1. Introduction

X-ray microscopes are valuable research tools for numerous applications [1–3]. Their short wavelengths combine high spatial resolution and element sensitivity. In classical microscopy, the spatial resolution is limited by the numerical aperture of the objective. In most X-ray microscopes, zone plate objectives are used. Their spatial resolution is directly proportional to the outermost zone width dr_N, which is indirectly proportional to the numerical aperture. Hence it is highly desirable to push dr_N as low as possible. However, for high diffraction efficiency, the zones ideally are kept at optimum height to achieve sufficient absorption or phase shift. This leads to the difficulty to fabricate extreme high aspect ratio zone structures for high resolution zone plates. This requirement is increasingly challenging for higher photon energies since they require much higher zones. In practice, the nanofabrication limitations of zone plate structures determine the achievable spatial resolution. To date, spatial resolutions of about 10 nm half-pitch were reported [4–6]. However, the diffraction efficiency under these conditions is still relatively low, in particular for imaging in the hard X-ray regime. For zone width of 25 nm the highest reported aspect ratio for hard x-ray zone plates is 22:1 and was achieved by using a zone-doubling technique [7]. Their calculated efficiency is 12% at 6.2 keV photon energy. With the same nanofabrication process zone plates with 15 nm outermost zone width and an aspect ratio of 23:1 are reported having a calculated efficiency of 5.7% at 6.2 keV. By using these zone plates in an X-ray microscope, they permit to resolve 15 nm features [7]. Additionally, in order to enhance the diffraction efficiency in the hard X-ray regime i.e. to increase the achievable aspect ratio of the zone plate structures, mechanical near field stacking of two separate zone plates to double the aspect ratio of the structures was reported for 24 nm wide zones [8]. Usually the zone plates have to be stacked very close to each other within the near-field. However, by decreasing the diameter of the second zone plate of the stack this restriction can be relaxed within certain limits [9, 10]. More than two zone plates with similar zone width were superimposed by on-chip stacking which overcomes the need for high precision alignment in the X-ray microscope [11]. Another approach for on chip stacking of two zone plates is described in [12].

Today almost all fabricated high resolution zone plates are based on e-beam lithography to generate the zone plate pattern. The minimum zone width and the maximum aspect ratio of the zones of a single zone plate are limited by the minimal structure width which can be fabricated with e-beam lithography and the subsequent nanostructuring processes to transfer the generated pattern into a suitable zone plate material. One of the main limitations to generate high aspect ratios is the poor stability of high aspect ratio structures which tend to collapse. For hard X-rays, a method based on atomic layer deposition (ALD) coating of e-beam lithography written zone plates was reported which extends the achievable resolution limitations and aspect ratios [13]. The method is based on the ALD deposition of a zone plate material (usually Ir or Pt) onto the sidewalls of a pre-patterned template structure usually made of Si or hydrogen-silsesquioxane (HSQ). It results in very high aspect ratios of the ALD deposited thin zone structures and a frequency doubling of the e-beam written zone period, thus improving the achievable resolution of X-ray microscopes [7, 13].

The ALD approach allows to decrease the zone period of the nanofabricated Si structures by a factor of two which makes them easier to fabricate than a full duty cycle zone plate. Subsequent ALD coating then provides the full duty cycle zone plate by coating both sides of the Si structures. In this paper, we investigate by simulation a method which has the potential to improve the resolution of high-efficiency ALD zone plates further by stacking multiple complimentary ALD zone plates [14]. With this approach current limits in zone plate fabrication like the limited e-beam resolution and aspect ratio limits due to zone structure collapsing can be overcome. For the first time, we investigate the performance of such stacked ALD zone plates and derive parameter for their experimental realization.

