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Abstract
In soft X-ray free-electron laser (FEL) beamlines, variable-line-spacing (VLS) gratings are often used as dispersive components of monochromators and spectrometers due to their combined dispersion and focusing properties. X-ray FEL pulses passing through the VLS grating can result in not only transverse focusing but also spatiotemporal coupling effects, such as pulse front tilt, pulse front rotation, and pulse stretching. In this paper, we present a theoretical study of the spatiotemporal response of concave VLS gratings to ultra-short X-ray pulses. The theoretical analysis indicates that the tilt angle of the non-zero diffraction orders varies with the propagation distance, and disappears at the focus, where the focal lengths and pulse stretching differ for different diffraction orders. The model demonstrates the pulse duration after the concave VLS grating is the convolution of the initial pulse duration and the stretching term induced by dispersion, while the beam size at the focus in x dimension is the convolution of the geometric scaling beam size and the dispersion term. This work provides a mathematical explanation for the spatiotemporal response of concave VLS grating to ultra-short X-ray pulses and offers valuable insights into the design of FEL grating monochromators, spectrometers, pulse compressors, and pulse stretchers.
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1. Introduction
In the past few decades, with the development of synchrotron radiation and free electron laser (FEL) facilities, X-ray monochromators and spectrometers have undergone rapid development. In extreme ultraviolet and soft X-ray regimes, variable-line-spacing (VLS) gratings are often used as high-resolution monochromators and spectrometers. The practical realization of the VLS principle was first achieved by Harada and Kita [1]. Hettrick and Bowyer brought attention to the advantageous characteristics of spectrograph designs employing a plane VLS grating [2]. Hettrick and Underwood expounded upon this principle, elucidating that the addition of a perpendicular mirror to mitigate astigmatism could enable the construction of scanning spectrometers and monochromators [3]. At present, VLS grating spectrometers [4–10] and VLS grating monochromators [11–16] have achieved sufficient development and have been widely used at synchrotron radiation sources and plasma diagnostics. VLS grating monochromators have also been adopted on X-ray FEL beamlines [17–22]. For synchrotron radiation facilities, the pulse duration is typically in the picosecond range, and in the design of VLS grating monochromators on beamlines, scientists often pursue high photon energy resolution. For FEL facilities, the pulse duration can reach the femtosecond range, and in the design of VLS grating monochromators, it is necessary to consider not only high photon energy resolution but also small pulse stretching. In other words, FEL beamlines with VLS grating monochromators belong to dispersive beamlines, and in the design process, we need to consider the spatiotemporal coupling effects caused by dispersion. It should be pointed out that the spatiotemporal response of ultra-short pulses passing through a VLS grating includes pulse front tilt, pulse front rotation, pulse stretching, and transverse focusing, while ultra-short pulses passing through a fixed line density grating only result in pulse stretching and a fixed tilt angle which has already been extensively investigated in references [23–27]. It requires a theoretical model to describe the spatiotemporal response of VLS gratings to ultra-short pulses.
Several software packages can be used to estimate X-ray beamline systems, such as Shadow [28], SRW [29,30], HYBRID [31], xrt [32], and MOI [33]. These packages can excellently describe the transverse distribution of X-ray pulses after going through non-dispersive beamlines. SRW [30] and WPG [34] also possess the capability to simulate dispersive beamline systems. Recently, we developed a method [35] based on 6-dimensional matrix which can be used to simulate non-dispersive and dispersive beamline systems. Although we can use these numerical simulation tools [30,34,35] to calculate the spatiotemporal coupling effects of ultrashort pulses passing through concave VLS gratings, there is still a lack of analytical models to describe the spatiotemporal coupling effects, including pulse front tilt, pulse front rotation, pulse stretching, and transverse focusing. Therefore, it is of great importance to establish an analytical model to explain the spatiotemporal response of concave VLS gratings to ultra-short X-ray pulses.
