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Abstract
Structural analysis of periodic samples with lensless imaging techniques in the extreme ultraviolet spectral range is an important and versatile tool for inspecting nanoscale structures including lithographic masks and quasi-crystals. More specifically ptychography is the method of choice for imaging such structures with high lateral resolution and an arbitrarily large field-of-view. This work reports on the reconstruction of highly periodic samples that occasionally contain isolated defects. For a realistic scenario scan map distortions, i.e. the actual scanning position does not exactly coincide with the position used as input for the reconstruction code, were taken into account. For such conditions it is well known, that the reconstruction will not provide always a physical reliable image. It will be shown, that the reconstructed illumination function is very sensitive to map distortions and small defects in otherwise periodic objects allowing to decide whether the reconstructed object is reliable or not. Applying this criterion the minimum detectable size of local defects in otherwise periodic structures will be compared to both the period length and the magnitude of the scan map distortion. As both quantities are usually much larger than the resolution given by the wavelength and numerical aperture of the imaging setup, they determine the resolution limit.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Lensless imaging techniques, such as coherent diffraction imaging (CDI) and its extension ptychography rely on solving the phase problem to reconstruct the complex-valued object and illumination function from measured diffraction patterns. With non-scanning CDI, only the exit wave defined by the combination of object transmission and illumination function, can be retrieved from a single diffraction pattern [1]. On the other hand, ptychography allows to reconstruct both the complex object transmission function and the illuminating field independently, if diffraction patterns are recorded from several overlapping illuminated spots, the so-called probes [2,3]. In the last decade, ptychography became of great interest due to the increased computing power and availability of coherent sources including electromagnetic radiation, ranging from the terahertz [4] to X-Ray [5] regime, and electron beams [6]. As ptychography allows imaging with wavelength-scale resolution in a wide spectral range, it has been successfully applied to image a huge variety of samples including periodic structures such as photonic crystals [7], to unwrap its lattice structures [8], and optical components like zone plates [9]. Due do the arbitrarily large field of view combined with the possibility of material sensitive reconstruction using wavelengths near absorption edges [10] or relative phase shifts [11], ptychography became also an interesting technique for imaging semiconductor devices [12]. Ptychography is not only useful to image the device but, more recently, it has been used to inspect the lithographic mask, looking for defects on the size of the mask’s smallest structures [13]. For all of these examples, it is necessary to reliably reconstruct periodic objects. It is well known in the literature, due to the quasi-periodicity of these structures, artifacts in the reconstruction can occur [14,15]. However, there exist techniques to get rid of these reconstruction artifacts [16] using additional knowledge either about the object or the illumination function. In the last year a tremendous growth of experimental work reporting on imaging of periodic samples with tabletop light sources in a wide spectral range has been noticed. All of them aim for high quality reconstructions requiring an estimation of the achievable resolution and criteria for judging the reliability of the reconstruction. Figure 1 point out the problem with periodic structures. The translation invariance of the Fourier transform causing identical diffraction pattern under a shift of the illumination function on highly periodic objects. Hence, the reconstruction converges quite fast under appearance of artifacts in the reconstructed probe and position refinement algorithms [17,18] are no longer capable of determining the real scan positions. In this work, the limitations of ptychographic reconstructions for identifying small defects within otherwise periodic samples under realistic conditions, such as deviations of the scan points from the assumed positions, are investigated. The validity of solutions are checked by cross correlating the reconstruction with the original object and compare the reconstructed probes with the original ones. It was found, that small defects are only visible in the reconstructions, if the size of the defect is larger than the structure’s period and the scan point deviations remains small in comparison to the defect size.
Fig. 1. Motivation for a closer investigation of periodic structures in ptychography. If the positions of the illumination function (red in 1,2c) on the sample differs from the presumed ones (blue) during the reconstruction, the transmission of aperiodic objects and the corresponding illumination functions begin to blur (1a). Position refinement algorithms like [17] are capable of adapting the presumed points to the real ones to ensure a successful reconstruction (1b) (the scan sequence is indicated with a gray spiral). The reconstructions of perfectly periodic objects however are independent from position displacements (2a and 2b) and position refining become not indicative (2c). Additionally, artifacts in the illumination function show up (insets in 2a,b). Here, the illumination functions (insets) are scaled down by a factor of 2.
