Comparison of CAST and CWT

FZP is a special kind of variable width grating (Fig. 1A), the FZP zone width dr_n is determined by Eq. (1) (Ref Reference Kirz25):

(1)\begin{equation}d{r_n} = \sqrt n \times {r_1}/2\end{equation}

(2)\begin{equation}{r_1} = \sqrt {\lambda f + {\lambda ^2}/4} \end{equation}

Figure 1. (A) Schematic structural diagram of FZP. (B) Local grating approximation.

where f, λ and r ₁ are the focal length, X-ray wavelength and zone width of the first ring, respectively. This satisfies the FZP principle that the incident light passes through the adjacent half bands with different refractive indexes and then reaches the focal point with optical path differences of light wavelength λ and phase differences of 2π, which realizes the focusing at the focal point (Refs Reference Maser35, Reference Rehbein, Heim, Guttmann, Werner and Schneider36).

The first-order DE is a key parameter as important as the resolution to evaluate FZP performance. According to CAST, the incident light passes through FZP and is focused according to the geometric relationship. There are some assumptions here that the zone width is sufficiently large or the FZP thickness t is extremely small, which can eliminate the need for consideration of the dynamic efficiency. Based on this, DE is calculated by comparing the superposed focus light intensity with incident light intensity as formula (3) (Ref Reference Wang37):

(3)\begin{equation}\begin{aligned} DE = \left( {1/{\pi ^2}} \right) &\times \left\{ \exp \left( { - 2k{\beta _1}t} \right) + \exp \left( { - 2k{\beta _2}t} \right) \right. \\ & \quad \left. - 2exp\left[ { - k\left( {{\beta _1} + {\beta _2}} \right)t} \right] \times {\text{cos}}\left[ {k\left( {{\delta _1} - {\delta _2}} \right) \times t} \right] \right\} \end{aligned} \end{equation}

where k = 2π/λ, and the complex refractive index of the material is expressed as n = 1-δ-βi which describes the absorption and attenuation of light as it passes through the material. It is shown that only the material refractive indexes and FZP t are considered in CAST.

CWT is an electromagnetic vector diffraction theory to calculate DE of periodic grating which combines Maxwell equations and boundary conditions to describe the coupling relationships between an incident and diffracted light (Refs Reference Takano, Sumida, Hirotomo, Koyama, Ichimaru, Ohchi, Takenaka and Kagoshima18, Reference Kirz25, Reference Kirz, Jacobsen and Howells33, Reference Yun, Viccaro, Lai and Chrzas34). According to Eq. (1) the local zone period 2dr_n changes slowly when zone number n is larger. Therefore, the FZP is approximated as an infinite periodic grating locally with a local period of 2dr_n. Figure 1B shows the FZP approximate grating model, through which the FZP DE could be estimated locally by CWT (Refs Reference Takano, Sumida, Hirotomo, Koyama, Ichimaru, Ohchi, Takenaka and Kagoshima18, Reference Mayer, Keskinbora, Grevent, Szeghalmi, Knez, Weigand, Snigirev, Snigireva and Schutz19).

The differences of the absorption coefficients and phases for X-rays between Al₂O₃ and HfO₂ are large, leading to high DEs for X-rays with Al₂O₃/HfO₂ material pairs (Refs Reference Sanli, Jiao, Baluktsian, Grevent, Hahn, Wang, Srot, Richter, Bykova, Weigand, Schutz and Keskinbora21, Reference Sanli, Keskinbora, Leister, Teeny, Grévent, Knez and Schütz38). Moreover, these two materials have high melting points and stable chemical properties, which enable them in a large ALD common temperature growth interval (Ref Reference Sanli, Jiao, Baluktsian, Grevent, Hahn, Wang, Srot, Richter, Bykova, Weigand, Schutz and Keskinbora21). Thus Al₂O₃/HfO₂ is selected as the multilayer film material for numerical analysis. ML-FZP DEs calculated by CAST and CWT is compared in Fig. 2. In all situations, the DE oscillates periodically with ML-FZP thickness t. The CAST DE is mainly dependent on t at a given photon energy and material pairs. As Fig. 2A shows, the maximum CAST DEs are above 25% at 8 keV no matter how the zone widths are. Actually, when X-ray pass through a grating with a smaller size, plenty of energy is exchanged between wave vectors in different directions (Refs Reference Toroglu and Sevgi23, Reference Luebbers, Kunz, Schneider and Hunsberger24), which leads to a decrease in diffraction light intensity and DE. While the DE calculated by CWT displays a sharp decrease as dr_N decreased below 25 nm. Take Al₂O₃/HfO₂ at 8 keV as an example, the calculated maximum CAST DE is 27.6% at the optimum t 6.1 μm, and the maximum CWT DE decreases to 14.8% at t 4.5 μm and 4.9% at t 2.9 μm when dr_N reducing to 15 and 10 nm, respectively.

