A Compact Magnetic Focusing System for Electron Beams Suitable with Metamaterial Structures

Size: px

Start display at page:

Download "A Compact Magnetic Focusing System for Electron Beams Suitable with Metamaterial Structures"

Berenice Long
6 years ago
Views:

1 A Compact Magnetic Focusing System for Electron Beams Suitable with Metamaterial Structures Ms. Kimberley Nichols University of New Mexico Advised by Dr. Edl Schamiloglu work performed in collaboration with Dr. Bruce Carlsten at LANL 1

2 Agenda 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 2

3 Challenges for High-Frequency Linear Sources (such as TWT s) 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 3

naturally deflect each other Necessary to balance the spread of

4 Typical TWT Device Typical TWT Interaction Circuits: Helical Coupled-Cavity Typical Coupled- Cavity Structure: Electrons naturally deflect each other Necessary to balance the spread of the electron-beam with magnetic confinement 4 Images from radartutorial.edu

5 Metamaterial TWT s 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 5 The focusing field from these PMQ studies will be compatible with these structures

6 E-beam Confinement Methods 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 6

7 PPM Focusing Lattice Typical PPM focusing lattice featuring a continuously varying magnetic field: 7 Invented by Mendel, Quate, Youkum

8 What is Quadrupole Strong Focusing? 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: 8

9 Motivation for PMQ Focusing for High Frequency TWT s 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 RFinteraction structure for ports, diagnostics, etc. 9 Image borrowed from NRL paper by Dave Abe

10 Importance of Emittance 2R c d dz 2 R k 2 o R 2 I a I o 3 1 R 2 R 3 0 2R b B-field Space Charge Emittance =0 Beam >0 beam envelope z x k x x Outermost trajectory of e-beam: Beam Envelope Equation R = radius of beam 10 Because the emittance term goes as 1/, the emittance term becomes very important as the beam radius becomes small. 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) Courtesy of Kevin Jensen, NRL

11 Scherzer s Theorem 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). 11

12 Development of Envelope Code - PMQ 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 12

13 13 PMQ Geometry Definitions

14 Quadrupole Field Model full fringe field model of 16 piece quadrupole - Halbach 14

15 Electron Equations of Motion Equations of motion are from Lorentz Force Law where is called the focusing strength parameter. Quadrupole Channel has two planes of symmetry Two equations of motion are needed 15

16 Solving the Diff EQ s 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 16

17 Single-Particle-Tracking Results Initial Beam and Lattice Parameters for Design: Beam Voltage Magnet Inner Radius 4mm Magnet Outer Radius 12mm Magnet Width variable Distance between Magnets variable What information does this give us? 17

18 Using Single-Particle-Tracking to determine Zero-Current Phase Advance, σₒ 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. 18

19 Include Lawson s Space Charge Term 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 19

20 Beam Profile with Space Charge Not Matched 20

21 Matching the Beam 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 21 *This allows us to determine the MAX current density transportable.*

22 Matched Beam Results Varying the Occupancy, L= 12mm Lattice # lz % Occupancy db/dx MAX db/dx RMS Zero-Current Phase Advance Max Current Density Transportable mm 4.16% T/m 5.42 T/m 4.46 deg A/cm mm 8.33% T/m T/m 9.82 deg A/cm mm 12.50% T/m T/m deg A/cm mm 16.60% 38.9 T/m T/m deg A/cm mm 20.83% T/m T/m deg A/cm mm 25.00% T/m T/m deg A/cm mm 29.16% T/m T/m deg A/cm mm 33.33% T/m T/m 58.7 deg A/cm mm 37.50% T/m T/m deg A/cm mm 41.60% T/m T/m deg A/cm mm 45.83% T/m T/m deg A/cm2 - unstable mm 50.00% drift unstable blows up 22

23 PPM Envelope Code 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 23

24 Results 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 24

25 Next Steps 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) 25

26 Summary 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 26

27 References 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 , Halbach, K., Physical and Optical Properties of Rare Earth Cobalt Magnets, Nucl. Instrum. Methods, vol. 169, pp , Humphries, S., Principle of Charged Particle Acceleration, John Wiley and Sons, Lawson, J. D., The Physics of Charged Particle Beams, 2d ed., Oxford University Press, Reiser, M., Theory and Design of Charged Particle Beams, Wiley-VCH Verlag, Weinheim, Germany,

28 28 Thank You for Your Attention

Physical Principles of Electron Microscopy. 2. Electron Optics

Physical Principles of Electron Microscopy 2. Electron Optics Ray Egerton University of Alberta and National Institute of Nanotechnology Edmonton, Canada www.tem-eels.ca regerton@ualberta.ca Properties