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Abstract
Modern X-ray free-electron lasers (XFELs) can generate pulses with durations ranging from femtoseconds to attoseconds. The numerical evaluation of ultra-short XFEL pulses through beamline systems is a critical process of beamline system design. However, the bandwidth of such ultra-short XFEL pulses is often non-negligible, and the propagation cannot be simply approximated using the central wavelength, especially in dispersive beamline systems. We developed a numerical model which is called Fourier optics based Ultrashort x-Ray pulse propagatION tool (FURION). This model can not only be used to simulate dispersive beamline systems but also to evaluate non-dispersive beamline systems. The FURION model utilizes Fresnel integral and angular spectrum integral to perform ultra-short XFEL pulse propagation in free space. We also present the method for XFEL pulse propagation through different types of dispersive gratings, which are commonly used in soft X-ray beamline systems. By using FURION, a start-to-end simulation of the FEL-1 beamline system at Shenzhen superconducting soft X-ray free electron laser (S3FEL) is carried out. This model can also be used to evaluate gratings-based spectrometers, beam splitters, pulse compressors, and pulse stretchers. This work provides valuable insights into the start-to-end simulation of X-ray beamline systems.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, X-ray Free Electron Laser (XFEL) facilities have undergone rapid development. At present, several XFEL facilities have been built or are under construction around the world, including LCLS [1], SACLA [2], PAL-XFEL [3], SwissFEL [4], European XFEL [5], and SHINE [6]. These facilities can provide ultra-short XFEL pulses with high brightness and high degree of transverse coherence. In the development and design of X-ray optics and beamline systems for FEL facilities, various types of dispersive X-ray devices will inevitably be employed, such as grating monochromators [7–10], crystal monochromators [11–13], multilayer monochromators [14,15], grating spectrometers [16–18], crystal spectrometers [19], grating beam splitters [20,21], multilayer beam splitters [22], crystal beam splitters [23], grating pulse compressors and stretchers [24], multilayer pulse compressors and stretchers [25,26], and crystal pulse compressors and stretchers [26,27]. In soft X-ray regime, gratings, variable-line-spacing (VLS) gratings, and multilayers are often used, while in hard X-ray regime, crystals and multilayers are usually employed. The aforementioned different types of dispersive devices, significantly increase the complexity of the optical system and introduce dispersion and spatiotemporal coupling effects, posing challenges for numerical simulations.
There is a significant need for simulation tools that can analyze how dispersion and spatiotemporal couplings occur along XFEL beamline systems. In the synchrotron radiation light source and FEL community, several software packages have been developed for numerical simulations of beamline systems, including SHADOW [28], SRW [29,30], XRT [31], HYBRID [32], and MOI [33]. These software tools demonstrate excellent performance in describing the evolution of the transverse beam profile in non-dispersive beamline systems. SRW [30], WPG [34], and OPC [35] also possess the capability to estimate pulse propagation in beamline systems. The WPG package is built on top of the SRW software and can be used to estimate XFEL pulse going through dispersive gratings by using the local stationary-phase approximation [36], and this package has been used to simulate the pulse propagation in SCS SASE2 beamline at European XFEL. The OPC package has been developed to simulate FEL pulse propagation in the development of the XFEL oscillator simulation framework. OPC is based on the expanded form of the Fresnel diffraction integral, combining the ABCD matrix. This method allows the calculation of propagation through a cascaded series paraxial optical system in one step, only requiring the overall ABCD matrix for that system. If there are apertures or a need to consider figure errors of mirrors in the system, this method is similar to the conventional Fresnel integral and requires a step-by-step approach for integration. For pulse propagation, there is a standard method based on Fourier optics. In this standard approach, the incident pulse is transformed from the time domain to the frequency domain. Then the modulation of the incident pulse by X-ray optics or by free space propagator is performed in the frequency domain. Finally, the output pulse is transformed from the frequency domain back to the time domain.
