This paper describes a new class of focusing crystal forms for the x-ray Bragg crystal spectroscopy of small, point-like, x-ray sources. These new crystal forms are designed with the aid of sinusoidal spirals, a family of curves, whose shapes are defined by only one parameter, which can assume any real value. The potential of the sinusoidal spirals for the design x-ray crystal spectrometers is demonstrated with the design of a toroidally bent crystal of varying major and minor radii for measurements of the extended x-ray absorption fine structure near the Ta-L3 absorption edge at the National Ignition Facility.

## I. INTRODUCTION

The experiments at the National Ignition Facility (NIF) and other high power laser facilities put challenging demands on the x-ray Bragg crystal spectroscopy. An example is measurements of the Extended X-ray Absorption Fine Structure (EXAFS), which are becoming increasingly important for studies of the state of matter under the extreme conditions in High Energy Density (HED) plasmas.^{1,2} The requirements for these measurements are as follows: (1) a high spectral resolution of E/dE = 10 000 for an extended energy range of 1 keV; (2) a high photon throughput to reduce the statistical errors; and (3) the elimination of source-size broadening effects, which deteriorate the spectral resolution.^{3,4} The last requirement asks for crystal forms with a well-defined Rowland circle at each crystal point, since source-size broadening effects can be minimized if the detector is positioned on a Rowland circle.

The requirements (1)–(3) cannot be satisfied with the presently existing spectrometers, which employ, as the x-ray diffracting element, crystals of standard geometrical forms, such as flat crystals, spherical crystals, cylindrically bent crystals, and crystals that are bent to a conventional toroidal form with a constant major radius **R** and constant minor radius **r**, so that entirely new focusing crystal forms are needed. These new crystal forms must exactly satisfy the conditions for a point-to-point imaging of the source onto the detector, for each wavelength in an extended spectral range of 1 keV, so that—in order to enhance the photon throughput—the area of the crystal can be increased without the introduction of imaging errors, which deteriorate the spectral resolution.

For the EXAFS spectroscopy at the NIF, we have therefore proposed a modified toroidal crystal form with a constant major radius, **R**, and a varying minor radius, **r**, where **r** was defined at each crystal point with respect to a local axis of rotational symmetry.^{5} We have, however, recently learned—from ray tracing calculations performed by Pablant with his XCISRT code and the SHADOW code^{6,7}—that the design described in Ref. 5 is not free of imaging errors and that in order to meet the requirements (1)–(3), an additional condition needs to be satisfied, namely, that the just-mentioned local axes of rotational symmetry must also align with each other. This is due to the fact that a rotationally symmetric surface cannot be generated by rotating a curve, or segments of a curve, about more than one axis. The deviations from rotational symmetry and the resulting imaging errors, for the design proposed in Ref. 5, are described in Ref. 6.

In this paper, we demonstrate that this additional condition can, indeed, be satisfied by a toroidal crystal form, which has both a varying minor radius, **r**, as described in Ref. 5, and a varying major radius, **R**, where **R** is identical with the locally varying radius of curvature of a certain sinusoidal spiral. The sinusoidal spirals constitute a family of curves, which are defined by a single parameter **b**, which can assume any real value. This family of curves includes many well-known curves, e.g., the straight line, the circle, and the logarithmic spiral, which have already found applications in x-ray spectroscopy. There is, however, an infinite number of curves, which are potentially useful for the design of crystal spectrometers.

The purpose of this paper is, therefore, twofold, namely, to draw attention to the sinusoidal spirals for a wider application in x-ray spectroscopy and to demonstrate their potential with the design of a crystal spectrometer for EXAFS measurements that satisfies all the afore-mentioned requirements.

This paper is organized as follows: Section II presents a brief derivation of the sinusoidal spiral, highlighting the important geometrical relation between the Bragg angle **Θ**, polar angle $\varphi $, and slope angle **α**. Section III describes two variants of a sinusoidal-spiral crystal spectrometer design, with different values for the parameter **b**, for EXAFS measurements at the NIF. The first design is not optimized and still similar to the design described in Ref. 5, although the crystal has already a varying minor radius and a varying major radius. The second design has been optimized by using an optimal **b**-value, and it satisfies the additional condition that the above-mentioned local axes of rotational symmetry are aligned with each other.

## II. SINUSOIDAL SPIRALS

We begin our introduction of the sinusoidal spirals to x-ray spectroscopy with Fig. 1, which shows a general curve in polar and Cartesian coordinates and an x-ray point source at the origin of the two coordinate systems.

The following equations can be inferred from Fig. 1:

Equation (5) can be immediately integrated if the Bragg angle, $\Theta $, is constant, yielding the solution

which represents the well-known logarithmic spiral.

To find additional solutions, we rewrite Eq. (5) as

and assume that $\alpha $ varies linearly with $\varphi $,

so that

Equation (7) can then be rewritten as

and

so that one finally obtains

where we also used the relation

Equation (12) describes a class of curves, which are known as sinusoidal spirals. Their shapes are determined by the parameter **b**, which can assume any real value.

We also need the following equation for the radius of curvature **ρ** of a sinusoidal spiral:

The sinusoidal spirals include many well-known curves, for example, the straight line, the logarithmic spiral, and the circle, which are obtained for **b** = 0, 1, and 2, respectively.

Figure 2 shows, for comparison, a logarithmic spiral and three sinusoidal spirals with **b**-values near **b** = 1.

Crystals that are bent to the shape of a logarithmic spiral have the unique property that the Bragg angle **Θ** is constant if a point source is placed at the proper location. This property can be used to maximize the photon throughput for a particular x-ray energy and a given crystal length. In most cases, it is, however, of interest to maximize the photon throughput for a wider spectral range, which comprises more than one x-ray energy. This goal can easily be achieved by bending a crystal to the shape of a sinusoidal spiral with an appropriately chosen **b**-value.