Figure 2. Theoretical ML-FZP DEs calculated by CAST and CWT with a decreasing dr_N for Al₂O₃/HfO₂ at (A) 8 keV and (B) 15 keV.

In addition, as shown in Fig. 2, we could obtain that the DE calculated by CWT is changed greatly with the zone width. Such a grating approximation method that only considers the dr_N is inappropriate and might cause great errors. The following two main reasons should be taken into account. Firstly, the central rings of FZP have larger zone widths (about >25 nm) and then higher DE, which should not be ignored. In addition, the DE descends greatly as the zone width decreases from 25 nm. We consider that the impact of all zones on DE should be calculated.

Furthermore, comparing Fig. 2A and B shows the optimal Al₂O₃/HfO₂ ML-FZP t is larger at 15 keV than at 8 keV for the same dr_N because of its strong penetrability in which thicker materials are needed to realize π phase shift. The optimal t increases from 4.5 and 2.9 μm to 8.7 and 5.1 μm when X-ray energy is raised from 8 to 15 keV at dr_N of 15 and 10 nm, respectively.

Given the above, in order to satisfy the application of X-ray microscopy imaging, all zone widths, thicknesses, X-ray energies and material pairs should be fully considered when calculating ML-FZP DE.

DE calculated by PCM

To calculate and analyse the FZP DE more reasonably and accurately, it is necessary to comprehensively consider the X-ray focusing and diffraction characteristics of all the zones, dr_N, t and material pairs of FZP. A PCM is proposed here. It derivates as follows:

Generally, DE is defined as the ratio of the energy E_{out_total} of diffracted light to the total energy E_{in_total} of the incident light, and then:

(4)\begin{equation}{E_{out\_total}} = \eta \times {E_{in\_total}}\end{equation}

Assuming that the FZP zone number is n, the incident light energy of each zone is expressed as E_in(n) with its corresponding efficiency of η_(n), then the diffracted X-ray energy can be expressed as follows:

(5)\begin{align}{E_{out\_total}} = \left( {{\eta _1} {\times} {E_{in1}} + {\eta _2} {\times} {E_{in2}} {+} \ldots {+} {\eta _{n - 1}} {\times} {E_{in\left( {n - 1} \right)}} {+} {\eta _n} {\times} {E_{in\left( n \right)}}} \right) \end{align}

The incident energy is defined as the total energy received by each zone of FZP. The area of each FZP zone is approximately equal based on the Fresnel criterion and the number of photons received per unit area is equal. It is reasonable to determine that the energy irradiated to the same area is equal, so the total energy of incident light can be expressed as follows:

(6)\begin{equation}{E_{in\_total}} = n \times {E_{in\_zone}}\end{equation}

The total DE of the FZP conforms to the following equation:

(7)\begin{equation}\begin{aligned} \eta &= \frac{{{E_{out\_total}}}}{{{E_{in\_total}}}} \\ & = \frac{{{\eta _1} \times {E_{in1}} + {\eta _2} \times {E_{in2}} + \ldots + {\eta _{n - 1}} \times {E_{in\left( {n - 1} \right)}} + {\eta _n} \times {E_{in\left( n \right)}}}}{{n \times {E_{in\_zone}}}} \end{aligned} \end{equation}

In the formula, the incident X-ray energy of the half band is equal, and the above formula can be rewritten as

(8)\begin{align}\eta = \frac{{{\eta _1} + {\eta _2} + \ldots + {\eta _{n - 1}} + {\eta _n}}}{n}\end{align}

Therefore, the FZP DE can be expressed by the average value of each zone DE as formula (8). Since the FZP zone number is generally large, such as 6200 rings for Al₂O₃/HfO₂ ML-FZP with dr_N 10 nm, diameter 0.25 mm, in order to improve the calculation efficiency, we can divide the FZP into several regions according to the fluctuation degree of the DE. Two principles are used here for the segmentation: first, the DE fluctuation in a region is not greater than 1%; second, the average DE fluctuation is not exceeding 1000th, where (the average DE fluctuation) = [DE fluctuation × (zone number in this region))/(total zone number). By this time, the DE could be assumed nearly equal in each region. Then, according to the characteristics of different regions, the CAST or CWT method is reasonably selected to calculate the DE for specific region. For regions with large dr_n near the centre, CAST can be used; with dr_n decreasing along the radial direction, CAST is no longer applicable, and CWT can play an advantage at this time. What follows is weighted averaging of the DE of all regions and obtaining the ML-FZP DE as reasonably and accurately as possible.

