368
Nuclear Instruments and Methods m Physics Research A272 (1988) 368374 NorthHolland, Amsterdam
LONGPULSE FELs AS SOURCES OF MONOCHROMATIC RADIATION Isidoro KIMEL * * and Luis R. ELIAS Quantum Institute, Umoersay of California, Santa Barbara, CA 93106, USA
A strong competition among modes m a longpulse FEL is shown to exist. Through that strong competition the dominant mode is able to suppress other modes which would otherwise be present. The theoretical analysis is based on a perturbation expansion of the transverse current driving the FEL. expanded m powers of the radiation field. The perturbation analysis was carried on m a way similar to the interaction representation treatment of quantum mechanics To third order m the expansion the crossed saturation between modes is twice as strong as the selfsaturation . Thus the intensity of the dominant mode decreases the gam of competing modes at a much faster rate than it decreases its own gain . The result is single mode operation . Besides discussing the selfsaturation in a single mode situation, the mode stability problem is treated analytically for the case of the two competing neighboring modes .
1 . Introduction and basic FEL equations The main purpose of the present paper is to study the optical mode competition in a longpulse FEL operating in the single particle regime . It is shown that such an FEL is a system with strong mode coupling, and only single mode states can be stable solutions of the dynamical equations . As the dominant mode approaches saturation, its power density depresses the effective gain of other modes . When their effective gain becomes negative these other modes disappear . Thus, the theoretical conclusion corroborates the experimental finding [1] that a longpulse lowgain FEL operates with a very narrow frequency bandwidth. For an undulator with constant period A o = 27T/k, length L and uniform peak field Bo , we can be write the vector potential A = inB0 e  ' k ° c T/2ko + c .c., where n is the polarization vector (n = x for linear and n = [x ± i y] for circular polarization) and T = z/L is the normalized longitudinal position coordinate. In such an undulator the electrons acquire the transverse velocity vT =in(cK/2y) e,k"c,+c .c .,
K=eBo/mcko .
(2)
In order to simplify the analysis we will consider the onedimensional problem in which the radiation field can be expressed as a superposition of plane wave modes . In such a case the electric field can be written as A
nmc 2 exp{i[kgLTcart+4~q(T)] mC Y_aq(T) q
I+ c.c.,
where a q (T) is the adimensional amplitude for mode q and k q =W q /c is its corresponding wave vector . In the particular case of an oscillator with angular frequency spacing between longitudinal modes ca, = car/Lo, all mode frequencies are integer multiples of car, as in wq = qwi . * Work supported by the Office of Naval Research under ONR contract N0001480C308 . * * Present address CREOL, University of Central Florida, Orlando, FL 32826, USA . 01689002/88/$03 .50 © Elsevier Science Publishers B .V . (NorthHolland Physics Publishing Division)
I. Kmel, L.R Ehas / Longpulse FELs as sources
of monochromatic radiation
369
Only Compton regime FELs will be considered neglecting space charge effects . In the slowly varying amplitude and phase approximation the wave equation for the vector potential (3) reduces to Yw q {exp[i(k g LT  wqt + Oq)] ) [ô r aq (T) q
+
iaq (ô,r4)q + a/2)] + (2 p)(c.c .)
  (ItoeLlmp)n * J,
where a is the mode loss per pass, and p = I n 1 2 is 1 for linear and 2 for circular polarization . J, the transverse electron current driving the FEL, will be discussed in detail in the following. In keeping with our onedimensional treatment let us consider an electron beam with an initial uniform density P = PLPT, i .e ., PL planes per unit length with P T electrons per unit area in each plane. The transverse current density driving the FEL is J(t, x) =  eVT(T)r_s3[xx , (t)]   ePTVTYS[LTz,(t)]> l
J
The ==> in eq. (4) indicates the transition from the general threedimensional situation to the simplified onedimensional problem . For K Z << 1 one can disregard harmonic radiation . In such a case the current can also be decomposed in the same way as the electric field (2), namely J= n(mle ~t 0 L)Y_wgJq exp[i(k g LTca g t+`f'q)] +c.c . q
(6)
With this definition, Jq are the dimensional sources for the amplitudes and phases as in ôzaq
is q (az)q  a/2) = Jq .
