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Abstract
High-resolution X-ray imaging diagnosis is a critical method for measuring Rayleigh-Taylor instability growth and hot spot interface morphology in inertial confinement fusion experiments. In this study, we develop a quasi-monochromatic elliptical Kirkpatrick–Baez microscope based on aberration theory, breaking the aberration limit of conventional Kirkpatrick–Baez microscopes. The microscope was characterized in the laboratory for spatial resolution performance and modulation transfer function before being implemented in cavity experiments at the SG-III prototype laser facility. The results demonstrate that the edge-based method achieves a spatial resolution of <2 µm in the central field of view and modulation of 800 lp/mm spatial frequency of >20%.
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1. Introduction
High-mode instabilities are significant factors limiting implosion performance in indirect drive inertial confinement fusion (ICF) experiments [1,2]. An incident laser energy can cause a deuterium-tritium (DT)-filled capsule to compress inward and reach ignition conditions. The pre-existing amplitude density perturbations in the capsule grow rapidly during implosion compression, resulting in distorted hot spots during the stagnation phase, limiting the temperature and density of the DT fuel at peak compression [3–5]. High-resolution imaging of the X-ray emission from the implosion capsule can accurately reflect Rayleigh–Taylor instability growth and hot spot interface perturbation [6–8]. X-ray imaging diagnostics are used in different stages of the implosion process for the study of high-mode instability, such as the emergence and growth of instability perturbations at the ablation front in the compression phase and the fine characterization of the hot spot interface morphology in the stagnation phase.
Currently, Kirkpatrick–Baez (KB)-type microscopy is a reliable diagnostic method with superior spatial resolution and collection efficiency over pinhole cameras, and it is commonly used for imaging laser-plasma X-ray emissions [9,10]. A 16-image KB microscope with a response energy range of 2–8 keV and a spatial resolution of ∼6 µm was used on the OMEGA laser system to measure the time-varying size and shape of the implosion hot spot [11]. The polar-view KB microscope used at the SG-III laser facility can image a stagnant hot spot with several discrete energy points, with a resolution of 3–5 µm [12]. The NIF’s KB microscopes detected high-mode perturbation in the low-convergence implosion by the inner surface of the capsule with tungsten [13,14]. Measuring hot spot high-mode perturbations with high convergence remains a challenge, which requires better resolving power, and the achievable spatial resolution of KB systems is dominated by spherical mirror aberrations. To improve imaging performance, Kodama et al. proposed an advanced Kirkpatrick–Baez microscope with two pairs of elliptic and hyperbolic mirrors [15]. This microscope satisfies Abbe’s sine condition, effectively correcting spherical aberration and field obliquity, and achieves a spatial resolution better than 3 µm at 2.5 keV energy points on the GEKKO-XII laser system. However, multiple reflections result in a significant reduction in the reflection efficiency, as well as significant uncertainty in adjustment and assembly.
In this study, we propose a novel elliptical mirror-based KB microscope to meet the requirements for the diagnosis of high-mode instability in ICF research. Observations of high-mode instability require better than 3 µm spatial resolution to obtain high-precision X-ray images of high-temperature plasmas in the relevant experiments [16]. The elliptical mirror’s optical design is used to achieve a breakthrough in spatial resolution, and a double-mirror KB configuration ensures that the adjustment accuracy is within a controllable range. Geometric aberration calculations and ray-tracing simulations were used to comprehensively evaluate the imaging performance of this microscope. A periodic multilayer technique was used to achieve a quasi-monochromatic response at an 8-keV energy point. Offline X-ray imaging and spatial resolution characterization of the elliptical KB microscope were conducted in the laboratory before being implemented in the SG-III prototype laser facility. Backlight X-ray imaging results show that the central field of view (FOV) can achieve a spatial resolution that is less than 2 µm and a modulation greater than 20% for a spatial frequency of 800 lp/mm.
