.. .

Development of a 3D FEL code for the simulation of a high-gain -harmonic generation experiment

S.G. Biedrona,H.P. Freundb,and S.V. Miltona‘AdvancedPhoton Source, ArgonneNational Laboratory,Argonne, IL 60439

bScienceApplicationsInternational Corp., McLean,VA 22102 .

SPIEpaper # 361417smm

0%0Free-Electron Laser Challenges II, pak of SPIE’S Photm.ics West ’99 (/) ~ ~

January 23-29,1999San Jose+CA

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The submitted manuscript has been created by the University of Chicago as Operator of kgonne NationalLaboratory (“Argome”) under Contract No. W-3 I-109-ENG-38 with the U.S. Department of Energy. TheU.S. Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocableworldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, andperform publicly and display publicly, by or on behalf of the Government.

DISCLAIMER

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Portions of this document may be illegiblein electronic imageproduced from thedocument.

products. Images arebest available original

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●✎

.

Development of a 3D FEL code for the simulation of a high-gainharmonic generation experiment

S.G. Biedron*a,H.P. Freundb,and S.V. Miltona‘AdvancedPhoton Source,ArgonneNational Laboratory,Argonne, IL 60439

bScienceApplications InternationalCorp., McLean,VA 22102

ABSTR4CT

Over the last few years, there has been a growing interest in self-amplified spontaneous emission (SASE)free-electron lasers (FELs) as a means for achieving a fourth-generation light source. In order to correctlyand easily simulate the many configurations that have been suggested, such as multi-segmented wigglersand the method of high-gain harmonic generation, we have developed a robust three-dimensional code. Thespecifics of the code, the comparison to the linear theory as well as future phms will be presented.

- ‘“. Keywords : FEL, Free-Electron Laser, SASE, Self-Amplified Spontaneous Emission, MEDUSA,numerical simulation, APS SASE FEL, High-Gain Harmonic Generation

1. INTRODUCTION

With the growing interest in free-electron laser (FEL) physics, particularly in the self-amplifiedspontaneous emission (SASE) regime for both a proof-of-principle experiment and an actual fourth-generation light source, many simulation codes have been developed.* Among these, the FEL simulationcode MEDUSA has evolved to handle a variety of applications.2 Most recently we have included Gaussianand waterbag distributions with energy spread, multiple-segment wigglers (with the capability of usingdifferent wiggler periods, if desired), multiple frequencies (harmonics and bandwidth), additional beamdiagnostics, as well as quadruple focusing and dipoles. Here we discuss MEDUSA’S capabilities in detail,as well as the results from the simulation runs based on the APS SASE FEL’s parameters compared to thelinear theory,4 the observation of nonlinear harmonic generation,5 and the plan to use MEDUSA in thesimulation of applications such as a high-gain harmonic generation experimented

2. MEDUSA’S CAPABILITIES

In the simulation code MEDUSA the electromagnetic field is represented as a superposition of Gauss-Hermite optical modes, whose vector potentiaI is defined as

k 18A(xSr)=‘x ~el,n,h (x>y #,hcos~h +‘~)i‘in~h . (1)l.n,h

Here, 1and n denote the transverse mode numbers, h denotes the harmonic number, and

elfl~(x, y) = exp (-~2/~:)~l@x/w* )~.(fiY/wh)$ (2)

where HIand H. are the Hermite polynomials of order 1and n, respectively, and w~is the spot size of the h*

harmonic component. The vacuum phase is given by

qh = A(koz- coot)+ahr2/wj, (3)

for the vacuum wavenumber k. (= c%/c), where

expansion in vacuum modes is accomplished bycurvatures. These solutions are then expressed as

ah is related to the curvature of the phase front. An

using the vacuum solutions for the mode waists and

● Correspondence: Email: biedron@aps.anl.gov, Telephone: 6302521162, Fax: 6302525703

.

....

w~ = w~(z =o~+zyz:h]uz (4) -

and,

ah= d% (5)

where ~= hkow~z(Z = 0)/2 is the Rayleigh range. The one drawback to this approach is that a large number

of Gauss-Herrrdte modes are required to accurately describe the optical guiding of the radiation. In anotherapproach, using a source-dependent expansion (SDE) technique re-derived for the Gauss-Hermiterepresentation of the three-dimensional radiation field,’ the radiation field is accurately described with asmail number of optical modes while still maintaining the existence of accuratdy represented opticrdguiding. This has significantly reduced MEDUSA’S run times.

