# Systematical design and three-dimensional simulation of X-ray FEL oscillator
for Shanghai Coherent Light Facility

Kai Li  Haixiao Deng<EMAIL_ADDRESS>Shanghai Institute of Applied
Physics, Chinese Academy of Sciences, Shanghai 201800, China  University of
Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Shanghai Coherent Light Facility (SCLF) is a quasi-CW hard X-ray free electron
laser user facility which is recently proposed. Due to the high repetition
rate, high quality electron beams, it is straightforward to consider an X-ray
free electron laser oscillator (XFELO) operation for SCLF. The main processes
for XFELO design, and parameters optimization of the undulator, X-ray cavity
and electron beam are described. The first three-dimensional X-ray crystal
Bragg diffraction code, named BRIGHT is built, which collaborates closely with
GENESIS and OPC for numerical simulations of XFELO. The XFELO performances of
SCLF is investigated and optimized by theoretical analysis and numerical
simulation.

###### keywords:

XFELO , SCLF , three-dimensional Bragg diffraction

###### PACS:

41.60.Cr

††journal: Nuclear Instruments and Methods in Physics Research A

##  1 Introduction

With the wide applications of X-ray in physics, chemistry and biology
researches, facilities which is able to generate X-ray are under development
in the last century continuously. Free electron laser (FEL) is a novel light
source which produces high-brightness X-ray pulse [1, 2]. For pursuing high
intensity and ultra-fast short wavelength radiation, some X-ray FEL facilities
have been completed or under construction around the world[3, 4, 5]. Almost
all of them take the advantages of self-amplified spontaneous emission (SASE)
scheme [6], which starts from the electron beam shot noise and generates
poorly temporal coherence light pulses. In order to obtain fully coherent
X-ray pulses, Linac Coherent Light Source employs self-seeding method [7, 8],
which is capable of improving the longitudinal coherence by a factor of 50.
Nevertheless, the large pulse-to-pulse energy jitters (around 50% r.m.s
fluctuation) prevent it from further improvements.

An alternative promising approach to obtain fully coherent stable FEL pulses
is X-ray free electron laser oscillator (XFELO) [9, 10]. Using the
relativistic electron beam as gain medium to amplify the radiation trapped in
an optical resonator is proposed conceptually and demonstrated experimentally
decades ago in long wavelength region, such as for infrared and ultraviolet
light [11, 12, 13], but no XFELO experiment has been conducted due to the
absence of X-ray high reflectivity mirrors. However, the X-ray high-
reflectivity sapphire crystal experiment which was first demonstrated in 2010
[14], paved the way for real construction of XFELO. Taking the advantage of
the crystal Bragg diffraction (BD), the XFELO scheme has been reconsidered and
some practical technical problems has been studied [15, 16].

Motivated by the successful operation of worldwide X-ray FEL facilities and
the great breakthroughs in observation and control of very fast phenomena at
the atomic time scale [17], the first hard X-ray FEL light source in China
named Shanghai Coherent Light Facility (SCLF) is under designing. SCLF plans
to utilize photocathode electron gun combines with the superconducting Linac
to produce 1 MHz repetition rate, 8 GeV relativistic electron beams. SCLF is
capable of accommodating six undulator lines, and three of them will be
constructed in the first phase project. After going through a beam switch
yard, electron bunches are separated to three undulator lines which cover the
photon energy of 0.4-25 keV. The undulator line that typically generates 12.4
keV X-ray, holds great potential for XFELO operation, as the layout shown in
Fig. 1. The cavity length is nearly 150 m, which ensures the X-ray pulse to
meet an electron beam after being reflected back by the upstream crystal
mirror at each round trip. The X-ray radiation overlaps with electron beam and
extracts energy from it in the undulator. The upstream crystal is thicker than
the downstream one, so that by transmission of the downstream mirror, part of
X-ray pulse energy is coupled to the subsequent X-ray beam line. And the total
reflectivity is assumed to be $R$. In order to maintain the growth of
radiation power, the number of undulator periods should be large enough for
single pass gain $G$ overcomes the round trip loss. According to [18], the FEL
gain is a constant at small signal regime, and decreases gradually due to the
over modulation of electron beam when XFELO approaches saturation. The light
pulse starts up from initial spontaneous radiation which acts as a seed and
grows exponentially until reaches equilibrium, when the round trip energy net
loss equals to the single pass gain. The light pulse power $P_{n}$ at the
entrance of undulator for the $n$-th round trip is

