Design and raytrace simulations of a multilayer-coated Wolter x-ray optic for the Z machine at Sandia National Laboratories

Wolter optics¹ offer major advantages over previously developed diagnostics for applications as x-ray diagnostics for high energy density physics, such as spherically bent crystals,² pinhole cameras,³ and basic grazing-incidence optics.⁴ A Wolter point-to-point imager enables coverage of a large field of view (mm to few cm) while obtaining simultaneously good throughput at (narrow or broadband) energies ≳15 keV and high spatial resolution (100-200 μm) overcoming limitations of the other above-mentioned technologies at the expense of requiring more elaborate calibration measurements, better alignment accuracy, and more complex image analysis techniques. This is due to the fact that the off-axis response of these optics is more intricate than that for most of the above-mentioned diagnostics.

The basic layout of this type of optic consists of two conical mirror surfaces to reflect incoming photons twice, focusing the image of the observed object on a detector positioned in the focal plane of the microscope (Fig. 1). There are three families of Wolter designs with the nestable version (Wolter I) being the one most commonly used. A Wolter I microscope basically acts like a thin lens and addresses two challenges of grazing-incidence optics: (1) the approach of nested designs allows for an increase in collection efficiency by adding additional shells (and therefore the effective area) to the optic and (2) the two conic surfaces of revolution nearly satisfy the Abbe sine rule, reducing the strongest aberration contributions resulting in sharper images.

Here we present the design and raytracing results for a multilayer-coated Wolter I optic (Z Wolter) devised and built for implementation in the Z machine at Sandia National Laboratories (SNL) to diagnose warm x-ray sources.⁵ The small-diameter Z Wolter is a replicated optic⁶ with W/Si multilayer (ML) coatings and consists of hyperbolic and ellipsoidal mirror surfaces for the first and second reflections, respectively. In this paper, we first introduce the Wolter layout and the basic concept of using multilayers in order to enhance the spectral performance of an x-ray optic in Sec. II, followed by an outline of the specific requirements for the Wolter based on the needs of the experiments on Z (Sec. III). Section IV details the design of the optic and the corresponding raytrace simulations completed to verify the performance expectations, especially in terms of spatial resolution and throughput. Before concluding, the fabrication process of the first Z Wolter optic is outlined in Sec. V and the usefulness of the raytrace for data interpretation will be shown.

The geometrical design of a Wolter aims at optimizing its performance for the intended use while taking into consideration any restrictions due to the experimental setup as well as fabrication limits. The shape of the Wolter I mirror surfaces is defined by four conic parameters: the semi-minor and semi-major axes for the elliptical surface and the semi-major axis along with the eccentricity for the hyperbolic mirror. These parameters are set to take into account the available space for the total throw of the system (the source-to-detector distance), magnification, grazing angle, and mirror length. Moreover the Wolter design parameters must allow for fabrication of the optic: Optic shells with radii smaller than about 20 mm are challenging to replicate and very steep angles between the two reflective surfaces make it impossible to release a replicated shell after fabrication.

While the geometric layout of the Wolter optic largely determines the field of view (FOV) and spatial resolution, the spectral response and throughput can be tuned and enhanced by depositing coatings onto the mirror surfaces. Generally the reflectivity for single-layer coatings is high up to a critical angle θ_c, but by depositing alternating layers of low- and high-mass materials, (reflectors like tungsten and spacer layers such as silicon) the energy range can be extended. These multilayers for the x-ray regime have typical bilayer thicknesses (d-spacings or periods) of the order of a few nanometers. ML coatings make use of Bragg’s law, mλ ≈ 2d sin θ, where m is an integer, λ is the wavelength of the incoming photon, and d is the bi-layer period. The grazing incidence angle is labeled θ. With a constant d-spacing for all bi-layers, the ML acts as a notch filter selecting a particular narrow energy range (see Fig. 2, top). In general, larger energies require smaller periods. The current practical limit on d is 1.5 nm, which has been demonstrated on flat substrates. If an application requires the coverage of a broader range in energy or angles, this can be achieved by depth-grading the ML or using aperiodic designs at the cost of lower reflectivity (Fig. 2, bottom). Additional experimental requirements might have to be considered in the design of a ML, such as a good rejection of photons with energies outside the region of interest as well as a throughput that is flat enough over the FOV to be compatible with the dynamic range of available detectors.

One of the major challenges for small-radii optics—as needed in the present case due to experimental restrictions—is the deposition of MLs on a mandrel before replication and thermal release of the mirror substrate together with the ML instead of simply coating a replicated substrate. This process has recently been developed,^7,8 and additional details on this technique can also be found in Sec. V.

