# A Simplified Model for Fast Optimization of Free-Electron Laser Oscillator

Kai Li1,2 Minghao Song1,2 Haixiao Deng 1<EMAIL_ADDRESS>1Shanghai
Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800,
China
2Graduate University of Chinese Academy of Sciences, Beijing 100049, China.

###### Abstract

A simplified theoretical model for free-electron laser oscillator (FELO)
simulation which reserves the main physics is proposed. In stead of using
traditional macro particles sampling method, the theoretical model takes
advantages of low gain theory to calculate the optical power single-pass gain
in the undulator analytically, and some reasonable approximations are made to
simplify the calculation of power growth in the cavity. The theoretical
analysis of single-pass gain, power growth, time-dependent laser profile
evolution and cavity desynchronism are accomplished more efficiently. We
present the results of infrared wavelength FELO and X-ray FELO with the new
model. The results is checked by simulation with GENESIS and OPC which
demonstrates the validity of the theoretical model.

PACS numbers

41.60.Cr

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Free-electron laser (FEL) is a new light source which uses relativistic
electron beam passes through the undulator and interaction with the radiation
field to generate high brilliant laser pulses. Due to its many advantages,
such as rapid and continuous tenability over a wide spectral range, a great
deal of interest is attached to it currently. Single-pass high-gain FEL,
especially self-amplified spontaneous emission (SASE) schematic, is able to
produce high brilliant laser pulses in X-ray region. With the great success of
Linear Coherent Light SourceEmma  _et al._ (2010) and Spring-8 Angstrom
Compact Free Electron LaserIshikawa  _et al._ (2012), several hard X-ray SASE
FEL have been built or under construction around the world. Although SASE FEL
provides fully transverse coherent and short temporal duration X-ray pulses,
it starts from shot noise and produces poorly temporal coherence lightKim
(1986). Numerous schemes, including external seeded FELYu (1991); Xiang and
Stupakov (2009); Feng  _et al._ (2014) and self-seedingAmann  _et al._ (2012);
Geloni  _et al._ (2011), have been proposed to solve this problem and improve
its performance.

Another promising operating mode is FELO, which works in the low-gain region
as well as employs electron beam to pass the undulator multiple times and
convert energy to radiation. There are many long wavelength FELO have been
established, e.g., the free-electron laser for infrared experimentsOepts  _et
al._ (1995) and Duke storage-ring based FELYan  _et al._ (2016). Also several
infrared and THz FELO are under construction, e.g., the first infrared free-
electron laser user facility in ChinaLi  _et al._ (2016). Recently, a
promising schematic X-ray free-electron oscillator (XFELO)Kim  _et al._ (2008)
has been reconsidered thanks to the development of high-reflectivity high-
resolution X-ray crystalShvyd ko  _et al._ (2010). XFELO can generate fully
temporal coherent and stable peak power laser pulses with the peak brilliance
comparable to SASE and the average brilliance several orders of magnitude
higher than SASE. However, there are still lots of challenges of FELO scheme,
including high repetition electron injectorOstroumov  _et al._ (2008), heat
loading of the Bragg reflection crystal mirrorSong  _et al._ (2016) and X-ray
optics.

In addition, unlike SASE FEL in which electron beam passes undulator only
once, FELO contains an oscillator in which electron beam and optical pulse go
through undulator hundreds of times before saturation. Thus tracking the
electrons motion and electric field evolution requires lots of calculation and
the theoretical analysis and design of FELO become another problem. Although
there are some conventional FEL simulation codes such as GENESISReiche (1999),
GINGERFawley (2001), by combine with optical codes OPCvan der Slot  _et al._
(2009) they can be used to simulate FELO process, these approaches are usually
relatively slow and time-consuming. For example, according toKim  _et al._
(2008); Dai  _et al._ (2012) a complete tracking from the initial spontaneous
emission to final saturation of XFELO took about one month. Thus there is a
strong scientific demand of a simpler and faster theoretical model, which is
able to obtain some primary results, basic performances of XFELO and optimum
parameters values with acceptable accuracy.

