## Abstract

Diffractive optics can be used to accurately control optical wavefronts, even in situations where refractive components such as lenses are not available. For instance, conventional Fresnel zone plates (ZPs) enable focusing of monochromatic radiation. However, they lead to strong chromatic aberrations in multicolor operation. In this work, we propose the concept of spatial entropy minimization as a computational design principle for both mono- and polychromatic focusing optics. We show that spatial entropy minimization yields conventional ZPs for monochromatic radiation. For polychromatic radiation, we observe a previously unexplored class of diffractive optical elements, allowing for balanced spectral efficiency. We apply the proposed approach to the design of a binary ZP, tailored to multispectral focusing of extreme ultraviolet (EUV) radiation from a high-harmonic tabletop source. The polychromatic focusing properties of these ZPs are experimentally confirmed using ptychography. This work provides a new route towards polychromatic wavefront engineering at EUV and soft-x-ray wavelengths.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Microscopy with extreme ultraviolet (EUV) tabletop sources is a promising diagnostic tool for nanoscience but remains challenging due to the relatively low photon flux available as compared to large-scale facilities such as synchrotrons or free-electron lasers. In addition, focusing and shaping polychromatic EUV radiation using refractive optics such as lenses is intrinsically difficult, as most elements across the periodic table exhibit high absorption and low refractive index contrast in this spectral region. Alternative imaging approaches are therefore required. Full-field tabletop EUV microscopes employ Fresnel zone plates (ZPs) to directly image a specimen onto a detector [1,2]. Since the focal length of Fresnel ZPs is wavelength dependent, EUV full-field microscopes are typically restricted to monochromatic radiation. Although cascaded systems of diffractive optical elements (DOEs) with achromatic focusing capabilities have been reported [3,4], a monolithic approach is preferred for EUV radiation, where phase-shifting optical elements are hard to realize and thus binary DOEs with lower diffraction efficiency have to be used [5]. The requirement for monochromaticity results in a loss of rich chemical information that can be extracted from spectrally resolved refractive index studies [6]. A direction that has been followed more recently for EUV microscopy is coherent diffraction imaging (CDI) [7,8], which computationally recovers specimen information from a single diffraction intensity. While this technique has been applied for EUV tabletop microscopy, it works reliably only when information about the specimen, such as finite support or sparsity [9,10], is known *a priori*. CDI requires spatially coherent EUV radiation, which is available from tabletop setups through high-harmonic generation (HHG) [11–13]. HHG sources have the additional benefit of producing extremely broad EUV and soft-x-ray spectra, making them intrinsically suited for spectrally resolved microscopy. However, CDI requires monochromaticity, which principally limits its range of application for EUV microscopy, although some methods have been reported that relax this shortcoming through algorithmic extensions [14,15] or experimentally via spectrally resolved diffraction measurements [16]. Both full-field microscopy and CDI have a finite field of view limitation. Attempts have been reported to extend the field of view by using a compact scanning transmission microscope (STM) based on an HHG source [17]. However, in STM the lateral resolution is directly coupled to the beam size. In addition, the results in [17] were obtained with a multilayer mirror that limited the illumination to a single harmonic. Grazing incidence mirrors may be used to focus polychromatic radiation, but these typically suffer from low numerical apertures and are sensitive to misalignment [18]. Thus all the aforementioned techniques lack spectral sensitivity and involve a compromise between field of view and resolution. A promising technique to solve these challenges is ptychography [19,20], a combination of STM and CDI. Ptychography enables simultaneous wavefront sensing and quantitative phase contrast microscopy at theoretically unlimited field of view [21–23]. Additionally, ptychography decouples spatial resolution from the illumination spot size. The large amount of information available in ptychography makes it possible to decode polychromatic wavefronts using only monochromatic detectors [24,25]. This is particularly useful for EUV microscopy, where polychromatic operation offers increased flux and enables chemical sensitivity, but wavelength-resolving cameras are not available. Most HHG-based ptychography applications to date have used multilayer mirrors that filtered the source spectrum to quasi-monochromatic radiation [22,26–30], and to our knowledge only a single result on multispectral EUV ptychography has been reported [31]. In the latter reference, the authors demonstrated the use of an ellipsoidal mirror in grazing incidence to produce a polychromatic focused beam. This approach is flux efficient, but it does not allow control over the structure of the incoming beam, which can be of great importance for reconstruction quality [32–35]. Here we demonstrate multispectral ptychography using structured EUV beams. To this end, we address the sub-problem of designing DOEs that are suitable for both confining and structuring polychromatic EUV radiation. This goal is achieved through a novel computational design principle based on spatial entropy minimization. While the designs reported in this paper are tailored to EUV radiation, the underlying concept may be used for both refractive and diffractive optics design in other spectral domains. In the first part of the paper, we introduce spatial entropy minimization, which formulates diffractive focusing as an optimization problem. It is shown that the approach generates conventional ZPs in the limit of monochromatic radiation. A wider class of designs is examined for polychromatic radiation, with the ability to balance spectral flux. In the second part, we report experimental results for multispectral EUV ptychography, where we apply entropy optimization to generate structured and focused beams. The proposed approach paves the way for utilizing the full potential of HHG sources for spectrally resolved EUV microscopy.