$P_{n}=P_{n-1}\times(1+G)\times R$ (1)

The main parameters of SCLF is presented in Table. 1. The bunch charge is
tunable between 10 pC to 300 pC and the peak current is able to achieve
several kA. Taking the advantages of superconducting accelerator technique,
the quasi-CW electron bunches are suitable for XFELO operation. This facility,
however, is basically designed for SASE and self-seeding, and the undulator
periods has been optimized to be 26 mm with 5 m for each segment. This paper
will discuss the XFELO operation on the condition of given electron beam and
undulator parameters. In the next section, the processes and major concerns
about XFELO design are discussed. And then the three-dimensional (3D) Bragg
diffraction code BRIGHT is built for numerical simulations. The XFELO
performances for the high peak current and low peak current mode are
illustrated in Section 4 and Section 5, respectively, and finally a brief
summary is presented.

Figure 1: Schematic view of XFELO for SCLF. Table 1: The main parameters of SCLF.  Parameter | Value
---|---
Beam energy | 8 GeV
Relative energy spread | 0.01 %
Repetition rate | 1 MHz
Peak current (low mode) | 10, 20, 30 A
Bunch charge (low mode) | 20 pC
Normalized emittance (low mode) | 0.2 mm$\cdot$mrad
Peak current (high mode) | 0.5, 1, 1.5 kA
Bunch charge (high mode) | 100 pC
Normalized emittance (high mode) | 0.4 mm$\cdot$mrad
Undulator period length | 26 mm
Undulator module length | 5 m
XFELO photon energy | 14.3 keV

##  2 XFELO design

The process of XFELO design mainly contains two parts: the transverse and
longitudinal profile optimization. On the one hand, electron beam transverse
size is shaped by FODO (focusing-drift-defocusing-drift) lattice and optical
transverse profile is determined by the cavity configurations. The goal is to
improve coupling factor between electron bunch and radiation pulse in order to
improve single pass gain. On the other hand, the longitudinal optimization
concerns the required undulator length for sufficient gain and the optimum
crystal mirror reflectivity for maximize the output pulse energy. For
illustration, the 1 kA peak current beam and the corresponding parameters are
utilized in the following subsections discussion.

###  2.1 Undulator lattice matching

For high energy electron beam, the transverse focusing from planar undulator
are trivial [19, 20], thus the external FODO structure is necessary. The
quadrupole magnets are installed at the nearly 1 m drift space between two
undulator modules and the Twiss parameters are matched to let the beta
function evolves periodically. For a given bunch charge, the smaller
transverse beam size is, the higher peak current as well as single pass gain
will be [21, 22, 23]. In addition, small beam cross section is also good for
preserving the X-ray fundamental transverse mode [24].

The lattice is matched by matrix method [25]. The electron beam transverse
beta function varies at different position of FODO, and its maximum and
minimum radius $r$ change with different quadrupole intensity are shown in
Fig. 2. The drift space between two modules and the quadrupole length is
assumed to be 1 m and 20.8 cm, respectively. When focusing strength is small,
the electron beam size expands and exhibits little fluctuation; as the
quadrupole intensity increases, the average beam size declines gradually,
however, the corresponding beam radius oscillation increases; and when the
focusing strength is too strong, no periodical solutions exist for the beam
dynamic functions, so the beam size explodes rapidly. For pursuing maximum
gain, we chose the quadrupole magnet gradient to be 20 T/m to get the small
average beam size with relatively low variation between 15 $\mu$m and 25
$\mu$m.

