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Abstract
X-ray grazing incidence microscopy has extensive applications in the fields of laser inertial confinement fusion and synchrotron radiation. Monte Carlo methods can be used to determine the optical performance of X-ray grazing incidence microscopes and predict the experimental results, which is of great significance for studying physical experiments and diagnostics. In this paper, we proposed a Monte Carlo method based on Geant4 for studying X-ray grazing incidence microscopy. We introduced the G4MultilayerReflection class to describe the physical processes of X-ray multilayer mirrors. We designed a dual-energy Kirkpatrick–Baez microscope that can operate at 6.4 and 9.67 keV simultaneously. Monte Carlo simulations of the spatial resolution and throughput efficiency of the microscope were performed using Geant4, which was assembled and characterized. The spatial resolution results obtained by the Geant4 laboratory simulations, the theoretical model, and the experiments were in good agreement. Additionally, we conducted throughput efficiency calibration experiments for the 6.4 keV imaging channel. The difference between the experimental and Geant4-simulated throughput efficiency was evaluated and resulted in root mean square error values of 8.7% and 9.5% along the Y- and Z-axes, respectively.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
X-ray grazing incidence microscopes based on ultra-smooth mirrors can achieve high-resolution imaging of objects. X-ray film can be deposited on the surface of ultra-smooth mirrors to achieve X-ray reflection [1,2]. Typical configurations of X-ray grazing incidence microscopes include the Kirkpatrick–Baez (KB) microscope [3], Montel microscope [4], advanced KB microscope [5,6], and Wolter microscope [7]. X-ray grazing incidence microscopes have a wide range of applications in the fields of laser inertial confinement fusion (ICF) and synchrotron radiation. In laser ICF, X-ray grazing incidence microscopes are commonly used to study plasma X-ray emitted from hot spot produced by the implosion of an ICF target with high time resolution, high spatial resolution, and high energy resolution [8–10]. In the field of synchrotron radiation, X-ray fluorescence microscopes can achieve high-throughput achromatic focusing imaging of biological samples [11]. By conducting Monte Carlo studies of X-ray grazing incidence microscopy, more comprehensive theoretical results can be obtained. This enables the prediction of experimental results in physical diagnostics and the optimization of system performance.
Currently, X-ray oriented programs(XOP) [12] and OrAnge SYnchrotron Suite(OASYS) [13] are commonly used for X-ray film reflectivity calculations and optical simulations of X-ray grazing incidence microscopy. They can perform Monte Carlo analyses of X-ray grazing incidence systems to evaluate their performance. In the research of X-ray grazing incidence microscopy, there is often a desire to achieve three-dimensional visualization of the optical system and to simulate non-sequential ray tracing that approaches the real-world optical system conditions. This includes analyzing stray light in the system and implementing stray light shielding to achieve improved optical performance. However, these requirements are challenging to fulfill using XOP and OASYS.
Geant4 (GEometry ANd Tracking) is a powerful software toolkit based on Monte Carlo methods, developed by the European Organization for Nuclear Research (CERN) for simulating high-energy particle interactions with matter [14,15]. It has a wide range of uses in fields such as high-energy physics, nuclear physics, accelerator physics, as well as in medical and space science research. Geant4 is written in the C++ programming language, and can be used on many different computing platforms. The modular design of Geant4 makes it easier for users to understand, develop, test, and maintain the program. In Geant4, modules are typically represented in the form of classes, which are a significant concept in the C++ programming language and act as templates for objects. Geant4 has a wide range of physics libraries that can simulate various types of particle interactions with matter, including ionization and excitation, Compton scattering, Compton effect, electron pairs, positron annihilation, and interactions between protons and neutrons [16]. Geant4 is suitable for modeling and analyzing X-ray grazing incidence microscopy. Its capabilities in geometric modeling, three-dimensional display, and the flexibility of customizing physical processes, allow researchers to simulate and analyze complex optical systems more accurately, and obtain Monte Carlo results that are closer to the actual situation.