In our previous work, we derived Kostenbauder matrices (K matrices) for various optics in the X-ray beamline [37], and applied this method to propagate ultrashort Gaussian X-ray pulses through dispersive systems. We also established a theoretical model for the spatiotemporal response of concave VLS gratings to ultrashort Gaussian X-ray pulses [38]. Although the two studies were based on Gaussian pulse propagation, they laid the groundwork for our development of a start-to-end simulation of XFEL pulses in dispersive beamline systems. In this paper, we establish a start-to-end simulation tool called Fourier optics-based Ultrashort x-Ray pulse propagatION tool (FURION) which is capable of calculating XFEL pulse propagation through both non-dispersive and dispersive beamline systems. The FURION model is based on the standard approach mentioned above. The paper is organized as follows. In Sec. 2, we briefly introduce the method of pulse propagation in free space. In Sec. 3, we discuss the approach of pulse propagation through mirrors. In Sec. 4, we investigate the method of pulse propagation through different types of gratings. In Sec. 5, we conduct a start-to-end simulation of the FEL-1 beamline from the source point to the Time-Resolved Spectroscopy and Coherent Diffraction Imaging (TR-SCI) endstation at S$^3$FEL using FURION.
2. Pulse propagation through free space
The FURION model is capable of simulating three-dimensional (3-D) Gaussian pulse propagating and 3-D XFEL pulse propagation. For the start-to-end simulation of XFEL beamline systems, 3-D XFEL pulses can be produced using simulation software such as GENESIS 1.3 [39]. In the numerical simulation, the optical field is stored in a 3-D matrix with sampling points of $N_x \times N_y \times N_t$, where $N_x$ and $N_y$ represent the number of sampling in the transverse coordinates ($x$ and $y$), and $N_t$ represents the number of sampling in the longitudinal coordinate ($t$). Based on the standard approach mentioned in Sec. 1, the 3-D optical field after propagating a distance $d$ can be expressed as
3. Pulse propagation through mirrors
In X-ray beamline systems, grazing incidence mirrors are often used. The types of mirrors in the Furion model include planar mirror, cylindrical mirror, spherical mirror, toroidal mirror, elliptical cylindrical mirror, ellipsoidal mirror, parabolic cylindrical mirror, and rotating parabolic mirror. The pulse propagation through mirrors in FURION is achieved by employing a local ray tracing method. This involves tracing rays from the input plane to the output plane. Both the input and output planes are positioned at the center of the mirror, as shown in Fig. 1. The input and output planes are oriented perpendicular to the incident and exit beam axes, respectively. Rays are initially generated on a regular grid at the input plane. These rays follow the direction determined by the local phase gradient of the X-ray field. They intersect with and reflect off the mirror’s surface before reaching the exit plane. During the ray-tracing process, a coordinate transformation is established between the input and output planes, along with the corresponding path length.
3.1 Ideal mirrors
An ideal mirror is characterized by a flawless and undistorted surface. The FURION model employs the local ray-tracing approach to describe the propagation of 3-D XFEL pulses through an ideal mirror. The optical path length from the incident plane to the exit plane is transformed into a phase modulation, and we have
Different types of mirrors yield different phase modulation, and the effects of modulation include reflection, collimation, focusing, and divergence. In practice, mirrors adopted in X-ray beamline systems are non-ideal due to machining errors and deformations which may introduce additional modulation to XFEL pulses, and non-ideal surfaces will degrade the beam quality. In the following, we will focus the discussion on non-ideal mirrors.
3.2 Non-ideal mirrors
For real mirrors, their surfaces are not perfect but instead deviate from ideal surfaces by a certain degree, which is typically described by using figure errors. The FURION model provides two methods to deal with figure errors, including the direct phase projection method (FURION option 1) and the power spectral density method (FURION option 2).
For option 1, the wavefront perturbation induced by figure errors $h(S,M)$ is described by a phase shift term
Furion option 2 deals with figure errors by separating low-frequency components and high-frequency components for processing. Here, the low-frequency components contribute to the effective slope error, while the high-frequency components contribute to the effective height error. In Susini’s work [40], scattering from rough surfaces can be conveniently categorized into three regimes, bounded by the extinction length $L_e$ and the coherence length $L_c$. Here, we use Susini’s approach to estimate the effective height error $h_e(S,M)$ and slope error $s_e(S,M)$.
