The broadband, 3D FEL code Puffin is presented. The analytical model is derived in absence of the Slowly Varying Envelope Approximation, and can model undulators of any polarisation. Due to the enhanced resolution, the memory and processing requirements are greater than equivalent unaveraged codes. The numerical code to solve the system of equations is therefore written for a parallel computing environment utilizing MPI. Some example simulations are presented

Campbell, L. T.

McNeil, B W. J.

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryCampbell, L. T. and McNeil, B W. J. (2013) Puffin : A three dimensional, unaveraged free electron laser simulation code. In: FEL 2012. JACoW, pp. 73-76. ISBN 9783954501236 , This version is available at http://strathprints.strath.ac.uk/57080/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  activities  or  any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any  correspondence  concerning  this  service  should  be  sent  to  Strathprints  administrator: strathprints@strath.ac.ukPUFFIN: A THREE DIMENSIONAL, UNAVERAGED FREE ELECTRONLASER SIMULATION CODEL.T. Campbell and B.W.J. McNeilSUPA, Department of Physics, University of Strathclyde, Glasgow, UKAbstractThe broadband, 3D FEL code Puffin is presented. Theanalytical model is derived in absence of the Slowly Vary-ing Envelope Approximation, and can model undulators ofany polarization. Due to the enhanced resolution, the mem-ory and processing requirements are greater than equivalentaveraged codes. The numerical code to solve the systemof equations is therefore written for a parallel computingenvironment utilizing MPI. An example simulation is pre-sented.INTRODUCTIONMost analytical and numerical models of the FEL usethe Slowly Varying Envelope Approximation [1] (SVEA)on the radiation field. This assumes a slow temporal andspatial evolution of the field envelope at the scale of theradiation wavelength, and allows an averaging of the fieldenvelope and some electron phase-space parameters over atleast one radiation period, removing the need to model anyfast oscillatory terms at the radiation frequency. Most of thecommonly used multi-dimensional FEL simulation codes(e.g. MEDUSSA [2], GINGER [3], GENESIS 1.3 [4] andFAST [5]), are based on averaged SVEA models. Whilethese have been used successfully to model basic FEL op-eration, and have been extensively benchmarked againstexperiment, SVEA means that resolution of sub-resonantwavelength scale processes are not possible. The particlesused to simulate the electrons are confined to localised re-gions of one radiation period within the electron pulse, sothat transport of particles over many radiation wavelengths,such as may occur as a result of the FEL interaction ore.g. a pre-imposed electron energy chirp, cannot be mod-elled easily. Furthermore, the correct simulation of electronshot-noise is only valid for a limited radiation wavelengthrange [6]. The minimum sampling period of the field en-velope imposed by SVEA also limits the range of radiationfrequencies able to be modelled without numerical alias-ing effects to ωr/2 < ω < 3ωr/2, where ωr is the radi-ation resonant frequency [7]. Thus, using SVEA, it is notpossible to model effects with a broader bandwidth usingthe same radiation field envelope. Such issues with SVEAconstrain the effective modelling of several more advancedFELmethods including designs to achieve shorter radiationwavelengths and pulse durations [8].This paper presents what the authors believe to be thefirst 3D unaveraged, broadband FEL computer simulationcode, named Puffin (Parallel Unaveraged Fel INtegrator).The primary aim of this code is to provide a flexible re-search resource that can be adapted to test new ideas andmethods for future FEL