1. Introduction

High-resolution hard x-ray imaging relies on optics with depths ranging from several micrometers for diffractive optics up to centimeters for reflective or refractive optics. In all cases, the depths are many orders of magnitude higher than the wavelength $\lambda$, resulting from the rather low interaction compared to other spectral ranges, notably visible light. This is the case for compound refractive lenses (CRL), reaching sub-50 nm focal sizes [1], waveguide (WG) optics, which most recently reached sub-15 nm resolution in holographic full-field imaging [2], as well as for multilayer optics, with focus sizes below 10 nm [3–5] and high focusing efficiencies [6,7]. The (graded) multilayer architecture can be separated in focusing by diffractive optics such as Fresnel zone plates (FZP), including its two modern sub-types multilayer zone plates (MZP) and multilayer Laue lenses (MLL), as well as in reflective x-ray optics, such as multilayer coated Kirkpatrick–Baez mirrors [8,9]. With improved fabrication and experimental capabilities comes the demand for accurate and performant wavefield simulations beyond the projection approximation, where propagation effects and volume diffraction within the optical element are neglected. In particular, the extended design and use of thick x-ray optics can often no longer be described by analytical solutions.

In particular, the simulation of x-ray optics is challenging for two reasons in particular. First, there is a large mismatch between the wavelength, typically around 0.1 nm, and the computational field-of-view, which has to encompass the full numerical aperture (NA) of the optical element in two (2D) or even three dimensions (3D). Second, thick optics are required because the interaction of x-rays with matter is intrinsically weak, resulting in small refraction and diffraction angles. To capture these small angles, a high accuracy in wavefield computation is necessary. Most software packages for numerical calculations of wavefields are designed for optical wavelengths and are therefore not applied for x-ray optics.

Several approaches have been developed to numerically calculate the propagation of hard x-ray wavefields interacting with x-ray optics. Examples are modified multi-slice (MS) propagation [10], methods of solving the parabolic wave equation eigenfunctions [11,12], the description by the coupled wave theory [13,14], an approach based on the Takagi-Taupin equations [15], and ray-tracing methods [16]. Although all these methods are used for the simulation of x-ray optics, the algorithms differ depending on the specific questions addressed. For example MS was used to simulate classical FZPs with zone widths down to 20 nm [10]. The Takagi-Taupin equations, the approach of parabolic wave eigenfunction, and the coupled wave theory were all used for MLLs, albeit only in 2D simulations [11–15]. Furthermore, ray-tracing was used to study focus shape and lens aberrations of MLLs [16]. Finally, finite element method were used to simulate dynamical diffraction problems in crystal and reflective optics [17–19]. Beyond these selected examples, many more applications of x-ray wavefield simulations can be found in the literature in the context of imaging. The reason is that the wavefield simulation is a prerequisite for many imaging schemes, including in particular iterative phase retrieval algorithms [20,21], which have revolutionized x-ray imaging since the invention of coherent diffractive imaging (CDI) and ptychography over twenty years ago [22–24].

In this work, we use finite-difference propagation (FD) to simulate optical elements based on multilayers. FD is well-established for the simulation of optical waveguides (see [25] and the references therein) but has to date been used only for a few x-ray optical studies [26–29]. We show that FD offers high accuracy at reasonable numerical complexity. We then investigate the volume diffraction inside (graded) multilayer structures in detail, as well as the focusing capabilities of different types of multilayer optics with large NA in 2D and 3D. Finally, we also show that FD can be combined with steps of free-space (FS) propagation in order to compute a complete imaging scheme.

The simulations of a complete imaging scheme demonstrates that the approach is not limited to single optical components but also well-suited for relevant imaging setups. We demonstrate this by simulating an experimental setup based on an MZP used as an objective lens and placed in the near-field of the sample. The experimental setup, at first sight, seems to be very similar to the classic transmission x-ray microscope (TXM) constructed by pioneers of x-ray microscopy over forty years ago in Göttingen [30]. However, diffractive optical elements (FZP, MZP, MLL), have never been exploited as an objective lens in an iterative phase retrieval approach, despite the potential to improve the resolution and contrast sensitivity. Importantly, the ability to simulate the setup promises to emancipate imaging from lens aberrations, as the real lens can be incorporated in the algorithmic processing, rather than an idealized assumption of a lens.

This manuscript is structured as follows. After this introduction, in section 2 the basic theory of diffractive focusing multilayer optics is outlined. In section 3 the numerical approach is presented. In the subsequent section 4 the convergence of the FD method for different sampling properties is studied, followed by the investigation of the diffraction inside representative multilayer structures and the impact of possible real structure effects in section 5. Section 6 presents a study of different multilayer focusing optics with an emphasis on the focus shape, intensity and efficiency. Finally in section 7 we introduce a new imaging approach based on off-axis MZP objective lenses. The manuscript closes with section 8, jointly presenting discussion, summary, and an outlook.

2. Diffractive focusing multilayer optics

We briefly introduce multilayer optics as a special case of FZPs. In fact, diffractive multilayer optics, such as MZPs or MLLs, are entire or partial FZPs that are thicker than classical FZPs along the optical axis. Further, the fabrication is based on thin film deposition [31,32], and subsequent dicing, rather than the lithographic fabrication scheme of FZPs [33]. A multilayer optical element is hence based on the same optical principle as a classical FZP: An incoming x-ray beam is diffracted by a zone, or in our case a layer structure, where the radial distance $r_m$ of the $m^{\mathrm {th}}$ layer is defined by:

The diffracted wavefield constructively interferes behind the optical element at a distance $f$, the focal length. $\lambda$ denotes the wavelength of the radiation. The layer width is defined as $\Delta _m = r_{m}-r_{m-1}$, with the cases of the innermost layer (largest width) defined as $\mathrm {max}(\Delta _m)$ and the outermost layer (smallest width) as $\mathrm {min}(\Delta _m)$. The NA of the optical element and thus the size of the focus is a function of the outermost layer width $\mathrm {min}(\Delta _m)$, respectively for off-axis optics defined by the focal length divided by the size of the illuminated area. A planar structure of layers can be used to focus to a 1D line focus, for a 2D point focus the diffractive element needs to have a radial symmetry around the optical axis. Alternatively a point focus can be generated by a pair of two planar structures arranged in a crossed geometry. For the case of multilayer optics, planar structures are referred to as MLLs and radial symmetric structures as MZPs. Given the low absorption of hard x-rays, the depth of the multilayer structure is used to modulate the wavefield (e.g shifting the phase between neighbouring layers by $\pi$), rather than creating a binary structure of opaque and transparent layers as is the case with FZPs.

The description of multilayer optics by the kinematic diffraction theory is insufficient when the layer width is reduced to few nanometers and the depth is increased to several micrometers [34,35]. For clarity, we use the term width for the transverse extent of the simulated and discussed layers and the term depth for their extent along the optical axis. In multilayer optics with higher diffraction efficiencies and smaller focus sizes than achievable with classical FZPs [11] one commonly encounters volume effects due to multiple scattering, such as waveguiding or the formation of standing waves resulting in the Pendellösung period [36].

