3.1.

Selection of Multilayer Materials

The first step in supermirror design is the selection of the materials for each layer pair. The multilayer consists of layer pairs with two different elements: heavy and light. For high reflectivity, the contrast between the refractive indices of two materials must be large, and their absorption edges should not appear in the target energy region. In addition, interfacial diffusion between two materials blurs the contrast of the refractive index at the boundaries, which is equivalent to an increase in the interfacial roughness and causes degradation of the multilayer reflectivity.

Because the refractive index $\delta$ is proportional to the electron density, high-$Z$ elements have been used for mono- and multilayer reflectors. These elements include W, Ir, Pt, and Au. Gold (Au) has often been used for total reflection mirrors of the ASCA and Suzaku x-ray telescopes because of its chemical stability. However, an island structure is sometimes found in thin Au layers of less than a few nm. Tungsten (W) is often used for the W/C and W/Si multilayers. Although the supermirror shows reasonable performance below its $K$ absorption edge at 69.5 keV, W cannot be our choice because we are pursuing hard x-rays up to 80 keV. Platinum (Pt) is most suitable for our purposes, because it is chemically stable and has higher (electron) density than Au. Its absorption edge at 79 keV is acceptable with respect to the high-energy end of the required energy range of $<80\mathrm{keV}$.

For light elements, carbon (C) and Si are often used in various multilayer designs. Pt/C and Pt/Si have almost the same calculated reflectivity, but the interfacial roughness of Pt/Si was several times higher than that of Pt/C in our fabrication system. Thus, our final choice was a combination of Pt and C for our multilayers.

3.2.

Fabrication of Pt/C Multilayers

The multilayer structure is created by vacuum deposition. In our case, Pt and C are sputtered using argon (Ar) with DC biases. The thickness of individual layers is controlled by the sputtering rate and deposition time. The DC bias voltage and current and the Ar pressure are controlled to maintain a constant deposition rate. For example, Ar pressure is set at 0.2 to 2.0 Pa and the bias voltage is adjusted between 500 and 900 V so that the plate current remains constant. The actual deposition rate on the sample is also determined by the distance from the sputtering target to the sample. Because the mirror sample is rotated in front of the sputtering target, the deposition time is defined by the rotation speed and the window width of a mask placed between them. The uniformity of the thickness distribution on the sample is better than 5% using a special mask pattern that is narrower in the middle and wider at both ends. This compensates for the fast deposition rate occurring in the middle of the 50-cm-long Pt and C targets.

It is well known that the reflectivity of a multilayer degrades as $\exp -{(2\pi \sigma /d)}^2$ when $d$ approaches $2\pi \sigma$, where $d$ is the thickness of one-layer pair and $\sigma$ is the interfacial roughness (Debye–Waller factor). For our typical Pt/C multilayer with $d$ of 5 nm, Fig. 1 shows simulated reflectivity curves of a typical multilayer with different $\sigma =0$, 0.3, 0.5, and 0.7 nm. Note that interfacial roughness higher than 0.3 nm degrades the multilayer reflectivity considerably. Here, $2\pi \sigma$ approaches or becomes greater than $d$. The solid line in Fig. 1 is the measured reflectivity of the multilayer produced by our DC sputtering system, which indicates that we are able to achieve $\sigma$ of 0.3 nm for a Pt/C supermirror.

Fig. 1

Measured reflectivity of typical multilayer (Pt/C multilayer with $2d$ of 5 nm, 30-layer pairs, $\mathrm{\Gamma }$ of 0.4) comparing actual grazing angles with simulated ones having interfacial roughness $\sigma$ of 0, 0.3, 0.5, and 0.7 nm.

Such $\sigma$ of 0.3 nm limits the minimum $d$-spacing $d_{\mathrm{mini}}$, as suggested above. From our experiments, the minimum $d$-spacing must be equal to or longer than 2.4 nm.

For the actual fabrication of the telescopes, the time we were able to spend on multilayer deposition was limited. We selected the upper limit of layer pairs as 140 for the Hitomi HXT. It was also found that $\sigma$ became worse when more than 200-layer pairs were deposited. This is because the interfacial roughness accumulates and grows significantly. The upper limit of 140-layer pairs prevents performance deterioration in our design.

4.1.

