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A kinetic small signal gain analysis of a planar wiggler FEL, operating in the high harmonic (strong wiggler) regime
192
Nuclear Instruments and Methods in Physics Research A250 (1986) 192-202 North-Holland, Amsterdam
A KINETIC SMALL SIGNAL GAIN ANALYSIS OF A PLANAR WIGGLER FEL, OPERATING IN THE HIGH HARMONIC (STRONG WIGGLER) REGIME E. JERBY and A. GOVER Faculty of Engineering, Tel-Aviv University, Ramat-Aviv, 69978 Israel
A kinetic linear analysis of a strong planar wiggler FEL operating at the fundamental or odd harmonic frequencies is presented in this paper. The gain dispersion equation in the high harmonic regime is shown to be an extension of the known general FEL equation, and converges to it in the weak pump limit. In the low gain, cold, tenuous beam regime we get the known gain expression with the Bessel function correction term . A similar term also appears in the gain expression of the high gain strong pump limit. When space charge effects are significant (Raman regime), the strong pump gain dispersion relation is more complicated than the corresponding weak pump relation. In the cold beam limit it leads to a quintic polynomial dispersion relation instead of the known cubic equation . This brings about gain curve dependences on the detuning parameter which are double humped and with a relatively high bandwidth. The higher the order of the harmonic frequency, the stronger is the effect of the axial velocity spread on the gain and the tighter are the beam acceptance parameters. The analytical model permits an arbitrary axial velocity distribution . The gain damping effect is computed, both with a simple model of a Gaussian electron velocity distribution and with a more accurate model, taking into account an analytical expression for the asymmetrical skewed velocity distribution produced by the combined finite emittance and energy spread effects .
1. Introduction The fundamental operating wavelength of a free electron laser (FEL) is determined by the energy of the electron beam according to the scaling law X=X w/(2yoZ ) (X W is the wiggler period, and yo, is the longitudinal relativistic Lorentz factor of the e-beam). One finds that a conventional wiggler FEL (X w - 0.01-0.1 m) operating in the visible range requires an electron beam energy of at least 100 MeV . Consequently, accelerators such as the Van de Graaff that are known to operate with high beam quality but relatively low energy can be used in FELs operating in the far infrared region only. High harmonic operation is one way by which such FELs can be made to operate at optical frequencies, using conventional period wigglers. Alternatively, this method can be used to make RF linac and storage ring based FELs to operate at ever decreasing shorter wavelengths down to the VUV-X-ray regime [3]. FEL operation in high harmonics has been considered by various workers as a method of extending the FEL range of operation towards shorter wavelengths. Madey and Taber [1] pointed out the feasibility of FEL interaction at high odd harmonic frequency as a unique property of the planar wiggler FEL, due to the periodic variation (at half the wiggler period) in the axial velocity of the electrons propagating along the wiggler. Colson [2] investigated nonlinear aspects of the higher harmonic operation of planar wiggler FELs in the cold, tenuous beam regime . Dattoli et al. [3] evaluated the dependence of the small signal gain on the electron beam quality factors in the low gain, tenuous beam regime. An experimental observation of gain in the third harmonic was reported by Barbini and Vignola [4] . A self-consistent kinetic description of the FEL instability in a planar magnetic wiggler was developed by Davidson and Wurtele [5], and a linear model including some collective and high gain effects was suggested by Murphy and Pellegrini [6]. The present paper presents a kinetic, linear analysis of a strong (relativistic wiggle velocity), planar wiggler FEL operating at the fundamental or odd harmonic frequencies. Space charge and axial velocity spread effects are included in this model. The resulting gain dispersion relation at fundamental and high harmonic frequencies is an extension of the known fundamental FEL equation [7], and is reduced to it in the limit of a small magnetic field wiggler. Analytical and numerical solutions of the gain dispersion relation are shown, and some unique aspects of FEL operation at higher harmonics in the space charge dominated regime are discussed.
E. Jerby, A . Gover / Kinetic linear analysis of a planar wiggler FEL
193
2. The theoretical model The development of the gain dispersion relation of the planar wiggler FEL operating at a high harmonic frequency is based on the solution of Maxwell equations simultaneously with the linearized Vlasov equation in one dimension. Assuming small-signal operation, the perturbation method is applied. Vlasov equations in the zero and first orders are solved by the method of the characteristic lines in the p, -z plane (the axial momentum and coordinate, respectively). After performing a Laplace transform in the z dimension, the gain dispersion relation is obtained by the assumption of a near-resonance operation at one of the harmonics. The transverse components of the e.m . wave under the 1-D assumption (ô/8x, â/8y = 0) are described by the reduced Maxwell equations: z
ôz2E`(z, t)
(1)
poEEr(z, t) = ,uOJ(z, t)
and B~(z, t) _ -z X
az E,(z, t).
The driving current J,(z, t) is given by : J, (z, t) = -e f vf(P, z, t) d 3p . P
(3)
The momentum distribution of the electrons f(p, z, t) is governed by the Vlasov equation :
AP, z,t ) + U=ô z f(P ,z , t)+F(P,z , t)' pf(P,Z,t)=0 .
(4)
The driving force F(p, z, t) is defined by the Lorentz equation : F(p,z,t)= -e[Et(z,t)+!E,(z,t)+vxBt(z,t)+vxBW(z,t) ] .
(5)
The longitudinal electric field EZ(z, t) is caused by collective effects and is the solution of the Poisson equation : âzEz(z,t) =-= ff (P,z,t)d 3p . P The "pump" force in eq . (5) (F,, = ev X B,,) is induced by the planar wiggler static magnetic field, which is approximated by : B, (z, t) =zB, cosk,,z. Eqs. (1)-(7) are a self-consistent complete set of equations. To solve them, we apply the standard perturbation method that is applicable with the small signal assumption . The electron momentum distribution is expressed in the form : f(P, z, t)
= fo(P, z)
+fi(P, z, t),
(8)
where fo(p, z) is the steady state e-beam momentum distribution in the wiggler field, and fl(p, z, t) is a perturbation due to the interaction with the e.m . wave, assuming that fo >> I f I exists . In a similar manner, the driving force F(p, z, t) is written as : F(P, z, t) = FO(P, z) +F,(P, z, t), where I Fo I >> I Fi I exists. IV . HIGH GAIN FELS
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E. Jerby, A . Goner / Kinetic linear analysis of a planar wiggler FEL
The steady state force
FO(p,
z) is :
(10) FO (P,z)=-e[2EOZ (z)+vXBW (z) ], where Eo,(z) is an internal space charge electrostatic field, caused by the periodic density variation along the electron beam. This periodic density modulation varies at half the period of the magnetostatic wiggler, and is due to the axial velocity modulation at the same periodicity [9] . The perturbation force in eq. (9) is: F,(P, (ll) z, t) = -e[E,(z, t) +îEl,( z, t) +vxB,(z, t)], where the e.m. wave components (Et , B,) are the TEM radiation fields that are the small signal first-order perturbation parameters and Elz (z, t) is the electron beam rf space charge field. Consequently, the Vlasov equation is separated into two coupled equations . The steady state (zero order) Vlasov equation is: VZ
ä z fo(P, z) +Fo(P, z) - ~p fo(P, z) = 0,
(12)
and the disturbance (first order) equation, transformed to the frequency domain, is : - lwf (P,
Z,
w) +
VZ
azf (P, z, w)Fl(Piziw) - ~pfo(P, Z) +FO(P,
Z) -
äp fi = 0 .
