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, 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

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 iris1) 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

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

• 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

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

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 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.

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)

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!

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:

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.

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

Numerical Solution of Er (C.S. Park)

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

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