Traditional simulation approaches mentioned above use macro particles sampling
method, which contains tens of thousands of macro particles in each slice, and
tracking all of them is laborious and time consuming. In this paper, we
propose a new theoretical model which takes advantages of electron
distribution function to solve the single-pass gain as a function of
electronic field intensity, and some assumptions are used to simplify the
calculation of radiation power in order to save time. The new approach reduces
the calculation time for a fully tracking of FELO from days to minutes by
analyzing single-pass gain, power growth, time-dependent laser profile
evolution and cavity desynchronism in a more efficient way. The passage is
arranged as following: the second section introduces the three main parts of
theoretical model. Then we show two examples: 1.6 $\mu\mathrm{m}$ infrared
wavelength FELO and 1 $\mathrm{\AA}$ X-ray FELO. Finally a brief conclusion of
this paper is given.

##  II Theoretical model of FEL oscillator

A FELO facility typically contains multiple mirrors with reflectivity $R$ to
form an optical cavity which captures the radiation emitted by relativistic
electrons traveling through undulator. In this paper we focus on the two-
mirror FEL oscillator and other multi-mirror cases are similar to it. The
initial optical field comes from the spontaneous radiation as the electron
beam passing through the undulator acts as a seed for the following
amplifying. With the carefully synchronism between electron beam and optical
pulse, the radiation starting from the shot noise overlaps with electrons and
is amplified on successive passes. In fact, the electron beam energy is
modulated and converted to radiation pulse in the undulator. In order to
ensure the increase of optical power, single-pass gain $G$ should overcome net
loss, i.e.,$(1+G)R>1$. The radiation field evolutes in the cavity and for the
$(n+1)$th pass at the entrance of undulator

$E_{n+1}(t)=\left[E_{n}(t)g(t)+\delta E(t)\right]R_{total}$ (1)

where $\delta E$ is the spontaneous radiation, $g(t)$ is the gain of optical
field and $R_{total}$ is the equivalent reflectivity of two mirrors. The laser
field experiences an exponential growth before the gain begin to drop off due
to too large energy spread, and the intensity approaches to saturation and
remains unchanged finally. After hundreds of passes through the undulator, the
laser pulse saturates. And the output power keeps steady while the gain in
undulator equals to round-trip total loss in the cavity. The main process
laser pulse undergoes during one round-trip can be divided into two main
parts: the interaction with electron beam in the undulator and the reflections
of mirror at the two sides of cavity. In order to clarify the new FELO model
three elementary procedures are analyzed and relative parameters are
calculated as following.

###  II.1 Gain calculation

Due to its relatively small number of undulator periods, FEL Low-Gain theory
is suitable for analyzing the increase of optical power. According to the Low-
Gain theoryBoscolo  _et al._ (1982, 1985), the evolution of electrons
distribution and thus the gain of laser pulse are solved analytically. We
derive formulas in one dimension approximation, and focus on the longitudinal
component of laser field and electrons coordinate. The motion of single
electron in the phase space $(\theta,\eta)$ is described by “pendulum
equation”Huang and Kim (2007)

$\displaystyle\frac{d\theta}{dz}=2k_{u}\eta$ (2)
$\displaystyle\frac{d\eta}{dz}=-\frac{\epsilon}{2k_{u}{L_{u}}^{2}}\sin\theta$
(3)

where we introduce the field strength parameter

$\epsilon=\frac{eE_{0}K[JJ]}{{\gamma_{r}}^{2}mc^{2}}k_{u}{L_{u}}^{2}$

The electrons distribution function $\rho(z;\eta,\theta)$ at point $z$ is
governed by the continuity equation

$\displaystyle\frac{\partial\rho}{\partial
z}+\dot{\theta}\frac{\partial\rho}{\partial\theta}+\dot{\eta}\frac{\partial\rho}{\partial\eta}=0$
(4)

where $\dot{x}=\frac{dx}{dz}$, substituting Eq. (2)(3) into Eq. (4) and using
scaled parameters

$\begin{split}z^{\prime}=\frac{\sqrt{\epsilon}}{L_{u}}z\\\
\eta^{\prime}=\frac{2k_{u}L_{u}}{\sqrt{\epsilon}}\eta\end{split}$ (5)

yield the following partial derivation equation

$\frac{\partial\rho}{\partial
z^{\prime}}+\eta^{\prime}\frac{\partial\rho}{\partial\theta}+\sin\theta\frac{\partial\rho}{\partial\eta^{\prime}}=0$
(6)

Assuming the initial distribution of electron beam fulfils Gaussian function
with scaled energy spread $\sigma_{\eta^{\prime}}$ and scaled energy deviation
$\eta^{\prime}_{0}$, the solution can be found by the method of
characteristics at the end of undulator is

