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Volume 53, number 3 OPTICS COMMUNICATIONS 1 March 1985

COLLECTIVE INSTABILITY OF A FRE E ELE CTRON LASERINCLUDIN G SPACE CHARGE AND HARMONICS

J .B. MURPHY and C. PELLEGRINIBrookhaven Nat ional Laboratory, Upton, NY I1 973, USAandR. BONIFACIOUniversity of Milan, Milan, Italy

Received 19 March 1984Revised manuscript received 19 October 1984

The effects of harmonics, space charge and electron energy spread on the collective instability regime of an electronbeam coupled to a planar undulator are analyzed. Both analytical and numerical results are presented.

Recently there has been interest in the collectiveinstability regime of the free electron laser (FEL) [ 1,2].There is also some work on the collective regime ap-pearing in the Russian literature [3,4]. Bonifacio,Narducci and Pellegrini [5] (referred to hereafter asBNP) have discussed the collective instability regimeof an FEL studying the growth of the radiation fieldfrom noise and the saturation level for the case of acold electron beam, ignoring the effects of space chargeand higher harmonics of the fundamental operatingfrequency, o = 2ck,y2/(1 + iK2). There has also beensome analytic work done on the effects of energyspread on the collective instability [6] _We shall ex-tend BNPs work to include the effects of higher har-monics, space charge and an energy spread in the elec-tron beam. A planar undulator is assumed throughoutthe discussion.We shall use the notation of BNP: z is the directionof propagation of the electron beam; x and y are thetransverse coordinates; the electron beam is character-ized by its energy, y, in units of the rest energy, nz,, c2,and n, the electron number density which are com-bined to give the relativistic plasma frequency fip =(4rmee2/my0) 3 j2 ; the undulator magnetic field strength,Bo, and period ho can be combined to yield the undula-tor parameter, K = eBoXo/2nmc 2 ; the undulator gives

0 030-4018/8.5/$03.30 0 Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

rise to a transverse velocity in the electron beam of &= (K/y) sin(2rrz/h0) the electron phase, @, s measuredrelative to the pondermotive wave and is taken as $ =2nzlho t 21rz/h - wt; the resonant energy, yr, is de-termined from the synchronism condition to be -rF =&,(l + ;K2)/2h; h e undulator frequency w. is w. =2nc&/~ ; the radiation electric field is assumed of theformE,(z,t)= CE,(z,t)exp[-(2nin/X)(z-ct)], (1)where En (z, t) is a slowly varying complex amplitude.

Using these definitions we can construct a set ofnormalized variables [ 11. For the electron phase wehavee =@-c&t ; (2)for the electron energy:F = YlPYg 3 (3)and for the complex electric field amplitude for thenth harmonic

A, = iE, exp (i40 t)(47m&yo n, p)1/2 (4)

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Volume 53, number 3 OPTICS COMMUNICATIONS 1 March 1985where y0 is the initial electron energy,P = (KY&/4Y&Jg) 2/S , (5)r = 2w()p(rR/ro)2 c and &, = wu(1 -y,/$).The starting point for our discussions are the BNPequations to which we add the effects of space chargeand generalize them to include higher harmonics.

The equation for the particle phase remains un-changede. = (1/2p)(l - lip2r.2)I I (6)wherej = 1: 2, . .N is the particle index. The dot in-dicates differentiation with respect to r.

The equation for the particle energy becomes+C ( nFnCK> iaPn 2rj + n (eXp(-tiei)) 1X exp(inOi)+c.c., n = 1,3,5 ,... (7)whereF, 0-0

= J(n -1)/2 ( $ s2, -Jcn+l)lI(+J(8)

the Jn are Bessel functions of the first kind and u2 =4p2 (1 t fK2)/K2. The second term in the above, con-taining u, is the space charge term. To derive the con-tribution of the space charge forces to the evolutionof the system we begin with the particle density as afunction of 0 : n (0) = (2nnJN) Zi 6 (0 - ej). We ex-pand this in a Fourier series in 0 and obtain n(0) =tie Z.n (exp(-tiei)) exp(ine) t C.C. The electric fieldassociated with this charge distribution is determinedfrom Poissons equation to be

E, = (9)The contribution of the space charge force to theenergy change of the jth particle is determined frommc2dyldt = -ev, E, to bei; -7 u2 (exp(-tiei)) exp(tie.)in J +c.c. (10)12

The equation for the nth harmonic of the complexelectric field amplitude in the slowly varying ampli-

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tude and phase approximation isF,, (K)A, = iF,A, + ~ P (exp(-inOJri), n = 1,3,5,(11)i.e., the nth harmonic of the electric field is driven by

the nth harmonic of the bunched beam current. Theangular brackets indicate averages over the particleinitial phases, i.e., ( ) = l/NCj where N is the numberof particles. F is the detuning divided by p.

