research papers
Online monitoring of the spatial properties of hard Xray freeelectron lasers based on a grating splitter
^{a}Institute of Shanghai Applied Physics, Chinese Academy of Sciences, Shanghai 201204, People's Republic of China, ^{b}University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China, and ^{c}Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China
^{*}Correspondence email: wangyuzhu@sinap.ac.cn, bianfenggang@sinap.ac.cn, wangjie@sinap.ac.cn
Xray freeelectron lasers (XFELs) play an increasingly important role in addressing the new scientific challenges relating to their high etc., and thus online monitoring and diagnosis of a single pulse are required for many XFEL experiments. Here a new method is presented, based on a grating splitter and bendingcrystal analyser, for singlepulse online monitoring of the spatial characteristics including the intensity profile, coherence and wavefront, which was suggested and applied experimentally to the temporal diagnosis of an XFEL single pulse. This simulation testifies that the intensity distribution, coherence and wavefront of the firstorder diffracted beam of a grating preserve the properties of the incident beam, by using the coherent mode decomposition of the Gaussian–Schell model and Fourier optics. Indicatively, the firstorder diffraction of appropriate gratings can be used as an alternative for online monitoring of the spatial properties of a single pulse without any characteristic deformation of the principal diffracted beam. However, an interesting simulation result suggests that the surface roughness of gratings will degrade the spatial characteristics in the case of a partially coherent incident beam. So, there exists a suitable roughness value for nondestructive monitoring of the spatial properties of the downstream beam, which depends on the specific optical path. Here, experiments based on synchrotron radiation Xrays are carried out in order to verify this method in principle. The experimental results are consistent with the theoretical calculations.
high coherence and femtosecond time structure. As a result of pulsebypulse fluctuations, the pulses of an XFEL beam may demonstrate subtle differences in intensity, energy spectrum, coherence, wavefront,Keywords: XFEL; Xray optics; grating diffraction.
1. Introduction
Hard Xray freeelectron lasers (XFELs) (McNeil & Thompson, 2010), like the LCLS (Bostedt et al., 2016), SACLA (Huang & Lindau, 2012) and the European XFEL (Tschentscher et al., 2017), enabled by developments in electron accelerator technology, generate nearly full spatial and temporal coherent and 1 fs ultrafast Xray pulses which is many orders of magnitude brighter than the brightest synchrotron source. The unique properties offered by the hard XFEL sources have led to a variety of groundbreaking, innovative experimental techniques, such as coherent diffraction imaging and serial femtosecond crystallography (Abbamonte et al., 2015). These experiments do not only rely on the unique time structure and the peak power of the Xray pulses but also on the high degree of coherence and the clean well defined wavefront (Barty et al., 2009; Pardini et al., 2017; Rutishauser et al., 2012). However, the XFEL pulses produced from selfamplified (SASE) or selfseeded operation exhibit intrinsic pulsebypulse fluctuations in temporal properties (pulse energy, spectral content) and spatial profile (intensity, coherence and wavefront, degree of polarization) (Abbamonte et al., 2015). Furthermore, potential performance degradations associated with deformation and vibration introduced by huge transient thermal loads on beamintercepting components also exist (Tschentscher et al., 2017). Since all these fluctuations can affect measurements performed at XFELs, both in situ diagnostics tools and coherencepreserving optical elements have been developed to monitor and to counteract these beam degradations.
Diagnostics devices for machine tuning can often be destructive to the beam whereas diagnostics for experimental operation must (nearly) transparently characterize the FEL pulses. The LCLS singleshot beam profile measurement based on Xray scintillation and optical imaging (albeit destructive) is capable of revealing imperfections of any upstream Xray optics such as mirrors and monochromators. The beam centroid position can be monitored nondestructively for hard Xrays at the LCLS using quadrant detection from Compton scattering (Feng et al., 2011). Coherence properties of individual femtosecond pulses of an XFEL beam have been typically measured using Young's experiment in `diffract and destroy' mode (Vartanyants et al., 2011) and the diffraction method based on Hanbury Brown and Twiss interferometry (Gorobtsov et al., 2018). Wavefront measurements have been achieved through traditional Hartmanntype sensors (Bernhard et al., 2010), grating interferometry (Rutishauser et al., 2012; Kayser et al., 2014, 2017) and speckle tracking (Berujon et al., 2017) or by measuring coherent scattering from well characterized nanoparticles (Loh et al., 2013), although these techniques are destructive, thus providing only typical or `average' information on the beam wavefront. As presented by Makita et al. (2015), an online spectrometer for hard XFELs has been developed based on a nanostructured diamond diffraction grating and a bent crystal analyzer. A beamsplitter grating was placed in the direction of the Xray beam to divert a small portion of the XFEL pulse onto a bendingcrystal spectrometer and to transmit the rest of the pulse to be used for experimental purposes. It provides high spectral resolution, interferes negligibly with the XFEL beam, and can withstand intense hard Xray pulses at high repetition rates of >100 Hz.