(13)
Eqs. (12) and (13) are linear, partial differential equations of the first order; thus the method of the characteristic lines [8] may be applied to solve them. The characteristic line of the steady state Vlasov equation (12) is [9] : Po(z) =mc[-ya,, sin(k,,z)+î
i2 -1-a,2,(1+8) sin2 (k,,z)1 1
y
(14)
where a, the wiggler normalized vector potential, is defined by: aW
(15)
mck w '
The axial quiver of the electrons, caused predominantly by the transverse periodic magnetic force, is enhanced by the static, periodic axial space-charge field that is caused by the periodicity of the steady state velocity vo,(z) . As shown in ref . [9], both effects of the transverse magnetic field and the axial space-charge field contribute to the axial velocity modulation with a period that is half the period of the wiggler . These effects add in a phase, causing an increased velocity modulation, as shown in eq. (14), where S is a positive small number, defined by: w2 8 = P (16) 4 kwyo(vo. ) The plasma frequency is defined in terms of the laboratory frame density parameter n o by: WP =
e 2no mE
(17)
,
and voz is the average along z of the axial velocity of the electron. The characteristic line in eq. (14) can be interpreted as the trajectory of an electron with an initial energy of yi mc2 and an average axial momentum The perturbation in the electron momentum distribution, 11(p, z, w), due to the interaction with the electromagnetic wave and the rf space charge field, is the solution of the first order Vlasov equation (13), given in [9] by :
MP, z , w) =
X
w
v ~pZfo(PZ)Y_Y,Jn(a)Jm(a)(-1)n expli 2nk w - f F (PZ, z, w) exp[i(2mk,+ dz', z
n
(
m
CJ
l
VZ )Z'J
VZ
)
z (18)
E. Jerby, A . Gover / Kinetic linear analysis of a planar wiggler FEL
195
where a, the argument of the modified Bessel functions, is defined by: a=
waW(1+S)
(19)
8kwYôßôzc
The axial driving force
il, (pz,
Plz
(18) is derived from eq. (11) :
z, w) = - eÈlz(z, w) + evoyÈ,(z, w),
(20)
where the longitudinal electric field Élz , related to Jlz by iweklz = Jlz, is : Elz( z , w) = -
if e
P~
vzil(pz, z, w) dpz,
(21)
and the ponderomotive force (FPon = ev oy B,,) can be written in the form [9] : e
FPon(PzI z, w) = iw Ym
eBw
]cw sin(kwz) azty(z' w) .
(22)
The radiation force (Prad = ePl y BwdYm) is 2YZ times smaller than FPon and therefore is considered negligible . Eqs. (18), (20), (21) and (22) describe the excitation of the perturbation to the momentum distribution 1l(pz, z, w) of the electrons, by the transverse electric field Éy(z, w) . To complete the self-consistent linear analysis, we should now use eqs. (1) and (3), which describe the excitation of the radiation wave by the density perturbation . The first order expansion of the transverse current (3) is: Jly(z w)=
-eJ poy Pz Ym
fjpz, z, w)
fo(pz, z) dpz . dpz-eJPz Ply Ym
(23 )
The second term in the left-hand side of eq. (23) describes the direct effect of the transverse electric field Ey (z, w) on the transverse current Jly (z, to) . Using the relation :
Ply(z, co) =
eÉy (z, w) iw
( 24 )
and eqs. (1) and (23), we get: a2 z y (z,w)= _iwtLoliy (z,w), i +kô )É
az
(25)
where koz , the wave number of the e.m . wave inside the electron beam, is : koz =
w2 w 2 _ _P c2 Yoc2
(26)
The FEL interaction current Jiy (the first term in eq . (23)), is : Ji y (z, w) =
e2Bw sin(kwz)I Y-111 (pz, z, w) dpz, mk w Pz
(27)
where we use eq. (14) for Poy (z) . Eqs. (25), (27), (18), (20), (21) and (22) are a closed set of linear, ordinary, integro-differential equations. Thus, a Laplace transform on the z dimension is called for. The wave equation (25), neglecting the excitation of a backward-going wave (i .e., assuming s = ik oz ), becomes: (s-lk oz )Ey (s,w)-Ey (z=0,w) =- 2k o Jly (s,w) . oz
(28) IV . HIGH GAIN FELS
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E. Jerby, A. Gover / Kinetic linear analysis of a planar wiggler FEL
The current Jiy (s, w) results from eqs. (27), (18), (20), (21) and (22) . After some tedious algebraic steps, Ji y (s, w) is found to be :
ii, (S w)=
wBW
Y_ j: X(s+i( 2n+ 1)kj 2mk~,Yo( ooz% n m J,,(a)Jm(a)( -1 ) n
Fpo (s+i(2(n+m)+1)k,) E 1 + J2(a)X(s +i(2(n +m +j) + 1)k,,)
FPo (s + i(2(m + n) - 1)k,,) y 1+ JJ2(a)X(s+i(2(n+m+j)-1)k,,)
(29)
where X(s, w), the longitudinal susceptibility of the beam, is defined by : X(
) _ ie s, w _
2
afo(pZ)/ap"
w
r
S-
iw/v.
(30)
d pZ .
The ponderomotive force (22), transformed to the s-plane, is : 2
Fpo (s,w)=
2mYB /CW
[ (s+ik,)Éy (s+ikw )+(s-ikw )Éy (s-ik w )] .
(31)
In practice, it is usually possible to obtain near-resonance operation with one harmonic . Thus, we assume the condition:
I koz + i(2n o + 1)kw -
lw VZ
I << kw,
(32)
where 2no + 1 (n o = 1, 2, - - - ) is the order of the resonating harmonic . This means that we require a small enough axial velocity spread of the beam : w
o. --- VZ
dvz VZ
« k,,
(33)
so that the condition: IX(s+i(2n o +1)k W ,w)I >> IX(s+i(2m+1)k,,w)1,
bm*n o
(34)
is satisfied. An explicit expression for the detuning spread parameter 9th ----- B1bL, is given later in eq. (58). Under conditions (32) and (33), we may assume that: I Éy(s, w) I >> I Éy (s ± i2mk w, w) I for any m =A 0; thus eq. (29) can be considerably simplified by taking into account only the resonating terms of its double series and neglecting the nonresonating terms. This results in : Jl y (s, w)
eB,
_
2 1
2YomkW ) X
I
vZ i
~Jn o (a) -J,, .+,(a) ] X2n 0 +1
Jn o+1 (a)
1
+Jo+l(a)X2n o +1
where we define : X2no+1 _-- X (S
+
i(2n o + 1)kW , w) .
