Bonifacio 1984 0075

Volume 50, number 6

OPTICS COMMUNICATIONS

15 July 1984

COLLECTIVE INSTABILITIES AND HIGH-GAIN REGIME IN A FREE ELECTRON LASER R. BONIFACIO *, C. PELLEGRINI National Synchrotron Light Source, Brookhaven National Laboratory, Upton, N Y 119 73, USA and L.M. NARDUCCI Physics Departmen t, Drexel University, Philadelphia, PA 19104, USA Received 5 April 1984

We study the behavior of a free electron laser in the high gain regime, and the conditions for the emergence of a collective instability in the electron beam-undulator-field system. Our equations, in the appropriate limit, yield the traditional small gain formula. In the nonlinear regime, numerical solutions of the coupled equations of motion support the correctness of our proposed empirical estimator for the build-up time of the pulses, and indicate the existence of optimum parameters for the production of high peak-power radiation.

Studies of the free electron laser (FEL) in the high gain regime have shown that with an appropriate selection of the electron density, detuning and undulator length, the radiation field and the electron bunching can undergo exponential growth as a result of a collective instability o f the electron beamundulator-radiation field system [ 1 - 8 ] . In this paper, we study the conditions for the onset of this instability using a new secular equation for the characteristic complex frequencies of the FEL system. On the basis of these results, we show how one can rederive the small-signal gain formula and establish the conditions for its validity. We also consider the problem of the initiation of laser action and o f the growth of the radiation field from noise, and propose a formula to evaluate the lethargy (build-up) time of the first pulse. Finally, we study the nonlinear regime of the FEL by numerical methods and obtain remits that suggest the existence o f an optimum efficiency of the device. In the derivation of our working equations we select the phase and the energy as the basic electron variables, and assume the slowly varying phase and * On leave from the University of Milano, via Celoria 16, Milano, Italy. 0 030-4018/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

amplitude approximation for the radiation field as done also in earlier developments [9,10]. In the remainder of the paper we shall adopt the following notations: z represents the direction of propagation of the electron beam and o f the electromagnetic wave; it also represents the undulator axis;x a n d y are the transverse coordinates; B 0 denotes the strength of the helical magnetic field and ~'0 and N O the period length and the number of periods of the undulator, respectively; the undulator parameter is K = eBoko/ (21rmc2), where m c 2 is the electron rest energy; k is the wavelength o f the radiation field, 3' is the electron energy in units o f m c 2,/3z ~ 1 is the longitudinal electron velocity and/~p = K/T the amplitude of the transverse velocity; the electron phase, 4, relative to that o f the electromagnetic wave, is connected to z and t by the relation ~ = 27rz/X 0 + 2n(z - cO~k; the resonant energy 3tR is related to k0, k and K by T2 = ~,0(1 + r 2)/2;k, and, finally, the undulator frequency 6o0 is given by 600 = 27rC13z/XO. With these notations, the FEL working equations can be written as [9,10] ~bi = ~0(1 - T 2 / 7 2 ) , Ti-

ecK [a exp(i~b/) + c.c.] , 2mc27/

(1) (2)

373

(o

1a

(e-i+/3'>

+c~7!

(3)

where j labels t h e / t h electron in the beam (J" = 1,2, ..., Are, with N e the total number of electrons); the average C ) is carried out over all electrons in a beam slice of length X at the position z - (l~z)Ct, where q~z) is the average longitudinal velocity. The remaining parameters have the following meaning: n e is the electron beam longitudinal density at position z (#z)Ct, F, is an effective beam transverse cross section describing the overlap of the beam with the radiation field whose amplitude E 0 and phase 00 have been combined in the complex amplitude a = iE 0 exp(i00). It is important to stress that in this discussion 3' is not restricted to be approximately equal to the resonant value 3'R, unlike earlier treatments of this problem. For the purpose of our subsequent analysis, it is convenient to rewrite eqs. (1.3) using the variables z' = z

-

([3z)Ct ,

(4)

t' = t ,

with the result:

(alat')Oj =

6o0(I

ecK

where 3'0 is the initial energy, and r e the classical electron radius, and the so-called Pierce parameter p

= (~K(3'0/3'R)2~2p/¢O0)2/3 .

