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Emittance limitations in the free electron laser
Optics Communications 93 (1992) 179-185 North-Holland
OPTICS COMMUNICATIONS
Emittance limitations in the free electron laser R. B o n i f a c i o 1, L. D e Salvo S o u z a a n d B.W.J. M c N e i l INFN- Sezione di Milano, Via Celoria, 16, 20133 Milano, Italy
Received 10 December 1991; revised manuscript received 2 June 1992
We present a simple generalized 1D model to take into account emittance and diffraction effects both in the linear and nonlinear regime. We show that the usual 1D model is recovered if the emittance has an upper or a lower limit. We show analytically that diffraction and eminance effects can be sensibly reduced both in the linear and nonlinear regime by properly detuning the electron beam energy.
1. Introduction In this paper we investigate the effects of emittance and diffraction on the performance of an FEL with a m a t c h e d electron beam. An electron beam is said to be matched when its transverse area does not vary as it propagates through the wiggler. Within the limits discussed here the major effect of emittance is to change an electron's resonant energy away from that of the "standard" FEL resonance relation thus introducing a resonant energy spread even into a mono-energetic beam. The other effects of emittance within a matched beam, namely the changes in the electron/radiation coupling as the electrons perform betatron orbits, are assumed to be negligible. Hence we model the electron beam emit° tance by using a 1D model with a resonant energy spread. We derive a general emittance criterium for both the low and high gain regimes which implies in an upper limit on the acceptable emittance in terms of the cold beam radiation gain linewidth. In the low gain limit and with the wiggler length equal to a betatron period this criterium is the one derived by Kim and Pellegrini [1]. In the high gain we show a relaxation o f a previous criterium derived from linear theory by Yu and Krinsky [2]. On the other hand the condition to neglect diffraction effects implies a lower limit in the real emitAlso Dipartimento di Fisica dell'Universit~l Statale di Milano.
tance. This lower limit and the higher limit described above combine to give a necessary condition which must be satisfied given that either the lower or upper limit is valid. In general diffraction effects in a guided mode can be represented with a detuning term in a 1D model [ 1 ]. We show analytically that diffraction and emittance effects can be sensibly reduced both in the linear [2] and nonlinear regime by properly detuning the electron beam energy. Finally we give an analytical criterium for an o p t i m u m detuning of the electron beam which sensibly reduces 3D effects of angular divergence o f the electron beam both in the linear and in the nonlinear regime.
2. The equivalence of energy spread and emittance It can be shown [3 ] that in a planar wiggler the transverse component of the velocity o f an electron in the absence of a signal field is given by 7 29--2 fltot = a w2 ( 1 -I- y ~ k 2 ) ,
( 1)
where ~ot is the average over a wiggler period o f the total transverse velocity (divided by c) due to both the wiggler and the betatron motion, aw is the RMS wiggler deflection parameter, kw is the wiggler wavenumber 2~/2w, ~ is the electron energy in dimensionless units m c 2 and YBis the m a x i m u m transverse amplitude o f the electron due to the betatron motion. Note that the transverse velocity o f an electron
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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is not a function of the position within its betatron orbit. Such behaviour is typical of wigglers with "natural focusing" [ 3 ]. Using the FEL resonance condition [4] fl_-= kr/(kw+kr), the definition of 7 and the approximation ( 1 - f f ~ : ) ~ 2(1 - f l : ) , it is easy to show that 2w ?)~ = ~ (1 +72flt2o,),
(2)
where ~)r defines the resonant electron energy for a given radiation wavelength 2r. Substituting eq. ( 1 ) into eq. (2) we obtain 2w
l+aw+awkwY~)
(3)
The last term in eq. (3) shows explicitly the dependence of the resonant energy of an electron on its particular betatron orbit. For a matched beam the maximum value of Y~ is the electron beam radius [ 3 ]
rb= EX/~~,
(4)
where e is the hard edge emittance of the electron beam, and k~ is the betatron wavenumber
k~=awkw/ f f .
(5)
In defining the hard edge emittance we mean that the electron beam is uniformly distributed in transverse phase space ( Y, d Y / d z ) within the matched ellipse of major axis (dY/dz)m~x=kprb and minor axis Ymax = rb. The area of the matched ellipse is then he. The factor f i s a focusing factor, which for a normal planar wiggler is equal to one. For a helical wiggler or a planar wiggler with curved pole faces [5] (in order to obtain equal focusing in both transverse directions ), f = , ~ . If additional focusing is added (e.g. plasma focusing) f may be made smaller than one [61. For a beam of given emittance equal in both transverse coordinates (ex= ey= e) the electrons will then have a resonant energy distribution between 2w
~2o-- ~ ( (1 +a~) and 2w 22~ 180
1 October 1992
where we have used eq. (3) to eq. (5). Hence we may write the normalized resonant energy spread due to emittance of: A - -
Ayr 7to
- -
-
kr~.wf 2 kr~aw.f 42 ~ 47r0 '
(6)
where we assume ATr << 7r0Within the assumption used in the model, in particular that of the matched beam condition eq. (4) and Ayr << 7m, by far the greatest effect of the emittance is that of the spread in the electrons' resonance conditions; such effects as changes in electron/radiation coupling via the changing deflection parameter a,~ and propagation angle [ 7 ] along an electron betatron orbit are negligible in the limit kwrb<< 1. This resonant energy spread will have minor effect on the FEL gain if its full width is smaller than the half width at half maximum (hwhm) of the natural FEL linewidth (e.g. in the low gain this corresponds to approximately half of the positive part of the Madey gain spectrum being populated with electrons) i.e.: A7___z_~< 1 (Aog)
Vr0
(7)
2 -7- hw.m"
Hence using eq. (6) we have kr{~ ~wwf-2 ~ -
hwhm"
(8)
Equation (8) is a general emittance condition valid in both the low and high gain regimes. In particular one must take:
_l co /hwhm -- 2Nw in the low gain and
(9)
(A~o) =2p [81 in the high gain . ~ hwhm
(10)
Here p is the fundamental FEL parameter [4] and may be written
p=O.1487-1B~/32~/3 II/3rg 2/3 ,
( 11 )
where I is the beam current and Bw is the RMS wiggler magnetic field strength. A rigorous proof of relations ( 8 ) - ( 1 0 ) will be given in the next section. Using eq. (4) we can write p for a matched electron beam as a function of the emittance:
Volume 93, number 3,4 p~
1.247-4/3Bw24/3iI/3f
OPTICS COMMUNICATIONS -- 1/3~--
1/3
.