In order to help scientists gain a more quantitative and intuitive understanding of this physical process, we conducted a theoretical study on the spatiotemporal response of concave VLS gratings to ultra-short X-ray pulses in this paper. Our analytical model indicates that ultrashort pulses undergo pulse front tilt after passing through VLS gratings, and the tilt angle changes with the propagation distance which is called pulse front rotation. The model also reveals the pulse duration after the concave VLS grating is the convolution of the initial pulse duration and the stretching term caused by dispersion, and the transverse beam size in $x$ dimension is the convolution of the geometric scaling beam size and the dispersion term. This model also can be used to estimate the photon energy resolution of concave VLS grating monochromators and spectrometers. This paper is organized as follows. In Sec. 2., we define the coordinate system and notation conversion. Then, in Sec. 3., we investigate the spatiotemporal response of the VLS grating to a $\delta$-pulse with a Gaussian distribution in the transverse dimensions, as well as the propagation characteristics of the $\delta$-pulse after passing through the concave VLS grating. In Sec. 4., we study the spatiotemporal response of the concave VLS grating to ultra-short pulses with Gaussian distributions both in transverse dimensions and time dimension. We also explore the propagation characteristics of ultra-short Gaussian pulses after passing through the concave VLS grating. In Sec. 5., we compare and validate the theoretical model, and apply this model to estimate photon energy resolution of concave VLS gratings. We also discuss the propagation properties of different diffraction orders. This analytical model can be reduced to describe the spatiotemporal response functions of toroidal VLS gratings, cylindrical VLS gratings, planar VLS gratings, toroidal gratings, cylindrical gratings, planar gratings, toroidal mirrors, cylindrical mirrors, and planar mirrors. This study can be used to evaluate the propagation characteristics of ultra-short X-ray pulses after passing through concave VLS monochromators and spectrometers, which is of great help for the design of VLS monochromators, spectrometers, pulse compressors, and pulse stretchers.
2. Definition of the coordinate system
In this section, we define the coordinate system. Here, we follow the definition of our recent work [35] which investigated ultra-short pulse propagation by using Kostenbauder matrices (K matrices) [27,36,37]. The coordinate systems of the X-ray beam and the concave VLS grating are independent, as shown in Fig. 1. The X-ray pulse is described in the six-dimensional phase space which is spanned by vector $\textbf {V} = (x, \theta _x, y, \theta _y , t , \nu )^T$. The reference point is at the origin, and its path is the optical axis. Here, $x, \theta _x, y, \theta _y , t,$ and $\nu$ are the deviation from the reference point. $x$ and $y$ are the spatial coordinates to describe the lateral position, and $t$ is the temporal coordinate. $\theta _x$ and $\theta _y$ describe the divergence angles relative to the optical axis, and $v$ represents the frequency deviation relative to the central frequency $f_0$, and $\nu = f - f_0 = (\omega - \omega _0)/2\pi$. The concave VLS grating’s coordinate system consists of the meridian direction $M$, sagittal direction $S$, and normal direction $N$ of the optical element surface. The grooves of VLS grating are perpendicular to the meridian direction $M$, and the groove density is $n = n_0(1 + b_2M)$, where $n_0$ is the central groove density, and $b_2$ is the VLS parameter. The K matrix of a concave VLS grating in rightward orientation, as shown in Fig. 1, is given by
Fig. 1. Schematic illustration of the spatiotemporal response of concave VLS grating to ultra-short x-ray pulses.
In the following sections, we investigate the spatiotemporal response of concave VLS grating to ultra-short X-ray pulses in $(x,y,t)$ space in which the spatiotemporal couplings occur after going through the concave VLS grating.
3. Spatiotemporal response to a $\delta$ pulse
In order to investigate the spatiotemporal response of concave VLS gratings to ultra-short pulses, it is imperative to conduct a preliminary exploration of the system’s response to a $\delta$ pulse, thereby acquiring the response function of the entire system. As shown in Fig. 1, the system consists of three parts: (1) The free space from the source to the concave VLS grating. (2) The concave VLS grating. (3) The free space after the concave VLS grating.
