Simulation Studies of the X-Ray Free-Electron Laser Oscillator Page: 1 of 4
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SIMULATION STUDIES OF THE X-RAY FREE-ELECTRON LASER
OSCILLATOR*
R.R. Lindberg t, Y. Shyd'ko, K.-J. Kim, ANL Advanced Photon Source, Argonne, IL 60439, USA
W. M. Fawley, LBNL Center for Beam Physics, Berkeley, CA 94720, USAAbstract
Simulations of the x-ray free-electron laser (FEL) oscil-
lator are presented that include transverse effects and realis-
tic Bragg crystal properties with the two-dimensional code
GINGER. In the present cases considered the radiation di-
vergence is much narrower than the crystal acceptance, and
the numerical algorithm can be simplified by ignoring the
finite angular bandwidth of the crystal. In this regime GIN-
GER shows that the saturated x-ray pulses have ~109 pho-
tons and are nearly Fourier-limited with peak powers in ex-
cess of 1 MW. We also include preliminary results for a
four-mirror cavity that can be tuned in wavelength over a
few percent, with future plans to incorporate the full trans-
verse response of the Bragg crystals into GINGER to more
accurately model this tunable source.
INTRODUCTION
First proposed by Colella and Luccio 25 years ago [1],
there has been renewed interest in an x-ray free-electron
laser (FEL) oscillator with the recent set of concrete, re-
alizable parameters put forth by Kim, Shvyd'ko and Re-
iche [2]. As shown in Ref. [2], the x-ray FEL oscilla-
tor can produce fully coherent x-ray pulses with MWs of
power by combining an ultra-low emittance, low charge
electron beam and a resonator cavity formed using high-
reflectivity, narrow-bandwidth Bragg crystals. This pa-
per presents some recent simulation results including the
full frequency-dependent reflectivity of the Bragg crystals
and the transverse effects of beam divergence and radia-
tion diffraction using the two-dimensional axisymmetric
code GINGER [3]. First, we discuss the two-crystal cav-
ity near-backscatter, including a summary of the relevant
new physics and simulation results from a number of pos-
sible designs that show spectrally pure pulses of ~' 109 pho-
tons and third harmonic generation with ~105 photons at
36 keV. Next, we present results relevant to a tunable, four-
mirror cavity for which similar pulse characteristics are ob-
served. Finally, we discuss future extensions and conclude.
TWO-CRYSTAL CAVITY
The two-crystal x-ray FEL oscillator cavity is a simple
extension of the stable two-mirror resonator familiar from
laser optics (see, e.g., [4]). In this geometry, x-rays are con-
tained by two Bragg crystals operating in near-backscatter
* Work supported by U.S. Dept. of Energy, Office of Science, Office of
Basic Energy Sciences, under Contract No. DE-AC02-06CH11357
t lindberg@aps.anl.govgeometry, while focusing is provided by grazing incidence
mirrors as shown in Fig. 1(a). In the next subsection we
briefly highlight the most relevant issues introduced by in-
clusion of transverse physics (i.e., beam divergence, radi-
ation diffraction, and x-ray focusing) and the complex re-
flectivity of the Bragg mirrors. We then present a set of
simulation results for the two-mirror cavity geometry over
a range of wavelengths using a variety of Bragg crystal,
electron beam, and undulator parameters. We find that third
harmonic emission may provide an interesting source of ra-
diation beyond 20 keV energy.
Simulations Including Transverse Physics and
Bragg Crystal Properties
Typical electron beam parameters for the x-ray FEL os-
cillator include a bunch length ac ~ 1 ps, peak current
Ipeak ~ 10 A, normalized emittance E , ~ 10-' mm
mrad, and normalized energy spread a- ~ 0.02%; specific
examples are listed in Table 1. These parameters are chosen
so that a low-charge electron bunch of energy Ebeam 7
GeV will give rise to a single pass FEL gain ginear > 0.3
over Nw ~ 3000 periods of undulator. In the low-gain
regime, the FEL gain is typically maximized when the
transverse spreading of the radiation matches that of the
electron beam, zR ~ zp, where zR is the x-ray Rayleigh
range while the (vacuum) beam focusing parameter zQ is
related to the emittance Ec and transverse size u- at posi-
tion z viaa-(z) Z x [i+ (z - zo)2/z .
(1)
The gain is maximized when the electron beam waist is lo-
cated at the middle of undulator length Lw, zo =L/2
NwX1/2, and when the focusing parameter zQ ~ Ls/2r
[5]. With zQ ~ zR, maximizing the gain for a fixed cav-
ity length in turn sets the grazing incidence mirror focal
length. Presently, GINGER approximates mirror focusing
by an ideal thin lens.
The reflective properties of near-perfect Bragg crystal re-
flectors are described by the theory of dynamical diffrac-
tion (see, e.g., [6]). The basic results of this theory show
that Bragg crystals coherently reflect radiation in a nar-
row spectral band near that defined by Bragg's Law: E
EH / cos 0, where E is the photon energy, 0 is the incidence
angle from normal, and EH is the Bragg energy, which is
related to the crystalline planar spacing d through the speed
of light c and Planck's constant h by EH _ hc/2d. The
weak angular dependence for 0 1 in near-backscatter
implies that the finite angular acceptance of the crystal can
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Lindberg, R. R.; Shyd'ko, Y.; Kim, K. J. & Fawley, W. M. Simulation Studies of the X-Ray Free-Electron Laser Oscillator, article, August 14, 2009; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc926862/m1/1/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.