$\rho(z^{\prime};\eta^{\prime},\theta)=\frac{1}{2\pi}\frac{1}{\sqrt{2\pi}\sigma_{\eta^{\prime}}}\times\mathrm{exp}\left\\{-\frac{1}{2\sigma_{\eta^{\prime}}^{2}}\left[\frac{\eta^{\prime}\mathrm{cn}(z^{\prime};C)-\sin\theta\,\mathrm{sn}(z^{\prime};C)\mathrm{dn}(z^{\prime};C)}{1-\cos^{2}\frac{\theta}{2}\,\mathrm{sn}^{2}(z^{\prime};C)}-\eta^{\prime}_{0}\right]^{2}\right\\}$
(7)

where $C^{2}=\frac{{\eta^{\prime}}^{2}}{4}+\cos^{2}\frac{\theta}{2}$. The
interaction between electrons and laser pulse meets the law of conservation of
energy, and thus the power gain of laser is

$G=\sqrt{m_{e}c^{2}K[JJ]{k_{u}}^{-1}}\frac{I}{c\beta}\frac{1}{2\pi\Sigma^{2}}\frac{1}{\varepsilon_{0}{E_{0}}^{3/2}}\langle\
\Delta\eta\rangle$ (8)

where $2\pi\Sigma^{2}$ is the cross section of electron beam,
$\varepsilon_{0}$ is the dielectric constant of vacuum, and
$\langle\Delta\eta\rangle$ is the average change of $\eta$ in one slice which
can be calculated by density function integration. Note that the influence of
electron beam emittance can be involved by replacing the energy deviation with
equivalent relative energy spread

$\frac{\sigma^{\prime}_{E}}{E_{0}}=\sqrt{\left(\frac{\sigma_{E}}{E_{0}}\right)^{2}+\left(\frac{\varepsilon\lambda_{u}}{4\lambda\beta}\right)^{2}}$
(9)

###  II.2 Cavity model

The FELO facility typically contains a mirror at each side of the FELO cavity,
which forms an oscillator to trap the optical pulse. For infrared wavelength
light, metal mirrors are utilized due to its broadband reflectivity and high
thermal conductivity. The interaction between radiation and mirror can be
approximated by a normal reflection without deformation of optical pulse.
However, it is more complicated for XFELO which exploits Bragg crystal
reflection. The high-reflectivity bandwidth for crystal mirror is relatively
much narrow so that the optical pulses are cut off in the frequency domain and
deformed in the temporal spaceShvyd ko and Lindberg (2012); Lindberg and Shvyd
ko (2012). Although the reflectivity of crystal mirror is depend on lots of
factors, such as X-ray incident angle, profile of pulse and thickness of
crystal, to illustrate the main properties of Bragg backscattering, we assume
the crystal to be semi-infinite and non-absorbed, and the symmetry Bragg
backward scattering is chosen. In this way, the complex reflectivity is
simplified asLindberg  _et al._ (2011)

$r(y)=\left\\{\begin{array}[]{r@{\quad}l}y-\sqrt{y^{2}-1}&\mathrm{if}\>y>1\\\
y-i\sqrt{1-y^{2}}&\mathrm{if}\>|y|\leqslant 1\\\
y+\sqrt{y^{2}-1}&\mathrm{if}\>y<-1\end{array}\right.$ (10)

where $y=\frac{1}{|\chi_{H}|}\left[\frac{2(E-E_{H})}{E_{H}}+\chi_{0}\right]$,
$E_{H}$ is the Bragg energy and $\chi_{0}$ and $\chi_{H}$ are Fourier
components of the dielectric susceptibility of the crystal. Fig. 1 shows the
reflectivity of diamond crystal C(4,4,4) in symmetry Bragg backscatter at
various incident photon energy deviation $\Delta E=E-E_{h}$. The shift of
curve cental from zero results from the fact that peak reflectivity located at
an energy slightly different from the Bragg energy.

Figure 1: The complex reflectivity of Bragg crystal at various incident photon
energy deviation from Bragg energy.