For higher focusing efficiency, the layers can be tilted into the direction of the focus. One has to differentiate between the tilted geometry, where all layers have the same tilt angle, and the wedged geometry, where all layers are tilted differently based on the Bragg angle $\theta _{\mathrm {Bragg}} = \lambda / 4 \Delta _\mathrm {m}$ [15]. The latter is more challenging in fabrication but results in higher diffraction efficiencies. For many experiments, a central beam stop in front of the MZP together with an order sorting aperture behind the MZP is required to block the 0^th order beam. Alternatively, with an off-axis illumination the focus generated by the +1^st diffraction order can be separated from other diffraction orders. This is the case for most MLLs, with layers only on one side of the optical axis. For MZPs, off-axis illumination can be realized by illuminating only one side of the MZP [5]. This configuration results in a point focus without the necessity of a pair of crossed MLLs, as later shown in Fig. 5(II).

The simulations presented in this manuscript are based on parameters of existing multilayer optics. As a representative example, we use the material combination and geometry corresponding to recent MZPs fabricated by pulsed laser deposition [31], with following parameters: The MZP has an outermost layer width $\mathrm {min}(\Delta _m)$ of 5 nm, a total number of layers $M_{\mathrm {layer}}$ of $784$ layers, and a depth depending on the experimental setup between 2 µm for a photon energy of 8 keV and up to 30 µm for photon energies of 100 keV [3,37]. For a photon energy of 13.8 keV, the focal length $f$ is about 1 mm. The material combination of the layers is $\mathrm {ZrO_2}$ and $\mathrm {Ta_2O_3}$. This combination of materials has shown advantageous growth properties in the manufacturing process (in particular concerning surface roughness), and a sufficiently large difference of the refractive indizes. As an example for a crossed pair of MLLs we take the values of [32], with an outer most layer width of $\mathrm {min}(\Delta _m)= {4.31}\,\textrm{nm}$, $M_{\mathrm {layers}}=2759$ layers and a focal length of 2.13 mm for a photon energy of 17.4 keV. It should be noted, that with pairs of one-dimensional multilayer optics wavefront aberration cannot be completely avoided [16]. This becomes of importance for focus sizes below 1 nm.

3. Simulation framework

In this section, we briefly recapitulate the fundamentals of the different simulation approaches. The methods under discussion are all scalar and frequency-domain approaches, so that the simulated quantity is one spectral component of the time-independent wavefield $\psi (x,y,z)$. In the following, $z$ denotes the optical axis and $x,y$ correspondingly span the transverse (lateral) plane. The simulation approaches solve the following propagation problem. Given some initial wavefield $\psi (x,y,z_0)$ and some distribution of the complex refractive index $n_\lambda (x,y,z)$, find the wavefield $\psi (x,y,z_1)$ for $z_1 > z_0$ or, more generally, for all $z \in [z_0, z_1]$. All simulation approaches discussed in this section are implemented and made available under an open license at https://gitlab.gwdg.de/irp/fresnel. Other numerical approaches of interest are referenced in section 1.

The starting point is the Helmholtz equation (or stationary wave equation) [38]

3.1 Free-space propagation

In free-space, or more generally in regions with constant refractive index, the solution $\psi _\lambda (x,y,z_1)$ can be calculated in a single step for any $z_1$. A plethora of methods exist that implement some discretized version of the Rayleigh-Sommerfeld or the Fresnel diffraction integrals, either as real-space convolutions or in Fourier space [39,40]. Different approaches can be chosen, depending on the propagation distance $z_1 - z_0$, the sampling, and the lateral size of the simulation domain. For a detailed review, we refer to [40,41]. In this work, we use the Fresnel transfer-function (Fresnel-TF) and the Fresnel impulse-response (Fresnel-IR) approaches for shorter and longer propagation distances, respectively.

For very large propagation distances, the Rayleigh-Sommerfeld integral simplifies to the Fraunhofer integral and can be computed by a single Fourier transform. The far-field detector images in the following applications are therefore calculated by a single fast fourier transform (FFT).

If both the initial field $\psi _\lambda (x,y,z_0)$ and the refractive-index distribution $n(x,y,z)$ are circularly symmetric with respect to the optical axis, then the propagated field will also have circular symmetry (CS). Since the diffraction integral can be conveniently expressed in cylindrical coordinates, we can omit the angular component in this case so that the 3-dimensional problem is reduced to a 2-dimensional one. For circular-symmetric optics, we use a CS Fresnel propagator, which is based on the discrete Hankel transform [42,43]. Note that this method can simulate normal illumination only, because otherwise the initial field would not be of circular symmetry.

3.2 Multi-slice propagation

The multi-slice (MS) method was originally developed for electron diffraction (see [44] and the references therein) but is well-established in x-ray optics nowadays [38]. To account for multiple diffraction, the propagation through an extended object is partitioned into small steps. Each step consists of (i) multiplication of the transmission function of the respective slice to account for its refractive index using the projection approximation and (ii) free-space propagation.

For (ii), the angular spectrum method (see [40] for a recent review; also known as exact transfer function approach [39]) is commonly used [10,38]. As a consequence, the MS method inherits some features of the angular spectrum method. First, the FFT-based diffraction implicitly imposes periodic boundary conditions. Second, it is very sensitive to the sampling intervals for both the longitudinal coordinate $z$ and lateral coordinates $x,y$, as well as the lateral extent of the computational domain as outlined in [40]. Finally, sub-step (ii) requires 2 FFTs per step, so that the computational complexity of a single step scales with $\mathcal {O}(N_\mathrm {px} \cdot \log N_\mathrm {px})$, with $N_\mathrm {px}$ being the total number of samples in each plane. The computational complexity of sub-step (i), on the other hand, scales with $\mathcal {O}(N_\mathrm {px})$ and is hence irrelevant.

3.3 Finite-difference propagation

Finite-difference propagation (FD) methods, an entirely different approach, solve the paraxial Helmholtz equation in real space by discretizing the differential operators and solving a sparse linear system of equations. These methods have been well-established for the simulation of optical waveguides for decades (see for example [25] and the references therein) and have proven reliable for x-ray optics [28,29,45,46].

In contrast to MS methods, FD methods work entirely in real space and, as such, do not inherit any boundary conditions. Instead, one has to impose explicit boundary conditions to obtain a unique solution. In fact, the solutions obtained by FD methods depend sensitively on the choice of boundary conditions. In the visible regime, often so-called transparent boundary conditions or absorbing boundary conditions in the form of perfectly matched layers are introduced, which minimize reflections at the boundaries of the computational domain. For hard x-rays, on the other hand, it usually suffices to specify approximate boundary values (Dirichlet boundary conditions) and extend the computational field beyond the region-of-interest, so that waves that are reflected at the boundaries are quickly dissipated due to the significantly higher absorption losses.

