1. Introduction

Over the last few centuries we have witnessed a tremendous progress in optical microscopy driven by the demand to image smaller and smaller objects. Nowadays we have reached the limit for the maximum achievable resolution given by half of the wavelength of the illuminating light source, if the imaging optics have a numerical aperture (NA) near its maximum magnitude in the order of 1. To increase the resolution of conventional optical microscopy, the wavelength of the light source must be reduced. Unfortunately, in the extreme ultraviolet (XUV) and x-ray range, imaging optics based on lenses become either highly absorbing or the refracting power is not sufficient for realizing high NA optics. Additionally, reflective optics in the XUV can only be efficiently used in near grazing incidence geometry, introducing huge aberrations for spherical surfaces. Diffractive optics such as zone plates require monochromatic illumination and are only available at low NA [1]. So exploiting the possible high resolution of short wavelength sources requires a fundamentally different approach, such as coherent diffraction imaging (CDI) [2,3]. CDI overcomes the shortcomings of imaging optics in the soft x-ray regime by retrieving the phase of a scattered field from a measured intensity distribution with iterative algorithms such as the hybrid-input-output algorithm [4].

In recent years CDI has been widely applied for imaging with XUV radiation. Synchrotrons [5] or free-electron lasers [6] are ideally suited, because they provide XUV radiation in a well collimated and spatially coherent beam with high average and/or peak brilliance. To make high resolution microscopy also available in a university scale laboratory, XUV radiation based on high harmonic generation (HHG) matches the requirements very well [7]. So far the samples imaged with HHG based CDI were mostly apertures with a custom designed shape [8]. For imaging a wider class of samples the approach should be further developed to image samples in the reflection geometry, too. This is especially important in the XUV, where biologic specimen must be placed on substrates, which are often opaque for XUV wavelengths. The reflection geometry solves the substrate problem, but the reconstruction of CDI patterns [9] is more challenging. Apart from the problem that biologic samples are also opaque at typical available HHG wavelengths, we believe that this method could still deliver important results for life science if the outer shape of a specimen can be retrieved with high resolution.

Here we present an experimental realization of CDI with a laser driven HHG source. From the broadband HHG radiation we select a single harmonic at 38nm, which is applied in all experiments reported in this paper. This single refocused harmonic was scattered off structures of gold on a silicon wafer patterned by e-beam lithography. The scattered light was recorded with an XUV camera which has to be placed far away from the sample, permitting only a low NA. Applying a hybrid-input-output algorithm (HIO), we reconstructed the real space image from the oversampled diffraction pattern [10]. This algorithm iteratively switches between the real and reciprocal space where the object support constraint and the measured diffraction modulus are enforced respectively. The paper is organized as follows. In Section 2 we describe the experimental setup and present a characterization of the HHG source. In section 3 we describe the applied reconstruction algorithm and present different methods for processing the experimental data prior to the reconstruction. Section 4 contains the results of the reconstruction, together with a detailed investigation of the influence of the distortions due to the reflection geometry and their correction, as well as pointing out the importance of centering the diffraction pattern on the computation grid prior to the reconstruction. Finally we summarize our most important findings in section 5.

2. Experimental realization

2.1 Overview of the experimental setup

The experimental setup [11] is depicted in Fig. 1. The commercial Ti:sapphire based femtosecond laser system delivers 1mJ, sub-30fs pulses at 800nm wavelength with 1kHz repetition rate. The pulses are focused (f = 400mm, f/# = 15) into a gas jet to a measured diameter of roughly 40µm (FWHM). From the spot size and the other pulse parameters we estimate the peak intensity to about 2 × 10¹⁴Wcm⁻². The gas target consists of a nickel tube with 2mm diameter which is pinched and soldered at the end and is backed with neon or argon. The next element in the beamline is a 200nm thin aluminum foil suppressing the fundamental light and the lower order harmonics. A toroidal mirror with a focal length of 32cm refocuses the filtered XUV radiation onto the sample in a 4f-setup. Between the toroidal mirror and the sample we put a blazed grating in grazing incidence to spatially separate the foci of different harmonics. In the focal plane of the harmonics a 50µm pinhole mounted on a translation stage allows to select the wavelength. Less than a millimeter behind the pinhole the sample to be imaged will be illuminated under an angle of incidence of 22 degrees. The scattered light is recorded with a 2048x2048 pixel XUV camera (13.5µm x 13.5µm pixel size, cooled to −70°C to minimize thermal noise) 43cm away. A beamstop with a diameter of 2mm is centered in front of the CCD suppressing radiation reflected into specular direction. From the geometry of our setup we can estimate the numerical aperture $NA=n\sin \alpha =0.032rad$. The current design of our vacuum chamber determines the NA, which is rather low but sufficient for a first proof of principle experiment. Moreover, the low NA together with the reflection under an angle of 45° has the advantage of implementing additional diagnostics in the beam path very easily to ensure reproducible results. E.g., putting the objective of an optical microscope close to the sample paves the way for multimodal microscopy, such as measuring in situ the sample’s fluorescence [12]. It is also worth mentioning that the whole setup (without the laser) is rather compact and fits onto a 3m² optical table.

