The effect of inhomogeneous broadening is also dis- cussed. I. INTRODUCTION. HE main troubles of the early generation of free-elec-. T tron laser (FEL) ...
IEEE JOURNQL OF QUANTUM ELECTRONICS, VOL 25. NO
2321
I I , NOVEMBER 1989
Free-Electron Laser Operation in the Intermediate Gain Region G. DATTOLI, A. TORRE, C. CENTIOLI,
Abstract-In this paper, we explore the modification induced on the well-known antisymmetric gain function of FEL when the gain coefficient go ranges up to 10. We point out that deviations from the linear regime are present for go 2 0.5, and also discuss a simple perturbative analysis, which accounts remarkably well for the corrections to the linear gain formula even for go = 10. The effect of inhomogeneous broadening is also discussed.
I. INTRODUCTION HE main troubles of the early generation of free-electron laser (FEL) experiments were related to the low gain. Low gain is a byproduct of low e-beam current, bad bezim qualities (namely, large energy spread and emittances), or a combination of both. Therefore, accelerators dedicated to FEL operation are carefully designed to provide an e beam with large current and good qualities. Gain values exceeding 100 percent can be therefore obtained with a FEL driven by RF Linacs [ 11, Van de Graaff [2], etc. Although the FEL gain has been carefully analyzed in the low-gain and high-gain regimes, the peculiar features of FEL operating in the intermediate regime (0.5 < go < 10) have not been exhaustively explored. The reasons for interest in this region is manyfold. Gain values exceeding 1 can be easily reached in actual experiments. For this range of values, the laser dynamics is significantly different from that predicted by the linear theory. The operating regime is not interferential, but not yet purely exponential. The aim of this paper is to bring insight into this gain region. We will show that the numerical results of the self-consistent theory [3]-[5] can be reproduced remarkably well by a perturbative analysis of the exact equations up to the third order in go, thus fixing the amount of the deviation from the linear theory with increasing gain coefficient. We also discuss the effect of inhomogeneous broadening on the gain and show that for large energy spread and emittances, the validity of the linear theory extends over a broader range of go. In this paper, we will limit the analysis to the case of a continuous e beam; we will not therefore include slippage and lethargy mechanisms which also may play a significant role and should be carefully analyzed (see, e.g., [ 6 ] ) .
T
Manuscript received February 28, 1989. The authors are with the Dipartimento TIB, U . S . , Fisica Applicata, ENEA, C.R.E. Frascati, C.P. 65-00044 Frascati, Rome, Italy. IEEE Log Number 89306 13.
AND
M . RICHETTA
as follows. In Section 11, we The -paper is organized review the single-mode self-consistent evolution equation of the FEL optical field and discuss its analytical and numerical solutions. In Section 111, we present a perturbative approach to the above-mentioned equation and analyze its range of validity. Section IV is finally devoted to the analysis of inhomogeneous contributions and to conclusive remarks. 11. THE SELF-CONSISTENT FEL EVOLUTION The main assumption of the linear theory of the FEL is that the laser field can be considered constant in one undulator passage. This is strictly true if the involved gain is not large enough. Otherwise, the field seen by the front part of the e beam is different from that experienced by the rear part. The equation, accounting for this self-consistent evolution, has been derived by many authors, and in the small-signal slowly-varying amplitude approximation reads [3]-[5]
(2.1) where z is the longitudinal coordinate, t is the current time, and w, is the frequency associated with the undulator. Assuming for the field amplitude a ( z , t ) the following form
u(z,t )
= a.
exp { h ( t - z / c ) } E ( t
+ z/c)
(2.2)
and introducing the dimensionless variable 1
7- = - (
At
t
+ z/c)
where At is the interaction time, (2.1) reduces to
d dr
E ( . - r ' ) e i 2 ' ' ' ~ ' d r r (2.4)
U = ( w - w,)At being the resonance parameter. Taking repeated derivatives of (2.4) with respect to r , we end up with the following third-order differential equation for E:
d3 d7
- E - 2iv
d2 d 7E - u2 - E dr dr
0018-9197/89/1100-2327$01.00 @ 1989 IEEE
+ hgoE = 0
(2.5)
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IEEE JOURNAL OF QUANTUM ELECTRONICS. VOL. 25, NO. I I . NOVEMBER 1989
which can be solved using the following trial form: E ( 7 ) = EOei(2'fst')'7
0.010
0 10
0.002
0.004
0.0 2 0.0 4
0.006
0.0 6
0 008
0.08
(2.6)
The unknown quantity 6 v can be determined by inserting (2.6) in (2.5), thus getting the cubic equation
6 v 2 ( v + 6 v ) = ng,.
(2.7)
As a consequence, (2.6) specializes as
0.010 -10 -8 -6 -4 -2
3
0
2
6
4
0.1 0 -10 - 8 -6 - 4 - 2
8 10
(ai
0
2
4
6
8 10
(bi
Fig. 1 . Gain spectrum for go
=
0.001 (a) and go = 0.1 (b).
The coefficients with 6vj,j = 1, 2, 3 are the solutions to (2.7). Ej are determined by the condition of an initially free incoming wave ( E o = 1 ) and by the requirement that (2.8) is a solution of (2.4), thus giving 3
C EJ j= 1
3
= 1,
3
C j= I
6vjEJ =
-U,
C j= I
6v;E,
= U'.