2. Numerical model and methods
Imaging of objects larger than the coherent part of the illumination function is not possible for non-scanning CDI due to the violation of the isolation constraint [19,20]. Ptychography imposes no limitations for the sample size because redundant data are obtained by multiple diffraction patterns recorded for different overlapping positions of the illuminating beam. However, new limitation can occur if the samples are periodic or the scan position is not precisely known and/or controlled For example, the computing grid can interfere with the scan pattern resulting in artifacts, summarized under the raster-grid-pathology [3]. Similar difficulties arises for reconstructing large periodic samples due to the translation invariance of the Fourier transform. For an arbitrary large object illuminated with a well-defined beam, the intensity distributions of the diffraction patterns remains nearly constant under shifting the object within the illuminated area [20], if the period is smaller than the probe. For a ptychographic measurement of periodic samples, this effect implies similar diffraction patterns independent of the scan position. Additionally, in real experiments, the coordinate of each scan point is a crucial input parameter. The real position can differ from the assumed one due to the limited position accuracy of the sample holder, thermal drifts and vibrations in the setup or beam jitter. In general, determining and holding the exact scan position can only be ensured with enormous technical efforts. To account for these statistical deviations in the following simulations map distortions as position shifts between the points used for generating the diffraction pattern and the points used for reconstruction were introduced. Especially aperiodic objects are highly sensitive to this deviation, seen in Fig. 1 as the algorithm tries to overlay non-matching parts of the structure. For the simulation presented in this work, the extended ptychographic iterative engine (ePIE) due to its robustness [2] was used. To compare the findings to previous experimental results [21], the following parameters were chosen: the photon energy is 68.6 eV, the distance from the sample to detector is about 23 mm, assuming a 2D detector consisting of 2048$\times$2048 pixel with a size of 13.5 $\mu$m, the reconstructed real space pixel size corresponds to 17.6 nm. Similar to experimental situations an illumination beam with a Gaussian intensity distribution and FWHM width of 5 $\mu$m and soft cutoff to zero outside of the probe’s area was used, to mimic a cutoff pinhole right in front of the sample. For creating the dataset, a spiral shaped scanning pattern was created seen in Fig. 2 with an overlap between consecutive scan points of 90%. The above mentioned map distortion was realized by shifting a scan point of the map to an arbitrary position within a circle with a fixed radius $\vec {r}$ around the original position with a randomly chosen angle $\theta$ as seen in Fig. 2. The reconstruction needs a first guess of the probe, which is in this case a disc function with a diameter of 7 $\mu$m, which enables the probe reconstruction to shrink down. For the algorithm, the feedback factors $\alpha$ and $beta$ [2] (see Eq. (2), 3) were set to 0.9 and claim the real space feedback between two subsequent iterations of object and probe reconstruction.
Fig. 2. Principle of simulated map distortion. Every point of the scanning map (blue dots) is shifted by a distortion vector $\vec {r}$ with a randomly chosen angle $\theta$ and a fixed length $|\vec {r}|$ and represent a distorted scanning point (red circle) for generating the diffraction pattern.
As a measure for the quality of a reconstruction, an sum squared error metric (SSE) was used and defined as follows:
2.1 Simulation results
To point out the reconstruction differences between aperiodic and periodic objects, a binary Siemens star as an example for a aperiodic object due to its wide range of structural varieties and the possibility to compare the finding with previous works was chosen. In the object, the structure size varies from 1 $\mu$m down to 10 nm over a total possible field of view of 40 $\mu$m. Figure 3 shows the reconstruction of a Siemens star sample recorded under the conditions described above. The presented results demonstrate the importance of the accurate knowledge of scan positions for the quality of the reconstructed object. With increasing map distortion, the reconstruction quality of the retrieved object decreases rapidly. It is also remarkable, that increasing the number of iterations will not improve the error anymore, pointing out a converged algorithm and therefore a stable solution.
Fig. 3. Reconstruction of a Siemens star with increasing map distortion. The error metric shows the decreasing convergence of the algorithm with increasing map distortion matching the structural decay of the object function.
The algorithm tries to enforce the reconstruction of a part of the object at a wrong place. In the next iteration, this part of the object will be overwritten by another misaligned part, which results in vanishing of the complex valued object function until a stable solution is reached. To conclude, already rather small map distortions around 100 nm lead to a different convergence behavior (Fig. 3. bottom graph) and to a lower image contrast. In a next step the influence of map distortions on the reconstruction of periodic objects are evaluated. As a periodic test pattern, a binary grid with a 200 nm period $p$ was assumed. Simulation results for the periodic sample using the same set of experimental parameters as for the aperiodic case are shown in Fig. 4. Due to the translation invariance similar diffraction patterns can be obtained and the algorithm is always capable of matching different parts of the objects, independent of the magnitude or direction of the actual map distortion. For a more general discussion, the map distortion is specified in multiples of the period $p$.
Fig. 4. Reconstruction of a grid with 200 nm period assuming map distortions in multiples of the grid period $p$. Due to the periodicity, the reconstruction is independent of the map distortion. The error metric shows a similar convergence behavior with minor variations.