development. It is not intended, atleast initially, as a design tool for FEL facility development.The radiation field is modelled using the Finite Elementmethod [9] and the electrons by a distribution of chargeweighted macroparticles that can model the effects of elec-tron shot-noise across a broad frequency bandwith [6].Electrons are not confined to localised regions of the beamso that electron transport throughout the beam is correctlymodelled. The main approximations applied are the neglectof the backward (counterpropagating) radiation field andthe paraxial approximation. Furthermore, space-charge ef-fects in the electron beam are neglected.The resulting parallelised numerical code is able to sim-ulate both CSE and spontaneous emission arising fromelectron shot-noise. The advantages are an enhanced andbroadband resolution of the radiation including a self-consistent modelling of variable radiation polarisation. Thedisadvantages are the increased computer memory require-ments and processing time when compared to the averagednumerical models.In the following, the derivation of the final workingequations and subsequent numerical solution is outlinedand the operation of the code is demonstrated using dif-ferent FEL configurations.MATHEMATICAL MODELThe electromagnetic field is given by:E =1√2(eˆξ0ei(krz−ωrt) + c.c.), (1)where the vector basiseˆ =1√2(xˆ+ iyˆ) (2)and ξ0(x, y, z, t) = |ξ0(x, y, z, t)| exp(iψ(x, y, z, t)) is thecomplex field envelope, with kr and ωr, the field wavenum-ber and angular frequency respectively, of a resonant wave.It is assumed the field propagates in vacuum and thatthe paraxial approximation applies, so that ωr = ckr and∂/∂z ≈ ∂/c∂t.The undulator magnetic field is defined as:Bu =Bu2(ueikuz + c.c.)(3)where ku = 2π/λu is the undulator wavenumber, λu is theundulator wavelength and Bu is the peak magnetic fieldProceedings of FEL2012, Nara, Japan MOPD12FEL TheoryISBN 978-3-95450-123-673 Copyrightc ○2012bytherespectiveauthorsof the undulator and the (generally non-unit) vector basisis u = (uxxˆ + iuyyˆ). The undulator polarisation is thendetermined by ux and uy , which are the relative peak mag-netic undulator field in x or y.The FEL or Pierce parameter determines the strength ofthe radiation/electron coupling in the 1-Dimensional limitand is given by [10] ρ = (a¯uωp/4cku)2/3/γr. Here, e andm are the charge magnitude and rest-mass of an electron,c is the speed of light, ωp =√e2np/ǫ0m is the (non-relativistic) electron beam plasma frequency, np is the peakelectron number density, a¯u = eB¯u/mcku is the RMSundulator parameter and γr is the electron energy in unitsof the rest-mass energy corresponding to the resonant FELwavelength λr.Starting from the coupled Maxwell-Lorentz equations,using the definitions of electromagnetic and undula-tor fields and above, and defining dimensionless scaledvariables:-p2j =1− βzjηβzj, η =1− β¯zβ¯z,p¯⊥ =u√2mca¯up⊥, A⊥ =eua¯ulg2√2γ2rmc2ρE⊥,z¯ = 2kuρz, z¯2 = 2kuρβ¯z(ct− z)(1− β¯z),x¯, y¯ =x, y√lglc,where u2 = u · u∗, lc = 1/2ρkr, and lg = 1/2ρku, givesthe final set of working equations as solved numerically bythe code:[12( ∂2∂x¯2+∂2∂y¯2)− ∂2∂z¯∂z¯2]A⊥ =− 1n¯p∂∂z¯2N∑j=1p¯⊥jLjδ3(x¯j , y¯j , z¯2j)(4)dp¯⊥jdz¯=12ρ[iU∗ − ηp2jf2k¯2βA⊥j]− u√η2√2fk¯βρLj×(k¯2β(x¯j − iy¯j) +η(1 + ηp2j)(dx¯jdz¯− idy¯jdz¯)dp2jdz¯∣∣∣F)(5)dp2jdz¯=2ρu2ηL2j[ηp2j(A∗⊥j p¯⊥j + c.c.)− i(1 + ηp2j)f2k¯2β(Up¯⊥j − c.c.)]