Jn u ( a )
1 +Jnô(a)X2no+1 I
S.L' (s, w),
(35)
E. Jerby, A . Goner / Kinetic linear analysis of a planar wiggler FEL
197
The gain dispersion relation of a planar wiggler FEL operating at the 2n + 1 harmonic (n = 0, 1,2 . . . ), can be written (using eqs. (28)-(35)) in the form : EE((w)) _ { [1 +J2 (a)X2n+1l [1 +Jn+1(a)X2n+11
x {(s - ik o ,)[1 +J2 (a)X2n+1] [1 +J+1(a)X2n+1, -1 K[Jn(a) - Jn+1(a)I 2X2n+1[ 1 - Jn(a)Jn+1(a)X2n+1~
(37)
where K, the coupling parameter, is :
K=
a
2 2(ko,+(2n+1)kw) .
4 1'0
(38)
The near-synchronism condition at the highly relativistic limit, assuming near-resonance operation with the 2n + 1 harmonic (33), is :
X w 1+a w(1+S) 2n + 1 2yô
(39)
hence a, the argument of the modified Bessel functions (19), becomes:
a
- aw(1 +8)/2 (2n + 1) = aW/2 (2n + 1). 1+aw(1+8) 1+aW
(40)
These results, (37) to (40), converge to the corresponding known ones in the weak pump limit [7] and in the strong pump, tenuous beam, low gain limit [3]. Eq . (37) does not agree in the collective regime with the results of ref. [6]. An analysis of the gain dispersion relation (37) in the various gain regime is presented in the next section.
3. Investigation of the gain dispersion relation The gain dispersion relation of the strong planar wiggler FEL (37) is analyzed in various gain regimes. These are the weak pump limit (a,, << 1) and strong pump limit (aw > 1), the collective and tenuous beam limits, and the cold and warm beam limits . The numerical solution of eq . (37) is described in the next section.
3.1. The weak pump limit In the weak pump limit aw << 1 and consequently a2n+1 (40), the argument of the Bessel functions in eq. (37), is very small. Thus, the Bessel functions converge to JO(al) 1 and to Jn ,, o(a2n+1) - 0. The gain dispersion relation (37) is reduced at the fundamental harmonic (n = 0) to : 4(s,w) __
Ejw)
1+X1
(s-iko,)(1+X1)-iKX,'
(41)
which is the general gain dispersion relation for the various kinds of FELs, as shown in ref. [7]. This equation is analyzed in ref. [10] . IV . HIGH GAIN FELS
198
E. Jerby, A. Gover / Kinetic linear analysis of a planar wiggler FEL
3.2. The strong pump cold beam limit In the cold beam limit the axial momentum distribution function function, fo(p,)
fo (p,)
is approximated by a delta
= n0s(pz - poz),
(42)
and the longitudinal susceptibility, X(s), becomes : X (s~ W)
Ewp
°
[1 + aw(1 + S)]
Y6vo (s-1W/voz
(43)
)2 -*
Therefore, eq . (37) is reduced in the cold beam limit to : Éy (s, W)
_ {S-lkoZ-1K[Jn(a)-J +1(a)1
Eyo(W)
~s+i(2n+1)kwX
((s+i(2n+1)kw-
Z
1w VOz
eP
2
+J)
(a)J +1(a)0 (44)
vo )Z+Jz(a)eP)((s+i(2n+1)kw- vw )Z+J+1(a)BP Oz
where the space charge parameter 6p is defined as : 9p
tip
vo=
/ 1 + aw(1 +8)
U
YO
(45)
,
In the space charge dominated regime, where Bp = OPLw > 7r, eq . (44) has five principal poles, instead of
Fig . 1 . Gain detuning curves of a linear wiggler FEL operating in the tenuous, cold beam regime ( Bth = fundamental and three odd harmonics are shown .
0, ip -+ 0,
Qj =10). The
E. Jerby, A. Goner / Kinetic linear analysis of aplanar wiggler FEL
199
G2n .j
e2n .1
Fig. 2. (a) Gain vs e2n+l of a linear wiggler FEL operating in the space-charge dominated cold beam regime. The fundamental (the double-humped curve) and three odd harmonics are shown. The operating parameters at the fundamental harmonic are K r = 0.1, op = 10, Btht = 0 and a , ~:> 1. The dashed line shows the gain curve of a helical wiggler FEL with the same operating parameters. (b) Same as (a) but Bth , =1 . (c) Same as (a) but Btht = 3.
three as in the a w << 1 limit. This may lead to a double-humpted gain curve, as shown in fig. 2a. The detuning parameter B2n+1 for the 2n + 1 harmonic is defined by :
#2n+1_w - koz - (2n+ 1)k,)L VZ
(46)
The double-humped gain curve, resulting in the strong-wiggler, space-charge dominated regime (in which Jn ( « 2 n+ 1)ep ~" 17), shows that the e.m. wave may be synchronized with one of two slow space-charge waves. One possible synchronism condition is [7,10] 02n+1 e~'
Jn(a2n+1)Bp5
(47a) IV. HIGH GAIN FELS
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E. Jerby, A . Goner / Kinetic linear analysis of a planar wiggler FEL
and the other is: 82n+1
(47b)
~Jn+1(a2n+1)Bp-
This phenomenon leads to a spurious homogeneous broadening of the gain curve, and results in a relatively high bandwidth. A single-humped gain curve of a helical wiggler FEL with the same parameters (ii, Bp ) is shown in fig . 2a for comparison with the double-humped gain curve of the linear wiggler FEL. The synchronism condition of the helical wiggler FEL is : (47c)
B = -op ,
and its single-humped gain curve is much narrower than the linear wiggler FEL gain curve. All curves were obtained by numerical inverse Laplace transforming of the gain dispersion equation using a computer numerical computation code which is described in section 4. An interesting property of the strong wiggler space-charge dominated regime is that in the cold beam limit the maximum gain at the high harmonics can be higher than the maximum gain at the fundamental frequency. In some cases, this can be used to extend the operating frequency range of the FEL without loss of gain, provided that the electron beam is cold enough . In the tenuous beam limit, where Bp << ir, the collective effects are negligible, and we get a three pole equation : Éy (s,
w) _ _
Eyo( w )
2
(s -
[s+i(2n + 1)kw+iw/ J ikoZ)[s+i(2n+1)k , +iw/VZ 1 2- iK[J,(a)
- Jn+l(a)] 2 Bp
.
(48)
Thus, the investigation of the gain regimes of the FEL gain dispersion relation carried out in ref. [10] is entirely valid in this case, whereas the gain parameter is modified to be: j2 2 3 Q2n+1-K2n+1[Jn(a2n+1)-Jn+1(a2n+1)J BPLw .
