1. DEVELOPMENT OF A STEADY STATE SIMULATION CODE FOR KLYSTRONS
Chiara Marrelli
22/06/2011

2. KLYSTRON STUCTURE AND DESIGN
Klystron electronic design
I0– V0 (μPerveance)
Cavity parameters (f0, R/Q, Q0 , Qext …)
Distances between cavities
Beam focusing field
Beam – cavities interaction
Inter – particles interaction
(high current)

3. Intensive use of simulation codes:
Design methodology
Intensive use of simulation codes:
“Disk” codes (e.g. SLAC AJDisk):
1D – only longitudinal motion allowed
Cavities represented by their impedence
Steady state simulations (no transient)
No information on the total cavity field
No information on beam focusing
No gun simulation
QUICK EXECUTION TIME design tool
Particle In Cell (PIC) codes (e.g. Magic):
2-3D simulation
Calculate fields and transient
Focusing system and gun simulation
TIME EXPENSIVE verify tool

4. New code development
Development of a new klystron design code in collaboration with SLAC - S. Tantawi
2D – steady state simulations
complete description of the interaction beam-field
focusing field simulation
no gun simulation
simulation of multiple beams interaction
Fast execution time

5. Beam-cavity interaction
Step 1: field on beam action
Cavity field
Electron Beam
Cavity field guess value
Step 2: beam on field action
Particles tracking
Current calculation
Evaluation of cavity field with the beam
NO SPACE CHARGE FORCES CONSIDERED (particle-particle interaction)
Check for convergence

6. Beam-cavity interaction
Step 1 (beam on field):
calculation of electromagnetic field in the cavity in the presence of beam
Expansion of the total field in terms of the cavity natural modes:
(1,2)
With:
(3)
Since only one term is considered in the expansions:
(4,5)
The field in the cavity is then equal to the field of the design mode multiplied by a complex coefficient, α, to be determined in amplitude and phase.

7. Beam-cavity interaction
S=S1+S2 S1 S2 V- V+
Determining α:
From the cavity power balance:
(6)
The first integral can be written as:
(7)
And, by using Leontovich-Schelkunoff:
(8)

8. Beam-cavity interaction
On S2 we can approximate:
And, by introducing the quality factor Q we get, for the integral over S2 :
(9)
With:
The balance equation becomes:
(10) V+ V- Ve
Vrefl
With:

9. Beam-cavity interaction
The left side of equation (10) contains the power flowing across the waveguide aperture. The incoming wave is given by:
(11)
While the outcoming wave is:
(12)
If the aperture is small we can assume Γ = -1 and obtain:
So that we have:
(13) V+ V- Ve
Vrefl
With:

10. Beam-cavity interaction
The amplitude of the emitted wave Ve depends only on the stored energy in the cavity |α|2u. Its phase can be obtained directly from the balance equation in the case without beam and then adding the phase of the field inside the cavity (phase of α):
(14)
(15)
We have an expression to obtain the cavity field in amplitude and phase in the presence of the beam current density provided the knowledge of the frequency shift (driving frequency).

11. Beam-cavity interaction
Input power from the waveguide
Beam-field interaction
The current density due to the electron beam can be represented by the sum of N individual electron currents:
(16)
The fundamental harmonic of this current density is then:
(17)
(18)
We get:
To get the velocity and position vectors we have to solve the relativistic equations of motion for every particle in the cavity (Step 2)

12. Beam-cavity interaction
Simulation algorithm:
Assume a set of electrons distribuited uniformly in transverse position, time and initial momenta (for first cavity);
Assume an initial value α0 for the field coefficient;
Integrate the equations of motion for each particle through the length of the cavity with the fields:
(plus focusing magnetic field); the cavity modes are obtained from a 2-D electromagnetic code, like SUPERFISH or the FEM code developed by Sami Tantawi.
Calculate the integral
Calculate the new value α1 of the field coefficient using eq. (15);
If α1=α0 (within a certain tolerance), then α=α1 ; otherwise assume a new value for α0 and go to step 3;
Step 2: beam on field action
with

13. Beam-cavity interaction
Step 2 (field on beam):
integration of the equations of motion
Time dependent relativistic Hamiltonian (non-autonomous system):
(19)
Pseudo time-independent problem by introducing two more variables, τ and Pτ :
(20)
Set of 8 equations:
The Hamilton’s equations have to be integrated numerically
Symplectic method (implicit for not separable Hamiltonian) in order to preserve the phase space structure

14. Example: pillbox input cavity
(no space charge)
Test of the code self-consistency:
initially uniform beam
no space charge
cavity without beam pipe (analytic field)
Outcoming power (W) as a function of the cavity resonant frequency
Pin=1 kW
BeamCurrent=100 A
BeamVoltage=100 kV
ω= GHz (driving frequency)
ω0= GHz
L=0.5 cm
Q0=5588
Qext=115
FocusingField=0.093 T (Brillouin field)
Outcoming power (W) as a function of the external Q
The cavity resonant frequency ω0 and the external quality factor Qext are chosen to minimize the outcoming power from the cavity when the beam is in.