(12)
Note that the Kim-Pellegrini criteria [ 1 ] is obtained immediately from eq. (8) by assuming that ;ta is equal to the wiggler length Lw, f = 1 and the low gain linewidth ofeq. (9). In the high gain, ref. [2] showed that the linear exponential gain is not adversely effected when (for f = x/~) k#
<<
2k~p/k#.
( 13 )
A relaxation of this limit is seen from eqs. (8) and
(10) to k 4 kwp ~E< 7 5 - - ,
f
kp
(15)
We now discuss further limitations on the electron beam emittance due to diffraction losses of the copropagating radiation in a high gain amplifier. Diffraction losses of radiation from the electron beam may be neglected provided that the Rayleigh range, Zr, is larger than some gain length, L v of the FEL under consideration. If we assume that the radiation and electron beam are initially of equal radius and that
Zr = 7~r2/J.r> Lg ,
(16)
using eq. (4) we obtain immediately:
kr~>ka/pkw.
of eq. ( 18 ), from which we obtain the condition, for both inequalities ( 15 ), eq. ( 18 ) to hold: 7I> 104( 1 +aZw).
This condition is necessary to neglect diffraction and emittance effects in a high gain FEL allowing a 1D description. The case of an unmatched beam has been discussed in ref. [9]. In this case, from eqs. (5), (14) and (17) one obtains that the necessary condition for the ID theory to be valid is 2 kwp 27p -> 1. f ka a,,,
(20)
3. A n a l y t i c t r e a t m e n t and g a i n o p t i m i s a t i o n
In the previous section we have given a qualitative discussion of the emittance effects in gain suppression. We now give a complete analytic treatment both of the low gain and high gain regime.
3. I. Low gain regime In the low gain (g< 1 ) the electrons do not interact cooperatively via the radiation field. Hence in order to obtain the small signal gain (SSG) for a distribution of electrons in 7r we may average over the individual single particle SSG: SSG= <&(6,, ~) >,
(17)
(18)
We stress that we do not claim that the gain goes to zero decreasing k~E below the limit of eqs. (17) and (18) but we claim, according to ref. [ 2 ], that the l D theory in this case is not valid since diffraction effects become relevant. In particular, it is possible that the gain increases even if krE becomes very small but, eventually, the gain is smaller than the one predicted by the I D theory. We see that in the high gain kr~ must now lie between two limits defined by eqs. ( 15 ) and ( 18 ). The RHS ofeq. (15) must necessarily be greater than that
(21)
where
Letting Lg=Xw/41tp, the gain length in the high gain regime and using eq. ( 12 ) we obtain, from eq. ( 17 ): kre> 184f -~7-WEI-W2( 1 + a 2 ) ~/2 .
(19)
(14)
or using eq. (12) to write eq. (14) in more physical units: kr ¢ ~<0.2 I f -171/411/4 ( 1 +a2w) -1/4
1 October 1992
l
N
4 & =~--7 ( 1 - c o s 6~e- 18~g sin 6~e), 70
6~ - -
70i - - 7ri
PTro is the electron detuning parameter with respect to its resonant energy, and where 7oi and 7ri are the initial electron energies and their resonant energies, respectively. For the case of a cold beam with 7oi= 70 and an uniform hard edge distribution of 7r as.described in the previous section, then in the limit of a continuous distribution in 7r eq. (21 ) becomes: 181
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7
SSG= J d S f ( 5 ) g ( 5 ) ,
1 October 1992
3:t
(22)
J
--oo
2.5J where 5 = (Yo-Yr)/PY,-o and f ( 5 ) is the normalised rectangular distribution of width A, defined in eq. (6) and centered in 50, in which 5o= (Yo-Yro)/PYro. We stress that the distribution can be applied generally to a beam with a square distribution f ( 5 ) due either to an emittance resonant energy spread in y~ or to a real homogeneous energy spread in Yo. Letting ~-= 50 - A , / 2 be the average detuning of the distribution, then by simple integration of eq. (22) we obtain SSG= - ( g2/ A,){sinc2[ ½( 5+ A,/2 )g] - s i n c 2 [ ½(c~-A,/2)g] }.