## 2. SPATIAL ENTROPY MINIMIZATION

Designing a multispectral ZP consists of two main ingredients, which we shortly summarize before proceeding to the mathematical formulation: First, polychromatic radiation needs to be confined into a preferably small focus. As detailed below, this constraint may be imposed by requiring the side lobes of the focal spot to decay with increasing distance from the optical axis. However, for the case of binary ZPs, we cannot prevent the presence of a zeroth-order diffraction contribution. Hence, the first constraint is enforced only inside the geometrical shadow of a central stop of the ZP, which is the interior of the purple circle in Fig. 1. In the final experiment, portions of the beam far away from the optical axis may be clipped using an order sorting aperture (OSA) complementary to the target focal plane region to suppress the zeroth order. Second, the ZP is required to be binary. This property facilitates nanofabrication and reduces losses due to absorption when used in transmission.

With regard to the first requirement, we consider the spatial entropy functional

Next, we consider a strategy that minimizes the entropy functional in Eq. (1). The complex gradient [37] of Eq. (1) with respect to ${\psi _k}$ is given by

*minimum entropy update step*.

To calculate the required ZP, we define the spectrally summed hologram

*concurrence*and

*binary constraints*, respectively. During the search process a small subset of pixels in $B({\boldsymbol q})$ is randomly selected and flipped from 0 to 1 and vice versa. We denote the binary ZP resulting from the latter step by $B^\prime ({\boldsymbol q})$. The number of random flips is gradually annealed to zero (see details below). The simulations presented in the next section indicate that random flips help prevent stagnation around the local minima of Eq. (1). As a final step, the updated focal plane spectral fields are given by

## 3. SIMULATION

We performed simulations of the entropy minimization algorithm proposed above and examined the influence of the source spectrum, the role of the entropy gradient, and the number of random flips following the concurrence and binary constraints. In each simulation we fixed the ZP diameter to $D = 159\;\unicode{x00B5}{\rm m}$, the smallest feature size to $\Delta x = 310\;{\rm nm}$, and the focal length to $f = 4.9\;{\rm mm}$, assuming radiation from an HHG source with a fundamental driving laser wavelength of 800 nm upconverted to the 17th to 23rd harmonic (34.8 nm to 47.1 nm). All values listed above were chosen similar to the experimental parameters below. We note that the feature size is defined here as the smallest spatial scale over which the ZP can switch from opaque to transmissive and vice versa. The bandlimited angular spectrum propagator ${\cal P}$ assumes the same pixel size in the zone plate and focal plane. All simulations below are performed with

Figure 2(a) shows in the upper left corner the entropy minimization result (white) for only the 17th harmonic present in the spectrum incident on the ZP. The fraction of random flips in the allowed ZP area, limited by the fixed ZP diameter and the central stop, was linearly annealed from 25% to 0% within a total of 1000 iterations. A standard ZP characterized by two parameters, namely, the wavenumber $\kappa$ and a constant phase offset $\phi$, was fit to the output of the entropy minimization algorithm by solving the nonlinear least-squares problem:

