## Abstract

We present an analytical study to provide guide lines to design a block structure hard X-ray supermirror. The block structure supermirror is a kind of layered structure consisting of several “blocks” of multilayer of different d-spacing to obtain broad energy bandwidth response. This structure has been widely applied in X-ray telescopes because it is easy to fabricate. To examine the propagation of X-rays in a supermirror structure, further simplified approximation of Kozhevnikov’s theory has been developed. The supermirror structure is described by a structure function. The spectral function of the structure, which is the Laplace transformation of the structure function, turns out to be proportional to the reflectivity profile against X-ray energy. By analyzing the expression of the spectral function, we found the reflectivity of the supermirror could be smooth due to the box-car shaped spectral function if the d-spacing and layer number of each block is arranged with appropriate constraints.

©2013 Optical Society of America

## 1. Introduction

Since imaging observations are essential in X-ray astronomy, X-ray telescopes have been developed with a single-layer coating of heavy elements from the 1970’s [1]. ASCA, Chandra [2] and Newton [3] telescopes are well known projects that covered the soft X-ray band (lower than 10keV) and have greatly affected current astrophysics in general. However, not only soft X-rays but also hard X-rays are very important for diagnosing physical processes in the universe because hard X-ray emission is mostly produced by high energy phenomena such as synchrotron radiation of high energy particles with magnetic fields, or gravitational energy release in the strong gravity of compact objects. Hard X-rays can penetrate absorbing material around these objects much more than soft X-rays. In addition, the emission line of some nucleons may exist in the hard X-ray band, which is critical for diagnosing the physical process. For example, the emission line around 78keV from Ti^{44} is a critical clue for understanding the evolution of SNR [4]. A mirror coated with a single layer no longer meets the requirement in hard X-ray bands above 10 keV, because the critical angle of the hard X-ray becomes much smaller than for soft X-rays. This situation inevitably reduces the effective area of the telescope for hard X-rays. In order to overcome the disadvantage of the performance of single-layer coating, several telescopes with supermirror coating have been developed. For example, the InFOCμS balloon telescope [5] developed by Nagoya University and NASA/GSFC was designed to cover the target band up to 45keV, and successfully observed celestial objects in balloon experiments. Meanwhile, several of the newest hard X-ray telescope projects are underway. The NuSTAR telescope [6] developed by NASA, covering the energy band from 1 to 80keV, was launched in 2012. The hard X-ray telescope (HXT) of the ASTRO-H mission [7] being developed by Nagoya University and collaborators, and scheduled to be launched in 2014, covers the energy band from 1 to 80keV. In order to meet the requirements of these hard X-ray projects, designing the hard X-ray supermirror is very important under boundary conditions. There are several design methods developed by several groups in the 1990’s. Joensen et al. concluded that the power law (ABC) method is an effective way to design the multilayer structure [8], while Kozhevnikov et al. considered the multilayer as a material in which the dielectric constant shows a step-function-like structure, and concluded that the thickness distribution should follow his theoretical equation [9]. Wang et al. introduced a simulated annealing method to design the supermirror, and demonstrated that this method is also an effective way to achieve the target reflectivity [10].

Although all of these design methods may successfully yield a supermirror structure with high and smooth reflectivity response, the X-ray multilayer thus designed requires accurate control of the thickness, which is hard to realize [11,12]. Usually, hundreds of mirror shells are needed in one telescope module. It is very important to design a supermirror structure which is easy to produce. Yamashita et al. introduced a new design method called “block method”, considering the supermirror as the combination of several periodic multilayer “blocks” [13]. Since such a supermirror is easy to fabricate and may be suitable for large production, this method has been successfully applied in the InFOCμS X-ray telescopes and is selected as the supermirror design method for the HXT of ASTRO-H mission. For the block structure supermirror, there are several parameters to be determined. Previous theoretical studies focused on the optimization of the parameters to get the highest integrated reflectivity [13–16]. However, in our experience, a slight difference of the d-spacing and layer number of each block may lead to serious oscillation of the response profile. In order to reduce oscillation, we need to understand the behavior of the X-ray propagation in the structure and the interference between blocks. In this paper, we begin with the theoretical study in [9] and an approximation called “vector model”, and finally we derive the reflectivity expression of the block-structure supermirror. By relating the “spectral function” of the multilayer structure to the reflectivity profile, a special combination of the multilayer structure parameters is determined for smooth reflectivity.