The option 2 deals with the effective slope error $s_e(S,M)$ by calculating the direct optical path lengths $\Delta P_{s_e}(x,y;x^\prime,y^\prime,\omega )$, and the corresponding phase modulation can be expressed as
The way of option 2 deals with the effective height errors $h_e(s,m)$ is the same with Eq. (5), and the corresponding phase shift can be expressed as
The XFEL pulse after passing through non-ideal mirrors by using option 2 can be expressed as
To validate the method through non-ideal mirrors, we adopt the optical system in Fig. 1. The distances $l$ and $l^\prime$ are 196 m and 98 m, respectively. The grazing incident angle is 12 mrad, and the radius of curvature of the mirror in meridian dimension is 10889.2 m. Here, the figure error profile adopted in the simulation is shown in Fig. 2(a). The coherence length $L_c$ is 0.124 m, and the high-frequency and low-frequency components of the figure error are presented in Fig. 2(b). The normalized intensity beam profile at the image plane estimated by Furion option 1, option 2, and SRW are plotted in Fig. 2(c). the blue curve is corresponding to the beam profile through ideal mirror. The results indicate that Furion makes great agreement with SRW.
Fig. 2. (a) Height error profile used in the simulation. (b) The high-frequency and low-frequency components of the height error. (c) Normalized intensity at the horizontal focus calculated by different methods.
4. Pulse propagation through dispersive X-ray optics
In the design of X-ray optical systems, various types of dispersive elements are used, such as gratings, multilayers, and crystals. Gratings and crystals are commonly employed as monochromators and spectrometers, while multilayers can also serve as monochromators or reflective elements. Additionally, gratings, asymmetry-cut multilayers, and asymmetry-cut crystals can be utilized as pulse expanders and pulse compressors. After going through these dispersive elements, the XFEL pulse undergoes various dispersion phenomena, such as pulse front tilt, pulse front rotation, pulse stretching, and pulse compression. In this work, we provide the method of 3-D XFEL pulses passing through different types of gratings.
Currently, the FURION model includes toroidal grating, spherical grating, cylindrical grating, planar grating, toroidal VLS grating, spherical VLS grating, cylindrical VLS grating, and planar VLS grating. In our recent work [37], we developed a unified grating model, a toroidal VLS grating, to describe the aforementioned gratings. As shown in Fig. 3, the coordinate system for the toroidal VLS grating comprises three directions: the meridian direction $M$, the sagittal direction $S$, and the normal direction $N$. The grooves on the toroidal VLS grating are arranged perpendicular to the meridional direction. The groove density is given by the formula: $n = n_0(1 + b_2M)$, where $n_0$ is the central groove density, and $b_2$ is the VLS parameter. The curvatures in the sagittal and meridian directions are $R_S$ and $R_M$, respectively. The angles between the incident direction, the reflection direction, and the normal are denoted as $\alpha$ and $\beta$, respectively. The coordinate systems of the XFEL pulse and the toroidal VLS grating are independent of each other.
When Ultra-short XFEL pulses pass through the aforementioned unified model of gratings, transverse beam profile magnification (demagnification) and angular dispersion occur in the $y$ dimension, and transverse focusing happens both in the $x$ and $y$ dimensions. Here, we complete the modulation of ultra-short pulses by the toroidal VLS grating in two steps. The first step is to perform the transverse coordinate scaling to achieve the transverse beam size magnification or demagnification which is caused by the asymmetric factor $C_{ff} = \cos \beta /\cos \alpha$. The relationship between the transverse coordinates of XFEL pulses before and after passing through the toroidal VLS grating can be expressed as
The second step is to achieve the phase modulation of 3-D XFEL pulses by the toroidal VLS grating in the $(x^\prime, y^\prime, \omega )$ space, and we have
The second method is based on the mentioned unified grating model developed in our previous work [37], and $\Phi _1$ can be expressed as
Here, we propose the second method to explain the extra phase shift $\Phi _2$ as angular dispersion modulation, and we have
Performing inverse Fourier transform on $E_{G1}(x^\prime,y^\prime,\omega )$, the 3-D XFEL pulse after passing through the VLS grating can be written as
By adjusting the parameters $R_M$, $R_S$, $n_0$, and $b_2$ in Eq. (17), $E_{G_1}(x^\prime,y^\prime,t)$ can be used to describe 3-D XFEL pulses passing through other types of gratings. Here, we have summarized different types of gratings and their corresponding parameters in Table 1.
