Ms. Kimberley Nichols

University of New Mexico

Advised by Dr. Edl Schamiloglu

work performed in collaboration with Dr. Bruce Carlsten at LANL

1

A Compact Magnetic Focusing System for

Electron Beams Suitable with Metamaterial

Structures

Agenda

2

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

Challenges for High-Frequency

Linear Sources (such as TWT’s)

3

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

Typical TWT Device

4

• 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:

Metamaterial TWT’s

5

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

E-beam Confinement Methods

6

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

PPM Focusing Lattice

7

Typical PPM focusing lattice featuring a

continuously varying magnetic field:

Invented by Mendel, Quate, Youkum

What is Quadrupole Strong Focusing?

8

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:

Motivation for PMQ Focusing for

High Frequency TWT’s

9

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

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

10

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

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).

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

PMQ – Geometry Definitions

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Quadrupole Field Model

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full fringe field model of 16

piece quadrupole - Halbach

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.

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

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?

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.

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”

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.*

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

PPM – Envelope Code

23

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

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

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)

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

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.

Thank You for Your Attention

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