2017 International Vacuum Electronics Conference
Application of Reciprocity to
the Design of Electron Guns*
T.M. Antonsen§, Jr., D. Chernin, and J. Petillo
Leidos, Inc. Billerica, MA
§Dept. ECE University of Maryland
* Work supported by DARPA/MTO. The views, opinions and/or findings expressed are those of
the authors and should not be interpreted as representing the official views or policies of the
Department of Defense or the U.S. Government.
Approved for Public Release, Distribution Unlimited
Global Beam Sensitivity
Function for Electron Guns
Goal
Derive and Calculate a function that gives the variation of
specific beam parameters to
- variations in electrode potential/position
- variations in magnet current/position
Can be used to
- establish manufacturing tolerances
- optimize gun designs
Should be embedded in gun code (e.g. Michelle)
Sensitivity Function
A
Basic question: How do small changes in
position or potential of anode affect the
properties of the beam leaving the gun?
K
C
Beam leaves here
Conventional approach: trial and error. Do many
simulations with different anode potentials or positions
select the best based on some metric measured at the
exit.
Reciprocity - Adjoint Appraoch
Problem #1
A
dF A ( x )
Due to wall displacement
K
C
Problem #2
A
Change in beam
radius
d En
Calculate and record
change in normal E.
K
C
Reverse and perturb
electron coordinates
Electrons run backwards
d En
Is the sensitivity function
EM Reciprocity
Example:
- Antenna sending and receiving radiation patterns are equal due to
time reversal symmetry of ME.
- Direct calculation of receiving pattern requires many simulations
- Instead, calculate sending pattern and invoke reciprocity
V
Receiving
V
Sending
Effective Area – Antenna Gain
dP
dW
I
V
V
Receiving
Power
PR = Ae (W)I
received
Effective
area
l 2G(W)
Ae (W) =
4p
Sending
Incident
intensity
gain
Power per
dP
G(W) =
/ PT unit solid
dW
angle
dP
PT = ò
dW
dW
Optimize shape
to minimize
drag.
Result is also
aesthetically
appealing.
Super Computer
1985 Volvo 240 DL
CODING ERROR !!!
1985 Volvo 240 DL
2017 Porche Panamera
That’s more like it !!!
Michelle: Petillo, J; Eppley, K; Panagos, D; et al., IEEE TPS 30,
1238-1264 (2002).
– HV
1 cm
Solid Model of Electrode
Emitter
Ground
Cut away view of trajectories
Focus
Electrode
Matching
Section
A
K
Code (Michelle) solves the following equations:
Equations of motion for N particles j=1,N
¶H
=
dt ¶p
¶H
=dt
¶x
dx j
dp j
Accumulates a charge density
Tj
r (x) = å I j ò dt d (x - x j (t))
j
0
Solves Poisson E
-Ñ F = 4pr
2
Iterates until converged
Beamstick: Gun Baseline Design
Thermal Beam
KT Nguyen et al. IVEC (2014).
Vk = -25 kV
Vma = -19 kV
Ik = 96.5 mA
Transmission = 100%
10X
Approved for public release; Distribution unlimited
Beamstick: Gun Baseline Geometry
Particle Trajectories at Actual Voltages
Vk = -25.1 kV
Vma = -20.2 kV
Ik = 39.8 mA
It = 16.7 mA
Transmission = 42%
10X
Approved for public release; Distribution unlimited
Hamilton’s Equations H(p,q,t)
Conserve Symplectic Area
(q(t), p(t))
(d q2 (t),d p2 (t))
(d q1 (t),d p1 (t))
perturbed orbit #1
dd q1 ¶2 H
¶2 H
=
×d q1 +
×d p1
dt
¶p¶q
¶p¶p
dd p1
¶2 H
¶2 H
=×d q1 ×d p1
dt
¶q¶q
¶q¶p
(
)
d
d p1 ×d q2 - d p2 ×d q1 = 0
dt
dq ¶H
=
dt ¶p
dp
¶H
=dt
¶q
perturbed orbit #2
dd q2
= ...
dt
dd p2
= -...