 

Fig. 1 Setup of the CDI experiment. The ultrafast infrared laser beam enters the vacuum chamber on the right hand side and is focused into a gas jet for HHG. After blocking the fundamental laser light with thin metal filters the XUV radiation is refocused with a toroidal mirror. With a grating and a pinhole (PH) a single harmonic line can be selected. The light scattered off the sample is detected with an XUV sensitive CCD camera (CAM).

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2.2 HHG focus and wavelength selection

The foci of the HHG radiation are spatially separated by approximately 100µm in the plane of the pinhole, depending on the actual wavelength, see inset in Fig. 2(a). The foci are actually located on a sphere centered at the grating. However, due to the high ratio between distance from the grating and spatial separation, the estimate to filter the wavelengths in a plane is justified. By choosing the gas in the target the wavelength range can be preselected as depicted in Fig. 2(a) for neon and argon, respectively. Since the toroidal mirror images the HHG source 1:1 to the pinhole plane, the focal spot size of the individual harmonics is between 30 and 40µm as shown in Fig. 2(b).

 

Fig. 2 (a) Typical normalized HHG spectra using argon (blue) or neon (red) as target gas. The number of photons using neon is roughly 5 times lower compared to the yield from argon. The inset shows the signal at the plane of the pinhole using argon. (b) The focal spot size in the pinhole plane is less than 40µm (FWHM). The resolution is limited by the pixel size.

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With the setup we can easily select a single harmonic. However, for a detailed evaluation of the scattering images, knowledge about the exact wavelength is necessary. For an absolute wavelength calibration we inserted a pinhole with a diameter of 5µm, calibrated by optical microscopy and SEM. The pinhole is small enough to form a well resolvable Airy pattern, see Fig. 3(a). The diffraction rings are characterized with the CCD located 43cm downstream of the pinhole. Imperfections of the Airy ring structure can be well explained by aberrations introduced by the toroidal mirror. By fitting the calculated Airy pattern to the measured one, we can estimate the wavelength of the XUV radiation very precisely. The best fit to the measured lineout is shown in Fig. 3(b), and the wavelength is estimated to 42.6nm and corresponds to the harmonic line from argon with the highest yield.

 

Fig. 3 (a) Measured Airy pattern in the far field 43cm behind the pinhole. Imperfections are induced by aberrations from the toroidal mirror. (b) By fitting (red line) an Airy pattern to a lineout of the measured data (position indicated by the white line in (a), the wavelength can be determined to 42.6nm, which fits well to reference measurements with a calibrated spectrometer. All plots have a logarithmic intensity scale.

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Due to the somewhat higher photon number we opted in all subsequent CDI measurements for the next higher harmonic from argon with a wavelength of 38nm. Also we switched back to the 50µm diameter pinhole and put the sample to be imaged as close as possible behind the pinhole.