In this paper, apart from the configuration of the online spectrometer, we present the analysis feasibilities of online monitoring of spatial properties of hard XFEL beams based on the grating splitting method. On the basis of theoretical simulation and experimental demonstration, the equivalency of the spatial properties of the zerothorder and the firstorder grating diffracted beams were proven; therefore monitoring the firstorder beam is equal to monitoring the transmitted zerothorder direct beam. The gratingbased method has advantages of high Xray beam transmission and radiation hardness, but in our beamline optical setup it may introduce beam degradation when the grating RMS roughness is >200 nm. Using this nondestructive online monitoring method can provide most of the spatial properties of the hard XFEL beam, including beam profile, position, intensity, coherence and wavefront distribution.
2. Methods
The highly coherent XFEL sources can be described by a finite number of transverse modes using the coherent mode decomposition (CMD) of the Gaussian Schell Model (GSM) (Vartanyants et al., 2010; Vartanyants & Singer, 2010; Singer & Vartanyants, 2014; Hua et al., 2013). For the propagation of the XFEL beam, we used a decomposition of the statistical fields into a sum of independently propagating transverse modes for the analysis of the beam properties of these fields at different distances from the source. The limited number of contributing modes significantly simplified the numerical calculations by reducing the number of variables.
A GSM beam is a particular type of partially coherent wavefield which is usually used to describe the XFEL radiation coherence properties as well as intensity distributions. The cross spectral density (CSD) of a planar GSM source is described by (Mandel & Wolf, 1995)
where
Here, I(r) describes the intensity distribution at the points r_{1} or r_{2} in the survey plane which is perpendicular to the z direction of the beam propagation. The degree of spatial coherence μ_{s} depends only on the difference of spatial coordinates r_{1} and r_{2}. I_{0x} and I_{0y} are the positive constants representing the maximum intensity in the respective directions that are set to 1 in this paper. The parameters σ_{Sx} and σ_{Sy} are the rootmeansquared source size in the x and y directions, respectively, and ξ_{Sx} and ξ_{Sy} give the coherence lengths of the source.
The CSD of a partially coherent, statistically stationary field of any state of coherence can be decomposed into the sum of independent coherent modes under very general conditions,
where E_{mn}(r) are eigenfunctions which describe the electric field of Xrays; they are known as the coherent modes and are orthogonal, and they are described by the Gaussian Hermite modes. β_{mn} are the eigenvalues that describe the occupancy in each mode. Accordingly, the modes E_{mn} and their corresponding eigenvalues β_{mn} can be found for the x and y directions, respectively. To simplify the simulation, few numbers of the Gaussian Hermite modes need to participate in the simulation process. The criteria for selecting the coherent mode is that the eigenvalue β_{mn} of the (m,n) mode is larger than 0.1% of the eigenvalue β_{00} of the fundamental mode. Our simulation demonstrated that the mode selection has little effect on the accuracy of the calculation results and has a greater contribution for improving the computational efficiency. That is, the more coherent the beam is, and the fewer coherent modes that are needed, so the CMD method of the GSM works better, especially for the highcoherent XFEL radiation.