+
(7)
As Fourier coefficients in the expansion (6), the current amplitudes Jq are obtained from eq. (4) as Jq_ 
ecp oL ~ I, {expi[wqt(kq+ko)LT`Yg]}n* " J (2pmLowq),~di e zp,ocxp T KL 4mywgLo ~Y{eXP1[wgtj(T)(kq+ko)LT ] } I
where the integration is over a fundamental period Tf= 2 alwf and the sum is over planes of particles with injection times uniformly distributed over this period. The electron that enters the undulator at time tl o will reach the position z = LT at the time tj (T)=tl o+(LT/Uo)
 ~J (T)L/c,
in which uo is the initial electron velocity [2] and the small timeshift ~l (t), due to the bunching, is obtained from the pendulum equation [3] dz
z
dT 2
c dt2 
iA
Y LwR a g~ q e",, + c .c.,
(10)
with wR=2 y 2ckol(I+pK z l2), RJq
=
AqT  wgtjO +'~ql
Pq = L[(Kq+ko)  wgluo] ,
A=7TpKNly 2 ,
4lq = exp(iLwq~,/c) .
(11)
The parameter p q measures the detuning from the resonant frequency w R. III(a). GENERAL THEORY
370
I. Kimel, L.R. Ehas / Longpulse FELS as sources of monochromatic radiation
Using eqs . (9) and (11) we can write the current amplitudes of eq. (8) as Jy= ~Jq ' >=iJ0 ( wQPw
Y_ 1~Jq(T)
LO l
a q T e ' j ( l
Jo=
Z
(12)
24mYWRK,
where we have indicated an expansion in powers i of the radiation field contained in the S,(T) . 2. Results from perturbation expansion We have previously obtained a perturbation expansion of the ~,,, in powers of the amplitudes aq. Here we will present the results up to third order and refer to our previous work for the derivation [4] . For an expansion of i~ according to c , ~=1+
J9
(13)
J9'
i=1
we obtained the terms 2iAY_ h
wh
TdT, dTZ ah sin R /,,( TZ ), f0 fo",
OR
wbwc f wh f dr, f i dTZ a h cos wâ iwq
(2) _ Ja  4112
T
T
T~
(14)
T
T_,
RJn(7z)~
/ dT3Jo dT4 a, sin R,,(T4 )
T_e
'T3
7p
dT,f dTZf dT3 a,, sin RJb(T3) f dT4 a, sin f ~c3) _ 8113 Jq
x
whwc wd bcd
~i
T
Z
w9
ç
7
7
T4
dT5f dT6 a d sin Rd (TI)
f0
0
T_
+ ~` dT,f dTZ f dT3 a n sin R,h(T3) f dr, a, cos y fo 0 0 0 c0
T
Wn o vf
+if 0
T3
T
.V, (
+
(15)
1 2 dT,f dTZ ah cos R,h(TZ)~ dT3 dr, a, cos R/,,( T4 ) f f
CJbO,
WR
R~~(T4 ) ,
T
TI
T7
dT,f dTZ
f0
0
7~
T~
dT 3 ah
0
4
7
T4
0
T~
R,h(T4)fo
Tç
dT5 f dr, a d sin R,d(T6) )f0 0
dT4f dT5 a, sin R,,
7~
dT,fo dTZf YdT3f dT4 ah sin 0
T7
cos RJ h(T3)f o
R~~(T4
72
(TS)
f0
dT6 a d sin Rd(T6)
T3
dT5 a, sin RJ , (TS)f dT6 a d sin RJd(T6) o
(16)
where wy = qca/Lo , wh = bc7/Lo, etc., b, c . are numbers of the same order as q. Using eqs . (14)(16) one can obtain the different terms in the power expansion of Jq of eq. (12). The first order term will be treated in the next section while the remainder of the paper will be devoted to the nonlinear effects . As in our previous work to which we refer for further details, the sum over electrons is performed in the continuum approximation according to w,t,o  0,
e,e(B),
R, y ( T)
 R,,(B, r),
Y_ > (PLL(,/77) fT
J
dB .
(17)
1. Kmel, L.R . Elias / Long pulse FELs as sources of monochromatic radiation
371
3. Nonlinear mode couplings When the transformation (17) is performed on the third order term (13), the following type of factor has to be calculated . . . b, c, d
aba,ad e
q
'(W ±Wn±W
<
± Wa)T
e 1(4Pg t$n± (><±$d)
2 1r f
(dO/2 ?t ) e'(gfb±ctd)8 , . .
(18)
The B integral imposes, of course, the equality q = ± b f c f d. This yields the eight cases q= bcd,
(19a)
q=b+c+d,
(19b)
q= bc+d,
(19c)
q= b+cd,
(19d)
q = b+c+d,
(19e)
q=bcd,
(19f)
q=bc+d,
(19g)
q=b+cd.
(19h)
Eq. (a) cannot, of course, be satisfied, while case (b) with b = c = d = p = 3q will be disregarded on grounds that this gives a p too far away from q and outside the gain bandwidth . For the type la terms in which b = c = d = q, only cases (e), (g) and (h) are possible, giving a factor 3 . Let us now consider type lß terms in which among b, c, d there is at least one q and one p =A q. Here again, only cases (e), (g) and (h) are possible. The common feature of these cases is that one of the signs is minus and the other two are plus. Let us consider one of these cases, (h) say . If d = q  b = c = q, we are back in type a. For ß then, d = p leading to q = b + c  p which is satisfied in the following two arrangements d= p' p,
b=q, {b=p,
c=p, c=q .