Figure 2: Transverse electron beam size inside the FODO lattice for different
quadrupole focusing strength. The upper (red) and the lower (blue) line show
the maximum and minimum beam size along the undulators, respectively.

###  2.2 Optical cavity design

Figure 3: The value of parameter $|m|$ as a function of CRLs focal length and
the distance between them.

As aforementioned, the $L=150$ m symmetry resonator is formed by two sapphire
crystal mirrors. According to [26], X-ray focusing elements are required for
the stability of optical cavity, thus two Be parabolic compound refractive
lenses (CRLs) [27] with focal length $f=57.7$ m are used, each with a radius
of 0.33 mm and a very high transmissivity $T=0.987$, assuming surface
microroughness less than 0.3 $\mu$m and an effective aperture of $0.66$ mm.
The rays propagation inside cavity is analyzed by ABCD matrix methods [28],
which defines the ray matrix for propagation through one period of the optical
system as

$M\equiv\begin{bmatrix}A&B\\\ C&D\end{bmatrix};\qquad m\equiv\frac{A+D}{2}$
(2)

For the periodic focusing system to be stable, the eigenvalues of $M$ should
be complex and have magnitude unity thus $|m|<1$. Figure. 3 presents the
contour plot of $|m|$ parameter as a function of CRLs focal distance $f$ and
the space $L_{1}$ between them. The red dashed line denotes the points where
$m=-1$, and the blue regions inside two critical lines are favorable values
for stable cavity configuration. According to it the distance between two CRLs
is 34.6 m$<L_{1}<115.4$ m. Note that different from the high-gain FEL in which
transverse radiation distribution is determined by gain guiding effect. The
XFELO optical eigenmodes depend on the configurations of resonator. For the
coupling benefit between electron bunch and X-ray pulse, $L_{1}$ is optimized
to be 100 m to match electron beam average radius by optical propagation code
(OPC) [29].

Figure 4: Single pass gain curve with different undulator segments.

###  2.3 Undulator length

For FEL oscillator, the undulator length is a trade-off between high power and
sufficient single pass gain. On the one hand, the maximum energy extracted
from a FEL oscillator is inversely proportional to the number of undulator
period [30]. On the other hand, the undulator should be sufficient long to
ensure single pass gain overcomes round trip loss. Thus the undulator is
chosen to be as short as possible once providing enough gain. Fig. 4
demonstrates the single pass gain as a function of $\Delta\gamma$, which is
the deviation of scaled electron beam energy from resonant energy. The curve
displays a typical small signal gain function, and for different numbers of
undulator periods, the maximum gain achieved at different magnitude of energy
deviation, and it approaches resonate energy as the undulator length enhances.
For a given beam energy, the undulator gap is adjusted to obtain proper energy
detune in practice. Considering the cavity passive loss ($\sim$10%), the
sapphire crystal finite bandwidth acceptance loss plus absorption ($\sim$40%)
and cavity output coupling ($\sim$20%), the optimized undulator length is
three segments.

###  2.4 Cavity crystal mirror

XFELO employs crystal Bragg diffraction, which enables high reflectivity at a
narrow bandwidth. According to [31], the response function is depends on
crystal thickness, X-ray incident angle, and excitation length. Assuming that
the interaction is Bragg symmetry diffraction, and the thickness of crystal is
designed for different reflectivity. For pursuing high reflectivity at normal-
incidence, low photon-electron effects and high Debye temperature, such as
sapphire crystal with (0 0 0 30) atom plane is used for 14.3 keV photon Bragg
exact backscattering. Figure. 5 shows the theoretical reflectivity for 70
$\mu$m thickness sapphire crystal with normal incidence. A high reflectivity
nearly 95% within 13 meV Darwin width was predicted. Note that the phase has
been manually scaled into $(0,2\pi)$, the phase deviation due to Bragg
diffraction insides Darwin width produces an additional delay to X-ray pulse,
which should be compensated by cavity detune to avoid FEL gain degradation.

