TITLESINGLE PARTICLE FEL EQUATIONS WiTH OVERTONES

~ulno~(s~ C. J. Elliott and M J. Schmitt

SUBM TTED TO

~“?-’ NOV08 KM!i

LA-uR--85-3862

DE86 002423

For Publication in Proceedings of Seventh International Free ElectronLaser Conference, Tahoe City, NV, September 8-13, 1985.

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LwwmilclsI!MS’

LosAlamos NationalLabOratorlmsAlamos,New Mexico 8754 i!

s=~STI?lttllTION OF THIS DOCUMHIIT 18 UNLIMITED

.Single Particle FEL Equations with Overtones

C.J. Elliott and M.J. SchmittLos Alamos National Laboratory

lJniversity of CaliforniaX- I,E531

Los Alamos, NM 87545

Abstraci

We report a generalization of the usual single-particle cquat.ions for a linear polarizedwiggler. Particle simuliitions show the presence of harmonics under conditions of low qand high magnetic fields. These harmonics are introduced into the singie particle equationsby separate field components fof each harmonic. The overtones are fed by the same phasespace and all their effects on the particle motion are treated simultaneously. The couplingof the magnetic field differs from the usual hessel function f’wrn because of the low electronenergy and thr high magnetic field. We also include longitudinal space charge fields.W~ compare t}Ie rwlucvd rquation results with a full 1D particle simulation. Finding goodagrcernent, wtI conclu(le that harmonic generation ha> a significant impact on the waveformunder these conditions.

1, Introduction

our tht-wrctical description of a planar FE1.( freo+lcctron laser) begins with a 1D3V (onedistance, three vclocitics) simulation of Maxwell’s equations and thr Lorcntz force equation,it er)ds with a reduced description that may br rcgiirdvd a% a generalization of the usualsinglr particle dcscriptioni This work is still in progrrss. ‘rho boundary conditions uwxi inthe simulation ar~ periodic in t}~t’dimw]sion coordinate that plays the roll of a phase inttw r(’ducw.l description. This roducvd dvsrription pvrmits timr stepx that are an order ofmagnitudv l~rgor than that uw’d in tho simulation, the number of particles was two ordersof rniign it udv 1(sss,Mnd thr (’quat ions urr sirnplor and more timenahle to interpretation,

‘:’hc periodic grid assull]ption oft IIC*simulation permits motion of the +bearn to occur,Thr grid is fixed ii] spacv hui Inay ho thought of as periodically extended ad tn~initum tothe right, Iii timr thv rlcct rorls n]ow’ at relativist.ir ~peed~ to the right, The optical pulsernovcs fmi[cr to t}w right hut uIs() is periodic, on t,ho cxtcndcd grid, if our eye were fixed ona Hir]gh’ partiriv, it would follow it continually to the right, Equivalently, we could watchthv motion within a solo prriod nnd as the particle moved out of the right boundary wewould W*Vit:, tx~uivaloilt tmtcr thv left, We use the economy of thi~ one period descriptionin particlr simulntionu wlwre the period is chosen to ho a multiple of both the optical

I

wavelength and tl~e wiggler wavelength.The reduced description path is to average the contribution to each harmonic as the

electrons slip back one optical wavelength. The !undarnental optical frequency componentis a sinusoidal waveform with phase varying from !I to 27r. Suppose a source were togive more energy to the phaw at. 7r/2 than to 3n/2, then it would distort the waveformby exciting overtones. To complete the description we consider each harmonic amplitudeseparately and sum those contributions at 7r/2 and 37r/2, among others, weighted by theharmonic phasor. In performing the averaging, we obtain a coupling coefficient that differsfrom that of Colson[2], Madey and Taber~3], and McNeil and Firth14,5].

2. Mathematical Description

2.1 Ill Maxwell Equations

First we describe the equations used in our WAVE simulation. They originate from those inthe chapter Fundamen.t al.s (JJ Particle Simufat ton, described by I). Forslund[l ];but theyare modified i, I two respects. First we use charge neutralization, with the average density<. p ‘.x subtracted out, and second WQusc only the time rate of change of the transversepart of the coulomb gauge vector potential A to determine the electric field. The latteroperation makes the rc.,ults of the code stable to unphysical and unmathematical temporaldrifts in the longitudinal part of A“and corresponds to voltage-short boundar) conditionsacross t.ht’ grid. The former condition is required mathematically with periodic boundaryconditions and corresponds to achievable physical conditions; eg. at physical beam center.

“rhe quantities we usc are in standard notation. The vector Z1 is a unit vector in thelongitudinal direction: The s~alar potential @and the vector potcr,tial ~ give the electricand magnetic fields E and B, The electron h~~ charge of magnitude e and rest mass mwith ~he ratio of the mass to resi mass designated q, here p is the charge density and J-wis a virtual current that produces thr static wiggler field.