Figure 4 shows the reconstruction behavior of the algorithm assuming map distortions. The convergence behavior does not change with the magnitude of the map distortion and reaches the minimal error after approximately 100 iterations. In comparison to the Siemens star sample, where the error metrics decrease with the deviation indicating a map distortion, for the periodic sample the evolution of the error metrics do not provide a hint on positioning faults. The indistinguishable reconstructions of periodic objects in ptychography leads to the question, whether the reconstruction reflects the correct image of the sample or not.
To answer the question above, a closer look will be necessary not only to the reconstructed object, but also to the reconstructed probe. Fortunately the ePIE algorithm is capable to provide the illumination function (probe), too. The probe ($P(\vec {r})$) and object ($O(\vec {r})$) reconstructions are coupled numerically due to the update functions [2]:
Fig. 5. Comparisons between reconstructed probes of Fig. 3 and the original one. This comparison shows a dependency on the map distortion which, in turn, correlate to the contrast decay in Fig. 3.
A comparison of reconstructed probes of Fig. 4 lead to an imperceptible decrease in the ICM (compare to Fig. 1). Since for periodic objects the reconstruction quality does not change when distorting the scan map, the question rises how meaningful ptychography with periodic objects is. To further address this problem, imperfections with increasing size were added to the otherwise purely periodic object to investigate if these small imperfections can be resolved. The addition of imperfection (defects) however may lead to small traces usable for the algorithm to converge to reliable solutions.
2.2 Defects
After the investigation of the influence of map distortions, it will be investigated whether small defects in otherwise periodic objects can be detected or not. After the first study regarding the influence of map distortions to periodic samples, the survey now is fixed on detection of defects with sizes in the order of the grid period. As shown above, the ePIE algorithm always converges to a stable solution for periodic objects. However, real life periodic structures often contain wanted or unwanted defects or imperfections such as dust particles on a lithographic mask or crystallographic perturbations. If the algorithm always retrieves a perfect grid, small defects may remain undiscovered. Especially for real life applications, it is indispensable to know the limitations of the chosen microscopy approach such as the minimum detectable size of defects with or without map distortions present. For testing the ability of the algorithm to detect defects, absorbing particles of different sizes on a periodic sample were assumed. Initially, disc like defects were placed on the grid and the diffraction pattern were recorded assuming an otherwise perfect experiment, i.e. no map distortions and all other parameters were stable. Figure 6 shows an absorbing imperfection on a grid with 200 nm periodicity. Defects smaller than the period of the grid lead to absorption artifacts, but are not correctly reconstructed as objects anymore. The line outs of Fig. 6 show the decreasing transmission of the grid, where the defect is located. Note that the resolution according to the simulations settings is 17 nm, which is high enough to resolve all used defects.
Fig. 6. Reconstruction of different imperfections on a grid with 200 nm period assuming no map distortions. Defects smaller than the grid period are barely visible. The insets in the upper reconstructions show the objects used for the simulation as a guide for the eye. The line outs (bottom) show a reduced transmission at the smaller defects positions. The insets in the top row show the placement of defects for the simulation. Note, that the offset was chosen to separate the normalized line outs for a better visibility.
As described in the previous section, a convincing reconstruction of the object corresponds to a correct reconstruction of the probe beam. To quantify the quality of the image, the retrieved probe was compared with the input probe. Full agreement between the reconstructed and the real probe is nearly impossible due to the fact, that the reconstructed probe is the average over all scanning points and the measurement of the diffraction pattern can be only performed with a limited bandwidth. Thus, even for a perfect scanning map, one cannot expect am ICM value of 1. Therefore, the relative change of the ICM need to be evaluated to quantify the reliability of the reconstruction.
Figure 7 shows the comparison between the reconstructed probes and the original one. Together with Fig. 6, one can see that the visibility of defects corresponds to the calculated ICM. If the defect is smaller than the grating period, the probes show modulations corresponding to the grating structure. For defects larger than the grating period (compare Fig. 6), the retrieved probes converge to the initial one without aforementioned modulations, implying a meaningful reconstruction of the object. Here, the strong diffraction signal of the periodic structure overlay the weak diffraction pattern of the defect, demanding a high signal-to-noise ratio (SNR) [22]. For real life samples, one can e.g. conclude that defects smaller than the wire size and period of an IC-bus are very difficult to detect with ptychographic methods or small variations in the crystalline grid may vanish during the reconstruction. Up to now the role of scan map distortion and local defects for the reconstruction of periodic objects were independently discussed. In a next step, the algorithms ability to converge if both of them are present simultaneously will be examined. Here, a defect with 200 nm diameter imposed on a periodic structure with 100 nm grid period is assumed. Note that his corresponds to a 400 nm defect on a 200 nm grid as discussed before. The smaller grid period was chosen because it enhances the visibility of reconstruction artifacts due to a higher grid density for a constant field of view. With this setup, the algorithm’s ability to retrieve images of defects assuming distorted scanning maps could be tested.
