1. Introduction

For soft X-rays the refractive index is close to unity with similar magnitudes for the real and imaginary part. Due to the small phase shift and the high absorption, classical refractive lenses are very inefficient in the photon energy range below 5 keV. For hard x-rays the absorption is small enough to utilize refractive lenses, but several such lenses have to be combined to create a compound refractive lens (CRL) [1].

Diffractive optics with radially increasing line density - Fresnel zone plates (FZP) – are hence the key elements for high resolution soft and hard X-ray imaging. They are mainly characterized by two parameters, their numerical aperture and their efficiency. As diffractive optics, they have many diffraction orders, but only one selected order can be used for imaging while all other orders must be blocked. With increased photon energy the ratio between the height of a zone and its width (aspect ratio) has to be increased to efficiently focus X-rays.

Today’s most successful fabrication method for FZP is electron beam lithography followed by planar etching techniques [2–4]. However, the smallest outermost zone period, which determines the zone plate resolution, is limited by the minimum electron beam diameter, electron scattering effects and the generation of secondary electrons in the electron beam resist during electron beam lithography. The achievable aspect ratio is limited by the dry etching process since the etch process is not fully anisotropic when constructing narrow nanostructures [5–7]. To overcome these limitations, so-called sputtered sliced or jelly roll FZPs have been proposed [8, 9].

These FZPs are produced by alternately coating a micro-wire with two different materials according to the required zone plate pattern. In principal, the width of the coated zone layers can be as small as a few atomic diameters and in addition there is no limitation on the achievable aspect ratio. As the outermost zone width determines the achievable resolution with FZPs, this manufacturing method is potentially suited to develop X-ray optics providing sub-10 nm resolution. However, the theoretical resolution limit given by the outermost zone width might be unattainable due to aberrations introduced by zone positioning errors influencing the focal spot. Several different types of systematic positioning errors were previously studied and tolerable upper bounds were given [10]. However, the manufacturing process for a jelly roll FZP is likely to introduce random zone positioning errors. The effect of such errors on the achievable resolution of zone-plate optics has not been described. Here we use scalar wave theory to determine the effect of random positioning errors by simulating the propagation of an incidental plane X-ray wave onto an erroneous FZP. The algorithms that we use are based on a numerical evaluation of the Rayleigh-Sommerfeld diffraction integral and will be further explained in the third section.

2. The focal spot of an FZP without positioning errors

For refractive lenses the Rayleigh criterion correlates the resolution and the numerical aperture (NA). The NA of a Fresnel zone plate depends on its smallest zone period, and is equal to the angle of diffraction of a transmission grating of this periodicity. According to the Rayleigh criterion, the spatial resolution in first order of diffraction provided by a full FZP without central stop is given by 0.61 times the outermost zone period. The manufacturing method of jelly roll FZPs requires a micro-wire substrate. As known from microscopy, blocking the inner parts of a lens reduces the FWHM of the central peak and increases the intensity of the side lobes of the focal spot pattern [11]. For X-ray microscopy, low-dose imaging is essential. The radiation dose required to detect an object detail is proportional to ${(C_{obj}\cdot MTF)}^{-2}$ where $C_{obj}$ denotes the object contrast and the modulation transfer function (MTF) describes how strongly a spatial frequency of an object is transferred into the image plane [12]. Therefore, the MTF needs to be optimized to ensure optimal X-ray imaging conditions.

The MTF is the real part of the optical transfer function (OTF). For linear shift invariant systems the OTF can be approximated by the Fourier transformation of the image of a point source. An experimental realization of such a system is a scanning transmission X-ray microscope (STXM) (see Fig. 1). The STXM works by scanning a sample through a focused X-ray beam and detecting the local change in transmission. In good approximation, the image formation can be described by the convolution of the object transmission with the FZP focal intensity distribution.

 

Fig. 1 X-ray optical setup of a scanning transmission X-ray microscope (STXM). An FZP with central stop forms a focal spot which is raster scanned in the object plane. The order sorting aperture blocks all orders except the diffraction order of the FZP used for imaging.

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In our theoretical analysis of the FZP performance, the focal spot is calculated by discretizing the FZP on an adapted polar grid and evaluating numerically the Rayleigh-Sommerfeld diffraction integral using an adapted algorithm described in [13]. From the intensity distribution of the focal spot, the MTF is computed using a Matlab Hankel transform [14]. Figure 2(a) shows the resulting focal intensity distributions for different central stop diameters. As expected the side lobes increase while the FWHM of the central peak decreases with increasing central stop size. Figure 2(b) shows the effect of the central stop on the MTF. Note that the MTF significantly decreases in the spatial frequency range of 0 - 0.13 nm⁻¹ with increasing central stop size.