Figure 5: The crystal mirror complex reflectivity for different incident
photon energy. Figure 6: The gain various at different radiation power. The
green line shows a typical XFELO round trip loss nearly 57% (including 20%
output coupling), and the saturation peak power inside the cavity would be
around 1.5 GW, thus the output power is 0.3 GW.

The cavity output coupling is optimized to generate highest X-ray pulse
energy. Fig. 6 shows a typical time-independent XFELO gain as a function of
light pulse peak power inside the cavity. When radiation intensity is small,
the single pass gain remains constant. While X-ray reaches saturation, portion
of electrons undergo strongly FEL interaction beyond 1/2 synchrotron
oscillation and energy extraction efficiency drops gradually. When single pass
gain equals to the round trip cavity net loss, the net gain is zero, and XFELO
reaches equilibrium and the output pulse energy remains steady.

The peak power inside the cavity is calculated by interpolation, like in Fig.
6, and the output power is predicted by multiplying mirror transmissivity.
Fig. 7 represents the output X-ray peak power as a function of mirror
reflectivity, in which the cavity round trip net passive loss, including
diffraction loss and CRLs absorption, is assumed to be 10%. On the one hand,
for the low reflectivity cavity, due to large amount of pulse energy leaks out
the optical cavity, effective positive feedback cannot build up thus small
radiation peak power; On the other hand, when the reflectivity is too high to
let the light power coupling out efficiently, the X-ray output power is small
as well. The optimum reflectivity is nearly 70% which predicts around 0.35 GW
output peak power.

Figure 7: The output peak power at different mirror reflectivity.

##  3 Three-dimensional effect of Bragg diffraction

The FEL oscillator simulations are normally carried out by combination of 3D
FEL codes GENESIS [32] and OPC. As demonstrated in [33, 34, 35], these codes
are efficient for long wavelength FEL oscillator simulations like ultraviolet
or infrared light. However, for XFELO which includes the interaction between
X-ray pulse and crystal, some extensions to the classical methods are needed.
In the past, as far as we know, the Bragg diffraction is traditionally
calculated in one or two dimensional way [31, 36], i.e., the 3D radiation
field is usually transformed into one dimension for crystal reflection and
then artificially transformed back to 3D in simulation, as seen in Fig. 8(a).
It smears out the transverse phase information totally and ignores some
important 3D effects including wave-front, which is essential for transverse
modes and cavity stability. The defects are not significant when cavity
configuration is primitive or single pass gain is overwhelming, but for more
advanced and delicate resonators [26], the self-consistent XFELO simulations
including all 3D effects are preferred [7].

Figure 8: The schematic of calculating Bragg diffraction. (a) Traditional one
dimensional process, which losses the transverse phase information; (b) The
directly three-dimensional way, which preserve the transverse phase
distribution and is supposed to be precise, while is much more time-consuming;
(c) The new approach which takes advantages of Fast Fourier Transform and
enjoys high efficiency and accuracy.

According to [31], Bragg diffraction of X-ray pulse was studied by taking
Fourier transform (FT), which decomposes the light pulse into monochromatic
plane-wave components, and multiply the corresponding Bragg diffraction
amplitude with each of them. This approach is available for 3D simulation: the
spatiotemporal radiation distribution function undergoes Fourier
transformation, then the related Bragg reflection spectral amplitude $R_{0H}$
are multiplied by each monochromatic planar wave, finally inverse Fourier
transform (IFT) is unitized to obtain the diffraction radiation.