Boundary Conditions and Design Goals

In this section, the design scheme of the depth-graded multilayers is described for the Hitomi HXT to provide a broad response at high energy above 20 keV, under the boundary conditions set by the geometry of the mission and by the deposition technology. First, boundary conditions are listed as below based on the scientific and technical requirements:

Under these boundary conditions, the goals are as follows:

In the following sections, our design scheme for the multilayers is shown step by step to optimize the multilayer parameters based on the above goals.

4.2.

Top Layer

One significant difference between this design and the balloon experiment appears in the soft x-ray region below 20 keV. In such a soft x-ray region, the total reflection is dominant, so it is necessary for each supermirror design to have a reflectivity profile smoothly connected to that of the total reflection.

At a given grazing angle $\theta$, the total reflection is effective for x-rays below the critical energy $E_c$. The first-layer thickness is set to be $d_{\mathrm{Pt}0}$ (thickness of the first Pt layer) so that the x-ray at an x-ray energy of $E_c$ will be attenuated to $1/e$ by the top layer. That is, $(d_{\mathrm{Pt}0}/\sin \theta )\times \sigma (E_c)\times n=1$, where $\sigma (E_c)$ is the absorption cross-section per atom and $n$ is the number density of atoms.

In Fig. 2, the reflectivity of a supermirror with a top layer is plotted against the energy for a grazing angle of 0.25 deg. The green line indicates that a reflectivity of 5.1 nm thus derived for $1/e$ fills the dip below 20 keV and allows sufficient enhancement around 25 keV, whereas a thicker top layer, such as 7.5 nm (red line), shows a significant dip near 25 keV. Figure 3 shows the reflectivity for a grazing angle of 0.11 deg. It indicates that the top layer of 10.8 nm (green line) derived for $1/e$ is too thick, so the dip at 50 keV is significant. The red line in Fig. 3 suggests that 7.5 nm is sufficient to recover half of the dip between 30 and 40 keV. This is because the ratio of optical constants $\delta /\beta$ becomes larger for the $E_c$ of 40 keV or higher, so the reflectivity below $E_c$ may be high. Therefore, the top layer should not be thicker than 7.5 nm for small grazing angles to avoid the large attenuation of hard x-rays that would be reflected by the lower part of the multilayers. We decided to set $d_{\mathrm{Pt}0}$ at 7.5 nm for $\theta <0.17\mathrm{deg}$ (Fig. 4).

Fig. 2

Reflectivity curves at grazing angle of 0.25 deg plotted against photon energy. Red, green, and blue lines correspond to top-layer thicknesses of 7.5, 5.1, and 3.5 nm, respectively.

Fig. 3

Reflectivity curves at grazing angle of 0.11 deg plotted against photon energy. Red, green, and blue lines correspond to top-layer thicknesses of 10.8, 7.5, and 5.5 nm, respectively.

Fig. 4

Thickness of top layer versus grazing angles. Solid line shows the attenuation length of Pt. Open circle illustrates the chosen thickness of a top layer in our design. Above 0.17 deg, the Pt top layer decreases with grazing angle. Below 0.17 deg, the thickness of the top layer should be 7.5 nm to avoid strong attenuation.

4.3.

First Block

Under the top-layer pair, the first multilayer block with a constant $d$-spacing of $d_1$ is designed to reflect x-rays optimally at $E_1$. The Bragg energy of the first block, $E_1$, is defined such that $1/e$ of the incident x-ray flux can emerge from the bottom boundary of the top layer at the energy $E_1$. Above $\theta \ge 0.17\mathrm{deg}$, the Bragg energy $E_1$ is equal to the critical energy $E_c$. However, some flux is transmitted for $\theta <0.17\mathrm{deg}$ and then $E_1$, at which the attenuation length is $\sim 7.5\mathrm{nm}$, slightly less than $E_c$. Thus, $d$-spacing of the first block is $d_1=hc/(2E_C\sin \theta )$ for $\theta \ge 0.17\mathrm{deg}$.