dt
Area conserved for
any choice of 1 and 2
A
K
Code (Michelle) solves the following equations:
Hamilton’s Equations for N particles j=1,N
¶H
=
dt ¶p
¶H
=dt
¶x
dx j
dp j
Accumulates a charge density
Tj
r (x) = å I j ò dt d (x - x j (t))
j
0
Solves Poisson Equation
-Ñ F = 4pr
2
Iterates until converged
Reference Solution + Two Linearized Solutions
( x ,p ) ® ( x ,p ) + (d x ,d p )
j
j
j
j
j
j
Two Linearized Solutions
r(x)® r(x)+ dr(x)
[d x j (t),d p j (t)]
true
F(x)® F(x)+ dF(x)
[d x̂ j (t),d p̂ j (t)]
adjoint
Reference Solution
Perturbation
subject to different BC’s
Can show
å I (d p̂ ×d x
j
j
j
j
- d p j ×d x̂ j
)
Tj
0
q
= - ò dan× éëdFÑdF̂ - dF̂ÑdF ùû
4p s
Can show
(
å I j d p̂ j ×d x j - d p j ×d x̂ j
j
)
Tj
0
=-
q
dan× éëdFÑdF̂ - dF̂ÑdF ùû
ò
4p s
Problem #1 (true problem) Unperturbed trajectories at cathode,
Perturbed potential on boundary.
d p j = 0, d q j = 0, dF(x) ¹ 0
0
(
0
)
q
å I j d p̂ j ×d x j - d p j ×d x̂ j Exit = - 4p ò dan× éëdFÑdF̂ - dF̂ÑdF ùû
j
s
Problem #2 (adjoint problem) Perturbed trajectories at exit,
Unperturbed potential on boundary.
d p̂ j = l x ^ j , d q j = 0, dF̂(x)= 0
T
(
l IRRMSd RRMS = l å I j x j ×d x j
j
)
Tj
T
(
q
= - ò dad F n×ÑdF̂
4p s
Sensitivity Function
)
Example of the Adjoint Method in Action
Problem: Compute the displacement of the beam in a sheet beam gun due to a
small change in anode potential or a small displacement of the anode:
MICHELLE Simulations of Sheet Beam Gun
‘Perfect’ case:
‘Perturbed’ case:
Beam centroid at gun exit is on axis
Beam centroid at gun exit is displaced
𝛿𝑥
The adjoint method gives us a way to compute the displacement of the beam without re-running
MICHELLE:
x  
𝛿𝑥 = Beam centroid displacement at gun exit
 = Small change or error in anode or other electrode potential
−𝒏 ∙ 𝛻Φ= Sensitivity (Green’s) function
q
ˆ
da
n






4I s
20
Task 1-4: Theoretical Study of Statistical Variations
Successful Test of Adjoint Method !
Vector plot of the ‘sensitivity’ or Green’s function
−𝛻𝛿 Φ
‘Direct’ MICHELLE Simulation with
Perturbed Anode Voltages
d x actual
d x pred = -
q
dan×d FÑd F̂
ò
4pl I s
Comparison: predicted displacement/actual displacement
d x pred /d xactual = 0.9969
21
Next Steps
•
Consider other goal functions
•
•
•
•
•
•
RMS radius
Emittance
Deviations from Brillouin Flow
Add magnetic Field
3D
Sensitivity to Magnet Strength/Placement
å I (d p̂ ×d x
j
j
j
j
- d p j ×d x̂ j
Thank You
)
T
j
sensitivity function
q
= - ò da d F A n×ÑdF̂ S - ò d 3 xd jm ×qd  s
4p s
Change in magnetization current
22
Can show
(
å I j d p̂ j ×d x j - d p j ×d x̂ j
j
)
Tj
0
=-
q
dan× éëdFÑdF̂ - dF̂ÑdF ùû
ò
4p s
Problem #1 (true problem) Unperturbed trajectories at cathode,
Perturbed potential on boundary.
d p j = 0, d q j = 0, dF(x) ¹ 0
0
0
Problem #2 (adjoint problem) Perturbed trajectories at exit,
Unperturbed potential on boundary.
d p̂ j = l x ^ j , d q j = 0, dF̂(x)= 0
T
(
l IRRMSd RRMS = l å I j x j ×d x j
j
)
Tj
T
(
q
= - ò dad F n×ÑdF̂
4p s
Sensitivity Function
)
Sensitivity Function G(x)
For a given change in potential along boundary
dF A ( x )
A
Due to wall displacement
K
Change in beam
radius
C
d R = ò dA d F AG(x)
2
Changes in beam radius due to
changes in electrode potential.
