A novel high-resolution x-ray spectrometer for point-like emission sources has been developed using a crystal shape having both a variable major and a variable minor radius of curvature. This variable-radii sinusoidal spiral spectrometer (VR-Spiral) allows three common spectrometer design goals to be achieved simultaneously: 1. reduction of aberrations and improved spectral (energy) resolution, 2. reduction of source size broadening, and 3. use of large crystals to improve total throughput. The VR-Spiral concept and its application to practical spectrometer design are described in detail. This concept is then used to design a spectrometer for an extreme extended x-ray absorption fine structure experiment at the National Ignition Facility looking at the Pb L_{3} absorption edge at 13.0352 keV. The expected performance of this VR-Spiral spectrometer, both in terms of energy resolution and spatial resolution, is evaluated through the use of a newly developed raytracing tool, xicsrt. Finally, the expected performance of the VR-Spiral concept is compared to that of spectrometers based on conventional toroidal and variable-radii toroidal crystal geometries showing a greatly improved energy resolution.

## I. INTRODUCTION

X-ray crystal spectrometers are routinely used in diagnostics for High Energy Density Physics (HEDP) experiments. In this application, the spectrometer is used to view a small x-ray emission source (typically less than a few mm) and provide high-resolution spectra (energy resolution of a few eV) that can be analyzed to determine the plasma state. An ideal spectrometer would have the following properties: high energy resolution, high throughput (meaning large étendue), minimization of source-size broadening, and one dimensional spatial resolution. Standard x-ray spectrometers at large scale laser facilities use simple crystal geometries, such as planar, elliptical,^{1} cylindrical,^{2} conical,^{2} spherical,^{3} or toroidally curved^{4} crystals, which cannot achieve all of these goals simultaneously; in particular, there is always a trade-off between throughput, which is gained by using a larger crystal, and energy resolution, which improves with smaller crystal sizes.

This paper presents a new crystal geometry, the Variable-Radii Sinusoidal Spiral (VR-Spiral) in which both the major- and minor-radii of the crystal shape are variable (see Sec. II). This novel shape provides near perfect imaging of a point-source for *arbitrarily large crystal dimensions* while also simultaneously minimizing the effect of source size broadening in the case of an extended source. The VR-Spiral geometry allows all four of the standard x-ray spectrometer design goals to be fulfilled simultaneously. This crystal geometry most closely resembles that of the conventional toroidal crystal while providing significantly improved performance in terms of higher throughput while maintaining very good energy resolution.

Using this new crystal geometry, a design for a new spectrometer for installation on the National Ignition Facility (NIF)^{5} is presented in detail in Sec. III. This spectrometer is to be used as a part of Extreme eXtended X-ray Absorption Fine Structure (EXAFS)^{6–9} experiments looking at the Pb L_{3} absorption edge. The expected performance of this spectrometer, as determined through raytracing, is then described in Sec. IV. Finally, the achievable energy resolution of the VR-Spiral concept is compared against two other geometries, the conventional toroidal crystal and the Variable-Radii Toroid (VR-Toroid),^{10} in Sec. V.

The analysis shown in this paper was enabled by the development of a new raytracing code, XICSRT. This code and the essential capabilities that allow the VR-Spiral shape to be raytraced are described along with the results of a validation study against SHADOW in Sec. VI.

## II. VARIABLE RADII SPIRAL GEOMETRY

The idea of using a crystal with a variable minor-radius to improve spectrometer performance was first introduced by Bitter *et al.* in Refs. 10 and 11. While these papers (along with Ref. 12) provided the essential groundwork for variable-radii designs, the VR-cone and VR-Toroid geometries still suffer from significant imaging aberrations that lead to degradation of the achievable energy resolution (this will be discussed further in Sec. V).

Producing a crystal geometry that is free from aberrations (while still minimizing source size broadening) requires the introduction of a variable *major*-radius in addition to the variable *minor*-radius that is part of those previous designs. The variable major-radius allows the spectrometer design to maintain a single axis of rotational symmetry for all points on the crystal surface. To achieve this symmetry, a geometry has been developed in which the major-radius is described by a sinusoidal-spiral. This variable-radii sinusoidal spiral concept has been described in a recent paper (see Ref. 13) along with a derivation of the essential equations. In this section, the VR-Spiral concept is expanded upon with an emphasis on optimization and adaptation for practical spectrometer designs.

