Murphy 1985 221

Volume

53, number

1 March 1985

OPTICS COMMUNICATIONS

3

COLLECTIVE INSTABILITY

OF A FREE ELECTRON LASER

INCLUDING SPACE CHARGE AND HARMONICS

J .B. MURPHY and C. PELLEGRINI Brookhaven

National Laboratory,

Upton, NY I1 973, USA

and R. BONIFACIO University of Milan, Milan, Italy Received 19 March Revised manuscript

1984 received

19 October

1984

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

Recently there has been interest in the collective instability regime of the free electron laser (FEL) [ 1,2]. There is also some work on the collective regime appearing in the Russian literature [3,4]. Bonifacio, Narducci and Pellegrini [5] (referred to hereafter as BNP) have discussed the collective instability regime of an FEL studying the growth of the radiation field from noise and the saturation level for the case of a cold electron beam, ignoring the effects of space charge and higher harmonics of the fundamental operating frequency, o = 2ck,y2/(1 + iK2). There has also been some analytic work done on the effects of energy spread on the collective instability [6] _We shall extend BNP’s work to include the effects of higher harmonics, space charge and an energy spread in the electron beam. A planar undulator is assumed throughout the discussion. We shall use the notation of BNP: z is the direction of propagation of the electron beam; x and y are the transverse coordinates; the electron beam is characterized by its energy, y, in units of the rest energy, nz,, c2, and n, the electron number density which are combined 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 undulator 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, @,is measured relative to the pondermotive wave and is taken as $ = 2nzlho t 21rz/h - wt; the resonant energy, yr, is determined from the synchronism condition to be -rF = &,(l + ;K2)/2h; t h e undulator frequency w. is w. = 2nc&/~ ; the radiation electric field is assumed of the form CE,(z,t)exp[-(2nin/X)(z-ct)],

E,(z,t)=

(1)

where En (z, t) is a slowly varying complex amplitude. Using these definitions we can construct a set of normalized variables [ 11. For the electron phase we have e =@-c&t

;

(2)

for the electron energy: F = YlPYg 3 and for the complex electric field amplitude nth harmonic A, =

(3) for the

iE, exp (i40 t) (47m&yo n, p)1/2 ’

(4)

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OPTICS COMMUNICATIONS

where y0 is the initial electron energy,

tude and phase approximation

2/S ,

P = (KY&/4Y&Jg)

(5)

c and &, = wu(1 -‘y’,/$). r = 2w()p(rR/ro)2 The starting point for our discussions are the BNP equations to which we add the effects of space charge and generalize them to include higher harmonics. The equation for the particle phase remains unchanged e’. I = (1/2p)(l

- lip2r.2) (6) I ’ wherej = 1: 2, . . ..N is the particle index. The dot indicates differentiation with respect to r. The equation for the particle energy becomes

AnFnCK> ia

+‘C

(

Pn

2rj

X exp(inOi)+c.c.,

+n

(eXp(-tiei))

n = 1,3,5 ,...

1

(7)

where

= 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, containing u, is the space charge term. To derive the contribution of the space charge forces to the evolution of the system we begin with the particle density as a function of 0 : n (0) = (2nnJN) Zi 6 (0 - ej). We expand this in a Fourier series in 0 and obtain n(0) = tie Z.n (exp(-tiei)) exp(ine) t C.C.The electric field associated with this charge distribution is determined from Poisson’s equation to be E, =

(9)

The contribution of the space charge force to the energy change of the jth particle is determined from mc2dyldt = -ev, E, to be -7

u2c 12

(exp(-tiei))

exp(tie.)

in

J +c.c.

