Finite-element simulation for X-ray volume diffractive optics based on the wave optical theory

Yuhang Wang; Yuhang Wang; Yuhang Wang; Lingfei Hu; Bingbing Zhang; Bingbing Zhang; Liang Zhou; Ye Tao; Ming Li; Quanjie Jia; Quanjie Jia

doi:10.1364/OE.409961

1. Introduction

To fully utilize the unique properties of hard X-rays, e.g., the coherence, brilliance, and beam stability, in the new generation of synchrotron radiation (SR) sources, many optical elements have been developed for X-ray optics to obtain the nanoscale focus over the past years [1–5]. To deeply understand the ultimate limit of focusing with X-ray diffractive optics, which generally have intrinsic periodicity, Yan et al. [6] developed a theoretical approach for volume diffraction optics based on the Takagi–Taupin (TT) dynamical X-ray diffraction theory developed for strained single crystals. They derived a set of partial differential equations (PDEs) describing the X-ray dynamical diffraction effects. This approach paved the way for analyzing the attributes of many volume diffractive optical elements, such as multilayer Laue lenses (MLL) [6], laterally graded multilayer (LGML) mirrors [7], and kinoform lenses [8]. In volume diffractive optics, the layer spacing may vary significantly with respect to the spatial location; thus, the variation of the diffractive wave vector is significant, which is the main difference from crystal optics and necessitates additional efforts. These equations mentioned above are typically solved via the finite-difference method (FDM) [6,7]. In most cases, however, the structures of these optical elements have curved shapes, e.g., parabolic or elliptical, for enhancing the optical performance. Therefore, the solution procedure is relatively complex. The FDM is a laborious method for solving curved structures, as it often employs a regular Cartesian mesh. In contrast, finite-element analysis (FEA) based on a variational numerical form of the differential TT equations has potential for solving all types of incident waves and diffractive optical elements of all shapes. Compared with the FDM, FEA allows greater flexibility for discretizing the structure to be solved and involves easy-to-use elements, accelerating the solution procedure. Furthermore, the boundary conditions are naturally included, which is often another key problem in solving PDEs. Another remarkable advantage of the FEA is that its solution procedure is well established [9]. Thus, in this study, we developed a more effective method based on FEA for dealing with complex situations related to volume diffractive optics.

The field of X-ray optics based on multilayer (ML) has seen tremendous development in recent years. The LGML mirror functioning as a nano-focusing optical element plays an important role in modern SR beamlines [10]. From a physical viewpoint, the LGML mirror systems can be divided into two main types: geometrical optics and wave optics. Many studies have proven that a full-wave optical method is necessary for better evaluation of the performance of LGML [11–13]. The original studies on LGML mirrors described by wave optical theory [7,14] were based on the FDM, where the aforementioned theory was used for reference. Different shapes were discussed, ranging from the ideal confocal elliptical shape to the modified shape, under the ideal incident condition of cylindrical waves. However, more general incidence situations, e.g., SR, are urgently needed with the rapid development of software for modeling X-ray beamlines [15]. Meanwhile, as a high-flux element, LGML mirrors still lack an effective way to evaluate the effects of polychromatic X-rays in the framework of wave optics. This is necessary in practical application, where the distribution of the reflected wave field is needed. Moreover, it is necessary to model X-ray beamlines, where the structure imperfections must be taken into account. Thus, in the present study, an easy and flexible application of FEA to LGML mirrors was simulated. In addition to the basic diffraction attributes of LGML mirrors, more general considerations are discussed based on an FEA.

The remainder of this paper is organized as follows. In Section 2, we present a strict derivation of the weak form of the TT equations for volume diffractive optics. Additionally, the general boundary conditions are discussed. This provides a new way of understanding the nature of LGML mirrors. In Section 3, in accordance with the general steps developed in Section 2, a detailed simulation for LGML mirrors is performed. It starts with the basic definition of the diffraction system. A simulation for a flat ML mirror was also performed, to validate the proposed method. Then, we performed calculations for different mirror shapes ranging from the ideal confocal elliptical shape to a modified shape for proving the necessity of modification. Based on the calculations, we constructed a basic modified structure for SubSection 3.4, where we focus on four aspects of the practical application of LGML mirrors. Finally, Section 4 summarizes the paper.

2. Theoretical method

We begin by deriving a general system of equations based on the TT dynamical theory of diffraction, which describes the scattering of X-rays in a synthetic layered structure. From the basic Maxwell’s equations, the following scalar wave equation can be derived:

(1)$${\nabla ^2}E(\textbf{r} )+ {k^2}[{1 + \chi (\textbf{r} )} ]E(\textbf{r} )= 0 ,$$

where E(r) represents the monochromatic scalar field with vacuum wave number k = 2π/λ, wavelength λ, and susceptibility χ(r). r is the position vector. Generally, in volume diffractive optics, χ(r) can be described by the following pseudo-Fourier series:

(2)$$\chi (\textbf{r} )= \sum\limits_{h ={-} \infty }^\infty {{\chi _h}\textrm{exp} [{i{\phi_h}(\textbf{r} )+ i{\varphi_h}(\textbf{r} )} ]} $$

with ${\phi _0}(\textbf{r} )= {\varphi _0}(\textbf{r} )= 0$. ${\phi _h}(\textbf{r} )$ describes the ideal structure (or specifically predesigned structure) with sharp boundaries. ${\varphi _h}(\textbf{r} )$ represents the distortion function accounting for the change in the structure, as introduced in a deformed crystal [16]. the incident wave with an arbitrary wavefront can be expressed as

(3)$${E_{incident}}(\textbf{r} )= E_0^{(a)}(\textbf{r} )\textrm{exp} ({ik{\textbf{s}_\textbf{0}} \cdot \textbf{r}} ),$$

where $E_0^{(a)}(\textbf{r} )$ represents the complex amplitude and is a function of the macroscopic variation. The incident amplitude $E_0^{(a)}(\textbf{r} )$ is constant only when the incident wave is a plane wave. In practice, it is usually space-dependent, similar to a three-dimensional spherical wave or a two-dimensional cylindrical wave. According to the basic assumption of TT dynamical theory [17] and Eq. (2), the trial solutions to Eq. (1) can be expressed in two ways:

(4)$$E(\textbf{r} )= \mathop \sum \limits_h E_h^{(1 )}(\textbf{r} )\textrm{exp} ({ik{\textbf{s}_\textbf{0}} \cdot \textbf{r} + i{\phi_h}} ) ,$$

(5)$$E(\textbf{r} )= \mathop \sum \limits_h E_h^{(2 )}(\textbf{r} )\textrm{exp} ({ik{\textbf{s}_\textbf{0}} \cdot \textbf{r} + i{\phi_h} + i{\varphi_h}} ) .$$

Here, trial solutions of two forms are introduced, similar to previous works [17,18]. In fact, these two formulas are equivalent at the point where $E_h^{(1 )}(\textbf{r} )= E_h^{(2 )}(\textbf{r} )\textrm{exp} ({i{\varphi_h}} )$. The second formula is physically transparent in that the local wave vector is expressed explicitly. Numerically, one advantage of introducing Eq. (5) is that it can enhance the stability in computation, when the distortion term is described analytically such that the coupled terms are represented by their derivative and some coefficients in the equation are constant numbers, as discussed below. For the distortion functions introduced by interpolation or numerically, their derivative cannot be obtained directly, and Eq. (4) is a more appropriate choice. Substituting Eqs. (4) and (2), Eqs. (5) and (2), respectively, into Eq. (1) and following the derivation steps in [6] yields