In MEDUSA, the field equations are integrated simultaneously with the three-dimensional Lorentz forceequations for a planar wiggler geometry for some number of electrons. The dynamical equations for thefield are

(6)

(i)where &zl,n,h= (i)I

e&ilmh m~c 2 are the normalized amplitudes (i= 1,2), the “prime” superscript denotes a z-

denvative,

and(‘~n,h =(l+n+l ah~-+--

W,

where @,2 = 4m?2nb I m, is the square of the plasma frequency, nb is the p....

(7)

(8)

icle density, and ox and v, are

the particle velocities in the x and z directions. For a beam with a uniform temporal profile ~d Gaus~ianenergy and phase space distributions, the source terms are

exp~ (yO-~0)2/2Ay2 ]

@Ay~+erf@O/fiAY)]

m

exp (/ /)-r: 20~ -p~o 2crP2x ~odyodp.OdpYo

(27r)%:op2(...) ~ (9)

where the average is over the initial be~ parameters(denotedby the subscript “O”) in ponderomotive

phase (VO = -@o~o ), energy (y&, and phase space (xo,yo,pa,p&; Y. and Aydenote the average beam energy

and the energy spread; and crrand crP describe the initial phase space parameters. For a matched beam, we

require that

(lo)

,

where P = y /aWkW, is the beam beta-function.

. ..

The utility of the SDE is that the diffractio~ of each harmonic component is governed by

w; 2a—=— - Yh,wh hkow~

where Xh and Yhare defined in terms of the source terms as

(11) “

(12)

(13)

(14)

2&(l) 2 +&&2 -and &zo,oJ = ~,oA , In the absence of the electron beam, Xh = Y~= O, we recover vacuum

diffraction given by Eqs. (4) and (5). ~

There are two wiggler models included in MEDUSA. Thedefined as

parabolic-pole-face (PPF) wiggler model is

‘w=Bw{cOskwz[’xsw4w+’ycO- @Zsinkwzcosh

(%)ifi(%)l ‘1’)

and the flat-pole-face wiggler (FPF) is expressed as

Bw=BW[6Ycosh(kwy)sin(kwz)+~z sinh(kwy)cos(kwz)]. (16)

Multiple segmented wigglers can be easily represented using these models, where the hinge fields aretreated in an approximate form. The quadruple field is defined by using the sharp-edge model

‘)BQ =~(j$x+xey ,a

(17)

where BQ is the magnetic field at the pole tip of the quadruple and a is the quadruple aperture radius. The

dipoles are defined as BD = BDGY,over some length as defined in the input.

As given above, the electron dynamics are governed by the three-dimensional Lorentz force equations.Although Maxwell’s equations are averaged over the harmonic period, no wiggler averaging is imposed onthe electron dynamics. This is a valid assumption since the Lorentz force equations are naturally S1OW1Y-

varying for frequencies near the natural resonances. AS a consequence, MEDUSA imposes the expressions

for the electromagnetic and magnetostatic fields at the position of each electron in the orbit equations as -follows: ,

Lp=.em. fvx(Bw+BQ+BD+m),‘z dz

(18)

where the wiggler, quadruple, and additionally imposed corrector magnet fields are given in their nativeform, aqd the electromagnetic field is determined by the Gauss-Hermite modes. Note that we must takeenough steps through the wiggler to resolve the wiggler motion. In practice, we find that ten steps perwiggler period is sufficient.

