## Abstract

Fresnel zone plates are frequently used as focusing and imaging optics in x-ray microscopy, as they provide the ease of use of normal incidence optics. We consider here the effects of tilt misalignment on their optical performance, both in the thin optics limit and in the case of zone plates that are sufficiently thick so that volume diffraction effects come into play. Using multislice propagation, we show that simple analytical models describe the tilt sensitivity of thin zone plates and the thickness at which volume diffraction must be considered, and examine numerically the performance of example zone plates for soft x-ray focusing at 0.5 keV and hard x-ray focusing at 10 keV.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Fresnel zone plates are employed widely as focusing and imaging optics in x-ray microscopy [1–3]. One important characteristic of a Fresnel zone plate is its finest, outermost zone width $ d{r_N} $, which limits the Rayleigh spatial resolution to $ {\delta _r} = 1.22d{r_N}/m $ [4,5] for the $ {m^{{\rm th}}} $ diffraction order. Another is its thickness $ {t_{{\rm zp}}} $, which limits the diffraction efficiency for x-ray focusing [6]. Ideally, the thickness should approach a value of

Zone plates with high aspect ratios, and finest zone widths $ d{r_N} $ approaching the x-ray wavelength $ \lambda $, have properties that begin to depart from those of thin zone plates. To maximize diffraction efficiency in thicker zones, the outermost zones ideally should be tilted relative to the incident beam direction by the Bragg angle $ {\theta _{\rm B}} $ [1,15] of

for the zone at a radius $ {r_n} $, though this has yet to be achieved in practice. In addition, as the aspect ratio increases, one must begin to consider waveguide effects, with coupled wave theory offering one approach to calculate these effects at a particular radius under the assumption that the small variation in the width of nearby zones means that locally one can use the solution for an infinite linear grating [15]. Subsequent coupled wave equation studies have shown that the diffraction efficiency can be improved for specific values of x-ray wavelength $ \lambda $, aspect ratio, and zone tilt [16–19]. While coupled wave theory provides an analytical solution for an infinite, constant-period volume grating so that one gains insight for specific subregions of a zone plate, multislice propagation methods [20,21] (also called the beam propagation method [22]) can be used to calculate the focusing properties of complete zone plates with volume diffraction effects included, as well as with fabrication errors accounted for [23–26]. Thus, the focusing properties of Fresnel zone plates in perfect optical systems are well understood.However, not all microscopes employing Fresnel zone plates are perfect. One of the challenges is to align a zone plate so that it is truly perpendicular to the incident x-ray illumination. This is different from considering the tilt angle of zones at specific radii so as to achieve the Bragg condition of Eq. (3); instead, this tilt of the entire circular zone plate introduces aberrations in focusing. Simple analytical estimates for tilt limits have been obtained for thin zone plates [4,27], as will be discussed below. However, apart from a brief summary of our early results [28], no systematic studies have been carried out to reveal the details of how the focal spot is degraded as a function of overall zone plate tilt, nor have volume diffraction effects for tilted, high aspect ratio zone plates been considered. We investigate this question so as to understand the tilt tolerances of Fresnel zone plates used for x-ray nanofocusing and nanoimaging, including in the case of hoped-for advances in finest zone width $ d{r_N} $ and thickness $ {t_{{\rm zp}}} $.

Zone plate tilting has been carried out in a deliberate fashion in specific prior studies. With a linear zone plate that focuses in only one direction, one can tilt the zone plate along the non-focusing direction in order to increase the projected thickness of the zone material and thus increase the focusing efficiency at higher photon energies [29]. When extreme tilts are used, zone positions can be tapered along that direction so that zones at all distances from the desired focus position satisfy the one-dimensional version of Eq. (4) below [30]. Multilayer Laue lenses (MLLs) can be thought of as thick one-dimensional zone plates that can be used in orthogonal pairs for 2D focusing [31], and both their focal spot properties and focusing efficiency are affected by the tilt of each 1D optic compared to the desired Bragg condition, either for constant-thickness zones [19,32] or for variable-thickness or wedged MLLs [33]. Our results here are instead for circular zone plates that provide 2D focusing in a single optic.