2. Zone plate fabrication – proposed method to reduce the effective zone period

The proposed zone plate fabrication method is based on the frequency doubling technique by ALD [7] combined with zone plate stacking [8, 11]. A simplified schematic of the process is illustrated in Fig. 1 for the case of two stacked zone plate layers [14]. In the first step, a circular grating pattern is exposed into an HSQ resist layer by e-beam lithography. After development, the HSQ pattern is coated in an ALD process with an iridium or platinum layer to form the active zones of the zone plate (in the following we only refer to iridium) [7]. The deposited thickness of the iridium layer corresponds to the outermost zone width of the final stacked lens composed of two zone plates. The metal coated sidewalls of the HSQ structures form a zone plate pattern with constant zone width and half the number of zones compared to the final composite zone plate pattern. In the second step, a second iridium zone plate pattern is fabricated (see first and second zone plate in Fig. 1). The ring pattern of this zone plate is designed such that when stacked on the first zone plate, the composite iridium zone plate has a two times smaller period than the individual iridium zones of either zone plate (see Fig. 1, right).Note that the zones of the equivalent composite zone plate (see Fig. 1, right, bottom) have a four times smaller period than the fabricated HSQ structures of single patterns and the smallest HSQ structures in a single zone plate of the stack are three times wider than the outermost zone width of the final zone plate. Thus to achieve a 15 nm outermost iridium zone width, the HSQ pattern would only require a smallest period of 120 nm and a smallest feature width of 45 nm. These dimensions can be routinely fabricated by e-beam lithography. Additionally, higher zones can be fabricated in the single patterns due to the relaxed manufacturing requirements leading to higher focusing efficiency zone plates.

 

Fig. 1 Illustration of two stacked zone plates fabricated by iridium ALD coated HSQ structures. The iridium layer is shown in blue color, the HSQ in gray color. The left panel shows the innermost-zone stacking of the two zone plates, while the right panel shows the outermost zone stacking. Note that the outer rings of the stacked iridium zone plates (right top and middle) are equivalent to a single zone plate with four fold smaller period (right, bottom). For illustration, parameters suitable for 8keV imaging were chosen to yield a highly efficient 15nm outermost zone width zone plate.

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2.1 Design of the zone plate patterns

The radii r_n of a normal zone plate pattern for the alternating opaque and transparent zones are given by the zone plate law. This law can be approximated by r_n² = n∙λ∙f, in which n is the zone number, λ the wavelength and f the focal length of the zone plate. The zone width dr_n = r_n-1 – r_n can be approximated for large zone numbers as dr_n = λ∙f/2r_n, which shows that the zone width shrinks inversely proportional to the zone plate radius. The outermost zone width of a zone plate with N zones is dr_N.

The iridium zones of the equivalent zone plate pattern shown in Fig. 1 (right, bottom) have a process-related constant zone width of dr_N given by the thickness of the coated iridium layer. The centerline of the zones is the same as in a normal zone plate pattern. The radii r_k of the first HSQ zone plate patterns to be fabricated are: r_k^ZP1 = (r_4k + r_4k-1)/2 + (−1)^kdr_N/2, k = 1, 3, 5,...,N/4 and k = 4n The radii r_i of the second HSQ zone plate pattern are described by: r_i^ZP2 = (r_4(i-1)-2 + r_4(i-1)-3)/2 – + (−1)ⁱdr_N/2, i = 2, 3, 4,...,(N + 2)/4 + 1 and i = (n + 2)/4 + 1.

3. Diffraction efficiency calculations of the stacked ALD zone plates

We applied electrodynamic theory based on rigorous coupled wave theory (RCWT) to calculate the diffraction efficiency of the ALD stacked zone plates as a function of their physical stacking separation. RCWT assumes periodic structures which is a good approximation for the local diffraction behavior of zone plate structures [15]. The X-ray optical material properties based on the atomic scattering factors were implemented into the GSOLVER software package to perform the RCWT calculations [16]. RCWT describes the local zone plate period as a Fourier series of infinite gratings. Therefore, the frequency doubling by the introduced second zone plate shifts the light from the 1st order of diffraction of the first zone plate into the 2nd order after propagating through the whole zone plate stack. Note that the spatial resolution achievable in an X-ray microscope using the 2nd order of diffraction of the zone plate stack for imaging corresponds to the 1st order of diffraction of a conventional zone plate with the same outermost zone width (Ir zone width).