3.1 From the source to the concave VLS grating
In the derivation process, the paraxial approximation assumption is followed. Here, we adopt the $\delta$ pulse with a Gaussian distribution in the transverse dimension, and the equation describing the pulse at the source point is given by
The expression of the $\delta$ pulse in Eq. (3) can be equivalently represented in Fourier space by the following formula
3.2 Properties after the concave VLS grating
When an ultra-short pulse propagates through a concave VLS grating, the duration of its non-zero diffraction orders is stretched, while the transverse beam size undergoes compression or broadening. Here, we will use K-matrix approach to derive the spatiotemporal response of VLS gratings to a $\delta$ pulse. According to the $\mathbf {K}$-matrix method, after going through the concave VLS grating, the output vector $\mathbf {V}_{\text {out}}$ can be expressed as
Then, we have
3.3 Propagation properties after the concave VLS grating
In this subsection, we investigate the propagation properties of the $\delta$ pulse in the free space after the concave VLS grating. Here, we use the model of the Fresnel propagator, and the Fourier transform of the response function for free-space propagation can be expressed as
where4. Spatiotemporal response to a Gaussian pulse
So far, we have obtained the response function of the concave VLS grating to the $\delta$ pulse, as presented in Eqs. (15). In this section, we first investigate the spatiotemporal response of the concave VLS grating to an ultra-short pulse with Gaussian distribution in both the temporal and spatial dimensions in Sec. 4.1. Then, we study the propagation properties of the Gaussian pulse after the concave VLS grating in Sec. 4.2. The propagation properties in $x$ and $y$ dimensions are discussed separately.
4.1 Properties after the concave VLS grating
For an ultra-short pulse with Gaussian distribution in both the temporal and spatial dimensions, its expression after going through a VLS grating is the convolution of a Gaussian pulse in the temporal dimension with the response function in Eq. (10), and we have
where the Gaussian pulse is expressed asSubstituting Eq. (10) and Eq. (18) into Eq. (17) yields
Equation (19) describes the spatiotemporal response of a Gaussian pulse going through a VLS grating. In the $y$ dimension, there is no dispersion, and the pulse is focused by the concave VLS grating, as shown in Eq. (20b). In the dispersion space $(x,t)$, the response function is presented in Eq. (20a). The first term represents the pulse front tilt, which is caused by angular dispersion. The second term represents lateral focusing, which refers to the VLS parameter. The third term represents lateral size broadening, which is caused by the asymmetric configuration. Here, the tilt angle is given by
4.2 Propagation properties of ultra-short pulse after the concave VLS grating
We have obtained the spatiotemporal response of the concave VLS grating to an ultra-short Gaussian pulse. In this subsection, we study the free space propagation characteristics of ultra-short pulses after passing through the VLS grating. The propagation properties in the $x$ dimension are investigated in Sec. 4.2.1, including pulse stretching, pulse front tilt, pulse front rotation, and transverse focusing. The propagation characteristics in the $y$ dimension are investigated in Sec. 4.2.2.
4.2.1 Propagation in the $x$ dimension
In the $x$ dimension, the distribution of an ultra-short Gaussian pulse that passes through a concave VLS grating and propagates a distance $r^\prime$ in free space can be described as the convolution of the Gaussian pulse in temporal dimension with the response function in Eq. (15a), and we have
Substituting Eq. (15a) and Eq. (18) into Eq. (22) and then performing a Fourier transform, the expression of the pulse after propagating a distance of $r^\prime$ after the concave VLS grating can be obtained
According to Eq. (23), we can find that the tilt angle of the pulse front and the projected beam size in the $x$ dimension change as a function of propagation distance $r^\prime$. It should be stressed that the pulse front tilt vanishes at the focus in $x$ dimension, and the propagation distance satisfies the following relation at the focus
The beam size at the focus in $x$ dimension is given by
When $r\gg z_x$, equation (26) can be simplified as
At the focus, the transverse beam size is the convolution of the geometric scaling beam size and the dispersion term. The first term of Eq. (27) is corresponding to the geometric scaling beam size (magnification or demagnification), and the second term is corresponding to the dispersion of the concave VLS grating. We also can find that, after the concave VLS grating, the pulse duration $\sigma _T$ is stretched and is the convolution of the initial pulse duration $\sigma _t$ and the stretching term induced by the dispersion. In addition, the pulse stretching also can be simply estimated with formulas in reference [38]. Here, we rewrite the pulse duration after the concave VLS grating
These results in Eq. (27) and Eq. (28) make great agreement with the conclusion in our recent work [35]. It needs to be emphasized that the beam size $\sigma _X$ and pulse duration $\sigma _T$ correspond to the amplitude, whereas in the reference [35], they correspond to the intensity. The theoretical analyses demonstrate that the spatiotemporal couplings occur in $x$ dimension, including pulse front title, pulse front rotation, transverse focusing, and pulse stretching.