(10)

The equation for the nth harmonic of the complex electric field amplitude in the slowly varying ampli198

A, = iF,A,

is

F,, (K) +~ (exp(-inOJri), P

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. The angular brackets indicate averages over the particle initial phases, i.e., ( ) = l/NCj where N is the number of particles. F is the detuning divided by p. =n6,

(12)

The equation for the electron energy change (7) and the field equation (11) contain only odd harmonics, in agreement with the characteristics of the spontaneous radiation on axis for a planar undulator. From the set of equations (6), (7), (11) we can obtain an invariant (13)

F, 0-0

i;

1 March 1985

which is the energy conservation relation. The set of equations (6), (7), (11) only contains two parameters, the detuning of the fundamental mode, and p. p is a measure of the electron beam density and is related to the Pierce parameter C of the microwave tube literature [7]. Typical values of p are 10e5 -1O-3 for a 0.5-l .O GeV storage ring electron beam and 10-l for a l-5 MeV, 1 kiloamp electron beam in a linac. The quantity qn = I(exp(inO)/rp)l,n = 1,3,5, . . . . is called the nth harmonic bunching parameter. It is a measure of the degree of bunching in the electron beam. For n > 1, qn is a measure of the harmonic content of the perturbed beam current. At the entry to the indulator an unbunched beam has nn = 0 for all n. As the system evolves the bunching parameter reaches its maximum value on the order of 1. The initial conditions that we will use with this set of equations for a monoenergetic beam are ri = 1/p, i=l , . . .. N and either A, # 0 and Bi = 2(i - l)n/N for i=l , . . . . N, i.e., no initial bunching, or A, = 0 and q, f 0, i.e., the electrons are displaced slightly from their uniform initial distribution. These 0 displacements can be produced by fluctuations in the longitudinal electron distribution and we will refer to it as bunching produced by noise. As was shown by BNP the state with particles uniformly distributed in phase in the range

Volume 53, number 3

OPTICS COMMUNICATIONS

1 March 1985

[0,2n], r. = l/p and A,,= 0 is an equilibrium state. In order for the system to evolve we must introduce a small initial field or some noise in the initial phases. If N, electrons are distributed randomly over one wavelength of the radiation the initial bunching parameter is (14) We want now to study our FEL equations; for simplicity we start from the case u = 0, ignoring the space charge effect. Following the procedure given by BNP we linearize this system about the equilibrium position Bi = 2n(i - 1)/N, F = l/p, and A, = 0,introduce collective variables and look for solutions proportional to exp(ihr). We then find that the modes are decoupled and the dispersion relation for each mode is a cubic. The dispersion relation for the nth harmonic is x3 - 6,Q

+ ;F;(K)@x

+ n) = 0 .

(15)

Since the coefficients in the cubic are all real the roots are either all real or there is one real root and a complex conjugate pair. For a fixed value of p there is a threshold value of 6, above which the nth mode is linearly stable. The system is unstable for all negative values of 6, but the growth rate decreases with increasing 16, I. Notice that since F,, (K)tends to zero for fixed K, p and 6, when n becomes larger, the roots of (15) tend to become all real in this limit. BNP have shown that the effect of the term proportional to p is to further destabilize the system. Both the threshold value of the detuning and the magnitude of the growth rate are increasing functions of p. In fig. 1 we plot the value of Im(A) versus 61 for p = 0.1. For p in the range lop4 < p < 10-l the maximum value of Im(h) is approximately Im(h) -(; nFn(K))li3 J-12 3 an d occurs at 61 = 0. It can be seen that the decrease in F,(K)as n increases results in both lower growth rates and threshold values of 6 1 for the harmonics. The curve labelled (a) in fig. 2 is a typical plot of IA11versus r for one field mode starting from a small initial field and some noise in the initial electron phase distribution. The wave exhibits an initial exponential growth at the growth rate computed from the dispersion relation, eq. (15). The field reaches a peak value after which no further exchange of energy between 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,respectively.

-I ‘4

Fig. 2. (a) IA 1I versus 7 for one mode starting from a small initial field. (b) IA 3 1versus 7 for the n = 3 mode of a competing two mode system. The initial conditions for both figures 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 trapped in the potential well of the wave. To determine the effects of harmonics, we integrated equations (6), (7) and (11) for two field modes, rz = 1 and 3 without space charge. A plot of the fundamental electric field versus time remains the same within the resolution of the figure as the single mode case (fig. 2a). Fig. 2b is a plot of the electric amplitude of then = 3 mode versus time. The peak field amplitude of the two modes differs by an order of magnitude, with the fundamental mode reaching a higher amplitude. Both modes saturate at about the same time indicating that once particles are trapped in the potential well of the fundamental mode net energy exchange between the particles and any of the modes ceases. Exactly how much energy appears in the harmonics depends on p, F and the initial conditions. Now we shall examine the effects of space charge on the system. If we linearize the system of equations including both the harmonics and space charge we obtain the following cubic to determine the linear growth rates for the nth harmonic