2.3 Object for diffraction experiments and measured diffraction pattern

The object for the CDI experiments were the characters “V 20”, see Fig. 4(a), etched out of a 30nm thick gold layer on a silicon substrate. The letters were written by e-beam lithography and fabricated in a lift-off process. The overall dimension of the characters is 33x11µm, whereas details of the letters are in the sub-micron level. A typical diffraction pattern is depicted in Fig. 4(b). To increase the dynamic range we have composed it from a series of individual images taken with different exposure times ranging from a few seconds to several hundreds of seconds ($t_{\mathrm{int}}=3s,6s,12s,24s,48s,96s,192s$). The individual images were merged together by replacing pixels close to the saturation of the CCD at a longer integration time with pixels from the next shorter integration time and a subsequent rescaling. By carefully processing all of the images we could increase the dynamic range by about 10dB. Further we have suppressed the signal in the center with a beamstop. In the image, the area behind the beamstop and the adjacent hotspots were set to zero. As seen in Fig. 4(b), the beamstop is not located in the center, which has to be corrected prior to the reconstruction as mentioned later on.

 

Fig. 4 (a) Image of the object taken with an optical microscope. The characters “V 20” are etched from a 30nm thick gold layer on silicon. (b) The measured raw diffraction pattern on a logarithmic scale. To increase the dynamic range, the image is composed of seven different images with different exposure times in the range of 3s to 192s. The axes for $q_x$ and $q_y$ are explained in detail in Fig. 5.

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The low numerical aperture of our experimental setup causes a high linear oversampling [13] ratio, which is given by $\sigma =\frac{z\lambda }{pD}=\frac{0.43m\times 38nm}{13.5\mu m\times 33\mu m}\approx 37$, where $z$ is the distance from the sample to the CCD, $\lambda$ is the wavelength, $p$is the pixel size of the CCD, and $D$is the largest spatial extent of the object. Since oversampling factors of $\sqrt{2}$ are sufficient for a 2D phase retrieval [14], binning of pixels on the CCD helped to reduce the exposure time while the oversampling ratio was kept at a sufficient level. In the presented measurement a binning of 8 by 8 pixels was applied, so the raw diffraction pattern in Fig. 4(b) consists of 256 by 256 pixels, reducing the oversampling rate to $\sigma \approx 4.6$, which is still very well suited for obtaining meaningful results with the phase retrieval algorithm.

2.4 Geometry for the diffraction experiment and picture preparation

The reflection geometry in a CDI experiment imposes several difficulties that are not present in transmission experiments [15]. First of all, the surface quality of the sample matters, i.e., highly reflecting surfaces with low surface roughness are essential. From the detailed geometry of the CDI experiment in reflection geometry as presented in Fig. 5, we can further conclude that the recorded diffraction pattern is stretched along the horizontal axis, and additionally the momentum transfer along this axis depends also on the height information (z-axis) of the sample. Thus the pattern contains three-dimensional information in contrast to the transmission geometry where the third dimension is only accessible by angular scanning of the sample [16] or in a very high NA regime [17]. In our geometry, the far-field distribution of the scattered wave corresponds to the projection of the 3D-Fourier transform (FT) of the object, and the image is taken in the reciprocal space. The phase information of the scattered field is lost and only the amplitudes (i.e., $\sqrt{I}$) can be recorded as a function of the momentum transfer $q$. The $q_y$ component is projected vertically onto the detector screen, while the $q_x$ and $q_z$ components are both projected horizontally, as indicated in Fig. 5. The components of the momentum transfer $q$ in reflection geometry can be written as [15]:

 

Fig. 5 Geometry and definition of axes as used in the experiment. The incident beam $k_i$ is scattered from the sample. In the detector plane the intensities are measured in the reciprocal space which is an interception with the Ewald’s sphere (a slice is indicated by the dashed line).

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Inspecting Eq. (1) immediately reveals the major problem for 2D image reconstruction from data taken in reflection geometry: the $q_x$ component is not mapped linearly onto the detector, requiring a nonlinear correction of the data grid. This correction becomes more and more important for angles of incidence approaching grazing incidence [15]. However, from the momentum transfer mapping for the actual geometry as shown in Fig. 6, it can be seen that the deviation from linear mapping of $q_x$ is small due to the low NA and the relatively large angle of incidence. So it will be a reasonable approximation to assume a linear mapping. Furthermore our sample is rather flat, so the achievable resolution, considering the highest momentum transfer for $q_x$ and $q_y$, is in the range of one micron. On the other hand, for $q_z$ the momentum transfer, see Fig. 6(c), suggests a height resolution in the range of 50nm, which is comparable to the thickness of our sample. As a consequence, no intensity normalization [15] is necessary, which is a prerequisite for good results of the reconstruction. Given the arguments above, it is well justified to neglect the $q_z$ contribution for the evaluation in the remainder of this paper.