Following the notation and deduction of two-dimensional spatiotemporally
response of ultrashort, laterally confined X-ray pulses Bragg diffraction in
[31, 37], we present here the derivation of a finite long X-ray pulse three-
dimensional Bragg diffraction, which provides theoretical support of a new
calculation method. The response for a monochromatic planar X-ray wave
$\varepsilon_{i}$ with a frequency $\omega_{0}$, incident angle $\theta$
respect to atom plane is

$\varepsilon_{H}^{(m)}(t,\vec{r})=\varepsilon_{i}e^{-i\left[\omega_{0}t-(\vec{K}_{0}+\vec{H})\vec{r}\right]}e^{i\left[\Delta_{H}(\vec{r}-\vec{r}_{H})\hat{z}\right]}R_{0H}(\omega_{0})$
(3)

where

$\vec{K}_{0}\approx\frac{\omega_{0}}{c}\hat{u}_{0}+\frac{\omega_{0}\,\delta\phi}{c}\hat{w}_{0}$
(4)

as defined in [31], $\hat{u}_{0}$ is the direction of optics propagating axis,
while $\hat{w}_{0}$ is the direction perpendicular to the dispersion plane.
The azimuthal angle $\delta\phi$, between $\vec{K}_{0}$ and dispersion plane,
is presumed to be small for a typical low divergence X-ray FEL. Thus the wave
number component in $\hat{w}_{0}$ is much smaller than along $\hat{u}_{0}$,
the influence of it on the additional momentum transfer $\Delta_{H}$ is
negligible. We eliminate all the higher order terms and follow the methods
used in [31], rewrite Eq. (3) as

$\varepsilon_{H}^{(m)}(t,\vec{r})=\varepsilon_{i}\,exp\left(i\,\omega\,\frac{\hat{w}_{0}\cdot\vec{r}}{c}\delta\phi\right)\,exp\left(-i\,\omega\tau_{H}\right)exp\left(-i\,\Omega\,\xi_{H}\right)R_{0H}(\Omega+\omega)$
(5)

where

$\tau_{H}=t-\frac{\hat{u}_{H}\cdot\vec{r}}{c},\quad\xi_{H}=t-\frac{\hat{u}_{0}\cdot\vec{r}}{c}-2sin^{2}\theta\frac{(\vec{r}-\vec{r}_{H})\hat{z}}{c\gamma_{H}},\quad\omega+\Omega=\omega_{0}.$
(6)

and we have assumed that $\left|\Omega/\omega\right|\ll 1$ and omitted
$\Omega\,\delta\phi$ term. Here $\omega$ is the Bragg frequency, $\omega_{0}$
is the frequency of incident photon.

For a finite long X-ray pulse with confined wave front, the radiation after
Bragg diffraction would be

$\displaystyle\varepsilon_{H}(t,\vec{r})$ $\displaystyle=$
$\displaystyle\varepsilon_{i}\int_{-\infty}^{+\infty}\frac{d\tilde{\theta}}{2\pi}exp\left[-i\,\omega(\tilde{\theta})\tau_{H}(\tilde{\theta})\right]\int_{-\infty}^{+\infty}\frac{d\,\delta\phi}{2\pi}exp\left[i\,\omega(\tilde{\theta})\frac{\hat{w}_{H}\cdot\vec{r}}{c}\delta\phi\right]$
(7)
$\displaystyle\times\int_{-\infty}^{+\infty}\frac{d\Omega}{2\pi}exp\left(-i\,\Omega\,\xi_{H}\right)f(\Omega,\tilde{\theta},\delta\phi)R_{0H}(\Omega+\omega)$

where $f(\Omega,\tilde{\theta},\delta\phi)$ is the Fourier transform of
incident pulse, i.e.