I -1.6

1985

: 0. 0

-0.8

0.8

1.6

DELTA

Fig. 3. + (pFj 6 1 for cluding

Plot of imaginary part of roots of h3 - 6,h* (K)/2 - o’/p)h + iFj(K)n + 026,/p = 0 versus the three modes (a) n = 1, (b) n = 3 and (c) n = 5 inspace charge, p = 0.1 and o = 0.156.

X3 - 6, X2 + (pF;(K)/2 - 02/p) h + nF,2(K)/2 + a26,1p= 0.

(16)

This cubic differs from the previous one by the terms proportional to u2. Recalling that BNP found that the term ph is destabilizing, it can be seen that the space charge has a stabilizing effect and competes for control of the system. Since the space charge term a2/p is proportional to p it will be negligible for systems with p < 10p2, such as storage ring devices. A more complete analysis of the cubic reveals that in contrast to the system without space charge the system with space charge has a lower bound on 6, below which the system is stable. In fig. 3 we plot Im(h) versus 6, for K = 3, p = 0.1. A comparison of figs. 1 and 3 clearly indicates the effects of space charge on the linear growth rate of the system. Large negative values of 6, no longer lead to an instability. In fact both the lower and the upper bounds on 6, (previously called the threshold) are increasing functions of 02/p. The region of 6, for which we have an instability has narrowed and the maximum growth rate now occurs for 6, > 0. In fig. 4 we plot IA II versus r for one mode with 200

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

0

I

d 0

4

8 TAU

Fig. 4. (a), (b) I.4 1 1versus 7 for a one mode system with and without space charge respectively. The initial conditions for both figures are: 50 particles, Eo = 0.1, p = 0.1, 6 1 = 0, K = 3. 0 = 0.156.

OPTICS COMMUNICATIONS

Volume 53, number 3

increases the time it takes for the field to peak. For a2/p > 1 the maximum growth rate for the fundamental mode occurs at 6, = (cJ~/~)‘/~. If the definitions for &I, a2 and p are substituted into this expression it can be shown that this is the stimulated Raman scattering synchronism condition [4,8]

2ck,$ w = 1+K2/2

2Yif+J - (1 + K2/2)l12

.

(17)

For the discussion to follow we shall ignore space charge and consider only the fundamental field mode, but we allow the electron beam to have an energy spread characterized by a distribution function F(+) =(l/&oJ

exp[-(+

- 1)2/2G~] ,

(18)

where i = y/(y) and G_r= u,/(r) is the r.m.s. energy spread. Since we have only modified the initial conditions the equations describing this system are still those given in eqs. (6), (7) and (11) with the modification that r. is taken to be (l’o). In fig. 5 we plot the electric field amplitude versus r for several values of G,r. The initial bunching parameter is taken to be TJo= 9.1 x 10-3 which is characteristic of the values obtainable in an electron storage ring which has been optimized to yield high electron densities, n - 1015 cmp3, suitable for short wavelength operation, h 100 A [9]. It can be seen that the system maintains a reasonable amount of gain for oT - p.

Y-[

+ b

ci

1 March 1985

We can attribute the reduction in gain as the energy spread in the beam increases to Landau damping. The collective properties of the system introduces a frequency shift of 6w = 2wo p(rR/ro)2 Re(h) to be resonant frequency of the system. Since Re(X) and (rR/yo) are on the order of unity the approximate frequency shift is 6w = 2wo p. When this frequency shift is on the order of the spread in frequencies caused by the velocity spread in the beam, the beam must be considered warm and Landau damping begins to eat away at the gain. The frequency spread due to the velocity spread in the beam, Aw, is determined from the synchronism condition to be, Ao = -(k t k,)Au - 2wo (AT/~). The limits on the energy spread for which the beam can be considered cold is then obtained from the condition that 60 = Aw which gives = P .