Sensitivity Function
Task 1-4: Theoretical Study of Statistical Variations
Example of the Adjoint Method in Action
Problem: Compute the displacement of the beam in a sheet beam
gun due to a small change in anode potential or a small
MICHELLE Simulations of Sheet Beam Gun
displacement of the anode:
‘Perfect’ case:
‘Perturbed’ case:
Beam centroid at gun exit is on axis
Beam centroid at gun exit is displaced
𝛿𝑥
The adjoint method gives us a way to compute the displacement of the beam without re-running
MICHELLE:
x  
q
ˆ
da
n






4I s
𝛿𝑥 = Beam centroid displacement at gun exit
 = Small change or error in anode or other electrode potential
−𝒏 ∙ 𝛻Φ= Sensitivity (Green’s) function
25
Task 1-4: Theoretical Study of Statistical Variations
Successful Test of Adjoint Method !
Vector plot of the ‘sensitivity’ or Green’s function
−𝛻𝛿 Φ
‘Direct’ MICHELLE Simulation with
Perturbed Anode Voltages
𝛿𝑥
x  
q
ˆ
da
n






4I s
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑎𝑑𝑗𝑜𝑖𝑛𝑡 𝑚𝑒𝑡ℎ𝑜𝑑 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙
= 0.9969 …
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑓𝑟𝑜𝑚 𝑀𝐼𝐶𝐻𝐸𝐿𝐿𝐸
26
Subject of Keynote Talk at IVEC 2017
Next Steps
•
Consider other goal functions
•
•
•
•
•
•
RMS radius
Emmittance
Deviations from Billouin Flow
Add magnetic Field
3D
Sensitivity to Magnet Strength/Placement
å I (d p̂ ×d x
j
j
j
j
- d p j ×d x̂ j
)
T
j
sensitivity function
q
= - ò da d F A n×ÑdF̂ S - ò d 3 xd jm ×qd  s
4p s
Change in magnetization current
27
Can show
(
å I j d p̂ j ×d x j - d p j ×d x̂ j
j
)
Tj
0
=-
q
dan× éëdFÑdF̂ - dF̂ÑdF ùû
ò
4p s
Problem #1 (true problem) Unperturbed trajectories at cathode,
Perturbed potential on boundary.
d p j = 0, d q j = 0, dF(x) ¹ 0
0
0
Problem #2 (adjoint problem) Perturbed trajectories at exit,
Unperturbed potential on boundary.
d p̂ j = l x ^ j , d q j = 0, dF̂(x)= 0
T
(
l IRRMSd RRMS = l å I j x j ×d x j
j
)
Tj
T
(
q
= - ò dad F n×ÑdF̂
4p s
Sensitivity Function
)
Task 1-4: Theoretical Study of Statistical Variations
Example of the Adjoint Method in Action
Problem: Compute the displacement of the beam in a sheet beam
gun due to a small change in anode potential or a small
MICHELLE Simulations of Sheet Beam Gun
displacement of the anode:
‘Perfect’ case:
‘Perturbed’ case:
Beam centroid at gun exit is on axis
Beam centroid at gun exit is displaced
𝛿𝑥
The adjoint method gives us a way to compute the displacement of the beam without re-running
MICHELLE:
x  
q
ˆ
da
n






4I s
𝛿𝑥 = Beam centroid displacement at gun exit
 = Small change or error in anode or other electrode potential
−𝒏 ∙ 𝛻Φ= Sensitivity (Green’s) function
29
Task 1-4: Theoretical Study of Statistical Variations
Successful Test of Adjoint Method !
Vector plot of the ‘sensitivity’ or Green’s function
−𝛻𝛿 Φ
‘Direct’ MICHELLE Simulation with
Perturbed Anode Voltages
𝛿𝑥
x  
q
ˆ
da
n






4I s
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑎𝑑𝑗𝑜𝑖𝑛𝑡 𝑚𝑒𝑡ℎ𝑜𝑑 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙
= 0.9969 …
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑓𝑟𝑜𝑚 𝑀𝐼𝐶𝐻𝐸𝐿𝐿𝐸
30
Subject of Keynote Talk at IVEC 2017
Task 1-4: Theoretical Study of Statistical Variations
Plans for Next Quarter
•
Continue work on adjoint method
•
•
Prepare additional examples
Draft summary journal article for publication (Phys Review Letters)
•
Continue work on the accurate calculation of Pierce’s parameters Q and C
from the exact tape helix TWT theory, for use in computing the forward gain
and backward wave oscillation threshold accurately.
