Pulsar Physics

Nucl. Instr. and Meth. in Phys. Res. B, Vol. 139, (1997) pp. 481

The General Particle Tracer code applied to the Fusion Free-Electron Maser

M.J. de Loos, S.B. van der Geer, Pulsar Physics
C.A.J. van der Geer, A.G.A. Verhoeven, W.H. Urbanus, FOM Institute for Plasma Physics "Rijnhuizen", Association EURATOM-FOM

Abstract

The fusion Free-Electron Maser (FEM) is the prototype of a high power, electrostatic mm-wave source, tunable in the range 130-260 GHz [1]. In order to achieve a high overall efficiency, the charge and energy of the spent electron beam, i.e. the beam which leaves the undulator after interaction with the EM-wave, has to be recovered. A 50% overall efficiency is achieved, even for the maximum energy spread of 320 keV generated in the undulator, using a collection system consisting of a decelerator and a depressed collector.
The General Particle Tracer code (GPT) [2] is being used as the major design tool for the whole Fusion FEM beam line, from the accelerator to the depressed collector. The high accuracy, ability to include FEL interaction and full 3D treatment make GPT the ideal choice for such a project. An overview of the separate sections and the use of GPT for each part of the FEM is presented. GPT is currently being applied to the design of the energy recovery system of the Fusion FEM. The first simulation results, including a 3D off-axis bending scheme and scattered incident electrons, are shown.
PACS: 07.05.Tp; 41.75.Ht; 41.60.Cr; 29.17

1 INTRODUCTION

The Fusion FEM has been operated without collection system at energies of 1.35 to 1.7 MeV and initial experiments have shown that low-loss beam transport is possible. Current losses below 0.02% for a 3 A electron beam have been achieved. The beam line has been shown to accept an energy variation of 5% with fixed beam optics. This would correspond to a rapid tunability of the microwave frequency of 10% [3].
The design of the FEM beam line up to the undulator has been performed mainly using the 3D General Particle Tracer code. The simulations included FEL interaction, space-charge effects, imported electro-magnetic field data and misalignments. The final set-up will also include a beam and energy recovery system using a decelerator and a depressed collector. The large energy spread after the decelerator, the investigation of an off-axis bending scheme and the required back streaming of less then 10 mA complicate this part of the design.
The rigorous GPT approach, full 3D particle tracking, will be used again for these simulations. The GPT kernel has been adapted to model scattering of incident electrons on the collector plates. While briefly discussing the simulation methods used in the design of the first part of the FEM, the design difficulties and first simulation results of the final set-up are presented.

2 GPT

Particle tracking is a very powerful tool in the design of accelerators and beam lines. The General Particle Tracer (GPT) code provides a new 3D simulation package to study relativistic charged particle dynamics in electromagnetic fields [2]. Because of its modern implementation, GPT can be conveniently customized without compromising its ease of use, accuracy or simulation speed. Due to the general character of the code and its flexibility, GPT is suited for a large number of purposes.
GPT is based on an embedded fifth-order Runge-Kutta driver with adaptive stepsize control [4], ensuring the highest simulation speed in a set-up with varying field gradients. A change in step-size of over a factor of 100 has been observed, greatly enhancing the overall efficiency without affecting the accuracy.
All beam line components can be modeled 3D and misalignments can be conveniently simulated using the built-in capability to reorient and move all elements arbitrary in 3D space. GPT runs on all platforms and a fully integrated Windows 95/NT graphical user interface is available.
GPT tracks macro particles through electromagnetic fields and offers complete freedom in the initial particle distribution. Two different space-charge models have been used for the FEM simulations. A 2D relativistic point-to-ray interaction model and a 3D relativistic point-to-point model. The 3D version makes no assumptions about particle distributions, and is therefore well suited for all design problems.
The 2D model interprets particles as moving rays with a homogeneous line charge density. It requires less particles than the 3D version for the same accuracy, but can obviously not be used for the simulation of 3D effects. This model is ideal for the simulation of cylindrical symmetric transport lines.
Undulators are simulated by importing the actual 3D magnetostatic fields into GPT. The radiation is modeled as a superposition of the excited modes. The particles are tracked through the combined magnetic fields of the undulator and the electromagnetic radiation pattern. Simultaneously the energy transfer from the particles to the various radiation modes, and vice-versa, is solved as additional differential equations to ensure self-consistent results.
The GPT code has been adapted to model scattered particles and the generation of secondaries. This has been used for the design of the FEM depressed collector.