Taking Al₂O₃/HfO₂ ML-FZP with dr_N of 10 nm and focal length of 16 mm as an example, the total zone numbers are 6200, of which 3444 rings have a zone width 10–15 nm, 1206 rings lie in 15–20 nm, 558 rings lie in 20–25 nm and 992 rings have the zone width larger than 25 nm, as Fig. 3A is shown. Based on PCM, the Al₂O₃/HfO₂ ML-FZP in Fig. 3A is divided into 158 regions as shown in Fig. 3B. It can be found that the region number increases linearly as dr_N decreases from 25 to 5 nm and fewer regions is required with dr_N above 25 nm, which depended closely on the DE fluctuation.

Figure 3. (A) Schematic diagram of the zone number and (r) region number calculated by PCM in different zone width intervals for Al₂O₃/HfO₂ ML-FZP with X-ray energy, dr_N and focal length f are 8 keV, 10 nm and 16 mm, respectively.

For outer rings with larger n, as the zone widths changed slowly, the FZP DE could be approximately calculated as a periodic grating with 2dr_N as a period through CWT (Refs Reference Mayer, Keskinbora, Grevent, Szeghalmi, Knez, Weigand, Snigirev, Snigireva and Schutz19, Reference Gao, Wu, Lu, Liu, Xia, Zhao, Li, Kong and Han39). That is, the ML-FZP DE is calculated here as an infinite periodic grating with a half-period of 10 nm.

As an illustration, Al₂O₃/HfO₂ ML-FZP DEs evaluated by CAST, CWT and PCM are compared in Fig. 4A for dr_N 5 nm and in Fig. 4B for dr_N 15 nm. It can be seen that the largest DE is estimated through CAST because the energy attenuation caused by the dynamic effect is ignored. Meanwhile, it is more undesirable to just consider the dr_N and based on this period to estimate the CWT DE of infinite periodic grating as that of the whole FZP, which causes the DE to be too low. The proposed PCM consideration of the focusing performance of all the FZP zones, which provides a reasonable way to calculate DE as accurately as possible.

Figure 4. Al₂O₃/HfO₂ ML-FZP DEs compared by CAST, CWT and PCM vs FZP t at 8 keV with (A) dr_N 5 nm, (B) dr_N 15 nm; DEs calculated by the proposed PCM vs FZP t with dr_N 10, 15, 20 and 25 nm, respectively at (C) 8 keV and (D) 15 keV.

Moreover, the Al₂O₃/HfO₂ ML-FZP DE curves with dr_N 10, 15, 20 and 25 nm at 8 and 15 keV are also calculated via PCM as shown in Fig. 4C and D, from which the theoretical maximum DE and optimal t could be estimated. It can be seen that, due to dynamic diffraction effects, the decrease of dr_N below a certain threshold makes an obvious influence on the ML-FZP DE significantly, especially at high energy. Besides, a higher t is desired at larger energy to achieve maximum DE of ML-FZP. For ML-FZP with extra small dr_N, such as 10 nm, Al₂O₃/HfO₂ ML-FZP has a maximum theoretical DE of 12.3% at 8 keV with t 4–6 μm, while as the X-ray energy increase to 15 keV, corresponding DE is below 1%, which indicates the material pair is selective for energy applied in X-ray imaging. In addition, the DE curves display multiple peaks at 15 keV which mainly depended on the complex relationship between the FZP structures, material pairs and X-ray.

From the results above, it can be seen that as the FZP aspect ratio increases, more factors and parameters should be considered, which have a great impact on the DE. CAST is a method that calculates FZP DE without reflecting on the zone width, while CWT evaluates DE using the outermost zone width dr_N only. Combining all zone widths, t, material pair and X-ray energy, the PCM is seriously proposed to estimate the FZP DE more exactly. It is of great significance for the design and fabricating of FZP with high DE and resolution, especially for the resolution below 25 nm.