(23)
Note that in the limit & - , 0 we obtain the Madey relation g3 d sinc2/gS-~ 2 d (g8-/2)
SSG=
(24)
In fig. 1 we plot the optimum value of the average detuning parameter clgto~ which maximises the SSG as a function of the low gain inverse normalized energy spread parameter (pNwA,)-I~ (NwAyJy~o)-J. In fig. 2 we plot the SSG for an FEL with gtot=0.5 for several values of the average detuning parameter ~gtot, as a function of (pNwd,)-~= (NwA~,J~,~o)-1. In fig. 2b the detuning is fixed and corresponds to the maximum of the Madey gain. In fig. 2a the de-
6z
or)
1.5. 1
/
f
0.5
°6{ i
-2
3 4 5 6 7 8
9
10
( N A'y/7 )4
Fig. 2. The small signal gain SSG (ztot=0.5) as a function of the low gain energyspread parameter (NwAT,/y~o)-~for (a) ~-i,o,= ~-ovtgto~and (b) ~ Z t o t =2.6056-or, riot (resonance). tuning 5 is chosen to be the optimum for a given value of Nw(Ayr/Yr0)according to fig. 1. The advantage of choosing the optimum detuning is evident. Notice that in both cases the gain begins to fall off around (NwAyr/Yro)-l~4, which, via eqs. (6), confirms the low-gain emittance limitations of eqs. (8) and (9). However of this critical value, the non-optimised gain is about one-half that of the optimally detuned case and it falls to zero more rapidly than the optimized gain. For example, for (NwATr/~,~o)-1=2 the first is already zero whereas the second is about 2.5%.
3.2. High gain regime In the high gain regime we modify the 1D model as follows:
35
dO/dg=p,
(25)
30
d p / d g = - [A exp(i0) + c . c . ] ,
(26)
25-
aA
20-
dg
15~__ 5-
0
+i~_A = ( e x p ( i 0 ) ) = f d0o dpof(Oo, Po)
× e x p [ - i 0 ( 0 o , Po, 2) ] .
10
2
4
6
8
1'0
12
( N.AT/7 ) -,
Fig. 1. Optimised detuning parameter b'~as a function of (N,~ryf/ y~o)-1. 182
/
2 0
/
(27)
Here we have used the universal scaling [4] and f(0o, Po) is the detuning distribution (1/2n)f(5) with Po= 50 and
o~=k~/ 4pkwkre
(28)
describes diffraction effects as in ref. [ 1 ]. The representation of diffraction by a simple de-
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tuning term suggested in ref. [ 1 ] is probably incomplete and it should be further investigated. However, the advantage of this representation with respect to 3D linear theory [ 10] is the simplicity and transparency of the 1D model which allows analytical understanding both in the linear and nonlinear regime. With easy calculations one finds that the linear solutions of eqs. ( 2 5 ) - ( 2 7 ) are exponentially diverging as the imaginary part of the characteristic equation: 2-
i
d6f(6)=0 (2_6)2
(29)
.
-oo
In the case of a square distribution centered on 60 and full width 20-~and changing 2-6--* - 2 in eq. (29) gives the simple cubic ( 2 - t~) ( 2 2 - 0 "2) -{- 1 = 0 ,
(30)
where 6=6o -0-~ - a
(31)
and A~
a~- 2 p -
ka(e/~) = a ( k r ~ ) 2 .
4pkw
(32)
The behaviour of the imaginary part of the roots of eq. (31 ) is proportional to the gain and is shown as a function of $ for different values of 0-~ in fig. 3. It can be shown from eq. (30) that the width of the instability region in 3-for a given 0-~is ~ 4 / v / ~ , for a, >/1. The optimum gain is for $ = 0-,, which using
7o-7m
ka ( 2 k ~ +
pTm -pkw
23-- 20"j]. 2 + 1 = 0 .
(33)
(34)
with the replacement 2 + 0-,~2. This is the standard FEL cubic relation in which the detuning parameter is substituted by the spread 20-,. The behaviour of the resonant gain as a function of the spread 0-~is shown in fig. 4a. It can be shown [ 11 ] that the severe limitation 0-,~< 1 is in good agreement with the best fit formula derived in ref. [10]. However, we stress that, according to our model, this limitation is relaxed if one uses the optimization of eq. (33). In this case, using $=0-,, the cubic eq. (34) becomes 2a+ 2o'~22 + 1 = 0 ,
(35)
where we have changed 2 - a c ~ 2 . This is the standard FEL cubic with the detuning parameter substituted by -20-o so that the imaginary part of the root has a behaviour shown in fig. 4b. The advantage
0.8
0.8
0.7-
0.7
~
0.6 0.5
•~ 0.4
0.4
--
0.3
0.3
0.2 0.1 0
0.20.1
o. 5
1
~ )"
An optimum gain criterium has been already proposed on a basis of a 3D numerical calculation in ref. [2] without the diffractive part 1/k# of eq. (33), whereas in ref. [ l ] only the refractive part has been considered choosing the optimization 60 = a. In this case one gets the strong emittance limitation a, < I. In fact, if 6o = a, from eq. (31 ) one has $ = - 0-~ so that the cubic eq. (30) reduces to
0.9
0.5-
E
the previous definitions, can be written as 60= 2a, + or. i.e.