(Arlr)

(19)

To design an experiment to explore the collective instability regime it would be very useful to know how long to make the undulator so that saturation is reached beyond or near the end of the device. We have obtained the same empirical formula as given in BNP for the time for the fundamental mode to reach the maximum of the first peak from an initial state of A, = 0 and TJ# 0. The lethargy time is given by rleth

=

-ln(rIO)/Im@)

I

Fig. 5. IA 11versus 7 for several values of the normalized

(20)

(2 1)

Typical values of r at which the field peaks are on the order of 10, for the example considered in fig. 5 [9], so that N, is related to p by

. TAU

,

where v. = (exp(iOi)) is the initial bunching parameter and Im(X) is the imaginary part of the root of the cubic dispersion relation. The additive constant is on the order of 2 and is typically in the range 1.5 -2.5. As a final note we would like to point out a relation between the maximum allowable energy spread and the number of periods in the wiggler, N, . The normalized time r for an electron to traverse the wiggler, can be written assuming yo/yR - 1, as r = 4rrpNw .

0

+ constant

I5

energy spread, a.,&-y). The initial conditions for the three figures are: 1000 particles, qo = 9.1 X 10e3, p = 3 X 10e3, 6 = 0 and (a) Or = O.lp, (b) $ = 0.5p, (c) I+ = p.

N, = l/p .

(22)

If we couple this result with eq. (19) we see that the energy spread should be less than l/N,, i.e., Arlr = l/N,

.

(23) 201

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For the general case one must use eq. (20) to determine rleth, but since the dependence of rfeth on v. is weak eq. (23) will only be modified by a multiplicative factor on the order of unity. When the conditions (22), (23) are satisfied the radiation field produced by the electrons starting from noise reaches its peak value IA, 12 - 1. The higher harmonics reach a peak at about this time but their maximum amplitude is lower by at least one order of magnitude. Using this result and the invariant (13) we see that, at the peak, the energy transfer from the electron beam to the radiation field is of the order of p. We have generalized the FEL equations (6), (7), (11) to include the effect of space charge and higher harmonics. Using the dispersion relation (15) one can evaluate a region of parameters p, F, where the system is unstable and a large energy transfer between the electron beam and the radiation field can take place. In this region we have followed the system evolution to saturation and have established the conditions (22), (23) on the wiggler length and electron beam energy spread to reach the maximum radiation intensity, and obtain a maximum energy transfer, of the order of p, from the electrons to the radiation field.

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1 March

Research supported by the U.S. Department Energy Contract #DE-AC02-76CH00016.

1985

of

References [l]

R. Bonifacio, L. Narducci and C. Pellegrini, Proc. Topical Meeting on Free electron generation of extreme ultraviolet coherent radiation, Brookhaven National Laboratory, Sept. 1983. AIP Conf. Proc. No. 118, Subseries on Opt. Sci. Eng. No. 4 (1984) p. 236. 12I A. Renieri, Proc. Topical Meeting on Free electron generation of extreme ultraviolet coherent radiation, Brookhaven National Laboratory, Sept. 1983. AIP Conf. Proc. No. 118, [3 Subseries on Opt. Sci. Eng. No. 4 (1984) p. 1. I L.A. Vainshtein, Sov. Phys. Tech. Phys. 24 (1979) 625, 629. [4 V.L. Bratman, N.S. Ginzburg and MI. Petelin, Sov. Phys. JETP 49 (1979) 469. [5 R. Bonifacio, L.M. Narducci and C. Pellegrini, Optics Comm. 50 (1984) 373. G.T. Moore and M.O. Scully, Free (6 J. Gea-Banacloche, electron generators of coherent radiation, eds. CA. Brau, SF. Jacobs and M.O. Scully, Proc. SPIE 453 (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. [9] J.B. Murphy and C. Pellegrini, Generation of high intensity cpherent radiation in the soft X-ray regime, 3. Opt. Sot. Am. B, 1984, to be published.