## 2. Mathematical description of the supermirror

In order to investigate the propagation of E-M waves in multilayer, the first step is to formulate the multilayer structure by introducing the distribution function of dielectric constants. The purpose is to solve the E-M wave propagation in such material, and to calculate the reflectivity, which is equal to the amplitude of the E-M wave emerging from the top boundary, as was described in Kozhevnikov’s theoretical work [9]. In his reference, the polarization between two boundaries was neglected because the grazing angle is small (< 0.5 degrees). In this paper, we introduce the equations of Eqs. (1) – (7) from ref. 9 and added more approximation to the equations to approach the analysis of block structure supermirror.

First of all, the mathematical expression of the material can be described as follows:

In order to formulate the propagation, the Fourier series expansion has been applied to describe the structure of the multilayer in [9]. The expression can be described as follows:

E-M wave propagating with the grazing incident angle in the material is given by the following equation:

U_{+} is the amplitude of the E-M wave propagating from the top to the bottom; and U_{-} is the amplitude of the E-M wave propagating from the bottom to the top. *k* is the wave number, which is equal to $2\pi /\lambda .$ $\kappa $ is the complex wave number equal to $k\sqrt{\mu -{\mathrm{cos}}^{2}(\theta )},$ which corresponds to the normal component of the wave vector in the material. B_{n} and y(z) comes from Eq. (4), and L is the total thickness of the multilayer. S_{+}(z) and S_{-}(z) are the terms which may oscillate rapidly as was described in [9]. In this paper, the last terms will be discussed in the next section and will be neglected eventually.

In Eq. (6), the constant 1 before the plus sign corresponds to the amplitude of the incident X-ray at the top. The term before the sign of integration corresponds to the reflectivity of each boundary. The integration term means the sum of the phase-shifted U_{-} from the top to position z. U_{-} is the amplitude of the E-M wave propagating upward. Equation (7) can be understood in the same way. Obviously, the reflectivity is equal to the amplitude U_{-}(0) of the E-M wave on the top of the structure.

Since Eqs. (6) and (7) are iterative with each other, the analysis of those equations is very complex, if at all possible. In this paper, the main goal of our approach is to analyze the propagation of the E-M waves in the multilayer and to quickly derive the reflectivity profile from the multilayer structure, and then give a qualitative guide of multilayer design especially for a block structure supermirror with small oscillation in the reflectivity curve, rather than to deduce accurate expression for the reflectivity of the block structure supermirror.

Then we introduce the “vector model” [17,18] of approximation where the incident X-ray may not lose intensity as it propagates through boundaries, i.e. U_{+}(z)≈1. Though this approximation leads to overestimation of the reflectivity, the reflectivity profile can be reproduced qualitatively,

Three points are summarized to apply this approximation in our block structure.

- 1. Amplitude reflectivity of each boundary is low for high energy X-rays at grazing angles larger than critical angles. Therefore the amplitude of the X-ray propagating in the structure is still comparable with unity.
- 2. The vector model is very simple and very suitable for qualitative analysis of the shape of the reflectivity (oscillation).
- 3. Reflectivity curves calculated by the exact method and the vector model will be compared with each other below to demonstrate the validity of this approximation.

According to the vector model, the reflectivity can be simplified as,

_{-}(z) will be discussed later. Considering the fact that $j\mathrm{exp}(-j\pi n\Gamma )$ may not affect the absolute value of $r(\kappa ),$ subsequently this term has been omitted from the Laplace operator. In Eq. (9), the integer n corresponds to the order of the Bragg reflection. In this paper, we only analyze the reflectivity contributed by the first order Bragg peak to simplify the analysis, i.e. n = 1. It is because the second order Bragg peak is negligible in amplitude.

The expression of the reflectivity can be derived as:

The first term in the reflectivity expression is the contrast of the material in grazing incidence geometry. The second term demonstrates the effect of the thickness ratio between two kinds of material and the third term is the spectral function. The phase shift and absorption information of the reflectivity is included in the third term. Γ is assumed at 0.5 throughout this paper. Thus, the expression of the reflectivity can be further simplified as follows,

## 3. Analysis of the block-structure supermirror

The supermirror designed by the block method consists of several “blocks” of the periodic multilayer. In this structure, the d-spacing and the layer pairs of each block are different. So the block structure may provide the broad energy bandwidth response as shown in Fig. 2 .

Compared with other kinds of supermirror structures such as the power law structure and computer optimized structure, the block structure supermirror is easy to fabricate and calibrate; fabrication of the constant d-spacing is usually more stable than the varying d-spacing. It is also easier to learn the information of the block structure from the measured reflectivity. However, when we design the block structure, the parameters of each block such as d-spacing and layer pairs are decided with many empirical implications and rules. It may sometimes lead to serious oscillation of the reflectivity profile because of the interference between blocks, which can be understood by Eq. (11).