0.9-
0.6-
1 October 1992
0 5
5
Fig. 3. Im 2 as a function of b"for (a) o~=0; (b) a,=2.0; (c) a,=5.0; (d) a,=8.0.
10 (Ay/py),'
Fig. 4. Im 2 as a function of a/-~ for (a) 6o=a; (b) 6o=a+2a,, 6obeing the normalized detuning. 183
Volume 93, number 3,4
OPTICS COMMUNICATIONS
o f the o p t i m u m detuning ( 3 3 ) is easily seen by comparing fig. 4a a n d b. F o r a,>/1 in the d e t u n e d case Im 2 ~ 1 / x / ~ , . Hence, using the detuning (32) the emittance l i m i t a t i o n o f a, ~< 1 m a y be significantly relaxed. This is no m o r e true if one uses the detuning d o = a , [2]. In fact, in this case the gain is given by fig. 3 in which, according to eq. ( 3 0 ) , ~ = - a = - ¢ / ( x e ) L Hence, for a,>_-2 the gain is zero, since the effective detuning 3-= -o~ is negative. We have p e r f o r m e d a tentative investigation for the case o f a gaussian d i s t r i b u t i o n o f width a. All previous conclusions a p p e a r to r e m a i n valid by substituting a, with a. The case in which one has both energy spread in Yo and effective energy spread in )% can be easily calculated for square distributions o f normalized widths do a n d a, centered on the average values #o and #r. In general one obtains a logarithmic dispersion relation for 2. H o w e v e r in the limit a,Ao << 1 one obtains a cubic dispersion relation as eq. ( 3 0 ) in which d = ( ~ o - # r ) / # r o a n d A=a~+do so that all previous conclusions can be easily generalized to this more general case. Notice that the limit on a, Ao << 1 does not require both a, and Ao to be small but just the product.
4. The nonlinear regime We now look at the n o n l i n e a r region o f evolution in the high gain by solving the 1D steady-state F E L equations o f evolution [4 ] with a resonant energy spread equivalent to that p r o d u c e d by the emittance, eq. (6), for a range o f values o f emittance spread and for ~ o = a for the o p t i m a l l y d e t u n e d cases ( 3 3 ) . We first verify emittance criterium eq. ( 8 ) in the nonlinear regime for the resonant case 8o = a . In fig. 5 we plot the saturated dimensionless intensity A=~t as a function o f the high gain inverse energy spread p a r a m e t e r a/-~ for ~o=C~ (fig. 5a) a n d in the o p t i m a l l y d e t u n e d case (fig. 5b) 8 o = a + 2 a , . As by eq. ( 3 2 ) e oc 6,, fig. 5 gives the saturation intensity as a function o f the emittance. The critical energy spread is a r o u n d 1 giving the high-gain e m i t t a n c e criterium eq. ( 1 4 ) i f f i o = a . The relaxation in the required emittance is clearly seen in fig. 5b where the electrons have been d e t u n e d 184
1 October 1992
2.0
~1
q
[
(b)
.o / / C.O
T --~
"0 1
I
~
I
~
i(}0
~ r
fl
i,~} 1 i
t AV/ey }
Fig. 5. The dimensionless saturation intensity as a function of the high gain energy spread parameter a; -~ for (a) 8o=a, (b) doa + 2 ¢ to the o p t i m u m for the linear gain given by eq. ( 31 ).
5. Conclusions The 1D generalized m o d e l described in this p a p e r is valid for a m a t c h e d electron b e a m / w i g g l e r under the a s s u m p t i o n that the m a j o r influence o f emittance on F E L performance arises from the emittance resonant energy spread o f the b e a m a n d diffraction can be roughly represented by a detuning [ 1 ]. This generalized linear theory indicates that both emittance and diffraction effects can be reduced both in the linear and nonlinear regime by properly detuning the electron beam.
References [ 1] See for example: J.B. Murphy and C. Pellegrini, Proc. of Joint US-CERN Particle Accelerator School, Lecture Notes in Physics, eds. M. Month and S. Turner (Springer, Berlin, 1988). [2 ] Li-Hua Yu, S. Krinsky and R.L. Gluckstern, Phys. Rev. 64 (1990) 3011. [ 3 ] See for example: E.T. Scharlemann, In high gain, high power FEL, eds. R. Bonifacio, L. De Salvo Souza and C. Pellegrini (Elsevier, Amsterdam, 1989) p. 95. [4] R. Bonifacio, C. Pellegrini and L. Narducci, Optics Comm. 50 (1984) 373. [ 5 ] E.T. Scharlemann, J. Appl. Phys. 58 (1985 ) 2154. [6]W.A. Barletta and A.M. Sessler, Radiation from Fine, Intense Self-Focused Beams at High Energy, UCRL-98767 (1988).
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OPTICS COMMUNICATIONS
[7] B.W.J. McNeil, Nucl. Inst. and Meth. A 272 (1988) 275. [8] R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo Souza, P. Pierini and N. Piovella, Rivista Del Nuovo Cimento Vol. 13, N. 9. [ 9 ] R. Bonifacio, Optics Comm. 81 ( 1991 ) 311.
1 October 1992
[10]Y.H. Chin, K.J. Kim and M. Xie, Proc. Thirteenth International Free Electron Laser Conference, Santa Fe, N.M., August 1991, LBL-30673. [ 11 ] W. Barletta, private communication.