## 2. ANALYTICAL LIMITS TO THE TILT OF THIN ZONE PLATES

A Fresnel zone plate with a focal length of $ f $ in first diffraction order has zone boundaries at radii

where $ n $ is the integer zone number (leading up to the outermost zone $ N $), and the second term corrects for spherical aberration when focusing a plane wave. When the total number of zones $ N $ is large, one can approximate the outermost zone width $ d{r_N} $ as which allows one to obtain equivalent expressions ofWe now consider the case of wavefield aberrations produced by optical path length differences in tilted zone plates, using the geometry shown in Fig. 1. The path length difference $ {\ell _u^\prime} $ between the upper marginal ray and the central axis {[27], Eq. (10)} is given by

As one can see from Eqs. (9) and (10), the coma term $ {\ell _c} $ can compensate for the opposite-sign astigmatism term $ {\ell _a} $ up to some tilt angle $ \theta $ for the upper marginal ray relative to the axial ray, but in the case of the lower marginal ray, both terms have the same sign. Therefore, the overall tilt limit of the zone plate is best characterized by the lower marginal ray calculation of Eq. (10). The ratio of the astigmatism term $ {\ell _a}{\theta ^2} $ over the coma term $ {\ell _c}\theta $ can be written as

Whichever term in the lower marginal ray calculation dominates, we can ask that it not exceed $ \lambda /4 $ so as to satisfy the Rayleigh quarter wave criterion for minimal aberrations in focusing. This gives rise to a tilt limit of

If the first term $ {\ell _c}\theta $ dominates (that is, $ \theta \ll {\rm N}{\rm .A.} $), we arrive at a tilt limit due to coma {[27], Eq. (14)} of If the second term $ {\ell _a}{\theta ^2} $ dominates (that is, if $ \theta \gg {\rm N}{\rm .A}. $), we arrive at a tilt limit due to astigmatism of {[27], Eq. (13)}## 3. APPROACHES TO SIMULATING ZONE PLATE FOCAL SPOTS

With a thin non-tilted zone plate, one can take a forward traveling plane wave $ {\psi _0}\exp [ - ikz] $ and modulate it by the transmission of the zone-containing regions of the optic. The modulation to the transmitted wave is

using the x-ray index of refraction $ n = 1 - \delta - i\beta $ given by Eq. (2), where $ k = 2\pi /\lambda $, and $ t $ is the thickness of the material along the x-ray beam direction. One can then use free-space Fresnel propagation to bring the zone-plate-modulated exit wave to the plane of the focus. If this calculation is done using a discrete Cartesian grid with transverse voxel size $ {\Delta _x} $ and longitudinal voxel size $ {\Delta _z} $, one must worry about aliasing effects with circular Fresnel zones. If a calculation grid voxel is fully filled with zone material, then one replaces $ t $ with $ {\Delta _z} $ in Eq. (16). If, however, only a fraction $ \chi $ of a calculation grid voxel is filled with zone material, one can make the substitution $ t \to \chi {\Delta _z} $ in Eq. (16) and obtain accurate results [25]. An example of this partial voxel filling approach for a zone plate with $ {t_{{\rm zp}}} \lt {\Delta _z} $ is shown in Fig. 3. One can obtain a good numerical approximation for $ \chi $ by first calculating zero-or-one filling of zones on a much more finely sampled 2D array and then downsampling to the desired array size for subsequent calculations.One has two options (Fig. 4) for carrying out the calculation of how a tilted zone plate modulates the incident plane wave to yield the wave exiting the zone plate:

- •
**Optic-aligned:**one approach is to work in a coordinate system aligned to the zone plate, and tilt the illumination by an angle $ \theta $ as illustrated in Fig. 4A. If one uses numerical wave propagation with a 2D input grid of $ {N_x} $ pixels each of size $ {\Delta _x} $, the incident wave phase change per pixel $ {\varphi _x} $ is given by where $ x $ is the coordinate of the pixel along the axis of tilt. This is implemented by creating a matrix that contains $ {\varphi _x} $ values for each pixel, followed by pointwise application of this phase using $ \exp ({\varphi _x}) $.If one uses a numerical wave propagation approach to take the exit wave to the $ {m^{{\rm th}}} $ order focal plane where one has the same calculation grid pixel size $ {\Delta _x} $ at the focal plane, one must limit the wavefield tilt angle to

in order to have the focus spot be within the calculation grid at the focal plane. The alternative is to increase the number of 2D pixels and thus surround the zone plate by a larger emptier area, but there are practical limits to array size $ N_x^2 $ as set by available computer memory if using a single compute node.We note that there are other approaches to calculating wavefield propagation with tilted plane wave inputs [34–36]. We became aware of these alternative approaches only after our main calculations were completed, so we did not consider their properties for this work.