3 FUSION FEM

Figure 1: Schematic of the Fusion FEM.

The Fusion-FEM is the prototype of a high power mm-wave source, rapid-tunable in the range 130-260 GHz [1], see Figure 1. The device is driven by a 2 MeV, 12 A dc electron beam and is designed to generate 1 MW microwave power. In the future FEM's may be used as power sources for electron cyclotron applications on magnetically confined plasmas in fusion research devices, such as ITER. The principal design parameters are presented in Table 1.
Presently the electron beam line consists of an 80 keV thermionic gun, a 2 MV dc electrostatic accelerator, an optical cavity around the wave guide in a step tapered undulator and a beam dump. In this set-up experiments are carried out for optimum beam transport and the first generation of microwaves at short pulses. In a later stage the beam dump will be replaced by a beam and energy recovery system, which is presently being designed. The design goal is a system efficiency of over 50%.
The energy recovery system of the Fusion FEM will consist of a decelerator and a depressed collector. The calculated energy distribution of the beam coming out of the undulator is such that a depressed collector with three electrodes will suffice to obtain the required efficiency.
In the depressed collector the electrons are collected on the backside of the electrodes. This assures that "true" secondary electrons fall back on the electrodes. Scattered electrons, however, have an energy close to the initial energy and can cause back streaming via multiple scattering. As all beam loss and back streaming current has to be delivered by the 2 MV power supply which can deliver only 20 mA, at least 99.8 % of the beam needs to be collected. An off axis-bending scheme is currently being investigated using the GPT code.

Table 1: Principal design parameters of the Fusion FEM

4 DESIGN UP TO THE UNDULATOR

Thus far the GPT code has been used successfully for the design of the beam line in the present set-up. This part of the beam line is optimized to produce a beam with a large radius, while not hitting any boundary. To avoid beam blow up in the long accelerator section, quite a large beam radius is required in front of it. This in turn requires a large solenoid to keep the filling factor small and thus to avoid emittance growth.
The field profiles of the accelerator and the subsequent iron-core lenses are calculated using the finite element package TOSCA. An analytical expression is fitted through the obtained electromagnetic fields and imported into GPT. An alternative would be to use a cylindrical symmetric 2D electrostatic field-map.
The simulated electron beam dynamics have been found to be in very good agreement with the experimental results. Table 2 lists a comparison between the calculated and actual settings, resulting in a remarkable average discrepancy of the order of a few percent [3].

Table 2: Lens settings in the first part of the FEM beam line. The experimental settings gave optimal transmission through the undulator.

5 UNDULATOR

The FEL code CRMFEL [5] has been used for the gain calculations of the FEM step-tapered undulator. Because the beam needs to be fully recovered, knowledge of the output beam is essential for the design of the downstream beam line. Therefore the simulations have also been performed using GPT, calculating the generated radiation and the particle trajectories simultaneously.
Additional differential equations for the amplitude and phase of the various longitudinal and transverse modes in the waveguide of the FEM are solved while tracking the particles. This enables multi-mode, multi-frequency examination of the modes in the cavity and their influence on the particle trajectories. The method was used to optimize both beam transport and output power. The obtained results are in good agreement with the FEL code.
Figure 2 shows the energy distribution of the particles after the undulator. The broad energy range makes this case the most difficult to transport. The resulting particle beam parameters were used as a starting point for the design of the recovery system.

Figure 2: Worst case energy distribution at the exit of the undulator, for overdrive at 250 GHz. Zero on the horizontal scale represents no energy loss in the undulator.