 

Fig. 2 (a). Impact of the central stop on the focal spot pattern of an FZP. Four different central stop diameters are considered: 0 (brown), 25 (green), 50 (purple) and 75% (blue). Figure 2(b). Corresponding MTFs for the focal spot pattern in Fig. 2(a). The calculations were performed for an FZP with 5 nm outermost zone width and a focal length of 50 µm at λ = 0.157 nm.

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The shape of the focal spot and its resultant MTF determines the imaging performance of an STXM. Since the Fresnel zone plate with central stop forms a focal spot with large side lobes, the convolution of the object transmission and the intensity of the focal spot creates a halo around each point. As a result, the imaging process is no longer linear in intensities and requires a careful interpretation of the images. For example, structures which are smaller than the distance between central peak and the first maximum of the side lobes are effectively doubled by this halo. For lenses without a central stop, the MTF drops continuously as a function of the spatial frequency. However, with increasing central stop size the MTF rises again and reaches its maximum value close to the cutoff frequency. Note that this increase is a consequence of the halo and does not necessarily help to improve the visibility of the object structures. For FZPs with random positioning errors the situation is even more complex.

3. Algorithms for simulating random positioning errors

Depending on the FZP manufacturing method, the zone positions might deviate from their set points. It is known that errors in the positioning of the zones reduce the efficiency and the imaging properties [10, 15]. In the following, we will describe the dominating positioning errors occurring in e-beam lithography and in the production of jelly roll FZPs. The main sources of positioning errors in e-beam lithography are thermal drift, beam position drift due to charging, non-linearities in the deflection unit and write field calibration errors. In state-of-the-art e-beam systems the positioning error is expected to be less than 5 nm for FZPs with diameters below 250 µm. The nano-structuring process, which is required to transfer the zone plate pattern into a suitable material, could potentially cause additional positioning errors. For example, a local shift of the zone structures due to insufficient adhesion of the zones to the underlying material layer could be introduced in very narrow zone structures with high aspect ratios during the development or electroplating processes. This error depends on the chosen process, the material parameters and the zone width. However, in state-of-the-art FZPs generated by electron beam lithography, this error is significantly less than half of the outermost zone width (see for example SEM micrographs in [16].

The main sources of positioning errors for jelly roll FZPs are the substrate and the deposition process. The accuracy of the FZP starts with the roughness and roundness of the micro-wire substrate that is used. The deposition process with its probabilistic nature leads to uncertainties in thickness and position of the deposited materials. In contrast to the errors occurring in e-beam lithography, these errors propagate within the FZP layer system. We simulate the influence of these errors on the imaging performance of a STXM.

For the simulations, we discretize the FZP with its possible errors and propagate into the focal spot by numerically solving the Rayleigh-Sommerfeld diffraction integral. For this numerical solution it is important that errors introduced by the discretization are negligible compared to the amplitude of the positioning errors. Figure 3 shows an overview of the different errors which are considered in this work. If the zones are perfectly centered rings with varying width a discretization on a polar grid is advisable (see Fig. 3(a)). We use this type of discretization to study the influence on the MTF of an erroneous deposition rate combined with an additional surface error. We assume that the fluctuations are independent and identically distributed, so that we can simulate them by normal distributions. For this type of computation, the same algorithm that we used to compute the effect of the central stop is also suitable.

 

Fig. 3 Illustration of the different types of positioning errors which are considered in this paper. a) Random fluctuations in the deposited zone width. b) Systematic shift of the zones due to an elliptical wire substrate. c) Random positioning errors caused by the roughness of the substrate. The erroneous FZP pattern is shown in color. The zone positions for the ideal FZP pattern are also indicated (cross-hatched).

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However, we also consider effects that are better suited for discretization on an orthogonal grid, for example the effects of non-perfect wire substrates (see Fig. 3(b)-(c)). The deviation of the substrate from its perfect circular shape can be divided into a slowly varying elliptical (Fig. 3(b)) and a fast oscillating roughness (Fig. 3(c)) part. We assume that the position of the n-th zone is shifted proportional to the deviation of the micro wire substrate due to its roughness. Furthermore, we include a damping of this error proportional to the inverse of the mean radius of the n-th zone. This damping simulates a smoothing by the deposited material as claimed in [17]. The roughness of the sputtering base is simulated by a realization of a random function of the angular position which is based on a random walk and includes some smoothing.

On an orthogonal grid a different algorithm is required, and so we adapted an algorithm described in [18]. It assumes that a monochromatic plane wave is incident on the FZP. Since the FZP consists of zones with different refractive indices, the incident wave is partially absorbed and phase shifted. The algorithm assumes that the FZP is a 2D pattern without extension of the zones along the optical axis. This approximation is valid for most currently used FZPs. For thick FZPs with very high aspect ratio zone structures, the scalar wave equation has to be solved taking volume diffraction effects into account [15, 19]. In this case, the diffraction behavior of the FZPs is much more complex and cannot be described by the methods presented in this paper.