Following ref. [10], the gain in the low-gain regime (where G2n+1(B) -
1+
d
Q2n+l de
(49) Q2n+l <,r) is
s ine (0/2)
G2n+1 B=0
( ~~
~ exp ~-'
(50)
(8/2)2
which is in agreement with ref. [3]. In the high-gain regime, where _ 1
approximated by:
1/3 1) . Q2n+
Q2n+l > 77,
we get: (51)
Eq. (51), which is valid in the tenuous beam, strong coupling regime, is obtained also in ref. [6], though our collective regimes results are in disagreement with ref. [6] . Examples of gain detuning curves at the fundamental and harmonics in the cold, tenuous beam limit (B1h = 0, Bp - 0) are shown in fig . 1. The peak gain at the fundamental frequency is higher than the peak gain at the harmonics and the shape of the gain detuning curve is the well known single-humped S shape. It should be noted that a recently published [15] correction for the known JJ term (49) neglects the small modulation on the axial electron velocity # I , c (in the transition between eqs. (9) and (10) of ref. [15]) and, therefore, leads to a different result . 3.3. The warm beam regime The considerations regarding FEL operation at higher harmonics are mainly affected by the e-beam quality limitations because higher harmonics are more sensitive than the fundamental to the axial velocity spread of the electrons [3] . The normalized axial velocity distribution of the electron beam due to the total energy spread and due to the finite emittance [11] of the beam was found to be [12]; f(x)=Rexp[R(R+ 2x)] erfc(R+x),
(52)
E. Jerby, A. Goner / Kinetic linear analysis of a planar wiggler FEL
201
where we define : x=
By = AY,/Yo Y0Z _ 1
UZ - VOZ
UoZsy
R = 8'Y
(53a, b, c)
and erfc(x) is the complementary error function : erfc(x)
= 2f FIT
1.
e-`2 dt .
(54)
The electron energy y and the initial opening angle (in the wiggler plane) cpy are assumed to be Gaussian distributed with standard deviations of Ay/ F2 and 440/ respectively . Another assumption on which eq. (52) is based is that the initial beam parameter at the entrance to the wiggler, LOY and Ayo, are optimized with regard to the betatron wavenumber k# in order to obtain minimal emittance effect [111, hence: (55) AOv = kpAyo,
r,
where
_ kwa , .
(56) Yo The high harmonic gain in the warm beam limit ( Bth2 +i > 17) can be calculated analytically, using the expression [7,10] : k8
Gen+1 = 27T
where
#then+1,
Blh2~+ 1
Q2n+1 #2n+1 z e~2n+I f,~ e~2n +,
(57)
the detuning spread parameter due to the energy spread, is,defined by :
= 4Tr(2n + 1)Nw Y ,
(58)
and N,,, is the number of wiggler periods (NW = L,,/Xw). The normalized distribution function f(x) (52) can be used in eq . (57) to evaluate the gain detuning curve in the warm beam limit. It should be noted that when the angular spread is dominant (r < 1) the gain curve is not symmetric and the net gain is lower than the net attenuation. When the energy spread is the dominant cause for the axial velocity 2spread (r >> 1), the normalized distribution function f(x) (52) reduces to a Gaussian (f(x) = 1/Fe -X ). The acceptance criteria for the energy spread Ay/y in the low-gain limit is [10] Ay/-y < 1/4(2n + 1)NW; thus, the highest harmonic "acceptance" conditions may be written as: 2n + 1 = 1 4N,,i y/Y In the high-gain regime higher acceptance may be used [10] .
(59)
4. The numerical solution of the gain dispersion relation In the general case where the collective effects are not negligible (9p > 7r) and the angular spread is significant relative to the energy spread (r > 1), the gain detuning curve should be calculated numerically. The longitudinal susceptibility of the electron beam X (w, s) (30), using the "exact" distribution function f(x) (52), can be expressed in the form [12]: z a 2R fow exp(-2Ry)Z'(~+y) (60) X(w, s) dy, sSY
IV. HIGH GAIN FELS
202
E. Jerby, A . Gover / Kinetic linear analyses of a planar wiggler FEL
where Z(~) is the known plasma dispersion function [13] and ~ is defined by [71 : iw/s - VO, oo ,6 Y
(61)
The program WARM [10,14] has been extended to include higher harmonics, taking into account the "exact" distribution function (52). The gain dispersion relation (37) is inverse Laplace transformed, using the numerical method of Gaussian quadrature for direct integration of eq. (37), as in the previous versions of the program [14]. The convolution integral for X(s, w) (60) is computed numerically, using a continued fraction approximation routine to calculate Z'(~). Examples of gain detuning curves that have been obtained by the extended version of the WARM program are shown in the series of figs. 2a, 2b and 2c for cold (6th, = 0), "lukewarm" (0th, = 1) and warm (6th, = 3) beams, respectively . This series demonstrates the effect of the velocity spread on the gain at various harmonics in the space charge dominated regime (Bp = 10). In fig. 2a the electron beam is considered cold in terms of all harmonics . In fig . 2b the electron beam may be considered cold for the fundamental harmonic but harmonics 5 and 7 suffer a significant reduction of the gain, due to the axial velocity spread. This effect of higher harmonic attenuation becomes stronger in fig . 2c. The double-humped gain curve at the fundamental harmonic is distorted and it appears that the additional peak of the gain curve (47b) is more sensitive to the axial velocity spread than the standard one (47a) . This result can be related to the different effective space-charge parameters that fulfill the different synchronism conditions, because the collective effect reduces the velocity spread effect on the FEL gain reduction [10]. References [1] [21 [3] [4] [5] [61 [71 [81 [9] [101 [111 [12] [131 [141 [15]
J .M .J . Madey and R .C . Taber, in : Physics of Quantum Electronics, vol . 7, eds ., S . Jacobs et al ., (Addison-Wesley, Reading, MA, 1980) chap . 30. W.B . Colson, IEEE J. Quantum Electron . QE-17 (1981) 1417, and Phys. Rev . A 24 (1981) 639 . G . Dattoli, T. Letardi, J .M.J . Madey and A. Renieri, IEEE J. Quantum Electron. QE-20 (1984) 1003, and F . Ciocci, G . Dattoli and A . Renieri, Lett . Nuovo Cimento 34 (1982) 341 . B. Barbini and G . Vignola, Proc . AIP Conf. 118 (1983) 103 . R.C . Davidson and J.S . Wurtele, Self consistent kinetic description of the FEL in a planar magnetic wiggler, to be published . IEEE Trans . Plasma Phys . 13 (1985) 464 . J .B . Murphy and C . Pellegrini, Opt. Commun. 55 (1985) 197. A. Gover and P . Sprangle, IEEE J. Quantum Electron QE-17 (1981) 1196 . F.B . Hildebrand, Advanced Calculus for Applications (Prentic Hall, London, 1962) . E. Jerby and A. Gover, A kinetic, linear analysis of a strong planar wiggler FEL, to be published. E. Jerby and A. Gover, IEEE J. Quantum Electron . QE-21 (1985) 1041 . A. Gover, H . Freund, V.L . Granatstein, J.S . McAdoo and C .M. Tang, NRL Rep. 8747 (1982) ; also in : Infrared and Millimeter Waves, vol. 11, Ed ., K.J. Button (Academic Press, New York, 1984). E. Jerby and A. Gover, "An exact electron momentum distribution kinetic analysis of the free electron laser", to be published . B.O . Fried and S .D. Conte, The Plasma Dispersion Function (Academic Press, New York, 1961) . Z . Livni and A . Gover, Linear analysis and implementation consideration of FELS based on Cherenkov and Smith-Purcell effects, Quantum Electron Laboratory, School of Engineering, Tel-Aviv, University, Ramat-Aviv, Israel, Sci . Rep . 1979/1 (AFOSR 77-3445) (1979) . B.W .J. McNeil and W .J. Firth, IEEE J . Quantum Electron . QE-21 (1985) 1034 ; also in these proceedings (7th Int . FEL Conf.) Nucl . Instr . and Meth . A250 (1986) 445 .