+dp2jdz¯∣∣∣F(6)dz¯2jdz¯= p2j (7)dx¯jdz¯=2√2fk¯βρu√ηLjℜ(p¯⊥j) (8)dy¯jdz¯= −2√2fk¯βρu√ηLjℑ(p¯⊥j). (9)Here, n¯p = lgl2cnp is the peak electron number densityat the start of the interaction in the scaling of (x¯, y¯, z¯2),U(z¯) = (ux cos(z¯/2ρ) + iuy sin(z¯/2ρ)), andLj ≡ γrβzjγj= γr√√√√√ηp2j(2 + ηp2j)1 + 2a¯2uu2|p¯⊥j |2(10)The equations (7..9) for the z¯-dependent electron co-ordinates (x¯j , y¯j , z¯2j) are simply derived from the scaledmomentum-energy relations.The scaled betatron wavenumber is given by k¯β =a¯u/2fργr, where the scaling is with respect to that of ‘nat-ural’ undulator focusing [11], when f =√2. For variablef the external focusing may be artificially strengthened orweakened. However, the undulator magnetic field (3) hasno transverse variation and so natural focussing is not in-cluded in the model. Thus the term:dp2jdz¯∣∣∣F= −k¯2β(1 + ηp2j)(x¯jdx¯jdz¯+ y¯jdy¯jdz¯)1 + η((dx¯jdz¯)2+(dy¯jdz¯)2) . (11)must be added in equations (5,6), to ensure the electronenergy remains constant over betatron period in the absenceof a radiation field.The field equation (4) is derived by neglecting the back-wards wave component of the radiation field [12]. Inthe 1D Compton limit, ignoring focusing and diffractionterms, equations (4-7) reduce to those of [6].In scaled units the Rayleigh range for a Gaussian beamof width of σxr is z¯R = krσ2xr/lg which, in the transversescaling above, may be written z¯R = σ¯2xr/2ρ. By definition,z¯R = πF where F is the Fresnel number for a gain length.For the case of a matched beam, so that the emittance ǫ =kβσ2xb, where σxb is the (constant) radius of the electronbeam and assuming the radiation beam radius is that of theelectron beam, the scaled Rayleigh range may be written:z¯R = ǫ¯/2k¯β where, the scaled electron beam emittanceǫ¯ ≡ 4πǫ/λr. Hence, the usual FEL emittance criterionbecomes: ǫ¯ = 2k¯β z¯R . 1All variables above have been scaled with respect to pa-rameters in the 1D limit. The work of [13] defines a set of3D scaled parameters: the FEL parameter ρ3D = B1/3ρ,where B = (2z¯R)3/2 is called as the ‘diffraction parame-ter’. The scaling can be related to that used here via ‘1D’Rayleigh range z¯R by B = (2z¯R)3/2. In a similar wayto the 1D scaling a 3D gain length was defined as l˜g =λu/4πρ3D so that the diffraction parameter B = 2zR/l˜g .MOPD12 Proceedings of FEL2012, Nara, JapanISBN 978-3-95450-123-674Copyrightc ○2012bytherespectiveauthorsFEL TheoryIf the working equations of (4..9) above are scaled withrespect to ρ3D, i.e. the substitution ρ = ρ3D/√2z¯R ismade into all dependent and independent variables andthese variables redefined so that z˜ = 2kuρ3Dz etc., thenall equations remain identical in form (i.e. with ‘bars’ re-placed by ‘tildes’ in the independent variables), with theexception of the wave equation (4) which is slightly modi-fied to become:[12(∂2∂x˜2+∂2∂y˜2)− ∂2∂z˜∂z˜2]A˜⊥ =− 1Bn˜p∂∂z˜2N∑j=1p˜⊥jLjδ3(x˜j , y˜j , z˜2j). (12)The 3D FEL parameter ρ3D is then dependent only uponthe longitudinal electron current, and the transverse be-haviour becomes explicit in the equations via theB param-eter. For a constant ρ3D, increasing the transverse radiusreduces the transverse electron density, increasing B andreducing the field source and generation.The numerical implementation in Puffin uses the equa-tions written in terms of the 1D scaling (4-9), mainly toallow easy comparison with 1D simulations and analysis.As has been shown, solutions in the 3D scaling of [13] aresimply obtained using the relations discussed above.NUMERICAL SOLUTIONThe parallel numerical algorithm used for the solu-tion of the working equations (4..9) has been developedfrom [14, 15, 16]. The coupled equations are advancedin z¯ by using the split-step Fourier method [17], in whichthe radiation field is described by the method of finite ele-ments [9] and the electrons are simulated by a distributionof charge weighted macroparticles [6].Puffin is written in FORTRAN 95, and uses the librariesFFTW 2.1.5 [18], used for the Fourier transforms, and Su-perLU DIST [19, 20] to solve the sparse linear system ofequations for the finite element description of the drivenwave equation. The initial electron