Further insight into the flux balancing mechanism provided by entropy minimization is obtained by examining the resulting axial beam cross sections around the focal plane. In Fig. 3 we compare axial beam cross sections obtained from the conventional ZP in Fig. 2(a) and the minimum entropy ZP in Fig. 2(j), both in polychromatic operation. The left column of Fig. 3 shows the focusing behavior of the conventional ZP. It is seen that a wavelength shift causes an axial displacement of the focal plane. This axial displacement is an immediate consequence of the exchangeable roles that the wavelength and the propagation distance play in paraxial diffraction [39]. The flux in a single observation plane (semitransparent line) is widely unbalanced as a result of the beam being dominated by a single diffraction order. The right column of Fig. 3 shows the focusing behavior of the ZP optimized for minimum entropy from Fig. 2(j). It is seen that the entropy minimization approach yields multiple and more balanced diffraction orders as compared to the conventional ZP. Changing the wavelength results in an axial displacement of the minimum entropy beam. However, for the minimum entropy beam the diffraction orders coincide upon axial displacement, in contrast to the beam generated from the conventional ZP. We conclude that entropy minimization achieves flux balancing through axial alignment of higher diffraction orders.

Two final notes concern the uniqueness of the simulation results and computational complexity: First, we have constrained the algorithm to focus into a relatively large target focal region with a diameter of 10 µm. However, the entropy minimization algorithm gives a solution with significantly smaller beam size. If the target focal region is constrained to be circular as given in Eq. (10), the entropy functional in Eq. (1) is invariant under translation of the focal spot intensity and the solution is nonunique. In particular, this can result in ZPs with off-axis focus. Practically, we omitted this by choosing a smooth apodization function

with a full width at half-maximum of ${\rm FWHM} = 2\sqrt {2\ln (2)} \sigma = 10\;\unicode{x00B5}{\rm m}$. This was used only for the entropy update step [Eq. (4)], while the renormalization step [Eq. (7)] and the evaluation of spectral efficiency [Eq. (13)] used the circular apodization function described by Eq. (10). Second, the computational complexity of a single iteration of the entropy minimization algorithm scales with ${\cal O}[{\Lambda \cdot P \cdot \log P}]$, where $P$ is the total number of pixels in the ZP. Each ZP optimization described in this section ($P = 2^{20}$, $\Lambda = 4$, ${10^3}$ iterations) required 160 s using GPU-accelerated computation on a NVIDIA Tesla K40 and a MATLAB software implementation.## 4. EXPERIMENTAL RESULTS

#### A. Zone Plate Design Considerations for EUV Ptychography

In this section we summarize considerations to match free parameters in the minimum entropy ZP design for operation in our EUV ptychography setup. We start with the diameter of the target focal plane. While the object field of view in monochromatic far-field ptychography is theoretically unlimited through transverse scanning, the probe field of view (pFOV) is restricted by the experimental geometry:

where $\Delta u$ is the detector pixel size, $b$ is the binning factor, and $z$ is the sample-detector distance [39]. In polychromatic ptychography, Eq. (17) is a function of wavelength, with the most restricted pFOV found at the minimum wavelength in the probe spectrum. Under optimized conditions for HHG in argon using an 800 nm driving laser, our source provides appreciable flux down to ${\lambda _{{\min }}} = 25\;{\rm nm}$. For a sample-detector distance of $z = 90\;{\rm mm}$, dictated by the geometry of our experimental vacuum chamber, and a binning factor of $b = 4$ (to reduce the computational overhead of the Fourier transforms involved in the multispectral ptychography solver, see Section 4.B below), this results in a minimum pFOV of 41.6 µm. We achieve an oversampling ratio better than 2 per dimension [40] by restricting the target focal region to 15 µm.Another important consideration is the desired beam shape. We demonstrated above that minimum entropy ZPs can be designed for multispectral operation. However, they are not necessarily optimal for ptychography, which benefits from spatially extended and structured beams [32–35]. While this has previously been shown for monochromatic radiation, the same observation holds for polychromatic ptychography (see Supplement 1). We thus decided to find a compromise between the need for probe localization as dictated by Eq. (17) (which is achieved by entropy minimization) and introducing spectrally dependent spatial structure into the polychromatic wavefront. Revisiting the simulation results in Figs. 2(d)–2(f) and the corresponding row in Table 1, we see that aberrations caused by the presence of bifurcations in the ZP design can introduce structure into the illumination while still providing a fair balance between the energy of each spectral field in the focal plane [35]. Our final ZP design was optimized using entropy minimization for six harmonics (the 17th to 27th order) simultaneously. We note that annealing with only 0.5%–0% random flips was used to promote the formation of bifurcations in the ZP, while searching for a solution with low entropy. Based on this a ZP was manufactured with additional support structure as shown in Fig. 4(a). The ZP has an outer diameter of 240 µm, a smallest feature size of 155 nm, and a focal distance of 4.2 mm. A central stop with a diameter of 75 µm was left opaque. This region can be used in conjunction with an order-sorting aperture of the same size to block the undiffracted portion of the radiation transmitted through the ZP. Our simulations indicate that the ZP has an estimated average spectral efficiency of better than 3%.