4.2.2 Propagation in the $y$ dimension
In the $y$ dimension, due to the absence of dispersion, there is no occurrence of spatiotemporal coupling effects. Therefore, the expression $\mathcal {E}_G(y;r^\prime )$ of the Gaussian pulse after the concave VLS grating and then propagating a distance $r^\prime$ in $y$ dimension is the same with Eq. (15b), and we have
Here, we discuss the propagation properties after the concave VLS grating in the $y$ dimension. According to Eq. (15b) and Eq. (16), we can find that the focus in the $y$ dimension is at
and, at the focus, we haveThese results given in Eq. (30) and Eq. (31) are consistent with the findings of the research work on focusing of Gaussian beam by a thin lens [39]. In the $y$ dimension, the theoretical analyses indicate that the concave VLS grating acts as a focusing element.
Now, we have derived the equations of the pulse in $x$ and $y$ dimensions after going through the concave VLS grating. Therefore, the full expression of the pulse in the free space propagation after the concave VLS grating can be expressed as
Equation (32) reveals a general property: the spatiotemporal response of the concave VLS grating to an ultra-short Gaussian pulse is given by a product of the envelope function $\mathcal {E}_G(y;r^\prime )$ in the $y$ dimension and the spatiotemporal envelope function $\mathcal {E}_G(x,t;r^\prime )$ in the $x-t$ domain where the spatiotemporal coupling occurs. The function $\mathcal {E}_G(y;r^\prime )$ represents the evolution of the transverse beam size in the $y$ dimension. The function $\mathcal {E}_G(x,t;r^\prime )$ describes not only the pulse front tilt whose tilt angle rotates as the change of the propagation distance but also the pulse stretching and the evolution of the transverse beam size in the $x$ dimension. It should be pointed out that, by adjusting the parameters $R_M$, $R_S$, $n_0$, and $b_2$, equation (32) can also be used to describe toroidal VLS gratings, cylindrical VLS gratings, planar VLS gratings, toroidal gratings, cylindrical gratings, planar gratings, toroidal mirrors, cylindrical mirrors, and planar mirrors.
5. Comparison and validation
Having obtained the spatiotemporal response of the concave VLS grating to an ultra-short pulse, we will compare and validate this analytical model in this section. We first use Shadow [28], SRW [29,30], and K-matrix approach [35] to benchmark the model developed in this work, and we also apply this model to estimate photon energy resolution of concave VLS gratings. Then, we validate the propagation properties of high diffraction orders.
5.1 Comparison with numerical simulation
To provide a quantitative comparison, we compare this analytical model with Shadow, SRW, and K-matrix approach. Here, it is important to note that the following discussions will utilize SRW’s single wavelength option and pulse propagation option. In our simulation, we consider an incident Gaussian pulse at 3 nm with pulse duration $\sigma _t$ of 21.02 fs, transverse beam size ($\sigma _x$, $\sigma _y$) of 10.69 $\mu$m. Here, a planar VLS grating is employed, where both the meridian and sagittal curvatures are infinite. The distance $r$ from the source point to the planar VLS grating is 30 m, and the central groove density $n_0$ is $1.5\times 10^5$ lines/m with the VLS parameter $b_2$ of 0.3714 lines/m. The incident angle $\alpha$ and the diffraction angle $\beta$ are 88.1241$^\circ$ and 87.450$^\circ$, respectively. Here, the simulation parameters are summarized in Table 1.
Table 1. Simulation parameters.