beam and radiation fieldparameters are read in from an input data file, and in the fu-ture it is possible that Puffin will be modified to read in anelectron beam distribution output by an external accelera-tor code. Data output is in the SDDS format [21], and thepost-processing in the following examples is performed us-ing MATLAB. The code is parallelized using MPI [22] ina method outlined in [12], and more detail will be given in[23].SIMULATIONIn this section, a full 3D simulation demonstrating SelfAmplified Coherent Spontaneous Emission is presented.An electron beam with a ‘top-hat’ current distributionof 6 cooperation lengths long in z¯2 and a Gaussian currentdistribution in x¯ and y¯, is propagated through an undula-tor of scaled length z¯ = 8 using the following parameters:ρ = 5.56 × 10−3, ǫ¯ = 0.6, a¯u = 1.0, σp2 = 0.002, γr =700, ux = uy = 1 and f =√2.The top-hat electron pulse should emit strong coherentradiation from its head and tail which have large currentgradients. In this high slippage/short pulse regime, the co-herent emission from the rear of the electron pulse willpropagate through the electron pulse and be amplified. CSEfrom the tail at the start of the undulator will propagatethrough and be amplified by the whole of the electron pulseby the end of the undulator.The electron beam of scaled emittance ǫ¯ = 0.6 ismatched to the focusing channel with betatron wavenum-ber k¯β ≈ 0.09. These parameters give a scaled Rayleighrange (z¯R = z/lg) of z¯R ≈ 3.3.The simulation results are shown in Figure 1 at propaga-tion distances z¯ ≈ 4 and z¯ ≈ 8. The scaled power P andthe transverse intensity distribution I⊥ are plotted at twofixed points in z¯2 for each propagation distance.In the earlier plots through the interaction, at z¯ ≈ 4,the electron pulse lies between 4 < z¯2 < 10. The radia-tion between 0 < z¯2 < 4 is the CSE from the front edgeof the electron pulse, having propagated forward into freespace. The region from 6 < z¯2 < 10 is in the ‘slippageregion’, and here the CSE emitted by the rear of the elec-tron pulse has begun to be amplified. The steady-state re-gion 4 < z¯2 < 6, has an evolution dominated by electronshot-noise as seen from the transverse plot of the intensitywithin this region which still has a poor spatial coherence.In contrast, the intensity near the peak of the amplified CSEdemonstrates a good spatial coherence.Later in the interaction, at z¯ ≈ 8, greater field amplifica-tion via the FEL interaction is observed. The electron pulsenow lies between 8 < z¯2 < 14, and the system is domi-nated by CSE emitted from the tail of the electron pulse atz¯2 = 14, propagating through and being amplified by theelectron pulse. The intensity at z¯2 = 4.7, which previouslyexhibited poor spatial coherence, now has a good spatialcoherence. This is not due only to the onset of the SASEtransverse mode selection process, but is also due to theinitial noisy spontaneous emission being dominated by theamplified CSE. The scaled intensity plotted near the peakof the amplified radiation pulse, at z¯2 = 11.8, is more nar-rowly focused than that at z¯2 = 4.7, which has undergonea greater free space diffraction having propagated outsideof the electron pulse.CONCLUSIONThe FEL simulation code Puffin has been presented.This is the first code which includes 3D modelling of theradiation field and electron beam using an unavaraged sys-tem of equations in a variably polarised, modular undulator.The equations allow a broad bandwidth of radiation fre-quencies up to a high harmonic of the resonant frequency tobe modelled using a single complex field variable. While arelatively simple electron beam focusing channel has beenimplemented, a more realistic FODO-type focusing latticeProceedings of FEL2012, Nara, Japan MOPD12FEL