\textbf{s}_\textbf{h}^{(m )} \cdot \nabla E_h^{(m )} = \frac{{ik}}{2}\mathop \sum \limits_l \;\chi _{h - l}^{(m )}E_l^{(m )} - E_h^{(m )}\left[ {\frac{1}{2}\nabla \cdot \textbf{s}_\textbf{h}^{(m )} + \frac{{({1 - s_h^{(m )\;2}} )k}}{{2i}}} \right] , \quad m = 1,2,

(6)$$ h,l = 0, \pm 1, \pm 2, \pm 3, \pm 4,\ldots ,$$

where

\textbf{s}_\textbf{h}^{(1 )} = {\textbf{s}_\textbf{0}} + \frac{{\nabla {\phi _h}}}{k}\textrm{,}\;\textbf{s}_\textbf{h}^{(2 )} = {\textbf{s}_\textbf{0}} + \frac{{\nabla {\phi _h}}}{k} + \frac{{\nabla {\varphi _h}}}{k}\textrm{,}\;\chi _{h - l}^{(1 )} = {\chi _{h - l}}\textrm{exp} ({i{\varphi_{h - l}}} )\textrm{,}\;\chi _{h - l}^{(2 )} = {\chi _{h - l}} .

Equation (6) represents the basic PDEs of the N-beam TT dynamical system, including distorted terms, where the vector form is retained to the greatest extent possible. It gives the distribution of the wave field inside the corresponding optical elements and is normally solved using the FDM with limited geometric shapes. In volume diffractive optics, ${\textbf{s}_\textbf{0}}$ and $\textbf{s}_\textbf{h}^{(m )}$ are typically space-dependent, and in crystal optics, they are often assumed to be constant vectors. This difference introduces considerable complexity in the FDM, particularly for solving the curved structure.

In the following part, we develop a novel numerical method for calculating Eq. (6) via the FEA. The first thorough discussion regarding the implementation of FEA on a TT system in crystal optics was presented in [19]. The constant propagation vector was handled, and many terms in Eq. (6) disappeared, simplifying the solution procedure. However, in volume diffractive optics, ${\textbf{s}_\textbf{0}}$ and $\textbf{s}_\textbf{h}^{(m )}$ are space-dependent; thus, all the foregoing terms are kept. Hence, in the present paper, a more general weak form of the TT system including any incident wave, any geometric shape, and any distorted function is developed.

FEA provides a general formalism for generating discrete algorithms for approximating the solutions of problems related to PDEs. It belongs to the Ritz–Galerkin method and uses the spline function method to convert the infinite-dimensional Hilbert space into a finite-dimensional function space. The basic procedure of FEA can be summarized as follows. First, the PDE problem is converted into the variational form (i.e., the “weak” form). Second, meshes are generated, and the element shape function is specified. Third, the algebraic linear equations are established and solved. Additional details regarding the mathematical theory of FEA were presented in [9]. In accordance with the standard procedure of FEA, we first obtained the variational form of Eq. (6). It is multiplied by the test function $f_h^{(m )}$, yielding

(7)$$\mathop \smallint \nolimits_V \left\{{\frac{{ik}}{2}\mathop \sum \limits_l \chi_{h - l}^{(m )}E_l^{(m )} - E_h^{(m )}\left[ {\frac{1}{2}\nabla \cdot \textbf{s}_\textbf{h}^{(m )} + \frac{{({1 - s_h^{(m )\;2}} )k}}{{2i}}} \right]} \right\}f_h^{(m )}{d^3}r = \mathop \smallint \nolimits_V ({\textbf{s}_\textbf{h}^{(m )} \cdot \nabla E_h^{(m )}} )f_h^{(m )}{d^3}r .$$

Next, we introduce a more general and important vector identity:

\nabla \cdot ({f_h^{(m )}E_h^{(m )}\textbf{s}_\textbf{h}^{(m )}} )= f_h^{(m )}E_h^{(m )} \nabla \cdot \textbf{s}_\textbf{h}^{(m )} + ({E_h^{(m )} \nabla f_h^{(m )} + f_h^{(m )} \nabla E_h^{(m )}} )\cdot \textbf{s}_\textbf{h}^{(m )} .

This general vector identity contains the special case used in [19], where $\textbf{s}_\textbf{h}^{(m )}$ is a constant vector. Thus, the weak form that we develop below has potential for handling all types of TT-related problems. By substituting this vector identity into Eq. (7) and using Gauss’s theorem (the divergence theorem), we obtain the following integral equations:

(8)$$\begin{aligned}&\mathop \smallint \nolimits_V \left\{{\frac{{ik}}{2}f_h^{(m )}\mathop \sum \limits_l \chi_{h - l}^{(m )}E_l^{(m )} + f_h^{(m )}E_h^{(m )}\left[ {\frac{1}{2}\nabla \cdot \textbf{s}_\textbf{h}^{(m )} - \frac{{({1 - s_h^{(m )\;2}} )k}}{{2i}}} \right] + E_h^{(m )}({\textbf{s}_\textbf{h}^{(m )} \cdot \nabla f_h^{(m )}} )} \right\}{d^3}r\\ &\qquad \qquad \qquad \qquad = \mathop \smallint \nolimits_S f_h^{(m )}E_\textbf{h}^{(m )}\textbf{s}_\textbf{h}^{(m )} \cdot \textbf{n}{d^2}r.\end{aligned}$$

Equation (8) is the basic variational formula of Eq. (6). It represents an N-beam self-consistent system of equations, and the boundary conditions are naturally included in the integral on the right-hand side. Equation (8) exhibits great flexibility in handling structures of all shapes, e.g., MLLs, LGML mirrors, and kinoform lenses. The general diffraction geometry is presented in Fig. 1 and was discussed according to the iterative method for crystal optics in [20]. The region of the illuminated material is identified by the susceptibility function χ(r). In two-beam approximation, the boundary conditions can be expressed as follows:

(9)$$E_0^{(m )}({{\mathrm{\Gamma }_1}} )= E_0^{(a)}({{\mathrm{\Gamma }_1}} )\textrm{,}\qquad E_h^{(m )}({{\mathrm{\Gamma }_2}} )= 0\textrm{,}$$

where Г₁ and Г₂ represent the boundaries for the incident and diffracted waves, respectively. The mainstream method distinguishes the Bragg- and Laue-diffraction geometries by the boundary conditions. As shown in Fig. 1, if Г₁ and Г₂ do not overlap, they have a Bragg geometry; if Г₁ and Г₂ are coincident, they have a Laue geometry; and if Г₁ and Г₂ overlap to some degree, they have a mixed geometry. These boundary conditions do not affect the implementation of FEA, whereas they complicate the FDM. After setting the specified boundary conditions in Eq. (9), a unique solution for the TT equations can be obtained by solving a system of algebraic linear equations generated by Eq. (8) via the standard finite-element modeling method. The commercial software COMSOL Multiphysics (https://cn.comsol.com) is used as the FEA solver in this study.