Following to Eq. (11), the expression of the reflectivity of a block structure supermirror can be written as:

Firstly, we analyze the Eq. (11) without S_{-}(0). The structure function of each block with periodic multilayer can be expressed as follows:

_{0}is the thickness of the block.

We looked up the table and found the Laplace transform of the structure function defined in [0, + ∞] is:

The total thickness of one block is limited, so that we are able to use the delay operator on the image function and subtract them from each other to cut out the block structure, as shown in Fig. 3 :

Here “Block_{i}” stands for the structure function of i-th block of the supermirror.

The Laplace transformation of Block_{0}’s structure function should be:

Secondly, with the “vector model” assumption, the S_{-}(0) of one block will be discussed as follows.

In [9], the original expression of U_{-}(z) is derived as follows,

_{+}(z) = 1 and z = 0 to obtain the reflectivity, then

Subsequently, the Eq. (19) can be derived as follows,

_{0}. Considering that the tail of the negative component drops so much at the position of the peak of positive component, it is concluded that S

_{-}(0) can be neglected in our reflectivity expression.

In order to prove the analysis described above, reflectivity curves calculated by the exact method and those by the vector model including the conclusion above will be compared with each other in an example below.

Eventually, the amplitude reflectivity expression of a block structure can be determined with the “vector model” assumption,

The spectral function of one block is derived in Eq. (17). In that case, the surface of the block is set at z = 0, as described in Fig. 3.

For the block structure supermirror, the blocks should be arranged sequentially because the starting point of the later block must be the end point of the former block. So the delay operator has to be added in the expression of the spectral function to shift the position of each block, as is shown in Fig. 4 .

So the expression of spectral function of the block structure can be derived by the following equation:

*L*

_{i}is the total thickness of

*i*-th block. L

_{-1}is defined as 0, which means the surface of the first block is at 0 point. N is the number of blocks.

Up to this point, the reflectivity expression of a block structure supermirror has been developed as Eqs. (21) and (22). However, the validity of those equations established with the assumption of the “vector model” still remains unclear.

In order to prove the validity of this assumption, calculated results for a model structure are compared in two cases with Eqs. (21) and (22) and exact method.

The above Fig. 5 depicts the sample structure consisting of five blocks. For each block, the thickness is set as 32nm and the layer number is incrementally increasing from 12 to 16. The material is Pt/C and the grazing angle is set at 0.3deg. Then Eqs. (21) and (22) are used to calculate the reflectivity,

In Fig. 6 , the blue line is the reflectivity calculated by Eqs. (21) and (22). The red line is the reflectivity calculated by exact method. Our result (blue) is higher because of the “vector model” assumption [17,18]. However, the reflectivity profiles (oscillations) are very similar, which suggests that the vector model is useful to analyze the interference behavior of the X-ray in block structure qualitatively.

In the case of the block structure supermirror, Eqs. (21) and (22) may help us to understand the interference behavior of the X-ray propagation. Usually, the expression of the spectral function should be quite complicated. However in our experience, the oscillation of the reflectivity can be small when the structure follows two special rules.

First, the total thickness of each block is equal. Second, the layer number of each block increases incrementally from the top to the bottom.

The structure following the above design rules is presented as Fig. 7 .

If we apply these two new rules, the Eq. (22) which describes the interference between blocks can be expressed as follows,

_{0}, and the inverse lattice is in the arithmetic series with common difference of $2\pi /{L}_{0}$ which suggests the layer number of each block increases with an increment of one.

Since the absolute value of the spectral function of each block follows the distribution as Sinc function, the structure constraints may lead to a very special case. According to “OFDM” theory which has been well developed in communication science [19], our conclusion is, in the case of the block structure supermirror, when the thickness of each block is equal to L_{0} and the inverse lattice between the adjacent block is $2\pi /{L}_{0},$ the structure function of each block should be orthogonal. Subsequently the coherence between each block should be minimized. Here, we omit the mathematical description of the orthogonal structure function. Instead, an example is given below to demonstrate this special case with a plot.

We assume a structure with 8 blocks. The thickness of each block is set as 50nm (L_{0} = 50nm). The layer number of each block is set incrementally from 17 to 24. The grazing angle is set at 0.28deg. Then, Eq. (23) has been calculated and the profile of the spectral function is shown as follows.

