Simulation codes for high brightness electron beam experiments Luca Giannessi ENEA, C.R. Frascati

Outline Effects of macroparticles based models Particle In Cells limits Injector –PIC codes –“Quasi static” codes –Point to point –Homdyn Compressor –Point to point relativistic codes –Line charge method FEL –SVEA Approximation –Lorentz equation in FEL’s –Quiet start –Harmonics –Time dependent simulations Conclusions

Maxwell’s equations & Lorentz force INJECTIONFEL SASE TRANSPORT COMPRESSION SELF CONSISTENT DYNAMICS General multibody problem in classical electrodynamics with  bodies !

Only a reduced number of particles can be included in a simulation N=10 6 is already a very large number to be handled numerically In Fourier Space Charge density Example: We assume a normal distribution for x j …. in the limit N  we have i.e. the FT of a Gaussian with rms = 

Noisy region EFFECT OF A REDUCED NUMBER OF PARTICLES ON THE ELECTRON DISTRIBUTION SPECTRUM The effects of noise must be suppressed in the bandwidth of self consistency Noise in the distribution Noise in the fields Self consistency

Specific solution for different problems 10 5 particles “Real” beam particles FEL GUN COMPRESSOR 10 2 particles

PDE METHODS Given charges and currents distributions Calculate new charge & current distributions with the known fields solve Maxwell’s Equations with a PDE Solver (e.g. LeapFrog)

The fields are evaluated on a mesh with spacing (  x,  y,  z ) Accuracy –The highest k vector supported by the mesh is given by –The highest frequency is given by Higher frequencies are “seen” as low frequency components (noise) Stability (Courant condition) Conditions

TIME / SPACE decoherence A LeapFrog algorithm induces decoherence: Magnetic and Electric Fields are not known in the same place at the same time n-1nn+1 m-1 m m+1 B n,m E n+1/2,m+1/2 x y At large  the Lorentz force equation is sensitive to relative errors between E and B components. Cancellation  1/  2 is not correctly reproduced. The requirements on the mesh becomes more strict at large 

Practical cases INJECTORCOMPRESSORFEL Transverse mesh size 10 cm1 cm1 mm Slippage length3 cm10 cm 1  m Cut off wavelength 100  m10  m 1Å1Å # of Mesh vertices (3D) # of Mesh vertices (2D) Integration time16 ps60 ps330 ps Maximum step length 0.3 ps0.03 ps s # of time steps

Specialize the problem INJECTION Space charge FEL Radiation COMPRESSION Space charge/radiation LOW FREQUENCY LOW ENERGY WIDE BAND LOW/MID FREQUENCY HIGH ENERGY WIDE BAND HIGH FREQUENCY HIGH ENERGY NARROW BAND NARROW k

INJECTOR External forces – RF – Solenoids Collective Self Fields Boundary conditions (cathode – walls) High gradient Short pulse  t <<T RF The beam becomes relativistic in the first half cell Transition from laminar to emittance dominaded motion typical  =1/I=100 A/ RF  10 cm/  th  0.3 mm-mrad the simulation length is  4 m (more with vel. bunching) “High Resolution” required to recover  th after acceleration RF-GUN LINAC SOLENOID

INJECTOR 2D PIC Codes: ITACA – Spiffe – etc. Point to point/Point to grid (retarded times): TREDI, Atrap Relax on boundary conditions which are“easily” treated only for flat, perfectly conducting walls (cathode). RF  bunch,  1 Point to point/Point to grid (quasi static): Parmela – GPT – ASTRA etc. + relax on velocity spread  /  <<1 no accel. fields Semi – analitic: HOMDYN + relax: Paraxial approximation  ’<<1 Small slice energy spread   <<1 Uniform bunch distribution & no wave breaking

Maxwell Eq. in 2D (Itaca, Spiffe …) Scalar potential formalism in a 2D axi-symmetric environment A closed sub-set of Maxwell equations, where the field propagation (driven by the source [ ,J] ) can be fully described in terms of a scalar pseudo-potential Wave equation for  calculate E r,E z,H  from  Scalar wave equation + ODE for electric field

SELF FIELDS & Retarded Potentials R(t’) Target Source Trajectories are stored and the fields evaluation requires bracketing of the retarded condition. Cpu – memory consuming Non trivial field reglarization by grid assignement (Parmela mode) “extended particles” requires careful treatment of ret. condition (TREDI, Trafic4, R.Li program for CSR)

Evaluation of fields in “Quasi Static” approx. (Parmela, GPT …) A space-charge mesh centered on a reference particle moves at the  of the particle. Particle coordinates and momenta of are transformed to the frame of this mesh and are assumed at rest. We have indeed The charge is assigned to mesh cells. The electrostatic field at the particles coordinates are given by the sum of Green functions of a charged ring (2D scheff) for each mesh node Apply kicks The coordinates and momenta of the new particle coordinates are transformed back to the lab frame.