Fig. 1. General diffraction geometry for a volume diffractive optical element with an arbitrary shape, which is denoted by χ(r). The boundary conditions should be satisfied on Г₁ (green) for the incident waves and on Г₂ (blue) for the diffracted waves. Г₃ (black) denotes the exit surface of the diffracted wave.

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In short summary, these integral equations provide a flexible, robust, and stable method to solve the diffraction problems for volume diffractive optics.

3. Application to LGML mirrors

In this section, a detailed simulation involving Bragg-type ML mirrors was performed on the basis of the theoretical method developed above. For simplicity, all the simulations presented in this section are based on materials W/B₄C, with a filling factor $\tau = 0.5$ at a photon energy of 21.5 keV. Generally, solving a TT dynamical system is equivalent to solving Eqs. (2), (8), and (9) in order; thus, the solving steps are clear, as described in the following parts.

3.1 Preparations for FEA

3.1.1 Basic diffraction system

In the TT description for volume diffractive optics, the key is to find the distribution of the susceptibility in a Fourier-series-like manner. In geometrical optics, confocal ellipses are used for focusing a diverging spherical wave, as shown in Fig. 2(a), where the optical path length of the entrance surface has been set as 2a₀. When the refraction is neglected, the optical path length of the same next material is 2a₀+λ as indicated in Fig. 2(b), indicating that the optical path length of the same material is periodic with a period of λ. Naturally, if we introduce the elliptical coordinates [see Appendix A], the distribution of the susceptibility of the LGML mirrors is a strict one-dimensional (1D) periodic system in the elliptical space, as shown in Fig. 2(b). After performing the Fourier transform in the elliptical space, we obtain the following Fourier series:

\chi (u )= \sum\limits_{h ={-} \infty }^\infty {{\chi _h}\textrm{exp} \left[ {ih\frac{{2\pi }}{d}({u - {a_0}} )} \right]} ,

(10)$${\chi _0} = \tau {\chi _A} + ({1 - \tau } ){\chi _B} , \quad {\chi _h} = \frac{{{\chi _A} - {\chi _B}}}{{2ih\pi }}[{1 - \textrm{exp} ({- 2ih\pi \tau } )} ] ,$$

where the filling factor τ is equal to the ratio of the thicknesses of the material A to d in the elliptical space. d is defined in elliptical space [see Appendix B]. For LGML mirrors, we only consider two wave fields: the incoming wave diverging from the source and the reflected wave converging to the focus; only the terms ${\chi _\textrm{0}}$, ${\chi _1}$ and ${\chi _{- 1}}$ are kept. On the basis of Eq. (10), a trial solution can be obtained:

(11)$$E({u,v} )= {E_0}({u,v} )\textrm{exp} [{ik({u + v} )} ]+ {E_{- 1}}({u,v} )\textrm{exp} [{- ik({u - v} )+ i2k{a_0}} ] .$$

The phase term in Eq. (3) is set as $\textrm{exp} [{ik({u + v} )} ]$ because we seek solutions to the wave equation in accordance with our a priori knowledge—confocal ellipses can focus a diverging spherical wave from a point source.

Fig. 2. (a) Geometric structure composed of a set of confocal ellipses serving as an ideal structure to focus a spherical wave. A and B denote the different filling materials. C₁ and C₂ represent the foci of the ellipses. (b) Material distribution in the space of “Optical path length” (τ represents the filling factor), which corresponds to a square wave.

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3.1.2 Boundary conditions

3.1.2.1 Point source

According to the foregoing analysis, a two-dimensional (2D) diffraction problem is considered. Therefore, we assume that the incident waves are cylindrical waves emitted by a line source, which appear as circular waves in the 2D plane from a point source. Thus, throughout this paper, the concept of a point source is defined in a 2D plane. When the source point is not at the left focus of the ellipse, obtaining the boundary conditions require additional effort. As shown in Fig. 3, we developed three coordinate systems to account for different types of incident waves. The global coordinate system is xOy, which provides the basic structural information regarding the LGML mirrors. The assistant system x₁C₁y₁ is designed to provide more detailed location information regarding the sources. The assistant system x₂C₂y₂ is used for Fresnel integration, which gives the distribution of the wave field in the image space. In the case of cylindrical waves, the radiated waves have an analytical expression. Thus, for a source point at any coordinate, e.g., D in x₁C₁y₁, we can always use the following vector relationship to convert the cylindrical wave radiated from this point D into a form represented by r_s:

(12)$$E_0^{(a)}({{r_D}} )\textrm{exp} ({ik{r_D}} )= E_0^{(a)}\left( {\sqrt {r_s^2 + {D^2} - 2{\textbf{r}_s} \cdot \textbf{D}} } \right)\textrm{exp} \left( {ik\sqrt {r_s^2 + {D^2} - 2{\textbf{r}_s} \cdot \textbf{D}} } \right) .$$

According to the boundary conditions defined in Eq. (9), we derive the following:

(13)$$E_0^{(m )}({{r_s}} )= E_0^{(a)}\left( {\sqrt {r_s^2 + {D^2} - 2{\textbf{r}_s} \cdot \textbf{D}} } \right)\textrm{exp} \left[ {ik{r_s}\left( {\sqrt {1 + \frac{{{D^2}}}{{r_s^2}} - 2\frac{{{\textbf{r}_\textbf{s}} \cdot \textbf{D}}}{{r_s^2}}} - 1} \right)} \right] .$$

This analytical expression makes the definition of the boundary condition for a wave radiating from a point source simple and convenient. It explicitly describes the incident wavefront from different source point in terms of the coordinate D, thus naturally including the global rotation of the LGML mirrors, which is discussed below. The factor $\sqrt {1 + \frac{{{D^2}}}{{r_s^2}} - 2\frac{{{\textbf{r}_\textbf{s}} \cdot \textbf{D}}}{{r_s^2}}} - 1$ is on the order of ${10^{- 8}}$ in the case of a micro scale light source; thus, $E_0^{(m )}({{r_s}} )$ is a function of the macroscopic variation. This condition satisfies the approximation requirement of the TT theory [18].

Fig. 3. Three coordinate systems handling the boundary condition due to the uncertainty of source point such as D, where x₁C₁y₁ takes C₁M as the x₁ axis and x₂C₂y₂ takes MC₂ as the x₂ axis. r_s represents the distance from C₁ to the incidence boundary. r_D represents the distance from D to the incidence boundary. M represents the middle point on the entrance surface, and θ represents the corresponding glancing angle. The red curve represents the trajectory of a single electron in the plane undulator.