Then, the propagation of the field from the source through free space to the observation plane for each mode can be calculated by utilizing the Huygens–Fresnel principle,
and the coordinate r is taken in the z_{0} plane of the source and the coordinate u is taken in the z_{1} observation plane. The integration is made in the source plane. The propagator P_{z} describes the propagation of radiation in free space. It is defined as
When the hard Xray beam passes through an optical element such as a grating splitter, the transmitted modes are given by E_{out} = TE_{in}, where T characterizes the complexvalued amplitude transmittance function of the optical element. After propagation, the CMD representing the beam properties in the plane of observation is determined by equation (3). Then, the intensity distribution I, coherence property μ and wavefront φ at the observation plane (z = z_{1} − z_{0}) can be obtained by
Consider the binary transmission grating splitter (Paganin, 2006) sketched in Fig. 1(a). The maximum and minimum projected thickness of the grating are taken to be A and B, respectively, with the grating period being equal to L. For simplicity, the widths of the grooves are equally distributed between the grooves (L/2). Adopting (x,y) Cartesian coordinates in the plane of the grating, the thickness function T(x,y) could be written as the projected thickness,
Assume that the grating is made of a single homogeneous isotropic nonmagnetic material with known complex δ_{ω} and β_{ω} quantify the refractive and absorptive properties of the material, respectively, as a function of angular frequency ω of the Xrays.
= . The real numbersWhen a zdirected (normal optic axis perpendicular to the x, y plane) monochromatic partialcoherent hard XFEL beam is incident upon a ruled grating splitter, a series of diffracted orders appears. Here, k = 2π/λ is the wavenumber associated with monochromatic scalar radiation of wavelength λ. Assuming the grating to be a sufficiently thin optical element and the grating period L ≫ λ, and the projection approximation to be valid, then, for z = 0 (i.e. the exit surface of the grating), the exit beam can be described as
To propagate the wavefield E_{out,ω} into the vacuumfilled half space z ≥ 0, equations (4) and (5) should be used for calculating the wavefield downstream of the grating. Then, the spatial properties of the partialcoherent beam can be obtained [equation (3)] and used to calculate the intensity, coherence and wavefront profile at the observation plane.
In light of the preceding analysis, it is clear that the diffraction grating can function as a beam splitter, as indicated in Fig. 1(b). Passing through the grating, a normally incident monochromatic beam is split into the various diffracted orders.
In addition to functioning as a beam splitter, equation (8) implies that the transmission grating can also function as a spectrometer. The grating dispersion effect is dependent on the grating period, wavelength and bandwidth. The CVD diamond grating period of 200 nm used by Makita et al. (2015) is much larger than the wavelength, 1.24 Å, so the grating dispersion effect is almost negligible when comparing with the crystal spectrometer and will not be discussed here.
3. Numerical analysis
The GSM method was used to simulate the hard XFEL source SHINE [Shanghai HIgh repetitioN rate XFEL and Extreme light facility (under construction)] (Zhu et al., 2017) whose parameters are specified in Table 1. The propagation of this hard XFEL beamline is sketched in Fig. 2(a). The hard XFEL radiation propagated through 100 m of free space, and was focused by a pair of ideal elliptical cylinder KB mirrors which can be considered as an ideal thin lens. For the sake of simulation and illustration, a beamsplitter grating was placed on the focus plane to diffract the beam; the divergence and wavefront of the incident beam and the position of the grating will not change the physical effects of the beam splitting. The downstream observation plane was placed 10 m downstream from the grating and was used to observe the intensity, coherence and wavefront distributions of the split diffraction beams. In this simulation, the selection of the source parameters and beamline layout were considered for typical hard XFEL beamlines and for computational efficiency, and all calculation data based on these will not influence the simulation results.

The beam size and coherence in the beamline expanded and contracted when the beam propagated and focused, and correspondingly the curvature of the wavefront also diverged and converged. The analytical calculation (AC) (Singer & Vartanyants, 2014) method based on the GSM can describe properties of the focused partially coherent Xray beam, as shown in Table 1. The AC method is based on the results of statistical optics and gives the beam size, wavefront and transverse coherence length at any distance behind an optical element. However, when the optical element is nonideal (with aberration, slope error and roughness) or unconventional (such as the grating splitter), the AC method is no longer relevant. Therefore, a twodimensional numerical simulation (NS) method should be developed for calculating the propagation of a hard XFEL beam for realistic and unusual use.
Based on optic imaging formulae, it should be noted that the focus plane was placed 100 m downstream of the ideal thin lens whose focal length was 50 m. Whereas, according to AC and NS results, the focal plane was moved forward to 93.59 m due to the high coherent beam and diffraction effects (Singer & Vartanyants, 2014).
3.1. Twodimensional partialcoherent XFEL beam propagation
The twodimensional NS method was developed based on the CMD method of the GSM and Fourier optics. According to the CMD method of the GSM, only ten coherent modes need to be used in the simulation of a partialcoherent hard XFEL source, as shown in Table 1, and their corresponding eigenvalues were found to be larger than 0.01% of β_{00} as shown in Fig. 2(b).