3.1 Diffraction efficiency and stacking distance

The local diffraction efficiency calculations presented in this section are performed for the outermost zones with a line to period ratio of 0.5 of the zone plate. Since the width of all iridium zones is a process-related constant and corresponds to the outermost zone width dr_N = const., the line to period ratio is reduced towards the zone plate center. This lowers the diffraction efficiency from the outer part to the inner part of the zone plate. To estimate the loss in integral efficiency due to the constant zone width dr_N, we estimated this effect for an optically thin zone plate neglecting volume diffraction effects. The efficiency η₁ in the first order of diffraction depends on the line w to period p ratio and is maximized for w/p = 0.5 [17]: η₁ ∝ sin²(π∙w/p). Using the approximations r_n² = n∙λ∙f and dr_n = λ∙f/(2∙r_n) to describe the zone plate pattern, we note that the area π∙r_n² ∝ n, i. e. Each zone has the same area with the local zone period given by p = 2∙dr_n,. Therefore, we can rewrite η₁ ∝ sin²(π∙dr_N/(2∙dr_n)) = sin²(π∙dr_N∙)$\sqrt{n/(\lambda \cdot f)}$, and so the ratio between the integral efficiency of a zone plate with constant zone width and a normal zone plate pattern with varying zone width is given by

Thus, the integral efficiency of a zone plate with constant zone width is 70% of a regular zone plate pattern. A similar moderate drop in integral efficiency is expected for the stacked zone plates discussed in this paper.

The diffraction efficiency of the outermost zones of a zone plate stack shown in Fig. 2 (right side) is calculated as a function of the distance Δd between the two stacked zone plate patterns. Fig. 3 shows the results for zone plate stacks with outermost zone widths of Δr = 15 nm and 28 nm and zone thicknesses of Δt = 900 nm and 1300 nm, respectively. Furthermore, the efficiency data for Δr = 7 nm and Δt = 900 nm are plotted. The range of the plotted distance Δd starts at a minimal negative value which corresponds to the assumed zone height. For example, the plot for the zone plate stack with the two zone plates with 28 nm outermost zone width and 1300 nm zone height starts at Δd = −1300 nm meaning that the iridium zones of both zone plate layers are in the same plane. For Δd > − 1300 nm the two zone plate layers are shifted along the optical axis. At Δd = 0 nm, the zone plate layers are directly positioned on top of each other.

 

Fig. 2 Illustration of the zone plate parameters used for the diffraction efficiency calculations. On the left, a zone plate with constant iridium zone width Δ ris shown (solid blue lines are the iridium zones). On the right, two stacked zone plates separated by a distance Δd are shown. Superposition of the two stacked zone plate patterns results in the zone plate pattern shown on the left. Efficiency calculations were performed for the outermost zones marked by the red box. These zones have a zone width of Δr and a zone thickness of Δt. The outermost zone period of the left zone plate is 2∙Δr and the period of the outermost structures of each of the two stacked zone plates on the right is 4∙Δr.

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Fig. 3 Calculated 2nd order diffraction efficiency as a function of the distance between two stacked zone plates. The curves show the diffraction efficiency of the outermost zones for three different zone plate stacks (green, red, blue) with the indicated outermost zone widths (Δr) and thicknesses (Δt).

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The plot in Fig. 3 shows for the zone plate stack with Δr = 28 nm (Fig. 3, green curve) a rather low efficiency drop from 32% at Δd = 0 µm to 27.5% at Δd = 3 µm. In case of the zone plate stack with Δr = 15 nm, the efficiency drops significantly from 22.5% at Δd = 0 µm to 5% at Δd = 3 µm. For a zone plate stack with 7 nm outermost zone width, the plot shows that the diffraction efficiency reaches a minimum of 3% at about Δd = 0 µm. That means even if the two zone plate layers are directly on top of each other, the efficiency is very low compared to the diffraction efficiency of about 16% of a regular zone plate pattern (at Δd = −0.9 µm). It can also be seen that there is a second maximum at about Δd = 1.5 µm for this zone plate stack. This maximum is even higher than the first maximum which will be discussed later. However, it is not clear if this second maximum can be used for imaging, i.e. if the same spatial resolution can be reached as for a regular zone plate with the same outermost zone width. A full electrodynamic calculation of the three-dimensional pattern with wave propagation into the focal point would be required to solve this question.