###  II.3 Initial start-up

The FELO laser start from the electron beam synchrotron radiation in the
undulator, which is chaotic in spectrum and randomly in phase. This initial
shot noise optical field acts as a seed which is trapped in the optical cavity
and amplified on the successive passes. According to Schmüser  _et al._ (2014)
an electron passing through an undulator with $Nu$ periods produces a wave
train

$E_{0}(t)=\left\\{\begin{array}[]{l@{\quad}l}E_{0}\exp(-i\omega
t)&\mathrm{if}\>-T/2<t<T/2,\\\ 0&\mathrm{otherwise}.\end{array}\right.$ (11)

where $T=N_{u}\lambda/c$ is the time for electron travel through the
undulator. And the power generated is

$P_{rad}=\frac{e^{2}c\gamma^{2}K^{2}k_{u}^{2}}{12\pi\varepsilon_{0}}.$ (12)

For an electron pulse contains $N$ electrons which is shorter than one wave
train (i.e., including only one wave packet), the total electric field is

$E(t)=E_{0}\exp(-i\omega t)\sum\limits_{j}^{N}\exp(i\phi_{j}).$ (13)

where $\phi_{j}$ is the initial phase of electric field which is related to
the relative location of electron in the bunch. The time-average field power
$U\varpropto E_{0}^{2}\left|b\right|^{2}$ with
$b=\sum_{j}^{N}\exp(i\phi_{j})$. The dimensionless variable
$\xi=b^{2}/\left<b^{2}\right>$ obeys the simple exponential probability
distribution

$P(\xi)=\exp(-\xi).$ (14)

which means the electric field intensity is proportional to square root of the
number of electrons in a coherent length Saldin  _et al._ (1998).

However, in the numerical model, we choose a Gaussian profile electron beam
which is far longer than the optical wave train and contains $M$ wave packets.
For this long electron bunch, the power probability distribution is more
complex, so we approximate the complex possibility distribution with normal
distribution and obtain the initial field power distribution by Gaussian
sampling with mean value and standard deviation equal 1 and $1/\sqrt{M}$
respectively. The electric field phase fulfils randomly distribution among
$[0,2\pi]$.

According to the FEL resonance condition, the optical field propagates ahead
of electron beam one resonant wavelength when travel through one undulator
period. By the virtue of several hundred times of pass through undulator and
continually slippage, the field in different slices “communicate” with each
other and develops into a high brilliant, monochromatic laser pulse. Due to
the chaotic phase at the beginning, the initial gain is relatively smallFeng
_et al._ (2013), the influence of slippage is roughly considered as phase
averaging of different slices within a coherent length. Besides slippage, the
field is shaped by Bragg crystal mirror reflection heavily for XFELO and this
is expected to improve longitudinal coherence of the laser. Finally, as the
laser power increases and single-pass gain drops off, FELO transforms to
saturation and maintain a steady state.

##  III Infrared FELO simulation

Figure 2:  The electron density distribution function in phase space of one slice. Figure 3:  The gain as a function of field intensity.(The dash line is the expected constant gain function curve in the low-gain theory at small electric field intensity.) Table 1: Simulation parameters for infrared FELO.  Parameter | Value | Unit
---|---|---
Beam energy $E_{0}$ | 80 | MeV
Slice relative energy spread $\sigma_{\eta}$ | 0.2% |
Normalized emittance $\varepsilon_{n}$ | 10 |  $\mu$m-rad
Peak current $I$ | 200 | A
Electron beam charge $Q$ | 100 | pC
Electron bunch length (FWHM) $\sigma_{e}$ | 0.5 | ps
Undulator period $\lambda_{u}$ | 45 | mm
Number of undulator $N_{u}$ | 16 |
Laser wavelength $\lambda$ | 1.6 | $\mu\mathrm{m}$
Rayleigh length $Z_{R}$ | 0.35 | m
Cavity loss | 1% |
Output coupling efficiency | 6% |
Figure 4:  The enhancement of output laser peak power with various passes
$N_{pass}$.

Figure 5:  Snapshots of output radiation pulse for a typical infrared FELO at
1.6$\mu m$. The top and the bottom row show the longitudinal pulse temporal
profile and corresponding spectrum respectively.

A typical infrared FELO is investigated using parameters shown in Table. 1
which are also presented inDeng (2016). The modulation of electron
distribution function in the phase space and the gain degradation as the laser
field increase are obtained by Eq. (7). When calculating the gain we assume
that the FELO is operated at its optimum electron energy deviation which
produces the maximum single-pass gain. And the electron bunch cross section is
considered the same as optical beam waist which is determined by Rayleigh
length $Z_{R}$. Fig. 2 shows the contour map of electrons distribution in the
phase space of one slice. As expected, the initial Gaussian profile is twisted
and the electrons rotates in the “bucket”, which means the electrons energy
are modulated and transformed to laser pulse. As a consequence of energy
conservation law, the growth of laser power is equal to the loss energy of
electron bunch which can be obtained by difference between initial and final
total energy using Eq. (8). The sing-pass gain as a function of optical field
is given by Fig. 3, gain decreases as laser power rises. However, it drops off
a little at the beginning which is contradict to the predicted constant gain
in low-gain theory. Further investigation reveals that is due to the relative
large value of $\eta^{\prime}$ leads to low accuracy of integration of
electron density function at the small radiation power. By refine the mesh
grids in the integration process, it can be mitigated.