 

Fig. 6 (a) The momentum transfer $q_x$ mapping on the detector for the presented geometry shows slight non-orthogonal mapping across the detector plane. E.g. $q_x\approx 0$- indicated by the dotted line - describes a curve on the detector. The momentum transfers $q_y$ and $q_z$ shown in (b) and (c) respectively are linear, as is necessary for a planar image reconstruction. It is assumed that the specular reflection ($q_x\approx 0$,$q_y\approx 0$) is centered on the detector.

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Several steps were taken to process the raw diffraction pattern, depicted in Fig. 4(b), prior to applying the reconstruction algorithm. In a first step, we corrected for the distortion caused by the angular projection under which the object is seen. The deviation from normal incidence results in a stretched diffraction pattern compared to what it would look like in a perpendicular projection, i.e., in a transmission measurement. This manifests itself also in the different axes scaling in Fig. 6(a) and (b). The influence of this distortion is essential for the quality of the reconstruction with a HIO algorithm. We investigate how the reconstruction is affected if the measured diffraction pattern is stretched in one dimension such that the scaling of the momentum transfer is ensured equal in both directions. In other words, we artificially remove the projection stretching due to observation of the object under an angle. The stretching factor $\kappa$ is a function of the “virtual angle of incidence” $\psi$, $\kappa =1/\sin (\psi )$. As the sample is close to the pinhole, the actual angle of incidence ${\alpha }_i$ is not always easy to measure in practice. We choose the initial value of $\psi$according to our experimental conditions to be roughly $\psi ={\alpha }_i\approx 22°$, so $\kappa =2.67$. To estimate the influence of an erroneous estimation of the angle of incidence on the quality of the reconstruction, we assumed different values for $\kappa$ and reconstructed the image. The results are reported in section 4.1.

In a next step the measured diffraction pattern is shifted in$q_x$ and $q_y$ direction to center it on the computation grid. The optimal translation was determined by symmetry aspects of the Fourier transform. Again, the influence of deviation from a perfect centering has been studied and is summarized in section 4.2. To maintain a quadratic computation grid, which is advantageous for Fast Fourier Transform (FFT), pixels of the measured raw diffraction pattern were removed. The final diffraction pattern on the computation grid after centering consisted of 235 by 235 pixels.

3. Image reconstruction algorithm

For image reconstruction many different phase retrieval algorithms have been reported in literature [18]. Among them is the HIO algorithm, which has further developed and improved since the initial implementation. The evaluations presented in this paper draw on the standard HIO algorithm [10] for planar image reconstruction with the following modifications. For all presented reconstructions we used the same implementation and parameters: For each reconstruction run we applied 10,000 iterations. The feedback parameter $\beta$ was ramped down from 3 to 0.1, in equal steps after 100 iterations. Within the 100 iterations the first 90 iterations were standard HIO iterations, while the remaining 10 iterations were error-reduction (ER) steps. All reconstruction runs started from random phases, and the constraint of positive numbers was implemented in the complex-valued real space $\rho$. The shadow of the beamstop, hotspots and pixels at the detector’s noise level, i.e., the dark blue regions in Fig. 4(b), were carefully masked out, and phases and amplitudes were allowed to develop freely within these regions.

First it was necessary to experimentally obtain a support function $S$for the optimized stretching of the recorded diffraction pattern, i.e., $\psi ={\alpha }_i=22°$, and a perfect centering of the pattern within the computation grid. A first support estimate was generated from the diffraction pattern’s autocorrelation [19]. Then the shrinkwrap method [20] was applied after every 10 iteration steps. We also allowed for a relaxation for the support (i.e. larger support), if the preceding shrinking of the support caused a further increase of the reconstruction error metric [14]. Also the amount of Gaussian blurring of the current result, before shrinkwrap was applied, was ramped down slowly. It should be noted that areas without any measured amplitude information (masked out areas behind the beamstop, hotspots, or the detector noise) severely spoil convergence of the HIO. However, after carefully tuning the parameters for the shrinkwrap we were able to achieve several decent reconstructions, which appear as mirrored images of the object as shown in Fig. 4(a). The details will be discussed in section 4.3. From this single reconstruction the experimentally determined support, depicted in Fig. 7, was taken for all subsequent analysis, while all other parameters for HIO were kept the same. To rule out erroneous reconstructions due to statistical fluctuations, we repeated each construction with the same parameters ten times and finally calculated the average of the ten reconstructed images.