The wave field and the effect of the FZP are discretized on an orthogonal grid. At each vertex a complex number represents the value of the field at that point. If a vertex lies in an opaque zone of the FZP, we modify the complex number at that vertex to allow for the absorption and phase shifting induced by the zone. All the examples presented in this paper are based on totally opaque-and-transparent zone plates. Finally, the modified wave is propagated into the focal plane by using a Fresnel diffraction based algorithm [18].

For the Fresnel diffraction based propagation of the wave field $u\left(x,y,z\right)$ from the plane $z´=z_0$ to the plane $z=z_1$ the following equation has to be solved:

The MTF of FZPs with position errors is calculated from the resulting focal spot intensity distribution by fast Fourier transformation (FFT) and finally by averaging over circular rings in the spatial frequency domain.

4. The impact of different errors

We already studied the impact of the central stop as shown in Fig. 2. With increased diameter the central stop suppresses certain spatial frequencies; this corresponds to the plateau seen in the MTF. Structures of these dimensions will appear blurry in the image. Hence the diameter of the central stop should be < 50% of the total diameter of the FZP.

The micro-wire substrate of a jelly roll FZP might deviate from the perfect circular shape. The impact of the roughness of the substrate on the focal pattern is shown for different rms values in Fig. 4. The increase of the side lobes in the focal pattern translates into a decreased image quality (see Fig. 4, middle row). The resulting MTFs are shown in Fig. 5. As expected the MTF decreases with increasing roughness of the substrate. It also shows that the roughness of the substrate should be significantly smaller than dr_N/2 to avoid a degradation of the MTF for high spatial frequencies. Furthermore, an elliptical substrate also introduces a systematic shift in the zones, and hence results in an FZP with an additional astigmatism (Fig. 3(b), see also [10]).

 

Fig. 4 Simulated images showing the impact of the substrate roughness (right column, see also Fig. 3(c)) on the focal pattern (left column) and hence the resolution (middle column) of the STXM. The simulation was performed for FZPs with an outermost zone width of dr_N = 5 nm and a diameter of 1.594 µm at a wavelength of 0.157 nm (see also [17]). Figure 4(a) shows the results for a perfect FZP without central stop. Figures 4(b)-(e) show the results for FZPs with central stop of 0.9 µm and different rms-values of 0 nm, 3 nm, 6 nm and 10 nm, respectively.

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Fig. 5 Plots showing the mean MTFs of the same FZPs as shown in Figs. 4(a)-(e). With increasing rms-values of the micro-wire substrate, the MTFs already decrease strongly at moderate spatial frequencies. Note the increasing standard deviation with increasing rms-values which is caused by the angular dependency of the MTF of non-circular FZPs.

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In the following, FZPs with production errors are investigated. We divide the FZPs into two types according to their manufacturing process: Deposited FZPs (d-FZP) with their micro wire substrate and e-beam lithography FZPs (e-FZP). In d-FZPs errors propagate over the zone plate layer system while the positioning errors from the different zones in e-FZPs are independent. Hence, for e-FZPs the acceptable error is largely independent of the diameter, while the acceptable errors of d-FZPs strongly depend on the diameter of the optic.

The impact of the variation of the zone width is shown in Fig. 6. For d-FZPs two effects are included in the simulation. First, the deposition of material is a random process. Hence, each zone width is produced with an inherent variation. Since each zone consists of many atomic layers independently deposited onto each other, the resulting zone width error is approximately normally distributed with a standard deviation equal to the square root of the numbers of layers times the standard deviation ${\sigma }_R$ of the thickness of a single layer.

 

Fig. 6 Plots showing the MTFs of d-FZPs and e-FZPs for different production parameters. The FZPs are assumed to consist of perfectly circular concentric rings that are alternately opaque and transparent. All FZPs have a diameter of 50 µm. The border of each ring is perturbed as described in the text. The percentage of the FZP area obscured by the micro-wire substrate is denoted by A. In the left plot (a), the standard deviation of the positioning error varied between 0.02 and 0.05 nm per each nm deposited. In this case, the condition $\frac{{\text{dr}}_N}{2}>\sqrt{{\text{σ}}_{\text{R}}{}^2\frac{\text{R}}{1 nm}}=0.05 \text{nm}*86.6=4.33 \text{nm}$ is not satisfied for ${\text{σ}}_{\text{R}}=0.05 \text{nm}$ (violet graph). In the green graph, the standard deviation is ${\text{σ}}_{\text{R}}=0.02 \text{nm}$ satisfying the condition $\frac{{\text{dr}}_N}{2}>\sqrt{{\text{σ}}_{\text{R}}{}^2\frac{\text{R}}{1\text{ nm}}}=0.05\text{ nm*}86.6=1.73\text{ nm}$. The brown line corresponds to a perfect FZP. The right plot (b) shows the MTFs of an e-FZP and two d-FZPs. The FZP-parameters for the FZP 4) and 5) are identical, but the simulated positioning errors correspond to a d-FZP in 4) and an e-FZP in 5). The fact that the d-FZP 6) does not satisfy the above given condition and has additionally a large central wire reduces the MTF below the level of the FZP 4) and 5) for a large part of the frequency range.

