The features of an FEL oscillator with a tapered undulator

OPTICS COMMUNICATIONS

Optics Communications 103 (1993) 297-306 North-Holland

F u l l length article

The features of an FEL oscillator with a tapered undulator E.L. Saldin, E.A. S c h n e i d m i l l e r Automatic Systems Corporation, Smyshlyaevskoe shosse la, Samara 443050, Russian Federation

and M.V. Y u r k o v Joint Institute for Nuclear Research, P.O. Box 79, Head Post Office, Moscow 10100, Russian Federation

Received 17 November 1992; revised manuscript received 4 June 1993

A one-dimensional analysis is presented of an FEL oscillator with a linear undulator tapering. Some principally new results have been obtained. Universal plots obtained with the help of similarity techniques, give the relations between the depth of the tapering, the quality factor Q of the resonator and FEL efficiency.

1. Introduction Tapering of undulator parameters, for the first time proposed by Kroll et al. in ref. [ 1 ], is widely used now to increase the efficiency o f a free electron laser. In the case of an FEL amplifier, this paper and papers o f other authors (see e.g. refs. [2,3] ) give a reliable way to describe the processes o f trapping and coherent deceleration o f the particles. Efficiency increase is achieved by an adiabatic change o f the resonant energy, for instance, with the undulator field decrease at a fixed undulator period. In case of the FEL amplifier all the parameters influencing the resonance condition (the undulator field and period, the frequency o f the amplified wave and the initial electron energy), are defined by experimenters. In comparison with the FEL amplifier, the situation with the FEL oscillator is more complicated because the lasing frequency is defined by the condition of the m a x i m u m amplification in the small-signal regime (it takes place when one can neglect the longitudinal mode competition and the sideband instabilities). The position of the amplification m a x i m u m depends on the depth o f the tapering (see e.g. ref. [4] ). This problem was not considered in papers [ 1,5 ], where the amplified wave

frequency was assumed to be known and was set closely to the resonant condition at the undulator entrance. Such an approach can not be recognized as a satisfactory one because in the real situation, the lasing frequency shift due to the tapering may lead to the situation when the particles are far from the exact resonance at the undulator entrance. At some values o f the depth of the tapering this effect leads to a significant decrease of the FEL oscillator efficiency when the undulator parameters are tapered in the same way as in the FEL amplifier (for instance, by a decrease of the undulator field at fixed period). We shall show below that in some cases quite a different way of tapering is more preferable, for instance, with an undulator field increase at a fixed period (so called "negative tapering"). Unfortunately, the approach proposed in refs. [ 1,5 ], ignoring the main features of the FEL oscillator with the taperd undulator, is dominating among the physicists till now (see, e.g. ref. [ 6 ] ). As a result, such a situation in the FEL oscillator theory had influenced seriously the state with experimental works and in some cases had led to nonoptimal design of experiments and incorrect interpretation o f the obtained results. In the present paper we consider a one-dimen-

0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

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sional theory of the FEL oscillator with a linear law of undulator tapering. The simplest case is under study: we do not take into account the space charge fields and energy spread in thc beam. The field amplification per resonator pass i,. assumed to be small, the electron pulse length is infinitely long and we do not take into account the longitudinal mode competition. The main goal of our paper is to attract the attention of the physicists to the novel features of the FEL oscillator with tapering i~arameters. We hope that the presented numerical results and universal plots, obtained with the h e b of similarity techniques, will be useful at the preparation stage of an experiment. The paper proceeds as follows. In sect. 2 we formulate the physical approximations of the model. Section 3 deals with the small-signal analysis of the FEL oscillator with the tapered undulator and in sect. 4 we analyze the nonlinear mode of the FEL operation and discuss the problen: of efficiency optimization. To illustrate our approach to the FEL oscillator analysis, in sect. 5 we present a numerical example and in sect. 6 discuss some experimental results [7-10].