 

Fig. 7 The support function $S$ as real space constraint (blue and yellow correspond to a 0 and 1, respectively) which was experimentally retrieved from a single reconstruction run with $\psi =22°$ from the measured diffraction pattern alone. The contour of the object, cf. Figure 4(a), is retrieved and then horizontally mirrored. The slightly different axes scaling is due to the weak nonlinear mapping of $q_x$. While the left part shows the computation grid, we provide the interesting area enlarged on the right. The scale bar for both corresponds to 20µm.

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4. Reconstruction results

4.1 Impact of the assumption for the virtual angle of incidence $\psi$

The virtual angle of incidence $\psi$, as defined in Eq. (4), was varied from 5° to 40°, causing different stretching of the diffraction pattern in the $q_y$ direction. A sample of such a transformed input pattern for a reconstruction run ($\psi =22°$) is shown in Fig. 8. For this stretching factor the momentum transfers in the $q_x$ and $q_y$ directions scale almost identically, with small nonlinear scaling in $q_x$ as explained in section 2.4.

 

Fig. 8 Diffraction pattern generated out of the measured raw diffraction pattern, see Fig. 4(b), for $\psi =22°$. The $q_x$ and $q_y$ momentum transfer scaling is slightly different due to the nonlinear scaling of $q_x$. From this diffraction pattern the support shown in Fig. 7 was reconstructed. Note that the beamstop was not perfectly centered on the specular reflection point, thus the beamstop region is not exactly centered at $q_x\approx 0$. The diffraction patterns were centered on the computation space by placing symmetric features of the pattern with respect to the center.

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The results of the reconstruction for varying $\psi$ are summarized in Fig. 9. If we assume a stretching deviating from the correct angle $\psi =22°$, we expect distorted real space images. Although all reconstructions were run with the fixed support as shown in Fig. 7, the object is still fairly well reproduced for angles in the range of $\psi =\mathrm{19...25}°$. This finding is surprising and implies that the quality of the reconstruction with the HIO is not overly sensitive to the exact knowledge of the angle of incidence. During our evaluation we also tried reconstructions for all angles in the mentioned range by using shrinkwrap and starting from the autocorrelation of the wrongly corrected diffraction pattern. If both axes had roughly the same momentum transfer scaling, we would obtain similar results. But then the reconstruction algorithm did not converge when using the raw measured diffraction pattern shown in Fig. 4(b) together with a stretched version of the support shown in Fig. 7. We assign this behavior to the great sensitivity of HIO algorithm to noise. If the real space object is strongly stretched, it contains more pixels having random phases at the beginning. Since real space is less defined at the beginning of the reconstruction (the support is only very roughly estimated by shrinkwrap at first, and phases are completely random). These increased noise and fluctuations make convergence less likely and can only be compensated by a high quality measured diffraction pattern. On the other hand, stretching the diffraction pattern in the described way reduces the resolution in one axis but results in a 1:1 aspect ratio in real space and thereby minimizes the real space occupied by the object. By stretching the diffraction pattern, no additional noise is imposed onto the HIO, improving the stability of the algorithm. In a recent publication [15] Sun and associates solved this problem in a different way by using an object which was strongly stretched along the horizontal axis. With their approach they were able to reconstruct an object (their institute’s logo) with 1:1 aspect ratio.

 

Fig. 9 Reconstruction results for different assumed values of the angle of incidence $\psi$. For convenience, all retrieved images were rotated by 90° (axis shown at $\psi =22°$) and cropped of residual whitespace around the object. The scale bar corresponds to 20µm and is strictly speaking only correct for $\psi =22°$. Each frame shows $\left|\rho \right|$ averaged over ten reconstruction runs with the fixed support shown in Fig. 7.