185
OPTICS COMMUNICATIONS
Emittance limitations in the free electron laser R. B o n i f a c i o 1, L. D e Salvo S o u z a a n d B.W.J. M c N e i l INFN- Sezione di Milano, Via Celoria, 16, 20133 Milano, Italy
Received 10 December 1991; revised manuscript received 2 June 1992
We present a simple generalized 1D model to take into account emittance and diffraction effects both in the linear and nonlinear regime. We show that the usual 1D model is recovered if the emittance has an upper or a lower limit. We show analytically that diffraction and eminance effects can be sensibly reduced both in the linear and nonlinear regime by properly detuning the electron beam energy.
1. Introduction In this paper we investigate the effects of emittance and diffraction on the performance of an FEL with a m a t c h e d electron beam. An electron beam is said to be matched when its transverse area does not vary as it propagates through the wiggler. Within the limits discussed here the major effect of emittance is to change an electron's resonant energy away from that of the "standard" FEL resonance relation thus introducing a resonant energy spread even into a mono-energetic beam. The other effects of emittance within a matched beam, namely the changes in the electron/radiation coupling as the electrons perform betatron orbits, are assumed to be negligible. Hence we model the electron beam emit° tance by using a 1D model with a resonant energy spread. We derive a general emittance criterium for both the low and high gain regimes which implies in an upper limit on the acceptable emittance in terms of the cold beam radiation gain linewidth. In the low gain limit and with the wiggler length equal to a betatron period this criterium is the one derived by Kim and Pellegrini [1]. In the high gain we show a relaxation o f a previous criterium derived from linear theory by Yu and Krinsky [2]. On the other hand the condition to neglect diffraction effects implies a lower limit in the real emitAlso Dipartimento di Fisica dell'Universit~l Statale di Milano.
tance. This lower limit and the higher limit described above combine to give a necessary condition which must be satisfied given that either the lower or upper limit is valid. In general diffraction effects in a guided mode can be represented with a detuning term in a 1D model [ 1 ]. We show analytically that diffraction and emittance effects can be sensibly reduced both in the linear [2] and nonlinear regime by properly detuning the electron beam energy. Finally we give an analytical criterium for an o p t i m u m detuning of the electron beam which sensibly reduces 3D effects of angular divergence o f the electron beam both in the linear and in the nonlinear regime.
2. The equivalence of energy spread and emittance It can be shown [3 ] that in a planar wiggler the transverse component of the velocity o f an electron in the absence of a signal field is given by 7 29--2 fltot = a w2 ( 1 -I- y ~ k 2 ) ,
( 1)
where ~ot is the average over a wiggler period o f the total transverse velocity (divided by c) due to both the wiggler and the betatron motion, aw is the RMS wiggler deflection parameter, kw is the wiggler wavenumber 2~/2w, ~ is the electron energy in dimensionless units m c 2 and YBis the m a x i m u m transverse amplitude o f the electron due to the betatron motion. Note that the transverse velocity o f an electron
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
179
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OPTICS COMMUNICATIONS
is not a function of the position within its betatron orbit. Such behaviour is typical of wigglers with "natural focusing" [ 3 ]. Using the FEL resonance condition [4] fl_-= kr/(kw+kr), the definition of 7 and the approximation ( 1 - f f ~ : ) ~ 2(1 - f l : ) , it is easy to show that 2w ?)~ = ~ (1 +72flt2o,),
(2)
where ~)r defines the resonant electron energy for a given radiation wavelength 2r. Substituting eq. ( 1 ) into eq. (2) we obtain 2w
l+aw+awkwY~)
(3)
The last term in eq. (3) shows explicitly the dependence of the resonant energy of an electron on its particular betatron orbit. For a matched beam the maximum value of Y~ is the electron beam radius [ 3 ]
rb= EX/~~,
(4)
where e is the hard edge emittance of the electron beam, and k~ is the betatron wavenumber
k~=awkw/ f f .
(5)
In defining the hard edge emittance we mean that the electron beam is uniformly distributed in transverse phase space ( Y, d Y / d z ) within the matched ellipse of major axis (dY/dz)m~x=kprb and minor axis Ymax = rb. The area of the matched ellipse is then he. The factor f i s a focusing factor, which for a normal planar wiggler is equal to one. For a helical wiggler or a planar wiggler with curved pole faces [5] (in order to obtain equal focusing in both transverse directions ), f = , ~ . If additional focusing is added (e.g. plasma focusing) f may be made smaller than one [61. For a beam of given emittance equal in both transverse coordinates (ex= ey= e) the electrons will then have a resonant energy distribution between 2w
~2o-- ~ ( (1 +a~) and 2w 22~ 180
1 October 1992
where we have used eq. (3) to eq. (5). Hence we may write the normalized resonant energy spread due to emittance of: A - -
Ayr 7to
- -
-
kr~.wf 2 kr~aw.f 42 ~ 47r0 '
(6)
where we assume ATr << 7r0Within the assumption used in the model, in particular that of the matched beam condition eq. (4) and Ayr << 7m, by far the greatest effect of the emittance is that of the spread in the electrons' resonance conditions; such effects as changes in electron/radiation coupling via the changing deflection parameter a,~ and propagation angle [ 7 ] along an electron betatron orbit are negligible in the limit kwrb<< 1. This resonant energy spread will have minor effect on the FEL gain if its full width is smaller than the half width at half maximum (hwhm) of the natural FEL linewidth (e.g. in the low gain this corresponds to approximately half of the positive part of the Madey gain spectrum being populated with electrons) i.e.: A7___z_~< 1 (Aog)
Vr0
(7)
2 -7- hw.m"
Hence using eq. (6) we have kr{~ ~wwf-2 ~ -
hwhm"
(8)
Equation (8) is a general emittance condition valid in both the low and high gain regimes. In particular one must take:
_l co /hwhm -- 2Nw in the low gain and
(9)
(A~o) =2p [81 in the high gain . ~ hwhm
(10)
Here p is the fundamental FEL parameter [4] and may be written
p=O.1487-1B~/32~/3 II/3rg 2/3 ,
( 11 )
where I is the beam current and Bw is the RMS wiggler magnetic field strength. A rigorous proof of relations ( 8 ) - ( 1 0 ) will be given in the next section. Using eq. (4) we can write p for a matched electron beam as a function of the emittance:
Volume 93, number 3,4 p~
1.247-4/3Bw24/3iI/3f
OPTICS COMMUNICATIONS -- 1/3~--
1/3
.
