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Ms. Kimberley Nichols
University of New Mexico
Advised by Dr. Edl Schamiloglu
work performed in collaboration with Dr. Bruce Carlsten at LANL
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A Compact Magnetic Focusing System for
Electron Beams Suitable with Metamaterial
Structures
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Agenda
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Challenges of High–Frequency Linear Devices
General Electron-Beam Confinement
Permanent Periodic Magnet (PPM) Focusing – Limitations
Motivation for Permanent Magnet Quadrupole (PMQ) Focusing
PMQ Envelope Code
PPM Envelope Code
Results
Next Steps
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Challenges for High-Frequency
Linear Sources (such as TWT’s)
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There is interest in higher-frequency vacuum tube
sources
Device dimensions scale inversely as frequency,
high frequency devices are very small
To increase the power of these devices, it is
necessary to either increase the current of the
electron beam or increase the voltage
Increasing the voltage is not practical
Small beams are more susceptible to emittance
issues
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Typical TWT Device
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• Electrons naturally deflect each other
• Necessary to balance the spread of the electron-beam with magnetic confinement
Images from radartutorial.edu
Typical TWT Interaction Circuits:
• Helical
• Coupled-Cavity
Typical Coupled-
Cavity Structure:
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Metamaterial TWT’s
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As part of this MURI grant several consortium members are studying and proposing novel electromagnetic interaction structures:
MIT - complementary split ring resonator-based structure
Ohio State and UC Irvine - structures with frozen modes (degenerate band edge modes)
LSU studying other novel structures
UNM plans on testing out the most promising of these structures as this program progresses
The focusing field from these PMQ studies will be compatible with these structures
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E-beam Confinement Methods
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Solenoid: Large B-fields
Bulky / Heavy Require external power supplies and cooling systems
Permanent Periodic Magnets (PPM’s): Compact
Light weight No power supplies / cooling systems Reduced confining fields
Permanent Magnet Quadrupoles (PMQ): (proposed) Even more compact Lighter weight Larger confining fields
Less emittance growth
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PPM Focusing Lattice
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Typical PPM focusing lattice featuring a
continuously varying magnetic field:
Invented by Mendel, Quate, Youkum
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What is Quadrupole Strong Focusing?
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Operates on FODO
principle
First quadrupole focuses in
the first plane, defocuses in
the second
Second quadrupole focuses
in the second plane,
defocuses in the first
Net effect of focus-defocus
is strong focusing
Focusing Channel
Magnet Configuration:
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Motivation for PMQ Focusing for
High Frequency TWT’s
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PMQ lattices present an alternative to PPM focusing:
lighter in weight
less expensive
transport more current density
reduce the emittance growth of the beam
Empty space in the lattice allows for easy access to the RF-interaction structure for ports, diagnostics, etc.
Image borrowed from NRL paper by Dave Abe
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2Rb
2Rc
B-field Space Charge Emittance
d
dz
2
R ko2R 2
Ia
Io
3
1
R
2
R3 0
>0 beam envelope
Beam =0 x
z x
kx
Importance of Emittance
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Emittance: measure of beam quality
• Transverse: reduced intensity at beam tunnel
• Longitudinal: increased E spread, reduced I
• Emittance at cathode stays w/ beam: cannot
be corrected by subsequent beam
manipulation (generally due to high
temperature of cathode)
Outermost trajectory of e-beam:
Beam Envelope Equation
R = radius of beam
Because the emittance
term goes as 1/ , the
emittance term becomes
very important as the beam
radius becomes small.
Courtesy of Kevin Jensen, NRL
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Scherzer’s Theorem
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In 1936, Otto Scherzer (Z. Phys., 101, 593 (1936)) showed that higher order radial terms always add in cylindrical magnetic lenses, leading to an unavoidable aberration in electron microscopes that limits resolution to 50 to 100 wavelengths (and, for us, cause an emittance growth). This is known as the Scherzer Theorem, and is commonly used to evaluate the emittance growth for the PPM model.
It is important to note that Scherzer also, in 1947 (Optic 2, 114, (1947)), showed that multipoles could be used to eliminate this aberration (and that focusing using only multipoles could be aberration free).
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Development of Envelope Code -
PMQ
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1) Develop Magnetic field models for PMQ and PPM lattices
2) Start with the equations of motion for a single particle (Lorentz Force Law). This is single particle tracking. Gives us the zero-current phase advance
3) Put the field model into the EOM’s, solve the equations
4) Add the Space-Charge term to the EOM’s. This accounts for a non-zero current density. Allows us to determine the maximum transportable current density
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PMQ – Geometry Definitions
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Quadrupole Field Model
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full fringe field model of 16
piece quadrupole - Halbach
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Electron Equations of Motion
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Equations of motion are from Lorentz Force Law
Quadrupole Channel has two planes of symmetry
Two equations of motion are needed
where
is called the focusing
strength parameter.