=n6, (12)The equation for the electron energy change (7) andthe field equation (11) contain only odd harmonics,in agreement with the characteristics of the spontane-ous radiation on axis for a planar undulator.

From the set of equations (6), (7), (11) we can ob-tain an invariant

(13)which is the energy conservation relation.

The set of equations (6), (7), (11) only containstwo parameters, the detuning of the fundamentalmode, and p. p is a measure of the electron beamdensity and is related to the Pierce parameter C of themicrowave tube literature [7]. Typical values of p are10e5 -1O-3 for a 0.5-l .O GeV storage ring electronbeam and 10-l for a l-5 MeV, 1 kiloamp electronbeam in a linac.

The quantity qn = I(exp(inO)/rp)l,n = 1,3,5, .is called the nth harmonic bunching parameter. It isa measure of the degree of bunching in the electronbeam. For n > 1, qn is a measure of the harmonic con-tent of the perturbed beam current. At the entry tothe indulator an unbunched beam has nn = 0 for all n.As the system evolves the bunching parameter reachesits maximum value on the order of 1.

The initial conditions that we will use with this setof equations for a monoenergetic beam are ri = 1/p,i=l , . N and either A, # 0 and Bi = 2(i - l)n/N fori=l , . N, i.e., no initial bunching, or A, = 0 and q,f 0, i.e., the electrons are displaced slightly from theiruniform initial distribution. These 0 displacements canbe produced by fluctuations in the longitudinal elec-tron distribution and we will refer to it as bunchingproduced by noise. As was shown by BNP the state withparticles uniformly distributed in phase in the range

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Volume 53, number 3 OPTICS COMMUNICATIONS 1 March 1985[0,2n],. = /p and A,, 0 s an equilibrium state.In order for the system to evolve we must introduce asmall initial field or some noise in the initial phases.

If N, electrons are distributed randomly over onewavelength of the radiation the initial bunching param-eter is

(14)We want now to study our FEL equations; for sim-

plicity we start from the case u = 0, ignoring the spacecharge effect. Following the procedure given by BNPwe linearize this system about the equilibrium positionBi = 2n(i - 1)/N, F = l/p, and A, = 0,ntroduce col-lective variables and look for solutions proportional toexp(ihr). We then find that the modes are decoupledand the dispersion relation for each mode is a cubic.The dispersion relation for the nth harmonic isx3 - 6,Q + ;F;(K)@x + n) = 0 . (15)

Since the coefficients in the cubic are all real theroots are either all real or there is one real root and acomplex conjugate pair. For a fixed value of p thereis a threshold value of 6, above which the nth modeis linearly stable. The system is unstable for all nega-tive values of 6, but the growth rate decreases withincreasing 16, I. Notice that since F,, K) ends to zerofor fixed K, p nd 6, when n becomes larger, theroots of (15) tend to become all real in this limit.BNP have shown that the effect of the term propor-tional to p is to further destabilize the system. Boththe threshold value of the detuning and the magnitudeof the growth rate are increasing functions of p.

In fig. 1 we plot the value of Im(A) versus 61 forp = 0.1. For p in the range lop4 < p < 10-l themaximum value of Im(h) is approximately Im(h)-(; nFn(K))li3-12 an occurs at 61 = 0. It can beseen that the decrease in F,(K) s n increases resultsin both lower growth rates and threshold values of6 1 for the harmonics.

The curve labelled (a) in fig. 2 is a typical plot ofIA 1 ersus r for one field mode starting from a smallinitial field and some noise in the initial electronphase distribution. The wave exhibits an initial expo-nential growth at the growth rate computed from thedispersion relation, eq. (15). The field reaches a peakvalue after which no further exchange of energy be-tween the beam and wave occurs. The period of the

DELTA

Fig. 1. Plot of imaginary part of root of h3 - 6, h2 + ;Fi (K)(~h+n)=O,n = 1,3,5 with p = lo-, labeleda, b,c,respec-tively.

4-I

Fig. 2. (a) IA 1I versus 7 for one mode starting from a smallinitial field. (b) IA 31versus 7 for the n = 3 mode of a com-peting two mode system. The initial conditions for bothfigures are: 150 particles,El = O.l,E3 = 0, q1 = 8.4 X 10-j,q3 = 6.1 x 10-2, p = 0.1 and 6 1 = 0.

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saturated field oscillations is that of particles trappedin the potential well of the wave.