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4.2 Centering the diffraction pattern for reconstruction

For a proper reconstruction it is necessary to center the diffraction pattern perfectly on the computation grid, otherwise the discrepancy between frequency space of the FFT during HIO iterations and the spatial frequencies in the recorded diffraction pattern will hinder convergence. In the following we studied the impact of a small deviation from the optimum centering. The diffraction pattern for $\psi =22°$ depicted in Fig. 8 was shifted pixel by pixel in either $q_x$ or $q_y$ direction. For each setting, ten reconstructions of the HIO with fixed support, shown in Fig. 7, were performed and the obtained images have been averaged and are shown in Fig. 10. Each displayed frame shows the modulus of the complex-valued real space $\left|\rho \right|$ after 10,000 HIO steps averaged over ten different reconstruction runs. One can clearly see that correct centering of the diffraction pattern is very important. Although we used a fixed support, see Fig. 7, which already resamples the object well, a small off-center deviation results in projecting more and more of the signal outside of the support and no convergence is observed. We also tried the same reconstructions with variable support using shrinkwrap and started from a support estimated by using the autocorrelation. The results are also unsatisfactory if the diffraction pattern is more than one pixel off center in any direction. We could not observe convergence, and the object would hardly be recognized. In terms of momentum transfer this implies that centering of the diffraction pattern must be better than $\delta q_x\approx \delta q_y\le 1.5\times {10}^{-5}n{m}^{-1}$.

 

Fig. 10 Amplitude $\left|\rho \right|$ obtained by averaging over ten independent reconstruction runs as a function of the position of the diffraction pattern. Between the different images the diffraction pattern has been shifted by $\Delta x$ and $\Delta y$ pixels in the computation space, which corresponds to a shift in $q_x$ and $q_y$ respectively. Perfect centering (i.e. $\Delta x=\Delta y=0$) was defined by placing symmetric features of the diffraction pattern on the computation grid with respect to the center. The scale bar corresponds to 20µm.

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The integrated modulus of the complex-valued real space inside of the support compared to outside of the support

 

Fig. 11 The logarithmic reconstruction error Γ is shown in false colors. For the optimum centering the highest amount of signal is mapped onto the support, whereas Γ continuously increases the further the diffraction pattern is shifted off center. The panels correspond to those in Fig. 10.

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4.3 Best reconstruction and estimated resolution

The complex-valued real space signal $\rho$ for the best reconstruction (for $\psi =22°$ and $\Delta x=\Delta y=0$) is depicted in Fig. 12(a). For convenience the result was horizontally mirrored allowing a better comparison with the microscopic image shown in Fig. 4(a). The contours of the object are very well reconstructed, whereas details inside of the letters are not fully resolved. The theoretical spatial resolution from the maximum momentum transfer of the measured diffraction pattern can be predicted to be $\delta x=\frac{\pi }{1.74\times {10}^{-3}n{m}^{-1}}\approx 1.8\mu m$. The obtained resolution has been estimated to $\delta x\le 3\mu m$ from the vertical bars of the reconstructed “V” (indicated by the white arrows in the magnified part of Fig. 12(a). Furthermore, the phase is relatively flat across the reconstructed object as expected.

 

Fig. 12 (a) Complex real space $\rho$ in an amplitude-phase plot for the best reconstruction assuming $\psi =22°$ and $\Delta x=\Delta y=0$. The brightness and hue indicate the amplitude and phase respectively. The scale bar is 20µm. For convenience the real space was horizontally mirrored allowing for a better comparison with the microscopic image, cf. Figure 4(a). (b) The diffraction pattern used for this reconstruction is the same as in Fig. 8 but with the same intensity scaling as (c) which is $\left|FT\right\{\rho \}{|}^2$. (d) The phase retrieval function PRTF represents the average over ten reconstructions with $\psi =22°$ and $\Delta x=\Delta y=0$. A magnitude above 0.5 indicates that the achieved resolution is not limited by noise. Note that the PRTF for $q<0.3\times {10}^{-3}n{m}^{-1}$ is not defined due to the beamstop.