Nuclear Instruments and Methods in Physics Research A250 (1986) 192-202 North-Holland, Amsterdam
A KINETIC SMALL SIGNAL GAIN ANALYSIS OF A PLANAR WIGGLER FEL, OPERATING IN THE HIGH HARMONIC (STRONG WIGGLER) REGIME E. JERBY and A. GOVER Faculty of Engineering, Tel-Aviv University, Ramat-Aviv, 69978 Israel
A kinetic linear analysis of a strong planar wiggler FEL operating at the fundamental or odd harmonic frequencies is presented in this paper. The gain dispersion equation in the high harmonic regime is shown to be an extension of the known general FEL equation, and converges to it in the weak pump limit. In the low gain, cold, tenuous beam regime we get the known gain expression with the Bessel function correction term . A similar term also appears in the gain expression of the high gain strong pump limit. When space charge effects are significant (Raman regime), the strong pump gain dispersion relation is more complicated than the corresponding weak pump relation. In the cold beam limit it leads to a quintic polynomial dispersion relation instead of the known cubic equation . This brings about gain curve dependences on the detuning parameter which are double humped and with a relatively high bandwidth. The higher the order of the harmonic frequency, the stronger is the effect of the axial velocity spread on the gain and the tighter are the beam acceptance parameters. The analytical model permits an arbitrary axial velocity distribution . The gain damping effect is computed, both with a simple model of a Gaussian electron velocity distribution and with a more accurate model, taking into account an analytical expression for the asymmetrical skewed velocity distribution produced by the combined finite emittance and energy spread effects .
1. Introduction The fundamental operating wavelength of a free electron laser (FEL) is determined by the energy of the electron beam according to the scaling law X=X w/(2yoZ ) (X W is the wiggler period, and yo, is the longitudinal relativistic Lorentz factor of the e-beam). One finds that a conventional wiggler FEL (X w - 0.01-0.1 m) operating in the visible range requires an electron beam energy of at least 100 MeV . Consequently, accelerators such as the Van de Graaff that are known to operate with high beam quality but relatively low energy can be used in FELs operating in the far infrared region only. High harmonic operation is one way by which such FELs can be made to operate at optical frequencies, using conventional period wigglers. Alternatively, this method can be used to make RF linac and storage ring based FELs to operate at ever decreasing shorter wavelengths down to the VUV-X-ray regime [3]. FEL operation in high harmonics has been considered by various workers as a method of extending the FEL range of operation towards shorter wavelengths. Madey and Taber [1] pointed out the feasibility of FEL interaction at high odd harmonic frequency as a unique property of the planar wiggler FEL, due to the periodic variation (at half the wiggler period) in the axial velocity of the electrons propagating along the wiggler. Colson [2] investigated nonlinear aspects of the higher harmonic operation of planar wiggler FELs in the cold, tenuous beam regime . Dattoli et al. [3] evaluated the dependence of the small signal gain on the electron beam quality factors in the low gain, tenuous beam regime. An experimental observation of gain in the third harmonic was reported by Barbini and Vignola [4] . A self-consistent kinetic description of the FEL instability in a planar magnetic wiggler was developed by Davidson and Wurtele [5], and a linear model including some collective and high gain effects was suggested by Murphy and Pellegrini [6]. The present paper presents a kinetic, linear analysis of a strong (relativistic wiggle velocity), planar wiggler FEL operating at the fundamental or odd harmonic frequencies. Space charge and axial velocity spread effects are included in this model. The resulting gain dispersion relation at fundamental and high harmonic frequencies is an extension of the known fundamental FEL equation [7], and is reduced to it in the limit of a small magnetic field wiggler. Analytical and numerical solutions of the gain dispersion relation are shown, and some unique aspects of FEL operation at higher harmonics in the space charge dominated regime are discussed.
E. Jerby, A . Gover / Kinetic linear analysis of a planar wiggler FEL
193
2. The theoretical model The development of the gain dispersion relation of the planar wiggler FEL operating at a high harmonic frequency is based on the solution of Maxwell equations simultaneously with the linearized Vlasov equation in one dimension. Assuming small-signal operation, the perturbation method is applied. Vlasov equations in the zero and first orders are solved by the method of the characteristic lines in the p, -z plane (the axial momentum and coordinate, respectively). After performing a Laplace transform in the z dimension, the gain dispersion relation is obtained by the assumption of a near-resonance operation at one of the harmonics. The transverse components of the e.m . wave under the 1-D assumption (ô/8x, â/8y = 0) are described by the reduced Maxwell equations: z
ôz2E`(z, t)
(1)
poEEr(z, t) = ,uOJ(z, t)
and B~(z, t) _ -z X
az E,(z, t).
The driving current J,(z, t) is given by : J, (z, t) = -e f vf(P, z, t) d 3p . P
(3)
The momentum distribution of the electrons f(p, z, t) is governed by the Vlasov equation :
AP, z,t ) + U=ô z f(P ,z , t)+F(P,z , t)' pf(P,Z,t)=0 .
(4)
The driving force F(p, z, t) is defined by the Lorentz equation : F(p,z,t)= -e[Et(z,t)+!E,(z,t)+vxBt(z,t)+vxBW(z,t) ] .
(5)
The longitudinal electric field EZ(z, t) is caused by collective effects and is the solution of the Poisson equation : âzEz(z,t) =-= ff (P,z,t)d 3p . P The "pump" force in eq . (5) (F,, = ev X B,,) is induced by the planar wiggler static magnetic field, which is approximated by : B, (z, t) =zB, cosk,,z. Eqs. (1)-(7) are a self-consistent complete set of equations. To solve them, we apply the standard perturbation method that is applicable with the small signal assumption . The electron momentum distribution is expressed in the form : f(P, z, t)
= fo(P, z)
+fi(P, z, t),
(8)
where fo(p, z) is the steady state e-beam momentum distribution in the wiggler field, and fl(p, z, t) is a perturbation due to the interaction with the e.m . wave, assuming that fo >> I f I exists . In a similar manner, the driving force F(p, z, t) is written as : F(P, z, t) = FO(P, z) +F,(P, z, t), where I Fo I >> I Fi I exists. IV . HIGH GAIN FELS
194
E. Jerby, A . Goner / Kinetic linear analysis of a planar wiggler FEL
The steady state force
FO(p,
z) is :
(10) FO (P,z)=-e[2EOZ (z)+vXBW (z) ], where Eo,(z) is an internal space charge electrostatic field, caused by the periodic density variation along the electron beam. This periodic density modulation varies at half the period of the magnetostatic wiggler, and is due to the axial velocity modulation at the same periodicity [9] . The perturbation force in eq. (9) is: F,(P, (ll) z, t) = -e[E,(z, t) +îEl,( z, t) +vxB,(z, t)], where the e.m. wave components (Et , B,) are the TEM radiation fields that are the small signal first-order perturbation parameters and Elz (z, t) is the electron beam rf space charge field. Consequently, the Vlasov equation is separated into two coupled equations . The steady state (zero order) Vlasov equation is: VZ
ä z fo(P, z) +Fo(P, z) - ~p fo(P, z) = 0,
(12)
and the disturbance (first order) equation, transformed to the frequency domain, is : - lwf (P,
Z,
w) +
VZ
azf (P, z, w)Fl(Piziw) - ~pfo(P, Z) +FO(P,
Z) -
äp fi = 0 .