A similar analysis holds for the other cases (e) and (g) which yield the arrangements ~ c=q, c=p,
d =p, d=q ;
=P, ~b=q, b=p, c
d=p, d=q .
b =p,
The conclusion is that for type lß, three cases contribute with two arrangements each for a total factor 6 (as compared to 3 for type la) . This twotoone ratio in favor of ß over a, resulting in a crossed saturation twice as large as selfsaturation, will prove to be very important for the operation of a longpulse FEL . According to the above discussion the cases a and ß of 1, contribute to the current amplitude Jq of eq. (12) with the third order term  2s(itq
r)a,
aq
 ah
b*q
I
(2~) III(a). GENERAL THEORY
I. Kimel, L R . Ehas / Longpulse FELs as sources of monochromatic radiation
372
where s by
= SR
SR4 ,
+ is,, simply related to the saturation function S = S R + iS, to be introduced later, is given
T) 
a [3x + (1/8  1 .25x) sin x + (1 .5
_X2 /8)
sin 2x
x(3 .13 + x 2/24) cos x  cos 2x] Itt , s, =a[  3  0 .75x2
+
+x sin 2x + (1 .5
(1 .5  1 .25x2 ) cos x  (3.88 +x2/24) x sin x _X2
/8) cos 2x] /t,6' x =
a=
/IT,
e e2mcfwq
(
Lo w e g) .
(21)
Eq. (16) is rather complicated and a large number of integrations have to be performed in deriving eqs . (20) and (21) . In order to simplify things, s in these equations was obtained under the assumption that all ,u's are approximately equal. This assumption is consistent with our purpose of analyzing, in this paper, the interaction among closeby modes . 4. Nonlinear single mode problem When only a single mode is present, the second term inside the square bracket in eq. (20) is, of course, absent. With the nonlinear remaining term the wave equation (7) yields for the amplitude and phase of the single mode daq(T)/dT
2[rR (A y,
d ,~q(T)/dT= 2[rl ( ily>

T) T)
IX]
a y (T)

2SR( 1 ye T)ay , ` 22 )
 SI(~q> T)a9~ .
Since in this section we are dealing with a single mode, we will drop the subindex q. The intensity or power density 1 is, of course, simply related to the amplitude by Z 2 I=Ca 2 , C= zcea(mcle) w .
(23)
From the first of eq. (22) and (23) we see that the intensity evolves during a pass in accordance with dI(T)/dT=1(T)[FR(T)aSR(T)1(T)],
S=S R +S,=(SR+S, )ICI
(24)
with C as in eq. (23), and s R and s, as in eq. (21). The solution of eq. (24) relates I(ij + 1) at the end of pass rl + 1 with I(,q) at the beginning by (25)
I(rl + 1) = 1(71) e", /[] + 1(rl)SJ . In this equation F, is  cos,L)  A sin ft, f(jA)=2(1 r~=ha=cof(ft)IA3 a ,
(26)
and S,=
f 1dTS R (T) expf a
o
dT'
( [TR(T ' )  al = 2Co e0
eL1l ) z me
F(lN~) > g,
F(it) = 6 .53 + 1u2/2  (5 .50  1 .13112) cos 1,  (5 .38 + t2 /24 )tt sin it + (,t 2 /16  1.03) cos 2it  0.56 sin 21, .
(27)
The expression after = in eq. (27) is valid for small F, . Up to now we have studied the evolution of the mode amplitudes (or intensities) for an amplifier, or within a single pass of an oscillator. It is a simple matter, however, to iterate eq. (25) over passes and
I . Kimel, L.R . Elias / Longpulse FELs as sources of monochromatic, radiation
373
obtain for I at pass q the expression
I(,1)=Io e Ir / [1+Io S,(e"F1)/GL1,
1o I(q=0),
GL +eF1 .
(28)
G L is, of course, the small signal gainminuslosses per pass . For low gain (exp Fc  1 = rc ), I as given by eq . (28) coincides with the solution of the differential equation dI(q)/d ,1=I(q)[rc S~I(71 )]
(29 )
.