Here, we essentially follow earlier work [28,29] but use a modern implementation in C++ and Python. We have implemented FD propagators for 2D, 3D, and CS geometries. The latter is based on earlier work by Melchior [46] but with improved treatment of the inner boundary. In each step, a system of linear equations (compare appendix A) is solved. Finding the solution requires only a small fixed number of passes, so that the computational complexity scales with $\mathcal {O}(N_\mathrm {px})$. The 3D geometry uses dimensional splitting to uncouple the two transversal derivatives, so that the computational complexity is only doubled.

To summarize, in comparison to MS, FD propagation does not require special sampling conditions, does not impose periodic boundary conditions, and is considerably less computationally complex.

3.4 Combined approaches

As illustrated in Fig. 1, many imaging setups consist of multiple elements (e.g. optics, samples, detectors) separated by free-space. To cope with these different geometries with different sampling constraints, the presented methods can be combined. This is especially relevant for the simulation of wavefields propagating through an optical element and subsequent in free-space to a distant focal spot or sample, as discussed in section 6 for the focus characterization of MZPs and MLLs, and in section 7 for the simulation of an entire imaging scheme.

 

Fig. 1. Experimental geometries and simulation schemes. (a) Objective x-ray microscope. A sample is illuminated by a parallel beam. The modulated wavefield is magnified by an MZP and propagates in free-space to the detector. (b) A pair of crossed MLLs focus an incoming beam onto a sample, the wavefield gets diffracted by the sample and propagates to the detector. (c) A WG generates a strongly divergent cone beam illumination. The sample is positioned close to the WG-exit in the divergent beam. The beam is modulated by the sample and propagates in free-space to the detector. (d) A stack of compound refractive lenses (CRL) focus an incoming beam. A sample is positioned in the vicinity of the focus, modulates the wavefield and subsequently propagates to the detector. In all cases the interaction of matter with the x-ray wavefield can be simulated using the FD method. For the free-space propagation (FS) the Fresnel-TF or the Fresnel-IR approach is used respectively. Furthermore, the propagation to the far-field, in the cases here the propagation to the detector, is computed using the FFT. As illustrated the FD propagations are performed in small sub-steps to cover multi-scattering. The FS propagations are performed in a single step.

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In this work we use FD to simulate the propagation through the optical elements. For propagation through free-space (FS), we use the strategy described in section 3.1, namely Fresnel-TF and Fresnel-IR for shorter and longer propagation distances, respectively. The far-field intensity distribution, where needed, is computed using FFT [39].

3.5 Computational consumption

The simulations were computed on a dual-CPU server with two Intel Xeon E5-2650v3 (Haswell microarchitecture from 2014, 10 cores each). We note that only FFT, FS, and 3D-propagation are parallelized, whereas the CS and 2D propagation were run on a single core. Memory consumption was insignificant, since only a few slices need to be kept in main memory. Propagating a wave-field in 2D was executed within minutes, and in 3D within hours.

4. Convergence properties

Before we simulate ‘real’ optical elements, we study the convergence properties of the FD propagation as a function of spatial sampling by simulating a thick multilayer structure under plane-wave illumination at the Bragg angle of the structure. The material composition of the structure is similar to recent experiments using MZP optics [31]. The multilayer structure is shown in Fig. 2(a) together with the intensity distribution of the propagating wavefield. The simulated multilayer structure approximates an infinite number of bilayers. The bilayers are composed of alternating high density $\mathrm {Ta_2O_3}$ and low density $\mathrm {ZrO_2}$ layers of equal depth and a layer width of $\mathrm {min}(\Delta _m)= {4.5}\,\textrm{nm}$. The depth in $z$ is $\Lambda = {15.1}\,\mathrm{\mu}\textrm{m}$. It is illuminated by a plane wave under an angle of $\theta _m = {5.0}\,\textrm{mrad}$ relative to the layer orientation, corresponding to the Bragg angle ($\theta _{\mathrm {Bragg}}$) of the multilayer at a photon energy is 13.8 keV.

 

Fig. 2. Convergence of the FD as a function of the simulation parameters. (a) The setup of the simulation shows a multilayer structure with a layer width $\mathrm {min}(\Delta _m) = {4.5}\,\textrm{nm}$, illuminated under an angle of $\theta _m = {5}\,\textrm{mrad} $ and a depth of 15.1 µm. The material density and the wavefield intensity modulation inside the structure is indicated. The varied parameters are depicted: the propagation step size $\Delta _z$, the lateral grid size $\Delta _x$, the size of the computational field $N_x \cdot \Delta _x$. (b) Simulation results of the multilayer exit intensity profile of the region-of-interest with a width of 409.6 nm used for the further analysis. The simulation parameters are: $\Delta _z= {0.5}\,\textrm{nm}$ and $\Delta _x= {0.05}\,\textrm{nm}$. (c) Center region of the multilayer exit intensity profile shown in (b). (d) RMS-error as a function of the propagation step size $\Delta _z$ and the lateral grid size $\Delta _x$. (e) RMS-error as a function of the computational field $\Delta _x \cdot N_x$.

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The simulation is set up in 2D as follows. The coordinate system is defined by the optical axis $z$, which is parallel to the layers, and the transverse axis $x$, which is normal to the layers. The incoming plane wave is described at $z=0$ with a unit amplitude and a linearly varying phase. The computational field-of-view is 8 µm. The boundaries are padded with 2 µm of $\mathrm {Pb}$. Furthermore the Dirichlet boundary is set to a zero amplitude. To mitigate the influence of the boundaries, only a central region-of-interest with an extent of 409.6 nm (or $2^{10}$ grid points with a 0.4 nm distance) is used for further analysis. We have simulated the system for different pixel sizes in longitudinal ($\Delta _z$) and transverse ($\Delta _x$) direction and simulated it for different computational field of views ($\Delta _x \cdot N_x$).

Figures 2(b)-2(e) shows the wavefield simulations. In Fig. 2(b) the region-of-interest of the compared intensity profile is shown and in Fig. 2(c) a magnified section of the profile. The simulation grid was sampled with a propagation step size of $\Delta _z= {0.5}\,\textrm{nm}$ and a lateral grid spacing of $\Delta _x= {0.05}\,\textrm{nm}$, corresponding to the finest sampling interval in the comparison shown below in Figs. 2(d) and 2(e).

Next, we investigate how the resulting intensity profiles depend on the pixel size and how the accuracy of the FD converges. To this end, we take the highest sampled simulations as reference and plot the RMS-error of the residual as a function of $\Delta _z$ and $\Delta _x$ (Fig. 5(d)). The residual is evaluated only in the central part of the computational field-of-view, in a region-of-interest of 409.6 nm, to avoid boundary effects. The effect of the computational field is evaluated in Fig. 5(e), by plotting the RMS-error as a function of the lateral computational field-of-view. Four different parameter sets were simulated from $\Delta _{z}= {0.5}\,\textrm{nm}$, $\Delta _{x}= {0.05}\,\textrm{nm}$ to $\Delta _{z}= {4.0}\,\textrm{nm}$, $\Delta _{x}= {0.4}\,\textrm{nm}$. The lead $\mathrm {Pb}$ padding was set to 25 % of the computational field-of-view on each side.