(12)
Note that the Kim-Pellegrini criteria [ 1 ] is obtained immediately from eq. (8) by assuming that ;ta is equal to the wiggler length Lw, f = 1 and the low gain linewidth ofeq. (9). In the high gain, ref. [2] showed that the linear exponential gain is not adversely effected when (for f = x/~) k#
<<
2k~p/k#.
( 13 )
A relaxation of this limit is seen from eqs. (8) and
(10) to k 4 kwp ~E< 7 5 - - ,
f
kp
(15)
We now discuss further limitations on the electron beam emittance due to diffraction losses of the copropagating radiation in a high gain amplifier. Diffraction losses of radiation from the electron beam may be neglected provided that the Rayleigh range, Zr, is larger than some gain length, L v of the FEL under consideration. If we assume that the radiation and electron beam are initially of equal radius and that
Zr = 7~r2/J.r> Lg ,
(16)
using eq. (4) we obtain immediately:
kr~>ka/pkw.
of eq. ( 18 ), from which we obtain the condition, for both inequalities ( 15 ), eq. ( 18 ) to hold: 7I> 104( 1 +aZw).
This condition is necessary to neglect diffraction and emittance effects in a high gain FEL allowing a 1D description. The case of an unmatched beam has been discussed in ref. [9]. In this case, from eqs. (5), (14) and (17) one obtains that the necessary condition for the ID theory to be valid is 2 kwp 27p -> 1. f ka a,,,
(20)
3. A n a l y t i c t r e a t m e n t and g a i n o p t i m i s a t i o n
In the previous section we have given a qualitative discussion of the emittance effects in gain suppression. We now give a complete analytic treatment both of the low gain and high gain regime.
3. I. Low gain regime In the low gain (g< 1 ) the electrons do not interact cooperatively via the radiation field. Hence in order to obtain the small signal gain (SSG) for a distribution of electrons in 7r we may average over the individual single particle SSG: SSG= <&(6,, ~) >,
(17)
(18)
We stress that we do not claim that the gain goes to zero decreasing k~E below the limit of eqs. (17) and (18) but we claim, according to ref. [ 2 ], that the l D theory in this case is not valid since diffraction effects become relevant. In particular, it is possible that the gain increases even if krE becomes very small but, eventually, the gain is smaller than the one predicted by the I D theory. We see that in the high gain kr~ must now lie between two limits defined by eqs. ( 15 ) and ( 18 ). The RHS ofeq. (15) must necessarily be greater than that
(21)
where
Letting Lg=Xw/41tp, the gain length in the high gain regime and using eq. ( 12 ) we obtain, from eq. ( 17 ): kre> 184f -~7-WEI-W2( 1 + a 2 ) ~/2 .
(19)
(14)
or using eq. (12) to write eq. (14) in more physical units: kr ¢ ~<0.2 I f -171/411/4 ( 1 +a2w) -1/4
1 October 1992
l
N
4 & =~--7 ( 1 - c o s 6~e- 18~g sin 6~e), 70
6~ - -
70i - - 7ri
PTro is the electron detuning parameter with respect to its resonant energy, and where 7oi and 7ri are the initial electron energies and their resonant energies, respectively. For the case of a cold beam with 7oi= 70 and an uniform hard edge distribution of 7r as.described in the previous section, then in the limit of a continuous distribution in 7r eq. (21 ) becomes: 181
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7
SSG= J d S f ( 5 ) g ( 5 ) ,
1 October 1992
3:t
(22)
J
--oo
2.5J where 5 = (Yo-Yr)/PY,-o and f ( 5 ) is the normalised rectangular distribution of width A, defined in eq. (6) and centered in 50, in which 5o= (Yo-Yro)/PYro. We stress that the distribution can be applied generally to a beam with a square distribution f ( 5 ) due either to an emittance resonant energy spread in y~ or to a real homogeneous energy spread in Yo. Letting ~-= 50 - A , / 2 be the average detuning of the distribution, then by simple integration of eq. (22) we obtain SSG= - ( g2/ A,){sinc2[ ½( 5+ A,/2 )g] - s i n c 2 [ ½(c~-A,/2)g] }.
