Spontaneous and amplified radiation at the initial stage of a SASE FEL. Page: 3 of 4
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3
1
0 0.8 '
0.6
0.
0 0o. 2 0 .4 0 .6 0 .8 i1
Wu/Wj
Figure 1. Fundamental undulator radiation spec-
trum (a-polarization, K < 1) as calculated from
Eq. (8) (solid line) and Eq. (9) (dashed line).
effects, the power spectrum of the fundamental
mode before saturation is
dPund pymc [ z (Aw) 1
d - =AOS 2v exp LG 2w1 a
(11)
where 9A is the input coupling coefficient, gs is
the start-up noise coefficient [7], p is the FEL scal-
ing parameter [9], LG is the power gain length,
and wia,(z) is the FEL bandwidth and is a func-
tion of z. Integration over the SASE spectrum
yieldsmc2
Pfund = OAOS wi-yez/LG
m(12)
In Ref. [8,7], the effective start-up power for the
fundamental mode is identified as the coherent
fraction of the spontaneous radiation in the first
two power gain lengths, which is a small fraction
of the total spontaneous radiation emitted. As-
suming that higher-order transverse modes have
negligible growth rates, the exponentially growing
fundamental mode will dominate over the sponta-
neous radiation emitted over all solid angles when
z > zt, where zt is determined by
zt I K2fy2 pr z1/LG
IA,,/Av = AgAgs v(z)ez .
(13)3. Comparison with simulation studies
Two SASE FEL codes GINGER [10] and GEN-
ESIS [11] are used for this study. Since GIN-
GER solves for the radiation field that is azy-
muthally symmetric in transverse coordinates, it
excludes most higher-order modes of the sponta-
neous emission. As a result, GINGER models the
central cone of the undulator radiation that pos-
sesses the azymuthal symmetry [6,4] and is found
to agree with the expected central-cone radiation
power at the start-up regime [13]. The three-
dimensional code GENESIS is used to examine
the SASE power emitted in all solid angles. For
a limited simulation bandwidth, the GENESIS
spectrum at start-up cannot reproduce Eq. (8)
due to the effect of aliasing [12]. Nevertheless,
we expect the initial GENESIS power within a
small bandwidth is comparable to that given by
Eq. (10). The output power of GINGER and
GENESIS should be comparable when the fun-
damental mode is dominant (i.e., when z > zt).
We compare zt determined from these codes with
that predicted from Eq. (13).
Figure 2 shows such a comparison using the low
energy undulator test line (LEUTL) FEL at the
Advanced Photon Source operated at 530 nm [1].
The electron beam energy is 217 MeV with a
0.1 % energy spread. The normalized emittance
is 9.3 pm in x and 7.6 pm in y. The bunch
profile is assumed to be Gaussian with the rms
bunch length 300 fs and the peak current 266 A.
The spontaneous undulator radiation, calculated
from Eq. (10) by taking Av, , 0.12 to be one
half of the simulated bandwidth (the other half
is above the fundamental wavelength), is found
to be somewhat higher than the initial power of
the GENESIS simulation. The results of GIN-
GER and GENESIS converge after 7 m (including
the drift spaces between undulators) as the fun-
damental mode dominates. Eq. (13) yields zt = 5
m using the above beam parameters.
Figure 3 shows another comparison using the
Linac Coherent Light Source proposed at the
Stanford Linear Accelerator Center for an x-ray
FEL at 1.5 A [14]. The electron beam energy is
14.3 GeV with a 6 x 10-5 energy spread. The
normalized emittance is 1.5 pm in both trans-
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Huang, Z. & Kim, K. J. Spontaneous and amplified radiation at the initial stage of a SASE FEL., article, November 1, 2002; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc739963/m1/3/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.