2. Elliptical KB optic
2.1 Optical design
The development of elliptical-cylindrical KB optics has been previously reported [17]. The elliptical mirror type can eliminate on-axis spherical aberration in the spherical mirror system, making it a better diagnostic scheme for improving central FOV spatial resolution. The theoretical spatial resolution of this system, including off-axis coma and field curvature, is discussed in the following section. A pair of orthogonally placed elliptical mirrors enable two-dimensional X-ray imaging by successive reflections in the elliptical KB microscope configuration. The dual-mirror design, as with conventional KB microscopes, corrects severe astigmatism caused by inconsistency in optical power with a single mirror. We used an optical design with a small magnification and low response energy point in previous studies of elliptical KB optics. This solution significantly reduces the precision requirements for assembly and adjustment, as well as makes mirror pair synthesis and X-ray optical path construction much easier in the laboratory. Meanwhile, the lower magnification limits the accuracy of the spatial resolution calibration in actual testing, and the working energy point of the system does not fully match the actual diagnostic application requirements. In order to be suitable for X-ray plasma diagnosis at the high-power laser facility, the working energy point is designed to be 8.0 keV (corresponding to Cu Kα1 line) to ensure a strong observation depth. The system magnification has been increased to ∼25.0×, considering the impact of detector pixel size on ultra-high spatial resolution testing. This design result in difficulties in optical path alignment, and improved system setup methods are discussed in section 3.
The schematic of the KB microscope based on elliptical mirrors is shown in Fig. 1, and the object point and image point of the optical system constitute the two focal points of the ellipse. Unlike spherical mirror imaging, different positions on the ellipse can be divided into various mirror shapes, so the ellipse parameters and the distance from the center of the mirror to the long axis of the ellipse must be provided to determine the surface shape of the mirror. The following equation can be obtained by determining the correlation between the geometrical optics principle and the ellipse mathematical formula:
where u0 is the object distance, M is the magnification, θ0 is the central grazing incidence angle, 2a is the major axis of the ellipse, 2c is the focal length of the ellipse, and h is the distance from the center of the mirror to the major axis. The initial structural parameters of the elliptical mirror can be obtained explicitly from the given object distance, magnification, and grazing incidence angle. The meridional and sagittal directions need to be calculated independently because the monolithic mirror only achieves one-dimensional focusing. The energy response of the system, distance to the implosion target, total limitation length, and expected optical performance are considered. The object distance is designed as ∼200 mm, the magnification is set to ∼25.0×, and the geometric solid angle is calculated as ∼2.7 × 10−7 sr, according to the actual conditions of diagnostic experiments. More detailed optical parameters are listed in Table 1.
Table 1. Parameters of the elliptical KB mirrors
2.2 Multilayer design
The elliptical fused-silica mirrors have peak-to-valley shape accuracy of ∼1.8 nm, a slope error of ∼0.05 µrad, and a root-mean-square surface roughness of ∼0.2 nm in the useful area of the reflective surface. The mirrors were deposited with W/Si multilayers and the grazing angle was designed to be 1.0° to respond at an 8.0-keV energy point. The W/Si multilayers are manufactured using magnetron sputtering technology and are deposited on a substrate made of alternating layers of high and low Z materials. The design bilayer number is 14, period thickness is 4.78 nm, thickness ratio is 0.535, and roughness is 0.55 nm. The simulated and measured W/Si multilayer reflectance curves are shown in Fig. 2. A customized iterative algorithm is used to generate the simulated reflection curve as a function of the grazing incidence angle, and the measured data is examined using an X-ray diffractometer. The measured reflectivity curve is consistent with the theoretically calculated reflectance waveform. Simultaneously, the peak reflectance is slightly lower than the theoretical value of 58.4%, which could be due to nonideal factors such as the interface roughness and material absorption in the working spectral region. The angular bandwidth of the W/Si periodic multilayers is ∼0.10°, which can transmit the FOV from the object surface up to 375 µm.