Figure 2(a) shows the pulse intensity distribution before the planar VLS grating in the $(x,t)$ domain. Figure 2(b) presents the distribution of pulse intensity at a distance of 7.5 m downstream of the planar VLS grating, and we can find the pulse front tilt effect. Figure 2(d) depicts the distribution of pulse intensity at the focus located 15 m downstream of the planar VLS grating, and the pulse front tilt vanishes. Figure 2(e) displays the pulse intensity after the focus located 22.5 m downstream of the planar VLS grating, and it is evident that pulse front tilt is reproduced and the tilt angle is opposite to that shown in Fig. 2(b). The simulation results illustrate that when an ultra-short pulse passes through a planar VLS grating, it generates pulse front tilt, and the tilt angle varies with the propagation distance. This phenomenon is called pulse front rotation. The simulation results in Fig. 2(a), (b), (d), and (e) indicate that when ultra-short pulses pass through the VLS grating, the pulse duration $\sigma _{t}$ broadens from 21.02 fs to 64.88 fs. Figure 2(c) and (f) present the projection intensities in $y$ and $x$ dimensions located 15 m downstream of the planar VLS grating. Since there is no dispersion occurring in the $y$ dimension, we can observe that our model makes great agreements with Shadow, SRW, and K-matrix approach. However, in the $x$ dimension, the projection intensity calculated by this work is broader than the results estimated by Shadow and the single wavelength option of SRW, and the result of this work exhibits a high degree of agreement with the result calculated by K-matrix approach and the pulse propagation option of SRW. The reason for the observed disparity lies in the limitations of Shadow and the single wavelength option of SRW simulations, which assume a single wavelength and can not account for spatiotemporal couplings. In contrast, this work, K-matrix approach, and the pulse propagation option of SRW enable pulse propagation analysis, including both temporal distribution and spectral information (Fourier transform limited bandwidth). Consequently, different wavelength components are dispersed in the focus of the planar VLS grating, resulting in a broader intensity width.
Fig. 2. The pulse intensity distributions at different positions. (a) Before the planar VLS grating, $\sigma _x$ and $\sigma _t$ are 1.34 mm and 21.02 fs. (b) After propagating 7.5 m from the planar VLS grating, $\sigma _x$ and $\sigma _t$ are 908.34 $\mu$m and 64.88 fs. (d) After propagating 15 m from the planar VLS grating (focus in $x$ dimension), $\sigma _x$ and $\sigma _t$ are 12.18 $\mu$m and 64.88 fs. (e) After propagating 22.5 m from the planar VLS grating, $\sigma _x$ and $\sigma _t$ are 908.34 $\mu$m and 64.88 fs. (c) At the focus in $x$ dimension, the projection intensity profiles in $y$ dimension estimated by Shadow, SRW, this work, and K-matrix approach. (f) At the focus in $x$ dimension, the projection intensity profiles in $x$ dimension estimated by Shadow, SRW, this work, and K-matrix approach. The "SRW $\sigma _{t}$" is corresponding to the calculation of the pulse propagation option of SRW.
Previous discussion has indicated that the beam size in the $x$ dimension is related to the bandwidth of the incident pulse: the larger the bandwidth, the larger the beam size. In Fig. 3(a), we compared the beam size at the focus calculated by Shadow, the single wavelength option of SRW, the pulse propagation option of SRW, and our method. By increasing the pulse duration (equivalent to reducing the Fourier transform-limited bandwidth), we discussed the change of the beam size at the focus. The calculations show that as the pulse duration increases from $\sigma _{t}$ to 20$\sigma _{t}$, the beam size calculated by this work and the pulse propagation option of SRW gradually approaches the results estimated by Shadow and the single wavelength option of SRW. Therefore, our model can be used to calculate the photon energy resolution by increasing the pulse duration (reducing the bandwidth). Figure 3(b) shows the projection intensities calculated by Shadow, the single wavelength option of SRW, and our model with pulse duration of 20$\sigma _{t}$ for wavelengths of 2.99985nm, 3nm, and 3.00015 nm. Figures 3(c), (d), and (e) present the intensity distributions in $(x,y)$ domain calculated by Shadow, the single wavelength option of SRW, and this work for wavelengths of 2.99985nm, 3nm, and 3.00015 nm, respectively. By using a slit, quasi-monochromatic light with a photon energy resolution of approximately $5\times 10^{-5}$ can be obtained. The calculation results indicate that using our method for photon energy resolution assessment can yield results consistent with Shadow and SRW.
Fig. 3. The photon energy resolution analysis. (a) At the focus in $x$ dimension, the projection intensity profiles in $x$ dimension estimated by Shadow, SRW and this work. (b) The projection intensity in $x$ dimension calculated by Shadow, SRW and this work with wavelengths of 2.99985nm, 3nm, and 3.00015 nm. (c), (d), (f) The intensity distributions in $(x,y)$ domain calculated by Shadow, SRW and this work for wavelengths of 2.99985nm, 3nm, and 3.00015 nm.