(2.9)
The relative energy variation after one interaction time is given by -10 -8 - 6
AI
- = 1E(1)12 - 1
I
= 0.84983
*
go
+ 0.19158
+ 0.42306 x
*
gi
4
0.8 I I I I I I I I I I ( -10 - E -6 - 4 -2 0 2 4 6 E IO
8 IO
6
Y
IOp2 . gi.
2.6 - 0.13972
+ 0.58141
X
*
(2.11)
go *
gi.
(bi
Fig. 2 . Gain spectrum for g,, = 0.5 (a) and go = 1.0 (b)
."
All1 I
All1 A ;m 8 6 4 2 0
-8
-4
-2 0
2
-8
8
4
-4 -2
0
2
8
4
V
V
(a) (b) Fig. 3 . Gain spectrum for go = 5 (a) and go = 10 (b).
L4 10
The peak value of the gain is fixed at a value of v also depending on the gain coefficient go and in the range ( 0 < go < 10) can be reproduced (within an approximation of 3 x l o p 3 )by =
2
(a)
and this defines the FEL gain. The results relevant to the numerical handling of the above relations are reported in Figs. 1-3. Fig. l(a) and (b) are relevant to A I / I of a FEL operating with go = lo-', respectively. The gain curve exhibits the wellknown antisymmetric shape and reproduces the same results of the linear theory; significant deviations can be inferred from the plots reported in Fig. 2 relevant to go = 0.5 and 1. More pronounced deviations from the linear theory are shown in Fig. 5 ( go = 5 , 10) exhibiting a gain curve in which the antisymmetric structure is completely lost and the maximum value has been significantly shifted towards smaller values of the detuning parameter. The deviations from the linear theory are better exemplified by Fig. 4 where the peak values of A I / I are reported versus go. This numerical scaling is well reproduced (within 3.5 X by the following simple relation :
go)
0
4 -2
(2.10)
(2.12)
0
2
4
6
8
1
0
Qo
Fig. 4. Maximum energy variation as a function of go.
As a conclusion to the present discussion, we stress that it is possible to set the limit of validity of the linear theory around go = 0.5, whereas larger values demand the inclusion of higher order corrections.
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DATTOLI et a / . : FEL OPERATION IN THE INTERMEDIATE GAIN REGION
111. PERTURBATIVE ANALYSIS OF THE SELF-CONSISTENT EQUATIONS Inspecting the relation (2.1 l ) , we may draw the following conclusions. 1 ) In the range 0 < go I 10, the “high” gain effect is well reproduced by a cubic scaling law versus go. 2) The coefficients of increasing powers of go have decreasing values. The above points suggest that we can treat (2.1) perturbatively, using go as a perturbative parameter even for values exceding the unity. We express the field as E(T) =
c En(7)g;;
-20
1:
20 V
(a) 0.30 I
(3.1)
which, once inserted in (2.4), yields the following recursive relation: d - E,, = - i r d7
IO
0
-10
0.10
0.05 0
E,-~(T-
T’)f?”“’T’ d T 1
(3.2)
-005 -20
-10
:I
IO
0
V
E,(O) = 6n,0, E P I = 0.
(3.3)
To get the third-order contributions to the gain, we are left with the evaluation of E 1 , 2 , 3Skipping . the trivial but tedious algebraic details, we end up with the following gain function:
(b) 0.0025 93 0.0020 0.0015
0.0010
0.0005
-0 0005
AI I
- = gogi(U> + g?ig2(~)+ g i h ( U ) .
(3.4)
-0.0010 -0.0015 -0.0020 -0.0025 -20
U ) functions are given by The g1.2,3( gl(v) =
2a 7 [ 2 ( 1 - cos U ) - U sin U ] H2
g2(v) = - [84( 1 - cos U ) - 602) sin v 3v6
+
1.5~’ cos v
0
-IO
10
20
(C) Fig. 5 . The function g , (a), g2 (b), and g3 (e) plotted against
21.
+ 3v2
+ v 3 sin U ]
20 15
lr3
g3(U) = 7 [11520(1 - cos U ) - 9 0 0 0 ~sin 602)
U
+ 3 6 0 ~+ ~2 8 8 0 ~cos~ U + 4 8 0 ~sin~U - 2oU4(1 + 2 cos U ) u 5 sin U ] . (3.5)
-10 - 8 - 6 - 4 - 2
0 2
-
Needless to say, g l ( u ) is the well-known FEL gain function and g2,3(U ) are the perturbative corrections. The above functions are plotted separately in Fig. 5 ; their relative magnitude decreases with increasing subscript order, thus justifying the perturbative procedure, even for large go. The perturbative solution (3.4) is compared to the exact solution in Fig. 6 for go = 1, 5 , and 10.
8
6
-10 - 8 - 6 - 4 - 2
V
(a)
0 2 4
6 V
(b)
Fig. 6 . Comparison between the exact (solid line) and approximate (dotted line) curves for go = 5 (a) and go = 10 (b).
The corrections due to the cubic term become significant for go 1 5 , whereas for smaller go, the second-order corrections ensure a remarkably good agreement with the exact curve. ’More quantitatively, by inspecting Fig. 5 , we can infer that the followmust be satisfied to avoid cubic corrections. This limit does not have a clear physical meaning and within the present framework is merely an upper limit for not including third-order corrections.
ing inequality go