Fig. 2. Experimentally obtained and simulated multilayer reflectivity curves versus grazing angle (8.0 keV).
2.3 Aberration theory
The aberration analysis of the grazing incidence imaging system is critical for setting appropriate optical parameters and predicting imaging performance. Geometric aberrations of the elliptical KB microscopes can be divided into two types. One is that the image points are not in the same image plane because of the difference in object and image distances in different FOV, which is a narrow beam aberration typically manifested as field curvature. Another type of aberration is due to the variation in focusing capabilities with the position of the aperture, resulting in an asymmetrical diffuse spot on the ideal image plane, which is a wide beam aberration that predominantly behaves as a coma. The linear superposition of field curvature and coma is a comprehensive manifestation of geometric aberrations, which can directly characterize the imaging performance of the elliptical KB imaging system [18].
Similar to traditional spherical mirror reflection, elliptical imaging can also calculate the aberration of one-dimensional focus directions. As shown in Fig. 3, the central ray from the left focal point F1 of the ellipse is reflected by the mirror and then imaged on the ideal image plane I0, with the image point located at the right focal point F2 of the ellipse. For the chief ray passing through the off-axis object point Q, the angle of view is $\sigma $, and the object distance $u^{\prime} = {u_0}/\cos \sigma $, which is finally focused on the point P of image plane I′. Because the vertical distance between two image planes is defined as the field curvature, the lateral aberration caused by the field curvature can be expressed as the product of the field curvature and the light collection angle $\eta$ of the image square
where $v^{\prime} = |{PM} |$ is the ideal image distance of point Q and ${v_0}$ is the distance from the mirror center to the image plane I0, which is written asTo explicitly analyze the impact of lateral aberration caused by field curvature, the discussion is conducted separately from the two mirror areas of BM and CM, and the field curvature calculation is performed first on the curved surface BM. According to the sine theorem, in $\Delta \textrm{QBM}$ and $\Delta \textrm{PBM}$, these equations are satisfied:
According to (4)(5)(8), we can obtain the field curvature expression of the mirror area BM, and the same method is used to derive the mirror area CM. The field curvature expression of the elliptical KB microscope is simplified after superposition to
Substituting the optical structure parameters in Table 1 into Eq. (6), the spatial resolution curve caused by field curvature is plotted as a function of the object FOV (Fig. 4). The FC curve maintains a linear increase as the FOV extends from negative to positive, while the aberration value at the central viewpoint is 0, demonstrating that the off-axis image point of the elliptical KB microscope constitutes an inclined ideal image plane.
Fig. 4. The theoretical calculation curve of field curvature and coma of the elliptical KB microscope. The calculated values are divided by magnification.
The elliptical coma can be characterized as the average value of the distance between the chief ray and edge rays on both sides of the image plane, then the coma of the elliptical KB system can be written as follows:
where ${Y_1}$ and ${Y_2}$ are the vertical distances from the intersection of the ray reflected from point B and point C on the mirror surface to the chief ray, respectively. The coma of the microscope is calculated based on the design parameters in Table 1, and the corresponding aberration curve also has a linearly increasing relationship with the FOV (Fig. 4). However, because the value of a coma is ∼10−2 µm, which is much smaller than the field curvature, its impact on the spatial resolution can be neglected, indicating that the elliptical mirror of the elliptical KB system can amend coma smoothly.The lateral aberration of the elliptical KB system is obtained by combining the above conclusions on field curvature and coma, which can directly characterize the imaging resolution, and the theoretical calculation results are shown in Fig. 5. The elliptical KB microscope eliminates on-axis spherical aberration and effectively reduces coma aberration compared to the spherical KB microscope according to Abbe's sine condition. Simultaneously, a ray-tracing program was used to simulate the resolution curve varying with the FOV, which is consistent with the theoretically derived curve. This simulation method is implemented by collecting the reflection information of a point source at different positions perpendicular to the optical axis in a specific optical system. Furthermore, a spherical KB microscope with the same object distance, magnification, and grazing incidence angle was designed for comparison, and the simulation results of its resolution are shown in Fig. 5. The spatial resolution curve shows that within 200-µm FOV, the spherical structure has a resolution effect level of ∼2.7 µm, while the elliptical structure can reach a better level of ∼2.0 µm. The elliptical design can form a high-resolution area of ∼1 µm in the range of 100 µm for the central FOV, which is an incomparable advantage over conventional spherical microscopes.