The sinusoidal spiral describes a family of curves that are represented by the following equations:

where *r* and *ϕ* are polar coordinates, *γ* is the angle between the tangent and radial directions, *b* is a parameter that controls the shape of the curve, and the subscript _{0} denotes the parameter values when *ϕ* = 0 (see Fig. 1). This family of curves is of particular interest for x-ray spectroscopy as it describes several common crystal shapes and can smoothly vary between them. In particular, values of *b* = 0, 1, and 2 correspond to a straight line, logarithmic spiral, and circle, respectively.

The radius of curvature, *ρ*, at any point on the curve is described by

which will correspond to the variable major-radius of our VR-Spiral crystal shape.

If the emission source is placed at the origin, then the parameter *γ* also describes the Bragg angle (*θ*_{B}) at every point along the crystal [matching Ref. 13, Eq. (12)]. However, this arrangement is not necessary. When designing a sinusoidal spiral spectrometer, any portion of the spiral curve can be used to describe the crystal shape and the source can be placed anywhere along the x-axis. Because of this freedom, two additional free parameters must be defined: *ϕ*_{C} describes the angle corresponding to the center of the crystal and *S*_{x} describes the distance of the source from the origin along the x-axis. (While, in principle, the source can be placed away from the x-axis, such a configuration is exactly equivalent to a rotational coordinate transformation with a change of variables and, therefore, does not actually provide additional design freedom.)

Construction of the three dimensional VR-Spiral crystal shape from the sinusoidal curve is done as follows (see Fig. 2): start with a point on the crystal (*C*) and determine the local Bragg angle (*θ*) by tracing a line from the source, *S*. Using this angle and the local radius of curvature, *ρ* [Eq. (2)], determine the detector location (*D*) at which the ray will intersect the local Rowland circle. The distance between *C* and *D* will be equal to *d*_{CD} = *ρ* sin *θ*_{B}. A line *SD* can now be drawn that forms the axis of rotation needed to provide sagittal focusing for this wavelength. The crystal surface is finally generated by rotating the crystal point *C* around this axis *SD*. This process is repeated for all points along the crystal to build up the three-dimensional surface. The extent of this rotation (defined by the rotation angle *β*) defines the sagittal crystal size. The perpendicular distance between the axis *SD* and the crystal defines the minor-radius, and will be different for each of the points along the crystal. This minor-radius, which is shown as *CP* in Fig. 2, is different from the vector *CQ*, which points toward the local center of curvature. The geometry described above is illustrated in Fig. 2(a) and the resulting 3D shape from this procedure is shown in Fig. 3.

The surface produced by the procedure above will only provide perfect focusing of a point source if all of the points along the crystal have a common axis of symmetry, meaning that all of the rotational axes, *SD*, are coincident (see additional discussion in Sec. V). This condition can be achieved (at a particular crystal location) by optimizing the parameter *b* in the sinusoidal spiral equation. To do this optimization, a minimization algorithm is used to determine *b* such that the derivative of the slope *SD* as a function of *ϕ* is equal to zero, *d*/*dϕ*(*SD*_{x}/*SD*_{y}) = 0. An exact solution can be found for any one point on the crystal (or equivalently for any one wavelength) but cannot be achieved over the entire crystal extent simultaneously. This principle is illustrated in Fig. 2(a), which has been optimized for the wavelength shown in bold colors. The inability to exactly align the rotational axis everywhere along the crystal also leads to the ideal detector surface having a slight curvature.

The extent to which the rotational axes can all be aligned over an extended crystal depends on the magnification of the spectro-meter. An optimized magnification can be achieved by introducing the second derivative of the slope *SD* as an additional constraint, *d*^{2}/*dϕ*^{2}(*SD*_{x}/*SD*_{y}) = 0, and then finding a solution for both *b* and *r*_{0}. The result of this further optimization is shown in Fig. 2(b). As before, this solution is only exact at one particular location along the crystal; however, in this case, the alignment is very close to perfect.

For a practical spectrometer, which will have a much smaller crystal than that shown in Fig. 2 (for example, see the NIF spectrometer design shown in Fig. 4), the deviation of the axis alignment across the crystal extent has a relatively minor effect on the overall performance despite using an un-optimized magnification (see Fig. 6). Some strategies for further improvement of performance at any magnification are discussed in Sec. VII.