However, Geant4 lacks a description of the physical process of X-ray grazing incidence reflection. To address this shortcoming, Buis and Vacanti developed the XRTG4 extension package based on Geant4, which is used to describe the physical processes of X-ray single-layer film grazing incidence reflection [17,18]. Currently, XRTG4 is commonly used to study the effective area and background radiation of X-ray astronomical telescopes [19,20].
The XRTG4 extension package is used to describe X-ray single-layer film grazing incidence reflection. In the application of X-ray microscopes in the fields of laser ICF and synchrotron radiation, it is often necessary to perform spectral selection of X-rays in a specific energy range. Therefore, it is necessary to establish classes in Geant4 to describe the physical processes of X-ray multilayer mirror reflection.
In this study, a Monte Carlo model of an X-ray grazing incidence microscope was established based on Geant4. We created a class called G4MultilayerReflection based on Geant4 to describe the physical processes of X-ray reflection and scattering on multilayer mirrors. We applied it to the Monte Carlo model of X-ray multilayer mirrors and evaluated the reliability of the G4MultilayerReflection class by establishing two sets of X-ray multilayer structures. In this study, we designed a dual-energy KB microscope that can operate at 6.4 keV (Fe $\rm {K}\alpha _{1}$ characteristic line) and 9.67 keV (W $\rm {L}\beta _{1}$ characteristic line) simultaneously. The XRTG4 package was introduced to describe the physical processes of X-ray reflection on a single-layer mirror surface, and the G4MultilayerReflection class was applied to study the X-ray reflection of X-ray multilayer mirrors, enabling Monte Carlo investigation of the microscope imaging process. Based on the visualization function of Geant4, three-dimensional visualization of dual-energy KB microscope imaging was achieved, and the instantiation function of Monte Carlo analysis of the dual-energy KB microscope was constructed. Based on Geant4, we performed Monte Carlo simulations of X-ray imaging and throughput efficiency for the dual-energy KB microscope. In the laboratory, we performed alignment of the dual-energy KB microscope and conducted X-ray imaging experiments for two image channels to calibrate the spatial resolution. We also performed throughput efficiency calibration for the 6.4 keV imaging channel. The spatial resolution results obtained by the Geant4 simulations, theoretical models, and experiments were all in good agreement. We used the root mean square error (RMSE) to evaluate the difference between the throughput efficiency results obtained by the Geant4 simulations and experiments for the 6.4 keV imaging channel, resulting in RMSE values of 8.7% and 9.5% along the Y- and Z-axes, respectively.
2. Implementation of X-ray grazing incidence reflection simulation in Geant4
2.1 Simulation of X-ray single-layer mirror reflection
In the extreme-ultraviolet and X-ray regime, the refractive index of matter is slightly less than 1, typically expressed in the following form [21–23]:
where $\delta$ is a small quantity representing the refractive index, typically on the order of 10$^{-5}$ to 10$^{-6}$, and $\beta$ is the extinction coefficient, which represents the absorption of X-rays by a material. Both quantities are related to the atomic scattering factor and wavelength of the material. The expressions for $\delta$ and $\beta$ are as follows [21,23]: where $\rho _{a}$ is the atomic number density, $r_{e}$ is the classical electron radius, $\lambda$ is the X-ray wavelength, $f_{1}\left ( \omega \right )$ is the real part of the complex atomic scattering factor, which is used to represent the phase shift of X-rays propagating through a material, and $f_{2}\left ( \omega \right )$ is the imaginary part of the complex atomic scattering factor, which is used to represent the attenuation of the X-ray amplitude. When X-rays are incident at the interface of two media at a small angle, reflection and refraction occur according to Snell’s law. When the incident angle is less than a critical angle, total external reflection occurs, and the refracted light does not penetrate the material but instead forms an evanescent wave that propagates along the material interface. The expression for the critical angle $\theta _{c}$ is as follows [24]:Because the atomic number density $\rho _{a}$ per unit volume varies slowly in natural elements, the critical angle factor mainly depends on the following function:
where $Z$ is the atomic number. Therefore, to obtain a larger critical angle, it is possible to increase the wavelength or deposit high-$Z$ materials on the surface of X-ray mirrors.High-$Z$ single-layer film, such as Pt and Ir, is usually deposited on a highly polished substrate of the reflection mirror to achieve high reflectivity and increase the critical angle. Figure 1(a) displays X-rays incident ($\mathbf {k_{i}}$) on the surface of a single-layer film (medium), with reflection ($\mathbf {k_{r}}$) and refraction ($\mathbf {k_{t}}$) occurring. Figure 1(b) presents the visualization interface displaying X-ray grazing incidence reflection on a single-layer film mirror, implemented in Geant4 based on the XRTG4 extension package. In Geant4, X-rays are represented by low-energy gamma photons. The mirror substrate was Si, and the single-layer film material was Pt with a thickness $d$ of 30 nm.