5.2 Propagation properties of high diffraction orders
In this subsection, we discuss the propagation properties of different diffraction orders. Due to the dispersion of the concave VLS grating occurring in the $x$ dimension, the following investigations in this study are focused on the $(x,t)$ domain. As previously indicated, the concave VLS grating exhibits a focusing capability, and its corresponding focal length $f_x = - A_x/C_x$. For a planar VLS grating, $R_M,R_s\rightarrow \infty$, and the focal length can be expressed as
The above equation indicates that for a planar VLS grating, the focal length varies with different diffraction orders. We now proceed to study high diffraction orders propagation properties of ultra-short pulses after passing through a planar VLS grating. In the calculations, the incident pulse has a central wavelength of 3 nm, a pulse duration $\sigma _t$ of 12.37 fs, a transverse beam size $\sigma _x$ of 1.13 mm, and an incident angle $\alpha$ of $87.186^\circ$. The central groove density $n_0$ is $6\times 10^4$ lines/m, and the VLS parameter $b_2$ is 8.646 lines/m.
When an ultra-short pulse passes through the planar VLS grating, it produces diffracted pulses of different orders. The snapshots in Fig. 4(a), (b), (c), and (d) show different diffraction orders after propagating down to the planer VLS grating at distances of 1.005 m, 1.5 m, 1.779 m, and 2.0 m, respectively. Figure 4(a) and (c) are located at the focuses of the +2 and +1 diffraction orders, respectively. We observe that diffracted pulses with positive orders undergo a process of first focusing and then diverging. The focuses for different diffraction orders are different, and during this process, pulse front rotation occurs, while pulse front tilt disappears at the focus. In contrast, diffracted pulses with negative orders remain in a diverging state throughout the propagation process, thus no focusing is observed. However, pulse front rotation still persists. The above discussion only focused on planar VLS gratings. For concave VLS gratings, such as spherical, cylindrical, and toroidal VLS gratings, they also possess the functions of focusing, dispersion, and wavefront rotation as discussed above. Therefore, a detailed discussion on each type of concave VLS gratings will not be conducted here. The different types of VLS gratings are often used as high-resolution soft X-ray monochromators and spectrometers.
Fig. 4. (a), (b), (c), and (d) are snapshots of different diffraction orders taken at positions 1.005m, 1.5m, 1.779m, and 2m down to the grating, respectively.
6. Summary
In this article, we conducted a theoretical study on the spatiotemporal response of concave VLS gratings to ultra-short pulses. Firstly, we derived the response functions of different diffraction orders of a $\delta$-pulse with a Gaussian transverse distribution passing through a concave VLS grating, and the propagation properties after the concave VLS grating were studied. Then, we derived the spatiotemporal distribution of a Gaussian pulse with a Gaussian transverse distribution passing through the concave VLS grating for different diffraction orders. The results showed that when an ultra-short pulse passes through a VLS grating, transverse focusing occurs, and the focal length of different diffraction orders is different. Non-zero diffraction orders generate pulse front tilt, and the tilt angle varies with the transmission distance (pulse front rotation), which disappears at the focal point. This model can be utilized not only for calculating non-dispersive devices including toroidal mirrors, cylindrical mirrors, and planar mirrors but also for evaluating dispersive devices including toroidal VLS gratings, cylindrical VLS gratings, planar VLS gratings, toroidal gratings, cylindrical gratings, planar gratings. Through numerical simulations, comparisons, and validations, for non-dispersion devices, this model makes great agreement with Shadow and SRW. For dispersive devices, the computational results of this model exhibit a high level of agreement with the results obtained from the K-matrix approach and the pulse propagationoption of SRW. This work can also be used to estimate the photon energy resolution of concave VLS gratings. By considering the spatiotemporal response of concave VLS gratings to ultra-short pulses, the design of soft X-ray monochromators, spectrometers, pulse compressors, and pulse stretchers can be optimized to achieve better performance.
Funding
National Natural Science Foundation of China (12005135, 22288201); 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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