•
Continue molecular dynamic simulation of beam emittance.
31
Jacobian Matrix – M(t)
(
)
d
d p1 ×d q2 - d p2 ×d q1 = 0
dt
(d q(t),d p(t))
2Nx2N
æ d q(t) ö
æ d q(0) ö
ç
÷ = M (t)× ç
÷
d
p(t)
d
p(0)
è
ø
è
ø
(q(t), p(t))
(d q(0),d p(0))
2N Eiegenvectors and Eigenvalues of M
æ d q(0) ö
æ d q(0) ö
L(t) ç
÷ = M (t)× ç
÷
d
p(0)
d
p(0)
è
ø
è
ø
Solutions come in N pairs- L1L2 = 1
Eigenvectors from different pairs orthogonal
(d p ×d q - d p ×d q ) = 0
1
2
2
1
Thank you
Tests so far.....
Running the beam backwards in the same fields as forward, the error in the beam position on
the cathode is…
Relative error:
(x,y) = (0.625%,
0.625%)
Absolute error normalized to the beam radius at the cathode: (x,y) =
(0.0002%,
0.0024%)
Running the beam backwards in newly converged fields, the error in the beam position on the
cathode is…
Relative error:
(x,y) = (24.3%,
24.3%)
Absolute error normalized to the beam radius at the cathode:
(x,y) =
(0.0046%,
0.1411%)
Conservation of Symplectic Area
Advisor Warning
Gratuitous Math Scene
This presentation rated M
Example
Our reading from the book of Jackson,
Problems 1.12 and 1.13
A charge q is placed at an arbitrary point, xo, relative to two
grounded, conducting electrodes.
q
q1
x0
q2
What is the charge q1 on the surface of
electrode 1?
Repeat for different x0
Solution - Green’s Reciprocation Theorem
Prob #1
Your
Problem
Prob #2
N.Y. P.
f B1 = f B2 = f (x ® ¥) = 0
Ñ2f = -qd (x - x0 ) BC:
q1 = ò d 2 x n ×Ñf
B1
Ñ2y = 0
Green’s
Theorem
BC:
y
B1
= 1, y
B2
= y (x ® ¥) = 0
3
2
2
d
x
y
Ñ
f
f
Ñ
y ) = ...
(
ò
V
When the dust settles:
q1 = -qy (x0 )
Features of Problems Suited to Adjoint
Approach
1. Many computations need to be repeated. (many
different locations of charge, q)
2. Only a limited amount of information about the
solution is required. (only want to know charge on
electrode #1)
Other Examples of Reciprocity
Electrostatics
Symmetry of the Capacitance Matrix
Electromagnetics
Symmetry of the Inductance Matrix
Symmetry of Scattering Matrix
Collisional Transport: Onsager Symmetry of off-diagonal elements of
transport matrices.
Temperature gradient
Electric field


Electric current
Heat flux
Hamilton’s Equations H(p,q,t)
(q(t), p(t))
(d q2 (t),d p2 (t))
(d q1 (t), d p1 (t))
perturbed orbit #1
dd q1 ¶2 H
¶2 H
=
×d q1 +
×d p1
dt
¶p¶q
¶p¶p
dd p1
¶2 H
¶2 H
=×d q1 ×d p1
dt
¶q¶q
¶q¶p
(
)
d
d p1 ×d q2 - d p2 ×d q1 = 0
dt
dq ¶H
=
dt ¶p
dp
¶H
=dt
¶q
perturbed orbit #2
dd q2
= ...
dt
dd p2
= -...
dt
Area conserved for
any choice of 1 and 2
Jacobian Matrix – M(t)
(
)
d
d p1 ×d q2 - d p2 ×d q1 = 0
dt
(d q(t),d p(t))
2Nx2N
æ d q(t) ö
æ d q(0) ö
ç
÷ = M (t)× ç
÷
d
p(t)
d
p(0)
è
ø
è
ø
(q(t), p(t))
(d q(0),d p(0))
2N Eiegenvectors and Eigenvalues of M
æ d q(0) ö
æ d q(0) ö
L(t) ç
÷ = M (t)× ç
÷
d
p(0)
d
p(0)
è
ø
è
ø
Solutions come in N pairs- L1L2 = 1
Eigenvectors from different pairs orthogonal
(d p ×d q - d p ×d q ) = 0
1
2
2
1
Conjecture #1: Among the set of people
from Nottingham
George Green > Robin Hood
Example
Our reading from the book of Jackson,
Problems 1.12 and 1.13
A charge q is placed at an arbitrary point, xo, relative to two
grounded, conducting electrodes.
q
q1
x0
q2
What is the charge q1 on the surface of
electrode 1?