#### B. Nanofabrication

The multicolor binary ZP as described in the previous subsection was fabricated on a 90 nm thick Au layer sputter coated on a 50 nm thick ${\rm Si}_3{\rm N}_4$ membrane (Ted Pella Inc.). The pattern was milled with a 30 keV focused gallium ion beam (FEI Helios Nanolab 600). The ion-beam current was set to 48 pA with a focal spot diameter of 50 nm. The ZP was milled in 12 cycles with a sputter step size of 35 nm and dwell time of 60 ms. The total time required for nanofabrication was approximately 6 h. In addition, we fabricated a binary sample for wavefront analysis of the multispectral wavefront produced by the ZP. Periodic structures were initially taken into consideration. However, periodic samples are not ideal for ptychography, as they give rise to localized diffraction peaks. These saturate a few detector pixels quickly while leaving others at a low level of exposure, allowing only a small total number of photons to be recorded. Moreover, self-calibration methods such as lateral position correction are challenging [42]. Hence we designed a sample [see Fig. 4(d) left side] that is aperiodic upon lateral translation. An analytical expression for the sample transmission function is given by

#### C. Experimental Setup

The EUV measurements have been carried out using a tabletop HHG source. A noncollinear optical chirped-pulse amplifier system amplifies infrared pulses with a central wavelength of 800 nm generated by a Ti:sapphire oscillator. This system delivers driving pulses with a duration of 25 fs at a repetition rate of 300 Hz. A pulse energy of 2 mJ is typically used for HHG. These pulses are focused into an argon (Ar) gas jet and are upconverted to EUV wavelengths through the HHG process as illustrated in Fig. 4(c). An aluminum (Al) membrane with a thickness of 200 nm is used to filter out the fundamental beam while the HHG beam is transmitted. This EUV beam is focused by the ZP onto a sample, which is mounted on a two-dimensional translation stage (Smaract SLC-1730) required to perform ptychography scans. In addition, a spectral Hartmann mask (SHM) is mounted next to the sample [41]. The SHM consists of multiple apertures with gratings inscribed, one of which is shown in Fig. 4(d). We note that the SHM cannot be used for wavefront sensing on focused beams and is used here only for the characterization of the HHG spectrum with the ZP removed from the beam path. The identified spectral lines, but not their relative weights, are used in the ptychography algorithm below as prior knowledge. Finally, a CCD camera (Andor Ikon-L 936 SO, 2048x2048 pixels, pixel size 13.5 µm) collects diffraction patterns at a distance of 92.5 mm downstream of the specimen and SHM. A typical multiwavelength HHG diffraction pattern with the described setup is shown in Fig. 4(b).