The reflectivity of the first block is calculated for different numbers of layer pairs (Fig. 5) at a grazing angle of 0.2 deg as a typical case for the HXT. The peak reflectivity $R_p$ rises as $\propto N^2$ for small $N$, where $N$ is number of layer pairs, and the slope gradually decreases for larger $N$. For higher x-ray energies, more layer pairs are necessary (smaller slope in Fig. 5), whereas the reflectivity is more than 90% at the saturation level for large $N$. At the same time, the bandwidth $W$ of the reflectivity will become narrower as a function of $1/N(W/E\propto 1/N)$ because more layer pairs place tighter constraints on interference. Therefore, the integrated reflectivity is given as the product of the peak reflectivity $R_p$ and the bandwidth $W$. Figure 6 is a plot of $R_p/N$, which is proportional to $R_p·W$. It peaks at seven layer pairs for 45 keV, for example. Adding more than seven layer pairs does not improve the integrated reflectivity. Reflectivity peaks are marked with a solid circle in Fig. 6. In Fig. 5, the optimum layer numbers are shown with circles on curves for different x-ray energies, corresponding to the peaks in Fig. 6. They essentially correspond to 60% of the saturated reflectivity with an infinite number of layers. We calculated such optimum layer numbers for various x-ray energies, as shown in Fig. 7. Most of the data can be fitted by a straight line from 30 to 60 keV. Below 30 keV, one pair is sufficient.

Fig. 5

Reflectivity of multilayer Bragg peak at a grazing angle of 0.2 deg plotted against the number of layer pairs $N$. It increases rapidly for small $N$ and begins to saturate for large $N$. Curves of different colored represent reflectivity curves for different x-ray energies. Solid circles on individual lines depict the $N$ that provides maximum integrated reflectivity ($=R_p/N$).

Fig. 6

Product of $R_p$ and $1/N$ plotted against x-ray energy $E$. Orange circles indicate peaks of $R_p/N$, which correspond to the circles in Fig. 5.

Fig. 7

Necessary layer numbers $N$ for various x-ray energies. They are given by the peaks in Fig. 6. Straight line is best-fit linear function to the data.

$\mathrm{\Gamma }$ is the ratio of thicknesses of the heavy elements $d_{\mathrm{Pt}}$ to the total pair thickness $d$. The reflectivity of the first and second Bragg peaks depends on $\mathrm{\Gamma }$. Because the reflectivity at the first Bragg peak becomes maximum for $\mathrm{\Gamma }$ of $\sim 0.4$, this is generally used for all designs in the following part of this paper.

However, $\mathrm{\Gamma }$ of 0.5 is intentionally adopted when the second Bragg peak energy is in the energy range to be covered by the bottom layer pairs with shorter $d$-spacing values. This is because the second Bragg peak reflectivity becomes negligible for $\mathrm{\Gamma }$ of 0.5. In turn, this alleviates the influence of the second Bragg peak to affect the block design scheme for the first block. Note that, for the $i$’th block, the same rule applies unless the second Bragg peak appears below 80 keV.

4.4.

Block Method

Our basic strategy is a block method, which is characterized by blocks with constant $d$.⁶ Figure 8 shows the structure of the supermirror. In our design, the block contributing to the lowest energy reflectance should be placed on the surface to reduce absorption. The deeper blocks contribute to the reflectance of higher energy x-rays.

Fig. 8

Structure of supermirror designed by block method (right). The supermirror consists of multiblocks, which have a constant $d$ spacing multilayer. The Bragg energy of each block is different, and reflectivity profile of the supermirror is determined virtually by superposition of each block.

As the block method consists of many parameters, $d_i$, $N_i$, and ${\mathrm{\Gamma }}_i$ ($i$ is the number of blocks counted from the top), we can construct various reflectivity profiles compared with other methods.

In Fig. 9, the dashed line represents the reflectivity of each block of the multilayer at $\theta =0.208\mathrm{deg}$. The solid line corresponds to the response of the supermirror with all blocks. The multilayer parameters are optimized in each block according to the scheme mentioned in Sec. 4.3. The energy steps between blocks will be determined in Sec. 4.5. In this scheme, all parameters are determined automatically without any assumed function to design the $d$-spacing $d_i$ or $d(z)$ ($z$ is the depth of the structure), which sometimes have been introduced in other research without any causality.

Fig. 9

Reflectivity of supermirror (solid line) and individual blocks (dashed lines) of supermirror designed for seventh group of mirror shells at incidence angle of 0.208 deg.

4.5.

