432

Nuclear Instruments and Methods m Physics Research A304 (1991) 432-435North-Holland

Hot electron beams for FELS

V .A . Bazy1ev and V .V . GolovizninI V Kurchator Institute of Atonuc Eneryt, Moscow 123182, USSR

We consider one of the FEL schemes in which the phase velocity of the electromagnetic wave (or the ponderomotive wave) isconstant, but the longitudinal electron velocity is changed due to an external magnetic field . There is also an additional magneticfield which turns the electrons back before the interaction . It is shown that the efficiency of utilizing hot electron beams of largecurrent density may be high enough m such systems with a refractive and an undulator medium.

1 . Introduction

In the well known FEL schemes [1-10] the requiredelectron beams must possess a small angular divergenceand a small energy dispersion . In designing powerfulFELs one encounters the problem of how to obtain ahigh efficiency, since the temperature of the electronbeam increases with increasing electron current density .

In this article we consider one of the proposed[11,12] FEL schemes . The basic idea behind theseschemes is to utilize hot electron beams of high currentdensity for powerful FELs . In our schemes the phasevelocity of the electromagnetic wave (or the pondero-motive wave) is constant, but the longitudinal compo-nent of the electron velocity is changed due to anexternal magnetic field . This is necessary for electronsto stay m phase with the wave for a sufficiently longtime . Though it is possible, the proposed scheme is nota surfatron FEL : the electrons are not captured by thewave . In its mathematical aspect this regime is close tothe regime of "empty bucket acceleration" [8] . However,the analogy takes place only for the first approach ;subsequent ones lead to new effects and possibilities .Because of simple considerations we draw our attentionto Cherenkov media [131 ; the generalization to undula-tor ones is given at the end of the article.

2. Influence of magnetic field H

2,1 Basin equations

Let a relativistic electron with charge e, energy e >> m(h = c = 1) and momentum p move through a mediumwith refraction index n = 1 + X/2 and interact with anelectromagnetic wave E = Eo sin(wt - k - r) and a mag-

0168-9002/91/$03.50 (1) 1991 - Elsevier Science Publishers B.V (North-Holland)

netic field H=const . Let the coordinates be x, y, z,with zl1k, x1JE, y11H, p = pr " p, =0, Then we have thefollowing set of equations :

dt Fep c sin 0,

dP =eE°

sin( P)d~ + eH,dt w dt

dO w p2+m2)dt - 2 (

E2

- X ,

where 0 = wt - kz denotes the phase of the wave . Letthe distribution of electrons over phases On at the initialmoment be uniform on the interval (0,21x) .

2.2 . Small-signal regime

It is suitable to investigate the set of eqs. (1) with thehelp of different methods depending on the value of theparameter EOX/H. Let the amplitude of the electncfield Eo of the electromagnetic wave be small enough,EOX/H « 1 (see below). In this case, the result can beobtained as an expansion in powers of EOX/H. Let usdenote ~ = EXt/2/m,

T = twX/2,

a = p/m,

m =2eH/mwX, a = eEo/mw. The set of eqs . (1) can berewritten in the form

d~ - 2aaT

sind

$,

d~

1 + a2dT =

~2

-1,

were the variable a, according to the second equation ofset (1), is of the form

a=as +mir +a(cos0a -cos0) .

V.A Bazyleo,

Let a("), ~("), 0(n) denote quantities, expanded inpowers of a to 0(a").For example, the quantities a( o)and 0(o) are equal to

a(o)(T)=ao+KT,

4P(0)=00+e~2f Td,[1+a(o)(T) 2 -eô1

.0

It is clear that (~), o = 0, where ( . . . ), o denotes theaveraging over phases 00 . Then we have for the variable~(2)

t

a(i) T(2)=S0 +2afOTd,' t(i)( r) sin lß (1)

(Ti) .

S T

If T - oo the integral in eq . (5) has a saddle point T,

which is defined by the equation

dO(0)(Ts)/dT=0 .

6

Using eq . (4) we have : T, = k-i [q 2 -1)12 - a0 ] . At this

point T, the phase synchronism condition is fulfilled, i .e.the phase velocity of the electromagnetic wave is equalto the longitudinal velocity of the electron (the electroncrosses the Cherenkov cone). One can get this crossingby the choice of initial conditions . So it is possible touse the saddle method to integrate over T .

As a result of the integration, the functions sinand cos ~P appear in eq . (5), where ~P = 0(0)( T, ) - 00 .With the help of the expression for T, and eq . (4) we canget the following value for the argument ~P :

IP= [3ao(~oo-1) -2(~o-1 ) 3/2 -aô,/3ic ô.