#### D. Multispectral Ptychography

Our reconstruction algorithm is based on ptychographic information multiplexing [24], which models the observed intensity on the monochromatic detector as an incoherent sum of multiple monochromatic diffraction patterns. All algorithmic details are given in Supplement 1. We collected a ptychographic scan consisting of 252 scan positions covering a field of view with a diameter of 50 µm. The average scan step was 2.5 µm, and the diameter of the binary object is 25 µm, resulting in a linear overlap of approximately 90% [43]. This relatively high degree of overlap in scan positions enabled us to reconstruct the spatial structure and the spectral weights of the beam. In Figs. 5(a1)–5(i1) we show the simulated beam assuming the ZP design from Fig. 4(a) under plane wave illumination. In Fig. 4(e) and in Figs. 5(a2)–5(i2) we show the ptychographic reconstruction of the object and the spectral beam profiles, respectively. We find good qualitative agreement between simulation, as shown in Figs. 5(a1)–5(i1), and experiment, in particular for the central harmonics. The largest discrepancy is observed at the extremal spectral lines, where the signal-to-noise ratio is lowest and defocusing is most severe. Moreover, differences between the simulated and the reconstructed beam profiles arise from fabrication errors in the ZP as seen in Fig. 4(a). In addition, the spectral wavefronts illuminating the ZP are not perfectly planar. In Figs. 5(a3)–5(i3) we show each spectral beam intensity integrated along the $y$ direction at $\pm500\;\unicode{x00B5}{\rm m}$ around the focal plane. This projected view is adopted here due to the lack of centrosymmetry in the beam.

Figure 4(e) shows the relative energy of the reconstructed beam (orange dots) as compared to the spectrum extracted using the SHM (turquoise line). We note that the spectrum measured by the SHM is acquired without ZP in the beam. Thus, the spectra obtained from the SHM are not expected to match in the weights of the spectral lines estimated by ptychography. It is emphasized that no *a priori* knowledge was used to obtain the spectral weights during the reconstruction process. This is possible because the object is binary (ptychographic reconstruction shown in Fig. 4(e), left). In cases where the object is dispersive, modulating each spectral wavefront in a different way, additional information about the source spectrum is required [24].

## 5. DISCUSSION AND CONCLUSION

In summary, we proposed a new route to polychromatic focusing based on spatial entropy minimization, enabling flux balancing in the design of polychromatic focusing optics. The ability to efficiently shape and focus the polychromatic flux offered by HHG sources is a key challenge in the realization of spectrally resolved tabletop EUV microscopes in the near future. Our proof-of-concept experiments demonstrate multispectral focusing and wavefront shaping are possible with binary DOEs, which were previously not considered for tabletop ptychographic scanning microscopes. Furthermore, we show that multispectral ptychography is capable of reconstructing a multitude of complex-shaped EUV field distributions in parallel and thus provide a method that can effectively use the generated wavefronts for spectrally resolved high-resolution imaging. Future studies may build on this work, precharacterizing polychromatic wavefronts and subsequently leveraging this information as *a priori* knowledge for EUV microscopy with chemical sensitivity.

Recent work suggests that multispectral ptychography may be used to characterize both the spatial and spectral structure of attosecond pulses [25]. We have seen in this study that entropy minimization yields DOE designs producing beams with multiple, flux-balanced diffraction orders, which axially coincide upon discrete wavelength changes. Consequently this approach may result in beams with extended depth of focus or even nondiffracting beams when applied to attosecond pulses [44]. However, the influence of our DOEs on the temporal structure of such attosecond pulses has not been investigated, and they may be expected to lead to significant spatiotemporal couplings that are common to diffractive optics [45].

Future challenges are computational in nature: first, while the entropy constraint in the focal plane allowed us to use continuous optimization techniques, the binary constraint imposed on the ZP renders the underlying optimization problem nonconvex. We proposed a heuristic search strategy based on random flips and annealing, which worked for the moderately sized binary ZPs used here but will require more efficient schemes for large-scale focusing optics with orders of magnitude more resolution elements per dimension. This would enable the design of DOEs with larger opening angles and smaller outer zone widths. Similarly, reflective DOEs require larger areas to be optimized but could reach higher efficiencies when used in grazing incidence.

The present work constitutes an approach to polychromatic wavefront sensing that offers orders of magnitude higher lateral resolution than traditional Hartmann–Shack sensors. This may provide new insights into the transfer of aberrations and their spectral dependence in the upconversion of high-harmonic generation [46,47].

## Funding

European Research Council (637476); Nederlandse Organisatie voor Wetenschappelijk Onderzoek (13934).

## Acknowledgment

The authors thank F. Campi (ARCNL) for useful comments on the manuscript. L.L. gratefully acknowledges the support of NVIDIA Corporation with the donation of the Tesla K40 GPU used for this research.

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement 1 for supporting content.

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