TheoryISBN 978-3-95450-123-675 Copyrightc ○2012bytherespectiveauthorsFigure 1: Upper plots: Scaled power P (z¯2) - the scaled intensity integrated over the transverse plane (x¯,y¯), as a functionof window scaled position z¯2 for two propagation distances z¯ ≈ 4 and 8. Lower plots: Transverse slices of the scaledintensity I⊥(x¯, y¯) at different window positions in z¯2 with propagation distances corresponding to the upper power plots.could be included by placing quadrupoles between undu-lator modules, and modelling the variation in off-axis un-dulator field components. Future work should also involvethe post-processing of data files which is currently not per-fomed in parallel. The output data files are also too large tobe easily portable and would benefit from being analysed‘on-site’ using local visualisation servers. These issues arecurrently being reviewed.REFERENCES[1] F.T. Arecchi and R. Bonifacio, IEEE J. Quantum Electron.QE-1, 169 (1965).[2] H.P. Freund, Phys. Rev. E, 52, 5401 (1995).[3] W.M. Fawley, Proceedings of FEL 2006, BESSY, Berlin,Germany, 218-221 (2006).[4] S. Reiche, Nucl. Instr. and Meth. in Phys. Res. A 429, 243-248 (1999).[5] E.L. Saldin et al, Nucl. Instr. and Meth. in Phys. Res. A 429,233-237 (1999).[6] B.W. J. McNeil, M.W. Poole and G.R.M. Robb, Phys. Rev.ST AB 6, 070701 (2003) .[7] B.W.J. McNeil, J.A. Clarke, D.J. Dunning, G.J. Hirst, H.L.Owen, N.R. Thompson, B. Sheehy and P.H. Williams, NewJournal of Physics 9, 82 (2007).[8] S. Reiche, FEL Simulations: History, Status and Outlook,Proceedings of FEL2010, Malmo¨, Sweden MOOCI1, 165-172 (2010)[9] K.H. Huebner, E.A Thornton and T.G. Byrom, The FiniteElement Method For Engineers, (1995) Wiley.[10] R. Bonifacio, C. Pellegrini and L.M. Narducci, Opt. Comm.50, 373 (1984).[11] E.T. Scharlemann, J. Appl. Phys. 58, 2154 (1985).[12] L.T. Campbell and B.W.J. McNeil, An Unaveraged Com-putational Model of a Variably Polarized Undulator FEL,MOPB30, Proceedings of FEL 2010, Malmo, Sweden(2010)[13] E.L. Saldin, E.A Schneidmiller and M.V. Yurkov, New Jour-nal of Phyics 12, 035010 (2010).[14] C.K.W. Nam, On the Theory and Modelling of the FourthGeneration Light Source, PhD Thesis, University of Strath-clyde, 2009.[15] C.K.W. Nam, P. Aitken and B.W.J. McNeil, UnaveragedThree Dimensional Modelling of the FEL, Proceedings ofFEL 2008, Gyeongju, Korea, 2008.[16] L.T Campbell, The Physics of a 4th Generation LightSource, PhD Thesis, University of Strathclyde, 2011.[17] R.H. Hardin and F.D. Tappert, SIAM Review 15, 423(1973).[18] M. Frigo and S.G. Johnson, FFTW 2.1.5 Manual, availablefrom http://www.fftw.org, 2003.[19] X.S. Li and J.W. Demmel, SuperLU DIST: A ScalableDistributed-Memory Sparse Direct Solver for UnsymmetricLinear Systems, ACM Trans. Mathematical Software, Vol29, 2, 110-140 (2003).[20] L. Grigori, J.W. Demmel and X.S. Li, Parallel SymbolicFactorization for Sparse LU with Static Pivoting, SIAM J.Scientific Computing, Vol 29 3, 1289-1314 (2007).[21] M. Borland, A Self-Describing File Protocol for SimulationIntegration and Shared Postprocessors, in Proceedings ofthe 1995 Particle Accelerator Conference, May 1-5, 1995,Dallas, Texas, 1995.[22] P.S. Pacheco, Parallel Programming with MPI, MorganKaufmann Publishers, Inc., 1997.[23] L.T. Campbell and B.W.J. McNeil, Submitted to Physics ofPlasmas.MOPD12 Proceedings of FEL2012, Nara, JapanISBN 978-3-95450-123-676Copyrightc ○2012bytherespectiveauthorsFEL Theory

Puffin : A three dimensional, unaveraged free electron laser simulation code

Name not available

http://accelconf.web.cern.ch/AccelConf/FEL2012/

Puffin : A three dimensional, unaveraged free electron laser simulation code

Abstract

Similar works

Full text

Available Versions

University of Strathclyde Institutional Repository

Name not available