## III. DESIGN OF A VR-SPIRAL SPECTROMETER FOR NIF

Utilizing the VR-Spiral crystal concept, a new spectrometer has been designed for installation at the National Ignition Facility to measure the spectrum around the Pb L_{3} absorption edge at 13.0352 keV.^{14} This spectrometer will be used as part of a set of planned EXAFS experiments. This is a challenging measurement due to low signal levels (as compared to K-edge EXAFS measurements for lighter elements) and the need for very high spectral resolution (a few eV). There are a number of constraints on the layout of this new spectrometer design including a requirement that the crystal and detector should be placed within an existing housing as shown in Fig. 4. For this design, the magnification of the system is, therefore, set by the external constraints rather than optimized for ideal performance.

The specific layout constraints that were developed are as follows: the distance from source to edge of the crystal *d*_{SC} = 0.3 m, lower energy bound *E*_{min} = 12.700 keV, optimization energy (for axis alignment) *E*_{opt} = 13.0352 keV, and distance from crystal to detector *d*_{CD} = 0.675 m (magnification of 2.25). The spectrometer will use a Si (422) crystal with a 2*d* lattice spacing of 2*d* = 2.2172 Å,^{15} leading to a Bragg angle at the Pb L_{3} energy of *θ*_{B} = 25.405°. To further constrain the layout, the crystal edge is defined to be at *ϕ* = 0. These parameters provide a complete set of constraints, and a unique solution has been found through a minimization algorithm providing values for *b*, *r*_{0}, *γ*_{0}, and *ϕ*_{C}. The final optimized parameters are summarized in Table I and the layout of the crystal and detector within the spectrometer housing is shown in Fig. 4.

The relation between the detector position and energy, and therefore the dispersion, is determined by using central rays. These are the rays that originate from the center of the source, impact the crystal mid-plane with the nominal Bragg angle, and then intersect the detector mid-plane. For all raytracing calculations presented in this work, a flat detector is used. The detector is placed so as to be at best focus for the optimization energy and oriented relative to the focus position at edges of the spectrum. This orientation leads to the second position of the best focus around 13.7 keV.

In the current work, we assume that all x-ray reflections are symmetric around the nominal Bragg angle and x-ray refraction effects at the crystal interface are not considered. This implies that the crystal lattice plains are perfectly aligned with the surface normal, and ignores any offset of the rocking-curve. These are reasonable approximations for a practical hard x-ray EXAFS spectrometer but may need to be reconsidered for other applications. In principal, reflection offsets and non-symmetric effects could be considered in both the raytracing and when calculating the optimized VR-Spiral shape of the crystal. However, to include these effects in a meaningful way, more information is needed with respect to the bent crystal rocking-curves and ultimately the manufacturing tolerances.

## IV. PERFORMANCE OF THE VR-SPIRAL SPECTROMETER

The expected performance of the VR-Spiral spectrometer is evaluated through the use of x-ray raytracing. Raytracing was performed using a newly developed code, XICSRT, which is described in Sec. VI. There were several motivations for developing a new code for this work including the ability to model complex optic shapes represented through the use of a mesh-grid, flexibility in 3D source distributions, and specification of the spectrometer layout in arbitrary 3D space.

To model the VR-Spiral within the XICSRT raytracing code, a mesh-grid representation (see Sec. VI) is used. Doing so avoids the need to analytically determine the intersection of rays with this complex 3D shape. While the numerical calculation of the *x*, *y*, *z* coordinates for the basic 3D geometrical shape as a function of *ϕ* and *β* is straightforward by following the recipe given in Sec. II (see Fig. 3), calculation of the surface normal vectors, needed for accurate raytracing, is analytically complex. To accurately and efficiently calculate these vectors, automatic-differentiation (AD) is utilized to find the gradient vector with machine precision at each grid point (implemented using the JAX library^{16,17} in Python).

Results from raytracing are shown in Fig. 5 for the VR-Spiral spectrometer described in Sec. III and Table I. Raytracing was performed with a finite size source modeled as a 1 mm cube. This cube represents the approximate source size expected in the EXAFS experiments at NIF. The crystal rocking-curve is modeled by a Gaussian with a full-width-at-half-max (FWHM) of 60 *μ*rad. The rocking-curve width was chosen to match the results from XOP^{18,19} for a doubly bent toroidal crystal of similar geometry and 100 *μ*m thickness. However, a symmetric Gaussian shape centered at the nominal Bragg angle was chosen instead of directly using the XOP reflectivity curve (which is highly asymmetric) to more clearly illustrate the effect of the crystal geometry on the shape of the instrumental response. A mono-energetic wavelength distribution was used, with all rays having an energy of 13.0352 keV (0.951 149 Å). A planar (flat) detector was used with the placement described in Sec. III.