$\displaystyle\varepsilon_{i}(\tau_{0},\nu_{0},w_{0})$ $\displaystyle=$
$\displaystyle\varepsilon_{i}\,exp(-i\,\omega\tau_{0})\int_{-\infty}^{+\infty}\frac{d\tilde{\theta}}{2\pi}exp\left[i\,\nu_{0}(\omega_{0}/c)(\tilde{\theta}-\theta)\right]\int_{-\infty}^{+\infty}\frac{d\,\delta\phi}{2\pi}exp\left[i\,w_{0}(\omega_{0}/c)\delta\phi\right]$
(8)
$\displaystyle\times\int_{-\infty}^{+\infty}\frac{d\Omega}{2\pi}exp\left(-i\,\Omega\,\tau_{0}\right)f(\Omega,\tilde{\theta},\delta\phi)$
$\displaystyle\equiv$
$\displaystyle\varepsilon_{i}\,exp(-i\,\omega\tau_{0})\int_{-\infty}^{+\infty}\frac{d\Omega}{2\pi}exp\left(-i\,\Omega\,\tau_{0}\right)\prod(\Omega,\nu_{0},w_{0})$

This formula could be used for 3D Bragg diffraction simulation directly as
shown in Fig. 8(b), which is expected to be accuracy theoretically. In
practice, however, it is proved to be time-consuming because the odd number of
grid points required by GENESIS is not consistent with fast Fourier transform
(FFT). With the absence of FFT, scheme (b) takes almost a whole week on a
personal computer to obtain the results after Bragg diffraction, which is
definitely not suitable for XFELO simulations. To simplify the Eq. (7), we may
introduce some reasonable approximations [31]:

$\displaystyle\tau_{H}(\tilde{\theta})$
$\displaystyle\approx\tau_{H}+\frac{\nu_{H}}{c}(\tilde{\theta}-\theta),\qquad\omega(\tilde{\theta})\approx\omega[1-(\tilde{\theta}-\theta)\,cot\theta],$
$\displaystyle\tau_{H}(\tilde{\theta})\,\omega(\tilde{\theta})$
$\displaystyle\approx\omega\tau_{H}+\frac{\omega}{c}(\nu_{H}-\tau_{H}\,c\,cot\theta)(\tilde{\theta}-\theta),\qquad\omega(\tilde{\theta})\,\delta\phi\approx\omega\delta\phi$
(9)

The spatiotemporal radiation after Bragg diffraction is rewritten as

$\varepsilon_{H}(t,\vec{r})\approx\varepsilon_{i}e^{-i\,\omega\tau_{H}}\int_{-\infty}^{+\infty}\frac{d\Omega}{2\pi}e^{-i\,\Omega\xi_{H}}\prod(\Omega,\,\nu_{H}-\tau_{H}\,c\,cot\theta,\,w_{H})R_{0H}(\Omega+\omega)$
(10)

where $w_{H}=\hat{w}_{H}\cdot\vec{r}$.

The results indicate that for an incoming high brightness X-ray pulse from
FEL, as shown in Fig. 8(c), the radiation after Bragg diffraction could be
obtained by: first, Fourier transform each longitudinal wavelet at a fixed
transverse coordinate; and then multiply the corresponding monochromatic wave
diffraction amplitude function $R_{0H}$, and get the radiation wavelet after
Bragg diffraction by inverse Fourier transform; finally, substitute the
transverse variable $\nu_{H}$ by $\nu_{H}-\tau_{H}\,c\,cot\theta$ which
represents spatial transverse shift. In practice, this could be taken into
account by a linear coordinates transform, since only first order small terms
are important and concerned in the derivation. For the XFELO discussed in this
paper, the incident angle $\theta$ is approximate to $\pi/2$, which means
$cot\theta\approx 0$ and there is not transverse deformation. According to the
strategies (b) and (c) mentioned above, the BRagg dIffraction alGoritHom in
Three-dimensional (BRIGHT) which is tailored for XFELO simulations has been
built. Due to the benefits of FFT in scheme (c), only several seconds is
required for obtaining the 3D spatiotemporal radiation after Bragg
diffraction.