2. Formulation of the problem We consider the problem o:~ the resonator excitation in the following way. We calculate the field amplification coefficient G per ur dulator pass assuming G< 0 and the saturation condition with relation G = K. We consider a one-dimensional model of an FEL oscillator. The electron bean: moves along the axis 298

15 November 1993

of the helical undulator with the length L. The magnetic field at the undulator axis is of the form

Hx+iHy=Hw(z) e x p ( - i f X w ( z ) d z ) . We assume the electron to move along the helical trajectories parallel to the z axis. The electron rotation angle 0 is considered to be small and the longitudinal electron velocity vz is close to the velocity of light c. We represent the electric field of the synchronous with the electron beam wave in the complex form: Ex+iEy--Eexp[im(z/c-t)]. To describe the electron motion we use the hamiltonian formalism with the "energy-phase" variables. In this representation the electron energy ~ is canonical momentum conjugated with the phase ~u=fXw d z + ~o(z/c-t) and z is an independent variable. When the electron energy deviation from the nominal one 8o is small, we can write the following expression for the hamiltonian (we neglect the space charge fields here) [ l l , 1 2 ] :

H(P, ~/, z) =C(z)P+o~p2/2c~,2 go - [ U exp (iq/) +c.c. ] ( 1 -P/go) ,

( 1)

where P = g - g o , C(z)=~Cw(Z)-[l+Q(z)2]to/ 2cy2 is the detuning of the particle with the nominal energy go, Yo=~o/racE, )J~2~y0-2"~-02, Q ( z ) = eHw(z)/~w(Z)mc 2 is the undulator parameter, m and ( - e) are the electron rest mass and charge, respectively, U = - e 0 o / ~ / 2 i is the complex amplitude of the effective potential and Oo=Q(O)/yo. We assume the relative value of the undulator tapering to be small ([AHw/H(O) l, IAXw/X(O)I<
dP/dz=-OH/Ogt,

and the Maxwell wave equation.

3. Small-signal gain The evolution of the electron beam distribution function f is described by the kinetic equation

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E( L ) / E( O) = 1 + i~zOZejoo~( c2y2:.o~o) - ~

Of/Oz+ ( OH/OP) Of/O~- (OHIO,) Of/OP=O . In the linear approximation we shall seek the solution for f i n the form

× jdz 0

f=fo(P) +fm (P, z) exp(i~,) +c.c.,

Of~/Oz+i[C(z)+ P~/CTZzoEo]f~ + i U Ofo/OP=O . When one has an unmodulated electron beam at the undulator entrance, the solution for f~ has the form

q~(z, ~ ) = e x p [ i C o ( ~ - z ) + i ~ ( ~ 2 - z 2 ) / 2 ] . Let us define the field amplification coefficient as G = R e [ E ( L ) / E ( O ) - 1 1 . Then after normalization of variables we get 1

i

..~dfo xp [ i F ( z ' ) ] ,

dz'lc~e

d~(z-C)~(z,~), 0

where

and then we have for the complex amplitude f~ the following equation:

~=exp[-iF(z)]

15 November 1993

~

( ~ = ~ d-~ f d ~ s i n ( O ° ~ + & ~ z - & ~ 2 / 2 ) ' (2)

0

(4)

0

0

where f ( z ) = .i [C(s) +Pco/c~'~o ~01 ds. 0

Further on we shall consider only the case with linear tapering when the detuning changes according to the law: C(z)=Co+o~z. The longitudinal component of the beam current is equal to

&-~ - 2gN[2w(L) - 2 w ( 0 ) l / 2 w ( 0 ) , where N is the number of undulator periods. At the linear change of the undulator field and fixed period we get

J~ = -Jo +J~ exp (iq/) + c . c ,

Jo ~- ecno, ], ~- - ec [. fl dP, where no is the beam density at z = 0 . In the framework of the one-dimensional model, from Maxwell's equations we get the following equation for the slowly changing a m p l i t u d e s / ~ ( z ) a n d j~ (z)(in CGS units) d~ff/dz = 2~0oC- iT1( z ) .

where (~s = GJfl, fl= 1rO2jo(.OL3/IAy2yo c is the gain factor (note that fl=j/2, where parameter j was introduced in ref. [4]), I A = m c 3 / e ~ - 1 7 k A , Co= Co L, & = o ~ L 2 and £ = z / L . Let us express the tapering parameter & through the undulator parameters. At the linear change of the undulator period 2w =27UXw and fixed undulator parameter Q, we get

(3)

In this paper we consider the case of the "cold" electron beam when the initial distribution function of the electron beam i s f o = n o 3 ( P ) , where c~( ) is the delta-function. Substituting this expression into eq. (2) and after integrating over P we get the expression forj~ (z). Then we substitute the obtained result into eq. (3) and after integration over z we get the following expression for the field amplification per undulator pass (we assume here a small change of field/~ per pass)