Maxwell’s equdtions are:

(?;A”-d~A”=-Fldl(j - J:,

d~tj:: d, J,,,

;):4 “’ p - p,

/i(L) A-(()),

4(L) - 4(()),

($(L)= #(()),

t?l Al = O,

L iMthe periodic boundary condition length, chwwn to be th: wig~ler wavelength2m/k,,, in the simulations, Also

2

i= –z~i31f#- f3*(i,xA)x&

E = al(;]xi).

These are coupled to the particle equations by;

df rn’yt.i - –e(l?+ tixg),

The convention here is using sin in J: rather than the cots convention we use in the reducedequations as does ref. [2].

2.2 Reduced Equations

Derivation of t]ie reduced equations follows ref. [2] with an important modification to thecoupling coefficient, a more complete resonant condition, and a sum over harmonics. Thestandard Madey and Taber[3] coupling coefficient that Co]son deri’”es without t~pograph-ical error is based upon the fourier transform arising from the axial location that ‘~ariesas sin(2w, ~t), or sin(2kt,,ir1 ). An incomplete description of a correction to this couplingcoefficient has been submitted as ref. 14,5] based upon the change in the axial componentof the electron velocity rat}wr than that to the total energy. Cur approach is different, itincludes treatment of the exact motion of an electron that has all even harmonics of kwzlpresent, and we fourim transform an elliptical integral function phasor to find the couplingcoefficient, but do not includv terms arising from non-exponetia] multiplicative variationsvia dt/dz1. our approach considers a 1D description in q and phase ; of the particle anddoes not lead to the ref. [4,5] corrections. In fact, rather than resulting in an enhancementof the fundamental, our corrcrtion reducm to the Madey-Taber-Colson coefficient at highq. At low q (and high magnetic field) when our correction is important, it decreases the

coupling of the fundamonta] rather than giving the ref. 14,5] enhancement, When we write

the expression for + in terms of P1 wr find additional terms that just cancel] the ref. [4,5]I// terms and thereby giv~’ the Madt’y-’~at)or-{;olso~l resul~ at large -y,

T}w coupling cocflicicnt is drrivtd rrltit}~t’lliatically by expanding the expression

using an elliptical intvgrai cxprwmion for t(~l ), the tirrw an electron reaches !ocation x 1,and c@ is thr avera~c w+xity in the xl direction. in the calculations r~ported here wcdrop the @c(dt/dxl ) in lieu of tlw expected r~pid variation of the pharw term, Hew we

take w. = k,c. The time t (Z1) can be represented by the sum of a term linear in Z] and aterm doubly periodic in z 1. The linear term combines with kW and ks to form the resonantcondition. The periodic term can be expanded in a fourier series in the even harmonics ofk.,zl . These ideas produce the mathematical identity

1OC

cc=–2 2( Cn -t Cn+I)COSlk&(] -- ~-’)z) + (2n -$ l)kwzl + @l,

-m

with the resonant condition obtained when the cos argument is stationary with respect toXl, and where we regard the parameters U&,~ and kU, as constants and start with

and end with

where for the numerical and analytical work here the ~c(cft /dzl ) term has been replacedby unity. Above, the function @ is given by

The quantities /3 and m are

fi ._ g!.: ~; V2F(:\m) ‘

arida;~t .

~2 -]’

where F is the cllipticr,l integral of the first kind defined in the }fandbook o/ MathematicalFunciions16] by ( 17,2.6). A special case that occurs for small m, evaluated at the harmonicresonance gives the Madcy-Tabcw-(hkm result at high -y

()J’j “’.-.O;’(1i .:.n;)4 -!--’-y 1- _!_. ] +!54

272 2’

and

(2n + l)a~—-— .. . . ..4 + 2a~

4

.i physical interpretation of the coupling coefficient identity is that the harmonics inthe t(z) motion produce an effectively higher harmonic wiggler field. That is, if we thinkof a prototypical linear wiggler as having a single resonan~e frequency with 109% couplingjust as a helical wiggler does, wc can view the cc identity as saying that an actual linearwiggler is, on average. a prototypical wiggler having a magnetic field that is a superpositionof fields with wave numbers that are the odd harmonics of kW. The amplitude of the 2n t 1hnrmonic of the effective magnetic field is then just the factor c. + Cn+ * multiplied by theactual fundamental field.

2.2.2 Coupling in the Particle Equations

The reduced equations describe the time dependence of the electromagnetic wave by asllperposition of the odd harmonics. The WAVE results reported in Section 3 show aconsistent time dependence because a uniform e-beam was injected and because the AWperiod for L was chosen. Were one to describe a problem where the local spectral widthwas small but finite compar~d to the harmonic separation, he could do so by generalizingthe approach of ref. 12] along the lines reported here. Such a case would model a timedependent injected current, for instance.