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Second, due to the change of material at each zone interface an additional error is produced. For simplicity, this interface error is also assumed to be normally distributed with a standard deviation ${\sigma }_N$. As long as the expected deviation of the position of the outermost zone is smaller than dr_N/2, an average d-FZP will produce an Airy pattern like focal spot. If this condition is not satisfied, some of the outermost zones most likely no longer satisfy the FZP condition for constructive interference. This idea leads to the following “rule of thumb”: Given the width (dr_N) of the outermost zone, the number (N) of zones and the radius (R) as well as the standard deviations of a single zone boarder ${\sigma }_N [nm]$ and the standard deviation ${\sigma }_R \left[nm\right]$ for a deposited distance (ΔR), the following inequality should be satisfied:

For e-FZPs it is included that the position of each interface is normally distributed around its prescribed position. As long as the standard deviation ${\sigma }_e$ is smaller than dr_N, most of the outermost zones most likely satisfy the condition for constructive interference. For relatively small d-FZPs such as the one used to simulate the effect of the roughness, the aberrations due to ${\sigma }_N$ and ${\sigma }_R$ are negligible. Our simulation shows that under these conditions the main errors arise from the deviation of the substrate from a perfect cylindrical geometry. In our simulations this deviation had to be smaller than the width of the outermost zone in order to produce a focal spot close to an Airy pattern.

For practical applications in X-ray microscopy, the working distance of the FZP X-ray objectives plays an important role. For example, tomographic applications require tilting of the sample in the X-ray beam. To minimize the missing wedge effects in the reconstruction tilt angles ≥ 60° are required. Therefore, in practice the focal length of the FZP should be at least 0.5 mm. Assuming FZPs with dr_N = 5 nm containing a micro-wire with half the diameter of the FZP diameter, operating at 1, 4 or 8 keV photon energy, the required number of deposited zones are 4700, 1164 and 582, respectively. The resulting deposition standard deviations ${\sigma }_R$ are 0.014, 0.028 and 0.040 nm. In other words, the required deposition accuracies for an inner zone of 9 nm width are 0.042, 0.084 and 0.12 nm at 1, 4 and 8 keV, respectively.

Even for d-FZPs which are ideally manufactured, their utility for applications depends heavily on the photon energy used. For example, in the soft X-ray region, the efficiency of d-FZPs is significantly lowered by the smaller difference in the absorption and phase shift of the two alternating zone materials compared to e-FZPs with their vacuum against zone material contrast. Additionally, high efficiency soft X-ray optics with dr_N ≤ 10 nm require tilted zone structures with radially increasing tilt angle (see [15, 19]). So far, no approach exists to manufacture such d-FZP optics.

For hard X-rays, efficient FZPs require very high aspect ratios, but zone tilting is less important due to the significantly lower numerical apertures. Furthermore, the absorption within the FZP is much lower than for soft X-rays and so phase shift is dominant. However, for most applications the focal length of currently produced d-FZPs is too short. For high resolution d-FZPs with significantly larger working distances, many more zones are required which makes the position accuracy progressively worse. Nevertheless, d-FZPs are ideal candidates for nanoscale hard X-ray imaging, but only if either the deposition accuracy can be extremely well controlled or the actual zone plate diameter can be measured and corrected during the fabrication process.

5. Conclusion

Our theoretical analysis of zone plates manufactured by deposition techniques shows the impact of the micro-wire substrate quality and the deposition accuracy on the achievable imaging performance. The substrate should be exactly circular and very smooth, which is fulfilled for glass wire substrates [9]. The diameter of the wire substrate should be smaller than 50% of the FZP diameter to avoid strong negative effects on the MTF. We also studied the influence of the deposition accuracy on the focal spot. We find that the deposition accuracy for FZPs with useful focal lengths for practical applications is a major challenge. This challenge could be overcome by in situ measurements of the deposited layer thickness during the manufacturing process. In this case, the zone thickness can be corrected during the fabrication process which helps to suppress the propagating positioning error and relaxes the deposition rate accuracy. However, the accuracy of the measurement of the zone plate layer thickness needs to be at least half the outermost zone width.