Fig. 1. (a) X-ray incidence, reflection, and refraction in a single-layer film with thickness $d$. The total reflection coefficient is the sum of infinite reflections. (b) Visualization of total reflection occurring when X-rays are incident on the X-ray single-layer mirror at grazing angles less than the critical angle, implemented in Geant4.
For the grazing incidence reflection of X-rays in the X-ray single-layer mirror illustrated in Fig. 1, we used a model for calculating the reflectivity, as expressed in Eq. (6) [25]:
Considering non-ideal factors such as roughness $\sigma$ between the X-ray single-layer film and the substrate in practical situations, we introduce the roughness formalism Eq. (7) developed by Névot and Croce to describe the reduction of the X-ray film reflectivity caused by roughness [26], which is discussed in greater detail in Section 2.2.
Therefore, the expression for the reflectivity $R$ of the X-ray single-layer film mirror in practical situations is $R=R_{0}\times M$.
Based on the model for calculating the X-ray single-layer film mirror reflectivity described above, we simulated the reflectivity curves of a 30 nm Pt thick single-layer film deposited on a Si substrate with respect to the grazing incidence angle and photon energy using Geant4 and IMD [27], as illustrated in Fig. 2. From Fig. 2, it can be seen that the reflectivity curves calculated based on our X-ray reflectivity model constructed using Geant4 and XRTG4 are in good agreement with those calculated using IMD. Therefore, our model for constructing the X-ray single-layer mirror in Geant4 is accurate.
Fig. 2. X-ray reflectivity curves calculated using Geant4 and IMD. (a) Variation of the X-ray reflectivity on the surface of a 30 nm thick Pt single-layer film mirror at energies of 6.4 and 9.67 keV and a roughness of $\sigma$ = 0.3 nm as a function of the grazing incidence angle. (b) Variation of the X-ray reflectivity on the surface of a 30 nm thick Pt single-layer film mirror with grazing incidence angles of $0.3358^\circ$ and $1^\circ$ and a roughness of $\sigma$ = 0.3 nm as a function of energy.
2.2 Simulation of X-ray multilayer mirror reflection
In the fields of laser ICF and synchrotron radiation, it is often necessary for X-ray imaging systems to have high spectral resolution. For X-rays above 10 keV, the critical angle of a metal single-layer film is very small, which is not conducive to the preparation and adjustment of X-ray microscopes and affects the imaging quality. X-ray multilayer mirrors have demonstrated advantages in addressing these challenges.
X-ray multilayer reflection has two main theoretical bases: dynamical diffraction theory [28] and optical multilayer theory based on the Fresnel equation [29]. The former is similar to the treatment of X-ray diffraction in natural crystals using the Bragg equation ($2d\sin \theta =n\lambda$) [30], while the latter extends the theories of visible and ultraviolet light to the X-ray wavelength range. The latter approach is commonly used to study the reflectivity of multilayers.