3'~I~- ),

(5)

[t~ exp(-i¢/) + c.c.],

(6)

[(1--(/3z))~z,+l~7]-27rne(Z')~(e-i¢/3') c a a-

z,

(7)

<#z>)a/az'

The propagation term (1 in eq. (7) is important to describe the evolution of the pulse in the FEL, especially when the accumulated path difference AL = Lph - Lel = (c - o)tin t between the photons and the electrons during an interaction time is comparable to the length of the electron bunch itself. Note that the path difference AL can also be expressed in the form Ctint(1 - (/~z)) = X0N0(1-(/3z)) = NOX. In this paper, we only consider situations where the length of the electron bunch is sufficiently larger than NOA; thus, we neglect the propagation term and assume the local electron density ne(g' ) to be constant. The linear stability analysis of eqs. (5)--(7) is greatly aided by the introduction of a suitable set of collective variables [8]. For this purpose, we first introduce the relativistic plasma frequency ~ p = (4rrrenoC2/3"30)l/2 ,

(8)

(9)

Furthermore, we introduce the quantity ~0

=

c°0(1

-

(10)

7213'2),

and rescale the time variable as follows: (11)

r = 2WOP(T R/3" o)2t.

In terms of the new scaled variables 4i=¢i-b0

t '

rj-3'i/(p3'0) ,

(12)

A - a e x p ( i ~ o t ) [ ( 4 , m c 2 3 " O n O p 2 ) 1/2 ,

the nonlinear equations of motion ( 5 ) - ( 7 ) take the form ( d / d r ) 4 ] = (112p)(1 - 1 ] 0 2 r 2 ) , (d/dz)rj =

-(llp)[(A/r i) exp(i4/-)+

dA/dr = iaA + -

(a/at')3'j - 2mc23".i

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Volume 50, number 6

(]/p)<e-i* Iv>.

(13) c.c.] ,

(14)

(15)

Note that in terms of eqs. (13)--(15), the dynamics of the FEL is controlled by only two parameters, the Pierce parameter p (eq. (9)) and 6 = A / p , where A is the usual detuning (3'02 - 3'2R)/(23'2R). Because we neglect space-charge forces, we shall assume in the following that p is sufficiently smaller than unity. It is also worth noting that eqs. ( 1 3 ) - ( 1 5 ) are consistent with the conservation law L = IA [2 + (F) = constant,

(16)

or also L = mc2no(3") + E2/4r¢ = constant,

(16')

which can be readily recognized as the conservation of energy for the electron beam-radiation field system. The method devised to analyze the stability of the system is based on the procedure suggested in ref. [8]. The equations are linearized around the equilibrium state A 0 = 0, F0j = l/p, (exp(-in40)) = 0 and perturbed by letting A = a, Fi = (l/p)(1 + 7/1-)and 4 i = 4 0 / + 6 4 i. The linearlized equations form the basis for a closed form linear system of equations for the collective variables x = (64 e x p ( - i 4 0 ) ) ,

(17)

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OPTICS COMMUNICATIONS

y = (1/p)(W exp(-i~k0)),

15 July 1984

/

(18)

and for the field perturbation a. These take the form dxldr --y

(19)

dy/dr = - a ,

(20)

da/dT" = - i S a

-

ix - py

0.5 J

(21)

.

Nontrivial solutions with a time dependence o f type exp(iXr) exist if and only if X is a solution o f the characteristic equation n5

X3 - 8X 2 + pX + 1 = 0 .

(22)

The results o f earlier analyses [ 1 - 8 ] can be recovered by setting formally p = 0 in eq. (22). Clearly, exponential growths, and thus, unstable behavior, results if the cubic equation (22) has one real and two complex conjugate roots. In this case, the imaginary part o f the eigenvalue measured the rate o f growth o f the unstable solution. The instability condition can be easily derived from eq. (22): in terms o f the parameters p and 8 it takes the form (fig. 1) O3 - - ~ p 2 6 2 * 9p6 - 6 + ~ - ~ > 0 .