(13)
Eqs. (12) and (13) are linear, partial differential equations of the first order; thus the method of the characteristic lines [8] may be applied to solve them. The characteristic line of the steady state Vlasov equation (12) is [9] : Po(z) =mc[-ya,, sin(k,,z)+î
i2 -1-a,2,(1+8) sin2 (k,,z)1 1
y
(14)
where a, the wiggler normalized vector potential, is defined by: aW
(15)
mck w '
The axial quiver of the electrons, caused predominantly by the transverse periodic magnetic force, is enhanced by the static, periodic axial space-charge field that is caused by the periodicity of the steady state velocity vo,(z) . As shown in ref . [9], both effects of the transverse magnetic field and the axial space-charge field contribute to the axial velocity modulation with a period that is half the period of the wiggler . These effects add in a phase, causing an increased velocity modulation, as shown in eq. (14), where S is a positive small number, defined by: w2 8 = P (16) 4 kwyo(vo. ) The plasma frequency is defined in terms of the laboratory frame density parameter n o by: WP =
e 2no mE
(17)
,
and voz is the average along z of the axial velocity of the electron. The characteristic line in eq. (14) can be interpreted as the trajectory of an electron with an initial energy of yi mc2 and an average axial momentum The perturbation in the electron momentum distribution, 11(p, z, w), due to the interaction with the electromagnetic wave and the rf space charge field, is the solution of the first order Vlasov equation (13), given in [9] by :
MP, z , w) =
X
w
v ~pZfo(PZ)Y_Y,Jn(a)Jm(a)(-1)n expli 2nk w - f F (PZ, z, w) exp[i(2mk,+ dz', z
n
(
m
CJ
l
VZ )Z'J
VZ
)
z (18)
E. Jerby, A . Gover / Kinetic linear analysis of a planar wiggler FEL
195
where a, the argument of the modified Bessel functions, is defined by: a=
waW(1+S)
(19)
8kwYôßôzc
The axial driving force
il, (pz,
Plz
(18) is derived from eq. (11) :
z, w) = - eÈlz(z, w) + evoyÈ,(z, w),
(20)
where the longitudinal electric field Élz , related to Jlz by iweklz = Jlz, is : Elz( z , w) = -
if e
P~
vzil(pz, z, w) dpz,
(21)
and the ponderomotive force (FPon = ev oy B,,) can be written in the form [9] : e
FPon(PzI z, w) = iw Ym
eBw
]cw sin(kwz) azty(z' w) .
(22)
The radiation force (Prad = ePl y BwdYm) is 2YZ times smaller than FPon and therefore is considered negligible . Eqs. (18), (20), (21) and (22) describe the excitation of the perturbation to the momentum distribution 1l(pz, z, w) of the electrons, by the transverse electric field Éy(z, w) . To complete the self-consistent linear analysis, we should now use eqs. (1) and (3), which describe the excitation of the radiation wave by the density perturbation . The first order expansion of the transverse current (3) is: Jly(z w)=
-eJ poy Pz Ym
fjpz, z, w)
fo(pz, z) dpz . dpz-eJPz Ply Ym
(23 )
The second term in the left-hand side of eq. (23) describes the direct effect of the transverse electric field Ey (z, w) on the transverse current Jly (z, to) . Using the relation :
Ply(z, co) =
eÉy (z, w) iw
( 24 )
and eqs. (1) and (23), we get: a2 z y (z,w)= _iwtLoliy (z,w), i +kô )É
az
(25)
where koz , the wave number of the e.m . wave inside the electron beam, is : koz =
w2 w 2 _ _P c2 Yoc2
(26)
The FEL interaction current Jiy (the first term in eq . (23)), is : Ji y (z, w) =
e2Bw sin(kwz)I Y-111 (pz, z, w) dpz, mk w Pz
(27)
where we use eq. (14) for Poy (z) . Eqs. (25), (27), (18), (20), (21) and (22) are a closed set of linear, ordinary, integro-differential equations. Thus, a Laplace transform on the z dimension is called for. The wave equation (25), neglecting the excitation of a backward-going wave (i .e., assuming s = ik oz ), becomes: (s-lk oz )Ey (s,w)-Ey (z=0,w) =- 2k o Jly (s,w) . oz
(28) IV . HIGH GAIN FELS
196
E. Jerby, A. Gover / Kinetic linear analysis of a planar wiggler FEL
The current Jiy (s, w) results from eqs. (27), (18), (20), (21) and (22) . After some tedious algebraic steps, Ji y (s, w) is found to be :
ii, (S w)=
wBW
Y_ j: X(s+i( 2n+ 1)kj 2mk~,Yo( ooz% n m J,,(a)Jm(a)( -1 ) n
Fpo (s+i(2(n+m)+1)k,) E 1 + J2(a)X(s +i(2(n +m +j) + 1)k,,)
FPo (s + i(2(m + n) - 1)k,,) y 1+ JJ2(a)X(s+i(2(n+m+j)-1)k,,)
(29)
where X(s, w), the longitudinal susceptibility of the beam, is defined by : X(
) _ ie s, w _
2
afo(pZ)/ap"
w
r
S-
iw/v.
(30)
d pZ .
The ponderomotive force (22), transformed to the s-plane, is : 2
Fpo (s,w)=
2mYB /CW
[ (s+ik,)Éy (s+ikw )+(s-ikw )Éy (s-ik w )] .
(31)
In practice, it is usually possible to obtain near-resonance operation with one harmonic . Thus, we assume the condition:
I koz + i(2n o + 1)kw -
lw VZ
I << kw,
(32)
where 2no + 1 (n o = 1, 2, - - - ) is the order of the resonating harmonic . This means that we require a small enough axial velocity spread of the beam : w
o. --- VZ
dvz VZ
« k,,
(33)
so that the condition: IX(s+i(2n o +1)k W ,w)I >> IX(s+i(2m+1)k,,w)1,
bm*n o
(34)
is satisfied. An explicit expression for the detuning spread parameter 9th ----- B1bL, is given later in eq. (58). Under conditions (32) and (33), we may assume that: I Éy(s, w) I >> I Éy (s ± i2mk w, w) I for any m =A 0; thus eq. (29) can be considerably simplified by taking into account only the resonating terms of its double series and neglecting the nonresonating terms. This results in : Jl y (s, w)
eB,
_
2 1
2YomkW ) X
I
vZ i
~Jn o (a) -J,, .+,(a) ] X2n 0 +1
Jn o+1 (a)
1
+Jo+l(a)X2n o +1
where we define : X2no+1 _-- X (S
+
i(2n o + 1)kW , w) .