For variations of the initial conditions for momentum8po and energy 8e o of the electrons we have the follow-ing variation of ~P :

8~P - 8poXw/eH,

8~P - 6EOX3/2w/eH .

Let us introduce the definition of a "hot" electronbeam :

8po >> eH/wX,

8e o >> eH/wX3/2 .

(8)

In this case, sin IP and cos 1k are the "quiver" func-tions, so, as a result of averaging over momenta po, allthe terms containing sin 1Y and cos k disappear, and wehave from eq . (5)(4(2)( T I oo)),.Po = o - 21Ta2 (4 2 - 1)/k~0.

(9)

Let the efficiency 71 be the fraction of the electronenergy transferred to the electromagnetic wave :(E(t ~ oo))~.P./E0 . Then i1 is given by

1/2ireEâ (EooX -

m2)

HwE17 =2

(10)

2.3 . Regime of powerful signal

Let us consider the regime of powerful signals E0 >>HX- ' . According to the results in ref. [111, q - E1/2

V V Golooizmn / Hot electron beams for FELS

11=1 -

5

instead of E2 in the small-signal regime. Here we shallget the required result with the help of a more rigorousmathematical method . Let us rewrite the set of eqs. (1)in the following form :

ds/did= -2(1 +2ßs)2(1 +gpC2X/,52) sin 0,

d4P/de=s + q(1 + 2ps)(1 +gaEÔX/2P2) ,

where

(Eoox - m2)1/2. Here we introduced new variables :

e= CoXpt,

s = (E2/E 2- l)/21ß,

q = qo + he +2,8(cos 4)o - cos 0), and new parameters : q0 =p( po -P)/XpE0 p = (eEoP12wXEÔ)1/2 , h=2H/EOX. If thevalue of p is of the order of 10 -1 -10-3 , the solution ofthe set of eqs. (11) can be found as a result of expand-ing in powers of p. However, at p << 1 the expressionfor the efficiency 11 dust contains ß as a factor :

71 - 1 - ((I + 2,8s( O GO)) -1/z )~opo

(s(e - -))0 oPu

(12 )So it is correct to restrict ourselves to the calculation ofonly the first term of the series expansion, s( o) . Hence,the set of eqs. (11) is reduced to

ds(0)/did= -2 sin 0,d oP/d e =s(o) + q0 +

lie,

with initial condition so"= 0, 00 E [0, 2m] .For the variable 0, from the second equation of the

set (13), we have an equation which describes the mo-tion of the material point with unit mass in the poten-tial U(O) = -hO - 2 cos (P :

d20/d*2 = h - 2 sin 9P .

(14)

Eq . (14) has the invariant w:

w

2 (dw )2+ U(0),

therefore

433

(13)

dO/di%= ±2 1 /2(w+hO+2 cos ot,)1/2 .

(15)

If h > 2 all the trajectories are aperiodic ones. If h < 2the potential field has an infinite number of localminima, one of which is shown in fig . 1 . In this case,there are two types of trajectories : periodic andaperiodic . For initial conditions of a "hot" electronbeam when we have only aperiodic trajectories crossingthe Cherenkov cone, the quantity s, according to eqs.(13) and (15), is given by

s(e - (Do) =-23/2 f

dO sin 0

1/ ,

(16)e, (w+h~+2cos~)

where 0, is the solution of the equation w = -h0* -2 cos 0., and w = qô/2 - hO0 - 2 cos 00 .Now it is clear that we can investigate our solution

only in the interval [W, W + 23rh] . It is also clear thatfor the values p0 - m and EOX - m the variation of w

V1 . 1D FEL THEORY

434

Fig. 1

Potential U(O) near one of the local minima (h < 2) .Different solid lines denote the motion of different electronsover the phase 0 The interaction occurs near the phasesynchronism points 0, (w - k - v= 0), where the electrons arereflected from the wall of the potential U(O). The shaded area

denotes the forbidden phase interval, where sin 0< 0.

due to variations in initial conditions 8p0, 8E0 exceedsthe width of this interval, 2 ,rrh :

8w - go89o -8P0/mß2 >> H/EOX -- I h 1 .

We can consider the electrons to be distributed uni-formly over w. By averaging we obtain the followingresult :S ( h) = (s("(,9 -, oc)%mwo

=2'/ZTr-i f~-de[U(02)-U(O) ]' /2,

(17)

where 01 and 02 are the boundaries of the region ofperiodic trajectories . The efficiency rl is given by

1/2eE0(E0 m2 )r/2

S(h) is shown in fig . 2. Here we encounter a peculiar"phase transition" in the system : at a value of E0 lower

Fig. 2. Function S(h) (see text) .