&~- { - 4 ~ z N e 2 ( o ) / [ 1

+Q2(0) ]}

X [Hw(L) - H w ( 0 ) ] / H w ( 0 ) . It should be noted that these expressions for & are valid for both positive and negative signs of &. In the case of an undulator with constant parameters, integration of expression (4) is performed analytically and we get the well known result (~s = [2(1 - c o s Go) - C'o sin (~o] ~ffg3. At & = 0 function (~s achieves its maximum (~p = 0.135 at Co = ~ ' ~ = 2 . 6 . Let us consider the behaviour of the small-signal gain at the linear undulator tapering. Figures 1 and 2 show the gain profiles for two different positive values of the tapering parameter &. It is clearly seen that the curves are antisymmetric with respect to the point ('o = - &/2. The value of G~ at the first max299

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15 November 1993

15 12 o.--a

9

\

<~ 6

\ \

0 -30

20

-10

0

10

. . 0

\

I . . . . .

8

i .

16

.~-~

E4

32

10

~o Fig. 1. The reduced small-signalgain versus the reduced detuning at the undulator entrance. Here &= 20, ( 1) is the first maximum, (2) the second maximum. The full curve is calculated with formula (4) and the crosses are the results of numerical calculations by formulae (6) and (7).

quency increases with respect to the case & = 0). A more precise formula for the first m a x i m u m has the form C ~ = 2 . 6 - & / 2 - y l ( l & [ ), and for the second m a x i m u m C~ = 10.6 - &/2 - Y2( ] ~ I ), where Yl and Y2 are positively valued functions of the absolute value of & giving a small contribution to the change of C'~ (Yt,2<< [&[ ). The values of the m a x i m u m gain ~m do not depend on the sign of& and are universal functions of the absolute value of &. The plots of these functions for two maxima u n d e r consideration are presented in fig. 3.

I <

~ 0

~,/

-1

-35

-25

15

-o

~o Fig. 2. The reduced small-signalgain versus the reduced detuning at the undulator entrance. Here &= 30, ( 1) the first maximum, (2) the second maximum. i m u m ( d o m i n a t i n g at & = 0 ) is decreasing rapidly when parameter & is increasing and at &-- 26 the first m a x i m u m becomes less than the second one. Further, the second m a x i m u m dominates till & ~- 38, and so on. We limit our consideration only with these two maxima, i.e. with the region 3 < l & [ < 38. The position of the m a x i m a is given approximately with the formula C ~ = C ~ I n = o - & / 2 (see ref. [4] ) which is valid for the both positive and negative values of parameter cE For instance, at the positive values of &, m a x i m a shift in the direction of the smaller values of the d e t u n i n g parameter Co (i.e. the lasing fre300

Fig. 3. The maximum reduced small-signalgain as a function of the tapering parameter; ( 1) the first maximum, (2) the second maximum.

4. The efficiency calculation We use a macroparticle method to simulate a nonlinear regime of the FEL oscillator operation• We simulate the electron beam with M macroparticles per interval (0, 2~r) of the phase ~. The beam current density Jz is represented as M

Jz=-2ztjo M-I ~

8(~-~u))

,

i=1

where ~ti) is the phase of the ith macroparticle and 6( ) is the delta-function. The amplitude j~ and the phase ~'1 of the first harmonic of the beam current density (jl exp(i~q ) = 2 j l ) are given by the expression

>

J~ t s i n ~/,/1

=2,o

l

i= ~~

sin ~u)

"

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We write down the equations of motion in the following reduced form ( i = 1, 2 ..... M)

dP(o/d2= Ocos(~'(/) +q/o) , d~(~)/d2=fi(i) + ~(2) +gO sin (~v(i) +~Vo),

(6)

where (~0/2) exp(i~vo)=iU, O=~oL2o~/cy~go, ['= PwL/cy~ ~o is the normalized energy deviation, and g=CTZzo/~oL=(4~rN)- ~. Parameter g is always small and we will neglect it in the second equation of the system (6). From equation (3) we get the reduced equations for the amplitude 0 and the phase ~to of the effective potential:

dO/d2=flf, cos (~Uo- ~u,) , d~o/d2=-flf~o

-~ sin(~Uo-~U, ) ,

(7)