Figure 9: The Bragg diffraction results of BRIGHT simulation for 1 kA XFELO at
10 passes. The first column shows the initial light pulse field longitudinal,
transverse, and phase profile; the second column plots the corresponding
figures after 3D Fourier transform; the third column shows the field
distribution after Bragg diffraction by directly 3D approach (b); and the
final column presents the results of new method (c).

The BRIGHT results of scheme (c) is compared with (b) in Fig. 9: the first
column represents the initial X-ray field distribution; the second column
shows the corresponding 3D FT; the third column demonstrates the final
radiation after Bragg diffraction of approach (b); while the final column is
the results calculated by approach (c). The validity is confirmed by the
similarity of the radiation power and phase distribution. Note that the small
discrepancy is mainly due to the limited range we have covered in the
frequency domain in scheme (b) which leads to the loss of fractional high
frequency signals. Fig. 10 shows the results of classical one-dimensional and
BRIGHT for 1 kA high peak current mode XFELO simulation of SCLF, the
reasonable agreement implies the validation of BRIGHT. The three-dimensional
simulation preserves transverse phase distribution of the radiation, and is
benefit for eigenmode formation as expected. The simulation reveals that 3D
simulation gives smaller radiation size and higher single pass gain, thus
generates larger radiation power. For low current mode, the 3D Bragg
diffraction effects proved to be indispensable, due to the much longer
undulator implemented, the drift space is suppressed and the optical cavity is
more sensitive. The conventional method undermines the resonator stable
conditions and prevent the X-ray power from growing. Thus the 3D simulation is
essential for low current XFELO simulation. With the discussions above, we
expect that 3D XFELO simulations by BRIGHT would be prevalent. Note that the
applications are not limited for XFELO, it is useful for simulations of other
X-ray Bragg diffraction such as hard X-ray self-seeding FEL as well.

Figure 10: The comparison of one-dimension traditional simulation results with
three-dimensional new method.

##  4 High current mode XFELO

Figure 11: The results of high peak current mode XFELO. The output energy
growth (a), output power profile (b) and spectrum (c) for different electron
beam peak current.

The aforementioned analysis is utilized for the design of XFELO in SCLF.
Candidate parameters are first calculated by a fast FEL oscillator code refer
to [16], in which the theoretical analysis produces approximately results
rapidly. Then the parameters are simulated and optimized by GENESIS, OPC and
BRIGHT. For high current modes, three undulator segments are employed, and
sapphire mirror reflectivity is 95% for the upstream and 80% for the
downstream with thickness equal to 70$\mu$m and 47 $\mu$m, respectively. The
output energy as a function of round trip pass is demonstrated in Fig. 11 (a).
As expected, the power grows exponentially before saturation, and presents a
stable X-ray output. The 1 kA XFELO generates largest pulse energy 128 $\mu
J$, which is followed by 1500 A, and then the 500 A. This is due to that high
peak current leads to short bunch duration, which broadens the corresponding
spectrum bandwidth. The fractional photons energy beyond sapphire crystal
Bragg reflection bandwidth cannot be accumulated. Thus the higher peak current
is, the larger single pass gain is, yet more power is cut off during the
interaction with mirrors. In addition, the normal peak current for SCLF is
1500 A with 100 pC charge, the results reveal that XFELO is able to operation
with lower peak current. Furthermore, the energy fluctuation demonstrated in
Fig. 11(a), might origins from the “respiration” of X-ray pulse length.

Fig. 11 (b) and (c) show the power and spectrum of different peak current
XFELO. For the 1 kA XFELO, the spectrum bandwidth is 15.6 meV (FWHM), which
results from the joint efforts of crystal Bragg reflection bandwidth and
electron beam duration. The X-ray pulse temporal length is around 0.09 ps,
which gives a 0.35 time-bandwidth product, nearly the value of the Fourier
transform limit (0.44) for a Gaussian pulse profile. In addition, with same
configurations of electron beams and undulators, SASE generates 111 $\mu J$
X-ray pulse with peak brilliance $3.4\times
10^{32}\,(photons/s/mm^{2}/mrad^{2}/0.1\%BW)$, while XFELO produces X-ray with
peak brilliance $2.1\times 10^{35}\,(photons/s/mm^{2}/mrad^{2}/0.1\%BW)$,
which is three orders of magnitude higher than SASE.