3. Experimental arrangement
Figure 6 shows the experimental arrangement of the elliptical KB microscope in the X-ray imaging laboratory. The elliptical mirror pair is mounted as an integral part on a six-axis electronically controlled manipulator, which was previously assembled synchronously using a coordinate measuring machine (CMM) to ensure orthogonal elliptical mirror alignment in the meridional and sagittal directions. The backlit X-ray source with a Cu anode emits Kα1 characteristic line emission at 8.0 keV, with an illuminated area of ∼1 × 1 mm2. The imaging object close to the backlight is a gold grid, and the replacement of the various periodic grids is performed under the monitoring of double optical lenses to ensure positioning accuracy of ±10 µm. A hard X-ray CCD detector (FDS 5.02 MP, Photonic Science), with a pixel size of 4.54 µm and a response area of 10 × 10 mm2 was used to capture images on the image plane. A 4.5-m long helium pipe was placed downstream of the mirror pair to reduce the attenuation of X-rays in the air environment and enhance the signal-to-noise ratio.
A pair of prisms with tightly controlled dimensions and inclination rests against the non-working area of the mirror plane, providing a calibration datum without affecting the imaging beam. The prisms have a premarked reticle on both sides that indicates the axial positions of the mirror pair as determined by the theoretical sag of the ellipsoid parameters. An internal focusing telescope was used to observe the reticle features and centered cross-wire intersections for alignment, with an estimated positional accuracy of ∼0.05 mm. Additionally, the well-polished surface of the prism on the image side is set to be highly perpendicular to the optical axis of the mirror pair, allowing the angular deviation of the mirror pair to be indirectly measured using collimated beams and a position-sensing detector located on the axis of the experimental platform through a prism-polished plane. The optical axis of the mirror pair is moved to coincide with the axis of the experimental platform by adjusting the position and rotation angle of the mirror pair with a precision manipulator, and a pitch and roll alignment accuracy of ∼0.5 µrad.
The alignment process has the following advantages: First, an identifiable optical axis is provided, which uses visible light for pre-aiming to minimize uncertainty. Specifically, the backlight grid can be precisely positioned on-axis using a fixed internal focusing telescope, the object distance can be set to a preset value using the CMM, and the theoretical image point can be quickly found at a 5-m image distance with the collimated beams. Second, when used in the subsequent implosion experiment at the SG-III prototype laser facility, the mirror module has a known datum for coupling to the target chamber center (TCC) and the record plate.
4. Results and discussion
4.1 Spatial resolution
Figure 7 shows the X-ray grid backlight image and the laboratory test results of the elliptical KB microscope’s resolution characteristics. The periodic multilayers set-up enables the imaging system to have a good reflection efficiency at 8.0 keV, and the pattern of the brighter center and darker edge regions is consistent with the angular response of the mirror pair. The 1000-mesh Au grid shown in Fig. 7(a) was imaged with a continuous X-ray source operating at 38 kV and 27 mA, with an exposure time of 30 min. The grid period is ∼25 µm, and the bar width is ∼6 µm. Clear and sharp grid lines are observed at the center FOV, gradually blurring toward the edges.
Fig. 7. Laboratory grid imaging results of the elliptical KB microscope. (a) Backlit image of a 1000-mesh Au grid collected on a CCD detector; (b) Spatial resolution across the FOV measured using a “10%–90%” criterion.
