Optimization of the parameters of a storage ring for a high power XUV free electron laser Page: 4 of 14
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2. FEL Issues
FEL Gain, Power, Saturation Length
An electron beam of energy ym c2 traveling through a magnetic undulator of period a
can interchange energy with a t.tansverse laser field of wavelength X propagating in the
same direction (ReEs. 6 and 7). The interaction becomes resonant when the following
condition is satisfied:
(1 + K2/2)
272 = u (2.1)
where K is the deflection parameter, characteristic of the undulator, given by
K - hu.. = 0.934 1. B. (2.2)
2ir ne c [cm] [Tesla]
An important quantity, which determines the FEL .haracteristics in the one-dimensional
theory, is the dimensionless parameter p, given by (Ref. 8)
S(K2(JJjrenb ) 1/3 (2.3)
[JJ] . [JO(() -J1(E)]2 with ( - K2 . (2.4)
4(1 + K2/2)
The . and 1 are ordinary vessel functions cf order zero and one, respectively. As
will emerge in the course of this paper, for the parameters of interest to us, p is
typically of the order of 10-3.
The characteristics of the powei growth of the laser wave, as obtained from the
one-dimensional theory, fall into three distinct regimes: the small-signal regime, the
exponential-growth regime and the saturation regime. Near the entrance of the undulator,
where the small-signal theory applies, the gain G (defined as the ratio of the laser power
at two points separated by a distance z along the undulator axis) is given by
G - 536 (pZ/Xu)3 (2.5)
Farther along the undulator. the laser power P grows exponentially (Refs. 9 to 14) with
distance, from the initial power PO at a rate proportional to:
P- Pe (2.6)
g - 4w V(p/u) (2.7)
The corresponding e-folding length for the growth of the radiation power in the
exponential growth regime is
t, . g-1 . l ku 2
4x p 2. S)
Eventually, the electrons are captured in the ponderomotive potential well and the
;rowcn of radiation power stops. The laser saturates at a distance z = z with a
:-araccteristic saturated peak power Psat. The saturation length and the 5ik power are
.ven approximately by (Ref. 8)
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Jackson, A.; Bisognano, J.; Chattopadhyay, S.; Cornacchia, M.; Garren, A.; Halbach, K. et al. Optimization of the parameters of a storage ring for a high power XUV free electron laser, article, October 1, 1985; United States. (https://digital.library.unt.edu/ark:/67531/metadc1212985/m1/4/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.