For on-chip stacking [11] of the zone plate patterns, the maximum diffraction efficiency values are shown in Fig. 3 at Δd = 0. For mechanical stacking [8] of the zone plate patterns, a distance Δd = 3 µm is more feasible than shorter distances but this results in lower diffraction efficiencies. In general, the two zone plates have to be longitudinally aligned to a proximity clearly within the optical near-field. The distance has to be less than 0.5 ∙ dr²_n / λ as reported in the literature [9, 18]. That means the required proximity of the two stacked zone plates corresponds to approx. 25% of the depth of focus of a zone plate. For the zone plates with 28 nm and 15 nm zone widths at 8 keV photon energy, we obtain a maximal distance of 2.5 µm and 0.7 µm, respectively. For the zone plate with 7 nm zones this criterion gives a maximal distance of 0.15 µm. However, this requirement can be relaxed, i.e. the drop in efficiency can be reduced by shifting the zone radii of the second zone plate pattern of the stack, but it needs to be matched to the envisioned distance.

This effect is shown in Fig. 4. The diffraction efficiency of the outer zones with Δr = 15 nm is plotted as a function of Δd and a reduction R of the zone radii of the second zone plate. A reduction of R = 100% of Δr means that the radii of the zones with width of Δr = 15 nm are reduced by 15 nm. If both zone plate patterns would be in the same plane (Δd = −900 nm) this would correspond to a zone plate with doubled (Δr = 30 nm) zone width and 1:1 line to space ratio. Referring to the same spatial resolution as obtained with the described stacked zone plates again the second order of diffraction has to be considered for this zone plate. It is known that the second order diffraction efficiency is zero in that case since the line to space ratio is 1:1 for the zones. However, if the distance between the two zone plate patterns is increased the efficiency increases slightly. Note that for Δd = 0 both zone plates are not in the same plane, but both patterns are directly on top of each other and separated by the height of the zones. By using an optimized reduction in radii in the second zone plate pattern the diffraction efficiency of the zone plate stack can be significantly improved for a certain distance Δd. The diffraction efficiency optimized in this way is indicated in Fig. 4 (magenta colored line in the contour plot). The upper plot in Fig. 4 shows the optimized efficiency, i.e. the optimized reduction in zone radii as a function of the distance Δd between both zone plate patterns. As can be seen by comparing the 2D-plot in Fig. 4 with the plot in Fig. 3, the diffraction efficiency can even be enhanced for a distance of Δd = 0 by a slight reduction in the zone radii of the second zone plate. Specifically, for the maximal plotted distance Δd = 3 µm the diffraction efficiency can be enhanced by a factor of about 3.4.

 

Fig. 4 Contour plot of the diffraction efficiency of 15 nm outermost zones as a function of both the distance Δd and the reduction in radii R of the second zone plate pattern within the stack. The magenta line shows the maximized efficiency in the colorized contour plot (bottom). The maximized efficiency is additionally shown in the 2D-plot at the top.

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The possibility to enhance the diffraction efficiency at certain stacking distances by shifting the zone radii of the second zone plate pattern of the stack can also be applied for the calculated zone plate with Δr = 7 nm and 900 nm zone height (see red curve in Fig. 3). However, calculations show that the diffraction efficiency corresponding to the first maximum of the red curve in Fig. 3 can only be reached for very small stacking distances 0 nm < Δd < 500 nm. For such small stacking distances the diffraction efficiency can be enhanced from less than 5% (see Fig. 3, 0 nm < Δd < 500 nm) to values between 7% (at Δd = 500 nm) and 13% (at Δd = 0 nm). Therefore, on-chip stacking would be best suited to enhance the diffraction efficiency of the zone plate with 7 nm wide zones to about 13% at Δd = 0 nm at a reduction of radii R = 60% of Δr as shown in Fig. 5. The stacking distance to obtain the second diffraction efficiency maximum in Fig. 3 could also be extended. However, since we did not calculate if the same spatial resolution can be expected, we do not show this case.

 

Fig. 5 Diffraction efficiency of 7 nm outermost zones as a function the reduction in radii R of the second zone plate pattern within the stack at the distance Δd = 0 nm. The maximal diffraction efficiency of 13% is obtained for R = 60%.