The spatial resolution is calibrated using a 10%–90% criterion for the grid’s shadow edges [19]. This measurement is repeated in the orange line (Fig. 7(a)) to obtain a spatial resolution across the FOV and a polynomial fit is then performed to estimate the resolution curve, which is shown in Fig. 7(b). The best spatial resolution of the elliptical KB microscope is estimated to reach ∼1.13 µm, and the spatial resolution is <2 µm within a 300-µm FOV, with a significant improvement compared to previously reported results [17]. Note that some unavoidable situations in practice can cause additional test errors in the spatial resolution of the microscope. Under current fabrication techniques, the bar edges of the Au grid are not strictly sharp, resulting in blurring at the black-to-white edges of the image plane, limiting high-resolution evaluations. Additionally, the pixel size of the CCD detector contributes to some estimation deviation. The grazing incidence angle varies on the mirror surface for the fringe FOV, and a multilayer narrow angular bandwidth of ∼0.1° reduces the reflectivity of a significant part of the mirror surface, sacrificing the partial reflection efficiency in exchange for a better on-axis and off-axis spatial resolution [20]. The characterization of optical system aberrations is based on geometric optics and assumes uniform reflectivity across the mirror surface, which ignores the impact of multilayer reflection efficiency, resulting in fringe FOV with a measured resolution that appears to be better than theoretical result.
4.2 Modulation transfer function
The modulation transfer function (MTF) was used to describe the image contrast value of different spatial frequencies over the original object contrast and can be used to evaluate the imaging performance of elliptical KB systems [21,22]. In the experiments, a 30-µm thick knife-edge Au foil was precisely placed on the object plane to measure the X-ray MTF of the microscope system. The MTF can also be calculated from the grid image, but this method has additional measurement errors due to the grid manufacturing process. A typical scanning electron microscope (SEM) photograph of a partially opaque Au knife-edge with a sharp and straight laser-cut edge is shown in Fig. 8(a). Double optical monitoring lenses can ensure the positioning and replacement accuracy of the knife-edge. The X-ray images of the vertical and horizontal knife-edges passing through the optical system recorded by the CCD detector are shown in Figs. 8(b) and 8(c), respectively. The microscope edge spread function (ESF) data were obtained from knife-edge radiographs and subjected to multiple noise reduction processing using filtering algorithms and parameter fitting. The corresponding line spread function is derived from the ESF, then Fourier transforms and normalizations are performed to obtain the MTF curve.
Fig. 8. (a) The SEM photograph of a 30-µm thick Au knife-edge; (b) and (c) are the X-ray images of the vertical and horizontal knife-edges with 30 min integration, respectively; (d) and (e) are the MTF curves of discrete horizontal and vertical FOV (0, +50, +100, +150), respectively.
The vertical knife-edge was scanned and photographed along the horizontal FOV using an electronically controlled displacement stage, and the MTF curves corresponding to different horizontal positions were obtained. The discrete MTF curves of horizontal FOV (0, +50, +100, +150) are shown in Fig. 8(d). The curves of MTF versus line pairs contained in each pixel (lp/pixel) are first obtained from the knife-edge images, then a coordinate transformation is performed based on the CCD pixel size and horizontal magnification to obtain horizontal MTF curves as a function of object spatial frequency (lp/mm). The MTF curves for the vertical FOV were calculated in the same way from the horizontal knife-edge radiographs shown in Fig. 8(e). The experimental results show that the elliptical KB microscope has a relatively consistent imaging performance in the horizontal and vertical directions. The central FOV has the optimal MTF in both dimensions, with the slowest decline in image contrast with increasing spatial frequency, implying a better ability to reproduce fine details. The modulation of the center FOV is greater than 20% for a spatial frequency of 800 lp/mm. Meanwhile, the elliptical microscope can achieve ∼20% modulation in a ±150 µm FOV for a spatial frequency of 500 lp/mm.