Energy Steps $\mathrm{\Delta }E$’s Between Blocks

After determining the multilayer parameters of the first block, the energy step $\mathrm{\Delta }E(=E_{i+1}-E_i)$ to the next block must be determined. It must be smaller than the bandwidth of $W_i$ to avoid gaps between the responses of adjacent blocks (Fig. 8). As $E_i$ increases, the necessary number of pairs increases, but the bandwidth $W_i$ becomes narrower, which is represented as $W_i/E_i\propto 1/N_i$. If we take into account the linear relationship between $N_i$ and $E_i$ in Fig. 7, $N_i$ is roughly proportional to $E_i$ at high energies $(N_i\propto E_i)$. Therefore, the energy step to the next block of the multilayer $\mathrm{\Delta }E(\sim W_i)$ has to be constant.

Once we assume the energy step $\mathrm{\Delta }E$ at a certain value, it is possible to design all parameters according to the above scheme and the boundary conditions. In Fig. 10, reflectivity curves are shown for different cases of $\mathrm{\Delta }E=2$, 3, and 4 keV. A small $\mathrm{\Delta }E$ of 2 keV (black line in Fig. 10), for example, has sufficient overlaps between neighboring blocks, but the energy band reaches only 60 keV. It cannot reach the upper bound of 80 keV because of the upper limit of the total number of pairs. Even if limited to no $<60\mathrm{keV}$, no significant increase of reflectivity is apparent in the reflectivity curve. By contrast, a large $\mathrm{\Delta }E$ of 4 keV causes gaps between neighboring blocks because $\mathrm{\Delta }E$ is larger than the bandwidth $W_i$. In Fig. 11, the integrated reflectivity is plotted against different energy steps $\mathrm{\Delta }E$ from 2 to 4 keV. It peaks at 3 keV in this design scheme. Hence, the energy step $\mathrm{\Delta }E$ of 3 keV is adopted for the Hitomi HXT. At a grazing angle of 0.208 deg, we need 138 layer pairs in 18 blocks spaced at 3-keV steps. Table 1 shows a set of the supermirror parameters optimized at a $\theta$ of 0.208 deg, which was actually used for the Hitomi HXT. The numbers in parentheses are the spacing $d_i$ and the number of layer pairs $N_i$ in the $i$’th block together with ${\mathrm{\Gamma }}_i$.

Table 1

Supermirror parameters for grazing angle of 0.208 deg.

Fig. 10

Reflectivity curves of supermirrors designed with different energy steps of $\mathrm{\Delta }E=2$ (black), 3 (red), and 4 (green) keV.

Fig. 11

Integrated reflectivity plotted against energy steps $\mathrm{\Delta }E$’s between blocks; obtained by integration of the reflectivity curves shown in Fig. 10.

4.6.

Grazing Angles from 0.07 to 0.27 deg

In the discussion above, a grazing angle of 0.2 deg is illustrated as a typical value to optimize the multilayer design. For the Hitomi HXT, the grazing angle ranges from 0.07 to 0.27 deg. Every set of multilayer parameters is, in principle, determined with the same design scheme. A total of 213 mirror shells are split into 12 groups so that the grazing angles of mirror shells in individual groups vary within 5% of each other. In each group, the same multilayer parameters are used for the mirror shells.

In Fig. 12, the $d$-spacing of the $i$’th layer pair is plotted against the layer number from the surface.

Fig. 12

$D$-spacing of $i$’th layer pair plotted against the layer number from the surface. Solid line shows the design for grazing angle of 0.208 deg shown in Table 1. The design for 0.169 deg (dotted line) and power-law type design (dashed line) also plotted for comparison.

For small grazing angles below 0.1 deg, a few more layers are added to each block. For small grazing angles, fewer layers are required by the design scheme. Hence, this result in the bandwidth of each block are broader than 3 keV, which is the optimal value derived earlier in the case of 0.208 deg. By adding extra layers, the bandwidth would be reduced to match the energy step $\mathrm{\Delta }E$ of 3 keV. Furthermore, these additional layers enhance the peak reflectivity $R_p$ so that the integrated reflectivity is improved within the boundary conditions.

The profiles shown in Fig. 12 are step functions obtained from the power-law type of supermirror design. The power-law profiles in this figure are designed for the largest grazing angle (0.174 deg) of the Pt/C multilayer in NuSTAR.¹² The profiles of the block method for a grazing angle of 0.169 deg are also shown for comparison. It is very interesting to find such similarity between two parameter sets derived based on two different design principles.