Analysis of Quadrupole Focusing Lattices for Electron Beam Transport in Traveling-Wave Tubes

Kimberley Nichols, Edl Schamiloglu Bruce Carlsten Department of Electrical and Computer Engineering, Accelerator Operations and Technology Division University of New Mexico Los Alamos National Laboratory Albuquerque, NM, 87106 Los Alamos, NM 87545 [email protected]

Table 1. Initial lattice parameters. Abstract: Analysis of a quadrupole focusing lattice for Magnet Inner Radius, 4mm high-frequency TWT’s is presented. Single particle 𝒊𝒊 tracking is performed to determine the phase advance, with Magnet Outer Radius, 𝒓𝒓 12mm the eventual goal of determining the advantageous Magnet Width, 𝒓𝒓𝒐𝒐 1.9 mm frequency regime for devices using this focusing method. This work is motivated by recent work which demonstrated Distance Between𝒍𝒍𝒛𝒛 Magnets, an advantageous case for strong focusing employing a 5 mm Halbach quadrupole lattice. 𝐝𝐝𝐝𝐝𝐝𝐝 PMQ Field Model Keywords: TWT; quadrupole strong focusing; permanent Magnet Analysis: Halbach derived an expression for the magnet; beam transport. fringing magnetic fields for the 16-piece quadrupole magnet [2]. Modifying his expression for a finite magnet Introduction and then using the principle of superposition, we developed A detailed study of the magnetic field profile produced by an expression for the gradient of the magnetic field for a permanent magnet quadrupole (PMQ) lattices was lattice of quadrupoles: performed and contrasted with the magnetic profiles of 𝒊𝒊 + 𝟏𝟏 permanent periodic magnet (PPM) lattices for the purpose 


of transporting high current density electron beams. It was

 =
𝐺𝐺
 ∗ 𝐹𝐹 −𝑖𝑖 ∗ 𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑧𝑧 −
− 𝐹𝐹 −𝑖𝑖 ∗ found that the root mean square (rms) magnetic field of the




PPM lattice suffers severe reduction in field strength as the 𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑧𝑧 +
−
𝐺𝐺
 ∗ 𝐹𝐹 −𝑖𝑖 ∗ 𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑧𝑧 − lattice period decreases, giving an optimum lattice period



for maximum current density transport. A PMQ’s main
− 𝐹𝐹 −𝑖𝑖 ∗ 𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑧𝑧 +
(1) field components are of higher order, and as such the field profiles are less prone to reduction as the lattice period decreases. Essentially this allows us to create much stronger magnetic field profiles in cases where a shorter lattice period is appropriate, specifically for devices operating at high frequencies. Reduction of PMQ field strength was also studied as a function both of the lattice period and the magnet length. Previous work by researchers at NRL [1] presented a comparison of PPM vs. PMQ for maximum transportable current as a function of lattice period. The detailed analysis of the magnet lattices that we have performed will allow us to develop a more accurate estimation of maximum current density transportable as a function of magnet dimensions. Figure 1. PMQ Field Profile from equation (1).

For verification of this analytic expression, we constructed PMQ Lattice the same model using the Ansoft code MAXWELL. The For this work, we have chosen lattice parameters MAXWELL model accounts for the complicated physics appropriate for transporting a beam in a 30 GHz coupled- of the permanent magnet interactions. The magnetic cavity travelling wave tube (TWT). The electron energy is material was chosen as , and the magnetic pole 16 keV and the mean beam radius is 0.5 mm. The lattice field was calculated as 0.9 T. The simulations agreed with parameters are chosen as indicated in Table 1. 𝟐𝟐𝟐𝟐 the analytic data within𝐒𝐒𝐒𝐒 5%𝐂𝐂𝐂𝐂 for a variety of magnet

978-1-4673-5977-1/13/$31.00 ©2013 IEEE parameters. The analytic model was used for the rest of where is well known function of . For this lattice, the this work. phase advance as calculated using equation (4) is 58.7 degrees,𝜃𝜃 and the phase advance derived𝜅𝜅 from the particle tracking based on the period fitting function is 64 degrees. This is in very good agreement.

