FEL2013 - List of Authors (Ellison, J.A.)

Paper	Title	Page
MOPSO06	Paraxial Approximation in CSR Modeling Using the Discontinuous Galerkin Method	32
	D. A. Bizzozero, J.A. Ellison, K.A. Heinemann, S.R. Lau UNM, Albuquerque, New Mexico, USA
	Funding: This work was primarily supported by DOE under DE-FG-99ER41104. The work of DB and SL was partially supported by NSF grant PHY 0855678 to the University of New Mexico. We continue our study* of CSR from a bunch moving on an arbitrary curved trajectory. In that study we developed an accurate 2D CSR Vlasov-Maxwell code (VM3@A) and applied it to a four dipole chicane bunch compressor. Our starting point now is the well-established paraxial approximation** with boundary conditions for a perfectly conducting vacuum chamber with uniform cross-section. This is considerably different from our previous approach* where we calculated the fields from an integral over history, using parallel plate boundary conditions. In this study, we present a Discontinuous Galerkin (DG) method for the paraxial approximation equations. Our basic tool is a MATLAB DG code on a GPU using MATLAB's gpuArray; the code was developed by one of us (DB). We discuss our results in the context of previous work and outline future applications for DG, including a Vlasov-Maxwell study. * See PRST-AB 12 080704 (2009) and Proceedings from ICAP2012 TUSDC2. ** See PRST-AB 7 054403 (2004), PRST-AB 12 104401 (2009) and Jpn. J. Appl. Phys. 51 016401 (2012).

MOPSO31	Quasiperiodic Method of Averaging Applied to Planar Undulator Motion Excited by a Fixed Traveling Wave	762
	K.A. Heinemann, J.A. Ellison UNM, Albuquerque, New Mexico, USA M. Vogt DESY, Hamburg, Germany
	Funding: The work of JAE and KH was supported by DOE under DE-FG-99ER41104. The work of MV was supported by DESY. We present a mathematical analysis of planar motion of energetic electrons moving through a planar dipole undulator, excited by a fixed planar polarized plane wave Maxwell field in the X-Ray FEL regime.* We study the associated 6D Lorentz system as the wavelength of the traveling wave varies. The 6D system is reduced, without approximation, to a 2D system (for a scaled energy deviation and generalized ponderomotive phase) in a form for a rigorous asymptotic analysis using the Method of Averaging (MoA), a long time perturbation theory. As the wavelength varies the system passes through resonant and nonresonant (NR) zones and we develop NR and near-to-resonant (NtoR) normal form approximations. For a special initial condition, on resonance, we obtain the well-known FEL pendulum system. We prove NR and NtoR first-order averaging theorems, in a novel way, which give optimal error bounds for the approximations. The NR case is an example of quasiperiodic averaging where the small divisor problem enters in the simplest possible way. To our knowledge the analysis has not been done with the generality here nor has the standard FEL pendulum system been derived with error bounds. * J.A. Ellison, K. Heinemann, M. Vogt, M. Gooden: arXiv:1303.5797 [physics.acc-ph]

Paper

Title

Page

Paraxial Approximation in CSR Modeling Using the Discontinuous Galerkin Method

32

D. A. Bizzozero, J.A. Ellison, K.A. Heinemann, S.R. Lau
UNM, Albuquerque, New Mexico, USA

Funding: This work was primarily supported by DOE under DE-FG-99ER41104. The work of DB and SL was partially supported by NSF grant PHY 0855678 to the University of New Mexico.
We continue our study* of CSR from a bunch moving on an arbitrary curved trajectory. In that study we developed an accurate 2D CSR Vlasov-Maxwell code (VM3@A) and applied it to a four dipole chicane bunch compressor. Our starting point now is the well-established paraxial approximation** with boundary conditions for a perfectly conducting vacuum chamber with uniform cross-section. This is considerably different from our previous approach* where we calculated the fields from an integral over history, using parallel plate boundary conditions. In this study, we present a Discontinuous Galerkin (DG) method for the paraxial approximation equations. Our basic tool is a MATLAB DG code on a GPU using MATLAB's gpuArray; the code was developed by one of us (DB). We discuss our results in the context of previous work and outline future applications for DG, including a Vlasov-Maxwell study.
* See PRST-AB 12 080704 (2009) and Proceedings from ICAP2012 TUSDC2.
** See PRST-AB 7 054403 (2004), PRST-AB 12 104401 (2009) and Jpn. J. Appl. Phys. 51 016401 (2012).

MOPSO31

Quasiperiodic Method of Averaging Applied to Planar Undulator Motion Excited by a Fixed Traveling Wave

762

K.A. Heinemann, J.A. Ellison
UNM, Albuquerque, New Mexico, USA
M. Vogt
DESY, Hamburg, Germany

Funding: The work of JAE and KH was supported by DOE under DE-FG-99ER41104. The work of MV was supported by DESY.
We present a mathematical analysis of planar motion of energetic electrons moving through a planar dipole undulator, excited by a fixed planar polarized plane wave Maxwell field in the X-Ray FEL regime.* We study the associated 6D Lorentz system as the wavelength of the traveling wave varies. The 6D system is reduced, without approximation, to a 2D system (for a scaled energy deviation and generalized ponderomotive phase) in a form for a rigorous asymptotic analysis using the Method of Averaging (MoA), a long time perturbation theory. As the wavelength varies the system passes through resonant and nonresonant (NR) zones and we develop NR and near-to-resonant (NtoR) normal form approximations. For a special initial condition, on resonance, we obtain the well-known FEL pendulum system. We prove NR and NtoR first-order averaging theorems, in a novel way, which give optimal error bounds for the approximations. The NR case is an example of quasiperiodic averaging where the small divisor problem enters in the simplest possible way. To our knowledge the analysis has not been done with the generality here nor has the standard FEL pendulum system been derived with error bounds.
* J.A. Ellison, K. Heinemann, M. Vogt, M. Gooden: arXiv:1303.5797 [physics.acc-ph]