In view of the results of the FD simulation accuracy, we adopted the following simulation strategy for the work presented below: For simulations in 2D, the grid spacings are kept constant, namely $\Delta _{x}^{\mathrm {2D}}= {0.1}\,\textrm{nm}$ in the lateral direction, and $\Delta _{z}^{\mathrm {2D}}= {1}\,\textrm{nm}$ along the propagation axis. Due to the larger computational effort in 3D, the lateral grid spacing will be increased to $\Delta _{x}^{\mathrm {3D}}= {0.35}\,\textrm{nm}$ and the propagation step size to $\Delta _{z}^{\mathrm {3D}}= {50}\,\textrm{nm}$. The parameters are summarized in Table 1.

Table 1. Parameters of the presented simulations depending on the dimension (D).

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5. Propagation through regular multilayers

5.1 Volume and dynamical diffraction effects

In this section, we present simulations of wave propagation in regular multilayers with plane waves coupling through the front. We compare a system with large periodicity ($\Delta _m= {25}\,\textrm{nm}$) with a system with small periodicity ($\Delta _m = {3}\,\textrm{nm}$). The latter corresponds approximately to the smallest layer widths currently fabricated for multilayer optics. The two periodicities demonstrate the transition from a regime that can be described by guiding modes to one that can be described by dynamical diffraction theory. The numerical calculations were performed in 2D with the simulation parameters shown in Table 1 and the parameters of the structure shown in Table 2.

Table 2. Parameters of the simulated multilayer structures in Fig. 3 and 4. The simulations were performed in 2D with a FOV of 8 µm and $80\cdot 10^3$ lateral grid points. In all cases the photon energy is $E= {13.8}\,\textrm{keV}$ and the layer materials are $\mathrm {ZrO_2}$ and $\mathrm {Ta_2O_3}$. $\Delta _m$ defines the layer width, $\theta _m$ the illumination incidence angle and $\Lambda _t$ the depth of the structure.

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Figure 3 shows wavefields propagating through multilayer structures illuminated by plane waves, comparing Figs. 3(a), 3(b) normal-incidence illumination ($\theta _m = 0$), and Figs. 3(c), 3(d) illumination under the Bragg angle ($\theta _m = \theta _{\mathrm {Bragg}}$), both for Figs. 3(a) ,3(c) large periodicity and Figs. 3(b), 3(d) small periodicity, respectively. The multilayer is in all cases 45.3 µm thick. The simulation includes 10 µm vacuum in front and behind the structure. The structure begins at the position $z=0$. The photon energy and material combination is the same as above in the section 4. The different rows in Fig. 3 show (I) intensity, (II) phase, (III) the angular spectrum, and (IV) the zero and first order diffraction intensity, as a function of the propagation distance $z$. The angular spectrum in Fig. 3(III) is calculated by performing a Fourier transformation for each wavefield with respect to the $x$ coordinate. This is equivalent to a far-field pattern of the multilayer exit wavefield, if it were cut at position $z$.

 

Fig. 3. Volume and dynamical diffraction effects of different multilayer structures and different illumination direction. The multilayer structures have layer widths of (a,c) $\Delta _m= {25}\,\textrm{nm}$ and (b,d) $\Delta _m= {3}\,\textrm{nm}$ and are illuminated by a plane under normal incidence (a,b) and under the corresponding Bragg angle (c,d). (I) Intensity modulation of the propagated wavefield, with the corresponding Poynting vectors (white). The layer edges are indicated by black lines. (II) Phase of the propagated wavefield. (III) Angular spectrum or far-field diffraction pattern as a function of the depth of the multilayer structure. (IV) Intensity modulation of the $0^{\mathrm {th}}$ and $1^{\mathrm {st}}$ diffraction peak as a function of the depth. The corresponding diffraction peak in (III) is indicated by arrows. In general the structure consists of a material combination of $\mathrm {ZrO_2}$ and $\mathrm {Ta_2O_3}$, and has a total depth of 45.3 µm padded by 10 µm vacuum before and after the structure. In (I)-(III) the material was considered as non absorbing ($\beta =0$). In (IV) $\beta =0$ and $\beta \geq 0$ is shown. The simulations were performed in 2D using FD. In case of (d) the half Pendellösung period is $\Lambda _{\mathrm {Pendel/2}}= {15.1}\,\mathrm{\mu}\textrm{m}$. The simulation parameters are shown in Table 2. Scalebars vertical: (I,II) 10 nm, (III) scattering vector $q= {0.25}\,\textrm{nm}^{-1}$, horizontal: 5 µm.

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We first discuss the results for Fig. 3(a) $\Delta _m= {25}\,\textrm{nm}$ and $\theta _m = 0$. The wavefield intensity, shown in Fig. 3(a,I), is attenuated in the layers of higher electron density ($\mathrm {Ta_2O_3}$) and intensity maxima (guiding modes) form in the layers of lower electron density ($\mathrm {ZrO_2}$). The formation of guiding modes in the material of lower electron density is well known from x-ray waveguides. The number of modes $N_{\mathrm {modes}}$ can be calculated by $N_{\mathrm {modes}}=\lceil \sqrt {2\delta _2 - 2\delta _1} k \Delta _m / \pi \rceil$, where $\lceil \boldsymbol {\cdot } \rceil$ denotes rounding up to the next integer. For normal-incidence illumination, only symmetric modes (even order) are populated. For the parameters of Fig. 3(a), this results in only the 0^th mode to be populated, in line with the simulation result. The phase modulation of the wavefield is shown in Fig. 3(a,II). The phase of the propagating wavefield is shifted differently in adjacent layers, due to the difference in the real part of the index of refraction $\delta$. The angular spectrum, plotted in Fig. 3(a,III), shows multiple diffraction peaks of even and odd diffraction orders. While the odd orders are formed at the beginning of the multilayer structure, the even orders build up after a propagation distance of $z> {250}\,\textrm{nm}$. This is in contrast to the analytical solution of an infinitesimally thin structure (projection approximation) [38], where only odd-order diffraction peaks are described. Even diffraction orders occur due to volume diffraction effects. This has been observed also experimentally for multilayer systems [34]. A first maximum of the first order diffraction peak is formed at $z= {10.1}\,\mathrm{\mu}\textrm{m}$, which is only slightly larger than an odd multiple of a $\pi$ phase shift between two adjacent layers with 9.6 µm. For no absorption ($\beta = 0$), the diffraction efficiency of the first maximum of the first order diffraction is 39.6 %, which is in line with the analytical solution of a multilayer system of 41.5 % [38]. Including the absorption of $\mathrm {ZrO_2}$ and $\mathrm {Ta_2O_3}$ ($\beta \ge 0$), the efficiency is reduced to 19.9 %.