Effects of Space Charge The maximum transportable current density can be found by adding space charge effects into the single particle tracking. Since the space charge term is a function of current density, we determine the maximum current density transportable for a given PMQ lattice by incrementally increasing the current density until the depressed phase advance, , goes to zero. The space charge effects were accounted for by adding the
 Figure 2. 16–piece quadrupole magnet model. space charge𝜎𝜎 term [4] to equations (2), (3). This gives us these two coupled equations: Single Particle Tracking
Single particle tracking is performed to ensure that the


∗
phase advance σ, is less than zero, this ensures the beam 𝑥𝑥[𝑧𝑧] + 𝜅𝜅
𝑥𝑥 𝑧𝑧 −



[
] = 0 (5)
∗

stays within the Mathieu stability conditions. [3] The is the

generalized perveance. where 𝑦𝑦[𝑧𝑧] + 𝜅𝜅
𝑦𝑦[𝑧𝑧] −




=Solving0, (5 ), (6) ( 6) allows equations of motion were solved with the complex us to calculate the depressed phase advance. When the
magnetic field profile of the quadrupole lattice from depressed𝐾𝐾 phase advance goes to zero degrees, the total equation (1) to analyze single particle tracking. These current density transportable for this lattice is 159 amps per equations were solved to give the motion of the particle in square centimeter. the XZ plane and the YZ plane: and Continuing Work



A detailed estimation of the maximum transportable current

w𝑥𝑥[here𝑧𝑧] − 𝜅𝜅 𝑥𝑥[𝑧𝑧] = 0 
𝑦𝑦[𝑧𝑧] − 𝜅𝜅 𝑦𝑦[𝑧𝑧] = 0 is (2), given (3 ) by density, using the above method, for various beam



∗
 
parameters and lattices will be presented to show the equation 1.
𝜅𝜅
=



 , 𝜅𝜅 = −𝜅𝜅𝑥𝑥, and
 advantages of PMQ focusing for TWT’s operating above These equations were solved with the standard differential 50 GHz. equation solver in Mathematica, and the periodic solution for the XZ plane is shown in Fig. 2. Acknowledgements The UNM authors wish to acknowledge funding received from TechFlow Scientific and the Air Force Research Laboratory. We also wish to thank John Petillo of SAIC and Rami Kishek of The University of Maryland.

References 1. 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. 2. Halbach, K., “Physical and Optical Properties of Rare Earth Cobalt Magnets,” Nucl. Instrum. Methods, vol.

169, pp. 109-117, 1981. Figure 2. The XZ particle trajectory, green, and a 3. Humphries, S., Principle of Charged Particle period fitting function, red. Acceleration, John Wiley and Sons, 1999. The period of the particle trajectory from Fig. 2 is used to 4. Lawson, J. D., The Physics of Charged Particle determine the phase advance per lens. This is contrasted Beams, 2d ed., Oxford University Press, 1988. with the phase advance which can be calculated 5. Reiser, M., Theory and Design of Charged Particle analytically from equation 3.34 in [5]: Beams, Wiley-VCH Verlag, Weinheim, Germany, 2008.

𝜎𝜎 = 𝐶𝐶𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶𝐶𝐶 𝜃𝜃 𝐶𝐶𝐶𝐶𝐶𝐶ℎ 𝜃𝜃 +

𝜃𝜃 𝐶𝐶𝐶𝐶𝐶𝐶 𝜃𝜃 𝑆𝑆𝑆𝑆𝑆𝑆ℎ 𝜃𝜃 − 𝑆𝑆𝑆𝑆𝑆𝑆 𝜃𝜃 𝐶𝐶𝐶𝐶𝐶𝐶ℎ 𝜃𝜃 (4)