From the foregoing equations it is easy to see that the third order theory leads to the limiting intensity (30)
I~  I(oc) = GL/Sc . For small F, this limiting intensity can be approximated by I~ .v
) S,  { 2 ,u 2 ( ec Z ~ I F(it)
1114
( PKLN
(31)
)2 '
The factor in braces is 3.518 x 10 7 W and the function inside the square bracket is 1 .55 for tt = 2 .6, at the maximum of the small signal gain . With the general definition of the gain per pass G as the ratio of the intensity change to the initial intensity, from eqs. (25) and (31) one derives
valid up to I = I,
5. Twomode problem Let us consider a situation with two modes having almost equal detuning j, and gainminuslosses FR . Due to the interaction of eq. (25) the power densities I, = Ca t and IZ = Ca 2 , where C is given in eq . (23), satisfy the system of coupled equations dI 2/dr=12 [ rR  SR(12+UI1) ], dIl/dr=I 1 [ FR SR(Il+u12) ],
(u
=2 ) .
(33)
In an FEL the above equations hold with u = 2, but it is worthwhile to continue the discussion with a generic u. As it was already remarked, the fact that u > 1 (crossed saturation larger than selfsaturation) is extremely important for the behavior of a longpulse FEL. The saturation function S R in eq . (33) is related, through the second of eq . (24), to SR of eq . (21) which was obtained with the simplification of taking the same detuning for both modes. This is a good approximation for closeby modes. Since for a single mode the inteapass eq. (24) leads to the interpass eq . (29), it is only natural to assume that corresponding to eq . (33), the two mode interpass evolution is described by dI,/dry =1,( 71)[Fc Sc (I,+UI2 )],
d12/drl=12 ( 71)[Fc  Sc ( 12+UI1)]
( 34)
It is convenient now to transform to a polar system according to a,(rl)=r(rl) cos 0(q),
a2( 7l) = r(rl) sin 0( ,l),
R=r 2 .
(35)
With this, the system of equations (33) turns into dR/d?,=R(q){rc S,:R( q )[l +
21 (u
d0/d,,= 8(u1)S,,R(71) sin 40(q) .
1) sin2 20(q)] (36) III(a) . GENERAL THEORY
374
1. Konel, L R. Elias / Longpulse FELs as sources of monochromatic radiation
At (or near) saturation the system is described by stable solutions (with constant O) of eq. (36). The boundary between weak and strong coupling is u = 1 for which any O is possible. For other a the only equilibrium points are O = 0, 77/4 and 7/2 . For weak coupling (u < 1) O = 0, 77/2 are unstable while O = g/4 is the stable point where the system saturates with equal amounts of the two modes . On the other hand, for strong coupling (u > l, case of the FEL with u = 2), O = 7/4 is unstable while O = 0 (pure mode 1) and O = 77/2 (pure mode 2) are the only stable equilibrium points . Thus near saturation a strongly interacting system like the FEL operates at a single mode which evolves according to eq. (28). For a given Il > 12 we can solve the second equation of (33) and obtain the gain for the intensity 12 of the weaker mode 2 as eG2 1> 1 + (Il/I2)G,t 
(i = ii/il .) 
(37)
The third order theory is expected to give a reasonable account of the behavior of a low gain longpulse FEL operating outside the deep saturation region . With sufficiently high losses the FEL operation can be kept outside that deep saturation region . Under those conditions the following picture holds : The strong crossed saturation among modes (for closeby modes it is twice the selfsaturation), results in an intense competition whereby the dominant mode is able to reduce the effective gain of other nearby modes to the point of extinction. The outcome is operation at a single mode. For FELs operating inside the deep saturation region, terms of higher order (than the third) in the radiation field have to be taken into account . These, as well as the effects they produce (sideband instabilities [58], for instance), are outside the scope and intent of the present paper. References [1]
L.R . Ehas et al , Phys . Rev. Lett . 57 (1986) 424. [2] We are considering an FEL operating in a homogeneously broadened regime with negligible initial velocity spread for the electrons . For early papers explaining the different FEL regimes see, for instance, F.A . Hopf, P. Meystre, M O. Scully and W.H . Louisell, Opt. Commun . 18 (1976) 413; N.M . Kroll and W.A . McMullin, Phys . Rev . A17 (1978) 300. [31 W.B . Colson, Phys . Lett. 59A (1976) 187. [4] I. Kunel and L.R. Elias, Phys . Rev. A35 (1987) 3818 . [5] N.M . Knoll and Rosenbluth, in  Physics of Quantum Electronics, Vol. 7, eds S Jacobs et al (AddisonWesley, 1979). [6] W.B . Colson, Proc . Int. Summer School of Quantum Electronics, eds. S. Martellucci and A.N . Chester (Plenum Press, 1980). [7] G. Dattoli, A. Marino, A. Remeri and F. Romanelh, IEEE J. Quantum Electron . QE17 (1981) 1371 . [81 J.C . Goldstein and W.B Colson, Proc . of the Int Conf. Lasers 1981, ed . C.B Collins (STS Press, McLean, VA, 1981).