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3.1.2.2 Undulator

However, in practice, the incident waves are provided by SR facilities. Modern SR facilities use insertion devices (wiggler and undulator) to generate intense X-ray radiation. Herein, we limit our discussion to the planar vertical undulator, where alternating magnets generate a sinusoidal magnetic field in the vertical direction. The theoretical frequency-domain formula for radiation from a moving charge with an arbitrary trajectory is [21]

(14)$${E_U}({\textbf{x},\omega } )= \frac{e}{{4\pi {\varepsilon _0}c}}\int\limits_{- \infty }^\infty {\left\{{\frac{{\textbf{n} \times \left[ {({\textbf{n} - \boldsymbol{\beta}} )\times \mathop {\boldsymbol{\beta}}\limits^. } \right]}}{{R{{({1 - \boldsymbol{\beta} \cdot \textbf{n}} )}^2}}} + \frac{c}{{{\gamma^2}{R^2}}}\frac{{({\textbf{n} - \boldsymbol{\beta}} )}}{{{{({1 - \boldsymbol{\beta} \cdot \textbf{n}} )}^2}}}} \right\}} {e^{i\omega ({t + R(t )/c})}}dt$$

where R(t) = x – r(t), R(t) = |R(t)|, n = R(t) / |R(t)|, β= $\dot{r}$(t) / c, r(t) represents the trajectory, x represents the observation point, ω represents the radiation frequency, c represents the speed of light, and γ represents the Lorentz factor. The center of the undulator is located at C₁ in Fig. 3, and the trajectory of ideal particles coincides with the axis C₁M. The basic trajectory of a single electron moving in a plane undulator is denoted by the red curve, which can be solved numerically using the computer code reported in [22]. Using this code all the necessary parameters in Eq. (14) can be calculated; thus, the strength of the electric field on the entrance surface of the LGML mirrors can be determined.

3.2 Diffraction from flat ML mirrors

To validate our finite-element method, we solve Eq. (8) for flat ML mirrors. For comparison, the well-established Parratt’s recursive method [23] is used to calculate the reflectivity of a flat ML structure with infinite width, under the assumption of plane-wave incidence. In finite-element method, the trial solution similar to Eq. (11) is assumed and solved in Cartesian coordinates. The propagation vector is constant. Using Eq. (8), the following integral equations can be derived:

\int_L {{f_0}{E_0}{\textbf{s}_\textbf{0}} \cdot \textbf{n}dl} = \int_S {\left\{{\frac{{ik}}{2}{f_0}[{{\chi_0}{E_0} + {\chi_1}{E_{- 1}}} ] + {E_0}({{\textbf{s}_\textbf{0}} \cdot \nabla {f_0}} )} \right\}} {d^2}r ,

(15)$$\int_L {{f_{- 1}}{E_{- 1}}{\textbf{s}_{- \textbf{1}}} \cdot \textbf{n}dl} = \int_S {\left\{{\frac{{ik}}{2}{f_{- 1}}[{{\chi_{- 1}}{E_0} + {\chi_0}{E_{- 1}}} ] - W + {E_{- 1}}({{\textbf{s}_{- \textbf{1}}} \cdot \nabla {f_{- 1}}} )} \right\}} {d^2}r ,$$

where $W = \frac{{ik}}{2}{f_{- 1}}{E_{- 1}}\left[ {{{\left( {\frac{{\nabla {\phi_{-1}}}}{k}} \right)}^2} + 2{\textbf{s}_\textbf{0}} \cdot \frac{{\nabla {\phi_{-1}}}}{k}} \right]$. The integrated reflectivity for an ML structure with finite dimensions is defined as follows:

(16)$$R = \frac{{\int_{{L_i}} {{{|{{E_{- 1}}(l )} |}^2}} dl}}{{\int_{{L_i}} {{{|{{E_0}(l )} |}^2}} dl}} ,$$

where ${L_i}$ represents the incident boundary. This definition is used throughout the paper. The parameters of the simulated flat ML structure are presented in Table 1. To minimize the disturbance caused by the edges of the structure [19], the incident plane wave is multiplied by a Gaussian window function with a full width at half maximum (FWHM) of 34.3 mm.

Table 1. Parameter values of flat and laterally graded ML mirrors for the simulations in Section 3

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The simulation results in Fig. 4(a) indicate that the two theories exhibited similar results, and the main characteristics of the Bragg peaks agreed well. Because the theoretical hypotheses were different, small differences were expected. In Fig. 4(b), the glancing angle is set as the nominal Bragg angle, and we clearly observe the damped oscillations or standing waves [24], which are also referred to as the Pendellösung effect in crystals [18]. Figure 4(c) shows the distribution of the reflected wave field at the peak reflectivity (glancing angle of 11.185 mrad). Clearly, the reflected waves were concentrated in the vicinity of the entrance surface, whereas at the nominal Bragg angle, not only was the reflectivity very low because of refraction but also the wave field was mainly distributed inside the structure, indicating that a high reflectivity necessitates a high concentration of the reflected wave field on the entrance surface. The simulation results validate the proposed method for the ML system.

Fig. 4. (a) Reflectivity curves of 100 bilayers simulated using Parratt’s method and the proposed method. (b) Distribution (arbitrary unit) of the inner reflected wave field at the nominal Bragg angle. (c) Distribution of the inner reflected wave field at the peak reflectivity (glancing angle of 11.185 mrad). The top boundary is the incident surface.

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3.3 Diffraction from LGML mirrors with ideal and modified shapes

In geometrical optics, the optical constant of the LGML is a complex number, which includes refraction and absorption. Both numerical and analytical ray-tracing simulations were performed [12,13,25] to account for the effect of refraction. All these simulations indicated that the ideal confocal elliptical structure cannot provide satisfying focusing and that a modified Bragg condition is necessary. In contrast, from the perspective of the TT dynamical theory [7], ${\chi _0}$ represents the mean susceptibility, which pertains to refraction and absorption, and yields the propagation of ${E_0}$ and ${E_{- 1}}$ in Eq. (11). The terms ${\chi _1}$ and ${\chi _{- 1}}$ account for the coupling between ${E_0}$ and ${E_{- 1}}$; thus, they form a dynamical system. In this section, we use FEA to obtain additional insights into the diffraction problem for LGML mirrors and construct a basic modified structure for SubSection 3.4.

3.3.1 Ideal confocal elliptical structure

We first study the focusing ability of an ideal confocal elliptical structure, as shown in Fig. 2(a). The distribution of the susceptibility and the trial solution given by Eqs. (10) and (11) provide the required Fourier coefficients and diffractive vectors in Eq. (8). The distortion function ${\varphi _h}$ is zero in this case. By combining these conditions and using the corresponding vector formulas in elliptical coordinates [see Appendix A], we obtain the following equations in the two-beam approximation from Eq. (8):

\int_L {{f_0}{E_0}{\textbf{s}_\textbf{0}} \cdot \textbf{n}d l} = \int_S {\left\{{\frac{{ik}}{2}{f_0}[{{\chi_0}{E_0} + {\chi_1}{E_{- 1}}} ] + \frac{1}{2}{f_0}{E_0}\nabla \cdot {\textbf{s}_\textbf{0}} + {E_0}({{\textbf{s}_\textbf{0}} \cdot \nabla {f_0}} )} \right\}} {d^2}r ,

(17)$$\int_L {{f_{- 1}}{E_{- 1}}{\textbf{s}_{- 1}} \cdot \textbf{n}d l} = \int_S {\left\{{\frac{{ik}}{2}{f_{- 1}}[{{\chi_{- 1}}{E_0} + {\chi_0}{E_{- 1}}} ] + \frac{1}{2}{f_{- 1}}{E_{- 1}}\nabla \cdot {\textbf{s}_{- \textbf{1}}} + {E_{- 1}}({{\textbf{s}_{- \textbf{1}}} \cdot \nabla {f_{- 1}}} )} \right\}} {d^2}r ,$$

where

{\textbf{s}_\textbf{0}} = \frac{1}{{{H_1}}}{\textbf{e}_\textbf{u}} + \frac{1}{{{H_2}}}{\textbf{e}_\textbf{v}} ,\qquad {\textbf{s}_{- \textbf{1}}} ={-} \frac{1}{{{H_1}}}{\textbf{e}_\textbf{u}} + \frac{1}{{{H_2}}}{\textbf{e}_\textbf{v}} ,

\nabla \cdot {\textbf{s}_\textbf{0}} = \frac{1}{{u + v}} ,\qquad \nabla \cdot {\textbf{s}_{- \textbf{1}}} ={-} \frac{1}{{u - v}} .