- •
**Wavefield-aligned:**an alternative approach is to use an incident plane wave aligned to a 3D calculation grid, and tilt the zone plate relative to the grid as shown in Fig. 4B. In this case, one fills the complex grid with per-voxel refractive index values using $ \chi {\Delta _z} $ for the material thickness $ t $ of Eq. (16), where now $ \chi $ includes the effects of tilting the zone plate relative to the calculation grid.

In the optic-aligned approach, the calculation grid can be one voxel deep in the beam propagation direction if the thickness of the zone plate $ {t_{{\rm zp}}} $ is less than or equal to the calculation grid voxel depth $ {\Delta _z} $, or $ {t_{{\rm zp}}} \le {\Delta _z} $. When $ {t_{{\rm zp}}} \gt {\Delta _z} $, or in the case of the wavefield-aligned approach (where one tilts the zone plate relative to the calculation grid), one must propagate the incident wavefield through the calculation grid over multiple voxels in depth. This can be done using the multislice propagation method as noted above; however, in this case, one must calculate the optical modulation produced by the first plane in the 3D grid using Eq. (16) but with $ t \to \chi {\Delta _z} $, and then free-space propagation over a distance $ {\Delta _z} $ is used to bring the wavefield to the next plane in the calculation grid. This process continues until one has obtained the exit wave at the back side of the calculation grid, after which one can use simple wavefield-aligned free space propagation methods to bring the wave to the focal plane.

For these numerical calculations, the voxel width $ {\Delta _x} $ of the calculation grid should be set to some small fraction $ { \epsilon _1} $ of the finest zone width $ d{r_N} $, such as with $ { \epsilon _1} = 0.1 $. Ideally, the pixel depth $ {\Delta _z} $ should be reduced in a series of numerical calculations to find how the results asymptotically approach the result obtained with very thin values of $ {\Delta _z} $, but a reasonable estimate can be found by setting the voxel depth to be a fraction $ { \epsilon _2} $ of the depth of focus of the non-tilted zone plate’s focus, with $ { \epsilon _2} = 0.1 $ giving good results [25]. One can obtain a similar estimate using the Klein–Cook parameter $ Q $ of [37]

where $ n $ is the mean refractive index [which is $ n \simeq 1 $ for x rays based on Eq. (2)]. Values of $ Q \lesssim 1 $ are adequately described using plane grating diffraction, while the condition $ Q \gtrsim 1 $ means that volume diffraction effects begin to come into play. If the grating half-period is $ b{\Delta _r} $, one can rearrange Eq. (19) in terms of the slice thickness to find The condition of having $ { \epsilon _1} = 0.1 $ and $ { \epsilon _2} = 0.1 $ corresponds to $ Q = 5\pi /(n{b^2}) $. As an example, if the pixel size is one-fifth the finest zone width $ d{r_N} $ in a Fresnel zone plate, one has $ b = 5 $ and $ Q = \pi /5 $, which indeed satisfies $ Q \lesssim 1 $.## 4. IMPLEMENTATION

We have implemented both the optic-aligned and wavefield-aligned approaches using the multislice method [20], with an implementation described in Algorithm 1. This was done for example zone plates for soft x-ray focusing at 0.5 keV and hard x-ray focusing at 10 keV, with parameters as given in Table 1. Propagation over the short distance $ {\Delta _z} $ between slices within the zone plate was accomplished using

For the optic-aligned approach, we use Algorithm 2, where we first apply a linear phase shift given by Eq. (17) to the illumination wave. In this case, the position of the focal spot shifts by a distance $ f\sin (\theta ) $ in the output plane, which must be corrected for when extracting the sub-array containing the focal spot as shown in Algorithm 2. In practice, this method is well suited to simulating soft x-ray zone plates, as the focal length and thus spot shift distance is relatively small.

The longer focal length $ f = (2{r_N} d{r_N})/\lambda $ of hard x-ray zone plates with small $ \lambda $ limits the maximum tilt that can be accommodated, as described by Eq. (18). Therefore, with hard x-ray zone plates, one is more likely to use the wavefield-aligned approach described above, using partial voxel filling (Fig. 3) to properly represent the tilted zone plate in the refractive index array. The implementation of the wavefield-aligned approach described in Algorithm 3 consists of three steps:

- 1. To simplify the task of rotation, we first reduce the size of the optic refractive index array to a small size containing the optic itself. Recall that we use a grid size much larger than the optic size because we want the output plane step size to match the input plane step size of Eq. (25). For the example paramters shown Table 1, this reduces the grid size from $ {55296^2} $ to $ \approx {15000^2} $. The rotation now has to occur on a three-dimensional grid of size $ 15 000 \times 15 000 \times {N_z} $, where $ {N_z} $ refers to the number of slices that will be used for propagaton.
- 2. We now have a dataset with dimensions $ 15 000 \times 15 000 \times {N_z} $, where the grid size in the plane of optic is different from the grid size along the direction of the beam propagation. Since rotating an isotropic grid is easier than rotating a non-isotropic grid, the grid is “expanded” along the direction of propagation to give $ {\Delta _z} = {\Delta _x} $. Rotation is now performed in small batches along the axis of rotation on the isotropic grid. We use libvips [40] to perform the rotation as an affine transform with bilinear interpolation due to its efficiency, parallelism, and low memory needs [41].
- 3. Finally, the grid is then “collapsed” back to the number of slices $ {N_z} $ to be used for propagation by a simple scaled summation. The size of the dataset is now back to $ 15 000 \times 15 000 \times {N_z} $. The only difference from the basic optic simulation algorithm is this: at each step of the multislice loop, we now extract the refractive index pattern from the HDF5 file and expand the grid size via zero padding. This is followed by the usual multislice method of scalar diffraction and near-field propagation. The exit wave is finally propagated to the focus.

We chose to implement the above in the Python3 programming language,
utilizing the scientific Python stack based upon
`NumPy` [42]
and `SciPy`. Additionally, the package
`numexpr` [43]
was used to speed up the pointwise multiplication required by propagation
function. The FFTs were performed using `FFTW` [44] via the Python bindings provided by
`pyFFTW` [45].
We use the HDF5 library [46] via
the Python interface h5py [47] to
store all datasets with chunking enabled to enable fast I/O. Owing to the
nature of the scientific Python ecosystem where many packages offer Python
interfaces for software packages (such as FFTW, HDF5, libvips, etc.) with
underlying code in C or Fortran, care was taken to install them with the
Intel Compiler Collection using the Spack [48] package manager. This was done to ensure optimal performance
on the workstation. Our code is available at [49].

All calculations were performed on a workstation with two Intel Xeon E5-2620 v4 processors and 512 GB RAM. Because the large array size exceeded their available memory, GPUs were not used. A typical optic-aligned simulation (for $ Q = 3.33 $, $ \theta { = 2.5^ \circ } $) took approximately 30 min, and a typical wavefield-aligned simulation (for $ Q = 3.33 $, $ \theta { = 2.5^ \circ } $) took approximately 150 min.

## 5. SIMULATION RESULTS

We have carried out the above procedures to evaluate the focusing properties of two zone plates with $ d{r_N} = 21\;{\rm nm} $ outermost zone width and a diameter of $ 2{r_N} = 58.9 \;{ \unicode{x00B5}{\rm m}} $: one for focusing 0.5 keV soft x rays (such as are used for imaging in the “water window” between the carbon and oxygen $ K $ absorption edges [50]), and one for focusing 10 keV hard x rays (such as are used in scanning fluorescence x-ray microscopes). Further parameters for these zone plates are given in Table 1. In each case, we carried out simulations for values of the Klein–Cook parameter $ Q $ of 0.33, 3.33, and 10 to see the role that waveguide effects play in tilted zone plate focusing. In addition to showing images (Figs. 5 and 6) and plots (Fig. 7) of the intensity around the focal spot region, we show in Figs. 8 and 9 two other measures of zone plate performance:

- 1. Diffraction efficiency: this is the fraction of energy in the vicinity of the $ m = 1 $ first-order focus, even if the focal spot is of poor quality. For a zone plate with a small value of the Klein–Cook parameter $ Q $, the expected value can be calculated using scalar diffraction theory and knowledge of the x-ray refractive index [6].
- 2. Strehl ratio: this is the ratio of the peak intensity in the focal region at a given tilt angle relative to the intensity at zero tilt angle, for a given zone plate thickness.