4.3 Implosion experiment imaged
The high-resolution X-ray microscope was implemented in the cavity experiment of the SG-III prototype laser facility. The metal cylindrical cavity had a diameter of 1700 µm and a height of 1375 µm, with two laser entrance holes. The cavity, which is located at the TCC, is equipped with a grid at the same azimuth as the microscope for diagnostic indication. Target aiming is achieved in the online alignment using the pitch and roll of the integral KB module. The positioning accuracy of the object-side target reaches ±30 µm to meet the requirements for the imaging FOV area and spatial resolution. This was achieved by adjusting the precise translation and rotary stages to coincide with the preset microscope object point and grid center under the online optical observation system. A diaphragm hole was placed in front of the KB module to effectively shield stray light from outside the FOV, while a 100-µm thick Kapton and 200-µm thick polycarbonate were installed for debris shielding. Two 632.8 nm lasers on the base provide synchronized image indication, which can eliminate the uncertainty of the implosion image.
Eight laser beams were injected into the cylindrical cavity from two laser entrance holes, each with a 3$\omega$ (351 nm) energy of 800 J in 1 ns square pulse, resulting in an X-ray radiation environment illuminating the grid. A static implosion image with a period of 12.5 µm grid, captured using a Fuji image plate, is shown in Fig. 9(a). The detected integrated signals are recorded as 104 order counts with a pixel size of 25 × 25 µm2, which is insufficient for resolution measurements. The imaging window is determined by the plasma radiation area and the angular response of the microscope multilayer. According to the grid target imaging results, the horizontal and vertical magnifications of the elliptical KB microscope are ∼24.1 and ∼22.5, respectively. Line profiles were drawn through the central 150 µm area of the image (Fig. 9(b)) to explore the details of contrast. Intensity distribution curves are plotted in the horizontal and vertical directions, with a perpendicular average of 10 pixels to reduce noise. The magnifications are corrected and the central peak positions are aligned for direct comparison. The observed peak-to-valley variation is evident, and the pixel size of the image plate limits the characterization of the system’s resolving performance, as only one-pixel width corresponds to approximately 1-µm distance on the object surface. A better way to record X-ray images is to use a vacuum CCD camera with a comparable spatial resolution, which currently suffers from insufficient signal strength. The line profiles in the two orthogonal directions are substantially close, indicating that the meridional and sagittal directions have consistent throughput and signal-to-noise ratios. The elliptical KB microscope provides a powerful observation method for diagnosing high-mode instabilities, and the higher spatial resolution can reveal more details not obtained with previous setups. Advanced optical configurations can be further explored to expand the range of high-resolution imaging based on aberration analysis methods. Additionally, the microscope enables dynamic imaging diagnostics of implosion processes combined with a streak camera.
Fig. 9. (a) Static implosion X-ray grid image with 12.5 µm period; (b) Lineout in the orthogonal direction of the image through the central 150 µm area.
5. Summary
In this study, we present the development of an elliptical KB microscope to address the critical need for high-modal instability diagnosis in ICF research. The breakthrough in spatial resolution enables a refined characterization of capsules in implosion diagnostic experiments. The elliptical KB microscope has a quasi-monochromatic response at 8.0 keV and an effective FOV of 400 µm. The laboratory backlit grid imaging results show that the microscope can achieve an optimal resolution of ∼1.13 µm and a spatial resolution of <2 µm within a 300-µm FOV. The knife-edge measurements demonstrate that the modulation of the central FOV is greater than 20% for a spatial frequency of 800 lp/mm, and the modulation within ±150-µm FOV is ∼20% for a spatial frequency of 500 lp/mm. The microscope was successfully used in the cavity experiment of the SG-III prototype laser facility and can be used to observe dynamic target high-mode instability in subsequent implosion experiments.
Funding
National Natural Science Foundation of China (12005157).
Acknowledgments
The authors acknowledge the support of the staff at the Research Center of Laser Fusion.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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