Figure 3(a) presents a schematic diagram of X-ray reflection in a multilayer mirror, while Fig. 3(b) presents a schematic diagram of the multilayer mirror constructed in Geant4. For a multilayer film with N layers of media, let the (N+1)-th layer be the substrate and the topmost layer be vacuum. The refractive index of the $j$-th layer is denoted as $n_{j}=1-\delta _{j}+i\beta _{j}$, and the thickness is $d_{j}$ ($j$=1,2,…,N). The grazing incidence angle of the $j$-th layer film is $\theta _{j}$, and the grazing incidence angle of the $(j+1)$-th layer film is $\theta _{j+1}$, as illustrated in Fig. 3. The X-ray multilayer film reflectivity recursion formula for N interfaces is expressed in Eq. (8) [25]:
Fig. 3. (a) Schematic diagram of an X-ray multilayer stack deposited on a reflective mirror substrate, where $k_{i,j}$ and $k_{r,j}$ denote the incident and reflected X-ray waves, respectively, in the $j$-th layer of the thin film. (b) X-ray multilayer reflective mirror constructed in Geant4, with N = 24 layers composed of W and C materials deposited on a Si substrate.
The actual preparation process of X-ray multilayer mirrors often involves non-ideal factors such as defects, impurities, interface roughness, and diffusion, which can significantly affect the reflection characteristics of the mirrors. Surface and interface roughness and the existence of interface mixing layers are particularly common factors. Therefore, the Névot–Croce roughness model was used to describe the X-ray thin film interface under actual conditions, as expressed in Eq. (7). As a result, the reflection calculation formula for non-ideal multilayer mirrors is $R=R_{0}\times M$.
We introduced the theoretical model of X-ray multilayer films into Geant4 and created a class called G4MultilayerReflection to describe the physical processes of X-ray multilayer mirror reflection. We designed two sets of periodic multilayer films, with relevant parameters presented in Table 1. Table 1 presents two sets of structural parameters for the X-ray periodic multilayers at X-ray energies of 6.4 and 9.67 keV with an incident angle of 1.1431$^{\circ }$. Period thickness $d$ in Table 1 refers to the sum of the layer thicknesses $d_{W}$ for the W layer and $d_{C}$ for the C layer.
Table 1. X-ray multilayer structural parameters.
The aforementioned multilayer parameters were introduced into the multilayer mirror model created in Geant4, and the reflectivity curves were calculated, as illustrated in Fig. 4. Figure 4(a) and (b) illustrate the variation of X-ray periodic multilayer reflectivity with grazing incidence angle and energy. The results are in good agreement with those obtained by IMD, demonstrating the reliability of the G4MultilayerReflection class established in Geant4 in describing the X-ray multilayer mirror reflection process.
Fig. 4. Reflectivity curves of X-ray periodic multilayer film at $E$ = 6.4 keV and $E$ = 9.67 keV, calculated using IMD and Geant4, respectively. (a) Reflectivity versus grazing angle curves and (b) Reflectivity versus energy curves.
3. Monte Carlo study of a dual-energy KB microscope
3.1 Geometric models
The focus equation of the KB microscope is presented below [31]:
where $u$ is the object distance, $v$ is the image distance, $f$ is the effective focal length, $R$ is the curvature radius of the cylindrical mirror, and $\theta$ is the grazing incidence angle. Therefore, the magnification is given by $M=\frac {v}{u}$[31].We proposed a dual-energy grazing incidence KB microscope for X-ray imaging diagnostics in laser ICF. The working energy points were 6.4 and 9.67 keV. The beam path schematic of the dual-energy KB microscope is shown in Fig. 5. Here, we define the YOZ plane as the object plane, with the Z-axis representing the horizontal direction, the Y-axis representing the vertical direction, and the X-axis representing the optical axis direction. The system comprised three concave cylindrical mirrors, with two imaging channels sharing a horizontal mirror M1. The horizontal mirror M1 was a Pt single-layer mirror, while the vertical mirrors M2 and M3 were two periodic multilayer mirrors. The object emits X-rays, which are reflected and focused by the horizontal mirror M1. Subsequently, the X-rays pass through M2 and M3, where they are separately reflected and focused for X-rays with energies of 6.4 keV and 9.67 keV, enabling spectral selection. As a result, two image spots with energies of 6.4 keV and 9.67 keV are simultaneously obtained on the image plane detector. This configuration of an X-ray single-layer mirror and two X-ray periodic multilayer mirrors arranged orthogonally has the advantages of achieving high energy resolution and improving the system tolerance to alignment errors.