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Now we address the open question of the second zone plate maximum noted above with 7 nm outermost zone width shown in the corresponding plot of Fig. 3. The zone plate or grating stack used for the calculations is illustrated in Fig. 5. The blue boxes illustrate the iridium zones of the two stacks. The two grating stacks are separated by the distance Δd. The stack can also be regarded as a volume zone plate [19] with a zone period of 28 nm and a line to period ratio of 7 nm : 28 nm = 1 : 4. The ideal shape of the volume zone plate with tilted zones (tilt angle Θ) is indicated by the grey boxes in Fig. 6. The zones drawn in blue approximate this shape to a certain degree depending on the distance Δd. The plot in Fig. 6 shows the 1st, 2nd and 3rd orders of diffraction of the blue iridium zones illustrated on the left side. The 2nd order of diffraction corresponds to the red curve of the plot in Fig. 3 and shows a maximum at about Δd = 1.5 μm. (Note that the plot in Fig. 6 starts at Δd = 0 and is plotted up to Δd = 5 μm). This maximum could be interpreted as the 2nd order of diffraction of a corresponding volume zone plate where Θ fulfills the Bragg condition, i.e. at the distance Δd = 1.5 μm the blue iridium zone best approximates the structures of a corresponding volume zone plate so that the diffraction efficiency is maximized. We also find at a distance Δd = 3.3 μm a maximum of the 1st order of diffraction, which could be interpreted as the 1st order of diffraction maximum of a corresponding volume zone plate with tilted zone fulfilling the Bragg condition for the corresponding geometry. Whether such a volume zone plate stack achieves the theoretical resolution requires additional studies.

 

Fig. 6 Calculated 1st, 2nd and 3rd order diffraction efficiency of the zone plate stack with Δr = 7 nm zone width as a function of the distance between two stacked zone plates. The zones of the stack are illustrated in blue color on the left.

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3.2 Influence of real zone plate material layers on the diffraction efficiency

The cross section of a stacked zone plate structure fabricated by an ALD process is illustrated schematically in Fig. 7. Such a stacked structure will generate not only the ideal zone profile B of Fig. 2, but in addition an absorbing iridium layer between the zones will cause absorption of the X-rays and the HSQ (SiO2) layers will cause local phase shifts and adsorption. To account for these effects we performed electrodynamic calculations of the diffraction efficiency of the stacked zones with the profile and material shown in Fig. 7.

 

Fig. 7 Illustration of a more realistic zone profile of a zone plate stack with zones fabricated by an HSQ/ALD process.

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Table 1 lists the corresponding results compared with the efficiency of an ideal profile as shown in Fig. 2, profile B. The calculation was performed for a photon energy of 8 keV and a distance Δd = 3 μm of the two stacked zone profiles. As before, zone widths of 28 nm and 15 nm were analyzed. Additionally, for the 15 nm wide zones, an optimized zone plate stack with reduced radii of the second zone plate pattern was calculated. This analysis shows that the loss in efficiency by the non-ideal cross section of the zones at 8 keV is negligible.

Table 1. Efficiency of iridium zone plate pair with distance Δd = 3 μm for the profiles shown in Fig. 4 and Fig. 7 at 8 keV photon energy

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4. Conclusions

In summary, we have analyzed the performance of a novel ALD and stacking-based fabrication method for high resolution zone plates with high aspect ratios. The method uses multiple zone plates stacked within the near field to divide the zone period in the resulting combined zone plate by the number of zone plate layers. Even with two zone plate layers, this method permits a reduction of the effective zone period by a factor of four compared to the initial-beam lithography exposed pattern. Furthermore, we performed a diffraction efficiency analysis of the near field stacked zone plates using rigorous coupled wave theory as a function of the zone plate distance. Our calculations performed for hard X-rays at 8 keV photon energy and outermost zone widths of 28 nm and 15 nm predict diffraction efficiency values of more than 20%.To achieve such high diffraction efficiencies the distance between the two stacked zone plates has to be about Δd < 0.5 ∙ dr²_n / λ. We found that this strict requirement can be relaxed, i.e. the drop in efficiency can be reduced for slightly larger distances by shifting the zone radii of the second zone plate pattern of the stack. Furthermore, we analyzed for this fabrication method the performance of very small zone widths of about 7 nm and 900 nm zone height. In this case the diffraction efficiency is significantly lowered from 16% to 3% for on chip stacked zone plates. We also identified a second maximum in the diffraction efficiency at Δd = 1.5 μm, but further studies are required to clarify whether these second maximum can be used for imaging. However, by adjusting the zone radii of the second zone plate pattern of the stack the diffraction efficiency can be increased to 13% at least for on chip stacked zone plates. In sum, we believe that such optics have great potential for nanoscale focusing in the hard X-ray photon energy range.