It is worthy to note that, even refining the mesh grids enhances the accuracy
of gain calculation, low gain theory is used to get the gain at low optical
intensity while solve the gain by the new model near the saturation. Through
the combination of these two approaches, single-pass gain for all electric
field intensity can be given effectively and correctly.

To evaluate the evolution of laser pulse profile and spectrum bandwidth in the
oscillator, we use Eq. (14) to get the initial shot noise signal, and adjusted
it to nearly 1W for typical infrared FELO case in this paper. The growth of
laser power is calculated by Eq. (1) until it approaches saturation and remain
constant. The metal mirrors reflectivity are assumed to be 100%. The output
coupling of the downstream mirror is 6% and extra 1% cavity loss due to the
mirror absorbing or deflection of light is considered. Fig. 4 displays the
growth of laser pulse after various number of pass. The evolution of both
spectral and temporal profile of radiation in the cavity from shot noise to
saturation are demonstrated at the upper and bottom row of panels respectively
in Fig. 5. These plots indicate significant temporal and spectral fluctuation
during the initial amplify period and become smooth as passing number
increases. The FELO finally generate a $1\mathrm{ps}$ laser pulse with narrow
spectral bandwidth at $1.6\mathrm{\mu m}$. Excepting a little faster purify of
spectrum and earlier boost of laser power, which are due to the roughly
treatment of slippage and initial shot noise power, the results are agree well
with Ref. Deng (2016) both in temporal and spectrum profile as well as peak
intensity. Thus it proves the accuracy of the new theoretical model. In
addition, the laser pulse temporal width first narrows and then broadens,
which is consist with the analysis in Ref. Kim (1991).

It is interesting to note that since the new theoretical model generates
results more efficiently, it can be adjusted to investigate the cavity
desynchronism in FELO. Due to slippage, the group velocity of optical field is
smaller than the speed of light $c$, which lead to the maximum gain appears to
lag behind and miss out the largest electron density location. This leads to
the degradation of output laser peak powerBakker  _et al._ (1994); Knippels
_et al._ (1995). It is hard to calculate the group velocity of light precisely
in our model, so we ignore the “lethargy” effect and assume the electron bunch
and laser pulse already at perfectly synchronism. However, we can deliberately
shift the electron profile of a different range and investigate its influence
to output laser energy. Fig. 6 shows the output energy declines when cavity
desynchronism increases. Fig. 7 demonstrates a typical light profile at
$0.6\lambda$ cavity desynchronism and it is obvious that the pulse tilts
towards electron beam slightly.

Figure 6:  The output laser energy as a function of desynchronism. Figure 7:
The laser profile at 0.6$\lambda$ cavity desynchronism.

##  IV x-ray FELO simulation

Table 2: Simulation parameters for X-ray FELO.  Parameter | Value | Unit
---|---|---
Beam energy $E_{0}$ | 7 | GeV
Energy spread $\sigma_{E}$ | 1.4 | MeV
Normalized emittance $\varepsilon_{n}$ | 0.2 |  $\mu$m-rad
Peak current $I$ | 10 | A
Electron bunch length $\sigma_{t}$ | 1.0 | ps
Undulator period $\lambda_{u}$ | 17.6 | mm
Number of undulator $N_{u}$ | 3000 |
Laser wavelength $\lambda$ | 1.0 | $\mathrm{\AA}$
Cavity loss | 5% |
Bragg mirror reflectivity $R$ | 94% |
Figure 8:  The electron density distribution in the phase space of one slice.
Figure 9:  The enhancement of output laser peak power with various passes
$N_{pass}$.