The substrate material for the three concave cylindrical mirrors is silicon. A combination of grinding, lapping, and polishing processes is employed to achieve a super-smooth mirror substrate, with the roughness of the working surface controlled to be below 0.3 nm. Subsequently, X-ray thin films were deposited on the working surfaces of these three concave cylindrical mirrors using the magnetron sputtering technique. The structural parameters of the dual-energy grazing incidence KB microscope are presented in Table 2. The lengths of the three mirrors along the optical axis were all 10 mm. For the horizontal mirror, we used an X-ray Pt single-layer mirror, M1, with an object distance of 200 mm, system magnification of $M$ = 20, and grazing incidence angle of 0.3358$^{\circ }$. For the two vertical mirrors, we used X-ray multilayer mirrors, M2 and M3, with grazing incidence angles of 1.1431$^{\circ }$, which were coated with X-ray periodic multilayer films and operated at energy points of 6.4 and 9.67 keV, respectively. The structural parameters of the corresponding X-ray single-layer and periodic multilayer films were the same as those described in Section 2, and the reflection curves are presented in Fig. 2 and Fig. 4(a)–(b).
Table 2. Optical parameters of the dual-energy KB microscope.
In order to achieve the integration and alignment of the dual-energy KB microscope, we utilize supporting cones as reference structures for the three concave cylindrical mirrors. This approach enables high-precision positioning of the mirrors by using the supporting cones as a basis. The material of the supporting cones is fused silica, with machining precision of $\pm$10 $\mathrm{\mu}$m and $\pm$5$^{\prime \prime }$. Based on the visualization capabilities of Geant4, we utilized the detector classes constructed in Geant4 to create a three-dimensional visualization of the dual-energy KB microscope proposed above. Figure 6(a)–(c) depict a geometric model of our dual-energy X-ray KB microscope built in Geant4. Figure 6(d) illustrates the imaging path of the dual-energy KB microscope, while Fig. 6(e) demonstrates the path of X-rays emitted from the object, undergoing continuous reflection through the single-layer mirror M1 and spectral selection through the multilayer mirror M3. We used the G4Tubs class to create the M1 X-ray single-layer mirror substrate and the M2 and M3 X-ray multilayer mirror substrates. A 30 nm thick single-layer film of Pt was deposited on the surface of mirror M1, while periodic multilayer were deposited on the surfaces of mirrors M2 and M3, responding to 6.4 and 9.67 keV, respectively. The periodic multilayer structural parameters are displayed in Table 1. The mirror frame and supporting cones, C1 and C2, were constructed using G4Box and G4SubtractionSolid. These cones were used for mirror positioning and blocking a significant amount of stray light. The working surface of M1 is positioned closely against the supporting cones C1, while the working surfaces of M2 and M3 are in close contact with cones C2 and C3, respectively, ensuring the absolute positioning relationship of the two imaging channels relative to the same object. In C1 and C2, we included 3 mm X-ray apertures to allow the passage of X-rays. We used G4Material to define the materials, where the materials for the three mirror substrates M1, M2, and M3, as well as the cones C1 and C2, were silica, and the material for the mirror frame was Al. The roughness of both the single-layer and periodic multilayer mirrors was set to 0.3 nm.
Fig. 6. Geant4 visualization interface and schematic optical diagram of dual-energy KB microscope. (a) Front view of the dual-energy KB microscope geometry model. (b),(c) Oblique views. (d),(e) Schematic optical diagram.
3.2 Theoretical models
The geometric spatial resolution $\delta _{geo}$[31] of the KB microscope can be described by Eq. (10), which consists of two main components. The first component is the axial spherical aberration caused by the effective space of the system, representing the optimal spatial resolution achievable in the central field of view (FOV) of the microscope. The second component is the lateral aberration, which is a linearly increasing quantity as the FOV deviates from the center. During the fabrication of mirrors, there are influences such as surface figure error and roughness. Therefore, to determine the spatial resolution of the KB microscope, it is necessary to consider the geometric resolution, figure error, roughness, and diffraction effects. As a result, Eq. (10) [32] is used to describe the spatial resolution of the dual-energy KB microscope.
