Multiparameter optimization of an ERL injector R. Hajima a, R. Nagai a a Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki 319 1195 Japan Abstract We present multiparameter optimization of an ERL injector. Two approaches, knowledge-based optimization with down-hill simplex and all-at-once optimization with simulated annealing are used. Both approaches show sufficiently good results. Key words: optimization, energy recovery linac, injector PACS: 29.27.Bd, 41.85.Ja 1 Introduction An ERL injector consists of an electron gun, a drive laser, buncher and booster cavities, solenoid magnets, quadrupole magnets and dipole magnets, etc. In order to obtain a high-brightness electron beam, these components should be operated with appropriate parameters. However, finding an optimum set of parameters for an ERL injector is rather difficult because of a large number of parameters. Corresponding author. e-mail: hajima@popsvr.tokai.jaeri.go.jp Preprint submitted to Elsevier Science 26 September 2005
We apply two kind of mathematical approaches to the optimization of an ERL injector, which has multiple design parameters. One is a knowledge-based approach, where the parameters are optimized from the upstream to the downstream step-by-step with a help of apriori knowledge of the beam dynamics such as bunch compression and emittance compensation. The other is an allat-once approach, in which all the parameters are optimized simultaneously without any apriori knowledge. 2 Optimization procedure A beam dynamics code with a capability of simulating the space charge force is required for designing an ERL injector. We employ the simulation code PARMELA[1] for this purpose. PARMELA is one of the standard simulation codes for electron accelerators and has many users in the world. However, a routine for parameter optimization is not included in PARMELA. This is because the particle tracking was a rather difficult task for computers in 1980s, when PARMELA was developed, and it was almost impossible to make simulations iteratively for optimization. Owing to the great improvement of computer performance for these twenty years, now we can run PARMELA simulations iteratively even on a notebook PC. In the following work, we treat PARMELA as a black box and all the optimization routines are integrated in a wrapper script and a post-processor. The wrapper script written in Perl manages the optimization processes. The wrapper script interprets a master input file and generates an input file for each PARMELA run. In the master file, optimization commands are embedded as comment sentences of a standard PARMELA input file, which specify an objective func- 2
tion and parameters to vary for the optimization. After each PARMELA run, a post-processor extracts simulation results from a binary output file. The post-processor written in Java originated from another simulation code [2]. The wrapper script receives the simulation results from the post-processor and generates another input file for the next run. This procedure is repeated till the objective function is minimized. 3 Example of an ERL injector We carried out optimization of a 5-MeV ERL injector as shown in fig.1. The injector is driven at 1.3 GHz and consists of a 500-kV DC-gun, a single-cell buncher, a booster cavity (3-cell x 5), two solenoid magnets, five quadrupole magnets and a 3-dipole merger. The bending radius of the dipole magnets is 1 m, the bending angle is 15 degrees for the first and third magnets and 22 degrees for the second magnet, and the drift between magnets is 0.316 m. The second magnet has edge angle rotation of 20 degrees. In the knowledge-based optimization, the parameters of the injector components are determined from the upstream to the downstream. Firstly, two solenoid magnets are tuned to obtain a small emittance at the booster entrance, that is emittance compensation. Secondly, the amplitude and phase of the buncher and the booster are optimized so that we have appropriate bunch length and small energy spread before the merger, that is bunch compression. Thirdly, the strength of the quadrupole magnets is varied to minimize the transverse emittance after the merger, where the emittance growth depends on a beam envelope in the merger[3]. These procedure can be repeated for a better solution. We employ the down-hill simplex method[4] in this procedure 3