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Solving the Diff EQ’s
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Use default numerical differential equation solver in Mathematica: “NDSolve[ ]”
To solve the differential equations on the previous page numerically we must set the initial conditions:
Initial conditions will need to be adjusted to match the beams when the space charge term is added
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Single-Particle-Tracking Results
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Beam Voltage 16000
Magnet Inner Radius 4mm
Magnet Outer Radius 12mm
Magnet Width variable
Distance between Magnets variable
Initial Beam and Lattice Parameters for Design:
What
information
does this
give us?
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Using Single-Particle-Tracking to
determine Zero-Current Phase
Advance, σₒ
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To determine σₒ
we can curve-fit
the particle
trajectory and
compare the
period of the
particle
trajectory
against the
period of the
magnet lattice.
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Include Lawson’s Space Charge
Term
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To add the space charge term we calculated the generalized
perveance:
where
Then our differential equations become:
x[z] and y[z] now represent the beam edge or the beam “envelope”
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Beam Profile with Space Charge –
Not Matched
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Matching the Beam
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By adjusting the initial conditions, we can minimize the ripples in the
beam edge that do not correspond with the field profile
This minimizes the beam size, matches the beam
*This allows us to determine the MAX current
density transportable.*
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Matched Beam Results – Varying the
Occupancy, L= 12mm
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Lattice #
lz
% Occupancy
dB/dx MAX
dB/dx RMS
Zero-Current Phase Advance
Max Current Density Transportable
1 0.25 mm 4.16% 9.818 T/m 5.42 T/m 4.46 deg 4.207 A/cm2
2 0.50 mm 8.33% 19.59 T/m 10.54 T/m 9.82 deg 15.78 A/cm2
3 0.75 mm 12.50% 29.30 T/m 15.76 T/m 16.11 deg 32.59 A/cm2
4 1.00 mm 16.60% 38.9 T/m 20.91 T/m 22.91 deg 53.82 A/cm2
5 1.25 mm 20.83% 48.36 T/m 25.97 T/m 30. 68 deg 77.79 A/cm2
6 1.50 mm 25.00% 57.64 T/m 30.93 T/m 39.19 deg 103.2 A/cm2
7 1.75 mm 29.16% 66.72 T/m 35.75 T/m 48.44 deg 130.25 A/cm2
8 2.00 mm 33.33% 75.58 T/m 40.42 T/m 58.7 deg 159.31 A/cm2
9 2.25 mm 37.50% 84.19 T/m 44.91 T/m 69.48 deg 186.33 A/cm2
10 2.50 mm 41.60% 92.53 T/m 49.22 T/m 82.24 deg 214.6 A/cm2
11 2.75 mm 45.83% 100.59 T/m 53.32 T/m 95.87 deg 225.47 A/cm2 - unstable 12 3.00 mm 50.00% 112.75 - drift unstable – blows up
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PPM – Envelope Code
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Similar to the PMQ Envelope Code an envelope
code has been developed for a PPM lattice
The PPM B-field model has been verified
There are still a few bugs in the envelope code
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Results
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The PMQ lattice was optimized by:
varying the magnet width
varying the focusing period
calculating the maximum current density
transportable for each case
The maximum current for this PMQ lattice:
Lattice period of L = 11 mm, lz = 3 mm
max current density is 220 A/cm2,
phase advance of 88.9 degrees
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Next Steps
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Verify and compare the results of the PPM envelope code with the PMQ results
Determine the range of beam parameters for which PMQ focusing is superior to PPM focusing
Add emittance and analyze emittance growth
Do simulations with more complex beam models (MICHELLE)
Model the beam interaction with the metamaterial interaction circuit (Metamaterial TWT) – (ICEPIC)
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Summary
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We have proposed using PMQ focusing for strong
focusing in TWT type devices, the advantages are:
Potentially transport more current density
Provides access to the EM interaction structures
for ports and diagnostics
Improve the beam-quality for small beams
Reduced size and weight compared to PPM
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References
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Abe, D.K., R.A. Kishek, J.J. Petillo, D.P. Chernin, and B. Levush, “Periodic Permanent-Magnet Quadrupole Focusing Lattices for Linear Electron-Beam Amplifier Applications,” IEEE Trans. Electron Dev., vol. 56, pp. 965-973, 2009.
Halbach, K., “Physical and Optical Properties of Rare Earth Cobalt Magnets,” Nucl. Instrum. Methods, vol. 169, pp. 109-117, 1981.
Humphries, S., Principle of Charged Particle Acceleration, John Wiley and Sons, 1999.
Lawson, J. D., The Physics of Charged Particle Beams, 2d ed., Oxford University Press, 1988.
Reiser, M., Theory and Design of Charged Particle Beams, Wiley-VCH Verlag, Weinheim, Germany, 2008.
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Thank You for Your Attention
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