Two additional raytracing runs are also plotted to illustrate the various broadening effects that affect the energy resolution. The first uses a point source while maintaining the same 60 *μ*rad rocking-curve width. The second additionally uses an artificially narrow rocking-curve width of 1 *μ*rad to isolate the broadening effects from the crystal geometry and resulting optical aberrations.

To understand the raytracing results, it is first necessary to distinguish between two different broadening effects related to the finite source size. The first effect is due to the source extent in the dispersion (meridional) direction, which can lead to broadening unless a curved crystal is used with the detector placed on the Rowland circle. This effect is what is typically meant by the term “source size broadening” and has been minimized in the VR-Spiral design. The second effect is from the source extent in the spatial (sagittal) direction. The optical focus and “perfect” imaging properties are only ideal for a point source and will degrade with the distance away from the mid-plane. For the remainder of this paper, we will describe these effects as meridional and sagittal source size broadening.

The two dimensional ray distribution on the detector is shown in Fig. 5(a). The energy dispersion is along the x-axis (meridional direction) and 1D imaging of the source is along the y-axis (sagittal direction). The hourglass shape, seen for raytracing with a finite size source, is primarily due to the effect of sagittal source size broadening. Near the mid-plane, the extent of the ray pattern in the spectral direction is nearly identical to that of the point source case, showing that the meridional source size broadening has been effectively minimized.

The histograms of the 2d ray patterns shown in Figs. 5(b) and 5(c) show the total distribution of rays in the spatial and spectral directions, respectively. This is equivalent to summing together all the rows or columns on a pixelated detector. The energy scale is determined from the central ray as described in Sec. III. To quantify the energy resolution, the following metric is used: a width within which 76.1% of the intensity is contained (76th-percentile width); this metric is exactly equivalent to the FWHM for a Gaussian profile and can be applied in a consistent way to highly asymmetric shapes (this is important for the results shown in Sec. V). The energy resolution of the VR-Spiral spectrometer, using the 76th-percentile width of the instrumental response shown in Fig. 5(c), is found to be 3.2 eV with a 1 mm source size and 1.6 eV with a point source (the FWHM value is the same within the numerical precision given).

To determine how much the energy resolution varies across the spectral range of the spectrometer, raytracing has also been performed for a series of source energies between 12.8 and 13.9 keV, as shown in Fig. 6. Near the energy at which the VR-Spiral was optimized (13.0352 keV), the broadening due to optical aberrations is entirely negligible (less than 0.1 eV) and the energy resolution is limited by the rocking-curve width and sagittal source size broadening. Away from the optimization energy, optical aberrations become significant and start to contribute to the overall energy resolution as well. For the experimentally relevant case of a 1 mm source size, the energy resolution across the full spectrum varies from 3.2 to 4.0 eV, a change of about 20%.

The VR-Spiral also provides 1D spatial imaging in the sagittal direction. To determine the spatial resolution, a series of raytracing runs were performed using point and line sources at different positions in the sagittal direction. The results are shown in Fig. 7. The achievable spatial resolution is strongly dependent on the meridional source size, as can be seen by comparing the results from the point and line sources. For a 1 mm source size, a spatial resolution of 30 *μ*m is achievable as determined by the 76th-percentile width on the detector divided by the magnification. The spatial resolution is nearly energy independent but is highly sensitive to the detector focus; to maintain the highest possible spatial resolution across the full spectral range, a curved detector is beneficial.

The VR-Spiral crystal shape is designed for a specific source location, and therefore, a spectrometer using this crystal must be fully aligned in three dimensional space with respect to the source to obtain optimal performance. This requirement is in contrast to the case of a conventional toroidal crystal, in which there are multiple degrees of freedom based on the rotational symmetries. Nonetheless, raytracing studies have shown that small misalignments of the source with respect to the crystal can be tolerated without a significant impact on the energy resolution. The spectrometer is most sensitive to misalignments in the spatial (sagittal) direction, in which an offset of ±0.3 mm will lead to a 20% broadening in the instrumental width (for the case of a 1 mm source size). This sensitivity can be readily understood as a consequence of the hour-glass ray pattern found in Figs. 5(a), 7(a), and 7(b). For the spectral (meridional) and focus directions, the required alignment tolerances are ±2.0 and ±10.0 mm, respectively.