(23)

The typical behavior o f the eigenvalues o f eq. (22) as a function o f detuning is shown in fig. 2. The eigenvalues are real when 6 exceeds a certain threshold value that depends on p according to eq. (23), while two o f the eigenvalues form a complex conjugate pair when 6 < 8th r. The small signal gain formula emerges in a natural way from our analysis in the limit p ~ 0, and for suf-

± 10

J

[

0

(~thr

~

10

Fig. 2. The behavior of three eigenvalues of the secular equation as a function of the detuning parameter a and for p = 0.1. The vertical axis labels both the real and imaginary parts. The real parts have been scaled by a factor of 10 to fit the display. For a sufficiently positive value of a (i.e., a > 6 thr "the eigenvalues are real (curves c, d, e). At threshold, two of the real eigenvalues degenerate into one, while, for the same value of a, the imaginary parts (curves b, b') become different from zero. The real part of the complex conjugate eigenvalues for 8 < a thr is labelled by a. ficiently large values of 16I. In this limit, the eigenvalues take the approximate form X1~8(1-1/6),

X2,3"~+1/6 1/2,

X1~6(1-1/6),

X2,3"+1/1611/2,

8>0, 8<0,

(24)

as one can conFLrrn qualitatively from fig. 2. The outp u t field A ( r ) in the linear regime can be calculated as a linear superposition o f elementary exponential func tions whose coefficients are to be FLxed from the initial conditions. A lengthy, b u t straightforward calculation yields the following expressions for the small signal gain: G = [lA(r)l 2 -- IA012]/IA012 a*

8

Fig. 1. Instability boundary in the ~o, a) plane. For 8 < ~ *, the solutions of eqs. (13)-(15) are unstable for all values of p. For selected values ofp (e.g., fi" in the figure) unstable behavior occurs for a < 6 thr"

= (4/8 3)(1 - cos 6 r cos r/x/r6 --~8 3/2

sinSrsinr/x/r6),

8 >0,

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Volume 50, number 6

OPTICS COMMUNICATIONS =

G = (4/63)(1 - cos fir cosh z / V ~

-~1613/2 s i n 6 r s i n h r / x / ~ ] ) ,

15 July 1984

6 <0.

(25)

IA

In order to make contact with the usual small-signal gain formula, it is not enough to require that 16[ be sufficiently larger than unity, but one also must impose the condition r [ x / ~ < 1. In this case, eq. (25) becomes 1

G "~ (4/63)(1 - cos 6T -- ~6"C fan 6 z )

(26)

which, in fact, agrees with the standard expression for G. In spite of the fact that the equations of motion of the FEL are nonlinear, some aspects of this problem can be handled accurately by analytic means. The evolution below threshold (6 > 6thr) is govemed by the linear approximation. In this regime, the eigenvalues are real (see fig. 2) and the output field displays small amplitude oscillations when plotted as a function of time. On varying 6, beat patterns or more complicated-looking modulation effects can be observed, whose origin can be understood entirely in terms of the eigenvalues of the linearized problem. A representative example is shown in fig. 3. It may be worth mentioning that while the trace in fig. 3 has been obtained by the appropriate superposition of exponential functions, the exact solution of eqs. (13)-(15) is indistinguishable on the scale of this graph. The system evolution above threshold (6 < 6 thr)

0

10

20

Z"

4O

Fig. 3. Output intensity IA12 for p = 0.01 and 8 = 4.0. The eigenvalues of the linearized equations are -0.519, 0.628, 3.066. The modulation is due to the beat of the different exponential terms in the solution. 376

20

Fig. 4. Output intensity 1,412versus time above threshold. The parameters used in this simulation are p = 0.0021, 8 = 1.86, no = 16.

is entirely different, and is shown in fig. 4 for the case of zero initial field and an initial bunching parameter I(exp(-i~k))l, small, but different from zero. Under unstable conditions, fluctuations in the electrons injection velocities, or the lack of uniformity in the initial distr~ution of the electron phase variables, or the presence of an initial field will trigger the growth of a signal. The signal will then grow to a peak value after which it oscillates. This behavior is very general and is independent of the initial triggering mechanism as long as this perturbation is small. This nonlinear regime require s numerical integration of the full equations of motion. This we have done for a number of values o f p and 6. Because of the nature of the triggering mechanism, intuitively, one would expect that the time required for the initial pulse to build up (lethargy time) should be a fairly sensitive function of the magnitude of the initial fluctuation. We have examined the dependence of the build up time of the first pulse on the initial value of the bunching parameter, and verified that (a) a significant fraction of the build up process is well described by the linearized equations; and (b) the arrival time of the first peak is well described by the formula: 7"peak = --(1/Im X) In I(exp(-i~k0)l + 1 .