VA Bazyleu, V V Golouizmn /Hot electron beams for FELS

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than some E«, the coefficient of the main term of theseries expansion is equal to zero .

It follows from the above consideration that we haveonly one restriction for the "temperature" of the elec-tron beam to this scheme, 80, < X, and only onethreshold condition for the energy, co > m/X. It isnecessary to keep in mind that this is a result of aone-dimensional model (see below) .

2.4 Increments

The increment y can be obtained from the energyconservation law: the increase of the electromagneticenergy density is equal to the decrease of the electronenergy density. Let N [cm -3 ] be the electron beamdensity. Then we have for the small-signal regime (E0<< HX-1)

2.5. Filling coefficient

3. Influence of magnetic field He

In the opposite case (E0 >> HX- ' ), the increment y isgiven by

_ me2NXhS(h)_EOXOl/2

)l/2

y

-

(

mf/22f/28P0

( eEW

11

E0X

f/4

(19)

(20)

It is clear from the above consideration that most ofthe usefulness of our scheme is contained in the amplifi-cation regime, not the generation one. It is clear alsothat there is a close mathematical analogy between ourscheme and the "empty bucket acceleration" scheme [8].However, this analogy disappears if it is necessary toaccount for a filling coefficient R. As a rule in FELschemes the laser beam and the electron beam arecolinear; hence, R = min(S,, Se)/max(SI , Se) where Sis the square of the cross section of the beam . In ourcase the beams are not colinear . Nevertheless, the coef-ficient R may be close to unity despite Se >> Si , if thereis an additional magnetic field Ha turning the electronbeam back before interaction .

There is a small region near the phase synchronismpoint TS where the interaction takes place:

Alesf -(mx-'/eHW)i/2 << L

(here L

is the amplificationlength,

L- m/eH). The transverse displacement ofelectrons due to the magnetic field H is equal to Xi/2leff= 8x . If (S, ) i/2 > 8x, one can propose that details ofthe electron trajectory do not play an important role

Fig. 3. Trajectories of electrons . Passing through the magneticfield range Ha the electrons move in opposite direction . Theshaded area covers the majority of phase synchronism points

0, .

and, for laser amplification, the laser beam need notcover the whole electron beam but a great number ofthe phase synchronism points .

The trajectory of an electron can be given in theform

x=x0+$Oz+eHz2/2c0,

where xo is the transverse coordinate of the electron atthe entrance of the interaction region, Bo =pole, is theincidence angle;

py= 0, px =po + eHz. At the phasesynchronism point

z,*the equality p2+ m2 = e2X

isvalid, hence

Let d be the diameter of the laser beam . Then thefollowing condition must be fulfilled :

(xmax-xmin) <d,* *s

SX.

mz(S( 0)2x* = Sxo-

-2eH 2eHcô

(22)

see fig. 3, the shaded area corresponds to the majorityof phase synchronism points (x*, z*). From eq. (21) wecan obtain the variation Sx' due to the variations Sxo,Seo, SBO. Then we can exclude the terms of first order inSBo and SE O , if Bo = 0 and the additional magnetic fieldHa turns the electrons back before interaction, as isshown in fig. 3. As a result we have (00 = 0)

(X +m2/co) _2

+

2eH

eHa~

Seo .

(23)

Thus the additional magnetic field Ha must be equal to4H/(X + m2/eô) . With the help of eq . (23), restriction(22) can be written as restrictions upon Sxo, Seo andSBo:

1 Sxo 1 < d/2,

(Sc0 ) z < 2eHdeô/m2,

(81t)2 < 2eHd/c0 .

V.A . Bazyleu, V V Goloorznin / Hot electron beams for FElz

If k =Sx� axse0ax is the emittance of the electron beam,we have the following restrictions :

Seo/c0 < (2eHeod/mz)11z,

As is seen from eq. (24), these restrictions are notstrong .

4. Generalization

First of all it is useful to emphasize that eq . (10),obtained for the small-signal regime, is the asymptoticlimit (at h oo) of the second term of the expansion inpowers of ß : rl = 4mh-t,ß

2. Taking into account eq . (18)for the powerful-signal regime (r1=#S(h)) one can getthe generalization for the case of an undulator mediumwith the help of the equalities

t/2eE.y

ß_{ 2w(mz+2pz)

2Hcâ (2dlc0/w - mz -2p2)1/z

h=Eow(m

z +2(w.z )

Here Tl and P are the constructive parameters of theundulator: if H°°a =Hy =Ho cos dlz, la = eHo/2S1.

References

435

k < (eHd3/2c0)t/

z .

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VI . 1D FEL THEORY