A similar tolerance study has been carried out to understand the requirements for parallelism of the crystal lattice to the crystal surface using the same criteria of less than 20% broadening of the instrumental response. As before, the instrumental response is most sensitive to non-parallelism in the spatial direction with a required tolerance of ±5 arc min, and largely insensitive to non-parallelism in the spectral direction. These requirements are well within the standard manufacturing tolerance of <1.0 arc min. Deviations of the crystal shape from the nominal design have also been examined using raytracing, leading to an estimation of the surface accuracy requirement of ±0.5 *μ*m. However, to obtain best results, particularly for smaller source sizes, deviations below ±0.05 *μ*m are recommended as a manufacturing tolerance.

## V. COMPARISON TO OTHER GEOMETRIES

To assess the performance improvement of the VR-Spiral over other crystal geometries, a comparative raytracing study has been carried out. For this comparison, three geometries are considered: the VR-Spiral, Variable-Radii Toroid (VR-Toroid), and conventional toroidal geometries. The VR-Toroid has a *constant* major-radius but a *variable* minor-radius, as was first proposed in Ref. 10; construction of the shape follows the same procedure as given in Sec. II except that there is no possibility of aligning the rotational axes. The conventional toroidal geometry has both a constant major- and a constant minor-radius and has a long history of use in x-ray spectrometers.

For this comparative study, a set of three matched spectrometer layouts were developed as shown in Fig. 8. All of these layouts use a Si (422) crystal and view the spectral range around the Pb L_{3} absorption edge as before. The VR-Toroid and conventional torus crystals both have a (constant) major-radius of 2093 mm, and the source is placed at a distance of 300 mm from the crystal. For this comparative study, the fully optimized VR-Spiral layout with the larger magnification was chosen [as described in Sec. II and shown in Fig. 2(b)]. The major-radii for the VR-Toroid and torus layouts were chosen to match this optimized VR-Spiral layout at the Pb L_{3} energy. This slightly larger spectrometer layout produces well optimized comparison targets for all three configurations. The details of each layout are summarized in Table II. In all cases, the major- and minor-radii describe the crystal surface curvature (as opposed to any underlying geometrical construct).

Crystal . | Si (422) . | . | . |
---|---|---|---|

Crystal size . | 60 × 30 mm . | . | . |

. | VR-Spiral
. | VR-Toroid
. | Torus
. |

Major-radius (m) | 1.975–2.633 | 2.0931 | 2.0931 |

Minor-radius (m) | 0.1902–0.2032 | 0.1926–0.1934 | 0.1979 |

Dispersion (eV/mm) | 6.62–6.32 | 21.55–30.15 | 28.85–39.31 |

Magnification | 2.89–2.92 | 3.06–2.33 | 3.18–2.53 |

Energy res. point (eV) | 1.6–1.9 | 15.0 | 12.5 |

Energy res. 1 mm (eV) | 3.1–3.3 | 15.1 | 12.5 |

Crystal . | Si (422) . | . | . |
---|---|---|---|

Crystal size . | 60 × 30 mm . | . | . |

. | VR-Spiral
. | VR-Toroid
. | Torus
. |

Major-radius (m) | 1.975–2.633 | 2.0931 | 2.0931 |

Minor-radius (m) | 0.1902–0.2032 | 0.1926–0.1934 | 0.1979 |

Dispersion (eV/mm) | 6.62–6.32 | 21.55–30.15 | 28.85–39.31 |

Magnification | 2.89–2.92 | 3.06–2.33 | 3.18–2.53 |

Energy res. point (eV) | 1.6–1.9 | 15.0 | 12.5 |

Energy res. 1 mm (eV) | 3.1–3.3 | 15.1 | 12.5 |

Raytracing was again performed with a 1 mm mono-energetic wavelength source and using a Gaussian rocking-curve with a 45 *μ*rad FWHM (appropriate for the major and minor radii in this scenario). A crystal size of 60 × 30 mm^{2} was used for all three geometries. The raytracing results are shown in Fig. 9. Both the toroidal and VR-Toroid geometries have highly asymmetric and significantly broadened instrumental responses due to optical aberrations. Using the 76th-percentile width metric, the energy resolution of the VR-Spiral, VR-Toroid, and torus for a 1 mm source size is 3.2, 15.1, and 12.5 eV, respectively. For the VR-Toroid and conventional torus, this resolution will depend strongly on the size of the crystal in the sagittal direction; the resolution of the VR-Spiral, on the other hand, is independent of the crystal size.