With this NS method, we can simulate the intensity, coherence and wavefront distribution at different places in the beamline, and the optical aberration and distortion can be introduced in arbitrary form. Based on this, we simulated the beam propagation through free space and through focusing lenses as shown in Fig. 3. Comparisons between the NS and the AC method are also specified in Table 1 and Fig. 3, which support the high consistency between the data and the two methods. When only a limited number of ten coherent modes were put into the simulation, a large distortion was found at large Δx,Δy in the coherence profile due to omitted higherorder coherent modes. So, we introduced a window truncation in the coherence and wavefront calculation, the truncation criteria being that the normalized intensity should be smaller than 1 × 10^{−4}, which has a small impact on the calculated results.
At the KB mirrors plane, the divergent wavefront was focused into a convergent wavefront, while the intensity and coherent profile behaved consistently across the theoretical ideal thin lenses. Very little wavefront distortion was found at the focal plane due to the nearfield Fresnel diffraction at the KB mirrors' focal plane.
As shown in Fig. 2(a), a CVD diamond black–white laminar grating splitter with a period of 4 µm and thickness of 4 µm was placed in the KB focal plane to diffract the hard XFEL beam. By employing the NS method, the beam profile diffracted by the grating was calculated and is shown in Fig. 4(a), while the transmitted zeroth and diffracted firstorder beam spatial properties [Figs. 4(c), 4(d)] were compared with the transmitted direct beam without the grating [Fig. 4(b)]. By using the numerical fitting method, the values of the intensity, coherence and wavefront distribution were kept consistent with each other between the zeroth and the firstorder beam as shown in Table 1. The effectiveness of this NS simulation was verified by the AC method. Here, the grating diffraction angle was taken into consideration when calculating the firstorder beam's wavefront. Also, the small oscillating parts of the coherence and wavefront profile in Figs. 4(d2), 4(d3) and 4(d4) may be caused by the diffraction angle.
3.2. Beam distortion
To verify the equivalence of the zeroth and the firstorder diffraction beam, amplitude and phase distortion should be introduced to the beam diffraction. As shown in Fig. 5(a), a normalized continuous bandlimit function was introduced just before the grating and worked as amplitude and phase distortions. Then, the diffracted hard XFEL beam was obtained using the NS method, and the wavy nature of both the amplitude and phase caused obvious distortions in the intensity distribution [see Figs. 5(b) and 5(c)]. Complete consistency was shown between the zeroth and the firstorder beam.
3.3. Grating roughness
While amplitude and phase distortions can introduce aberrations in the hard XFEL beam, the grating roughness cannot be omitted due to its direct participation in the optical process (TorcalMilla & SanchezBrea, 2011). As demonstrated by Makita et al. (2017) and David et al. (2011), micrometresized phaseshifter gratings contain a rough bottom and smooth top which is caused by the inductively coupled plasma (ICP) assisted reactive ion etching process. The roughness of the bottom of a grating can be represented as a Gaussian random distribution; the RMS roughness of the grating bottom surface is about 100 nm and the correlation lengths are about 500 nm in both twodimensional orthogonal directions.
To build a similar grating roughness, a twodimensional random surface height profile was constructed based on the Monte Carlo method,
where ϕ is a random phase map in normal distribution and satisfies , and f is a coordinate in the spatial frequency domain. The F{…} and F^{−1}{…} symbols in (9) represent the fast Fourier transform (FFT) and inverse FFT routines, respectively. The power spectral density (PSD) of the grating height describes the spectral content of the rough surface in the space frequency domain and L is the surface width. For a Gaussian random distribution, the PSD can be described as , where δ is the RMS value of the surface roughness and l is the correlation length.
The simulated rough grating is shown in Fig. 6, where the spatial correlation length and roughness were set to 500 nm and 100 nm, respectively. To investigate the influence of the degree of roughness on the transmitted and diffracted beams, different roughness values (δ = 400 nm, 200 nm 100 nm) were calculated by comparing the spatial properties between the zeroth and firstorder beams (see Fig. 7). Simulation results demonstrated that small distortions can be found in the coherence and wavefront profiles when RMS roughness δ > 200 nm. As a result, the RMS roughness of this CVD diamond grating splitter used in this beamline layout should be smaller than 200 nm. According to the optical path, there exists a suitable roughness value. When the roughness is lower than this value, the influence on the spatial characteristics can be basically ignored; the roughness and other requirements of the CVD diamond grating and the rest of the optical elements can also be calculated using the NS method.