15. Example: two cavity klystron
(no space charge)
Test of the code self-consistency when applied to a cavity with PIN=0
velocity modulated beam coming from the input cavity
no space charge
cavity without beam pipe (analytic field)
Pin=1 kW
BeamCurrent=100 A
BeamVoltage=100 kV
ω= GHz (driving frequency)
ω01= GHz (input cavity - pillbox)
L1 =0.5 cm
Q01=5588
Qext1=115
ω02= GHz (output cavity - pillbox)
L2 =0.5 cm
Q02=5576
Qext2=55
FocusingField=0.093 T (Brillouin field)
Ldrift=30 cm (space between cavities)
The output cavity resonant frequency ω02 and external quality factor Qext2 are chosen to maximize the output power.

16. Comparison with klystron kinematic theory
No space charge
Input cavity represented by V1 modulating the particles momenta
passive cavities represented by their equivalent parallel circuit
M = cavity coupling coefficient due to finite transit time V2 V1
t0, p0
t0, p1
t1, p1
t1, p2 I1

17. Comparison with klystron kinematic theory
Pin=250 W
BeamVoltage=100 kV
BeamCurrent=10 A
ω= GHz
L drift=0.1 m
FocusingField=0.093 T
Qext2=∞ zn
pzn
Particles normalized longitudinal momentum after the drift space and after the second cavity zn
pzn
new code
Differences due to the fact that the kinematic theory does not take in account the effect of the beam back to the cavity
kinematic theory

18. Comparison with 1-D simulation code
(AJDisk)
1 D code currently used at SLAC for the design of round and sheet beem klystrons
beam splitted into a series of disks of charge moving only in the longitudinal direction
the disks are acted by both the cavity fields and the space charge fields
Outputs:
cavity voltages
beam current in the cavities
particles minimum β
gain
efficiency
maximum output electric field

19. Comparison with 1-D simulation code
Low current simulations to minimize the space charge effects
Pin=250 W
BeamVoltage=100 kV
BeamCurrent=0.5 A
ω= GHz
L drift=5 cm
Qext2=∞
AJDisk
Particles normalized longitudinal velocity (vz/c) – function of z
new code

20. Comparison of results The klystron kinematic theory & AJDisk
Two cavity klystron
Pin=250 W
BeamVoltage=100 kV
ω= GHz (driving frequency)
L drift=0.1 m
FocusingField=0.093 T (Brillouin field)
Maximum δβz after the 2nd cavity vs beam current
AJDisk
new code
Kin. Theory
Voltage in the 1st cavity vs beam current
Voltage in the 2nd cavity vs beam current
High current simulations require to take in account the repulsive forces between particles:
SPACE CHARGE FIELDS

21. Inter-particles interaction
We search a steady state solution by taking in account the Coulomb repulsion between macroparticles
Simulation algorithm based on two main steps :
Calculation of the total space charge field inside the drift tube as a function of time t ;
Particles tracking inside the drift tube in presence of this field.

22. Inter-particles interaction
Approximations:
Calculation of the space charge field only inside the drift tubes (cavity gaps small with respect to the drift lengths);
Free space solution for the Laplace partial differential equation when calculating the potential due to a point charge in the particle frame
Solution in the circular pipe leads to the Green function:
(21)
Where the ξsl are the zeros of the Bessel functions
And:
n=1
n=2,3,4,…
The evaluation of expression (10) has to be performed for every particle at every iteration; this can be very slow in case of a big number of particles
To speed up the simulation
Free space fields

23. Inter-particles interaction
From the potentials produced by particle i in free space (particle frame)
Electromagnetic fields (particle frame)
Electromagnetic fields (laboratory frame)
The total field (lab frame) is then given by the sum if all the particles fields
Once that we have the space charge field at the location of particle j due to the presence of all the other particles in the laboratory frame, we can integrate the equations of motion for the considered particle in presence of the total (space charge and focusing) field.

24. Inter-particles interaction Main development issues:1)
Since we want to perform steady state simulations only the evolution of one set of particles distributed over an RF period is evaluated, BUT
To calculate the space charge field at time t* we need to take in account ALL the particles that are in the region of space around the considered particle (± 0.5 of the beam wavelength) at that time (and not only particles of the set), i.e. all the particles that satisfy the condition:
(22)
Where the contributions for k>2 can be neglected 2)
Particles in the cavities and pipes before and after the drift space give a contribution to the space charge field;
They will be used as sources for the space charge fields but their trajectories will not be modified during the iterative procedure.

25. Inter-particles interaction First (partial) results:
Drift space after input cavity
Z normalized momentum for on axis particles
pzn
Pin=250 W
BeamCurrent=15 A
BeamVoltage=100 kV
ω= GHz (driving frequency)
ω0= GHz
Ldrift= 10 cm
Q0=5588
Qext=95
FocusingField=0.093 T (Brillouin field)
0.5 plasma wavelength zn
Further work to be done:
test of the results for the plasma frequency
optimization (and speeding up) of the space charge routine

26. Work on klystrons at Cern
High efficiency klystrons for the CLIC study:
Efficiency goal: 80%
Very low μperveance (<=0.25)
Use of higher harmonic cavities (not only second but also 3rd and maybe 4th )
More systematic optimization methods required (Evolutionary Algorithms)