where f~ =J~/Jo and ~/~ are calculated with formula (5). The simulation is performed as follows. At the moment of the time t t~) we have the unmodulated electron beam and electromagnetic field with amplitude E ~) at the undulator entrance. Then the initial conditions at z = 0 are as follows: ~(,)=27~(i1/2)/M, fit/)=0, j , ( 0 ) = 0 , O(O)=E(~)/Eo, and Eo =c7~ go/eOo~oL~. It should be noted that the field amplitude increment per undulator pass does not depend on the initial field phase ~o(0) because an unmodulated electron beam is fed to the undulator entrance. Here we study only the field amplitude evolution, so, we let ~ ( 0 ) = 0 in our simulations. Equations (6) and (7) are solved numerically with the Runge-Kutta scheme and we obtain the field amplitude increment A0 ("). The radiation losses in the resonator are taken into account introducing the field damping factor K (see sect. 2) which gives the following initial conditions after the ( n + 1)th resonator round-trip: ~/(~)= 2 ~ r ( i - 1/2)/M, P(o =0, j~(0)=0, 0(0)=(0(")+A0("))(1-K). Thus, this algorithm allows one to calculate the field evolution in the resonator in time. The saturation regime is achieved asymptotically when the field amplification coefficient per undulator pass G= A 0 ( ~ ) / 0 (~) becomes equal to the field damping factor K. The code has been tested in the regime of small signal (see fig. 1). The FEL oscillator efficiency is defined as q= ce[EAEI/2~zJo¢o. It is convenient for further consideration to introduce the reduced efficiency

15 N o v e m b e r 1993

fl=rllg=~OZ /2 ,

(8)

where g = (47rN) -1, N is the number of undulator periods and ~ = G/fl is the reduced field amplification coefficient. It should be noted that the reduced efficiency 0 is equal to the mean normalized beam energy deviation at the undulator exit: 0= - (/5). At the saturation we have f/=/~(0 (°°)) 2/2, w h e r e / ~ = K/fl is the reduced field damping factor. In the case of an undulator with constant parameters, 0 (°°) depends only on/~, so the FEL efficiency ~ at the saturation is a universal function of the only parameter /~ and achieves its maximum f/m~x= 3.62 at Ropt = 0.028 (see refs. [12,13]). In the case of a linear law of the undulator parameter tapering, at ~ ( 2 ) = ~ + 6 t 2 (where ~ is defined by the condition of the maximum gain in the small-signal regime), the FEL efficiency ~ at the saturation is a universal function of two parameters: f/= F(/£, &). At each value of the tapering parameter dt there is always an optimal value/~opt(&) when efficiency achieves its maximum qmax(&). In figs. 4 and 5 we present the plots of the maximum FEL efficiency f l m a x ( & ) and/~opt(&) in the range 0 < la] <25 when the lasing takes place at the first maximum of the gain curve. The complicated behaviour of these plots at the positive values of parameter & needs some explanation. The rapid decrease of the efficiency at & ~ 2 0 is explained as follows. Let us consider the field amplification coefficient G as a function of the field amplitude 0 in the resonator. When

15 12

E

---

9

- -

\

6 b 31 a

0

a

5

10

15

20

25

IC(I

Pig. 4. The m a x i m u m reduced efficiency at s a t u r a t i o n as a function o f the tapering p a r a m e t e r for the first m a x i m u m ; curve (a):

~i> 0, curve (b): 6~<0. 301

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the field a m p l i t u d e is increasing, coefficient achieves its m i n i m u m at first, and at large values o f ~b it achieves the m a x i m u m value. The dependencies o f G(~b) and ~ ( 0 ) at the value o f & = 1 5 are presented in fig. 6. It is seen from the plot o f (~(~b) that a transition from the range o f the small field to the range of the strong field (where efficiency achieves its m a x i m u m ) is possible only at / ( < ( ~ i n . At & = 15 the m a x i m u m F E L efficiency is achieved at ~op~=58 and conditions / ~ < G r n i n and /('=C~(0op~) are fulfilled simultaneously at / ( = / ( o p t = 0 . 0 0 3 8 . Thus, at a given value o f the undulator tapering parameter c~, there is always an optimal value o f the field d a m p i n g factor /( when the FEL efficiency

:LO L'.5

\ ....

2.0

o

[

1.5 1.0 O.5 0,0 0

5

10

15

20

95

Fig. 5. The optimal reduced damping factor as a function of the tapering parameter for the first maximum; curve (a): d~> 0, curve (b):~<0.