To determine the effects of harmonics, we inte-grated equations (6), (7) and (11) for two field modes,rz = 1 and 3 without space charge. A plot of the fun-damental electric field versus time remains the samewithin the resolution of the figure as the single modecase (fig. 2a). Fig. 2b is a plot of the electric amplitudeof then = 3 mode versus time.

The peak field amplitude of the two modes differsby an order of magnitude, with the fundamental modereaching a higher amplitude. Both modes saturate atabout the same time indicating that once particles aretrapped in the potential well of the fundamental modenet energy exchange between the particles and any ofthe modes ceases. Exactly how much energy appearsin the harmonics depends on p, F and the initial con-ditions.

Now we shall examine the effects of space chargeon the system. If we linearize the system of equationsincluding both the harmonics and space charge we ob-tain the following cubic to determine the linear growthrates for the nth harmonicX3 - 6, X2 + (pF;K)/2 - 02/p) h

+ nF,2(K)/2a26,1p 0. (16)This cubic differs from the previous one by the

terms proportional to u2. Recalling that BNP foundthat the term ph is destabilizing, it can be seen thatthe space charge has a stabilizing effect and competesfor control of the system. Since the space charge terma2/p is proportional to p it will be negligible for sys-tems with p < 10p2, such as storage ring devices.

A more complete analysis of the cubic reveals thatin contrast to the system without space charge the sys-tem with space charge has a lower bound on 6, belowwhich the system is stable. In fig. 3 we plot Im(h) ver-sus 6, for K = , p = 0.1. A comparison of figs. 1 and3 clearly indicates the effects of space charge on thelinear growth rate of the system.

Large negative values of 6, no longer lead to an in-stability. In fact both the lower and the upper boundson 6, (previously called the threshold) are increasingfunctions of 02/p. The region of 6, for which we havean instability has narrowed and the maximum growthrate now occurs for 6, > 0.

In fig. 4 we plot IA II versus r for one mode with

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I :-1.6 -0.8 0 . 0 0.8 1.6

DELTAFig. 3. Plot of imaginary part of roots of h3 - 6,h*+ (pFj (K)/2 - o/p)h + iFj(K)n + 026,/p = 0 versus6 1 for the three modes (a) n = 1, (b) n = 3 and (c) n = 5 in-cluding space charge, p = 0.1 and o = 0.156.

and without space charge for K = 1, p = 0.1. Spacecharge causes the reduction in the initial exponentialgrowth, reduces the peak field amplitude and slightly

0d I0 4 8

TAUFig. 4. (a), (b) I.4 11 ersus 7 for a one mode system with andwithout space charge respectively. The initial conditions forboth figures are: 50 particles, Eo = 0.1, p = 0.1, 6 1 = 0, K = 3.0 = 0.156.

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Volume 53, number 3 OPTICS COMMUNICATIONS 1 March 1985increases the time it takes for the field to peak.

For a2/p > 1 the maximum growth rate for thefundamental mode occurs at 6, = (cJ~/~)/~. If thedefinitions for &I, a2 and p are substituted into thisexpression it can be shown that this is the stimulatedRaman scattering synchronism condition [4,8]

2ck,$ 2Yi+Jw = 1+K2/2 - (1 + K2/2)l12 . (17)

For the discussion to follow we shall ignore spacecharge and consider only the fundamental field mode,but we allow the electron beam to have an energyspread characterized by a distribution functionF(+) =(l/&oJ exp[-(+ - 1)2/2G~] , (18)where i = y/(y) and G_r= u,/(r) is the r.m.s. energyspread. Since we have only modified the initial condi-tions the equations describing this system are stillthose given in eqs. (6), (7) and (11) with the modifi-cation that r. is taken to be (lo). In fig. 5 we plotthe electric field amplitude versus r for several valuesof G,r. The initial bunching parameter is taken to beTJo= 9.1 x 10-3 which is characteristic of the valuesobtainable in an electron storage ring which has beenoptimized to yield high electron densities, n - 1015cmp3, suitable for short wavelength operation, h -100 A [9]. It can be seen that the system maintains areasonable amount of gain for oT - p.

Y-[

+bci

.I

0 TAU I5

Fig. 5. IA 11 ersus 7 for several values of the normalized ener-gy spread, a.,&-y). The initial conditions for the three figuresare: 1000 particles, qo = 9.1 X 10e3, p = 3 X 10e3, 6 = 0 and(a) Or = O.lp, (b) $ = 0.5p, (c) I+ = p.