Repeat for different x0
Solution - Green’s Reciprocation Theorem
Prob #1
Your
Problem
Prob #2
N.Y. P.
f B1 = f B2 = f (x ® ¥) = 0
Ñ2f = -qd (x - x0 ) BC:
q1 = ò d 2 x n ×Ñf
B1
Ñ2y = 0
Green’s
Theorem
BC:
y
B1
= 1, y
B2
= y (x ® ¥) = 0
3
2
2
d
x
y
Ñ
f
f
Ñ
y ) = ...
(
ò
V
When the dust settles:
q1 = -qy (x0 )
Features of Problems Suited to
Adjoint Approach
1. Many computations need to be repeated. (many
different locations of charge, q)
2. Only a limited amount of information about the
solution is required. (only want to know charge on
electrode #1)
George Green 1793-1841
Physics Today Dec. 2003
•
•
•
•
•
•
Born in Nottingham (Home of Robin Hood)
Father was a baker
At age 8 enrolled in Robert Goodacre’s Academy
Left after 18 months (extent of formal education pre 40)
Worked in bakery for 5 years
Sent by father to town mill to learn to be a miller
• Fell in love with Jane, the miller’s
daughter.
• Green’s father forbade marriage.
• Had 7 children with Jane.
• Self published work in 1828
• With help, entered Cambridge
1833, graduated 1837.
• “Discovered” by Lord Kelvin in
1840.
• Theory of Elasticity, refraction,
evanescence
• Died of influenza, 1841
Green’s Mill: still functions
Task 1-4: Theoretical Study of Statistical Variations
‘Adjoint’ Method Approach to Tolerance Analysis in Electron Guns
Basic question: How do small changes in position or potential of anode affect
the properties (radius, emittance, …) of the beam leaving the gun?
A
K
C
R
Gun exit
Adjoint method: Compute a sensitivity (Green’s) function that
permits calculation of the effect of any perturbation on a beam
property (radius, emittance, centroid displacement, ..). The
Green’s function depends on the particular property chosen.
• The Green’s function is computed once, from two runs of
MICHELLE, with all surfaces and potentials at the nominal
48
(design) values, but different initial conditions for the
Electromagnetic Reciprocity
Example:
- Antenna sending and receiving radiation patterns are equal due to
time reversal symmetry of Maxwell’s Equations.
- Direct calculation of receiving pattern requires many simulations
- Instead, calculate sending pattern and invoke reciprocity
V
Receiving
V
Sending
Effective Area – Antenna Gain
dP
dW
I
V
V
Receiving
Power
PR = Ae (W)I
received
Effective
area
l 2G(W)
Ae (W) =
4p
Sending
Incident
intensity
gain
Power per
dP
G(W) =
/ PT unit solid
dW
angle
dP
PT = ò
dW
dW
Other Examples of Reciprocity
Electrostatics
Symmetry of the Capacitance Matrix
Electromagnetics
Symmetry of the Inductance Matrix
Symmetry of Scattering Matrix
Collisional Transport: Onsager Symmetry of off-diagonal elements of
transport matrices.
Temperature gradient
Electric field


Electric current
Heat flux
Courtesy, Elizabeth Paul
Optimize shape
to minimize
drag.
Result is also
aesthetically
appealing.
Super Computer
1985 Volvo 240 DL
CODING ERROR !!!
1985 Volvo 240 DL
2017 Porche Panamera
That’s more like it !!!
Adjoint Problems I Have Known
• Calculation of RF current drive in magnetic confinement
plasma configurations, TMA and KR Chu, PoF 25, (1982)
• Calculation of RF induced transport in magnetic confinement
plasmas, TMA and K. Yoshioka, PoF 29, (1986), Nucl.
Fusion, 26 (1986).
• Spontaneous poloidal flow spin-up, Bull Am Phys Soc.
(1994).
• Shot noise on gyrotron beams, TMA, W Manheimer, and A
Fliflet, PoP (2001).
• Beam optics sensitivity function TMA, D. Chernin, J.
Petillo, TBP