15 November 1993

achieves its absolute m a x i m u m . Then, if p a r a m e t e r & is increasing, the value o f Gmi, is decreasing and at d~> 17 the equilibrium state in the region of the strong field is possible only at (/3> ~bopt and the FEL efficiency is decreasing rapidly. At &_~21 the decrease of Gm~, leads to a leap o f the function /(opt(&) and a break of the function 0 m a x ( l ~ ) . The further increase o f p a r a m e t e r & leads to the situation when the FEL saturation is achieved at small field and low efficiency. Thus, in the region 0 < I&l < 25, the tapering with a positive value of & (i.e., at the decreasing o f the undulator field and fixed undulator p e r i o d ) becomes ineffective. This is explained by a shift o f the lasing detuning C ~ to the region o f negative values. As a result, at the initial part o f the undulator the particles are bunched at the accelerating phase o f the effective potential and take away the energy from the radiation field, and only then the field amplification process takes place. So, in this case the u n d u l a t o r tapering with & < 0 , when the field amplification is p e r f o r m e d from the very beginning o f the undulator, has an advantage against tapering with &>0. Figures 7 and 8 show the dependencies of ([max(If~) and /(op,(&) in the range 2 6 < 1 & l < 3 8 when the lasing takes place at the second m a x i m u m of the gain curve. In this case the m a x i m u m FEL efficiency increases significantly but achieves at small values o f p a r a m e t e r / ( (which corresponds to very strong fields).

l i

21

~7~f

,,,e

~

l)

<~- 5.0

I 0.0

20

30

0

I 40

50

26 60

~

~

i

I

I

29

32

35

38

70

Fig. 6. The reduced gain (curve 1) and the reduced efficiency (curve 2) versus the reduced field amplitude for 6z= 15. 302

~

Fig. 7. The maximum reduced efficiency al saturation as a function of the tapering parameter for the second maximum; curve (a): &>0, curve (b): &<0.

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OPTICS COMMUNICATIONS

0.5

i

slightly shifted to the negative values of the detuning parameter. Almost the same results (the increase of the efficiency by the factor of 2-3.5 with respect to the case of an u n d u l a t o r with constant parameters) can be achieved using "negative tapering" in the region of the tapering parameter & = ( - 20 ) - ( - 25 ).

\ a

0.4 ~O 0.3 0.2

15 November 1993

b

0.1 5. Numerical example 0.0 26

29

32

35

38

A Igl

Fig. 8. The optimal reduced dumping factor as a function of the tapering parameter for the second maximum; curve (a): &> 0, curve (b): 6~<0

Table 1 Tapering parameters. 10~x g:

~/I,~1

I,~1

0.1-0.15 0.15-0.2 0.2-0.3 0.3-0.5 0.3-0.5 0.5-0.7 0.5-0.7 0.7-1.0 0.7-1.0 1.0-1.3 1.0-1.3

- 1 - 1 +1 +1 - 1 +1 - 1 +1 - 1 +1 - 1

26-38 26-29 26-32 26-32 23-25 26-30 20-25 26-28 18-25 26-28 18-25

20-27 17-23 9-12 10-13 10-13 12-13 10-12.5 8-12.5 8-10.5 6-10 7-8.5

With the help of the plots in figs. 4, 5, 7 and 8 one can find the optimal value of the resonator Q-quality and m a x i m u m FEL efficiency corresponding to a given value of the tapering parameter &. In m a n y practical situations, the problem rises how to find the optimal value of the tapering parameter and m a x i m u m FEL efficiency at a given value of the resonator losses. This can be done with the help of table 1. It is seen from table 1 that in a wide region of the reduced d a m p i n g factor values, good results may be achieved using tapering within & = 26-30. In this region of the tapering parameter &, lasing happens at the second m a x i m u m of the gain curve which is only

To illustrate a general approach to the analysis of the FEL oscillator with a tapered undulator, in this section we present a numerical example. The FEL parameters are presented in table 2. The FEL undulator is assumed to be a planar one (see appendix). To make the exposure more clear, we do not take into account such effects as the energy spread of the electrons in the beam, emittance, the slippage effects, etc. The radiation mode excited in the resonator is assumed to be a gaussian TEMoo mode and its parameters change insignificantly along the undulator length. First, we should calculate the gain parameter ft. Assuming the transverse electron beam dimensions to be much less than the gaussian mode waist Wo, we use in our calculations an effective value of the beam current density jo = 21/(n3/2w~). As a result, we find fl= 2, K = 0.01 a n d / £ = 5 X 10-3. We consider the case of the u n d u l a t o r tapering at a constant u n d u l a t o r period, so we have AH/Ho= [Ht(L)-Ht(O)]/HI(O) - ~ - 3 . 4 X 10 -3 6l. The physical efficiency q and reduced efficiency ~ are connected by the relation q=fl/4nN~-2× 10-3 fl. The lasing frequency shift Table 2 FEL parameters. Electron beam Energy, go Peak current, I Undulator Period, 2 w Number of periods, N Peak magnetic field, H0 Maximal taper, AH/Ho Optical resonator Wavelength,2 Optical waist, Wo Total power losses