Table 1. Parameters of different types of gratings. Here, $\checkmark$ denotes the relevant parameters in Eq. (17) remaining unchanged.
5. Applications and validations
In this section, we apply the FURION model to simulate pulse propagation through the FEL-1 beamline from the source point to the TR-SCI endstation at S$^3$FEL. The FEL-1 beamline operates in SASE mode, and the photon energy range is from 400eV $\sim$ 1240eV. This beamline aims to construct three experimental endstations, including TR-SCI, surface ambient-pressure X-ray photoelectron station (AP-XPS), and resonant soft X-ray scattering station (including RIXS and REXS), respectively. As shown in Fig. 4, we present the preliminary optical layout of the TR-SCI FEL-1 beamline.
Fig. 4. The preliminary optical layout of TR-SCI FEL-1 beamline system. The locations of the optics from the source point are marked by dashed arrows.
When the beamline system operates in SASE mode, M4 and M5c are in the optical path, while M3 and the planar VLS grating are removed from the optical path. When the beamline system operates in monochromatic mode, M3 and the planar VLS grating are in the optical path, while M4 and M5c are removed from the optical path. M2c focuses the XFEL pulse in the horizontal dimension and generates a horizontal focus at a distance of 294 m from the source point. The planar VLS grating and M5c focus the XFEL pulse in the vertical dimension and generate a vertical focus at the exit slit. The exit slit is located at 325 m from the source point and selects the monochromatic light from the input XFEL beam that has been diffracted by the gratings. The K-B mirrors focus the input XFEL pulse to the sample point. Table 2 summarizes the main parameters of the optics in the TR-SCI FEL-1 beamline. The central groove density and the VLS parameter are 300 lines/mm and 2.8829$\times 10^{-5}$ mm$^{-1}$, respectively.
Table 2. TR-SCI FEL-1 optics specifications.
In the following simulation, the 3-D XFEL pulse is produced using Genesis 1.3, and pulse propagation through beamline systems is implemented by FURION. In the FEL simulation, the relevant simulation parameters are summarized in Table 3. At the source point, the properties of the 3-D XFEL pulse are shown in Fig. 3. Figure 5(a) represents the transverse-longitudinal $(y,\tau )$ distribution of the XFEL pulse, and we can observe that the SASE pulse has multiple longitudinal modes. Figure 5(b) shows the transverse-spectral $(y,\mathcal {E})$ distribution of the XFEL pulse, and we can find that the SASE pulse has multiple spikes in the frequency (photon energy) domain. Figure 5(c) shows the integrated transverse intensity distribution of the SASE pulse, and it can be seen that the transverse profile of the SASE pulse closely resembles a Gaussian distribution. Figure 5 (d), (e), and (f) present the transverse distributions of the XFEL pulse at different photon energies. The simulation results illustrate that the transverse distribution of frequency components in the spectrum is not uniform. Specifically, not all frequency components exhibit their highest spectral intensity along the central axis. Therefore, the central wavelength of the optical field can not be used for the propagation. The FURION model offers the capability to calculate pulse propagation considering different frequency components.
Table 3. The simulation parameters of Genesis.