The electric field is taken to be a superposition of harmonics; allowing oI~ly resonantinteractions that occur for the f’th wave vector given by

(21 , “)k,,,ksl - (2/ + l)k.q= ------- ,

f)-1..]

we have

2.2.3

[Ising

Here,

The Reduced Equation Set

complex notation, /’1 - el exp ir$l, we have the square pulse equations,

dy (&i “t ~1+1) ~,~k eEIPl

& ‘“ 2-- cxpi(2f + l)G?J- ------,-Y m(bc2k8

we have taken the propagating wave to bc planar, and p reprcstmts the averageelectron density in th~ wiggler measured in w~/u~ units; 8? indicates the real part of theexprwwion, The electrostatic te~r~l follows refermce z. In it~ implemel~tation, numerical

!?)

noise problems are avoided by subtracting out the initial densityvector just before integration of h’]. The conservation of energybox follows mathematically from the above equations with ~ = 4

vector from the densitywithin a computationaland is expressed as

That is, the loss of electron energy density is compensated by the gain in energy densityon one or more of the harmonics.

3. Computational Results

3.1 Wave Resutls

The WAVE particle simulations used 10000 particles with 188 cells in Z1 ranging from Oto 1&r. In this work k~/kw = 6.0, fll~ = 3.775 at injection just outside and also just insidethe wiggler, rl = 47.34 where ~1 is t}~e injection time; also we have ~ = 12n/T1; the scaleddensity is (w~/u~)O = 8.710”’6 outside the wiggler and /?-1 times larger inside. The scaledvector potential magnetic field is ak = 2.82. We used 70000 time steps of At = 0.15. Atthe fundamental frequency, the shape factor of ref. [7] is S1 = 0.9985, indicating morethan adequate spatial resolution. The first run shown in Fig. 1 had magnetic permeabilityof unity; all other runs had p = 0,99893. All runs reported here were started from noise.

Simple charge/current neutralization and propagation in free space did not give ade-quate representation of the expected results. Fig. 1 shows a plot of the optical field energyand magnetic field energy averaged over the computational grid. These results show adrift in the electric field energy at long times that arises from a small non-physical timedependent A 1 component, before the second modification was made to Maxwell’s equa-tions. Also the rapid oscillation at 7’<2000 is attributed to a generated diamagnetic fieldthat slightly altered the presumed equilibrium between the initialized static magnetic fieldand the initialized virtual current.

The calculation shown in Fig. 2 shows similar quantities but here the above mentionedproblems do not appear. The electrical energy grcwth rate q = 0.0055 in the code unitsdiffers substantially from 0.0043 of Fig. 1.

The final wave form of the Zz component of the electric field is shown in Fig. 3 alongwith a harmonic decomposition, The significant components are the 1‘st, 3’rd, and 5’thharmonics of the fundamental frequency, Theoretical considerations suggest that thesecomponents propagate to the right.

The electrons are shown in the ~1 q, xl phase space in Fig. 4a and their density inFig. 4b. We can clearly see the bunching occurring in both figures. Most of the variationin the density peak amplitude is attributed to the variation of /11, but the peaks are notequally separated in time or space.

3,2 Reduced Equatkm (Maculations

T},e coupling coefficient for the fundamental frequency that is used in the reduced equationsi~ shown in Fig, 5, it differs substantially form a variation of 0.755 to 0.745 that occurs by

6

approximating @ by a sin2kW z approximation ,sans resonance assumption, over the samerange, but yet at high ~ the two results agree with each other.

The magnitude of the total electrical field is shown in Fig. 6a in the same time units asthe WAVE calculation. The harmonics are shown in Fig. 6b. These harmonics are plottedon a time and magnitude scale that requires the subplot to be blown up to the full grid sizeto read the values. As in the simulation, the harnlonics are seen to be appreciable. Herethe 100 particles were placed on the grid with equal spacing but, then each particle wasdisplaced a small random amount that reproduced the expected noise on the fundamentalfor 10009 completely randomized particles. This gives more noise on the harmonics thata complete randomization of 10000 particles would give, but may be thought of as beingrepresentative of some realizations of 10000 random particles.

The growth rate for this case was computed by finding the largest growth root insolving the cubic equation that arises in the small signal limit of these equations. Thissmall signal theory was used as a means of checking the reduced equation description, givingoverlaying plots for the cases compared. The growth rate was 0.00553 in good agreementwith 0.0055 from WAVE. This comparison was serendipitous for severai reasons: (1) theharmonic modes contribute an undetermined amount to the growth rate in the WAVE case(2) the /?dt/dzl term may alter the coupling coefficient; (3) the reduced description maybe slightly more complicated than that shown.