(23)
Note that in the limit & - , 0 we obtain the Madey relation g3 d sinc2/gS-~ 2 d (g8-/2)
SSG=
(24)
In fig. 1 we plot the optimum value of the average detuning parameter clgto~ which maximises the SSG as a function of the low gain inverse normalized energy spread parameter (pNwA,)-I~ (NwAyJy~o)-J. In fig. 2 we plot the SSG for an FEL with gtot=0.5 for several values of the average detuning parameter ~gtot, as a function of (pNwd,)-~= (NwA~,J~,~o)-1. In fig. 2b the detuning is fixed and corresponds to the maximum of the Madey gain. In fig. 2a the de-
6z
or)
1.5. 1
/
f
0.5
°6{ i
-2
3 4 5 6 7 8
9
10
( N A'y/7 )4
Fig. 2. The small signal gain SSG (ztot=0.5) as a function of the low gain energyspread parameter (NwAT,/y~o)-~for (a) ~-i,o,= ~-ovtgto~and (b) ~ Z t o t =2.6056-or, riot (resonance). tuning 5 is chosen to be the optimum for a given value of Nw(Ayr/Yr0)according to fig. 1. The advantage of choosing the optimum detuning is evident. Notice that in both cases the gain begins to fall off around (NwAyr/Yro)-l~4, which, via eqs. (6), confirms the low-gain emittance limitations of eqs. (8) and (9). However of this critical value, the non-optimised gain is about one-half that of the optimally detuned case and it falls to zero more rapidly than the optimized gain. For example, for (NwATr/~,~o)-1=2 the first is already zero whereas the second is about 2.5%.
3.2. High gain regime In the high gain regime we modify the 1D model as follows:
35
dO/dg=p,
(25)
30
d p / d g = - [A exp(i0) + c . c . ] ,
(26)
25-
aA
20-
dg
15~__ 5-
0
+i~_A = ( e x p ( i 0 ) ) = f d0o dpof(Oo, Po)
× e x p [ - i 0 ( 0 o , Po, 2) ] .
10
2
4
6
8
1'0
12
( N.AT/7 ) -,
Fig. 1. Optimised detuning parameter b'~as a function of (N,~ryf/ y~o)-1. 182
/
2 0
/
(27)
Here we have used the universal scaling [4] and f(0o, Po) is the detuning distribution (1/2n)f(5) with Po= 50 and
o~=k~/ 4pkwkre
(28)
describes diffraction effects as in ref. [ 1 ]. The representation of diffraction by a simple de-
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tuning term suggested in ref. [ 1 ] is probably incomplete and it should be further investigated. However, the advantage of this representation with respect to 3D linear theory [ 10] is the simplicity and transparency of the 1D model which allows analytical understanding both in the linear and nonlinear regime. With easy calculations one finds that the linear solutions of eqs. ( 2 5 ) - ( 2 7 ) are exponentially diverging as the imaginary part of the characteristic equation: 2-
i
d6f(6)=0 (2_6)2
(29)
.
-oo
In the case of a square distribution centered on 60 and full width 20-~and changing 2-6--* - 2 in eq. (29) gives the simple cubic ( 2 - t~) ( 2 2 - 0 "2) -{- 1 = 0 ,
(30)
where 6=6o -0-~ - a
(31)
and A~
a~- 2 p -
ka(e/~) = a ( k r ~ ) 2 .
4pkw
(32)
The behaviour of the imaginary part of the roots of eq. (31 ) is proportional to the gain and is shown as a function of $ for different values of 0-~ in fig. 3. It can be shown from eq. (30) that the width of the instability region in 3-for a given 0-~is ~ 4 / v / ~ , for a, >/1. The optimum gain is for $ = 0-,, which using
7o-7m
ka ( 2 k ~ +
pTm -pkw
23-- 20"j]. 2 + 1 = 0 .
(33)
(34)
with the replacement 2 + 0-,~2. This is the standard FEL cubic relation in which the detuning parameter is substituted by the spread 20-,. The behaviour of the resonant gain as a function of the spread 0-~is shown in fig. 4a. It can be shown [ 11 ] that the severe limitation 0-,~< 1 is in good agreement with the best fit formula derived in ref. [10]. However, we stress that, according to our model, this limitation is relaxed if one uses the optimization of eq. (33). In this case, using $=0-,, the cubic eq. (34) becomes 2a+ 2o'~22 + 1 = 0 ,
(35)
where we have changed 2 - a c ~ 2 . This is the standard FEL cubic with the detuning parameter substituted by -20-o so that the imaginary part of the root has a behaviour shown in fig. 4b. The advantage
0.8
0.8
0.7-
0.7
~
0.6 0.5
•~ 0.4
0.4
--
0.3
0.3
0.2 0.1 0
0.20.1
o. 5
1
~ )"
An optimum gain criterium has been already proposed on a basis of a 3D numerical calculation in ref. [2] without the diffractive part 1/k# of eq. (33), whereas in ref. [ l ] only the refractive part has been considered choosing the optimization 60 = a. In this case one gets the strong emittance limitation a, < I. In fact, if 6o = a, from eq. (31 ) one has $ = - 0-~ so that the cubic eq. (30) reduces to
0.9
0.5-
E
the previous definitions, can be written as 60= 2a, + or. i.e.
