps, n = (0.1-2) mm-mrad , Bn= (1x1013-1017) A/m2rad2
•Laser Wakefield Accelerators: I = 1000 A, tFWHM = 0.2 ps, n = 3 mm-mrad , Bn = 2x1014 A/m2rad2
Future experiments demand high-brightness electron beams from photoinjectors:
*G. Suberlucq, EPAC 2004
![Page 3: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/3.jpg)
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).
![Page 4: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/4.jpg)
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
![Page 5: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/5.jpg)
Code Development Path
2) Cathode with iris1) Cathode
3) Cathode with irises IRPSS can currently simulate geometry 1)
We are developing methods for simulating geometry 2)
![Page 6: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/6.jpg)
Theory
• IRPSS solves the fields in the Lorentz Gauge
AB
AE
t
Field-Potential Relations
JA 0
02
2
22 1
tc
Potential-Source Relations in Lorentz Gauge
0s
0|| s
A
0|| s
E
0 sB
Metallic Boundary Conditions
![Page 7: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/7.jpg)
• 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
zwallsidezboundary z
AABoundary
Conditions:
*M. Hess and C. S. Park, submitted to PR-STAB.
tttGdtdtt
,,;,1
, 3
0
rrrrr
tJttGdtdtA zA
t
z ,,;,, 30 rrrrr
![Page 8: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/8.jpg)
Time-Dependent Green’s Functions (Pipe w/ Cathode)
20
20
21
22
nn
mn
im
mnm
mnm
mnm
A
kJkJ
ejJ
arj
Jarj
J
a
c
G
G
222 zzttc
Solution:
Where:
and a is the cavity radius
![Page 9: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/9.jpg)
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.
![Page 10: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/10.jpg)
Space-Charge Fields Calculation in IRPSS
Specify z’’i(t) and i(t) for each slice
Compute E and B due to space-charge
Simulate trajectories of test particles
Current Method :
N
iii tzztrtr
1
,,
N
ii
iiz tzz
dt
zdtrtrJ
1
,,
• Bunches are divided into slices, each having a transverse charge density and zero thickness longitudinal distribution, i.e.
![Page 11: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/11.jpg)
Computational Criteria
bn
bn r
aj
r
aj
60~,
2max,0max,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 nmax~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
rt
cj
at 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)
![Page 12: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/12.jpg)
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
bslice
b
bbslice r
lN
r
llengthbunchl
slicefor
ofFWHMN
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/
No
rmal
ized
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!
![Page 13: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/13.jpg)
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
![Page 14: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/14.jpg)
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:
zVdt
zd tzkeE
dt
dPz sin)cos(0
Where:E0=100 MV/m , f=2.856 GHz
a=0.04 m , rb=0.001 m , φ=68°
tzzr
rrr
r
Qtr
bb
b
2
2
21
2,
tzzr
rrr
dt
zd
r
QtrJ
bb
bz
2
2
21
2,
Charge Density:
Current Density:
Equations of motion:
*K. Batchelor et al, EPAC’88.
![Page 15: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/15.jpg)
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
~ 9.0
![Page 16: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/16.jpg)
Numerical Solution of Er (C.S. Park)
![Page 17: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/17.jpg)
Benchmark Results
vvQ-Q
vvQ-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
![Page 18: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/18.jpg)
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%
![Page 19: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/19.jpg)
Benchmark Results
For times after reflection from the side wall, the side wall image charge reduces the simulation potentials (IRPSS - Blue, Model - Red)
![Page 20: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/20.jpg)
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)
![Page 21: IRPSS: A Green’s Function Approach to Modeling Photoinjectors](https://reader036.vdocument.in/reader036/viewer/2022062323/5681531b550346895dc14062/html5/thumbnails/21.jpg)
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