SNL is currently developing non-thermal wire arrays⁵ that can be used as x-ray sources for energies greater than 15 keV. This requires a diagnostic with better spatial resolution and higher signal-to-noise than what is currently available to image the structure of this type of load. For the inital Wolter optic, wire arrays made of molybdenum and silver were considered. The competing state-of-the-art imaging diagnostic at Z for this purpose is a series of pinholes with a Ross-pair filter³ (Time-integrated pinhole camera, TIPC) providing a spatial resolution of about 900 μm with a solid angle of 10⁻⁷ sr. To study small features, the requirement for the Z Wolter was to achieve a resolution better than 100-200 μm (in the source plane) over a field of view of 5 × 5 × 5 mm³ with a collection solid angle of 1 × 10⁻⁴ sr. The experimental constraints for the Wolter required a minimum optic-to-source distance of 600 mm in order to avoid damage from debris as well as the shock during a Z-pinch experiment. The maximal source-to-detector distance was limited to 3050 mm, such that optic and detector could be mounted on a 0°-port at Z. To reduce cost, a single geometric layout was designed to be used for both Mo K_α (17.5 keV) and Ag K_α (22.3 keV) energies. This requires different MLs but only one set of fabrication mandrels for replication. The upper limit on the bandpass was set to 1 keV full width at half maximum (FWHM), with a requirement that the contrast ratio (in-band versus out-of-band) of reflected photons was greater than 5. In order to end up with enough photons per resolution element, the two-bounce reflectivity of the multilayer needed to exceed 1%.

To meet the design requirements both the geometric layout of the Wolter as well as the ML recipe (composition of the ML) were optimized using IDL (Interactive Data Language) based algorithms together with Interactive Raytrace (IRT) libraries to verify the performance of the design. In this process, the total throw and magnification were fixed at 3.05 m and 3.5, respectively. This was feasible from both an experimental as well as a fabrication standpoint. In order to allow for the use of the same geometric layout, we optimized for the more restrictive case of 22.3 keV with a grazing angle of about 0.6°. The performance degradation for using the same Wolter geometry for the 17.5 keV Wolter was minor. The final design required an intersection radius between the two mirror surfaces of 23.3 mm and the length of the hyperbolic and the elliptical mirror were set as 30.0 mm and 30.8 mm, respectively.

The multilayer was chosen out of several candidate recipes from a multi-parameter optimization process to obtain the most suitable spectral response of the optic for the Z pinch experiment at 17.5 keV: maximal throughput, good out-of-band photon rejection, and acceptable dynamic range. Two final design recipes were determined for the ML, both W/Si: one with a constant d-spacing and one with a depth-grading of d.

To verify the design, raytracing was performed to simulate the detailed spatial resolution of the Wolter optic as a function of the off-axis position (radially symmetric in the ideal case) as well as a function of the source-to-optic distance (depth of focus). In order to characterize the spatial resolution, a half-power diameter (HPD) and a 2-dimensional Gaussian (elliptical HPD) were calculated. The HPD is the diameter within which half of all focused x-rays are enclosed and is generally the most suitable measure if highest accuracy spectroscopy and flux measurements are needed. The elliptical HPD, i.e., a 2-dim Gaussian encompassing half of the total flux, is closer to the FWHM and generally provides better, i.e., smaller, resolution values. It is more suitable than a HPD for high resolution imaging, for which a HPD provides a conservative estimate. The results for the simulated resolution in terms of HPD are shown in Fig. 3 and include the initially expected figure error as well as the ML response (as designed) for the Z Wolter. While the off-axis degradation of the spatial resolution in the plane perpendicular to the optical axis is relatively slow, the spatial resolution worsens quickly with the change of the source-to-optic distance. The required FOV for the Z Wolter is indicated by the dashed lines. Overall the simulated Wolter performance indicated that the required 100-200 micron resolution over the desired FOV was achievable with the final Wolter design.

Multilayer performance was optimized for narrow bandpass and high efficiency. The throughput together with the dynamic range over the FOV was calculated from the raytrace simulation and found to satisfy the requirements for the proposed experiments on Z (see Fig. 4). The variations in throughput along the source-to-optic distance are less pronounced than those from off-axis source positions. Here the input photon numbers are based on an anticipated flux of 100 J/cm² in the spectral band of interest (17-18 keV) and resolution elements of 100 μm³ within the wire to determine the number of photons entering the Wolter optic for the raytrace simulations. The issue of the high dynamic range over the field of view was addressed by using a stacked image plate (IP) setup with both IPs covering the entire FOV. This avoids the loss of information due to saturation in high intensity locations while preserving the overall IP sensitivity.

To demonstrate the capabilities of this type of diagnostic, we also performed simulations of 3-dimensional sources. More specifically, 41 spherical sources were uniformly distributed along the surface of a cylinder of 20 mm length and 0.375 mm in radius, similar in dimensions to the wire arrays used in Z. The sources were uniformly filled spheres with a radius of either 0, 25, 50, 100, 150, or 200 μm, and the simulations included various degrees of the figure error for the Wolter. The figure error in the simulations was initially based on the best estimates from similar Wolter optics and later updated to reflect the actual values for the fabricated optic. An example of these raytrace simulations for a source radius of 50 μm with a 450-μm spacing between sources is shown in Fig. 5. The optical axis is centered on the helix-shaped array of sources. The required FOV for the Z Wolter is indicated in yellow. For these simulations, the source-to-optic distance was set at the design value and the raytrace demonstrated the achievable spatial resolution in the region of interest as well as how the point spread function (PSF) evolves if the observed sources are far off-axis (bow-tie shape due to the complicated Wolter response in Fig. 5).