The numerical aperture $NA$ of a KB microscope is given by the following equation:
The intensity of X-rays emitted from the object point $(y,z)$ in the object field is denoted as $I_{0(z,y)}$. After X-rays pass through the dual-energy KB microscope, calculation of the intensity $I_{(z^{'},y^{'})}$ obtained at image point $(z^{'},y^{'})$ must take into account the throughput efficiency of the system and the response efficiency of the detector. The intensity $I_{(z^{'},y^{'})}$ obtained in the image plane can be expressed as follows:
3.3 Monte Carlo simulation
Based on the aforementioned geometric model established in Geant4, we studied the spatial resolution of a vertical mirror in the horizontal direction (Z-axis) for the dual-energy KB microscope. We placed Geant4 X-ray point sources every 20 $\mathrm{\mu}$m in the horizontal direction, covering a FOV of $\pm$200 $\mathrm{\mu}$m. We simulated the resulting dispersed image spots at each FOV point after the X-rays passed through the dual-energy KB microscope. To evaluate the resolution, we used the 80% diameter of the spot diagram in the image plane. We obtained the spatial resolution curve as a function of FOV, as illustrated in Fig. 7. The solid red line in Fig. 7 represents the spatial resolution calculated by the theoretical model in Section 3.2, while the dashed blue line represents the results obtained from the Geant4 simulation. The figure illustrates that the results obtained by the theoretical model and Geant4 simulation are in good agreement. The seven heatmaps in the figure display the point spread patterns at seven different FOVs. It should be noted that the direction of increasing FOV corresponds to the direction of increasing grazing incidence angle.
Fig. 7. Spatial resolution of the dual-energy KB microscope as a function of the FOV. The solid red line represents the spatial resolution calculated based on the theoretical model Eq. (10), while the dashed blue line represents the spatial resolution calculated by the Geant4 simulation.
We present the Geant4 visualization process of X-ray imaging in the dual-energy KB microscope using spontaneous emission imaging in the 6.4 keV channel. We placed a circular source spot with a radius of 0.5 mm in the object plane and simulated its distribution in the image plane after the X-rays passed through the 6.4 keV channel using XRTG4 and our proposed G4MultilayerReflection class, as illustrated in Fig. 8(a). An imaging detector was placed in the image plane and obtained four image spots on the detector. Image 1 represents the two-dimensional (2D) image of the system, formed by the reflection from the M1 X-ray single-layer mirror and the M2 X-ray multilayer mirror. Images 2 and 3 correspond to the vertical and horizontal one-dimensional(1D) images, respectively, after reflection from M2 and M1. Image 4 represents the straight-through X-rays of the system without reflection. Figure 8(b) displays the shapes and distributions of the four image spots on the detector, while Fig. 8(c) displays the 2D image (Image 1) of the 6.4 keV imaging channel.
Fig. 8. Geant4 image visualization of dual-energy KB microscope. (a) Geant4 X-ray grazing reflection visualization interface. (b) Image distribution heatmap. (c) Two-dimensional image 1 formed by reflection and focusing through M1 and M2.
To assess the spatial resolution of X-ray grazing incidence microscopes, it is common to perform backlit grid imaging experiments on laboratory setups or laser ICF devices. Monte Carlo imaging simulations are crucial as they allow for the prediction of the imaging performance of a system. To this end, we set up a circular backlight source with a radius of 0.5 mm in Geant4. In the object plane, we placed a four-quadrant and a #1000 mesh grid, with the mesh grid parameters presented in Table 3. We performed simulations of X-ray backlit grid imaging in Geant4 using these geometric setups. We treated the mesh grid as an absorber with gold in Geant4. The resulting imaging results for the four-quadrant and the #1000 mesh grid in the image plane are presented in Fig. 9. From Fig. 9, it can be seen that at positions near the edge of the FOV, the grid lines are blurred due to the decrease in spatial resolution. In Fig. 9, the decrease in edge FOV intensity in the horizontal direction (Z-axis) is due to the angular bandwidth of the X-ray multilayer mirror. In contrast, the intensity in the vertical direction (Y-axis) remains relatively uniform across the entire FOV due to the consistent intensity response of the X-ray single-layer mirror over a wider range of angles.