In Fig. 8 , the red line shows the power spectrum of the whole block structure. The power spectrum of each block is shown by the profile with a different color. Obviously, the nodes of all the profiles coincide. Moreover, the center position of each Bragg peak is located exactly at the position of the node position of adjacent block, which suggests the orthogonal phenomena described in “OFDM” theory [19]. As is shown in the figure above, the power spectrum of the whole block (red line) shows a boxcar profile. This is because all the side lobes are periodically distributed between the Bragg peaks, which allows the Bragg peak of each block to independently decide the final response.

One major point in this paper is the fact that the real reflectivity profile with oscillations can be directly related to the spectral function of the supermirror structure. If a smooth profile of reflectivity response is required, the boxcar type spectral function shown in Fig. 8 should be very promising. Eventually, we can give one possible solution to guide the parameter design of a block mirror.

Though we have started from the supermirror structure with boxcar type spectrum, the obtained reflectivity profile may decline with the increasing energy. This is because the contrast of the optical constant in high energy is usually lower than that of the constant in a low energy band. As a result, the reflectivity designed by our schema inevitably declines quickly with the increasing energy. In some cases, smooth response is very important, while flat response is not the first priority. For the hard X-ray telescopes, for example, the reflectivity of the supermirror is usually designed to be smooth but decreasing with X-ray energy because of the scientific requirements (soft band data is more important than hard band to get continuous spectra from data below 10 keV). Moreover, the block structure is easy to fabricate, so this structure may provide a possible way to achieve smooth reflection in high energy band up to 60keV. In the next section, we will introduce an example to achieve the goal.

## 4. An example of the block structure supermirror in high energy band

In this chapter we demonstrate a simple example with selected application in a high energy band (>30keV) with our design rules.

Before the designing process, several premises are set for possible missions with our current technology.

- 1. Total number of the layer pairs is limited for ease of fabrication (N~150-200).
- 2. Roughness is set at 0.4nm, which corresponds to our fabrication ability and materials (Pt/C).
- 3. All the blocks are far from saturated (Details will be published in a future work).

Then, we assume a selected application in which the target band is set from 45keV to 60keV with the grazing angle at 0.28deg. The material is Pt/C.

The design process to achieve smooth reflectivity is shown as follows:

- 1. The d-spacing of the first block is set at 2.94nm, corresponding to 45keV.
- 2. The number of the first block is arbitrarily set at 17. So, the thickness of each block is 50nm
- 3. Following our design rules introduced in Fig. 7, the d-spacing and layer pairs of all other blocks can be decided sequentially.
- 4. Add the blocks in the bottom of the structure until all the target band is covered.
- 5. If the total layer pairs N does not achieve 150-200, return to step 1 and add 1 layer pair to the first block and do step 2-4 and let the number of layer pairs meet the requirement. (In this example, the number of layer pairs is 168.)

When the structure is constructed, we compare the design result with another design method. Here we use the power law structure [8] (ABC method) with the same total layer pairs of the block structure and compare the design results (Fig. 9 ).

The number of layer pairs was decided at 168. The d-spacing of the first block is 2.94nm. The gamma is 0.4. The upper red line shows the carbon thickness and lower one shows that of platinum. The layer thickness of the power law structure shown with the blue line seems to be mostly the same with the block structure. The definition of the MF (Merit Function) to evaluate the ripples of the reflectivity curve is shown as the insert (b) in Fig. 9, where E_{i} is the discrete energy point defined within [45keV, 60keV]. R_{0} is the target reflectivity which is set at 30% (without the roughness).

The power law structure is optimized by the simplex algorithm to maximize the integrated reflectivity in target band (N = 164, Grazing angle 0.28deg, Roughness 0.4nm). As is shown in Fig. 10 , the block structure thus designed provides comparable integrated reflectivity with reasonably small ripples as the power law model structure. This result suggests that the design rules introduced in this paper are a reasonable guideline to design a block structure supermirror in the high energy band that is easy to be fabricated for real applications, or as the initial structure for computer optimization. So, it is important for hard X-ray telescopes and other potential application in future.

## 5. Conclusion

In this paper, we have examined the propagation of the electromagnetic wave in the block structure supermirror, beginning with Kozhevnikov’s theory. With the assumption of the “vector model”, the block structure has been described by the structure function and its Laplace transformation (i.e. spectral function). The reflectivity expression of the block structure has been derived and the validity has been proved by an example. The shape of reflectivity profile (oscillation) is directly related to the spectral function of the supermirror structure. Two special constraints of the design have been introduced to get a box-car type spectral function. Such kinds of constraints are applied to get small oscillation of the reflectivity. An example of supermirror structure has been designed to demonstrate the potential application of the block structure design rules in the high energy band.

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