Next, we discuss the results for Fig. 3(b) $\Delta _m= {3}\,\textrm{nm}$ and $\theta _m = 0$. The intensity modulation is much smaller Fig. 3(b,I), and a phase modulation is not noticeable Fig. 3(b,II). The angular spectrum, shown in Figs. 3(b,III), 3(b,IV), shows only the 0^th diffraction order. If absorption is taken into account ($\beta > 0$), the intensity decays approximately with the mean absorption coefficients of $\mathrm {ZrO_2}$ and $\mathrm {Ta_2O_3}$. In contrast to the regime characterized by the formation of well-separated modes, as in Fig. 3(a), here in Fig. 3(b) the guided modes from adjacent 'guiding layers' have significant overlap and the approximation of separate waveguides is no longer valid. As a result, multilayer structures of such small layer widths can not be used as focusing optics since no phase modulation occurs.

We now turn to oblique incidence under the Bragg angle, shown in Fig. 3(c) for $\Delta _m= {25}\,\textrm{nm}$ and $\theta _m = {0.9}\,\textrm{mrad}$. In Fig. 3(c,I) the formation of two modes can be observed in the intensity field. This is different from the parallel illumination case and results from the non-symmetric overlap integral in coupling the incident beam to the modal profiles [47]. This less symmetric pattern is also observed in the phase modulation Fig. 3(c,II) and the angular spectrum Fig. 3(c,III). In Fig. 3(c,IV) the formation of a strong maximum of the +1^st order diffraction peak can be seen. However, the propagation distance after which this peak forms sensitively depends on the ’zone width’ $\Delta _m$, which makes it impractical to design focusing multilayer optics based on this reflection peak, since this would require different depth for each layer.

Finally, Fig. 3(d) shows a multilayer structure with $\Delta _m= {3}\,\textrm{nm}$ under Bragg incidence $\theta _m = {7.4}\,\textrm{mrad}$ ($\theta _{\mathrm {Bragg}}$). Panel Fig. 3(d,I) exhibits the formation of intensity maxima that can be described by standing waves, similar to the guiding modes shown at Fig. 3(a,I). The anti-nodes of the standing waves are positioned at the material boundaries and, in contrast to guiding modes, not in the center of a layer. This formation is well known from dynamical diffraction theory and is denoted as Pendellösung [36] for thick crystals illuminated in Laue geometry. The Pendellösung describes the oscillation of a wavefield by multiple diffraction inside a crystal. It has also been observed experimentally in a system similar to the simulated one [35]. Here, it can be most clearly observed by following the Poynting vectors in Fig. 3(d,I) and by the phase modulation inside the multilayer in Fig. 3(d,II). In the angular spectrum, this phase modulation results in the formation of two diffraction peaks oscillating exactly out of phase Figs. 3(d,III), 3(d,IV). The Pendellösung period depends on the interfacial reflectivity, but not on the layer depth (for the layers of $\Delta _m= {25}\,\textrm{nm}$ the Pendellösung effect is not prominent enough to be observed). The exit wavefront after half a Pendellösung period ($\Lambda _{\mathrm {Pendel/2}}$) is opposite to the angle of incidence. For multilayers, one finds $\Lambda _{\mathrm {Pendel/2}} = \pi \lambda /2 \left |\delta _2 - \delta _1 \right |$ [35]. Which is $\Lambda _{\mathrm {Pendel/2}}= {15.1}\,\mathrm{\mu}\textrm{m}$, for the given parameters and is in perfect agreement with the simulations. The independence of $\Lambda _{\mathrm {Pendel/2}}$ on the layer width $\Delta _m$ and the high diffraction efficiency of nearly up to 100 % (here: 99.6 % in the absence of absorption) makes it the optimal depth for multilayer focusing optics. When absorption is taken into account Fig. 3(d,IV), the maximum diffraction efficiency of the Bragg peak decreases, for the material combination here down to 32.1 % (see dashed line). Unlike for the guiding modes of $\Delta _m= {25}\,\textrm{nm}$, the effective absorption is for the Pendellösung with $\Delta _m= {3}\,\textrm{nm}$ no longer dominated by the layers with low electron density, but by a weighted mean of both layers resulting in an higher absorption.

In conclusion, we have shown that multilayer structures with large periodicity (here $\Delta _m= {25}\,\textrm{nm}$) could be used for focusing of incident wavefields (shown in Fig. 3(a)). The diffraction efficiency can be further increased when the multilayer structure is illuminated under the corresponding Bragg angle (shown in Fig. 3(c)). Contrarily, multilayer structures with small periodicity (here $\Delta _m= {3}\,\textrm{nm}$) illuminated in normal-incidence ($\theta _m = 0$) can not be used as focusing multilayer optics, since the phase modulation of the wavefield in the multilayer structure is negligible (shown in Fig. 3(b)). But, if the multilayer structure is illuminated under the corresponding Bragg angle the formation of standing waves can be observed, this effect is known as Pendellösung. When this effect is exploited, diffraction efficiencies of (nearly) up to 100 % are achievable (shown in Fig. 3(d)).

5.2 Real-structure effects

In this section, we consider the sensitivity to imperfections of multilayers with small periodicity, so called real-structure effects. Figure 4 illustrates the effect of different non-idealities that occur in non-perfect multilayer structures. We have simulated the following imperfections: in Fig. 4(a) the perfect binary material is replaced by a sinusoidal density profile; in Fig. 4(b) the interface between the low and high density layer has a small inter-layer (constant gradient) representing for example inter-diffusion or a smooth transition regime in the growth process; in Fig. 4(c) a layer sequence with random and uncorrelated width errors of 0.4 nm standard deviation has been simulated, and in Fig. 4(d) randomized surface height fluctuations representing interfacial roughness, here with RMS-roughness of $\sigma = {0.45}\,\textrm{nm}$. In Fig. 4, row (I) shows a zoom into the different multilayer structures where the gray value encodes the decrement of the local refractive index $\delta$. For all cases, the periodicity is $\Delta _m= {3}\,\textrm{nm}$ and the illumination angle is $\theta _m = {7.4}\,\textrm{mrad}$ ($\theta _{\mathrm {Bragg}}$). All other simulation parameters are identical to Fig. 3. A list of the basic simulation parameters can be found in Table 2.

 

Fig. 4. Real structure effects. Multilayer structures with different layer transitions with a layer width of $\Delta _m= {3}\,\textrm{nm}$ and an illumination under the Bragg angle ($\theta _m= {7.4}\,\textrm{mrad}$). (a) Layer transition with a sinusoidal profile, (b) a linear gradient layer transition over a length of 1.5 nm, (c) a multilayer with a mean error of 0.35 nm in the width of the layers and (d) a multilayer with a surface roughness $\textrm {RMS}= {0.45}\,\textrm{nm}$ [48]. (I) Zoom in of the decrement of the multilayer structure. Color scaling indicates the phase shifting part $\delta$ of the refractive index. (II) The angular spectrum as a function of the depth of the multilayer structure. (III) The intensity modulation of the $0^{\mathrm {th}}$ and $1^{\mathrm {st}}$ diffraction peak as a function of the depth. The corresponding diffraction peak in (II) is indicated by arrows. The half Penedellösung period is for (a) 19.2 µm, (b) 16.7 µm, (c) 16.5 µm and (d) 16.3 µm which was 15.1 µm for the perfect multilayer structure in Fig. 3(d) 15.1 µm. The difference of the Pendellösungs periods are indicated by the grey vertical dotted lines. The parameters of the multilayer structure are equivalent to the simulations shown in Fig. 3. The parameters are listed in Table 2. Scalebars vertical: (I) 10 nm, (II) scattering vector $q= {0.25}\,\textrm{nm}^{-1}$, horizontal: 5 µm