We performed calculations for an LGML structure, whose parameters are presented in Table 1. According to the simulation results, the distribution of the total wave field was plotted explicitly, as shown in Fig. 5(a). Clearly, the distribution of the wave field comprised a slowly varying term and a fast-changing term, and the damped oscillations were clearly observed because of the extinction and absorption. With an increasing number of bilayers, the focus profile became increasingly distorted, and the peak intensity decreased severely, as shown in Fig. 5(b). Furthermore, after the Fresnel integration was performed for the structure of 100 bilayers, its focus profile deviated strongly from the pattern of the diffraction limit, as shown in Fig. 5(c). These results agree with those derived from geometrical optics [13]. Alternatively, from the viewpoint of wave optics, they can be interpreted in terms of the phase change of the reflected wave field. As shown in Fig. 5(d), the 20-bilayer structure had a relatively smooth phase distribution because it had fewer reflecting interfaces, whereas the 100-bilayer structure had distinct phase oscillations, leading to the lower diffractive efficiency, which coincide with the damped oscillations shown in Fig. 5(a). Note that adding pairs leads, however, to reduce peak intensity.

Fig. 5. (a) Distribution of the inner total wave field, where the bilayer is set as 100. (b) 1D focus profiles with different numbers of bilayers at the best focal line. (c) Intensity distribution in the image space with a 100-bilayer structure. (d) Phase (radian) distributions of the reflected wave fields inside LGML structures with 20 and 100 bilayers.

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Traditionally, we could rock the structure to compensate for the effect of refraction. Such global rocking was studied in [7] by introducing a distortion function. However, according to the boundary conditions presented in SubSection 3.1.2, a more practical method for simulating such global rotation was developed. In practice, when the LGML structure in Fig. 3 is rotated, we normally choose a fixed point, e.g., M in xOy. Thus, rotating a structure about M is equivalent to rotating the axis C₁M about M, implying that the source point is on a circle with a radius equal to the length of C₁M. Then, the LGML structure can be rotated easily in accordance with Eq. (13). We calculated the integrated reflectivity as a function of the glancing angle for the aforementioned 100-bilayer structure. As shown in Fig. 6(a), the peak integrated reflectivity occurred at approximately 0.26 mrad relative to the nominal Bragg angle. However, after Fresnel integration was performed at this position, the reflected wave field at the best focal line was negligible, as shown in Fig. 6(b). Moreover, the focusing ability was hardly improved by the global rotation. This phenomenon agrees with the conclusion of [7], where refraction effects could not be removed completely by global rotations, because the glancing angle was space-dependent.

Fig. 6. (a) Global rocking curve of the LGML structure under the incidence of the point source. (b) 1D focus profiles at the best focal line with different rocking angles plotted in a height map.

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3.3.2 Modified structure

As discussed previously, the ideal confocal ellipse structure cannot provide a satisfying focus profile; thus, a modified structure must be used. A detailed study on a modified structure based on the TT dynamical theory was performed using the FDM in [14]. Herein, according to FEA, we present a brief discussion on the construction of a basic modified structure for subSection 3.4. Generally, we design an ML structure according to the Bragg equation:

(18)$${d^B} = \frac{\lambda }{{2\sin \theta }} ,$$

where d^B represents the bilayer spacing for a given wavelength λ and glancing angle θ. When the effect of refraction is considered [5,25], the modified Bragg equation can be obtained:

(19)$${d^{mod}} = \frac{\lambda }{{2\sqrt {{n^2} - {{\cos }^2}\theta } }} ,$$

where n represents the average refractive index for two types of filling materials, and the real part of n is equal to $1 - \delta $. A pseudo-Fourier series to describe such a modified structure can be derived according to Eqs. (18) and (19). The structure composed of ideal confocal ellipses is selected as a reference structure; then, we obtain the following:

{d^{mod}} = \frac{\lambda }{{2\sin \theta }}\frac{1}{{\sqrt {1 + \frac{{{n^2} - 1}}{{\sin {\theta ^2}}}} }} \approx \frac{\lambda }{{2\sin \theta }}\left( {1 + \frac{\delta }{{\sin {\theta^2}}}} \right) ,

(20)$${\chi _{mod}}({{u_{mod}}} )\approx \sum\limits_{h ={-} \infty }^{+ \infty } {{\chi _h}} \textrm{exp} \left[ {ih\frac{{2\pi }}{d}({{u_{mod}} - {a_0}} )} \right]\textrm{exp} \left[ {- ih\frac{{2\pi }}{d}({{u_{mod}} - {a_0}} )\frac{\delta }{{\sin {\theta^2}}}} \right] .$$

Moreover, a parameter p can be introduced for further optimization [14]:

(21)$$d(p )= {d^B} + p \times ({{d^{mod}} - {d^B}} ) .$$

From the perspective of Eq. (2), we obtain the distortion function ${\varphi _h} ={-} h\frac{{2\pi }}{d}({{u_{mod}} - {a_0}} )\frac{{p\delta }}{{\sin {\theta ^2}}}$, where p is a real number. By substituting the distortion function into Eq. (8), we can scan for a modified structure easily.

As indicated by the integrated reflectivity curve with p plotted in Fig. 7(a), the peak reflectivity occurred at approximately p = 0.73. At this position, we performed Fresnel integration and obtained the wave field in the image space, as shown in Fig. 7(b). The modified focus profile was better than that of the ideal confocal elliptical structure with a focus size equal to the diffraction limit [5], and the modification process is detailed in Fig. 7(c), where the 1D focus profile with different modification factor is calculated. Additionally, the peak intensity was enhanced by a factor of approximately 12 compared with the results in Fig. 5(c). Other proofs are presented in Fig. 7(d), where the reflected wave field is mainly concentrated on the entrance surface and the modified phase distribution is far smoother than that in Fig. 5(d). This ensures effective focusing. The results indicate that the capacity of LGML mirrors for focusing can be significantly improved by introducing such a modified Bragg condition. In the TT dynamical theory, the modification effects are based on the distorted term, which determines the phase correction used to compensate for the phase lag caused by refraction [14]. In summary, both the phase and intensity are consistent with the modified objectives, which necessitate a diffraction-limit focus profile [5].