We can attribute the reduction in gain as the energyspread in the beam increases to Landau damping. Thecollective properties of the system introduces a fre-quency shift of 6w = 2wo p(rR/ro)2 Re(h) to be res-onant frequency of the system. Since Re(X) and(rR/yo) are on the order of unity the approximatefrequency shift is 6w = 2wo p. When this frequencyshift is on the order of the spread in frequencies causedby the velocity spread in the beam, the beam must beconsidered warm and Landau damping begins to eataway at the gain. The frequency spread due to thevelocity spread in the beam, Aw, is determined fromthe synchronism condition to be, Ao = -(k t k,)Au- 2wo (AT/~). The limits on the energy spread forwhich the beam can be considered cold is then ob-tained from the condition that 60 = Aw which gives(Arlr) = P . (19)

To design an experiment to explore the collectiveinstability regime it would be very useful to know howlong to make the undulator so that saturation isreached beyond or near the end of the device. We haveobtained the same empirical formula as given in BNPfor the time for the fundamental mode to reach themaximum of the first peak from an initial state of A,= 0 and TJ# 0. The lethargy time is given byrleth = -ln(rIO)/Im@) + constant , (20)where v. = (exp(iOi)) is the initial bunching parameterand Im(X) is the imaginary part of the root of the cu-bic dispersion relation. The additive constant is on theorder of 2 and is typically in the range 1.5 -2.5.

As a final note we would like to point out a rela-tion between the maximum allowable energy spreadand the number of periods in the wiggler, N, . The nor-malized time r for an electron to traverse the wiggler,can be written assuming yo/yR - 1, asr = 4rrpNw . (2 1)Typical values of r at which the field peaks are on theorder of 10, for the example considered in fig. 5 [9],so that N, is related to p byN, = l/p . (22)If we couple this result with eq. (19) we see that theenergy spread should be less than l/N,, i.e.,Arlr = l/N, . (23)

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For the general case one must use eq. (20) to deter- Research supported by the U.S. Department ofmine rleth, but since the dependence of rfeth on v. is Energy Contract #DE-AC02-76CH00016.weak eq. (23) will only be modified by a multiplicativefactor on the order of unity. When the conditions (22),(23) are satisfied the radiation field produced by the Referenceselectrons starting from noise reaches its peak valueIA, 12 - 1. [l] R. Bonifacio, L. Narducci and C. Pellegrini, Proc. Topical

The higher harmonics reach a peak at about thistime but their maximum amplitude is lower by at leastone order of magnitude. Using this result and the invar-iant (13) we see that, at the peak, the energy transferfrom the electron beam to the radiation field is of the 12order of p.

We have generalized the FEL equations (6), (7), (11)to include the effect of space charge and higher harmon-

[3Iics. Using the dispersion relation (15) one can evaluatea region of parameters p, F, where the system is un- [4stable and a large energy transfer between the electron [5beam and the radiation field can take place. In this re-gion we have followed the system evolution to satura- (6tion and have established the conditions (22), (23) onthe wiggler length and electron beam energy spread to

Meeting on Free electron generation of extreme ultra-violet coherent radiation, Brookhaven National Labora-tory, Sept. 1983. AIP Conf. Proc. No. 118, Subseries onOpt. Sci. Eng. No. 4 (1984) p. 236.A. Renieri, Proc. Topical Meeting on Free electron gener-ation of extreme ultraviolet coherent radiation, BrookhavenNational Laboratory, Sept. 1983. AIP Conf. Proc. No. 118,Subseries on Opt. Sci. Eng. No. 4 (1984) p. 1.L.A. Vainshtein, Sov. Phys. Tech. Phys. 24 (1979) 625,629.V.L. Bratman, N.S. Ginzburg and MI. Petelin, Sov. Phys.JETP 49 (1979) 469.R. Bonifacio, L.M. Narducci and C. Pellegrini, OpticsComm. 50 (1984) 373.J. Gea-Banacloche, G.T. Moore and M.O. Scully, Freeelectron generators of coherent radiation, eds.CA. Brau, SF. Jacobs and M.O. Scully, Proc. SPIE453 (1984) p. 393. . .[7] J.R. Pierce, Traveling-wave tubes (Van Nostrand, 1950).

[8] T.C. Marshall, S.P. Schlesinger and D.B. McDermott,Advances in electronics and electron physics, Vol. 53(Academic Press (1980) p. 47.

reach the maximum radiation intensity, and obtain amaximum energy transfer, of the order of p, from theelectrons to the radiation field.

[9] J.B. Murphy and C. Pellegrini, Generation of high inten-sity cpherent radiation in the soft X-ray regime, 3. Opt.Sot. Am. B, 1984, to be published.

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