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Strictly speaking, the sinusoidal material profile shown in Fig. 4(a) is not an unwanted real-structure effect, as a perfect sinusoidal profile would offer optimum focusing properties without higher orders and focus side-maxima [38]. The resulting far-field diffraction pattern Figs. 4(a,II), 4(a,III) is similar to the perfect binary structure shown in Fig. 3(d), with the difference that $\Lambda _{\mathrm {Pendel/2}}$ is increased by 4.1 µm. The presence of absorption ($\beta \geq 0$) reduces the maximum diffraction efficiency to 24.1 % compared to 32.1 % for the perfect binary structure. The same applies for the inter-layer with linear profile Fig. 4(b). Such interfacial gradients can occur in deposition processes and result in lower effective reflectivities, increasing $\Lambda _{\mathrm {Pendel/2}}$ by 1.6 µm and reducing the diffraction efficiency to 28.6 %.

For Fig. 4(c) the layer positioning error and Fig. 4(d) the interfacial roughness, $\Lambda _{\mathrm {Pendel/2}}$ is also increased, namely by 1.4 µm and 1.2 µm, respectively. The layer positioning error and the interfacial roughness result in more diffuse far-field pattern in between the Bragg peaks Figs. 4(c,II), 4(d,II). Both, position errors and the roughness, only have small effects on the diffraction efficiency, namely 97.3 % and 99.2 %, respectively without absorption ($\beta = 0$), and 27.3 % and 28.8 %, respectively including absorption ($\beta > 0$). Interestingly, in all cases the effects are relatively small and the diffraction efficiency seems robust at the level of realistic ’real-structure’ parameters.

6. Multilayer focusing optics

We present simulations of focusing multilayer optics to show the possibilities and constraints to generate small focal spot sizes. We simulate fully illuminated MZPs, off-axis-illuminated MZPs, single MLLs, and a pair of crossed MLLs. We are particularly interested in the shape and size of the respective focus.

All simulated optics have a wedged geometry (the layers are matched to the local Bragg angles), a depth corresponding to $\Lambda _{\mathrm {Pendel/2}}$, and were illuminated by normally incident plane waves. The outermost layer widths $\mathrm {min}(\Delta _m)$ for the optics was set such that all have similar NAs. The parameters are tabulated in Table 3. The fully illuminated MZP, the off-axis illuminated MZP and the crossed MLLs were simulated in 3D. Additionally, we performed simulations with even larger NAs of a single MLL and an MZP in 2D and CS, respectively.

Table 3. Parameters of the simulated multilayer optics in Figs. 5 and 6(a,b). The simulations were performed in 3D with a FOV of $9.45 \times {9.45}\,\mathrm{\mu}\textrm{m}^2$ and $27 \times 10^3$ grid points in each lateral direction. The photon energy is $E= {13.8}\,\textrm{keV}$, the focal length is $f= {1}\,\textrm{mm}$ and the multilayer depth is $\Lambda _t= {15.1}\,\mathrm{\mu}\textrm{m}$. NA defines the numerical aperture, $\mathrm {max}(\Delta _m)$ the largest and $\mathrm {min}(\Delta _m)$ the smallest layer width. The total number of layers is given by $M_{\mathrm {layers}}$.

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Figure 5 shows the intensity of the diffracted wavefields propagating in 3D of (I) an MZP, (II) an off-axis MZP, and (III) a pair of crossed MLLs. Figure 5(a) Shows the intensity distribution in the plane directly behind the structure, Fig. 5(b) the integrated longitudinal focus profile, and Fig. 5(c) the transverse focus profile. The intensity distribution behind the different lenses Fig. 5(a) shows characteristic differences: Firstly, we have full radial symmetry with respect to the optical axis for Fig. 5(a,I) the MZP. On the other hand, the center is shifted to the top and the top right of the FOV, for Fig. 5(a,II) off-axis MZP and Fig. 5(a,III) crossed MLL, respectively. Regarding Fig. 5(II), the multilayer structure itself was clipped, i.e. corresponding to only a section of a complete MZP. The same results (apart from apodization effects) could also be obtained by clipping the illumination. Regarding Fig. 5(III), two subsequent MLLs are needed to achieve a point focus. The lateral focus position differs in all three cases. The exit wavefield intensity for the pair of MLLs is lower than in Figs. 5(a,I), 5(a,II) due to absorption in two subsequent multilayer structures instead of only one. The insets in Fig. 5(a) display a small magnified region of the respective exit wavefield. Note that the curvature of the layers in Figs. 5(a,I), 5(a,II) is not visible on these small scales. The exit wavefield of the pair of the crossed MLLs Fig. 5(a,III) shows a grid-line intensity pattern, as both MLLs focus in one direction only. In Fig. 5(b), we show the longitudinal intensity profiles integrated over $x$ and $y$, respectively. The on-axis and off-axis focusing shows different propagation directions. In Fig. 5(c), the corresponding lateral focal spot is shown in a linear and a logarithmic color scaling. In contrast to the circular-symmetric focal distribution of the MZP Fig. 5(c,I), and the axis-symmetric focal distribution of the MLL Fig. 5(c,III), for the off-axis MZP Fig. 5(c,II) no 2D symmetric focal distribution is observed. More interestingly, also the pattern of side maxima in the focal plane differs, according to the different pupil geometry.

 

Fig. 5. Simulations in 3D of the multilayer optics. The diffracted wavefields of (I) an MZP with $\mathrm {max}(\Delta _m) = {50}\,\textrm{nm}$, $\mathrm {min}(\Delta _m) = {10}\,\textrm{nm}$ corresponding to an $\mathrm {NA}= {4.5} \times 10^{-3}$, (II) an off-axis MZP with $\mathrm {max}(\Delta _m) = {50}\,\textrm{nm}$, $\mathrm {min}(\Delta _m) = {5}\,\textrm{nm}$ corresponding to an $\mathrm {NA}={4.1} \times 10^{-3}$ and (III) two crossed MLLs with $\mathrm {max}(\Delta _m) = {50}\,\textrm{nm}$, $\mathrm {min}(\Delta _m) = {5}\,\textrm{nm}$ corresponding to an $\mathrm {NA}= {4.1} \times 10^{-3} $. The focal length is in all cases $f= {1}\,\textrm{mm}$. (a) Intensity of the exit wavefield directly behind the multilayer optics, with an inset showing a zoom in, the position is indicated by the white square. (b) Focus profiles in propagation direction integrated over the lateral axes. (c) Lateral focus shape in logarithmic (lower left) and linear (upper right) color scaling. A list of the basic simulation parameters can be found in Table 3. Scalebars: (a) 1 µm, inset 10 nm, (b) vertical 0.1 µm and horizontal 10 µm, (c) 10 nm