Fig. 7. (a) Integrated reflectivity with the factor p. (b). 2D focus profile at the peak reflectivity (p = 0.73). (c) 1D focus scanning for the modification factor p at the best focal line (d) Intensity (arbitrary unit) and phase (radian) distribution inside the best modified structure. Top: the reflected wave field has been modified to concentrate it on the entrance surface. Bottom: the uniform phase distribution is in accordance with the optimization targets.

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3.4 Further investigations

SubSection 3.3 constructed a basic modified structure (p = 0.73). In this Subsection, this structure was used to perform some further investigations.

3.4.1 Extended sources

3.4.1.1 Point sources

A fully incoherent Gaussian source distributed over the y₁ axis with the center fixed at C₁, as shown in Fig. 3, was simulated based on the basic modified structure (p = 0.73) constructed in SubSection 3.3. According to the linearity of the Maxwell equations, by utilizing the boundary conditions in Eq. (13), we calculated the wave field generated by these point sources independently.

The size of the simulated source is 49 µm (FWHM). The focus profiles at the best focal line of the point source and extended sources are plotted together for comparison in Fig. 8(a). For the point source, the focus shape is of the diffraction-limit size [5], with an FWHM of approximately 4.315 nm and a $\sin {c^2}$-function shape, whereas for the extended sources, the shape appears Gaussian and the FWHM is approximately 23.8 nm. Additionally, the side lobes disappeared, indicating that the focus profile was mainly affected by the finite source. The 2D intensity distribution of the incoherent sum of the reflected wave in Fig. 8(b) indicates that most of the energy of the wave field was concentrated at the nominal focus, and the focus profile was not distorted, because the source was micros. In general, such a simulation for extended sources can provides guidance on the source size in simple way.

Fig. 8. (a) 1D focus profile comparison between the point source and extended sources. (b) Focus profile around the nominal focus plotted in a height map with extended sources.

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3.4.1.2 Undulator

Simulation of modern X-ray beamlines is necessary prior to their design and construction [15]. Such simulation is equivalent to performing a computer experiment in a virtual environment, where the optical system can be evaluated quantitatively. In practice, the X-ray sources are well-described by statistical optics. In particular, partial coherence has been an important topic [26]. Here, we consider a simplified incidence situation defined in SubSection 3.1.2. The basic modified structure (p = 0.73) constructed in SubSection 3.3 was used. The parameters of the simulated undulator are presented in Table 2, and a filament beam with no emittances was assumed.

Table 2. Parameters of the undulator simulated in this study. The first harmonic energy was used.

View Table | View all tables in this article

The simulation results are summarized in Fig. 9. The focus in the image space plotted in Fig. 9(a) has similar behavior to that in Fig. 7(b), although there are differences around the peak. The nominal focus is well-kept, including the focus size and position, as shown in Fig. 9(b). A direct calculation of the integrated reflectivity indicates that the result from undulator is 9% lower than that from point source. This can be explained by the phase distributions of the incident wave field plotted in Fig. 9(c). For the point source, the phase distribution is uniform, which ensures effective focusing. However, for the undulator, the phase distribution is slightly distorted, because the incident wavefront radiated from undulator is not a perfect spherical wavefront, on the basis of the geometric arrangement defined above. Consequently, a small part of the standing wave arises in the reflected wave field, as shown in the Fig. 9(d), and its phase distribution is again slightly distorted compared with that plotted in Fig. 7(d). Thus, the reflectivity is reduced. Importantly, the structure calculated here is modified for the point source, which was derived in SubSection 3.3. Nevertheless, according to the boundary conditions in Fig. 3, an additional modification procedure can be performed by scanning the source position.

Fig. 9. (a) Intensity distribution in the image space with the 100-bilayer structure under the incidence of the undulator. (b) 1D focus at the best focal line under the incidence of the point source and undulator. (c) Phase (radian) distribution of the incident wave field inside the LGML mirror in the case of the point source or the undulator. (d) Intensity (arbitrary unit) and phase (radian) distributions under the incidence of the undulator. Top: small standing waves arise in the reflected wave field, and the energy is mainly concentrated on the right part. Bottom: the phase distribution is slightly distorted owing to the incidence of the undulator without a perfectly spherical wavefront.

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3.4.2 Energy scanning

It is well-known that a flat ML mirror has a relatively large energy bandwidth ($\Delta E\textrm{/}E \cong {10^{- 2}}$) to allow a high flux compared with a crystal monochromator. In this part, we examine the LGML structure to check whether it has a similar energy bandwidth to flat ML mirrors and determine how different energies impact the focus profile. According to the foregoing optimization results, we calculate the energy resolution at the fixed best modified structure; i.e. p = 0.73. The key idea is that for an energy in the vicinity of 21.5 keV, a new distortion function ${\varphi _h}$ is introduced to account for the change in energy. It is described as follows. For a predesigned energy, e.g., 21.5 keV, it fulfills the modified Bragg condition (Eq. (21) with p=0.73), whereas for a different energy like 21.4 keV, it deviates from this modified Bragg condition, indicating that an additional phase is introduced. Furthermore, from the standpoint of 21.4 keV, the structure calculated here can be mapped into the structure best fitting for 21.4 keV by introducing a new distortion function. Accordingly, we set the wave number for the energy of 21.5 keV as k₀. For an incident wave with different energy, the distribution of the susceptibility for the known modified structure (p = 0.73) can be expressed as:

{\chi _k}({{u_{mod}}} )\approx \sum\limits_{h ={-} \infty }^{+ \infty } {{\chi _h}(k )} \textrm{exp} [{i2h{k_0}({{u_{mod}} - {a_0}} )} ]\textrm{exp} \left[ {- i2h{k_0}({{u_{mod}} - {a_0}} )\frac{{0.73\delta }}{{\sin {\theta^2}}}} \right]

Then the distortion function can be introduced by

(22)$${\chi _k}({{u_{mod}}} )\approx \sum\limits_{h ={-} \infty }^{+ \infty } {{\chi _h}(k )} \textrm{exp} [{i2hk({{u_{mod}} - {a_0}} )} ]\textrm{exp} \left[ {ihk({{u_{mod}} - {a_0}} )\left( {\frac{{2{k_0}}}{k} - 2 - \frac{{2{k_0}}}{k}\frac{{0.73\delta }}{{\sin {\theta^2}}}} \right)} \right]\textrm{.}$$

Using Eq. (2), we obtain the distortion function ${\varphi _h} = hk({{u_{mod}} - {a_0}} )\left( {\frac{{2{k_0}}}{k} - 2 - \frac{{2{k_0}}}{k}\frac{{0.73\delta }}{{\sin {\theta^2}}}} \right)$. This distortion function indicates the phase distortion due to the changed wavelength. By substituting it into Eq. (8), we can calculate the integrated reflectivity for different energies.