50 MeV 50A 4 cm 40 4.5 kG 13% 5 lam 0.15 cm 2%

303

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OPTICS COMMUNICATIONS

with respect to the u n t a p e r e d case is equal to A~o/~oo = ( ( ~ - 2 . 6 ) / 2 n N ~ _ - 4 × 10 - 3 ( ( 2 ~ - - 2 . 6 ) , where

C~ n is a function

of

& (see

sect.

3).

Figures 9 a n d 10 p r e s e n t the d e p e n d e n c i e s o n the t a p e r i n g d e p t h o f the lasing f r e q u e n c y c o r r e s p o n d i n g to the m a x i m u m gain a n d the m i n i m u m gain, respectively. Both cases are illustrated: positive tapering with AH/Ho < 0 a n d negative with AH/Ho > O. T h e m a x i m u m gain is calculated by m u l t i p l i c a t i o n o f the m a x i m u m r e d u c e d gain (see fig. 3) a n d the gain p a r a m e t e r v a l u e f l = 2. T h e leap o f the lines in fig. 9 a n d the b r e a k o f the curve in fig. 10 at I M-I/Hor ~-8.5% c o r r e s p o n d to the t r a n s i t i o n o f the lasing o n the s e c o n d m a x i m u m . It s h o u l d be n o t e d here that such a b e h a v i o u r o f the F E L oscillator with a t a p e r e d u n d u l a t o r could n o t be suppressed by the 6.0 J 1

2.0 ~ _ _ _ ~ _ -~-

/-/

J-~

/

2.5

1

2.0

0.0 -e.o

~-'~

-6.0 g.0 -lO.O

I

!

-

[

L

i v

{

~

"~ i

.

!

,

1.5

-4.0

<1

change of the electron b e a m energy. It will only change the lasing frequency ~Oo c o r r e s p o n d i n g to the u n t a p e r e d case b u t the d e t u n i n g C ~ will stay unchanged. Figures 11 a n d 12 illustrate the d e p e n d e n c i e s of the s a t u r a t i o n efficiency on the t a p e r i n g depth. The leaps o f the d e p e n d e n c i e s at [AH/Ho] ~-8.5% corr e s p o n d to the t r a n s i t i o n of the lasing on the second m a x i m u m . The leaps o f the curve in fig. 11 at A H / H o ~ - - 4 . 5 % a n d AH/Ho~_-ll°/o are conn e c t e d with the n o n m o n o t o n i c d e p e n d e n c e o f the gain on the a m p l i t u d e of the r a d i a t i o n field stored in the r e s o n a t o r (see fig. 6 a n d discussion in sect. 4). It is seen from fig. 11 that at the generally accepted m a n n e r o f the u n d u l a t o r t a p e r i n g (AH/Ho < 0 a n d & > 0 ) , a significant efficiency increase m a y be

3.0

4.0

be

15 November 1993

~

,

1.0

i

k5

0

2

o.5

4

,.,

i ...... , , , <-'~, , 6 8 10 12 14 I A H / H o l (7.)

~

o.o

.....

i

i

0

2

I

,-?----+

4

G

g

10

. . . . . 12

11

- all/no(Z) N

Fig. 9. The dependence of the lasing frequency shift versus the tapering depth for AH/Ho < 0 ( 1) and Att/Ho > 0 (2). '?,0.0

3.0

20.0

2.0

15.0

1.5 1.0

I

~

4

J

05

i

i

i

6

8

t0

. . . . . .

2

/

¸ _

,5.0 0.0

m--I

I

be 10.0

,

2.5

25.0

-

J

J

i i

I , , , ] , , , i , 12

I A H / H o i (7.) Fig. 10. The small-signal gain versus the tapering depth. 304

Fig. 11. Saturation elTlciencyversus tapering depth (AH/Ho<0).

O.0

14

0

. . . . . .

2

] . . . . . . . . . . . . . . .