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In Figs. 6(a), 6(b), the corresponding line profiles of the focal planes are plotted. In Figs. 6(a), 6(b) all optics simulated in 3D – MZP, off-axis MZP and crossed MLL – show a point focus of comparable size (full width at half max (FWHM) of $ {10} \pm {0.1}\,\textrm{nm}$). This is a result of the similar NAs of all three multilayer optics. Still, they differ in peak intensity and focusing efficiency, which is 21.4 % for the MZP, 23.7 % for the off-axis MZP, and 6.5 % for the MLLs (if absorption is neglected, $\beta =0$, the focusing efficiencies are 52.9 % for the MZP, 75.7 % for the off-axis MZP, and 61.9 % for the MLLs). The efficiency is calculated using the intensity of the focus in the region between the first minima divided by the intensity in the structured area of the multilayer optics. The difference of the focusing efficiencies between the MZP and the off-axis MZP results from the smaller outermost layer width $\mathrm {min}(\Delta _m)$ of the off-axis MZP. Small layers have the highest focusing efficiency for the given depth of $\Lambda _{\mathrm {Pendel/2}}$ due to the Pendellösung effect discussed above. Regarding the MLLs, the area of a single layer is decreasing with decreasing layer width, contrarily the area of a single layer of the MZP and off-axis MZP is constant with decreasing layer width. Since the highest focusing efficiency is given for the smaller layer widths, the difference in the geometry has an effect on the overall focusing efficiency of the MLL. Nevertheless, the main difference between the efficiency of the MZPs and the crossed MLLs results from the absorption by two subsequent optics.

 

Fig. 6. (a) Focus profiles of the 3D simulations shown in Fig. 5 and (b) the cumulative sum of the focus profiles. (c) The normalized 2D focus profiles of the MLLs and an MZP simulated in a 2D Cartesian and circular-symmetric (CS) coordinate system, respectively. The results show simulations for one-sided (conventional) MLLs and symmetric (two-sided) MLLs. The latter is often used in the theoretical description of MLLs but has not been experimentally realized. (d) The normalized cumulative sum of the focus profiles. The simulation parameters are listed in Table 3, and 4 respectively.

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Table 4. Parameters of the simulated multilayer optics in 2D shown in Figs. 6(c), 6(d). The simulations were performed with a FOV of $ {60}\,\mathrm{\mu}\textrm{m}$ and $0.6 \times 10^6$ lateral grid points, an exception is the MZP with a FOV of $ {30}\,\mathrm{\mu}\textrm{m}$ and $0.3 \times 10^6$ lateral grid points. The photon energy is $E= {13.8}\,\textrm{keV}$, the focal length is $f= {1}\,\textrm{mm}$ and the multilayer depth is $\Lambda _t= {15.1}\,\mathrm{\mu}\textrm{m}$. NA defines the numerical aperture, $\mathrm {max}(\Delta _m)$ the largest and $\mathrm {min}(\Delta _m)$ the smallest layer width. The total number of layers is given by $M_{\mathrm {layers}}$.

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Finally, Figs. 6(c), 6(d) show the focus profiles of multilayer optics with even larger NAs of up to 8.6×10^-3 for sub-5 nm focusing. To simulate the large numerical apertures with sufficient accuracy at reasonable numerical expense, we have performed the simulations in 2D. In total, 4 multilayer optics have been simulated: an MZP, two MLLs with different NAs and one symmetric MLL with layers on both sides of the optical axis. Although the latter has been discussed previously [15], to our knowledge it has never been realized experimentally. The parameters are tabulated in Table 4. In Fig. 6(c), the different normalized focal intensity profiles are shown for comparison. As expected, the FWHM is a function of the numerical aperture. The often-used approximation that the outermost layer width $\mathrm {min}(\Delta _m)$ is about equal to the focus size is only valid for the on-axis optics (MZP, symmetric MLL). For on-axis MZPs, the side maxima can also increase if large areas of the inner layers are blocked, e.g. by a central beam stop [49]. Within the simulations, the smallest focal points with a FWHM of approximately 5 nm are generated by the MLL with $\mathrm {min}(\Delta _m)= {2.5}\,\textrm{nm}$ and the MZP with $\mathrm {min}(\Delta _m)= {5.0}\,\textrm{nm}$. Both show similar focus profiles. This demonstrates that specific geometries of multilayer optics for point-focusing can be approximated by simulations in the 2D or CS coordinate system.

In conclusion, we have shown that different kinds of optics can be used for point focusing. However, for two crossed MLLs, the absorbing material is doubled since two subsequent optics are needed. Further, the relation between the focus size and the outermost zone width is only valid for on-axis illuminations. For off-axis illuminated optics, on the other hand, the illumination NA is the decisive parameter.

7. Coherent diffractive imaging with a multilayer zone plate

In Fig. 7, we present the simulation of an entire imaging setup. The setup, sketched in Fig. 7(I), shows an x-ray microscope with an MZP as an objective lens to magnify the object. An incident Gaussian beam is modulated by a phase-shifting and absorbing sample, diffracted by the MZP, and detected in the far-field.

 

Fig. 7. 3D simulations of an x-ray microscope using an MZP as an objective lens. An incident Gaussian beam is modulated by a sample, diffracted by the MZP, and detected in the far-field. (I) Illustration of the experimental setup. MZP diffracts the beam in multiple diffraction orders, most prominent 0^th, +1^st, and -1^st. The samples are a pure absorption object ’A’ and a pure phase object ’P’. The FWHM of the incident beam is denoted by $\Delta _{\mathrm {beam}}$, the depth of the sample by $\Lambda _{\mathrm {sample}}$, the depth of the MZP by $\Lambda _{\mathrm {MZP}}$ and the focal length of the MZP by $f$. A list of the simulation parameters can be found in Table 3. (II) Diffraction pattern of the wavefield and (III) empty-beam divided diffraction pattern, highlighting the wave modulation. The corresponding diffraction order is indicated on the right side. (a-d) different positions $z_{\mathrm {sample-MZP}}$ between the sample in units of focal length. In case of (c) with the sample positioned in the focal plane the pure phase object is in the +1^st diffraction order almost not observable, only a light edge enhancement. The hologram of both objects is visible in every other distance and within the other diffraction orders. Scalebars: 200 px corresponding to a scattering vector $q= {0.125}\,\textrm{nm}^{-1}$.