From these raytracing results, it is immediately clear that the VR-Toroid, despite having a variable minor-radius, has an energy resolution and instrumental response that are comparable to those of a conventional toroidal crystal. The reason for these large aberrations is that without a single axis of rotational symmetry, the surface normal vector deviates from the direction *CQ* in Fig. 2. This deviation can be readily illustrated by considering a ray pattern for the VR-Toroid and performing a rotation around the axes *SD* of 180°. This separation of the minor-radius vector and the surface normal vector in the VR-Toroid case highlights the importance of the rotational axis optimization for the VR-Spiral.

## VI. XICSRT RAYTRACING SOFTWARE

Raytracing for all geometries considered in this work was performed using a newly developed x-ray raytracing code: XICSRT.^{20–22} The XICSRT code allows general purpose optical and x-ray raytracing, with special attention to the preservation of photon statistics. In addition to the calculations here, this software has also been successfully used to model diagnostic performance for tokamak and stellarator plasmas, including the emission from large extended 3D x-ray sources. The code is written in the python programming language and released under an open-source MIT License.

Raytracing in XICSRT is performed in full three dimensional geometry and can handle arbitrarily complex volumetric ray sources and optical components. Raytracing utilizes probabilistic filtering to model the reflectivity of optics (including Bragg-reflection). As an example, for a Bragg crystal, this means that each ray is either reflected or lost based on its probability of reflection for the given wavelength, incidence angle, and crystal rocking-curve. The use of probabilistic filtering allows direct one-to-one comparisons with photon counting detectors, simplifies error analysis as all outputs exactly follow Poisson statistics, is well suited for parallelization during computation, and is efficient in terms of computational performance and memory demands. This is different from the approach taken by SHADOW, which instead uses a ray-weighting scheme.

Basic ray sources emit an isotropic distribution of rays into a specified emission cone. Rays can be mono-energetic or drawn from any given wavelength distribution. For example, line emission can be drawn from a Voigt profile to model plasma emission, in which a natural line width and temperature are provided. In addition, XICSRT is capable of modeling of plasma sources with 3D geometry and spatially varying parameters in real units, for example, emissivity (photons/m^{3}/sr/s), temperature (eV), and plasma flow (m/s). This modeling is accelerated through the use of ray bundles allowing reduced sampling of the plasma parameters while still producing large numbers of rays (see Ref. 22 for details).

Several types of optical components are currently available, including Bragg-reflection crystal optics. Crystal optics use a rocking-curve that can be defined either through a pre-calculated rocking-curve file, for example, from XOP^{18,19} or X0H,^{23,24} or by a simple function such as a Gaussian or step-function. Mosaic crystals (such as HOPG) are also supported, though not used in the current work.

Important for this work, and one of the motivations for the development of XICSRT, is the ability to construct optics through the use of mesh-grids. A mesh-grid optic is defined by a set of points and surface normal vectors on a three dimensional grid. These points are turned into a triangular (isometric) grid using Delaunay triangulation.^{25,26} Intersections of rays with this grid are determined using the Möller–Trumbore ray-triangle intersection algorithm.^{27} Two-dimensional Clough–Tocher interpolation^{26,28} is then used to determine the surface normal vector at the intersection point (interpolation is also used to refine the intersection coordinates). Once the intersection location and surface normal vectors have been determined, probabilistic ray reflection is performed normally including consideration of the Bragg-condition as needed.

A mesh-grid refinement option has also been implemented, in which a course mesh is used to pre-select faces on the full mesh for raytracing. Use of the course mesh critically improves the performance and decouples the computational time from the density of the full mesh-grid.

### A. Validation against SHADOW

Since the XICSRT code is quite new, a validation exercise has been performed against the SHADOW3^{29} code. This validation is meant not only to verify that the basic geometry and Bragg-reflection calculations are implemented correctly but also to validate the mesh-grid implementation. The SHADOW is a mature and widely used x-ray raytracing code that has been thoroughly validated against the experiment,^{30} thus providing an ideal target for code validation.