Fhmlly, it is interesting to note that one can place diagnostics anywhere along the specified undulator line.This includes at the entrance, interior, or exit of a quadruple magnet, dipole magnet, wiggler, and/or driftspace. The current list of diagnostics in MEDUSA now incfudes:

+— .-. t

. .+

the position in meters for each harmonicthe ~wer in Watts for each harmonicthe radiation spot size in centimeters for each harmonicthe curvature of the radiation phase front in radians for each harmonicthe rms beam radiusthe particle numberoverall efficiencyenergy conservation testbeam-centroid in the x-directionbeam-centroid in the y-directionaverage beam-width in the x-directionaverage beam-width in they-directionrms beam radius

kknormalized momentum px / m~cnormalized momentum pY I m.c

pondromotive phase in radiansnormalized axial momentum pZ / m~c

3. SIMULATIONS AND COMPARISON TO THE LINEAR THEORY

To test the upgraded MEDUSA, we simulated the APS SASE FEL. This FEL is a proof-of-principleexperiment for a possible pathway toward a fourth-generation light source. Although it is capable ofradiation wavelength tuning via the electron-beam energy over many wavelengths, the scope of theexperiment will include three basic phases: SASE at 517 nm, 120 nm, and shorter. The APS SASE FELwill employ the BNL GUN IV photocathode gun driven by a Nd:Glass laser system;g the APS Iinac, whichis capable of producing 650-MeV e]ec~ons;9 and a series of APS Type A fixed gap undulatory.]o Here wewill focus only on the simulation of the first experimental phase and have chosen previously achievableelectron-beam pararneters*land the specifications for the undulatory. These values maybe found in Table 1.

In all of the simulations, we have neglected the wiggler errors. In the first case, we have chosen to representthe undulator beamline as a single-segment parabolic-pole-face wiggler. The actual undulator bearnline,however, will be made up of a series of 2.726-m sections, or cells, that are composed of 2.4-m undulatory,drift space, diagnostics, and combined-function quadmpole/corrector magnets. In the second case, werepresent the undulator line with parabolic-pole-face wigglers. In the third case, we build up the undulatorline with flat-pole-face wigglers and the corresponding quadruples between these segments. We havechosen to employ a Gaussian beam distribution. Also, to find the optimal wavelength, we performed

.

wavelength scans at a fixed energy. These results may be found in Figure 1. From this scan, we found the ‘‘optimum wavelength of 518.819 nm. To verify the performance of MEDUSA in the single-frequencymode, we have found comparable results in the single-segment and multiple-segment cases to simulationsperformed in RON, GENESIS, GINGER, and TDA3D. All codes also agree well with the lineartheory.12-17Figure 2 demonstrates the strong correlation of gain length between MEDUSA and the lineartheory for various cases of energy spread. The results of the power gain and waist sizes along the length ofthe undulator for the above three cases, all running in the single-flequency mode, may be found in Figures3,4, and 5, respectively. Here we chose to perform these simulations at the resonant wavelength; however,later we will perform multiple harmonic simulations at the optimal wavelength to demonstrate thedifferences. The four simulations corresponding to Figures 2-5 used an input seed of 10 W, a waist of w(O)= 0.05 cm, three modes, and 5832 particles.

ParameterBeam EnergyNormalized Emittance

-.. . Peak CurrentEnergy SpreadOn-Axis Wiggler StrengthWiggler PeriodRadiation WavelengthMultiple-Segment tieCell Length (includes up and down tapers)Undulator LengthQuadruple Length (FPF)

Value219.5 MeV5nmmmrad150 A0.10%10.06 kG3.3 cm516.8 nm

2.726 m2.4 m5.0 cm

Table 1: APS SASE FEL Parameters

0.60

0.58

I \

0.56I

0“s4b==-0.52

I I

1’ t 8 1 I I I

k~P,= 518.819 nm1

o.50t’’’’’’’’’’”’’’’’’’’’’’*’” “’’’’’”j514 515 516 517 518 519 520 521

Wavelength (rim)

Figure 1: Power versus distance along undulator for wavelength scans at a fixed energy for the single-segment, PPF case.

. ...

1.2

1.0

0,8

0.6

0.4

0.2

+

L] 1 # [ I 1 1 1 1i

I 1 1 1 I 1 1 1 1 J

0.0 ~0.00 0.05 0.10 0.15 0.20

Ay/yo (%)

Figure 2: Gain length versus energyspreadfor the linear them-yand the single-segment, PPF case.

.A 810

l“’’’”’’’ 1’’8’’’’’’’’ 7-’--+ 0.10

107

106

105

104

103

102

0.06 ~

%’3

0.04 w

0.02

10’ ~ ()(-)()

o 5 10 15 “z (m)

Figure 3: Power at the resonant wavelength, radiation spot-size, and rms beam radius versus distance along theundulator for the single-segment, PPF case.