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We use the combined approach introduced in section 3.4 to simulate the imaging setup. The parameters are tabulated in Table 5. In an exemplary setup, where the detector is positioned in a distance of $z_{\mathrm {detector}} = {5}\,\textrm{m}$ to the sample, the simulated diffraction patterns correspond to a detector pixel size of 45.25 µm. The sample is modeled as a pure absorption object in the shape of the letter ‘A’, and a pure phase object in the shape of the letter ‘P’, positioned at four Figs. 7(a)-7(d) different distances in front of the MZP. The corresponding diffraction patterns are shown in Fig. 7(II) and the empty-beam divided patterns are shown in Fig. 7(III). In all diffraction patterns, the parallel non-diffracted beam — the 0^th order — illuminates only a few pixels in the center, whereas the rest of the image corresponds to the diffraction signal. The diffraction signal of the +1^st and -1^st order can be observed in the upper and lower half of Fig. 7(II), respectively, and clearly shows a magnified copy of the incident beam including the modulation by the sample. The modulations of the sample can be observed even clearer in the empty-beam divided images shown in Fig. 7(III). These kind of modulations are well known from in-line holography, and referred to as holograms. In classical TXM [30,50] with zone plate lenses to magnify the object, the sample is positioned at a distance of about one focal length, here position Fig. 7(c). With the off-axis MZP simulated here, the +1^st diffraction order then shows a sharp magnified image of the pure absorption object (A) while the pure phase object (P) is barely visible in this position. Shifting the object along the optical axis (defocus position), enables propagation-based phase contrast at selectable propagation distances, or Fresnel numbers, up to the deeply holographic regime. Importantly, at each position, holographic images can be recorded at two different Fresnel numbers, given a sufficiently large detector, as simulated here. It can also be noted as a curious consequence of the holographic self-interference in the +1^st and -1^st diffraction orders of the off-axis MZP, that high spatial frequencies are encoded in the diffraction signal around the 0^th order in the center of the detector, while lower and medium spatial frequencies are well encoded in the holographic signals appearing at higher angles.

Table 5. Parameters of the simulated imaging setup in Fig. 7. The simulation was performed in 3D with a FOV of $10.15\times {10.15}\,\mathrm{\mu}\textrm{m}^2$ and 29×10³ grid points in each lateral direction. The pixel size in the sample plane is 0.35 nm and in the detector plane 44.25 µm. The photon energy is $E= {13.8}\,\textrm{keV}$, the focal length is $f= {1}\,\textrm{mm}$. $\Delta _{\mathrm {beam}}$ defines the FWHM of the Gaussian illumination, $\Lambda _{\mathrm {sample}}$ the depth of the sample and $z_{\mathrm {detector}}$ the distance of MZP to the detector. NA defines the numerical aperture, $\mathrm {max}(\Delta _m)$ the largest and $\mathrm {min}(\Delta _m)$ the smallest layer width. The total number of layers is given by $M_{\mathrm {layers}}$ and the depth of the MZP by $\Lambda _t$.

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At first, this configuration seems identical or similar to the classical TXM, but some important differences apply. Instead of recording the intensity pattern on the detector as ’the image’, the coherent illumination allows for quantitative phase retrieval as in CDI. If the transfer function of the MZP is known, based on simulations and/or experimental measurements, one could develop a phase retrieval approach for an aberration free image reconstruction, similar to [51,52]. Compared to the classic (incoherent) TXM, coherent imaging with an MZP objective lens, also allows for better image quality concerning both resolution and contrast. Resolution can be increased, since the full diffraction pattern can be included in phase retrieval. Concerning contrast, both phase and absorption can be retrieved quantitatively, without additional elements such as a Zernike phase plate. The relative contrast strength of absorption and phase can be systematically varied by the defocus distance. In contrast to pure CDI or ptychography, the holographic nature of the signal within the +1^st and -1^st diffraction order encodes especially the low and moderate spatial frequencies, and results in a particularly well posed problem for phase retrieval [53]. These spatial frequencies are therefore easily phased based on the enlarged holographic signal in the +1^st and -1^st diffraction orders. At the same time, high spatial frequencies are encoded within the diffracted signal around the center 0^th order. Interestingly, the presence of the +1^st and -1^st diffraction orders also means that near-field holographic images at two different Fresnel numbers are recorded simultaneously. Furthermore, compared to the classical TXM scheme, this approach makes much more efficient use of the diffracted photons, by using the full diffraction pattern, and not only the +1^st diffraction order. An image reconstruction approach could be kept similar to the algorithm presented in [2], which jointly phases holographic and diffractive components. Advantages of using a known transfer function (modulator) in a three plane CDI setup has also been shown in [54], an has been denoted as coherent modulation imaging . A modulator consisting of randomly distributed equal sized pillars is positioned between the sample and the detector, equivalent to the MZP. The purpose of the modulator is to overcome inherent ambiguities in the phase retrieval process. By deploying an MZP, we redefine the purpose of the modulator by exploiting the magnifying capabilities of the MZP, as we already suggested in [51]. Thereby high spatial frequencies of weakly scattering samples can be recorded. Furthermore, by recording also the low spatial frequencies in the holographic components of the +1^st and -1^st diffraction orders, the usual sampling constraints can be overcome extending the possible field of view, similar to [2]. The detailed implementation of the reconstruction approach using an MZP and its experimental verification exceeds the scope of the present manuscript, and will be pursued in forthcoming work.

8. Conclusion

In summary, we showed that the FD approach is well-suited for numerical wavefield propagation through multilayer optics with large aspect-ratios of layer width to depth. The FD approach yields accurate wavefields, with no artifacts resulting from the periodic boundary conditions. Further, the computational field of view can be kept small, reducing the computation time, which is especially relevant for 3D simulations. Note that the results obtained here, by FD simulations of multilayer optics, are in line with previous investigations of the FD for the case of WG propagation [46].

Further, we studied the transition from a regime of guiding modes, observed at larger layer width, to the pronounced coupling of layers and formation of standing waves at small layer width observed in particular for oblique incidence. The latter phenomenon is well known as Pendellösung from the analytical theory of dynamical diffraction. By the present numerical simulations, the Pendellösung effects can be studied beyond approximations. Next, we have provided detailed 3D simulations comparing three different multilayer implementations, MZP, off-axis MZP and crossed MLLs. All three configurations are relevant in view of the ongoing developments in this field and the respective constraints of fabrication. The simulations clearly show that in all cases small focal points can be achieved. Particularly high focusing efficiencies are achieved when the fraction of the illuminated area structured by small layers is highest, such as in off-axis MZP. This results from the fact that the incidence angle and Pendellösung effect can then be optimized to a smaller range of layer widths. In addition when using a pair of MLLs for point focusing, the focusing efficiency is reduced by the doubled absorption since two subsequent optics are needed. Despite these differences, we can conclude that in principle all focal spots presented in this work are suitable for imaging applications, either in scanning or in full field.

After studying specific properties of isolated multilayer optics, we have extended the simulation to an imaging scheme combining multilayer optics with free-space propagation and diffraction. Different coherent imaging schemes can be conceived and treated by the present simulation approach. Correspondingly, Fig. 7 can be considered as just one example of many, where the optical design requires simulation of illumination, object, objective optics, and detection. Almost always, the simulation of the forward problem is an useful if not indispensable setup before a full phase retrieval and imaging scheme can be implemented. However, it is a further goal of this work to also propose a specific coherent imaging scheme, based on an off-axis MZP (or FZP) objective lens, as presented in Fig. 7. This example also shows that full experimental setups can be simulated by FD, which hence can help to identify new imaging schemes for high resolution and optimized contrast. Altogether, this opens a brilliant perspective for coherent x-ray imaging, well suited for the current developments of multilayer focusing optics with higher NA and advanced iterative reconstruction algorithms.