0

"C'

(27)

A test of this equation is provided in fig. 5, where we have plotted the arrival time of the first pulse calculated from the nonlinear equations of motion ( 1 3 ) (15), as a function of the initial bunching parameter I(exp(-i~b0))l. One aspect of considerable interest for

Volume 50, number 6

OFrlCS COMMUNICATIONS

20

10

I

I

o

,o

I

~,,1< ,,o)1

Fig. 5. The arrival time of the first peak (lethargy time) is plotted as a function of the logarithm of the initial bunching parameter (dots). The solid curve corresponds to eq. (26). The parameters used in this scan are no = 8, p = 0.4, = 1.25.

2

PIAImax

0.4

15 July 1984

the purpose of optimizing the system's parameters is the existence o f a maximum peak power output as a function o f p and 5. We have verified that while the maximum growth rate is obtained for ~ ~ 0, the maximum peak amplitude o f the first pulse occurs for ~ ~ t~th r. Thus, we have scanned the (/9, 6) plane in the neighborhood of, but above, the threshold line and for A 0 = 0, and recorded the peak output intensity p[A[2ax as a function o f p (fig. 6). Notice that it follows from eq. (16) that p]A[ 2 = ((Tf - 70)/70), so that p[A[ 2 gives the energy transfer from the electrons to the radiation. The scatter o f the points is almost certainly due to the slight variations o f the conditions from run to run. The solid line, which is only a qualitative average through the points, suggests the existence o f an optimum gain-detuning condition such that the efficiency o f the system is maximum for operation just above threshold. It is clear that in the presence o f efficiencies as large as, in principle, 40%, the old approximate treatments [ 1 - 8 ] in which the electron momentum is assumed to vary only by small amounts cannot be adequate to describe situations where such large energy exchanges take place between the electron beam and field. On the other hand, it is intuitively obvious that for sufficiently small values of the Pierce parameter, the electron energy will suffer only a limited depletion so that earlier treatments should be sufficiently accurate.

Acknowledgements

0.2 I.-

I O01

One of us (LMN) wishes to acknowledge the support of the Army Research Office and the Research Laboratories o f the Martin-Marietta Corporation. We wish to thank Dr. J. Murphy for many useful discusions. The help of H. Sadiky with some of the numerical computations is also gratefully acknowledged. This work has been partially supported by the U.S. Department o f Energy.

/ •

I

] 0.1

I p

Fig. 6. Dependence of the peak output intensity pPilmax on p in the neighborhood and just above the instability boundary line of fig. 1. The solid line is only a qualitative average of the points.

References

[ 1] N.M. Kroll and W.A. McMullin, Phys. Rev. A17 (1978) 300. [2] A. Gover and Z. Livni, Optics Comm. 26 (1978) 375. [3] I.B. Bemstein and J.L. Hirschfeld, Phys. Rev. A20 (1979) 1661. 377

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OPTICS COMMUNICATIONS

[4] C.C. Shil and A. Yariv, IEEE J. Quantum Electron QE17 (1981) 1387. [5] P. Sprangle, C.M. Tang and W.M. Manheimer, Phys. Rev. A20 (1980) 302. [6] G. Dattoli, A. Marino, A. Renieri and F. Romanelli, IEEE J. Quantum Electron. QE17 (1981) 1371. [ 7] A. Gover and P. Sprangle, IEEE J. Quantum Electron. QE17 (1981) 1196.

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[ 8] R. Bonifacio, F. Casagrande and G. Cascati, Optics Comm. 40 (1982) 219. [9] W.B. Colson and S.K. Ride, in: Physics of quantum electronics, Vol. 7, eds. S.F. Jacobs et al. (AddisonWesley, Reading, MA 1980) p. 377. [ 10] C. Pellegrini, in: Free electron lasers, eds. A.N. Chester and S. Martellucci (Plenum Press, NY 1983) p. 91. [ 11] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover Publications, NY, 1970) eq. (3.8.2), p. 17.