Fig. 9. X-ray backlit gold mesh grid imaging simulation. (a) Four-quadrant backlit mesh grid imaging simulation. (b) #1000 backlit mesh grid imaging simulation.
Table 3. Parameters of the four-quadrant and #1000 mesh grid.
To determine the system throughput efficiency of the dual-energy KB microscope, we assumed that the X-ray intensity in the object plane was uniformly distributed, that is, $I_{0(z,y)}$=1. We simulated the X-ray intensity distribution in the image plane of the dual-energy KB microscope using the Monte Carlo model based on Geant4. We obtained Fig. 10(a) and (b) for 6.4 and 9.67 keV, respectively, which represent the throughput efficiency distribution of the KB microscope with respect to the object FOV. The calculation principle of the above process is as follows: for each FOV point in the object plane, after the X-rays pass through the dual-energy KB microscope, we obtain a diffuse spot in the image plane. Therefore, we statistically analyze the X-ray intensity of each object FOV point in the diffuse spot in the image plane and calculate the average intensity. The calculation formula is presented in Eq. (13), which allows us to obtain the distribution map of the throughput efficiency of the dual-energy KB microscope as a function of the object FOV.
where $R_{(z,y)}$ represents the throughput efficiency of the dual-energy KB microscope at FOV point $(z, y)$, $N$ represents the number of X-rays within the diffuse spot in the image plane corresponding to each object FOV point, and $R_{i1}$ and $R_{i2}$ represent the reflectivity of each ray when it undergoes reflection on the single-layer film mirror and multilayer film mirror, respectively.It can be seen from Fig. 10(a) and (b) that along the Y-axis, the throughput efficiency remains relatively constant as it is predominantly influenced by the reflection from the X-ray single-layer mirror M1. However, along the Z-axis, where the X-ray multilayer mirrors M2 and M3 are responsible for the reflection, the throughput efficiency exhibits a peak.
As illustrated in Fig. 4(b), X-ray multilayers exhibit high reflectivity in the low-energy X-ray range. Therefore, it is necessary to add metal filters in front of the dual-energy KB microscope to obtain clean energy spectra. For this reason, we designed X-ray filters for the two energy points of the dual-energy KB microscope. In the above Geant4 Monte Carlo model, we added the Livermore low-energy electromagnetic model to calculate the transmission of the filters. The obtained transmission curves of the metal filters and the response curves of X-rays passing through the metal filters and dual-energy KB microscope are presented in Fig. 11. The filter pair consisted of 3 $\mathrm{\mu}$m of Al and 80 $\mathrm{\mu}$m of Be. It can be seen that after adding the filters, reflection in the low-energy range of the X-ray multilayer mirror’s reflectivity curves was almost completely eliminated.
Fig. 11. Transmission curves of 3 $\mathrm{\mu}$m Al and 80 $\mathrm{\mu}$m Be filters calculated using Geant4 (purple line), the calculated response efficiency curves of single-layer and multilayer mirror pairs (black and red dashed lines), and the response efficiency curve with filters (blue and green lines).
4. Experimental results
4.1 X-ray backlit imaging results
The experimental results of the four-quadrant X-ray backlit imaging in the dual-energy KB microscope are presented in Fig. 12. Two imaging channels were used with iron- and tungsten-anode X-ray tubes as backlit sources, operating at 40 kV and 20 mA. An X-ray Charge-Coupled Device(CCD) was used to record the images in the image plane, with a pixel size of 4.54 $\mathrm{\mu}$m, 2750 $\times$ 2200 pixels, and an exposure time of 1800 s. Figure 12(a) and (d) present the results of backlit grid imaging at 6.4 and 9.67 keV, respectively. Figure 12(b) and (e) display the intensity distribution along the green lines in Fig. 12(a) and (d), respectively. Boltzmann fitting is performed for each black-to-white edge response, and the 20%—80% intensity criterion is used to evaluate the spatial resolution [34,35]. This corresponds to the distance in the object space as the image intensity changes from 20% to 80% in an edge response function [31]. The measured and simulated spatial resolutions are presented in Fig. 12(c) and (f). It can be observed that the spatial resolution in the central FOV was 3.63 $\mathrm{\mu}$m and 2.72 $\mathrm{\mu}$m for the 6.4 and 9.67 keV imaging channels from Fig. 12, respectively. At a $\pm$200 $\mathrm{\mu}$m FOV, the spatial resolution reached 7.2 $\mathrm{\mu}$m and 6.8 $\mathrm{\mu}$m. Furthermore, the experimental and Geant4 simulation results were in good agreement.