to find a global optimum of the objective function without falling into local optima. In this optimization, the drive laser parameters and position of each component are fixed. Figure 2 shows the result of optimization of transverse focusing parameters that is one of the steps in the total optimization procedure. Seven parameters (2 solenoids and 5 quadrupoles) are varied to minimize the sum of the transverse emittances, ε x + ε y, after accelerating to 72 MeV by main linac. The bunch parameters are chosen as follows: the effective cathode temperature is 35 mev, the laser spot radius on the cathode is 0.5 mm, the bunch length at the cathode is 14 ps (rms, Gaussian), and the bunch charge is 7.7 pc. The calculated thermal emittance at the cathode is 0.065 mm-mrad. The buncher and booster are tuned so that the bunch is 3 ps (rms) long at 72 MeV. The optimization process converges within 50-80 iterations as shown in fig.2. The best result after 5 sequences of optimization with a different seed of random numbers is ε x = 0.11 mm-mrad and ε y = 0.095 mm-mrad. From similar optimization for a 77 pc bunch, we obtain ε x = 1.2 mm-mrad and ε y = 0.99 mmmrad. In these optimization, an initial set of the focusing parameters is determined from a matched envelope in the merger, which gives the minimum emittance growth in the merger[3]. This matched envelope is not consistent with a condition minimizing total emittance growth from the cathode to the main linac because an electron bunch suffers the emittance growth immediately after the bunch generation at the cathode. However, the matched envelope obtained from the merger calculation is a reasonable condition to start the optimization of transverse focusing from the gun to the main linac. 4
For multiparameter optimization in accelerator applications, various algorithms have been employed including simulated annealing (SA)[5], genetic algorithm (GA)[6], etc. Recently, optimization of an ERL injector using GA was reported[7]. In the present work, we utilize SA for the all-at-once optimization of an ERL injector, in which 20 parameters are varied to achieve an optimum emittance and bunch length at the main linac. Although SA can be applied to multiobjective optimization[8], we use a single objective function, which is a weighted sum of multiple objective functions. Figure 3 shows optimization results obtained by SA, where the transverse emittance, bunch length and energy spread are plotted from the gun to the main linac exit (72 MeV). The target values of emittance and bunch length, 0.1 mm-mrad and 3 ps, are attained successfully after 9000 iterations. The energy spread grows rapidly in the main linac because we have no constraint on the energy spread in this optimization. The energy spread can be made sufficiently small by tuning the phase of the main linac manually. 4 Summary Two approaches, knowledge-based optimization by down-hill simplex method and all-at-once optimization by simulated annealing, showed sufficiently good results. After the optimization of a 5-MeV injector, normalized emittances of 0.1 mm-mrad for a 7.7 pc bunch and 1 mm-mrad for a 77 pc bunch were achieved. These results will be further improved by optimization involving parameters fixed in the present study: drive laser profile, position of beam transport elements, etc. 5
Acknowledgment The authors acknowledge JPPS-15360507 for financial support. References [1] J. H. Billen, Los Alamos Accelerator Code Group, LA-UR-96-1835 2002, version 3.28. [2] R. Hajima, Proc. of the 1998 International Computational Accelerator Physics Conference, SLAC-R-580, C980914 (1998). [3] R. Hajima, in these Proceedings. [4] W.H. Press et al., Numerical Recipes in C second edition, Cambridge University Press (1992), p.408. [5] M.S. Curtin et al., Nucl. Instr. Meth. A272 (1988) 187; R. Baartman et al., Proc. PAC-1997, 2778. [6] R. Hajima et al., Nucl. Instr. Meth. A318 (1992) 822; D. Schirmer et al., Proc. PAC-1995, 1879. [7] I.V. Bazarov and C.K. Sinclair, Phys. Rev. ST Accel. Beams 8 (2005) 034202. [8] P. Czyzak and A. Jaszkiewicz, J. Multi-Criteria Decision Analysis, 7 (1989) 34. 6
gun buncher solenoid Quad. x 5 booster (3-cell x 5) 3-dipole merger Fig. 1. A 5-MeV ERL injector. 7
2 ε x + ε y (mm-mrad) 1.5 1 0.5 0 0 10 20 30 40 50 60 70 80 90 iterlation Fig. 2. Optimization of transverse focusing parameters (2 solenoid magnets and 5 quadrupole magnets). Results for 5 sequences with a different seed of random numbers are plotted. 8
ε x, ε y (mm-mrad) 0.5 0.4 0.3 0.2 0.1 0 ε x ε y 0 2 4 6 8 10 12 14 16 s (m) σ t (ps) 15 12 9 6 3 0 0 2 4 6 8 10 12 14 16 0 300 200 100 σ E (kev) s (m) Fig. 3. Results of optimization by SA. Transverse emittance, bunch length and energy spread are plotted from the gun to the main linac exit (72 MeV). 9