Suppression of noise Frequency cut-off with Filter functon Unfiltered Filtered In configuration space where with  x 1  k A charge distribution extended in space reduces high frequency components 3Dim: Gaussian charges (Tredi) – Uniform charges (GPT) 2Dim: ITACA: Rician distribution - Uniform Rings: Parmela – GPT (& CSR codes: 2D Gaussian in R. Li CSR code – “pencil” Gaussian in Trafic4)

HOMDYN (M. Ferrario) multi-envelope model: time dependent space charge of a uniform charged bunch Paraxial approximation  ’<<1 Small energy spread   <<1 Uniform bunch distribution, no wave breaking

Sensitivity to number of  beam s 2D simulation 40 – 300  beam 10 meters on a 2GHz P-IV - execution in << 1’ LNF-00 / 004 (P) M. Ferrario, J. E. Clendenin, D. T. Palmer, J. B. Rosenzweig, L. Serafini “Homdyn Study for the LCLS RF Photo-Injector” - 3 marzo 2000 Courtesy of M. Ferrario

Sparc Injector – Parmela/Homdyn Final emittance: Parmela 0.38 mm-mrad – Homdyn 0.28 mm-mrad (courtesy of M. Ferrario - C. Ronsivalle) SIMULATION 1 nC, 10 psec, r=1 mm, Eth=0 1.6 cell rf-gun. 140 MeV/m 1.5 m drift 2 TW linac 25 MeV/m PARMELA: Np=10000,step=0.1° (GUN)step=1° (elsewhere). Mesh Nr=20,Nz=400. Uniform distribution

Comparison Itaca – Parmela - Homdyne sigr (mm) emitnx (mm.mrad) Z (mm) ITACA PARMELA HOMDYNE Courtesy of L. Serafini

SPARC run with TREDI/Homdyn Homdyn Tredi Tredi z m +2cm Tredi z m +1cm WG1: M. Quattromini – Today 11:00

COMPRESSION R  bb WIDE BAND FEEDBACK Review of S. Anderson Complicate structure of phase space  Sensitive to number of macroparticles Longitudinal & transverse nonlinear forces. Low energy – Space charge High energy - Radiation HIGH FREQUENCY microbunching amplification & uncorrelated Emittance Growth LOW FREQUENCY Correlated energy spread & emittance growth

CSR Simulation Point to point/point to mesh: TRAFIC4, TREDI, R.Li Code … Cartesian geometry in 2D or 3D to correctly evaluate ret. times Extended charges to reduce the noise Effect of shielding included via image charge (TRAFIC4) Line charge method: Elegant, CSR_CALC (Emma), M. Dohlus 1D prog… CSR via line charge treatment (Saldin et. Al. - Extended by Stupakov, Emma EPAC 2002 to include the post-dipole region.) In a dipole: where (s) is the charge density vs long. coordinate in the bunch

Line charge method: Elegant (Borland), CSR_CALC (Emma), M. Dohlus 1D prog. … Neglect transverse beam size No transverse internal forces Neglect Coulomb repulsion Before compression After compression Assume 1D beam for CSR long. field calculation (projected beam) Neglect deformation of the line charge distribution due to ret. evaluation of fields (overestimation of longitudinal fields / underestimation of transverse fields) FAST! particles – < 1 hour (CSR CALC - EMMA) SIMULATION of MICROBUNCHING

ICFA Beam Dynamics mini workshop Coherent Synchrotron Radiation and its impact on the dynamics of high brightness electron beams January 14-18, 2002 at DESY-Zeuthen (Berlin, GERMANY) 3D EE EE  TRAFIC TREDI DProgram by R.LI DElegant CSR_CALC (P. Emma) Program by M. Dohlus