For the validation exercise, we have chosen to use a spectrometer with a conventional toroidal crystal. The geometry is similar, but not identical, to that shown in Sec. V. The differences in the geometry make it easier to setup the matched configurations, provide a better target for validation, and are better suited to experimental validation in the future. The spectrometer that is modeled has the following parameters: *R*_{maj} = 2200 mm, *R*_{min} = 200 mm, *d*_{source} = 300 mm, crystal: Ge (400), central energy: 9818.8 eV (tungsten *Lβ*_{1}), and crystal size: 60 × 30 mm^{2}. In this scenario, the detector is placed perpendicular to the central ray; this means that the detector is not exactly aligned with the focal plane. This misalignment of the focal plane was done purposefully to provide a more interesting and broadly applicable validation target. The rocking-curve was taken from XOP^{19} for a bent crystal with the given major- and minor-radii using a multi-lamellar model;^{18} the full asymmetric sigma-polarization reflectivity curve was used. For XICSRT, a mesh-grid model with 121 × 61 points is used for the toroidal crystal. The surface normal vectors were specified at each grid-point and interpolated at the ray intersections as described in Sec. VI.

Results from the validation exercise are shown in Figs. 10 and 11. A point source with a “flat” emission distribution profile was used to launch rays with three energies, each separated by 20 eV: 9778.8, 9788.8, and 9798.8 eV. For each energy, 1 × 10^{7} rays (for a total of 3 × 10^{7} rays) were launched with an angular divergence of (half-angle) of 5.0°. The final intensity on the detector in both cases was ∼5300 rays.

Excellent agreement is seen in both the spatial distribution of rays on the detector and the total intensity. The spatial distribution shows systematic differences between the two ray patterns on the detector at the level of about 5 nm. This level of agreement is dependent on the mesh-grid resolution used within XICSRT; for example, if the grid resolution is halved to 61 × 31, then larger differences of about 50 nm are observed (which is still more than sufficient for any practical use). Excellent agreement in the absolute intensity is also found in Fig. 11, and the total integrated intensity on the detector agrees within statistical error bars. This agreement is especially encouraging since XICSRT utilizes probabilistic filtering, while SHADOW uses ray weighting (see the discussion at the beginning of Sec. VI).

## VII. CONCLUSIONS

A new x-ray spectrometer design, the VR-Spiral, based on a crystal with variable major- and minor-radii, has been developed and its performance is evaluated through the use of x-ray raytracing. The VR-Spiral design is shown to effectively provide a spectrometer that simultaneously achieves high energy resolution, high throughput, minimization of source size broadening, and 1D spatial imaging. The performance of this new design has been compared against two other crystal geometries, the conventional torus and the VR-Toroid, showing greatly improved energy resolution.

The VR-Spiral geometry has been applied to the design of a new spectrometer for Pb L_{3} EXAFS experiments at NIF. The crystal for this spectrometer is currently in the process of being manufactured. The ultimate performance of this spectrometer will depend greatly on the accuracy in which this complex crystal geometry can be constructed. The crystal shape is achieved by first machining and polishing a substrate with the desired VR-Spiral geometry and then attaching a thin crystal to the substrate through optical contacting, presumably due to van der Waals forces (see Ref. 31).

As discussed in Sec. II, the optimized sinusoidal spiral, while producing an optical geometry with excellent performance, is unable to *exactly* align all of the rotational axes across the entire energy range (or, equivalently, the crystal extent). The misalignment is negligible at the optimized magnification but important for other magnifications such as that required for the NIF spectrometer. While this small misalignment is perhaps not an issue for practical purposes (see Fig. 6), it is still interesting to consider that a more complex curve may be able to further improve the performance. An improved crystal shape at any magnification can be straightforwardly achieved by expanding the parameter *b* in Eq. (1) as a function of *ϕ*. For the particular case of the NIF design, an even polynomial expansion, *b* = *b*_{0} + *b*_{2}*ϕ* + *b*_{4}*ϕ*, could have provided a solution with the same level of axis alignment as the optimized magnification case while retaining the essential spectrometer dimensions and magnification. It is likely possible to develop a crystal shape that can achieve a true “perfect” solution; however, any such solution is unlikely to have the simplicity of the sinusoidal spiral.

## ACKNOWLEDGMENTS

This work was performed under the auspices of the U.S. Department of Energy by Princeton Plasma Physics Laboratory under Contract No. DE-AC02-09CH11466 and by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.

## DATA AVAILABILITY

The data that support the findings of this study are openly available at https://arks.princeton.edu/ark:/88435/dsp01x920g025r.