4. Experiment
The demonstrated experiment was performed on beamline BL19U2, an undulator beamline dedicated to biological smallangle Xray scattering at the Shanghai Synchrotron Radiation Facility (SSRF), China (Li et al., 2016). A monochromatic beam of 12 keV was provided by a 1.6 m U20 undulator and a Si(111) doublecrystal monochromator (DCM). Downstream horizontal (at 31.2 m) and vertical (at 34 m) mirrors focused the Xray beam onto the detector plane. A secondary source slit at 41 m was used to cut the beam. The spatial coherence properties of the Xray beam have been measured using a combined method with a pinhole and a grating, and the coherence length was 3.44 µm × 4.76 µm (horizontal × vertical) with a square secondary source opening of 100 µm × 100 µm (Hua et al., 2017). Here a 5 µm pinhole (Zeiss) drilled into a piece of platinum–iridium alloy sheet was used to select the coherent beam which was located 50 m downstream from the source, and a JJ Xray scatteringfree slit, placed 1 m further downstream, was used to block the parasitic scattering. The beamsplitter grating followed closely behind. The experiment used a onedimensional transmission grating with a period of 282 nm, a line width of 141 nm (see Fig. 8) and an active area of about 120 µm × 120 µm. The grating was fabricated on a lowstress 100 nm Si_{3}N_{4} membrane using lithography (Crestec CABL9500C). An 80 nmthick Au layer was deposited by electronbeam evaporation after lithography. A Hamamatsu ORCAFlash 4.0 LT C1144042U sCMOS camera (2048 × 2048 pixels with an effective pixel size of 6.5 µm × 6.5 µm) was placed 3.44 m downstream from the grating to allow for a sufficiently large Δy of 1.23 mm, and to separate the diffraction peaks at the camera plane. To obtain lowbackground data, an evacuated flight tube (sealed by kapton windows) was placed between the grating and the detector to reduce the additional signal due to air scattering. A 200 µmdiameter tungsten bar was used as a beamstop to protect the camera from being damaged and to allow a longer exposure time.
At 12 keV photon energy, the direct beam was almost identical with and without the grating for high transmission and low diffraction, as shown in Figs. 9(a1) and 9(a2). To calculate the diffraction efficiency, the diffracted ±firstorder beam [see Figs. 9(c1) and 9(c2)] contained about 2.17 × 10^{−8} of the incident beam intensity [see Fig. 9(a1)]. Higher diffraction orders are negligible, resulting in nearly 100% transmission of the zerothorder diffraction.
To verify the equivalence of the zeroth and the ±first diffracted beam, an emery paper with fine particles was used to introduce random phase. As shown in Fig. 10, strongly distorted speckle was obtained with coherent incident beam, but the diffracted ±firstorder beams showed nearly the same profile as the transmitted zerothorder beam. The visible difference of the firstorder diffraction beam may stem from the large scattering range of the zerothorder beam. Furthermore, the clean speckle pattern with good contrast also proved the high coherence of the incident beam.
5. Conclusion
In summary, we have developed a twodimensional NS method to calculate the propagation of a highly coherent hard XFEL beam. Compared with the developed AC method, the NS method has almost the same computational accuracy, and it could introduce nonideal and unconventional optical elements into the beamline. Using this NS method, we calculated the performance of a grating splitter used on a hard XFEL beamline, and verified the equivalence of the spatial properties of the zeroth and firstorder diffraction beam. All the simulation results give a theoretical basis for nondestructive online monitoring of a hard XFEL beam's properties, including intensity distribution, coherence and wavefront profile. The demonstrated experiments have also been performed on a SSRF undulator beamline to validate its effectiveness and its versatility, while the intensity distribution, coherence and wavefront profile of the hard FEL Xray beam were monitored with high transmission, and there is negligible interference with the diffracted beam for analysis.
Acknowledgements
The authors would like to thank all colleagues at beamlines BL19U2 and BL13W1 at the SSRF. The grating was manufactured by the XIL Group of the SSRF.
Funding information
The following funding is acknowledged: National Natural Science Foundation of China (Nos. 11675253, 11505278, U1732123, U1432115).
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