Since the diameter of the Z Wolter is small, it is impossible to coat the mirror shells with the MLs after they have been replicated from a precision mandrel. A special technique^7–9 in which the ML is deposited directly on the mandrel and then released together with the replicated mirror shell (shell thickness ∼264 μm) was used. The procedure to fabricate the mandrel (Fig. 6) was developed by the NASA Marshall Space Flight Center (MSFC) and has been extensively used for optics in astrophysics,¹⁰ medical,¹¹ and neutron imaging.¹² 

The first step in fabrication is to use computer numerical control (CNC) technology to machine the mandrel from an aluminum bar. It is then chemically cleaned and activated before undergoing electroless nickel plating. In the next step, the mandrel is diamond-turned to about 2 nm surface finish, with submicron figure accuracy. The last step in the mandrel preparation is superpolishing it to 0.3-0.4 nm surface finish. Metrology is then used to verify that the mandrel meets all specifications and requirements. If not, the mandrel is returned to the polishing state and then re-inspected. At the Smithsonian Astrophysical Observatory (SAO), the mandrels were first coated with a TiN layer for hardness and a carbon layer for smoother release (both stay on the mandrel when the optic with the ML is released). Then the coating is sputtered onto the pre-coated mandrel and a Pt layer is added at the end, before the ML-coated mandrel is placed in a NiCo plating tank for replication of the shell. Once completed, the optic is thermally released from the mandrel in a cold water bath.

For the Z Wolter optic, a W/Si ML was selected since it was able to fulfill all requirements while being a well-studied material system. Only the constant-d recipe was implemented for the first two production optics for Mo to reduce the risk during the deposition. The Wolter optic that has been fielded on Z can be fairly well approximated as a double-stack W/Si ML of 10 + 30 layers with each stack having a constant d, although there is an (unintentional) variation of d along the length of the optic for both 17.5-keV Wolters. Figure 7 shows the high-precision mandrel (left) and one of the final two, unmounted Mo Wolter optics (middle). Before calibration and fielding in Z, the optic was mounted and carefully aligned in its housing as shown in the right part of Fig. 7.

Both Mo Wolter optics underwent extensive x-ray calibration^13,14 in dedicated facilities at the Lawrence Livermore National Laboratory (LLNL) and SNL to determine the best focus position and characterize both the spatial response (the point spread function) and spectral behavior (throughput, bandwidth) of the instrument. Calibration measurements include on-axis and off-axis behavior as well as variation of spatial and spectral response at various source-to-optic and optic-to-detector distances. Results were compared to raytrace simulations, and Fig. 8 illustrates this comparison: Raytrace simulations of the point spread function for the Z Wolter are shown on the right and calibration data on the left. The image shows an area of 5 × 5 mm² and is a composite image of 25 measurements, in which the source was rastered in x and y perpendicular to the optical axis in steps of 1 mm. While there are some differences apparent between simulation and data, especially on-axis (central region of images in Fig. 8), the overall agreement is good and the calibration measurements confirm the expected Wolter performance in terms of spatial resolution over the field of view, throughput, and spectral response.

The Wolter optic acquired data for two different Z shots to date (twisted Mo target wire array and straight Mo wire array), and these data indicate a spatial resolution of 150 μm. Details on the first data acquisition campaign can be found elsewhere.¹⁵ 

In summary, we have optimized the geometry and multilayer of a Wolter microscope for the SNL Z machine to obtain images with high spatial resolution (100-200 microns) over a large FOV (5 × 5 × 5 mm³) with high-throughput and narrow energy bandpass (∼1 keV FWHM). Raytrace simulations were used to predict and test the microscope performance accurately. A first set of two Wolter optics tuned for the detection of Mo K_α photons was then fabricated using a novel approach in which a multilayer is deposited directly on a high-precision mandrel and released together with the replicated shell. Both custom optics were calibrated extensively at dedicated calibration facilities set up at LLNL and SNL. One of the two Wolter optics was aligned in SNL’s Z machine and acquired data during two shots in February and March 2018, respectively. First data demonstrate the Wolter capabilities are superior to the best current competing diagnostic, TIPC. This is the first time a Wolter optic has been implemented and successfully used in Sandia’s Z machine. The next step is the fabrication of two additional Wolter optics optimized for imaging of Ag K_α emission which will use the same mandrels for shell replication but will feature a ML optimized for 22.3 keV.

This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 (LLNL-JRNL-750575). SNL is a multimission laboratory managed and operated by NTESS, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE’s NNSA under Contract No. DE-NA-0003525.