Jn u ( a )
1 +Jnô(a)X2no+1 I
S.L' (s, w),
(35)
E. Jerby, A . Goner / Kinetic linear analysis of a planar wiggler FEL
197
The gain dispersion relation of a planar wiggler FEL operating at the 2n + 1 harmonic (n = 0, 1,2 . . . ), can be written (using eqs. (28)-(35)) in the form : EE((w)) _ { [1 +J2 (a)X2n+1l [1 +Jn+1(a)X2n+11
x {(s - ik o ,)[1 +J2 (a)X2n+1] [1 +J+1(a)X2n+1, -1 K[Jn(a) - Jn+1(a)I 2X2n+1[ 1 - Jn(a)Jn+1(a)X2n+1~
(37)
where K, the coupling parameter, is :
K=
a
2 2(ko,+(2n+1)kw) .
4 1'0
(38)
The near-synchronism condition at the highly relativistic limit, assuming near-resonance operation with the 2n + 1 harmonic (33), is :
X w 1+a w(1+S) 2n + 1 2yô
(39)
hence a, the argument of the modified Bessel functions (19), becomes:
a
- aw(1 +8)/2 (2n + 1) = aW/2 (2n + 1). 1+aw(1+8) 1+aW
(40)
These results, (37) to (40), converge to the corresponding known ones in the weak pump limit [7] and in the strong pump, tenuous beam, low gain limit [3]. Eq . (37) does not agree in the collective regime with the results of ref. [6]. An analysis of the gain dispersion relation (37) in the various gain regime is presented in the next section.
3. Investigation of the gain dispersion relation The gain dispersion relation of the strong planar wiggler FEL (37) is analyzed in various gain regimes. These are the weak pump limit (a,, << 1) and strong pump limit (aw > 1), the collective and tenuous beam limits, and the cold and warm beam limits . The numerical solution of eq . (37) is described in the next section.
3.1. The weak pump limit In the weak pump limit aw << 1 and consequently a2n+1 (40), the argument of the Bessel functions in eq. (37), is very small. Thus, the Bessel functions converge to JO(al) 1 and to Jn ,, o(a2n+1) - 0. The gain dispersion relation (37) is reduced at the fundamental harmonic (n = 0) to : 4(s,w) __
Ejw)
1+X1
(s-iko,)(1+X1)-iKX,'
(41)
which is the general gain dispersion relation for the various kinds of FELs, as shown in ref. [7]. This equation is analyzed in ref. [10] . IV . HIGH GAIN FELS
198
E. Jerby, A. Gover / Kinetic linear analysis of a planar wiggler FEL
3.2. The strong pump cold beam limit In the cold beam limit the axial momentum distribution function function, fo(p,)
fo (p,)
is approximated by a delta
= n0s(pz - poz),
(42)
and the longitudinal susceptibility, X(s), becomes : X (s~ W)
Ewp
°
[1 + aw(1 + S)]
Y6vo (s-1W/voz
(43)
)2 -*
Therefore, eq . (37) is reduced in the cold beam limit to : Éy (s, W)
_ {S-lkoZ-1K[Jn(a)-J +1(a)1
Eyo(W)
~s+i(2n+1)kwX
((s+i(2n+1)kw-
Z
1w VOz
eP
2
+J)
(a)J +1(a)0 (44)
vo )Z+Jz(a)eP)((s+i(2n+1)kw- vw )Z+J+1(a)BP Oz
where the space charge parameter 6p is defined as : 9p
tip
vo=
/ 1 + aw(1 +8)
U
YO
(45)
,
In the space charge dominated regime, where Bp = OPLw > 7r, eq . (44) has five principal poles, instead of
Fig . 1 . Gain detuning curves of a linear wiggler FEL operating in the tenuous, cold beam regime ( Bth = fundamental and three odd harmonics are shown .
0, ip -+ 0,
Qj =10). The
E. Jerby, A. Goner / Kinetic linear analysis of aplanar wiggler FEL
199
G2n .j
e2n .1
Fig. 2. (a) Gain vs e2n+l of a linear wiggler FEL operating in the space-charge dominated cold beam regime. The fundamental (the double-humped curve) and three odd harmonics are shown. The operating parameters at the fundamental harmonic are K r = 0.1, op = 10, Btht = 0 and a , ~:> 1. The dashed line shows the gain curve of a helical wiggler FEL with the same operating parameters. (b) Same as (a) but Bth , =1 . (c) Same as (a) but Btht = 3.
three as in the a w << 1 limit. This may lead to a double-humpted gain curve, as shown in fig. 2a. The detuning parameter B2n+1 for the 2n + 1 harmonic is defined by :
#2n+1_w - koz - (2n+ 1)k,)L VZ
(46)
The double-humped gain curve, resulting in the strong-wiggler, space-charge dominated regime (in which Jn ( « 2 n+ 1)ep ~" 17), shows that the e.m. wave may be synchronized with one of two slow space-charge waves. One possible synchronism condition is [7,10] 02n+1 e~'
Jn(a2n+1)Bp5
(47a) IV. HIGH GAIN FELS
200
E. Jerby, A . Goner / Kinetic linear analysis of a planar wiggler FEL
and the other is: 82n+1
(47b)
~Jn+1(a2n+1)Bp-
This phenomenon leads to a spurious homogeneous broadening of the gain curve, and results in a relatively high bandwidth. A single-humped gain curve of a helical wiggler FEL with the same parameters (ii, Bp ) is shown in fig . 2a for comparison with the double-humped gain curve of the linear wiggler FEL. The synchronism condition of the helical wiggler FEL is : (47c)
B = -op ,
and its single-humped gain curve is much narrower than the linear wiggler FEL gain curve. All curves were obtained by numerical inverse Laplace transforming of the gain dispersion equation using a computer numerical computation code which is described in section 4. An interesting property of the strong wiggler space-charge dominated regime is that in the cold beam limit the maximum gain at the high harmonics can be higher than the maximum gain at the fundamental frequency. In some cases, this can be used to extend the operating frequency range of the FEL without loss of gain, provided that the electron beam is cold enough . In the tenuous beam limit, where Bp << ir, the collective effects are negligible, and we get a three pole equation : Éy (s,
w) _ _
Eyo( w )
2
(s -
[s+i(2n + 1)kw+iw/ J ikoZ)[s+i(2n+1)k , +iw/VZ 1 2- iK[J,(a)
- Jn+l(a)] 2 Bp
.
(48)
Thus, the investigation of the gain regimes of the FEL gain dispersion relation carried out in ref. [10] is entirely valid in this case, whereas the gain parameter is modified to be: j2 2 3 Q2n+1-K2n+1[Jn(a2n+1)-Jn+1(a2n+1)J BPLw .