Now, we present the numerical simulation of FEL pulses through the FEL-1 beamline to the TR-SCI endstation in the monochromator operation mode. As indicated in Fig. 4, we can observe that the dispersion of XFEL pulses after passing through the VLS grating occurs in the $(y,\tau )$ space, while no dispersion occurs in the $(x,\tau )$ space. As predicted in our previous theoretical work [37,38], when an XFEL pulse passes through the VLS grating, due to the angular dispersion and focusing effects of the VLS grating, the XFEL pulse in the $(y,\tau )$ space exhibits pulse front tilt. Moreover, the tilt angle of the pulse varies with the propagation distance (pulse front rotation). As the XFEL pulse reaches the exit slit, the pulse front tilt disappears, and components of the optical field with different photon energies are arranged in the $y$ direction. At this point, the optical field distribution in the $y$ direction is essentially the spectrum of the XFEL pulse. Therefore, the exit slit can selectively filter out monochromatic light. In Fig. 6, we present the process of the evolution of XFEL pulses during propagating in the beamline system. Subplots Fig. 6 (a), (b), (c), and (d) respectively represent the 3-D distributions of the XFEL pulse propagating through the VLS grating to the horizontal focus, before the exit slit, 20m after the exit slit, and the sample point. Subplots Fig. 6 (e), (f), (g), and (h) respectively depict the projected intensity distribution of the XFEL pulse in the $(y,\tau )$ space at the four positions mentioned above. Subplots Fig. 6 (i), (j), (k), and (l) represent the projected intensity distribution of the XFEL pulse in the $(x,\tau )$ space at the four positions. Subplots Fig. 6 (m), (n), (o), and (p) respectively show the projected intensity distribution of the XFEL pulse in the $(x,y)$ space at the four positions mentioned above. The simulation results indicate that the pulse front tilt angle changes with distance from the VLS grating to the exit slit. The pulse front tilt at the horizontal focus is shown in Fig. 6(e). At the exit slit, the pulse front tilt disappears and is manifested in the $y$-direction as the spectrum of the XFEL pulse, as shown in Fig. 6(f). After passing through the slit, the XFEL pulse is monochromatized, but the pulse rotation effect still exists. As shown in Fig. 6(g), at a distance of 20m downstream of the exit slit, we can observe a pronounced pulse front tilt and a significant diffraction effect in the $y$-direction. It should be noted that the slit does not eliminate the pulse front tilt effect, and it only disappears at the focal point. The monochromatic XFEL pulse is focused to the micrometer scale by the KB mirror, as shown in Fig. 6(p). According to the longitudinal projection intensity distribution of the XFEL pulses shown in Fig. 6, we can find that the pulse duration is broadened to approximately 80 fs after passing through the VLS grating monochromator. As demonstrated in this example, the FURION model is capable of performing three-dimensional simulation evaluations of dispersion beamline systems. This capability provides a more quantitative and intuitive assessment for the design of XFEL beamlines. Moreover, the FURION model can be utilized to construct a start-to-end simulation framework for XFEL experiments. Of course, this requires integration with specific experimental computation modules.
Fig. 6. Snapshots of a 3-D FEL pulse propagation in the TR-SCI FEL-1 beamline system (monochromator operation mode) at different locations.
6. Summary
The FURION model is a simulation tool used to assess the propagation of partially coherent XFEL pulses. This model is applicable for both evaluating XFEL pulses passing through non-dispersive beamlines and assessing the propagation of XFEL pulses through dispersive beamlines. In this paper, we first introduced the propagation of XFEL pulses in free space. We also discussed the methods for XFEL pulses passing through ideal and non-ideal mirrors. Furthermore, we conducted a study on XFEL pulses passing through different types of gratings, such as toroidal grating, spherical grating, cylindrical grating, planar grating, toroidal VLS grating, spherical VLS grating, cylindrical VLS grating, and planar VLS grating. Finally, we applied the FURION model to perform a 3-D start-to-end simulation of XFEL pulse propagation in the monochromator mode of TR-SCI FEL-1 beamline at S$^3$FEL. In this simulation, we provided the evolution of the 3-D XFEL pulse in this dispersive system, including pulse stretching, pulse front tilt, and pulse front rotation after passing through the VLS grating. We also simulated the propagation and focusing of 3-D XFEL pulses after narrowband monochromatization and the diffraction effect generated by the exit slit is observed. The FURION model can also be utilized for start-to-end simulations of other dispersive soft X-ray beamline systems, such as grating-based spectrometers, beam splitters, pulse compressors, and pulse stretchers. The FURION model can also be employed as part of the multiphysics simulation framework for conducting source-to-experiment simulations of coherent diffraction imaging experiments. The user interface for FURION is currently under development, and we will update the software development status on the FURION homepage [41].
Funding
National Natural Science Foundation of China (12005135, 22288201); Shenzhen Science and Technology Program (JCYJ20230807112409019); National Key Research and Development Program of China (2018YFE0203000); Scientific Instrument Developing Project of the Chinese Academy of Sciences (GJJSTD20190002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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