6 DESIGN AFTER THE UNDULATOR

The second transport line, starting after the undulator, consists of four iron-core solenoids to guide the beam into the electrostatic decelerator. The lens system and decelerator are comparable to the first transport line and the accelerator.
The beam energy in the transport section between the decelerator exit and the collector entrance ranges from 55 keV to 375 keV. This high energy spread requires a transport system with a high energy acceptance.
The necessary guiding field is provided by four solenoid lenses. This magnetic field must be strong enough to guide particles with the lowest energy, while not being a magnetic mirror to particles with high transverse velocity. Therefore, iron-core lenses are not used because they provide a too localized field. An additional lens at the entrance of the collector, strongly eases the requirements for the settings of the other four lenses. Figure 3 shows the beam envelope in the transport line and decelerator for the minimal energy of 55 keV. Higher energies produce slightly different envelopes, always having a smaller maximum radius.

Figure 3: Beam envelope in the beam line after the undulator till the collector, for a fixed particle energy of 55 keV. The radius of the beam pipe is 40 mm, indicated are the lenses and the decelerator structure.

7 DEPRESSED COLLECTOR DESIGN

A beam transmission exceeding 99.9% has been demonstrated for the entire beam line [3], except for the energy recovery system, which has not yet been built. Thus to realize a high transmission for the entire beam line, the current back streaming from the collector must be less than 10 mA.
Basically the depressed collector is a shadow-type collector. Electrons are collected on the backside of electrodes, such as to keep secondary electrons on the collector surfaces. However, when electrons impinge on a (copper) surface, both secondary particles are generated (so called true secondaries) and primary particles scatter from the surface. The true secondaries have low energies, < 100 eV, and are forced back to the electrodes by the electrical fields of the collector. Scattered primaries can have energies as high as the energy of the primary particle, and are more difficult to keep inside the collector [6].
The beam is bent off-axis into the collector using a set of coils around the whole collector to further reduce back streaming, at a slight expense of collector efficiency. This scheme requires a 3D simulation code. Modifications to the GPT kernel were necessary to be able to model scattering within the collector accurately. The new kernel (version 2.40) uses 3D ray-tracing techniques to detect boundaries and to calculate scattered particle directions. Correct statistics are maintained by enlarging or reducing the number of elementary particles that a macro particle represents according to the scattering probability.
The electrostatic fields of the collector were calculated using the SUPERFISH set of codes and imported as a dense rectangular 2D electrostatic field map. Figure 4 shows the trajectories of the particles inside the depressed collector, both primary and scattered particles are shown.

Figure 4: Particle trajectories in the FEM depressed collector for a beam energy of 55-375 keV.

8 CONCLUSION

The design of the energy recovery system of the Fusion-FEM is a challenging project. Major difficulties to overcome are to transport the electron beam loss-free from the decelerator into the depressed collector and to prevent back streaming from the collector due to electrons scattering on the collector electrodes. The transport problem for the beam with the high energy spread due to the FEM interaction is solved by an array of five lenses, a compromise between the concept of a guiding field and a focusing system. For the collector, a 3D solution has been found with GPT which includes a beam bending system.
The high accuracy, the possibility to add custom code, 3D calculation method and the user friendly interface, all make GPT an attractive design tool for particle accelerators and beam lines. The possibility to add self-consistent differential equations makes GPT even a powerful tool for FEL's. GPT has been used very successfully for the design of the FEM. The current status of the GPT project can be found on the web at http://www.pulsar.nl.

9 ACKNOWLEDGEMENTS

This work was performed as part of the research program of the association agreement of EURATOM and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support of the "Nederlandse Organisatie voor Wetenschappelijk Onderzoek" (NWO) and EURATOM.

10 REFERENCES

[1] W.H. Urbanus, et al, Nucl. Instr. and Meth. A375, (1996) p. 401.
[2] M.J. de Loos, et al, Proc. 5th Eur. Part. Acc. Conf., Sitges, (1996) p. 1241.
[3] M. Valentini, et al, Nucl. Instr. and Meth. A390, (1997) p. 409.
[4] W.H. Press, et al, Numerical Recipes in C, Cambridge Univ. Press, 2nd edition, (1992) p. 714.
[5] P.J. Eecen, Thesis (1996).
[6] C.A.J. van der Geer, et al, Proc. FEL'97 Conf., Beijng (1997).