## III. DESIGN OF A SINUSOIDAL SPIRAL CRYSTAL SPECTROMETER

In this section, we discuss three sinusoidal-spiral crystal spectrometer designs for EXAFS measurements near the Ta-L_{3} absorption edge at E = 9.877 keV in the energy range: 9.750 ≤ E ≤ 10.750 keV. The three designs employ, as the x-ray diffraction element, a Ge400 crystal, with a 2d-spacing of 2.828 68 Å, which is bent to a toroidal form with continuously varying major and minor radii. The major radii **R** are identical with the radii of curvature of a sinusoidal spiral and, therefore, equal to the diameters of the local Rowland circles at each crystal point. The minor radii, **r**, are defined as described in Ref. 5. The dimensions of the spectrometer layouts are compatible with the experimental conditions at the NIF.

### A. Non-optimized design

Figure 3 shows the calculated ray paths for three x-ray energies, E = 9.75, 10.25, and 10.75 keV, at the center and the limits of the spectral range of interest for a design with a toroidally bent crystal of varying major and minor radii. The major radii **R** are identical with the radii of curvature of a sinusoidal spiral with the parameter values **b** = 0.7 and **r**_{0} = 300 mm. The major radii **R** are, therefore, equal to the diameter of the Rowland circle at each crystal point. The source is at the origin of the x,y-coordinate system for the sinusoidal spiral. The rays, which emanate from the source, are Bragg reflected at a crystal point and focused onto a detector point on the local Rowland circle; this detector point is the point of intersection of the Rowland circle with a tangency circle for the reflected rays with the radius of **R** cos(Θ) about the local center of curvature, **M**. The minor radii, **r** (*green lines*), are normal to the connecting lines between the source and the detector points.

### B. Optimized design

The design, shown in Fig. 3, can be optimized so that the connecting lines between the source and the various detector points are aligned with each other. As shown in Fig. 4, this is accomplished by choosing a sinusoidal spiral with **b** = 0.356.

The value of 0.356 for the parameter **b** is an optimum value. It was inferred from Fig. 5, which shows a plot of the ratios, yd/xd, of the coordinates (xd, yd) of the 11 detector points for the 11 x-ray energies considered vs the sinusoidal spiral parameter **b**. The ratios yd/xd represent the slopes of the connecting lines between the source and detector points, as is evident from Fig. 3.

The energy dispersion and spectral resolution of this optimized design are obtained with the aid of Fig. 6, which shows a plot of the 11 x-ray energies vs the distances of the corresponding detector points from the source.

We infer from Fig. 6 that the energy dispersion, dE/ds, where s denotes the coordinate along the detector, is approximately linear and equal to dE/ds = 0.0052 keV/mm. If we assume that the spatial resolution of image plate detectors is Δs = 0.1 mm,^{8} we obtain for the energy resolution the very high value of E/ΔE = 20 000. This value exceeds the limits that are set to the achievable spectral resolution by the crystal properties and vertical source-size broadening.^{6} Thus, we can fairly say that the achievable spectral resolution is no longer limited by the spectrometer design.

The optimized design, shown in Fig. 4, satisfies all the requirements for EXAFS measurements. (1) It provides perfect imaging for each wavelength so that the crystal area can be increased without the introduction of imaging errors to enhance the photon throughput. (2) It provides a very high spectral resolution that is no longer limited by imaging errors due to the spectrometer design. (3) It minimizes source-size broadening effects by placing the detector point for each wavelength on a Rowland circle.

We conclude this section by pointing out the striking similarities and one essential difference that exists between our optimized design and the von Hamos cylindrical x-ray crystal spectrometer and the Extreme Luminosity Imaging Conical Spectrograph (ELICS).^{9,10} The three designs have two obvious features in common: (1) They provide perfect images of a point source for each wavelength, and (2) the point source and its images are on an axis of rotational symmetry. Another common feature, which is less obvious but noteworthy, is that the crystal forms in all three designs make use of a sinusoidal spiral. In fact, the cylindrically bent crystal shape of the von Hamos spectrometer can be obtained from Eq. (12) by setting **b** = 0 and Θ_{0} = 90° and demanding that the source and its images are on the y axis and that the y axis is an axis of rotational symmetry. Similarly, ELICS’s conical crystal form can be obtained from Eq. (12) by setting **b** = 0 and Θ_{0} = 90° ± α, where α is the half-aperture angle of the cone, and by defining the y axis as the axis of rotational symmetry and choosing it for the location of the source and its images.

An essential difference between our optimized design and the von Hamos and the ELICS spectrographs is, however, that in our design, the positions of the detector points and the axis of rotational symmetry cannot be freely chosen, since the detector points must also be on Rowland circles in order to minimize source-size broadening effects.

Fortunately, the sinusoidal spirals offer solutions where the source and its images are on a common axis of rotational symmetry and where the images of the source are also on Rowland circles. We point out that the von Hamos and the ELICS spectrographs do not have Rowland circles.

### C. Optimized design for a shifted source

It is also possible to optimize the spectrometer design for a shifted source position.^{6} Figure 7 shows an optimized design for a case where the crystal-to-source distance was reduced from 300 to 200 mm. In this case, the design optimization required us to set the sinusoidal-spiral parameter to be **b** = 0.67. Comparing the optimized designs in Figs. 4 and 7, we find that the required detector lengths of 192 mm and 76.7 mm, respectively, differ significantly. The energy dispersion and energy resolution of the design, shown in Fig. 7, are dE/ds = 0.013 keV/mm and E/dE = 7670, respectively. These values are smaller than those obtained for the design shown in Fig. 4.

## ACKNOWLEDGMENTS

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

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.