A typical $1\,\mathrm{\AA}$ X-ray XFELO is investigated with the parameters in
Table. 2. The optical cavity is built up with diamond crystal mirrors by
C(4,4,4) Bragg reflection with the Bragg energy at $12.04\,\mathrm{keV}$ and
reflectivity reaches 94%. The Bragg crystal mirrors filter out the optical
frequency beyond its high-reflectivity spectral bandwidth, and in order to
maintain enough gain to overcome the round-trip loss and magnify the optical
field, the number of undulator periods is chosen to be 3000. In this case the
single-pass gain is near 39% which is given by the theoretical gain
calculation method mentioned above. The electron density distribution in the
phase space of a slice, when pulse power inside the cavity equals to 0.2MW, is
shown in Fig. 8. Due to the relative larger electron energy, the bucket which
traps the electron becomes flatter, and the energy modulation is smaller.
After a shot periods of struggling as the initial shot noise, the laser power
inside the cavity increases as it goes through the undulator again and again
and becomes a stable 0.6MW laser pulse output at saturation, which is shown in
Fig. 9. The final output power from GENESIS simulation presented with dashed
line is nearly twice as large as that from the new model, which is mainly due
to the inaccurate evaluation of electron beam and laser cross section as well
as the coupling factor between them. Given the approximations used in the
analysis process, it is easy to understanding this discrepancy.

Figure 10:  Snapshots of output radiation pulse for a typical X-ray FELO at
1.0 $\mathrm{\AA}$. The top and the bottom row show the longitudinal pulse
temporal profile and corresponding spectrum respectively. Figure 11:  The
output laser energy as a function of electron beam shift distance in each
pass.

The XFELO laser power evolution inside the cavity is solved in the same method
as for infrared FELO except that the reflectivity of metal mirrors is replaced
by complex reflectivity in Eq. (10) and the snapshots of output pulse profile
and the corresponding spectrum are displayed in Fig. 10. The evolution of the
laser power and profile are quite similar with the infrared wavelength case.
However, due to the spectrum purifying of the crystal mirrors, the output
laser spectral bandwidth is much smaller than the previous infrared FELO. And
the complex reflectivity of the crystal mirrors causes an extra phase shift of
optical field and leads to the pulse slides backward. The “lethargy” from the
crystal mirrors reflection are much larger than that of the infrared FELO,
thus cannot be ignored. In the theoretical model, the electron beam is
constantly delayed a distance to overlap with the optical field and the shift
distance is equal to 20$\mu$m to maintain the overlap between electron bunch
and laser pulse in Fig. 10. The output laser FWHM duration is nearly 2ps.

The output total energy as a function of this shift distance is demonstrated
in Fig. 11. The energy increases sharply when the shift distance larger than
$18\mu m$ and reduces gradually when it is bigger than $50\mu$m. The
fluctuation of laser energy when the electric field struggles to grow up at
edge of over-desynchronism is due to the different initial shot noise at each
case. Our result shows that the energy remain constant when the shift distance
is 30$\mu$m larger than the optimum desynchronism. The flat-top profile of the
curve is due the relative narrow high-reflectivity spectral bandwidth of
crystal mirrors which leads to the corresponding broad temporal optical
profile which acts as a long seed to be amplified. Thus the high intensity
output power covers larger desynchronism range in XFELO.

##  V conclusion

In this paper we proposed a novel theoretical model which is useful for fast
optimization of FELO and obtain some results in a shorter period of time. The
model solved the partial differential equation theoretically in order to
obtain the single-pass gain when FELO is approaching saturation. The
oscillator mirrors reflection is considered simply by multiplying the
reflectivity. The gain calculation as a separate part takes a few minutes and
once the gain function is established as a data base, the oscillator cavity
simply calls correspond gain and spends tens of seconds to reaches saturation
and returns the needed results.

We investigated the performance of a 1.6 $\mu$m infrared and a 1
$\mathrm{\AA}$ X-ray FELO by the new approach. The agrement between our
results and those from GENESIS simulation proves that the new model is
feasible and reliable. Taking the advantages of the higher efficiency of the
new model, it is easily adjusted to investigate the cavity desynchronism
quickly. The electrons initial distribution function is assumed to be Gaussian
function, the following work would be to using the truly electron distribution
from accelerator tracking to obtain some practical results and to enhance the
accuracy of XFELO theoretical model as well as taking into account the
influence of laser pulse heating effects on the Bragg crystal mirrors.

###### Acknowledgements.

The authors are grateful to Xiaofan Wang, Han Zhang and Mengying Deng for
helpful discussions and useful comments. This work was partially supported the
National Natural Science Foundation of China (11322550) and Ten Thousand
Talent Program.

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