Fig. 7. Comparison between the reconstructed probes used for Fig. 6 and the original probe. The probe reconstruction for small defects smaller than 100 nm (the grating period) show modulations, which results from the grid structure itself.
The simulation results as shown in Fig. 8 reveal that the grid structure in the vicinity of the defects remains unchanged, whereas the defect becomes almost invisible, if the map distortion reaches the size of the grid defects. The observation can be understood as follows. The algorithm matches patterns by using existing structures of the previous iteration. If the map misalignment reaches the size of the defect, there are no patterns from the previous iterations to match, resulting in a mixture between the grid structure with the defect and the misaligned scanning points, which appears like a defect-free grid due to the reduced contrast. Thus, under realistic conditions with no precisely known scan point positions, such defects cannot be detected with standard ptychography algorithms. A test with position refinement also stops to converge (see Fig. 1), if the map distortion reaches the size of the defect. Figure 9 shows the similarities between the probes of the reconstructed dataset of Fig. 8 to the original one. The ICM starts to decrease, if the map distortion reaches the size of the object, matching the loss of defects information in Fig. 8.
Fig. 8. Reconstruction of the 100 nm period grid with a small defect of 200 nm with different map distortions. The defects starts to vanish when the map distortion reaches the defect size (2$p$), as a result, the reconstructed object appears quasi defect-free for large map distortions.
Fig. 9. Comparison of the reconstructed probes with the assumed input probe for generating the dataset. The ICM starts to decrease, if the map distortion reaches the size of the defect. This behavior matches the structural decay of the objects in Fig. 8.
3. Conclusion
Ptychography is a powerful state-of-the-art lensless imaging method for short wavelengths illumination which is applicable to many different real world objects in science and technology. However imaging samples containing periodic structures can end in erroneous reconstruction results which can be identified by the appearance of modulation in the retrieved probe beam. Considering the coupled update functions Eq. (2) and Eq. (3), a reconstruction of the object is valid, if the reconstructed probe beam is comparable to the real probe beam, i.e. it should have the same intensity distribution and curvature of the wave front. In this way these findings are similar to previously reported approaches, like in [16]. There, a far field measurement of the probe was used as a-priori knowledge, to converge the algorithm on periodic objects. However, perfectly ptychographically reconstructed periodic samples suggest highest resolutions due to the apparently high contrast, despite scan map distortions. If the reconstruction is not reliable, a resolution measurement could also be corrupted. Therefore, a comparison of reconstructed and measured probe has to taken into account as a criterion for validating the reconstruction.
This work highlighted the role of scan map distortions, due to e.g. thermal drifts and/or imperfect positioning of the sample, for the correct reconstruction. Aperiodic objects show immediate degradation of the image quality for scan map distortions. On the contrary, periodic objects can be well reconstructed which can be understood by similar scan-to-scan-diffraction patterns. For real life quasi-periodic samples, a reconstruction seems always possible, but the validity of the reconstruction must be verified independently. This has been demonstrated by placing imperfections with different sizes in an otherwise periodic structure. To derive a criterion for a valid solution, the above-mentioned method of comparison for evaluating the quality of the reconstructed probe was applied. For different defect sizes, it was shown that the size of the distortion has a direct influence on the validity of the retrieved image. Despite the algorithms capability of resolving structures down to 17 nm for the mentioned experimental setting [21], defects within periodic structures have to be larger than half of the period to appear in the reconstructed object. The situation becomes even worse, if defects in periodic structures are imaged in the presence of scan map distortions. For scan map distortions in the range of the periodicity, the defect becomes immediately invisible and only the defect-free periodic structures were reconstructed. In this case, even a map refinement algorithm like in [18] would not find a correct solution anymore and the object would not be reliable.
Summing up, imaging periodic structures with ptychography need particular attention regarding imperfections and map distortions due to the algorithms inclination towards physically invalid solution. A comparison between the reconstructed probe and a known one gives a criterion for valid object reconstructions. As a rule of thumb, defects have to be larger than the periodicity to be properly reconstructed and map distortions have to be minimized to a fraction of the periodicity to observe the aperiodic features in experimental setups like [21]. Real-life application often demand imaging of objects embedded in highly periodic structures. Here, an approach for imaging small imperfections could be using a dual-wavelength setup for separating the scattering signal of periodic and aperiodic parts. High-dynamic-range (HDR) imaging can also used to improve the SNR to amplify weak scattering signals of imperfections [22] under stable illumination.
Funding
European Social Fund (Project 2018 FGR 0080).
Acknowledgments
The authors acknowledge support from the Federal State of Thuringia and the European Social Fund (ESF) Project 2018 FGR 0080
Disclosures
The authors declare no conflicts of interest.
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