Consider first the case of the 0.5 keV soft x-ray zone plate with no tilt ($ \theta { = 0^ \circ } $). Because the values of $ \delta = 0.00474 $ and $ \beta = 0.00465 $ for gold are quite similar at 0.5 keV, Eq. (1) does not lead to the maximum scalar diffraction efficiency: a thickness of $ {t_{{\rm zp}}} = 0.191 \;{\unicode{x00B5}{\rm m}} $ (with $ Q = 1.67 $) gives a scalar diffraction efficiency of 0.117, whereas $ {t_{{\rm zp,opt}}} = 0.096 \;{\unicode{x00B5}{\rm m}} $ (with $ Q = 0.86 $) gives a calculated efficiency of 0.087. Thus, the result shown for $ Q = 0.33 $ has reduced diffraction efficiency (Fig. 8), but has an Airy focus profile as expected from scalar diffraction theory (Figs. 5 and 7). At higher $ Q $ values, Fig. 8 shows that the diffraction efficiency with waveguide effects included exceeds the scalar efficiency, with $ Q = 3.33 $ providing higher efficiency than $ Q = 10 $. At the same time, the waveguide effects at high $ Q $ mean that the focus profile is degraded, as can be seen in Figs. 5 and 7.

The behavior of the 0.5 keV soft x-ray zone plates upon tilt depends on the thickness and thus the value of $ Q $. For the thin zone plate with $ Q = 0.33 $, the diffraction efficiency is unaffected by tilt (no waveguide effects are involved), but the focal profile degrades as the tilt $ \theta $ is increased (Fig. 5). For the thicker zone plates, the diffraction efficiency begins to drop off as one approaches the Bragg angle $ {\theta _{\rm B}}({r_N}) $ for the outermost zones. In addition, the net path length error $ {\ell _l} $ of Eq. (10) between axial and lower marginal rays matches the Rayleigh quarter wave limit at an angle of $ {\theta _l} = {0.548^ \circ } $, which is the tilt angle where one starts to see a reduction in the Strehl ratio.

The behavior of the 10 keV hard x-ray zone plate upon tilt is rather different. Here the diffraction efficiency shows a sharp increase, as the tilt angle $ \theta $ matches the outermost zone Bragg angle $ {\theta _{\rm B}}({r_N}) $ [Eq. (3)] for the $ Q = 3.33 $ case, whereas the $ Q = 10 $ case shows a decrease in focusing efficiency for any angle other than normal incidence, with a dropoff characterized by $ {\theta _{\rm B}}({r_N}) $. Also, Fig. 2 shows that there is very little $ {\ell _c} $ offset to the $ {\ell _a}{\theta ^2} $ term in the expression for the net path length error of Eq. (10), so that the tilt limit of $ {\theta _l}{ = 1.191^ \circ } $ from Eq. (13) nicely describes the decrease in diffraction efficiency and Strehl ratio observed as a function of zone plate tilt $ \theta $.

## 6. CONCLUSION

We have systematically studied the effect of tilt at different x-ray energies as a function of thickness and number of zones on a zone plate. The tilt limit of $ {\theta _l} $ [Eq. (13)] that one arrives at by using the axial and lower marginal rays and the Rayleigh quarter wave criterion provides a good estimate of the tilt misalignment tolerance of thin zone plates. This analysis indicates that zone plates used for focusing 0.5 keV soft x rays are more demanding of proper tilt alignment than are zone plates used for 10 keV hard x rays. As the zone plate thickness is increased to the point where waveguide effects begin to be important [that is, as the Klein–Cook parameter $ Q $ of the outermost zones approaches values of 1 or more; see Eq. (19)], then the allowable tilt is greatly reduced, and is better characterized by the Bragg angle $ {\theta _{\rm B}} $ of the outermost zones as given by Eq. (3).

The calculations shown here have been for the properties of a focal spot produced by plane wave illumination, which is representative of the case of scanning x-ray microscopes. In full-field or transmission x-ray microscopes (TXMs), high optical magnifications are employed due to the large pixel size of common x-ray image detectors [1]. As a result, a point object on axis essentially produces a plane wave from the objective zone plate to the detector. In other words, the principle of reciprocity [51–53] relates the light source in scanning microscopy to the detector in full-field microscopy. Because of this, the results shown here are directly applicable to TXM imaging of objects near the optical axis. Further studies should be carried out to consider the degree to which the tilt of 2D objective zone plates affects off-axis imaging points, which could degrade the usable field of view beyond what one would expect for a non-tilted zone plate [1,54].

## Funding

DOE Office of Science (DE-AC02-06CH11357); NIH National Institute of Mental Health (R01-MH115265).

## Acknowledgment

This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facilities operated for the DOE Office of Science by Argonne National Laboratory. We thank the National Institute of Mental Health, National Institutes of Health. The authors would like to thank Kenan Li and Michael Wojcik for helpful discussions.

## Disclosures

The authors declare no conflicts of interest.

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