Following ref. [10], the gain in the low-gain regime (where G2n+1(B) -
1+
d
Q2n+l de
(49) Q2n+l <,r) is
s ine (0/2)
G2n+1 B=0
( ~~
~ exp ~-'
(50)
(8/2)2
which is in agreement with ref. [3]. In the high-gain regime, where _ 1
approximated by:
1/3 1) . Q2n+
Q2n+l > 77,
we get: (51)
Eq. (51), which is valid in the tenuous beam, strong coupling regime, is obtained also in ref. [6], though our collective regimes results are in disagreement with ref. [6] . Examples of gain detuning curves at the fundamental and harmonics in the cold, tenuous beam limit (B1h = 0, Bp - 0) are shown in fig . 1. The peak gain at the fundamental frequency is higher than the peak gain at the harmonics and the shape of the gain detuning curve is the well known single-humped S shape. It should be noted that a recently published [15] correction for the known JJ term (49) neglects the small modulation on the axial electron velocity # I , c (in the transition between eqs. (9) and (10) of ref. [15]) and, therefore, leads to a different result . 3.3. The warm beam regime The considerations regarding FEL operation at higher harmonics are mainly affected by the e-beam quality limitations because higher harmonics are more sensitive than the fundamental to the axial velocity spread of the electrons [3] . The normalized axial velocity distribution of the electron beam due to the total energy spread and due to the finite emittance [11] of the beam was found to be [12]; f(x)=Rexp[R(R+ 2x)] erfc(R+x),
(52)
E. Jerby, A. Goner / Kinetic linear analysis of a planar wiggler FEL
201
where we define : x=
By = AY,/Yo Y0Z _ 1
UZ - VOZ
UoZsy
R = 8'Y
(53a, b, c)
and erfc(x) is the complementary error function : erfc(x)
= 2f FIT
1.
e-`2 dt .
(54)
The electron energy y and the initial opening angle (in the wiggler plane) cpy are assumed to be Gaussian distributed with standard deviations of Ay/ F2 and 440/ respectively . Another assumption on which eq. (52) is based is that the initial beam parameter at the entrance to the wiggler, LOY and Ayo, are optimized with regard to the betatron wavenumber k# in order to obtain minimal emittance effect [111, hence: (55) AOv = kpAyo,
r,
where
_ kwa , .
(56) Yo The high harmonic gain in the warm beam limit ( Bth2 +i > 17) can be calculated analytically, using the expression [7,10] : k8
Gen+1 = 27T
where
#then+1,
Blh2~+ 1
Q2n+1 #2n+1 z e~2n+I f,~ e~2n +,
(57)
the detuning spread parameter due to the energy spread, is,defined by :
= 4Tr(2n + 1)Nw Y ,
(58)
and N,,, is the number of wiggler periods (NW = L,,/Xw). The normalized distribution function f(x) (52) can be used in eq . (57) to evaluate the gain detuning curve in the warm beam limit. It should be noted that when the angular spread is dominant (r < 1) the gain curve is not symmetric and the net gain is lower than the net attenuation. When the energy spread is the dominant cause for the axial velocity 2spread (r >> 1), the normalized distribution function f(x) (52) reduces to a Gaussian (f(x) = 1/Fe -X ). The acceptance criteria for the energy spread Ay/y in the low-gain limit is [10] Ay/-y < 1/4(2n + 1)NW; thus, the highest harmonic "acceptance" conditions may be written as: 2n + 1 = 1 4N,,i y/Y In the high-gain regime higher acceptance may be used [10] .
(59)
4. The numerical solution of the gain dispersion relation In the general case where the collective effects are not negligible (9p > 7r) and the angular spread is significant relative to the energy spread (r > 1), the gain detuning curve should be calculated numerically. The longitudinal susceptibility of the electron beam X (w, s) (30), using the "exact" distribution function f(x) (52), can be expressed in the form [12]: z a 2R fow exp(-2Ry)Z'(~+y) (60) X(w, s) dy, sSY
IV. HIGH GAIN FELS
202
E. Jerby, A . Gover / Kinetic linear analyses of a planar wiggler FEL
where Z(~) is the known plasma dispersion function [13] and ~ is defined by [71 : iw/s - VO, oo ,6 Y
(61)
The program WARM [10,14] has been extended to include higher harmonics, taking into account the "exact" distribution function (52). The gain dispersion relation (37) is inverse Laplace transformed, using the numerical method of Gaussian quadrature for direct integration of eq. (37), as in the previous versions of the program [14]. The convolution integral for X(s, w) (60) is computed numerically, using a continued fraction approximation routine to calculate Z'(~). Examples of gain detuning curves that have been obtained by the extended version of the WARM program are shown in the series of figs. 2a, 2b and 2c for cold (6th, = 0), "lukewarm" (0th, = 1) and warm (6th, = 3) beams, respectively . This series demonstrates the effect of the velocity spread on the gain at various harmonics in the space charge dominated regime (Bp = 10). In fig. 2a the electron beam is considered cold in terms of all harmonics . In fig . 2b the electron beam may be considered cold for the fundamental harmonic but harmonics 5 and 7 suffer a significant reduction of the gain, due to the axial velocity spread. This effect of higher harmonic attenuation becomes stronger in fig . 2c. The double-humped gain curve at the fundamental harmonic is distorted and it appears that the additional peak of the gain curve (47b) is more sensitive to the axial velocity spread than the standard one (47a) . This result can be related to the different effective space-charge parameters that fulfill the different synchronism conditions, because the collective effect reduces the velocity spread effect on the FEL gain reduction [10]. References [1] [21 [3] [4] [5] [61 [71 [81 [9] [101 [111 [12] [131 [141 [15]
J .M .J . Madey and R .C . Taber, in : Physics of Quantum Electronics, vol . 7, eds ., S . Jacobs et al ., (Addison-Wesley, Reading, MA, 1980) chap . 30. W.B . Colson, IEEE J. Quantum Electron . QE-17 (1981) 1417, and Phys. Rev . A 24 (1981) 639 . G . Dattoli, T. Letardi, J .M.J . Madey and A. Renieri, IEEE J. Quantum Electron. QE-20 (1984) 1003, and F . Ciocci, G . Dattoli and A . Renieri, Lett . Nuovo Cimento 34 (1982) 341 . B. Barbini and G . Vignola, Proc . AIP Conf. 118 (1983) 103 . R.C . Davidson and J.S . Wurtele, Self consistent kinetic description of the FEL in a planar magnetic wiggler, to be published . IEEE Trans . Plasma Phys . 13 (1985) 464 . J .B . Murphy and C . Pellegrini, Opt. Commun. 55 (1985) 197. A. Gover and P . Sprangle, IEEE J. Quantum Electron QE-17 (1981) 1196 . F.B . Hildebrand, Advanced Calculus for Applications (Prentic Hall, London, 1962) . E. Jerby and A. Gover, A kinetic, linear analysis of a strong planar wiggler FEL, to be published. E. Jerby and A. Gover, IEEE J. Quantum Electron . QE-21 (1985) 1041 . A. Gover, H . Freund, V.L . Granatstein, J.S . McAdoo and C .M. Tang, NRL Rep. 8747 (1982) ; also in : Infrared and Millimeter Waves, vol. 11, Ed ., K.J. Button (Academic Press, New York, 1984). E. Jerby and A. Gover, "An exact electron momentum distribution kinetic analysis of the free electron laser", to be published . B.O . Fried and S .D. Conte, The Plasma Dispersion Function (Academic Press, New York, 1961) . Z . Livni and A . Gover, Linear analysis and implementation consideration of FELS based on Cherenkov and Smith-Purcell effects, Quantum Electron Laboratory, School of Engineering, Tel-Aviv, University, Ramat-Aviv, Israel, Sci . Rep . 1979/1 (AFOSR 77-3445) (1979) . B.W .J. McNeil and W .J. Firth, IEEE J . Quantum Electron . QE-21 (1985) 1034 ; also in these proceedings (7th Int . FEL Conf.) Nucl . Instr . and Meth . A250 (1986) 445 .