0.9-
0.6-
1 October 1992
0 5
5
Fig. 3. Im 2 as a function of b"for (a) o~=0; (b) a,=2.0; (c) a,=5.0; (d) a,=8.0.
10 (Ay/py),'
Fig. 4. Im 2 as a function of a/-~ for (a) 6o=a; (b) 6o=a+2a,, 6obeing the normalized detuning. 183
Volume 93, number 3,4
OPTICS COMMUNICATIONS
o f the o p t i m u m detuning ( 3 3 ) is easily seen by comparing fig. 4a a n d b. F o r a,>/1 in the d e t u n e d case Im 2 ~ 1 / x / ~ , . Hence, using the detuning (32) the emittance l i m i t a t i o n o f a, ~< 1 m a y be significantly relaxed. This is no m o r e true if one uses the detuning d o = a , [2]. In fact, in this case the gain is given by fig. 3 in which, according to eq. ( 3 0 ) , ~ = - a = - ¢ / ( x e ) L Hence, for a,>_-2 the gain is zero, since the effective detuning 3-= -o~ is negative. We have p e r f o r m e d a tentative investigation for the case o f a gaussian d i s t r i b u t i o n o f width a. All previous conclusions a p p e a r to r e m a i n valid by substituting a, with a. The case in which one has both energy spread in Yo and effective energy spread in )% can be easily calculated for square distributions o f normalized widths do a n d a, centered on the average values #o and #r. In general one obtains a logarithmic dispersion relation for 2. H o w e v e r in the limit a,Ao << 1 one obtains a cubic dispersion relation as eq. ( 3 0 ) in which d = ( ~ o - # r ) / # r o a n d A=a~+do so that all previous conclusions can be easily generalized to this more general case. Notice that the limit on a, Ao << 1 does not require both a, and Ao to be small but just the product.
4. The nonlinear regime We now look at the n o n l i n e a r region o f evolution in the high gain by solving the 1D steady-state F E L equations o f evolution [4 ] with a resonant energy spread equivalent to that p r o d u c e d by the emittance, eq. (6), for a range o f values o f emittance spread and for ~ o = a for the o p t i m a l l y d e t u n e d cases ( 3 3 ) . We first verify emittance criterium eq. ( 8 ) in the nonlinear regime for the resonant case 8o = a . In fig. 5 we plot the saturated dimensionless intensity A=~t as a function o f the high gain inverse energy spread p a r a m e t e r a/-~ for ~o=C~ (fig. 5a) a n d in the o p t i m a l l y d e t u n e d case (fig. 5b) 8 o = a + 2 a , . As by eq. ( 3 2 ) e oc 6,, fig. 5 gives the saturation intensity as a function o f the emittance. The critical energy spread is a r o u n d 1 giving the high-gain e m i t t a n c e criterium eq. ( 1 4 ) i f f i o = a . The relaxation in the required emittance is clearly seen in fig. 5b where the electrons have been d e t u n e d 184
1 October 1992
2.0
~1
q
[
(b)
.o / / C.O
T --~
"0 1
I
~
I
~
i(}0
~ r
fl
i,~} 1 i
t AV/ey }
Fig. 5. The dimensionless saturation intensity as a function of the high gain energy spread parameter a; -~ for (a) 8o=a, (b) doa + 2 ¢ to the o p t i m u m for the linear gain given by eq. ( 31 ).
5. Conclusions The 1D generalized m o d e l described in this p a p e r is valid for a m a t c h e d electron b e a m / w i g g l e r under the a s s u m p t i o n that the m a j o r influence o f emittance on F E L performance arises from the emittance resonant energy spread o f the b e a m a n d diffraction can be roughly represented by a detuning [ 1 ]. This generalized linear theory indicates that both emittance and diffraction effects can be reduced both in the linear and nonlinear regime by properly detuning the electron beam.
References [ 1] See for example: J.B. Murphy and C. Pellegrini, Proc. of Joint US-CERN Particle Accelerator School, Lecture Notes in Physics, eds. M. Month and S. Turner (Springer, Berlin, 1988). [2 ] Li-Hua Yu, S. Krinsky and R.L. Gluckstern, Phys. Rev. 64 (1990) 3011. [ 3 ] See for example: E.T. Scharlemann, In high gain, high power FEL, eds. R. Bonifacio, L. De Salvo Souza and C. Pellegrini (Elsevier, Amsterdam, 1989) p. 95. [4] R. Bonifacio, C. Pellegrini and L. Narducci, Optics Comm. 50 (1984) 373. [ 5 ] E.T. Scharlemann, J. Appl. Phys. 58 (1985 ) 2154. [6]W.A. Barletta and A.M. Sessler, Radiation from Fine, Intense Self-Focused Beams at High Energy, UCRL-98767 (1988).
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OPTICS COMMUNICATIONS
[7] B.W.J. McNeil, Nucl. Inst. and Meth. A 272 (1988) 275. [8] R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo Souza, P. Pierini and N. Piovella, Rivista Del Nuovo Cimento Vol. 13, N. 9. [ 9 ] R. Bonifacio, Optics Comm. 81 ( 1991 ) 311.
1 October 1992
[10]Y.H. Chin, K.J. Kim and M. Xie, Proc. Thirteenth International Free Electron Laser Conference, Santa Fe, N.M., August 1991, LBL-30673. [ 11 ] W. Barletta, private communication.
185