The integrated reflectivity was calculated for different incident energies, and a flat ML structure was calculated for comparison, as shown in Fig. 10(a). The reflectivity was nearly identical between the two types of MLs, indicating that the bandwidth of the LGML mirrors was nearly identical to that of the flat ML mirrors. Further, the 1D focus profile with the energy at the best focal line is presented in Fig. 10(b), and it exhibits a dependence similar to that of the integrated reflectivity. The incoherent sum of the wave field around the nominal focus for all the considered energies is illustrated in Figs. 10(c) and (d). It can be concluded that for different energies in the vicinity of the predesigned energy, the focus profile is well maintained, except for a small peak shift. In the view of the TT dynamical theory, the energy effects are introduced in terms of the distortion function φ_h. The simulation results indicate that LGML mirrors can serve as high-flux focusing optical elements. In practice, this pure wave optical description makes beamline designs that take the dispersion effects into account more accurate and efficient when combined with the general boundary conditions developed above.

Fig. 10. (a) Reflectivity curves with different energies for flat ML and LGML structures. (b) 1D focus profiles scanning for different energies at the best focal line. (c) Sum of 2D focus for all the energies calculated. (d) 1D focus profile comparison between the designed energy and the sum of all the energies.

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3.4.3 Figure error

In general, imperfections always exist in real MLs for unavoidable reasons, e.g., polycrystalline growth and compound formation at the layer boundaries [27]. Different scattering theories [27–31] have been developed for the flat ML system to account for various imperfections. In early studies [27,30–32], the scattering from periodical MLs with rough interfaces was split into the coherent and incoherent parts. The coherent part only includes the root-mean-square (RMS) roughness, whereas for the incoherent part, the in-plane and vertical correlations must be considered. However, to fully exploit the novel numerical method developed above, we consider two simplified situations: fully correlated figure errors and fully uncorrelated roughness. Herein, we consider the former one. In the framework of the TT dynamical theory, the key is to obtain the distribution of the susceptibility of the filling materials in a Fourier-series-like manner. We still utilize the mapping method and select the ideal LGML structure as a reference structure. For a substrate with a randomly generated profile (or a detected experimental profile [10]), we assume that all the LGML interfaces have the same profile so that we can map the imperfect structure to an ideal reference structure. By interpolating, we introduce a known function $f(v )$ [see Appendix B] to represent such figure errors imposed on the substrate. Thus, the distribution of the susceptibility of the new structure (including figure errors) can be described as follows:

(23)$${\chi _{error}}(u )= \chi ({u - {a_0} - f} )= \sum\limits_{h ={-} \infty }^{+ \infty } {{\chi _h}} \textrm{exp} \left[ {ih\frac{{2\pi }}{d}({u - {a_0}} )- ih\frac{{2\pi }}{d}f} \right] .$$

From the perspective of Eq. (2), we find that ${\varphi _h} ={-} h\frac{{2\pi }}{d}f$, which inherently includes the effect of figure errors and causes a phase shift to the reflected wave field. The function $f(v )$ is arbitrary and not analytical, and the trial solutions of Eq. (4) were selected.

To quantitively examine the effect of the figure errors, the ideal 20-bilayer confocal elliptical LGML structure was simulated. A set of figure error functions with different RMS roughness values ranging from 0 to 0.5 nm were simulated. For an X-ray total reflection mirror, these figure errors with a spatial period in the millimeter range significantly affect the focus profile [33]. A similar situation occurs for a fully correlated roughness in an LGML structure in which the peak intensity has decreased severely and energy has been increasingly transferred into the side lobes, as shown in Fig. 11(a). Figure 11(b) shows the distribution of the wave field in the image space with the RMS roughness equal to 0.5 nm. Here, the nominal focus is almost indistinguishable, and compared with Fig. 11(c), the energy distribution is more disperse, and the peak intensity is four times smaller. These results can be explained by the phase distortion introduced by the figure errors. From the perspective of Eq. (8), the coupled terms (χ_h) multiplied by these error functions cause a phase shift to the reflected wave field. As shown in Fig. 11(d), compared with the perfect structure, the reflected wave field on the exit surface has a significantly distorted phase distribution; thus, the focus profile in the image space diffuses severely. The fully correlated roughness is a relatively ideal model for describing a real LGML structure, but it is reasonable to use this model to evaluate the quality of a substrate. Because the extent of the propagation varies with respect to the spatial frequency, the low-frequency components should be easily replicated, while the high-frequency components are likely to be planarized [30] and can be polished via a conventional process relatively easily [34]. In this regard, evaluating the effects of such replicated low-frequency components is important.

Fig. 11. (a) 1D focus profile at the best focal line for different σ values. (b) Intensity distribution around the nominal focus with σ = 0.5 nm. (c) 2D focus profile for the ideal structure with 20 bilayers. (d) Phase distribution on the exit surface.

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3.4.4 Stochastic roughness

For the fully uncorrelated roughness where the ML structure does not have propagation of the roughness in the horizontal and vertical directions [30], it is assumed that all the interfaces throughout the LGML structure have the same Gaussian normal distribution with an RMS roughness of $ \sigma $. This assumption implies that the mean interface position does not deviate from the predesigned structure and that the fabrication process is a stationary stochastic process. After the statistical average over ensembles is performed for all possible height functions [see Appendix B], the following pseudo-Fourier series can be derived:

\chi (u )= \sum\limits_{h ={-} \infty }^\infty {{{\mathop \chi \limits^ - }_h}\textrm{exp} \left[ {ih\frac{{2\pi }}{d}({u - {a_0}} )} \right]} ,

(24)$${\chi _0} = \tau {\chi _A} + ({1 - \tau } ){\chi _B} , \qquad {\bar{\chi}_{h}} = \frac{{{\chi _A} - {\chi _B}}}{{2ih\pi }}[{1 - \textrm{exp} ({- 2ih\pi \tau } )} ]\textrm{exp} \left( {- {h^2}{{\left( {\frac{{2\pi }}{d}} \right)}^2}{\sigma^2}/2} \right) .$$

By comparing Eq. (24) with Eq. (10), a new decay factor is obtained. From the standpoint of the coupled system of Eq. (8), this decay factor affects the energy transfer between ${E_0}$ and ${E_{- 1}}$, which causes the reflected wave field ${E_{- 1}}$ to obtain less energy form wave field ${E_0}$. This reduces the diffraction efficiency, as shown in Figs. 12(a) and (b). However, in Fig. 12(c), a different situation occurs: the phase distribution of ${E_{- 1}}$ retains a relatively smooth shape and does not change significantly; thus, the focus profile is well maintained despite the decreasing intensity. This can be explained as follows: the decay factor does not have a phase factor, because it is derived from the ensemble average of all the possible interface profiles. Nevertheless, when we examine the intensity in the vicinity of the nominal focus, a similar result is found for the figure errors and stochastic roughness: the peak intensity of the wave field is significantly attenuated for RMS roughness values of >0.2 nm. This is because the glancing angle was larger than the critical angle, which led to larger reciprocal vector. Thus, evaluating such the roughness in this manner can place restrictions on the fabrication process.

Fig. 12. (a) 2D focus profile at $\sigma $ = 0.5 nm. (b) 1D focus profile at the best focal line for different $\sigma $ values. (c) Phase distribution on the exit surface for the stochastic roughness.