.‘

. .

t

. .-.

108

107

106

105

104

103 -.J

102 z 0.03

101

10°0 2 4 6 8 10 12 14 16

z (m)

Figure 4: Power at the resonant wavelength, radiation spot-size, and rms beam radius versus distance along theundulator for seven multiple PPF undulatory without quadruples.

107

1061

108 0.12

0.10

0.08

1050.06

104

~03

102 0.02

10’ 0.00

40.04 -

0 5 10 15 20z(m)

Figure 5: Power at the resonant wavelength, radiation spot-size, and rms beam radius versus distance along theundulator for seven mukiple FPF undulatory with quadruples.

.>.

After these initial cases were performed, we further tested MEDUSA by running with multiple harmonics. ‘‘Again, we performed these simulations with a waist size of w(O)= 0.05 cm and a start-up signal of 10 W.We kept the initial waist size constant for each of nine harmonics, but only seeded the fundamental, i.e., theharmonics were allowed to start-up without any input signal. Also, we included a total of 221 modes in allharmonics and 34,992 particles. We performed these simulations for the single- and multiple-segment casesdescribed above. The results of the gain lengths and the harmonic saturation powers for the single-segmentPPF case may be found in Table 2. In Figures 6 and 7, the single-segment (PPF) nine-harmonic case isplotted for the odd and even harmonics, respectively. In Figures 8 and 9, the multi-segment (PPF) nine-harmonic case is plotted in the same fashion as the single-segment case. The power in the harmonics isquite significant.

Harmonic Number

12

— . .. . 3.... 45678Q

Wavelength (rim) Gain Length (m) Harmonic SaturationPower

516.8 0.592 ~ 118MW258.4 0.335 4.28 kW172.3 0.201 999 kW129.2 0.165 1.57 kW103.4 0.124 47.5 kW86.13 0.098 220 w73.82 0.089 43.5 kW64.59 0.065 144 w57.42 0.072 32.9 kW

Table2: Variation in gain length and output power with harmonic number in the single-segment, PPF case

108E 1 1 I I I 1 I 1 I I & I I I # I I I I I I I I s 3

106

10°

1(J#J t

/ [f

10-8 1 # I t 1 I I 1A 1 I I 1~1 I I I 1 1 1 # 1 t I IO 2 4 6 8 10 12 14

— Ist— — -3rd-----5th~ 7th-- + -- 9th

Figure 6: Power versus distance along the undulator for the fundamental and odd harmonics in the single-segment,PPF case. The simulation was run at the optimal wavelength.

..

108

104 -

102

10°1’-’”

10-2AF

. ‘/f I — 1st I‘i/“ “

/ .J

,.

/f

,[,g~

— — -2nd/ /l -- ---4th

/ ● 1 ~ 6th

10+ / ft ; --+--8th

/ t f

1n+ 1 1 n 1/, # # 1 I 1 # I IflAv

O 2 4 6 8 10 12 14z (m)

Figure 7: Power versus distance aIong the undulator for the fundamental and even harmonics in the singIe-segrnent,PPF case. The simulation was run at the optimal wavelength.

108

106

- 104~ ~02

8> 10°0k 10-2

104

104

10-8

B 1 , 1 @ 1 I I & 1 I I , , # I 1 , , I a

&

.“

o

/ L-JSt

{ t’I

-- +--f

9th

+ !’, ,:, ,;t,1 8 I 1

5 10 15 20

z (m)

Figure 8: Power versus distance along tie undulator for the fundamental and odd harmonics in the multiple-segment,PPF case. The simulation was run at the optimal wavelength.

.-

106

104

102

10°

10-2

10-”

104

10-8

E. /r~ ,- ‘,,~,,n,~d

— Ist/ .—

) - 2nd/ f -----4th

/-t ~ 6th

4 1 -- +-- 8th/ t ●1

/ tr# /, 9 1 * I 1

o 5 10 15 20z (m)

Figure 9 Power versus distance along the undulator for the fundamental and even harmonics in the multiple-segment,PPF case. The simulation was run at the optimal wavelength.