Acknowledgements

K, Lee suggested attacking the problem of low ~ and high a~ by using WAVE. D. Forslundgenerously answered many questions about modifications to conventions in WAVE. Thiswork was supported by the lJ,S. Department of Energy.

Refere] Ices

[l\

12]

[3]

[4]

[5]

[6]

[7]

D. Forslund,Fundamentals of Particle Simulation, in Space Plasma Simulations, M.Ashour-Abdalla and D, A, Dutton, Eds., Dordrecht: D. Reidel Publishing Company,1985;~also to appear in Space Science Review, 1985.W, B, Colson, “The nonlinear wave equation for higher harmonics in free-electronlasers, ”lEEE J, o~ Quantum Electron,, vol. QE-17, pp. 1417-1427, 1981.J.M.J. Madey and R.C, Taber,” Equations of motion for a free-electron laser with atransverse gradient, “ in Physics OJ Quantum ~fectronics, vol. 7, Jacobs et al., Eds.,Reading, MA: Addison-Wesley, 1980, ch, 30,

B. W.J. McNci! and W .J. Firth, “ Effect of the radiatkn force on free-electron lasergain,” IEEE J. Quardum Electron,, vol. QE-21, pp. 1034-1036, 1985.B,W,J, McNeil and W,J. Firth, “Effect of the radiation force on free-electron laser,”Proceedings of this conferencelfarzdbook o/ Mathematical Functions, M, Abramowitz and I, Stegun, Eds., WashingtonDC: lJS Government Printing OffIce, 1964, ch, 17,2.6,C,K, 13irdsall and A.13, Langdon, Plasma Ph~sics Via Computer Simulation, New YorkNY: McGraw-Hill Book Company, 1985, ch. 4,6,

7

ROUGHDRAR.

z .y.1] F HISTI?Y10

10 ‘

la 0

10-4

10 ‘sI

------

—

. - .-’ ?,. n‘---- .>. z ,~>q-@

. /’

Jo”g

10-’

3iL*L L-

u ,!—

I

“J

1

-J

0,00

wo4/J@2/:4$:31

0.s0 1 *OOlIHE

T=10500,OO

Fig. 1 WAVE plot of energy of field versus time. The drift in the syn-chrotrons bumps is caused by ZltA 1 and initial oscillations are caused

by a diamagnetic effect. The upper line is the total magnetic field en-mgy, the lower line the total electric field energy, Fit to the straightline portion of the lower curve gives energy = &Oexp(,0043T).

ROUGHMm.L1

10

10

2 F HISTRY

I I1

[

/

o . r

l=

10”1

10-2I/#

0.00 0.50 1 ●OO 1.50TIME *1O 4

05/06/19

02:11:15 T=10500,00

Fig. 2 Wave plot of energy versus time as in Fig. 1. Diamagnetic oscillations

do not appear. Removal of the d~A1 term has accounts for the lackof drift in the synchrotrons bumps, Fit to the straight portion of thelower curve gives energy c &o exp(.~~ss~).

610-3

‘(: 0.4I.Of+.

4

0.50

~-o.53 —

0.00 0,94 1.88 2.83 ~ 3.77x *1O

&v

10-310-a10-510”610-710-e

● 10 -9

~ ,~lo

1611

16’2

1613,;14

●

✌✎“.’

.“.

In -z EY1

I I I

~-

1 I Ibu P I I If-l= I

0.00 0.20 30,40 0.60 0,80 ~ 1.00

KX 010

Fig. 3 Waveform and harmonic decomposition of the propagating fieldEv = E2 at time T = 10500.

!

0.00 ~.q4 t, 1.08 2003 , 3.77\. ., i’ x alo

Olo“5r= 0.7

0.00 0,94 1000 2.03 , 13,77x 41O

os/M’J:21:41:31

Fig. 4 (a) The (Pi?, xl) phas[? space plot o! the electrons at time ?’ = 10500.(b) The density corresponding to the phase space plot.

Q,lo-1

7.00i ,=

6.90

1~‘lJ

II

S*OO

4.00

COWL

=3*OO 3*5U 4.00 4.$0

GAMIIA05/05/0116:J3:44

Fig. 5 The coupling parameter co + c1 versus ~ for the conditions of thecalculation.

,.-] SUMEW?GV= 9.S46&-08t 4.73SE-16

I—

I

JU-4

u-: ,.-3u ~r{k

-3.37

-4.6s

-5.94

0,00 0.$0 1.00 J ●SO 0,00 0.26 0052 0,79TIM

J ●O~ojo 4 71ME OJo 4

Fig, 6 (a) ~ lef12 in the reduced equations starting frori] noise. Comparr to

~rig. 2. (b) I’he 1’s{, 3’rd, 5’th , ,.,19’th harmonic fields versus time,appearing as one reads, To use the scale, the subplot is expanded tofit the labeled axes, The first subplot repwts (a.).