Fig. 12. Results of X-ray backlit imaging. (a),(d) Results for gold mesh in the 6.4 keV imaging channel using an iron-anode and 9.67 keV imaging channel using a tungsten-anode. (b),(e) Intensity profiles along the green lines in (a) and (d), respectively. (c),(f) Experimental and Geant4-simulated spatial resolutions for the 6.4 keV and 9.67 keV imaging channels.
4.2 Throughput efficiency calibration results
After the X-ray backlit imaging experiments, we conducted a throughput efficiency calibration experiment for the 6.4 keV imaging channel. The iron-anode X-ray tube was operated at 30 kV and 15 mA. A Si-PIN detector (Amptek, XR-100CR) was placed at a distance of $s$ = 1000 mm from the system to measure the input and output X-ray spectrum, with a counting time of 60 s. The calibration principle is illustrated in Fig. 13. The gold mesh grid was replaced with a 15 $\mathrm{\mu}$m pinhole, and the detector was placed at the position of the 2D image of the system to measure the output X-ray spectrum. When the input X-ray spectrum was measured, the dual-energy KB microscope was removed from the optical path, and a 1$\times$1 $\rm {mm}^{2}$ rectangular pinhole was added in front of the Si-PIN detector. The throughput efficiency of the dual-energy KB microscope was calculated using the following formula:
Figure 14 illustrates the measured throughput efficiency curves for the 6.4 keV imaging channel as a function of FOV. The theoretical curves are derived from the profile data extracted from Fig. 10(a). Measurements of the input and output X-ray intensities were performed at intervals of 20 $\mathrm{\mu}$m (object plane) in the range of $\pm$160 $\mathrm{\mu}$m FOV. It can be observed that the measured throughput efficiency curve follows a similar trend to the simulated curve. We used the RMSE to measure the difference between the simulated and experimental throughput efficiency values. The RMSE for Fig. 14(a) and (b) is 8.7% and 9.5%, respectively. The reason why the measured curve is lower than the simulated curve is the degradation of the X-ray multilayer due to exposure to air.
5. Conclusion
In this study, Monte Carlo models were constructed for X-ray grazing incidence mirror elements and an X-ray grazing incidence microscope system. We introduced the XRTG4 extension package to describe the physical processes of X-ray single-layer mirror grazing incidence reflection and developed the G4MultilayerReflection class to describe the grazing incidence reflection of an X-ray multilayer mirror. We designed a dual-energy KB microscope for laser ICF, which operates at 6.4 and 9.67 keV. Furthermore, we performed Monte Carlo simulations using Geant4 to investigate the spatial resolution and throughput efficiency of the microscope to predict the system performance. We performed simulations of X-ray backlit grid imaging and spontaneous emission based on Geant4. The spatial resolution results obtained by the Geant4 Monte Carlo simulations, theoretical model, and X-ray backlit imaging experiments were in good agreement. We also conducted throughput efficiency calibration experiments for the 6.4 keV imaging channel and evaluated the difference between the experimental and Geant4-simulated throughput efficiencies using the RMSE. The obtained RMSE values for the dual-energy KB microscope were 8.7% along the Y-axis and 9.5% along the Z-axis. The obtained experimental results were in good agreement with the Geant4 simulation results, thus demonstrating the reliability and accuracy of the proposed Geant4 Monte Carlo analysis for determining X-ray microscopy performance.
Funding
National Natural Science Foundation of China (12005157).
Acknowledgments
The authors acknowledge the researchers and technicians of China Academy of Engineering Physics for their support and suggestions for the experimental scheme design.
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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