It is interesting to note that not only are the harmonic powers quite high but the harmonic power arisespurely from the interaction of the fundamental with the nonlineax component of the source-current. Thisnonlinear term occurs due to the beam micro-bunching. As seen in Table 2 and Figures 6-9, the gain length

varies approximately as LG = (2hr)-1, which is characteristic of this nonlinear component. The scaling of

the nonlinear gain-length decreases linearly with the harmonic number, whereas the linear gain lengthsincrease with harmonic number. This phenomenon has been previously seen in traveling-wave tubes.Notice also in Table 2 and Figures 6-9 (7 and 9 specifically) that the even harmonics do not experience ashigh a power as compared with the odd harmonics. This is due to the fact that the natural electron motion inthe parabolic-pole-face-wiggler configuration does not lead to extraordinary interactions at these evensuperpositions.

We have also exercised, the added beam diagnostic portionsof the upgraded MEDUSA. For the twoparabolic harmonic runs, single and multi-segmented (both PPF), it is interesting to observe the mode waistalong the length of the wiggler and to exempli~ the existence of optical guiding only in the fundamental.This may be found in Figures 10 and 11, respectively. nis diagnostic as well as those described above willbecome mandatory for examination of such things as bunching in a high-gain harmonic generationexperiment, as is briefly described in the next section.

. -.

1-

0.20

0.15

0.10

0.05

—lst— — -2nd-----3rd~ 4th~ 5th—~ -6th-. * -- 7th—m- -- 8th--- ---9th+

.,

0.00~ ‘ I 9 * I 1 # a 1 1 t 1 I * I 1 1 I 1 10 5 10 15 20

z (m)

Figure 10 Mode waist versus distance along the undulator for the first nine harmonics in the single-segment, PPFcase. The simulation was run at optimal the wavelength.

1St——-2nd---- - 3rd~ 4th~ 5th+- - 6th-. * -- 7th—8- -- 8th~ 9th

/’

O.oof‘ I I I I 1 1 I a I I # * I I I I 1 1 45 10 15 20

z (m)

Figure 11: Mode waist versus distance along the undulator for the first tine harmonics in the multi-segment, PPF case.The simulation was run at the optimal wavelength.

4. CONCLUSIONS AND FUTURE PLANS

. .

A three-dimensional code has been developed to handle Gaussian and waterbag distributions, energyspread, multiple-segment wigglers (with the capability of using different wiggler periods, if desired)multiple-frequencies (harmonics and bandwidth), as well as quadruple focusing and dipoles. Whh thismodified version of MEDUSA, we have simulated the AI% SASE FEL, a proof-of-principle experiment fora fourth-generation light source, and have shown the existence of nonlinear harmonic generation. We havefound MEDUSA to be in good agreement to other existing FEL codes and to the linear theory.

Future plans for MEDUSA include simulation of another possible pathway to a fourth-generation lightsource, the high-gain harmonic generation experiment defined in reference 4. In this experiment an electronbeam and seed-laser beam, whose wavelength is at the fundamental, pass simultaneously “through a firstwiggler for energy modulation (the modulative section). Just after this first wiggler is a dispersive sectionmade up of a dipoles chicane that induces bunching, i.e., a longitudinal phase-space rotation. The finaI stepin this process involves a second wiggler (the radiative section), which is in resonance with the electronbeam energy and the second harmonic of the input seed radiation. We also plan to complete similar. ..

. simulation runs with the inclusion of a tapered radiative section. Also, the ability to run with a bandwidthdefined around each harmonic will be included in both of the above cases.

Finally, we plan to further upgrade the code with the following options:

+ off-axis injection into the fist wiggler and between wigglers using “automated” correctormagnets

+ inclusion of the spontaneous radiation for the given parameters+ output in APS-standard self-describing data set (SDDS) format, for easy manipulation using

the Unix platform]8+ short beam-pulse interactions+ preprocessor for making multiple sequential runs and corresponding repetitive changes to the

input file parameters and the output data file names

ACKNOWLEDGEMENTS

This work was supported by the U.S. Department of Energy, OffIce of Basic Energy Sciences, underContract No. W-31 -109-ENG-38.

Special thanks to Li-Hua Yu and John Nicolas Galayda for encouraging me (SGB) to perform this work.

1.2.

3.4.5.

6.7.8.

9.

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