FEL Narrow bandwidth resonant process Paraxial approximation (k=k z ) Slowly Varying Envelope Approximation The slowly varing field equation may be solved with a PDE solver (as in TDA3D – GENESIS – GINGER …) Or by projection on Gauss or Laguerre modes: SDE* expansion (as e.g. in MEDUSA) * P.Sprangle, A. Ting, C.M. Tang Phys. Rev A 36 (1987)

Tapering/magnetic errors have “low frequencies” in z Scale length of e-beam variations is large, e.g. large betatron wavelength Small wiggle amplitude Lorentz force equation KMR assumption (FRED, GINGER, PROMETEO): “external parameters” change adiabatically Many time steps per period required for integration of particles dynamics (MEDUSA)  Pendulum equation (TDA3D - GENESIS)

Quiet Start BANDWIDTH SHOT NOISE: 1 nC -  t=200 fs sn  0.01 SHOT NOISE SIMULATED BY

Noise at saturation Nyquist theorem: we need at least 2 equally spaced electrons per wavelength to suppress the relative frequency component. In principle 2*h to simulate the n th hrmonic. FEL dynamics “moves” particles – bunching brakes the symmetry and many more particles are needed to simulate correctly the evolution at saturation The number of particles depends on the local energy spread, the emittances and on the number of space dimensions for the simulation. Typically 10 3 particles per wavelength for the fundamental harmonic. More particles for higher order harmonics

FEL Codes Comparison S.G. Biedron et al. NIM A445 (2000) p.110 MEDUSA appears more sensitive to UM gaps LEUTL parameters

Harmonics Emission on high order harmonics represents a consistent resource to extend the wavelength tunability of a free electron laser. S.G. Biedron et al. NIM A p.101 MEDUSA PROMETEO GINGER S.G. Biedron et al. NIM A p.94

Harmonics and electron dynamics Sparc 13.5 nm K=4.8  = with Perseo:

Time dependent simulations The beam is sliced longitudinally Each slice has input and output fields Slippage is applied at discrete time steps, by translating electrons over radiation (GINGER) Slices simulation can be done “almost” independently starting from the bunch tail and passing fields up to the bunch head (GENESIS) Easily parallelized (large grain problem) “Quasi” independent simulations Slippage length  Slippage as a function of z:

Time dependent simulation with Perseo (1D – in MathCad) Slice length 0 Slices 1524 = 2/  Slippage applied every period zend = N 0

Start to end simulation (Sparc phase space) Projected beam at The UM entrance (Sparc)

Start to end II SLICE TWISS PARAMETERS

SASE simulation “DESIDERATA” Startup from shot noise Gain and exponential growth Transverse effects: emittances detuning - gain focusing Longitudinal effects: pulse propagation – spectrum Saturation Power Linewidth Harmonics (power & linewidth) Sensitivity to input parameters “Realistic” input electron beam phase space “Realistic” beam dynamics in the undulator (wakes, ISR) Reasonable integration time We are close !

Conclusions We limited the analysis to Injector - Compression - FEL families of codes In these three families we found –“First principles” codes including most of the physical aspects of the problem. This completness is usually paid in terms of CPU time. –Semi-analytic codes based on some “smart” theory, which hiding some physics allows fast relaxation of the usually large number of parameters involved. These are not necessarily less physically insightful. Both classes of codes are required. The modellization aspects have been privileged, but several other features of the codes have great importance –Interface (SDDS – HDF5 – XML … other) –“Ease of use” –LAST BUT NOT LEAST: availability & sources availability (peer review)

S. Biedron, J. Billen, R. Bartolini, M. Borland, G. Dattoli, M. Dohlus, P. Emma, B. Faatz, W. B. Fawley, M. Ferrario, H. P. Freund, A. Kabel, J.W. Lewellen, R.Li, T. Limberg, L. Mezi, S. Milton, M. Quattromini, S. Reiche, C. Ronsivalle, L. Serafini Thanks

TIME / SPACE Decoherence II Emittance growth produced by unphysical head tail correlation (Gaussian bunch with  t =  z /10  = 4 - courtesy of L. Serafini)

Courtesy of M. Dohlus, Presented at ICFA Workshop Jan.2002 CSR effect on longitudinal Phase space