##  5 Low current mode XFELO

Figure 12: The results of low peak current mode XFELO. The output energy
growth (a), output power profile (b) and spectrum (c) for different electron
beam peak current.

The same methods are used to investigate the XFELO performance at 10, 20, 30 A
peak current. To obtain enough single pass gain, the corresponding undulator
segments demanded are 10, 8, 6. The optimum mirror reflectivity is nearly 93%
for downstream with thickness equals 65 $\mu$m and 95% for upstream, with the
crystal Darwin bandwidth nearly 13 meV. Fig. 12 (a) shows the output light
pulse energy evolves pass-by-pass of different peak current. It demonstrates
that the pulse power increases exponentially before saturation. The maximum
final output pulse energy is nearly 5.5 $\mu$J for 30 A.

For the low current mode, electron beams charge in SCLF is 20 pC, which means
the corresponding electron bunch length (FWHM) for different peak current are
2.00, 1.00, 0.67 ps, respectively. The output light pulse power profiles and
spectrum are shown in Fig. 12 (b) and (c). The pulse duration reduces as the
peak current increases, the spectrum, however, does not change much, which
indicates that the output X-ray is not at the Fourier transform limit. The
output light pulse duration is nearly the same as the corresponding electron
bunches length, while the spectrum bandwidth is about 9.4 meV, with peak
brilliance nearly $3.6\times 10^{33}\,(photons/s/mm^{2}/mrad^{2}/0.1\%BW)$ for
30 A XFELO.

##  6 Conclusion

A systematical process for designing and optimization of XFELO is proposed. On
the one hand, for optimization of transverse beam size, FODO lattice is used
for average electron beam transverse size suppression; while the CRLs are
employed to preserve the stability of optical cavity, whose transverse mode
matches to electron bunch. On the other hand, suitable mirror reflectivity is
chosen to couple out X-ray effectively and to feedback enough power to ensure
the cavity pulse energy grows up. Meanwhile, the undulator should offer
sufficient single pass gain to compensate the cavity loss.

SCLF is expected to be the first hard X-ray FEL facility in China. The quasi-
CW high quality electron beam is appropriate for XFELO operation. XFELO
examples based on SCLF is studied with high current mode ($\sim$ 1 kA) and low
current mode ($\sim$ 10 A). We developed a novel simple three-dimensional
Bragg diffraction codes BRIGHT, and 3D self-consistent numerical simulations
are carried out by the combination of GENESIS, OPC and BRIGHT. The simulation
results prove the design approaches to be feasible and efficient. According to
our results, SCLF XFELO is able to generate 128 $\mu$J, 1 GW fully coherent
stable X-ray with 1 kA electron beam peak current; and nearly several $\mu$J,
10 MW X-ray with a few dozens of A peak current. The peak brilliance of high
current mode XFELO is nearly 3 orders of magnitude larger than SASE. It is
worth to note that the design process is suitable for other FEL machines, and
the results is useful for construction of XFELO test facility in SCLF as well
as the facilities around the world. In addition, the parameters are useful for
the investigation of other advanced proposals based on XFELO, such as cascaded
single-pass XFELO [38]. The following work would be to considering practical
technical problems like the crystal thermal loading tolerance of X-ray
interaction.

## Acknowledgments

The author would like to thank B. Liu, D. Wang and Z. Zhao for helpful
discussions on SCLF project; K. J. Kim and W. L. Qin for enthusiastic
discussions on XFELO physics; and Y. Shvyd’ko, R. Lindberg for providing
information and related parameters of crystal. This work was partially
supported by the National Natural Science Foundation of China (11775293) and
Ten Thousand Talent Program.

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