4

6 8 AH/Ho(Z)

i0

12

14

Fig. 12. Saturation efficiency versus tapering depth (AH/Ho>0).

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achieved inside a narrow tapering depth band, IzSJ-//nol~8.8-11%. Otherwise, at I A n / n o l ~ 8.5% "negative tapering" (AH/Ho > 0 and & < 0) is more preferable and a significant increase of the FEL efficiency is achieved at AH/Ho ~-7-8.5% (see fig. 12). These results are in good agreement with table 1 of optimal relations between the the reduced field damping factor/~, the reduced tapering depth & and the reduced FEL efficiency r). In conclusion we should note that the value of K = 5 × 10 -3, chosen for this numerical example, is not optimal for the case ofuntapered undulator. It results in a saturation efficiency at A H / H o = 0, equal to r/= 0.48% which is less than the maximal efficiency ~/max= 2 × 10-3× 3.6-~ 0.72% [ 12 ]. This maximal efficiency can be achieved at a higher value of the resonator losses and the gain in the efficiency due to the tapering should be compared with ~/max.

6. Discussion In this paper we have presented a detailed study of a one-dimensional model of the FEL oscillator with a linear tapering of the undulator parameters. Analyzing the obtained results, we can make the following conclusions. (i) There is a principal difference between an FEL amplifier with a tapered undulator and an FEL oscillator with the same undulator. In the FEL oscillator, the detuning of the particle ~ at the undulator entrance is defined by the condition of maximum small-signal gain and depends on the method and the depth of the tapering only. In the FEL amplifier, the detuning of the particle at the undulator entrance is defined by the experimenter and it may be optimized on a maximum efficiency. (ii) There are a series of maxima in the small-signal gain lineshape of the FEL oscillator. The relative scale of these maxima depends on the tapering depth and lasing may occur on the frequency corresponding to the 1st, 2nd, etc. maximum. (iii) The dependence of the FEL oscillator efficiency on the tapering depth is nonmonotonic and discontinuous. At the generally accepted way of tapering with ~ > 0 (for instance, by a decreasing undulator field at constant undulator period), a significant efficiency increase takes place in the narrow

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region of the tapering depth values corresponding to the lasing on the second maximum. (iv) When lasing takes place on the first maxim u m the so called "negative tapering" with & < 0 (for instance, by an increasing undulator field at constant undulator period) is more preferable. It should be noted that there is a rather popular tendency among the experimenters to increase the FEL oscillator efficiency. First, they perform an amplifier experiment with the tapered undulator. Then, using the results of the amplifier experiment (efficiency increase due to the trapping and deceleration of a significant fraction of electrons), they suppose that the FEL oscillator with the same undulator will provide the same results. But such an assumption is incorrect because, as we have shown above, there is a significant difference between an FEL oscillator and an FEL amplifier with tapered undulator. Performed at LANL, experimental works [ 7-9] are the example of such an approach. At first the physicists from LANL performed an amplification experiment with a tapered undulator and experimental results were in good agreement with theoretical predictions [7]. Then they performed an oscillating experiment with the same undulator and did not obtain a significant increase of the FEL oscillator efficiency [ 8]. In paper [ 9 ] they wrote: "In past lasing experiments with tapered wigglers [7], the extraction efficiency has been less than expected. We have not observed the behaviour expected from a simple model of FELs, i.e., the trapping of a significant fraction of electrons and their deceleration by an amount consistent with the taper of the wiggler. This behaviour was observed in amplifier experiments with the same wiggler [ 8 ]". Now let us discuss the experiment by the Rocketdyne/Stanford group [ 10]. In this experiment the undulator field was decreased along the undulator length by a linear law. The tapering depth could be varied in the range - 9 . 6 % ~ < A H / H o ~ 0 (it corresponds to the case of &> 0). According to the theoretical predictions of the authors of paper [ 10 ], the efficiency must grow with the tapering depth but experimental results are presented only for AH/Ho= -9.6%. The obtained efficiency was equal to 1.2%. We may explain the obtained results in the following way. In this experiment the undulator parameter was equal to QI(0) =0.86, the number of the undulator 305