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4. Summary and outlook

We developed a general integral system of equations of the Takagi-Taupin dynamical theory for volume diffractive optics based on the vector nature of wave propagation. On the basis of the integral form, we successfully used finite-element analysis to solve the Takagi-Taupin dynamical system. By virtue of this novel numerical method featuring flexibility, robustness, and stability, we performed a comprehensive and highly efficient study of the laterally graded multilayer mirror. Parratt’s recursive method [23] was used to validate the proposed method. Simulations for the ideal confocal shape and a modified shape yielded results similar to those obtained in previous studies [7,14]. However, our method is more general and efficient in handling diffractive optical elements of arbitrary shape. Moreover, we developed a global coordinate system that can handle any incidence situation and naturally includes the global rocking of the LGML structure. The simulation of the illumination from the undulator and the analysis of the energy resolution allow evaluation of the beamline design with regard to pure wave optics. The figure errors were calculated to provide guidance for designing LGML mirrors. Additionally, the uncorrelated stochastic roughness was examined, although it is a simplified approach [35].

The proposed method is not limited to the scope of the present work. The method is capable of solving not only 2D but also three-dimensional X-ray dynamical diffraction problems related to diffractive optical elements. Moreover, modern SR facilities and free-electron laser sources provide X-ray beams with high brilliance, which can cause deformations of the optical elements, degrading the performance of optical elements [15]. To realize the potential performance of these light sources, the study of thermal deformation is imperative. FEA is widely used in the field of engineering, e.g., structural mechanics and thermal analysis. Therefore, in future research, heat problems can be combined with the TT dynamical system via the multiphysics coupling simulation method, which is an advantage of FEA.

Appendix

A. Elliptical coordinate system

Here, we comprehensively explain the elliptical coordinate system. The elliptical coordinate system forms a 2D orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae [36]. As shown in Fig. 13, the two foci C₁ and C₂ are taken to be fixed at (c, 0) and (–c, 0), respectively, on the x-axis of the Cartesian coordinate system. According to the basic definition of curved coordinates, after using basic mathematical techniques for vector analysis [36], the following important relationships can be derived:

(25)$$P = \sqrt {{{({x + c} )}^2} + {y^2}} \quad Q = \sqrt {{{({x - c} )}^2} + {y^2}} , \qquad P + Q = 2u\; P - Q = 2v ,$$

(26)$${H_1} = \sqrt {{{\left( {\frac{{\partial x}}{{\partial u}}} \right)}^2} + {{\left( {\frac{{\partial y}}{{\partial u}}} \right)}^2}} = \sqrt {\frac{{{u^2} - {v^2}}}{{{u^2} - {c^2}}}} , \quad {H_2} = \sqrt {{{\left( {\frac{{\partial x}}{{\partial v}}} \right)}^2} + {{\left( {\frac{{\partial y}}{{\partial v}}} \right)}^2}} = \sqrt {\frac{{{u^2} - {v^2}}}{{{c^2} - {v^2}}}} ,$$

where ${H_1}$ and ${H_2}$ are the corresponding scale factors in elliptical coordinates. Thus, the vector formulas can be expressed as follows:

(27)$$\nabla f = \frac{1}{{{H_1}}}\frac{{\partial f}}{{\partial u}}{\textbf{e}_\textbf{u}} + \frac{1}{{{H_2}}}\frac{{\partial f}}{{\partial v}}{\textbf{e}_\textbf{v}} , \quad \nabla \cdot ({F_1}{\textbf{e}_\textbf{u}} + {F_2}{\textbf{e}_\textbf{v}}) = \frac{1}{{{H_1}{H_2}}}\left( {\frac{{\partial {H_2}{F_1}}}{{\partial u}} + \frac{{\partial {H_1}{F_2}}}{{\partial v}}} \right).$$

Fig. 13. Basic coordinate system describing LGML mirrors, where u and v are elliptical coordinates.

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B. Fourier series representation

The distribution of the susceptibility in Fig. 14(b) in elliptical coordinates and its Fourier transformation are given as follows:

(28)$$\chi (u )= {\chi _{ideal}}(u )+ \delta \chi (u ) , \quad F(H )= \int {\chi (u )\textrm{exp} ({- iHu} )} du , $$

(29)$$< F(H )> = {F_{ideal}}(H )+ < \int {\delta \chi (u )\textrm{exp} ({- iHu} )du} > . $$

The ideal part is the normal Fourier series used in Eq. (10), and the distorted part is given as follows:

< \int {\delta \chi (u )\textrm{exp} ({- iHu} )du} >

(30)$$= < \sum\limits_h {\left\{{\frac{\Delta }{{iH}}[{\textrm{exp} ({- iH {z_{{A_h}}}} )- 1} ]+ \frac{\Delta }{{- iH}}[{\textrm{exp} ({- iH {z_{{B_h}}}} )- 1} ]\textrm{exp} ({- iH\tau d} )} \right\}\textrm{exp} ({- iH{u_h}} )} > , $$

where

\Delta = {\chi _A} - {\chi _B} , \qquad \sum\limits_h {\textrm{exp} ({- iH{u_h}} )} = \frac{{2\pi }}{d}\sum\limits_h {\delta \left( {H - h\frac{{2\pi }}{d}} \right)}

Calculating the statistical average for the Gaussian normal distribution yields

(31)$$< F(H )> = \textrm{exp} ({- {H^2}{\sigma^2}/2} ){F_{ideal}}(H ). $$

Fig. 14. Three types of distributions of the susceptibility in elliptical coordinates. (a) Ideal LGML structure. (b) Stochastic roughness. (c) Figure error (or fully correlated roughness), which is discussed in SubSection 3.4.3.

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Funding

National Natural Science Foundation of China (11905242); Beijing Municipal Science & Technology Commission (Z191100001619005); High Energy Photon Source (HEPS).

Acknowledgements

We acknowledge Ari-Pekka Honkanen (University of Helsinki) for fruitful discussions on numerical implementation of the TT system.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Structure (W/B₄C)	Photon energy	Source distance	Focal distance	Glancing angle	Mirror length	Bilayers
LGML	21.5 keV	115.1 m	75 mm	10.925 mrad	70 mm	10–100
Flat ML	21.5 keV	-	-	10.925 mrad	70 mm	100

Structure (W/B₄C)	Photon energy	Source distance	Focal distance	Glancing angle	Mirror length	Bilayers
LGML	21.5 keV	115.1 m	75 mm	10.925 mrad	70 mm	10–100
Flat ML	21.5 keV	-	-	10.925 mrad	70 mm	100

Finite-element simulation for X-ray volume diffractive optics based on the wave optical theory

Abstract

1. Introduction

2. Theoretical method

3. Application to LGML mirrors

3.1 Preparations for FEA

3.1.1 Basic diffraction system

3.1.2 Boundary conditions

3.1.2.1 Point source

3.1.2.2 Undulator

3.2 Diffraction from flat ML mirrors

3.3 Diffraction from LGML mirrors with ideal and modified shapes

3.3.1 Ideal confocal elliptical structure

3.3.2 Modified structure

3.4 Further investigations

3.4.1 Extended sources

3.4.1.1 Point sources

3.4.1.2 Undulator

3.4.2 Energy scanning

3.4.3 Figure error

3.4.4 Stochastic roughness

4. Summary and outlook

Appendix

A. Elliptical coordinate system

B. Fourier series representation

Funding

Acknowledgements

Disclosures

References

Cited By

Figures (14)

Tables (2)

Equations (44)

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