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In a next step the measured input diffraction pattern, depicted in Fig. 12(b), is compared to the squared modulus of the Fourier transform of the complex-valued real space $\left|FT\right\{\rho \}{|}^2$ shown in Fig. 12(c). It is evident that the main features of the pattern are reproduced, and also that the signal inside the shadow of the beamstop is fairly smooth and well defined. A ring structure present in the measured diffraction pattern cannot be observed in the reconstruction, implying a measurement artifact, and not part of the real object. These rings originate from diffraction of the 50µm pinhole used to select the wavelength as described in section 2.2. Therefore for future experiments, a larger pinhole or a tighter HHG focus should be chosen. However, these rings can be considered as distortion to the desired diffraction pattern of the object and we showed that HIO is capable to cope with such if the data is prepared properly.

The phase retrieval transfer function (PRTF) [21] is defined as

5. Summary and perspectives

We have demonstrated a compact table-top setup for coherent diffraction imaging based on a laser driven coherent XUV light source. The illuminating wavelength can be selected in a wide range out of the HHG spectrum. For the first time, we presented results obtained from recording scattering patterns in a reflection geometry illuminated with coherent radiation at a wavelength of 38nm. The reconstruction with the HIO algorithm provided the best results, if the momentum transfer scaling was made equal in both axes, resulting in a reconstruction of the object with an aspect ratio of 1:1, which was horizontally mirrored. We attribute this to the lowest possible noise that is induced in real space if the HIO iterations start from random phases. Furthermore, we have investigated how off-centering of the diffraction pattern on the computation grid influences the reconstruction and found that perfect centering is essential for a good convergence. Finally we have shown that for a low numerical aperture geometry flat structures can be reconstructed easily even in reflection without any a priori knowledge of the object, and a spatial resolution in the micron range was reached. On the other hand, our studies reveal that knowledge of the exact reflection angles is not so critical, and a correct reconstruction is possible even if only rather simple corrections are applied to the recorded diffraction pattern. This allows for a fast reconstruction of the object without the need of recording diffraction patterns of the same object under different angles to correct the $q_z$ momentum transfer.

The reflection geometry allows imaging of nanostructured samples mounted on an opaque substrate. So with the setup it will be feasible to study the evolution of the electron density in plasmonic nanostructures not only with high spatial resolution but also with an unprecedented temporal resolution in the attosecond range provided by the illumination with ultrashort XUV pulses generated via HHG. Further, the reconstructed phase is able to provide information on very small fluctuations of the height profile of the object. This very high sensitivity opens the way to single-image three dimensional sensing with a height resolution on the order of the wavelength of the illuminating XUV radiation.

To fully exploit the potential of CDI in the reflection geometry for high resolution imaging we have to increase the NA of this system, i.e. we have to bring the detector closer to the sample. For placing the detector 10cm away from the sample, the NA will be around 0.3, suggesting a spatial resolution of around 200nm. However, the higher NA requires a more careful remapping of the momentum transfer. For the given geometry we must calculate the momentum transfer as shown in Fig. 6(a). For a higher NA we expect for the $q_x$ momentum transfer a substantial bending of equi-color lines in the contour plot, which can easily be corrected by a horizontal shifting of the pixels, i.e. the equi-color must be straight and parallel lines similar as in Fig. 6(a). Such a correction can be easily implemented in the reconstruction algorithms. For real world application the exposure time should be further reduced, i.e. the flux on the target must be increased. One way to increase the flux is illumination with the full HHG spectrum rather than a single harmonic [22], though it still remains an open question whether the currently applied reconstruction works also for spatially complex objects such as biologic specimen. More straightforward is a reduction of the exposure time by replacing the kHz laser source by the recently demonstrated few cycle laser sources with MHz repetition rate [23].

The CDI reflection geometry imposes many challenges on the reconstruction, but paves the way for applying it routinely to high resolution microscopy capable of imaging biologic specimens. The preparation of samples for the reflection geometry is very simple and consists of pipetting them onto a substrate only. Biologic samples in their natural environment are highly absorbing, so only with HHG sources in the water window we will be able to gain insight into cells. For longer wavelengths it will still be possible to study the morphology of specimens, i.e. their silhouette, which will also have a major impact on many branches of life science [24,25]. Together with a state-of-the-art laser driven HHG source the proposed setup has the potential to become a versatile tool for cell identification and classification in many laboratories for addressing scientific problems as well as for daily routine applications.