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periods were equal to N = 80 and at the value o f the tapering depth A H / H o = - 9 . 6 % we obtain & _ 2 6 (see a p p e n d i x ) . It follows fro:~a our theory that in this case lasing takes place on lhe second m a x i m u m and in a wide range of the reso]lator losses/(, the reduced efficiency is equal to ~:-~ 10-13 which corresponds to the efficiency q ~ 1-1.3%. This result is in good agreement with the e x p e r i m e n t a l results [ 10]. Unfortunately, the authors of paper [ 10] have not presented experimental results :rot the whole range o f the tapering depth. According to our theory, at smaller values o f [ ~-I/Ho [ the lasing takes place on the first m a x i m u m which leads to a deep fall in the efficiency. We suppose that it was observed in this experiment (in this case it would be better to use a "negative tapering" to o b t a i a the efficiency increase) and the theoretical predLictions of the authors of paper [ 10 ] should be recogn:ized as unsatisfactory. In conclusion we should emphasize that despite the present p a p e r deals with a o n e - d i m e n s i o n a l model, the o b t a i n e d novel features o f the F E L oscillator with the tapered u n d u l a t o r will take place in the three-dimensional theory, too. We hope that the presented analysis of the FEL, oscillator o p e r a t i o n m a y be used to explain existing experimental results and help physicists to plan future experiments.

Acknowledgements The authors are extremely grateful to V.P. Sarantsev for support in our work. We also want to thank A.N. Lebedev, A.V. Agafonov a n d V.A. P a p a d i c h e v for their kind interest to our work, and Yu.N. Ulyanov for m a n y useful discussions a n d r e c o m m e n dations.

Appendix FEL oscillator with a planar undulator All the results o b t a i n e d above m a y be generalized to the case of a linearly polarized r a d i a t i o n a n d a p l a n a r u n d u l a t o r having magnetic field: Hy=0, 306

Hx=H,(z) cos( I lc,(z) dz )

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by the following r e d e t e r m i n a t i o n o f the p r o b l e m parameters:

fl~ fl'= nO~joo~L 3A~j( 2cp~ 7olA ) - ~ , Eo ~ E'o = 2Co~oy2 ( eOlcoL ZA jj ) -1 , g-~g'=cTZ( o~L ) - l = ( 4 n N ) - ' , C(z) ~ C ' ( z ) = tot(z) - o J [ 1 +Q~(z) /2 ]/2cy~ , where Qt(z)=eHt(z)/[xt(z)mc2], Ol=QAO)/po, p/-2 =pff2 + 02/2, A s j = J o ( p ) - J l ( P ) and v= O~co/8cKt. W h e n the u n d u l a t o r p e r i o d is tapered by a linear law at the constant factor Qz, the tapering p a r a m e t e r & is equal to & - ~ - 2 n N [ 2 t ( L ) 2 l ( 0 ) ] / A t ( 0 ) , where 2 l = 2 n / x t . At the linear tapering o f the u n d u l a t o r field and fixed u n d u l a t o r period we have

&~_{-2nNQ~(O)/[ I +Q~(O)/2]} × [Hz(L) - H t ( 0 ) ] / H t ( O ) .

References [ 1] N. Kroll, P. Morton and M. Rosenbluth, SRI Rep. JSR-7901; IEEE J. Quantum Electron QE-I 7( 1981 ) 1436. [2] P. Sprangle, C.M. Tang and W.M. Manheimer, Phys. Rev. A21 (1980) 302. [3] D. Prosnitz, A. Szoke and V.K. Neile, Phys. Rev. A 24 (1981) 1436. [4] W.B. Colson, in: Laser Handbook, Vol. 6, Free Electron Lasers, eds. W.B. Colson et al. (North Holland, Amsterdam, 1990) p. 115. [5] C.A. Brau and R.K. Cooper, Phys. Quantum Electron 7 (1980) 647. [6 ] A. Serbeto, B. Levush and T.M. Antonsen Jr., Phys. Fluids B 1 (1989) 435. [ 7 ] R.W. Warren et al., Nucl. Instrum. Methods A 259 ( 1987 ) 8. [8] R.W. Warren et al., IEEE J. Quantum Electron. QE-19 (1983) 391. [ 9 ] D.W. Feldman et al., Nucl. Instrum. Methods A 285 (1989) 11. [ 10 ] M. Curtin et al., Nucl. Instrum. Methods A 296 (1990) 69. [ 11 ] E.L. Saldin, E.A. Schneidmiller and M.V. Yurkov, Sov. J. Particles & Nuclei 23 ( 1992 ) 104. [ 12] E.L. Saldin, E.A. Schneidmiller and M.V. Yurkov, Optics Comm. 102 (1993) 360. [ 13] B.W.J. McNeil, NucL Instrum. Methods A 296 (1990) 388.